On the arc length method: combining ideas and implementations aspects

1 Introduction

In non linear finite element analysis, traditional incremental or iterative method like the standard Newton-Raphson method (NR) is widely used to trace the right equilibrium path for nonlinear problems due to large geometry change and/or material behaviour. Common examples appear in softening and buckling instability problems involving snap-through or snap-back phenomena. Typical equilibrium paths associated with snap-through and snap-back behavior are illustrated in Figure 1 where appears the critical points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K}
  commonly categorized into limit points and into bifurcation points respectively. However, a further disadvantage of the NR method is that without some addition of special techniques  a falling load path cannot be handled and hence no convergence at specific locations along the trajectories of load-deflection is reached.  In other words, in such situations, the standard load-controlled NR method in which  incremental displacement vector is calculated by a prescribed incremental load factor lead to error  near the peak point ( see point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F}
of Figure 1). To overcome these kinds of difficulties,  often, the NR method is accompanied with some load or displacement controlled strategy and a line searches methods. However, is well known that similar failures happens when equilibrium paths encounter the limit point of snap-back  (see points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H}
of Figure 1).

Figure 1: snap-back and snap through response [7].

To overcome such problems, methods and techniques of switching between load and displacement controls (Sabir and Lock, 1972), of using the artificial springs (Wright and Gaylord, 1968), of abandoning the equilibrium iterations in the close vicinity of the limit point (Bergan and Soreide, 1978; Bergan et al., 1978), or of allowing load reductions within load control (Cope and Rao, 1981; Bergan and Holand, 1979; Crisfield, 1982; Phillips and Zienkiewicz, 1976) have been successively proposed. However, these methods and techniques usually require great care and cannot always give satisfactory results. Ramm (1981) reviewed all the available methods and suggested the most versatile is the arc-length method which was originally developed by Riks (1972, 1979) and Wempner (1971) and later modified and widely used by Ramm (1981) and Crisfield (1981,1982a, 1983,1984)[1]. After that more and more attentions (Crisfield, 1986; Lam and Morley, 1992) were paid to improve and apply this method.

The AL method is the most powerful path following method in the solution of equilibrium paths, bifurcation points and limit points by introducing a constraint condition which establishes the nonlinear equations in which the unknown load parameter is determined[2]. The AL method has been accepted as the most effective numerical method for the post-buckling problems with NR method type algorithms. For such situations the AL method is useful to avoid bifurcation points and track unloading. The AL method method causes the Newton-Raphson equilibrium iterations to converge along an arc, as can be seen in Figure 2, thereby often preventing divergence, even when the slope of the load vs. deflection curve becomes zero or negative. The constraint equation is forced to be satisfied at each iteration.

In this dissertation we described the basic aspect of the AL method and integrated ideas to improve the standard AL method . This combinations are based on the Crisfield arc-length method, but in this case using a spherical arc length and the strategies developed by [1] to avoid complex roots. For analysis of softening material, an isotropic damage model is introduced with the objective energy mesh dissipation. Benchmarking examples are tested for proving the implemented algorithm.

2 Basic formulation of Arc-Length solution procedure

The Arc-Length method is described within the category of continuation methods and it is applied to obtain solution paths[3]. In the non-linear finite element analysis, the system of non-linear algebraic equations to be solve take the form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {r}(\boldsymbol {u},\lambda ) = \boldsymbol {f}_{int}(\boldsymbol {u}) - \lambda \boldsymbol {f}_{ex}

(1)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {r} = \boldsymbol {r}(\boldsymbol {u},\lambda )}

is a function of the displacement vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {u}}
and the load parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda }

, and is the out-of-balance force vector which vanishes at equilibrium. The internal force vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {f}_{in}(\boldsymbol {u})}

is a function of the displacements and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {f}_{ex}}
is the prescribed external force vector and is scaled by the load intensity parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda }

. Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda \boldsymbol {f}_{ex}}

represents the load applied at the current load increment.  Basically, the arc-length method first consider the load factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda }
in Eq.1, as a variable in the residual equation. Then, an extra new constraint equation is added to the residual equilibrium equation for defining unequivocally  the next equilibrium point solution at an intersection between the solution path and the constrains equation. This representation enables introducing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda }
as an additional  "degree-of-freedom", thereby expanding the dimensions of the solution space by one, for tracing the snap-through and snap-back behaviour of the curves of load against displacements.  This constrains equation can be written in differential form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s = \int \sqrt{d\boldsymbol {u}^T d\boldsymbol {u} + \psi ^2 d\lambda ^2\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex}}

(2)

or increment form as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Psi = \boldsymbol {\Delta u}^T\boldsymbol {\Delta u} + \psi ^2\Delta \lambda ^2\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex}-\Delta l^2 = 0

(3)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\Delta u}}

is vector of incremental displacement, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {f}_{ex}}
is specified external load vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda }
is incremental load-factor, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta l}
is fixed radius of desired intersection,  and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi }
is the scaling parameter. In load-displacement space, it is designed to search the solution  of the next iteration on a small ellipsoidal surface surrounding the last converged point,  of size determined by the prescribed arc length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta l}

. Another constraint equations can also be found in the literature, e.g, Bathe and Dvorkin (1983).

Assuming that displacement increments Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\Delta u}}

 satisfying constraint Eq 3, is possible to obtain a simplified linearized version of this constraint equation   adopting the idea of Riks and Wempner(1971) for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (i+1)th}
iteration, i.e

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\delta u_i}^T\boldsymbol {\Delta u_i} + \psi ^2\delta \lambda _i\Delta \lambda _i\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex} = 0

(4)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \boldsymbol {u}_i}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda _i}
are the required corrections to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda _i}
respectively. This results in this so called fixed normal plane constraint, with  the solution iterates all lying on the hyperplane normal to the tangent at last converged point in load-displacement space[1]. However, the Risk and Wempner method was not suitable for standard finite  element analysis even with modified Newton-Raphson (mN-R) procedure, because equations proposed  by Riks destroy the banded nature of the stiffness matrix [4].

Therefore, we can directly used a matrix form (Crisfield 1991), that with some simplifications in Eq 1 and 3, we compute the iterative change in displacement vector and load-factor in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (i+1)th}

iteration,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[\begin{array}{c}\delta \boldsymbol {u} \\ \delta \lambda \end{array} \right]_{i+1} = -\left[\begin{array}{cc}\boldsymbol {K}_t & -\boldsymbol {f}_{ex} \\ 2\Delta \boldsymbol {p}^T_i & 2\Delta \lambda _i\psi ^2\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex} \end{array} \right]^{-1}_i \left[\begin{array}{c}\boldsymbol {r} \\ \Psi \end{array} \right]_i

(8)

where we define the effective general matrix of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n+1}

dimension

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\hat{K}}_T = \left[\begin{array}{cc}\boldsymbol {K}_t & -\boldsymbol {f}_{ex} \\ 2\Delta \boldsymbol {p}^T_i & 2\Delta \lambda _i\psi ^2\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex} \end{array} \right]

(10)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {K}_T}

is the tangential stiffness matrix, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {r}_i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Psi _i}
are the previous values of out-of-balance load vector and arc-length. Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\hat{K}}_T}
is a non-symmetric matrix. However, as Crisfield (1981) stated, the direct full assembling and solving of this matrix  reduces the solution efficiency, since the symmetric  banded nature  of the system stiffness matrix can no longer be exploited. After the iterative change Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \boldsymbol {u}_{i+1}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda _{i+1}}
have been computed, the displacement vector and load-factor  are updated, i.e

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \boldsymbol {u}_{i+1} = \Delta \boldsymbol {u}_{i} + \delta \boldsymbol {u}_{i+1} (11) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \lambda _{i+1} = \Delta \lambda _{i} + \delta \lambda _{i+1} (12)

Alternatively, instead of solving directly Eq 8, constraint equation can be introduced by following the technique of Baltoz and Dhatt (1979) for displacement control at single point. Ramm (1981) and Crisfield (1981, 1982, 1983, 1984) applying this technique, gave the modification of the method and introduced an indirect solution scheme for the constraint equation 3. According to this technique, the iterative change of displacement for the new unknown load level Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda _{i+1} =\Delta \lambda _{i} + \delta \lambda _{i+1}}

  is written as [4]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \boldsymbol {u}_{i+1} = \boldsymbol {K}_T^{-1}\boldsymbol {r}_i + \delta \lambda _{i}\boldsymbol {K}_T^{-1}\boldsymbol {f}_{ex} (13) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): = \delta \boldsymbol {u}_{ri} + \delta \lambda _{i}\delta \boldsymbol {u}_{f} (14)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \boldsymbol {u}_{ri}}

is the displacement vector resulting from eliminating the out-of-balance forces  and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \boldsymbol {u}_{f}}
is the parallel displacement vector computed and stored after every update of the stiffness matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {K}_T}

. Substituting values from 11 and 14 into the constraint equation yields in a quadratic equations in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda _i}

 so that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a\delta \lambda _{i}^2 + b\delta \lambda _{i} + c = 0

(15)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a = \delta \boldsymbol {u}_{f}^{T}\delta \boldsymbol {u}_{f} + \psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} (16.a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): b = 2\delta \boldsymbol {u}_{f}(\Delta \boldsymbol {u}_i + \delta \boldsymbol {u}_{ri}) + 2\Delta \lambda _i\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} (16.b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c = (\Delta \boldsymbol {u}_i + \delta \boldsymbol {u}_{ri})^T(\Delta \boldsymbol {u}_i + \delta \boldsymbol {u}_{ri})-\Delta l^2 + \Delta \lambda ^2\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} (16.c)

This method is graphically shown in 2. Note that the parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi }

convert everything to the same dimension and order of magnitude. Many analyst set this Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi=0}

. This procedure is known as the indirect spherical AL method Crisfield(1891). A typical value for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi ^2}

is taken as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {u}_0^T\boldsymbol {u}_0^T/(\lambda _0^2\boldsymbol {f}^T_{ex}\boldsymbol {f}^T_{ex})}
[1].   Eq.15 is solved to get the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda }

, and so to define the iterative change completely. This equation leads to two results of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda } . Despite the sophistication of the arc-length procedure of Ramm and Crisfield certain difficulties remain in its applications. The first concerns to the choice of proper roots for the quadratic equations, which the method for selection of proper value will be discussed in subsequent sections; and secondly, complex roots of a quadratic equations something occurs and the reason are not clear yet, but is strongly related to the arc-length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta l }

chosen.

Figure 2: Arc-length procedure for specific iteration [8].

3 Modify arc-length procedure

Lam and Morley suggested decomposing the out-of-balance load Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {r}}

into a  parallel component Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g_i\boldsymbol {f}_{ex}}

, which is in the specified applied load direction; and orthogonal component Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {r}^\perp = P^\perp (g_i\boldsymbol {f}_{ex})} , with respect to fixed nodal load vector in case of divergence, where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g_i = \frac{\boldsymbol {r}_i^T\boldsymbol {f}_{ex}}{\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex}} (17) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {r}^\perp = \boldsymbol {r}_i-g_i\boldsymbol {f}_{ex} (18)

In the equations above the, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P^\perp (\cdot )}

is the orthogonal operator introduce by convenience. In its work, they proposed a new strategy to deal with the complex roots arising from the solution of the quadratic equation, which will be be discussing later.  Given the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_i}
and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda _i}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_{i+1}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda _{i+1}}
have the formation represented as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \boldsymbol {u}_{i+1} = \Delta \boldsymbol {u}_{i} + \delta \lambda _i\boldsymbol {K}^{-1}\boldsymbol {f}_{ex} + \eta _i\boldsymbol {K}^{-1}\boldsymbol {r}^\perp (19.a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): = \Delta \boldsymbol {u}_{i} + \delta \lambda _i\delta \boldsymbol {u}_f + \eta _i\delta \boldsymbol {u}_{ri} (19.b)

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \lambda _{i+1} = \Delta \lambda _i + g_i + \delta \lambda _i

(20)

where was introduced a scaling parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta }

in Eqn 19.a  which will be discussing later. Substituting   Eqns 20 and 19.b into the constraint  equations 3, gives a new quadratic equations  in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda _i}

, similar to Eqn 15 written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a(\delta \lambda _{i})^2 + b(\delta \lambda _{i}) + c = 0

(21)

where, in this case

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a = \delta \boldsymbol {u}_{f}^{T}\delta \boldsymbol {u}_{f} + \psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} (22.a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): b = 2\delta \boldsymbol {u}_{f}(\Delta \boldsymbol {u}_i + \eta \delta \boldsymbol {u}_{ri}) + 2(\Delta \lambda _i-g_i)\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} (22.b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c = (\Delta \boldsymbol {u}_i + \eta _i\delta \boldsymbol {u}_{ri})^T(\Delta \boldsymbol {u}_i + \eta _i\delta \boldsymbol {u}_{ri})-\Delta l^2 + (\Delta \lambda ^2-g_i)\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} (22.c)

This procedure becomes general arc-length procedure proposed by Crisfield when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta=1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi ^2=0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g_i=0}

. Thus far, the basic formulation of arc-length solution procedure is completed. However, during calculation of practical problem, there still are certain difficulties remain in application of arc-length method. The main problem is that quadratic equation 21 sometimes has complex roots that cause failure of convergence. Methods to solve this problem are discussed in the next part. Even we can found in the literature that is possible to get the correct answer of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda }

without resolving any quadratic equations and avoiding the trace-back solution path when a wrong solution of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda }
  is found and choose.

4 Adjustment for complex roots

As we commented before, the main problem arises in this arc length procedure came from the complex solution obtained by the solutions of the quadratic equations 21. Particularly, we get this situations in analysis reinforced concrete when a severe strain localizations occurs and unloading of adjacent elements on cracking, leading difficulties in solution convergence still persist, while the procedure repeatedly gives complex roots. Crisfield(1983,1984) for avoiding this problem, introduced a line searches, by multiplying the both terms of Eqn 14 for by a factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta \neq{1}} , determining by minimization of potential energy, obtaining a considerable improvement. However, exist in the literature another procedures to avoid this problem, that will be described in the following.

4.1 Method 1. Lam and Morley procedure

Lam and Morley (1992) proposed a method in a pseudo-line search based to avoid the complex roots by introducing a scaling parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta }

 in the usual constraint equation, that is, however, computed without introducing energy considerations.  According to Eqn 21, in order to get the real root, the parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c}
in Eqn 22.a, 22.b and 22.c respectively   have to satisfy that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): b^2 - ac \geq 0

(23)

Which lead to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \vartheta (\eta ) = [\psi ^2(\Delta \lambda _i-g_i) + \delta \boldsymbol {u}_f^T(\Delta \boldsymbol {u}_i + \eta _i\delta \boldsymbol {u}_{ri})]^2 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): - (\psi ^2 + \delta \boldsymbol {u}_f^T\delta \boldsymbol {u}_f)\times [\psi ^2(\Delta \lambda _i-g_i)^2 - \Delta l^2 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): + (\Delta \boldsymbol {u}_i + \eta _i\delta \boldsymbol {u}_{ri})^T(\Delta \boldsymbol {u}_i + \eta _i\delta \boldsymbol {u}_{ri})\geq{0}

(24)

After some algebra, we get the following quadratic equations in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _i}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \vartheta (\eta _i) = \breve{a}\eta _i^2 + \breve{b}\eta _i + \breve{c} \leq 0

(25)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \breve{a} = (\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} + \delta \boldsymbol {u}^T_{f}\delta \boldsymbol {u}_{f})\delta \boldsymbol {u}_{ri}^T\delta \boldsymbol {u}_{ri} - (\delta \boldsymbol {u}_{f}^T\delta \boldsymbol {u}_{ri})^2 (26.a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \breve{b} = 2(\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} + \delta \boldsymbol {u}^T_{f}\delta \boldsymbol {u}_{f})(\Delta \boldsymbol {u}_i^T\delta \boldsymbol {u}_{ri}) - 2(\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex}(\Delta \lambda _i-g_i) + \delta \boldsymbol {u}^T_{f}\Delta \boldsymbol {u}_i)\delta \boldsymbol {u}_{f}^T\delta \boldsymbol {u}_{ri} (26.b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \breve{c} = (\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex} + \delta \boldsymbol {u}^T_{f}\delta \boldsymbol {u}_{f})(\Delta p^T_i\Delta p_i-\Delta l^2) - (2\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex}(\Delta \lambda _i-g_i) + \delta \boldsymbol {u}^T_{f}\Delta \boldsymbol {u}_i)\times \delta \boldsymbol {u}^T_{f}\Delta \boldsymbol {u}_i Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): +\psi ^2\boldsymbol {f}^T_{ex}\boldsymbol {f}_{ex}(\Delta \lambda _i-g_i)^2\delta \boldsymbol {u}^T_{f}\delta \boldsymbol {u}_{f} (26.c)

Note that all this terms can be computed at the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} th iteration in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda _i} . Newly computed the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _i}

and chosen the more convenient value,  is then fed back into original constraint equation which is solved again. Suitable value of this scaling parameter are chosen close to 1. Zhou and Murray (1994) suggested that suitable value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _i}
should be chosen from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (0,1)}

. A suitable value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _i}

proposed by Lam and Morley [1]  is chosen as below:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \eta _i = \eta _2 - \xi \hbox{ if }\eta _2<1 (27.a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \eta _i = \eta _2 + \xi \hbox{ if }-\breve{b}/\breve{a}<1<\eta _2 (27.b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \eta _i = \eta _1 - \xi \hbox{ if }\eta _1<1<-\breve{b}/\breve{a} (27.c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \eta _i = \eta _1 + \xi \hbox{ if }1<\eta _1 (27.d)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _2}
are roots of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \vartheta (\eta _i)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _2\geq \eta _1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi = \beta (\eta _2-\eta _1)}

. Usual, a value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta }

is take as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0.05}

. While we found in some certain situation in the practical calculations, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _i}

should be chosen carefully to avoid problems of numerical precision 1. The value of parameter is initially set to 1; with the consequent divergence of solution process due to complex. This way the problem of complex roots can be avoided successfully. The authors further report that generally the quadratic equation solved for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta }
gives real values, but in case it lead to complex values, the following strategy explain in Method Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2}
most be adopted.

(¹) By author's experiences, it necessary to take care about computer precision, due to sometimes the value of the quantities in quadratic equations are really small and can be lost significant digits.

4.2 Method 2. Elimination of r^⊥

Generally the quadratic equation 25 solved for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta _i}

gives real roots. Since it is not possible to find a real roots to give the chosen arc length,  Lam and Morley (1992) suggested that the main problem is that the orthogonal  error Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {r}^{\perp }}
is too large. In case that Eqn 25 leads to complex values,  the follow procedure proposed by Lam and Morley (1992) is adopted [4].

1. Compute the displacement due to the orthogonal component of out-of-balance load vector (orthogonal to applied load vector) and update the displacement

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \boldsymbol {u}^{cr} = \Delta \boldsymbol {u}_i + \delta \boldsymbol {u}_{ri} </li>

   (28)

   where superscript Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle cr}

   is related to the complex roots.
   

2. Compute associated nodal forces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {f}_{in}^{cr}} .
3. Compute incremental loading parameter and the arc-length from orthogonal component of out-of-balance force vector.

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lambda ^{cr} = \frac{(\boldsymbol {f}_{in}^{cr})^T\boldsymbol {f}_{ex}}{\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex}} (29.a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \lambda ^{cr} = \lambda ^{cr}-\lambda _0 (29.b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\Delta l^{cr})^2 = (\Delta \boldsymbol {u}^{cr})^T\Delta \boldsymbol {u}^{cr} + \psi ^2(\Delta \lambda ^{cr})^2\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex} </li> (29.c)

4. Reduce the loading parameter by multiplying the loading parameter by the ratio of original arc-length to arc-length associated with complex roots, and iterate the solution process.

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mu = \frac{\Delta l}{\Delta l^{cr}} (30.a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \lambda = \mu \Delta \lambda ^{cr} (30.b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \boldsymbol {u}_{i+1} = \mu \Delta l^{cr} </li> (30.c)

Compared with reducing arc-length directly, this method is more targeted to eliminate residuals and hence should be more effective.

4.3 Method 3. No solution of quadratic equations

Forde and Stiemer (1987) proposed another arc-length method in which the direct solving of quadratic equation 21 is not required, and thus the problem of dealing with complex roots is avoided. These kind of arc-length methods are called arc-length orthogonality methods. In the standard incremental iterative solution of equilibrium equation for proportional loading, the incremental displacement can be written in two components.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \boldsymbol {u} = \Delta \lambda \Delta \boldsymbol {u}_1 + \Delta \boldsymbol {u}_2

(31)

The first is the displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_1}

due to a unit load factor multiplied by the incremental variation  in the load level Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda }
. The second Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_2}
is the displacement update for a conventional  load controlled procedure due to the unbalanced loads. Therefore, A general arc length procedure can be derived from orthogonality  principles:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {t}_i = \boldsymbol {u}_i + \beta \lambda _i (32) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {n}_i = \Delta \boldsymbol {u} + \beta \Delta \lambda (33)

in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {t}_i}

is the tangent of the current incremental load-displacement configuration, while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {n}_i}
is an arbitrary update direction chosen with reference to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {t}_i}

, see Figure 3 .

The scalar product of these vectors yields a residual Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {g}_i} . Form the scalar product using equations 31,32 and 33 a general expression can be derived for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \lambda = \frac{\boldsymbol {g}_i-\boldsymbol {u}_i^T\Delta \boldsymbol {u}_2}{\beta ^2\lambda _i + \boldsymbol {u}_i^T\Delta \boldsymbol {u}_1} (34) (35)

The expression can be simplified for particular cases of orthogonality. The use of the method simplifies the solution process. The method reveals exactly similar results for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda }

as obtained by Crisfield (1981) but without solving quadratic equation and selection of proper root [4].

Figure 3: arc-length orthogonality method.

5 The choice of the appropriate root

The iterative load factor is normally chosen as the solution to the quadratic equation that yields the minimum angle between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_{i-1}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_{i}}
(Crisfield, 1991),  i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda _i}
is the solution of 21 which gives the maximum product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_{i-1}^T\Delta \boldsymbol {u}_{i}}
[5]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \lambda _i = arg\left[\max _{\delta \lambda |(\delta \lambda _{i})^2 + b(\delta \lambda _{i}) + c = 0}(\Delta \boldsymbol {u}_{i-1} + \eta \delta \boldsymbol {u}_{ri} + \delta \lambda \boldsymbol {u}_f)^T \Delta \boldsymbol {u}_{k-1} \right]

(36)

6 Solution formulation at beginning of the iteration

The arc-length increment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta l_i}

for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i-th}
loading step is determined by the following procedure:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta l_i = \Delta l_{i-1} \frac{I_d}{I_{i-1}} (37) (38)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta l_{i-1}}

is the arc-length increment of the previous loading step, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I_d}
is the desired number of iterations for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i-th}
loading step before convergence and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I_{i-1}}
  is the number of iterations required to converge in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (i-1)th}
loading step.

The first iteration from the previous converge point is a predictor solution in which the appropriate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda }

cannot be obtained by solving the quadratic equation showed above.  Lam and Morley (1992) proposed a formation of  and  by introducing a scalar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mu = \frac{\Delta l}{\sqrt{\Delta \boldsymbol {u}_{pr}^T\Delta \boldsymbol {u}_{pr} + \psi ^2\Delta \lambda _{pr}^2}} (39.a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \boldsymbol {u}_1 = \mu \Delta \boldsymbol {u}_{pr} (39.b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \lambda = \mu \Delta \boldsymbol {\lambda }_{pr} (39.c) (39.d)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ()_{pr}}

corresponds to quantities of previous converged step. Another formation is also widely used in the cylindrical arc-length method  (in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi ^2}
is set equal to zero),   written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mu = \frac{\Delta l}{\sqrt{\Delta \boldsymbol {u}_{pr}^T\Delta \boldsymbol {u}_{pr}}} (40) (41)

While in both formation, the sign of the should be choose carefully to avoid track back on the current path (Crisfield, 1991). Many procedures are developed to predict the continuation direction, i.e. to choose the sign of that carries on tracing the current solution path. Some criteria are listed below:

1. Stiffness determinant. Follow the sign of the actual tangent stiffness determinant:

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): sign(\delta \lambda ) = sign(det(\boldsymbol {K}_T(\boldsymbol {u}))) </li>

   (42)

2. Incremental work. Follow the sign of the predictor work increment:

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): sign(\delta \lambda ) = sign(\delta \boldsymbol {u}^T\boldsymbol {f}_{ex}) </li>

   (43)

3. Secant path (Feng et al., 1995, 1996). The sign of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda } is determined as:

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): sign(\delta \lambda ) = sign(\Delta \boldsymbol {u}^T\delta \boldsymbol {u}) </li>

   (44)

Criteria Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (c)}

are implemented in program NOLM. Procedure (a) is widely used in commercial finite element codes and works well in the absence of bifurcations. In the presence of bifurcations, however, it is known not to be appropriate and fails in most cases. As pointed out by Crisfield (1991), its ill-conditioned behaviour stems from the fact that the sign of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {K}_T}
changes either when a limit point or when a bifurcation point is passed. In this case, the predictor cannot distinguish between these two quite different situations, unless further (usually computationally expensive) analyses are undertaken. In the presence of a bifurcation, instead of following the current equilibrium path, the solution will oscillate about the bifurcation point. This property is mathematically proved by Feng et al. (1997). Procedure (b), on the other hand, is blind  to bifurcations and can continue to trace an equilibrium path after passing a bifurcation point. However, this criterion proves ineffective in the descending branch of the load-deflection curve in snap-back  problems, where the predicted positive slope  will provoke a back tracking  load increase.  One important feature shared by the criteria (a) and (b) is the fact that they rely exclusively  on information relative to the current equilibrium point (at the beginning of the increment).  The decision on the sign of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda }
is made without considering the history of the currently  traced equilibrium path. In situations such as the ones pointed out above, this may result in wrong direction prediction. In contrast, a key point concerning criterion Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (c)}
is the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \boldsymbol {u}_n}
carries with  it information about the history of the current equilibrium path [5].

6.1 Summary Arc Length Algorithm Procedure

A basic arc length algorithm implemented in NOLM program is summarized in the following box depicted in 6.1. This algorithm correspond to the spherical arc-length method, however, with small change in the code, a cylindrical arc-length procedure can also be used .

Arc length scheme.

1. In step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n+1} , assuming that we converged in step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} , initialize iteration counter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i:=0} , and set initial guess for displacement and incremental load factor.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mu = \frac{\Delta l_{n+1}}{\sqrt{\Delta \boldsymbol {u}_{n}^T\Delta \boldsymbol {u}_{n} + \psi ^2\Delta \lambda _{n}^2}} \Delta \boldsymbol {u}_{n+1}^0 = \Delta \boldsymbol {u}_{n} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lambda _{n+1}^0 = \lambda _{n} \boldsymbol {r}^0_{n+1} = \boldsymbol {f}_{in}(\boldsymbol {u}_n)-\lambda _{n+1}^0\boldsymbol {f}_{ex}

2. Assemble the tangent stiffness matrix or a projection of it: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {K}_T = \boldsymbol {K}_T(\boldsymbol {u}_{n+1})} . It is important to calculate it because the quadratic convergence is ensured and also the snap-back point and bifurcation points can be localized.
3. Set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i= i + 1} . Solve the linear systems for perpendicular solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {u}_{ri}} and for the tangential solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {u}_{rf}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g_i = \frac{\boldsymbol {r}_i^T\boldsymbol {f}_{ex}}{\boldsymbol {f}_{ex}^T\boldsymbol {f}_{ex}} \boldsymbol {r}^\perp = \boldsymbol {r}_i-g_i\boldsymbol {f}_{ex} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {u}_{ri} = \boldsymbol {K}_T^{-1}\boldsymbol {r}^{\perp } \boldsymbol {u}_{f} = \boldsymbol {K}_T^{-1}\boldsymbol {f}_{ext} </li>

4. Find iterative load factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda ^i}
   1. if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1} (predictor solution) then compute:

      Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \lambda _{n+1}^1 = sign(\Delta \boldsymbol {u}_n^T\delta \boldsymbol {u}_f)\mu \Delta \lambda _{n} </li>

   2. if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i\neq{1}} then solve the spherical arc-length constraint equation

      Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a(\delta \lambda _{i})^2 + b(\delta \lambda _{i}) + c = 0 </li>

      with coefficients defined by 22.a, 22.b and 22.c and choose root Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda _i}

      according to  36 if real roots was found.
      

      1. if complex root was found solve the quadratic equations for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } in 25 and choose a suitable value for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } according to 27.a, 27.b 27.c,27.d. If real roots was found, come back to b and try again to solve the quadratic equation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \delta \lambda } in 21.
      2. Is a complex value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } was found apply Method Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2} , i.e, eliminate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {r}^{\perp }} and update the new value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \lambda } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {u}} according to 30.a, 30.b and 30.c.
5. Apply correction to incremental load factor

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \lambda ^{n+1}_{i+1} = \Delta \lambda _i - g_i + \delta \lambda _i </li>

6. Compute iterative displacement

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta \boldsymbol {u}^{n+1}_{i+1} = \eta \delta \boldsymbol {u}_{ri} + \delta \lambda _{i}\delta \boldsymbol {u}_{f} </li>

7. Correction to total and incremental displacements

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta \boldsymbol {u}_{i+1}^{n+1} = \Delta \boldsymbol {u}_{i} + \delta \boldsymbol {u}_{i+1} \boldsymbol {u}_{i+1}^{n+1} = \boldsymbol {u}_{i}^{n+1} + \delta \boldsymbol {u}_{i+1} </li>

8. Update residual

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {r}_{i+1}^{n+1} = \boldsymbol {f}_{in}(\boldsymbol {u}_{n+1}^{i+1})-\lambda ^{n+1}_{i+1}\boldsymbol {f}_{ex} </li>

9. Check for convergence with a appropriate norm. IF Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \| \boldsymbol {r} \| /\| \boldsymbol {f}_{ex} \| <\epsilon _{tol}} THEN set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\cdot )_{n+1}=(\cdot )^{n+1}_i} and EXIT. ELSE GO TO (ii).
10. If not convergence was not reached during the iteration, reduce the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta l_{n+1}} with a suitable user factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta } and try again.

    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta l_{n+1} = \beta \Delta l_{n+1} </li>

11. If not convergence was not reached during the reduction of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta l_{n+1}} a bifurcation analysis must be done, for getting the new equilibrium path (Future work).

7 Smeared Isotropic Rankine Damage Models

7.1 Constitutive Equation

In the last years, the so-called continuum damage model have been widely accepted as an alternative to deal with complex constitutive behavior. This approach has proved simple and versatile, as well a rigorously based in fundamental constitutive theory. The continuum damage model explains the process of gradual material deterioration induced by nucleation and grown of multitude of micro-defects. It is based on the definition of effective stress, which is introduced in connection with the hypothesis strain equivalence. In others words, the strain associated with a damage state under the applied stress Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\sigma }}

 is equivalent to the strain associated  with its undamaged state under effective  stress Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{\boldsymbol {\sigma }}}

. This damage state is monitored trough a single internal scalar variable, called damage or degradation , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d} . This variable is a measure of the lost of secant stiffness of the material, and it ranges from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0}

for undamaged material to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1}
for fully degradated one Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0 \leq d \leq 1}

.

In the present work, the effective stresses Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{\boldsymbol {\sigma }}}

are computed in terms of the total strain tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\varepsilon }}
as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{\boldsymbol {\sigma }} = \boldsymbol {C} : \boldsymbol {\varepsilon }

(45)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {C}}

is the secant fourth order constitutive tensor and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (:)}
denotes double contraction. The constitutive equations for the scalar isotropic damage model used in this work is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\sigma } = (1-d)\overline{\boldsymbol {\sigma }} =(1-d) \boldsymbol {C}_0 : \boldsymbol {\varepsilon }

(46)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {C}_0}

is the linear elastic constitutive tensor. This equations reveals that the material stiffness  is only affected by a scalar factor and therefore isotropic is preserve and the explicit integration of the constitutive equation is achieved, as only current values of strains and damage.

7.2 Isotropic damage model

The model defined in equation (45) is fully determined if the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d}

can be evaluated at every time of the deformation process. To that extends one must define a equivalent effective stress  or a suitable norm, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau }

. It is used to compare different states of the stress, so that is possible to define such concepts as loading , unloading and reloading , well known as Kuhn-Tucker conditions. The norm is a scalar and positive function and have a zero value for the undeformed state. In this work, the equivalent stress will assume the following form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tau = \left(\theta + \frac{1-\theta }{n}\sqrt{\boldsymbol {\sigma }_0:\boldsymbol {C}_0^{-1}: \boldsymbol {\sigma }_0} \right)

(47)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}

is defined as ratio compressive strength/tensile strength and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta }
is a weighting factor depending on the state of stress Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _0}

. A satisfactory definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta }

is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \theta = \frac{\sum _{i=1}^3\langle \sigma _0^i \rangle }{\sum _{i=1}^3 | \sigma _0^i |}

(48)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \langle \cdot \rangle }

are the Macaulay brackets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\langle  x \rangle  = x \hbox{, if } x\geq{0} \hbox{ and } \langle  x \rangle =0\hbox{, if } x<0)}

. Note that the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \theta }

range from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0}
for triaxial compression to one for triaxial tension and for intermediate stress a value betweens Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0<\theta{<1}}

. With the above definitions for the equivalent effective stress, the damage criterion, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{F}(\tau , r)\leq{0}} , formulated in the undamaged stress spaces, is introduced as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{F}(\tau ^t, r^t) = \tau ^t - r^t \leq{0} \hbox{ } \forall t \geq 0

(49)

where the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau ^t}

is the norm described before and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r^t}
is the current damage threshold at the time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}

. This last value is an internal stress-like variable whose controls the size of damage surface. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_0=\sigma _0}

is the initial threshold value, in this case, the initial uniaxial damage stress(tensile strength),  it must be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r^t\geq r_0=\sigma _0}

. Damage occurs when the norm Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau }

exceeds the current threshold value. Fig 4 shows the elastic domain of the function defined in Eqn 49.

Figure 4: Elastic domain selected for the isotropic damage model.

The evolution of the damage bonding surface for loading, unloading and reloading conditions is controlled by the Kuhn-Tucker relations and the damage consistency conditions, which are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot{r}\geq 0 \mathcal{F}(\tau , r) \leq{0} \dot{r}\mathcal{F}(\tau , r) = 0 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hbox{if } \mathcal{F}(\tau , r) = 0 \hbox{ then } \dot{r}\dot{\mathcal{F}}(\tau , r) = 0

(50)

leading, in view of equation (49) , to the loading condition

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot{\tau } = \dot{r}

(51)

This, in turn, the evolution of the internal variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}

may be explicitly integrated to render:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r = \hbox{max}\{ r_0, \hbox{max}(\tau (t))\} 0\leq t< \infty

(52)

Note that (51) allows to compute the current values for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}

in terms of the current value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau }

, which depends explicitly on the current total strains. Stress softening is controlled by the softening function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q = q(r)} . In this work, the following exponential softening law is used:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): q(r) = r_0 e^{(-2H_S(\frac{r-r_0}{r_0}))} \hbox{ , } r\geq r_0

(53)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_S}

is the softening parameter which defines the softening response, dependent on the element size and  also sets a maximum size for the elements that can be used in the analysis.  The figure shows a schematic representation of this function.  Finally, the damage index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d = d(r)}
is explicitly defined in terms of the corresponding current value of damage threshold,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d(r) = 1-\frac{q(r)}{r} \hbox{ , } r\geq r_0

(54)

so that it is monotonically increasing function such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq d \leq 1} .

7.3 Mechanical Dissipation

The mechanical free energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Psi }

for an isotropic damage model is defined as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Psi = (1-d)\Psi _0(\boldsymbol {\varepsilon }) = (1-d)[\frac{1}{2}\boldsymbol {\varepsilon }:\boldsymbol {C}_0:\boldsymbol {\varepsilon }] \geq 0

(55)

Thus, the rate of energy dissipation may be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot{\mathcal{D}} = \Psi _0 \dot{d} = \geq 0

(56)

Consider now an uniaxial experiments in which the tensile strain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varepsilon }

increases monotonically and quasi-statically from an initial unstressed state to another in which full degradation takes place. In this case, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Psi _0 = \frac{\sigma _0^2}{2E} = \mathcal{U}_0}

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E}

is the Young's modulus and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _0 = E\varepsilon }

. Therefore, for any process the total specific dissipated energy (per unit volume) is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{D} = \int _0^\infty \dot{\mathcal{D}} dt = \int _0^\infty \Psi _0 \dot{d}dt Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{D} = \frac{1}{2E}(\int _{r_0}^\infty q(r)dr-\int _{q_0}^0 rdq)

(57)

Combining equation (57) with (54) results:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{D} = (1+\frac{1}{H_S})\frac{\sigma _0^2}{2E} = (1+\frac{1}{H_S})\mathcal{U}_0

(58)

Finally, as it is usual in smeared crack models, it is possible to relate the specific dissipated energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{D}\mathcal{L}_e = \mathcal{G}_f

(59)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{G}_f}

is the mode I fracture energy per unit area (assumed to be a material property)  and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{L}_e}
is the computational width of the fracture zone well knows as element characteristic length, that is computed depending on the geometric dimension of the element. Therefore, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_S}
parameter can be expressed

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H_S = \frac{\overline{H}_S\mathcal{L}_e}{1-\overline{H}_S\mathcal{L}_e} \geq 0

(60)

where the specific softening parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{H}_S = \mathcal{U}_0 \mathcal{G}_f}

measures the brittleness of the material and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_S}

, the elemental softening parameter measures the brittleness of the element. For linear simplex element, the characteristic length can be taken as the representative size of the element, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{L}_e = l_e} . In this work, and assuming that the elements are almost equilateral, the size of the element can be computed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l_e^2 = \sqrt{A_e}}

for quadrilateral elements  being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_e}
the area of the finite element. It is interesting to comment that for a linear irreducible formulation, the discrete localization band Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{w}}
is only one element across. The box 7.3 shows the very simple algorithm has to be used to evaluate stresses using this smeared crack models. Note the updating of the threshold value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau ^t}
as the nonlinear behavior proceeds:

boxed BOXBox

Smeared Isotropic damage model.

1. Initial Data for time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t+1}
   1. Material Properties Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _t} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _c} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nu } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{G}_f}
   2. Current values: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\varepsilon }^{t+1}} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle d^t} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r^t}
2. Determine Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_S} . Eqn 60.
3. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=0} the initialize Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau _0=\sigma _0} .
4. Evaluate undamaged stresses.

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\sigma }_0^{t+1} = \boldsymbol {C}_0:\boldsymbol {\varepsilon }^{t+1} </li>

5. Evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau ^{n+1}} . Eqn 47.

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tau = \left(\theta + \frac{1-\theta }{n}\sqrt{\boldsymbol {\sigma }_0:\boldsymbol {C}_0^{-1}: \boldsymbol {\sigma }_0} \right) </li>

6. Update internal variables

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r^{n+1} = max(r^t,\tau ^{n+1}) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d^{n+1} = d(r^{t+1}) </li>

7. Update stresses

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\sigma ^{t+1}} = (1-d^{n+1}) \boldsymbol {\sigma _0^{t+1}} </li>

8 Validation Examples

The application of the arc-length method is presented in this work to the problem of softening material and catch the equilibrium path of the structure. The arc length method was integrated in NOLM program, together to a smeared isotropic damage model and no linear truss element. We begin to solve a very simple problem using this non linear truss element as a benchmarking. Results reveals the strong snap through and snap back in the equilibrium path of this examples.

Other examples concerns to the continuum models discretized in quadrilateral elements using the smeared crack model coded in NOLM. Another constitutive laws, implemented in NOLM is used. However, no mechanical dissipation and objectivity exist, so a very difficult convergence is found is some situations. We proposed an objective mechanical dissipation for future work to take account the dissipated energy, related strongly with the plastic work of the model. For quadratic convergence for the described isotropic smeared crack model is necessary to apply the tangent stiffness matrix. However, it is really difficult to compute and other alternatives need to be adopted. The most common way to get a projection of the tangent stiffness matrix is applying the perturbation method. But, in the present work we used the secant stiffness matrix defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {C}_{sec} = (1-d)\boldsymbol {C}_0} . This can be effectively used, but not quadratic convergence is guaranteed. In any case is more faster than using the non-damage stiffness matrix. However, in presence of snap back points, sometimes is encountered some difficult to converge.

It was coded in NOLM subroutines allow connect the results for future reading in Gid, the Pre and post-processing developed at CIMNE [6].

8.1 Two 2D nonlinear truss element

The first numerical example consist of two-member nonlinear truss element shown in the figure 5 and has been widely used as a benchmark problem for comparison of numerical solution algorithm. The following material are assumed: Axial rigidity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle EA=1884.694lb} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=25.874in}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha=63.4}
degrees.  The same example was tested by Yang and Shieh (1990). Three loading cases will be studied herein. In the first case (Case I), only a vertical load is imposed; secondly (Case II),  the horizontal load is considered as an imperfection, i.e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_u=0.05P_v}
and the third one (Case III), the vertical load is treated as an imperfection, i.e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_v = 0.05P_u}

. The results obtained by the present method for the three considered cases are shown in Figs 6, fig:Sec4-3 and 9 respectively, which are according with solution presented by Pecknold et al (1985).

Note that, in the first case an snap-through is obtained. As can be seen in the figure fig:Sec4-3, there are four limits points and two snap back points. The present analysis has demonstrated the self-adaptive capability of this implemented arc-length method. At the third case, the observation for the first load can also be applied here.

Figure 5: Two non-linear truss element. Geometry and boundary conditions.

Figure 6: Analysis result for case I.

(a) Vertical Load Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_v vs horizontal displacements.	(b) Vertical Load Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_v vs vertical displacements.
Figure 7: Analysis results for Case II.

(a) Vertical Load Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_u vs horizontal displacements.	(b) Vertical Load Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_u vs vertical displacements.
Figure 8: Analysis results for Case III.

8.2 A simple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2D benchmarking example8.2 A simple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2D benchmarking example

In this example, we test our code of arc-length method with a problem of square non-linear material under tension. A simple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2D}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\times{1} m\times m}
 block discretized in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 4}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 16}
quadrilateral bilinear element and drown in figure 9, 10a and 10b respectively, are analyzed both Von-Mises linear/exponential softening model and Drucker-Prager linear/exponential model. A plain strain analysis is performed. Material are assumed and is depicted in table 1. The softening modulus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H}
for linear softening was taken Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1MPa}

. For exponential softening, we take the following formula for evolution of shear stress

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tau _s({\kappa }) = \tau _{s0}e^{-h\kappa }

(61)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \kappa }

the equivalent plastic strain and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}
is a constant that controls the softening curve, here assumed to be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=3000}

. Note that this models, is not objective due to no dissipation of energy is controlled. For define a DP-Model, the parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha } , that controls the pressure is assumed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha=0.15} . Take into account that the objective of this examples is testing the AL method implemented in NOLM.

Table. 1 Material data of 2D benchmarking example.

Material Data	Value(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N-m} )
Young's Modulus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E}	10 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MPa}
Poisson's ratio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \upsilon }	0.33
Shear strength Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau _{s0}}	1 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle KPa}

Figure 9: Mesh Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): one .

(a) Mesh Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): two .	(b) Mesh Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): three .
Figure 10: Analysis meshes cases.

The results of the analysis is plotted in figures 11 and 12. Observe that for both material model, identical responses is obtained for the three different meshes when is used the same softening low. However, although not was used a energy dissipated criteria, all element have the same geometric properties making the softening responses objective.

(a) Vertical Load vs vertical displacement using DP-model with exponential softening.	(b) Vertical Load vs vertical displacement using DP-model with linear softening.
Figure 11: Analysis cases for DP-Model.

(a) Vertical Load vs vertical displacement using Von-Mises model with exponential softening.	(b) Vertical Load vs vertical displacement using Von-Mises model with linear softening.
Figure 12: Analysis cases for Von Mises -Model.

8.3 A Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2D pull test benchmarking example8.3 A Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2D pull test benchmarking example

In this example, analysis of a cracked flat under surface tension is taken to test our code of arc-length method. We selected three different sizes of meshes of the same model depicted in figure 13 and calculated with perfect plastic Drucker-Prager model, perfect plastic Von-Mises model and exponential softening Von-Mises model. We also tested the influence of different softening parameters with exponential softening Drucker-Prager model. The same material used in the previous example are assumed again. A plain strain analysis is performed. The analysis cases are

1. For Drucker-Prager model with exponential softening taken Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha=0.15} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_1=100} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_2=300} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_3=180} . Case I.
2. For Von-Mises model with perfect plastic. Case II.
3. For Von-mises model with with exponential softening Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=1000} . Case III.

The Case I is represented is the figure 14. Observe how the AL method obtain the softening behavior and thre different responses and post-pick load are obtained when is used different values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} . Case II represent a perfect plastic behavior and the analysis results is plotted in fig 15. Note how the AL method implemented obtained the expected perfect plastic behavior. The Case III analysis is depicted in figure 16. Note how the results is not objective due to no dissipation is controlled. As a future work, is necessary to implement a mesh objective dissipation in the standard model material implemented in NOLM.

(a) Mesh one. 50 nodes and 34 elements.	(b) Mesh two. 605 nodes and 544 elements.
(c) Mesh three. 167 nodes and 136 elements.
Figure 13: Finite element meshes for analysis.

Figure 14: Responses of pull test using three different parameter in exponential softening.

Figure 15: Perfect plastic response for three meshes using Von-Mises model.

Figure 16: Non objective response in Von-Mises material with exponential softening.

8.4 A Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2D notched beam8.4 A Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2D notched beam

In this example, we have selected for analysis a notch beam tested by Petterson(1981) . The major adantages of this experiments are that it has been repeated several times and that necessary material parameter, such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{G}_f} , has been carefully specified. Finite element meshes, dimension and properties are shown in figure 17. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 702}

nodes and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 612}
quadrilateral bi-linear element was used in the analysis. We are assuming a plane stress state for this model. The beam is subjected to a point load in the top center, producing a direct traction in the notch. An exponential softening curve was used according to the average element size. The analysis were performed by full arc-length method implemented and the smeared  isotropic crack model described. The figure 18 compare experimental results and numerical load deflection curves. Note  how the softening behavior can be satisfactorily simulated  together with the maximum load.

(a) Geometry and material properties.	(b) Mesh for analysis.
Figure 17: Finite element meshes for analysis and basic properties.

(a) Experimental and numerical results reported by Rots et al (1985).	(b) Numerical result using smeared crack model with objective dissipation.
Figure 18: Experimental and numerical result reported and numerical result using smeared crack model with objective dissipation.

8.5 Double cantilever beam benchmark

Fig shows 19 a Double cantilever beam (DBC) tested by Sock, Baron and Fracois(1979). In this work, we used the smeared crack models using two different fracture energy to guarantee objective dissipation. A total number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 608}

nodes and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 548}
linear quadrilateral elements was used for discretization.    This example is selected  because represent a pure mode I of fracture. However, some simplifications  is done in the model with respect to the original model, due to this example is aimed to test the arc length method.  Material properties is drown in table 2. Note that for define the smeared crack model is only required Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3}
parameter. In fig,  the load is plotted against the crack mouth opening displacement (CMOD).  The computation was performed for exponential softening and assuming a plane stress state.Development of the fracture Zone is represented in fig. 4.8  which represent the evolution of damage (fracture process ) For three different process in the calculation.

(a) Geometry properties.	(b) Mesh for analysis.
Figure 19: Finite element meshes for analysis and basic properties.

Table. 2 Material data of 2D benchmarking example.

Material Data	Value(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N-m} )
Young's Modulus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E}	40 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle GPa}
Poisson's ratio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \upsilon }	0.2
Tensile strength Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _0}	4 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MPa}
Fracture Energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{G}_{f1}}	350 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N/m}
Fracture Energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{G}_{f2}}	250 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N/m}

Due to the boundaries condition apply, the forces tends to open the beam in front of the notch and all damage process begin in this place. Damage progresses gradually as the loading process increases and finally the beam is separated in two part. The evolution of damage to the end of simulation is potted in 21b . Note that when the damage reaches the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1} , means that this zone is totally smeared cracked.

(a) Experimental and numerical results reported by Rots et al (1985.	(b) Numerical result using smeared crack model with objective dissipation.
Figure 20: Experimental and numerical result reported for DCB and numerical result using smeared crack model with objective dissipation.

(a) Contour Sxx Cauchy Stress field at the end of simulation. (Gid pre-post processor[6]).	(b) Evolution of damage for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{G}_{f1} at the end of simulation. (Gid pre-post processor[6]).
(c) Direccion of principal stress at the end of simulation. (Gid pre-post processor[6]).	(d) Deformed mesh Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\times{500)} . (Gid pre-post processor[6]).
Figure 21: Double cantilever beam analysis.

9 Conclusion, recommendations, actual and future works

9.1 Conlcusion

In this work we introduced and implemented the AL method in NOLM program. Basically the actual implementation in NOLM is based in the work proposed by Lam and Morley (1992) and was coded with take into account the sign for no tracking back the solution. An efficient method was included for avoid complex roots and if the problem persist, two method is available to get the convergence. In case of no convergence is reached, we introduce a user factor, for reducing the arc-length, that should be taken betweens Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (0.5-0.75)}

per cent of the actual arc length.

A linear truss-element, together to the isotropic smeared crack model was introduced for testing and benchmarking the AL method . During the work, we found several difficulties to converge in the standard constitutive law implemented in NOLM when a softening parameter was introduced. It is caused by the no mechanical dissipation in the model. However, for a perfect plastic an hardening they get the corresponding results.

All meshing process and post-processing was made with Gid[6], proving that it is possible to connect the developed owner software and this Pre-and post processing program.

9.2 Recommendations

By author experience, it is recommended, as a future work, the redesign of the basis of the NOLM code. Difficulties are found in debugging process when several static variables, jumping ( GO TO sentences) and writing process was used. Some bugs was fixed, one of them related to the solver and solution flag processes.

Its really necessary the checking and testing of the constitutive models implemented in NOLM with its corresponding benchmarking. Also recommended the change of the data base of NOLM (Common Block A), due to lost of memory and a lot of them are not using in a real run time calculations and the difficulties that could be encountered for parallelization.

The storage of the global stiffness matrix can be done in a parallel way. However, the needed to know the exactly position of its terms make difficult a future parallelization. In other words, not only be considered the parallelization of the solver using MKL Pardiso, but also all looping in elements, conditions and nodes. A suitable way to reaches this objectives is a modular or object programing inside NOLM, now available in the new INTEL COMPILER versions.

9.3 Actual and future works

The actual and future works are aimed to checking standard constitutive laws integrated in NOLM and coded and objective dissipation in the finite element. This can be done with a small relative change in the softening parameter curve adding the characterized length of the elements.

Integrating an analysis of bifurcations points when convergence can not reaches using the AL method for stopping or tracing another possible equilibrium path for the next iteration step. It is done by checking the determinant of the global stiffness matrix using the LU decomposition.

BIBLIOGRAPHY

[1] Lam, W. and Morley, C. (1992) "Arc‐Length Method for Passing Limit Points in Structural Calculation", Volume 118. Journal of Structural Engineering 1 169-185

[2] Cinthia A. G. Sousa and Paulo M. Pimenta. (2010) "A new parameter to Arc-Length method in non-linear structural analysis", Volume XXIX. Mecánica Computacional 1841-1848

[3] J.V. Ferreira and A.L. Serpa. (2005) "Application of the arc-length method in nonlinear frequency response", Volume 284. Journal of Sound and Vibration 1–2 133–149

[4] MEMON Bashir-Ahmed, SU Xiao-zu. (2004) "Arc-length technique for nonlinear finite element analysis.", Volume 5. J Zhejiang Univ Sci 5 618-28

[5] de Souza Neto, E. A. and Peri, D. and Owen, D. R. J. (2008) "Computational Methods for Plasticity". John Wiley & Sons, Ltd 1–15

[6] GiD. (2009) "The Personal Pre and Post Processor"

[7] M. A. Crisfield. (1986) "Snap-through and snap-back response in concrete structures and the dangers of under-integration", Volume 22. International Journal for Numerical Methods in Engineering 3 751-767

[8] Kyoung Soo Lee and Sang Eul Han and Taehyo Park. (2011) "A simple explicit arc-length method using the dynamic relaxation method with kinetic damping", Volume 89. Computers & Structures 1–2 216–233