Discretización por elementos finitos del problema termo-difusivo-mecánico con grandes deformaciones
Ángel Ortiz-Toranzo and Ignacio Romero Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): { }^{1,2,}
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/":): { }^{1}
Departamento de Ingeniería Mecánica, Universidad Politécnica de Madrid, Madrid, 28006, Spain
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): { }^{2}
IMDEA Materials, Madrid, 28906, Spain
Resumen
En algunos problemas relevantes de la ingeniería, el campo mecánico está fuertemente acoplado con el de temperatura y el de transporte de masa. La solución de estos problemas es compleja sobre todo cuando las deformaciones son grandes. Por ello, a menudo se acude a aproximaciones numéricas que pueden encontrar todos los campos involucrados en el problema acoplado, sus efectos y sus interacciones. En este artículo se describe una discretización por el método de los elementos finitos del problema acoplado de difusión, temperatura y deformación incluyendo el rango no lineal de deformaciones. La formulación tiene en cuenta todos los acoplamientos posibles entre los tres campos de estudio y se proporcionan todos los detalles necesarios para su completa implementación llenando un hueco de la literatura de estos métodos. Asimismo, el artículo describe la fundamentación termodinámica de los problemas acoplados termo-difusivo-mecánicos, insistiendo en la derivación de las ecuaciones de balance y restricciones que siguen de la segunda ley de la termodinámica. Los resultados del artículo serán de interés para investigadores que necesiten implementar las ecuaciones de problemas acoplados en códigos de elementos finitos y, en particular, para aplicaciones en el modelado de baterías, del comportamiento de metales bajo los efectos del hidrógeno, de geles, etc.
In certain relevant engineering problems, the mechanical field is strongly coupled with the temperature and mass transport fields. The solution to these problems is complex, especially when deformations are large. Therefore, numerical approximations are often used to find all the fields involved in the coupled problem, their effects, and their interactions. This article describes a finite element method discretization of the coupled
problem of diffusion, temperature, and deformation, including the nonlinear range of deformations. The formulation takes into account all possible couplings between the three fields of study, and all the necessary details for its complete implementation are provided, filling a gap in the literature of these methods. Additionally, the article describes the thermodynamic foundation of coupled thermo-diffusive-mechanical problems, emphasizing the derivation of balance equations and constraints that follow from the second law of thermodynamics. The results of the article will be of interest to researchers who need to implement the equations of coupled problems in finite element codes, particularly for applications in battery modeling, the behavior of metals under the effects of hydrogen, gels, etc.
(1)
(2) #1
Finite Element Discretization of the Thermo-Diffusive-Mechanical Problem with Large Deformations
OPEN ACCESS
Received: 21/07/2024
Accepted: 29/10/2024
DOI
10.23967/j.rimni.2024.10.56364
Keywords:
Problemas acoplados elementos finitos problema termo-difusivo-mecánico grandes deformaciones implementación termo-mecánica tensión-difusión
Keywords:
Coupled problems finite elements thermo-diffusive-mechanical problem large deformations implementation thermo-mechanics stress-diffusion
1 Introducción
Los problemas de acoplamiento de múltiples campos que involucran la transmisión de calor, el transporte de masa y la deformación mecánica son trascendentes en diversas aplicaciones de ingeniería y tecnología avanzada, tales como las relacionadas con materiales compuestos reforzados con fibras [1,2], dispositivos de conversión y almacenamiento de energía [3,4], tejidos biológicos [5], mezclas porosas de geomateriales como el hormigón [6-8], procesos de fragilización del acero por el hidrógeno, relevante en aplicaciones de ingeniería como las relacionadas con el almacenamiento y transporte de petróleo, gas e hidrógeno [9], estudios relacionados con los geles poliméricos [10-12], en el desarrollo de portadores para la administración de fármacos y en el estudio de tejidos [13], y en otras aplicaciones en el ámbito de la biología y medicina [14]. El acoplamiento de los campos térmico, difusivo y mecánico es la característica común de todos estos problemas y, por lo tanto, constituyen el objetivo de multitud de estudios, tanto para poder predecir el comportamiento y posibles problemas en el desempeño de estructuras y componentes de máquinas como para el desarrollo de nuevos materiales.
Durante los últimos años se han realizado multitud de investigaciones centradas en los problemas de acoplamiento entre tensiones mecánicas, transporte de masa y campo térmico. En líneas generales estos problemas se pueden conceptualizar como un sólido deformable poroso que actúa de huésped y que se encuentra embebido en líquidos, mezclas gaseosas o átomos o iones de soluto que pueden migrar dentro o fuera del sólido huésped induciendo dilatación o contracción y modificación del campo de temperatura. Existe una extensa literatura focalizada en los problemas de acoplamiento de dos o más campos, desde los estudios de Gibbs sobre la difusión de fluidos en sólidos, las investigaciones de Biot sobre el asentamiento de suelos bajo consolidación (similar en muchos casos al proceso de exprimir agua en un medio poroso elástico, seguido de sus estudios relativos a problemas termoviscoelásticos acoplados) hasta los estudios recientes sobre geles poliméricos y materiales elastoméricos [11,12,15,16], materiales porosos [17,18], fenómenos difusivos en los procesos de fragilización de aceros por hidrógeno [19,20] donde se presenta una implementación totalmente acoplada del campo mecánico y fenómenos de difusión y tejidos biológicos [5,21].
Mientras tanto, promovido por el desarrollo de baterías y celdas de combustible, el problema de la difusión en sólidos acoplado con el comportamiento térmico y mecánico ha sido ampliamente estudiado [22-24]. Con la llegada de los vehículos de tecnología eléctrica, las baterías de ion litio se han convertido en el estándar tecnológico en la motorización de este tipo de vehículos [25,26],
además de ser los principales sistemas de almacenamiento para equipos eléctricos y electrónicos de tamaño reducido [27], implantes biomédicos y componentes microelectrónicos autoalimentados [28], aplicaciones aeroespaciales [29]. Además, las baterías de ion litio, tienen aplicaciones en redes de distribución, nivelación de carga y energía renovables [30], pudiendo considerarse una tecnología con un papel fundamental en el camino hacia un futuro energético más limpio. Por todo ello, desarrollar baterías con alta capacidad energética y larga vida útil es un desafío tecnológico trascendente, siendo el daño producido en los electrodos por el proceso de litiación el cuello de botella en el desarrollo de baterías de alta densidad y uno de los retos tecnológicos presentes.
El objetivo de este estudio es presentar de manera completa el modelo y la discretización por elementos finitos del problema acoplado de tres campos, mecánico, térmico y transporte de masa, en el rango no lineal de deformación, teniendo en cuenta todos los acoplamientos posibles entre los campos pero limitando la respuesta mecánica al comportamiento elástico. Un paso previo para alcanzar el objetivo propuesto es la definición completa de los fundamentos termodinámicos de dichos problemas no lineales. Para ello, el artículo presenta de manera resumida, pero completa, estos fundamentos y los emplea para formular métodos numéricos que resuelven los tres campos, manteniendo en todo momento una conexión de la discretización y la formulación continua de estos problemas fuertemente acoplados.
El presente artículo está organizado de la siguiente forma: en la sección 2 se describirán las ecuaciones que rigen el comportamiento acoplado de los cuerpos deformables elásticos bajo los efectos de la temperatura y el transporte de masa, considerando un modelo de difusión de una sola especie. El modelo tendrá en cuenta los acoplamientos bidireccionales existentes entre los tres campos: mecánico, térmico y difusión de masa. En dicha sección, se presenta un conjunto de relaciones constitutivas para la energía libre de Helmholtz, el flujo másico de acuerdo con los postulados del principio de disipación y el flujo de calor. En la sección 3, se presenta la discretización espacio-temporal de las ecuaciones diferenciales parciales resultantes y una vía para su resolución numérica basada en la formulación de elementos finitos tipo Galerkin. En la sección 4, se presentarán algunos ejemplos de aplicación del modelo planteado. Para ilustrar la generalidad del modelo se resolverán dos problemas completamente diferentes: por un lado un ejemplo de geles y por otro se mostrará una simulación relacionada con el comportamiento de las baterías de ion litio; ambos ejemplos utilizarán modelos donde los campos termo-difusivo-mecánicos están fuertemente acoplados. Por último, se finalizará con unas conclusiones en la sección 5.
2 Planteamiento del problema
En esta sección se presenta el problema mecánico acoplado con el fenómeno de transporte de masa y con el de conducción de calor. El propósito, por tanto, es desarrollar un modelo matemático, termodinámicamente consistente, para el medio continuo con grandes deformaciones que permita describir la interacción termo-difusivo-mecánica para sólidos capaces de absorber especies químicas. Posteriormente, la solución del modelo se aproximará mediante un modelo numérico cuya implementación se describirá en detalle.
En lo que sigue, se trabajará con un cuerpo que se considerará medio continuo, definido matemáticamente como un conjunto de partículas Failed to parse (MathML with SVG or PNG fallback (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{B}=\left\{P_{1}, P_{2}, P_{3} \ldots \right\}} , de tal manera que existe un conjunto de aplicaciones inyectivas Failed to parse (MathML with SVG or PNG fallback (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{K}=\{ \chi \} }
que transforman el conjunto Failed to parse (MathML with SVG or PNG fallback (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{B}}
en conjuntos abiertos del espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}^{3}}
. Esto es, para toda partícula Failed to parse (MathML with SVG or PNG fallback (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 \in \mathcal{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 \boldsymbol{x}=\chi (P) \in \mathbb{R}^{3} ; \quad \chi (\mathcal{B})=\mathcal{O} \subseteq \mathbb{R}^{3}}
, abierto, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y \chi ^{-1}(\mathcal{O})=\mathcal{B}}
,
cumpliéndose que para cualquier pareja de aplicaciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{1}, \chi _{2} \in \mathcal{K}}
su composición
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{12}: \chi _{1}(\mathcal{B}) \subseteq \mathbb{R}^{3} \rightarrow \chi _{2}(\mathcal{B}) \subseteq \mathbb{R}^{3}, \quad \varphi _{12}=\chi _{2} \circ \chi _{1}^{-1}} ,
es diferenciable. Se denomina configuración a cada una de las infinitas aplicaciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi } . Como es habitual, se tomará como configuración de referencia, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{{ref }}} , la configuración sin deformar. Con frecuencia se utiliza la expresión "configuración" para referirse al conjunto imagen de una de ellas, es decir, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi (\mathcal{B})}
- su uso es tan extendido que aquí también se utilizará.
Las configuraciones de referencia y una cualquiera deformada definen una aplicación entre Failed to parse (MathML with SVG or PNG fallback (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{B}_{{ref }}}
y Failed to parse (MathML with SVG or PNG fallback (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{B}_{{def }}}
que se denomina deformación (véase figura 1):
|
Se define un movimiento como una familia de configuraciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{t}}
que depende de un parámetro Failed to parse (MathML with SVG or PNG fallback (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}
, el tiempo, de forma diferenciable. Esto es, si Failed to parse (MathML with SVG or PNG fallback (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{K}(\mathcal{B})}
es el espacio de configuraciones posibles de un cuerpo continuo Failed to parse (MathML with SVG or PNG fallback (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{B}}
, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{t}}
es una curva en Failed to parse (MathML with SVG or PNG fallback (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{K}(\mathcal{B})}
.
Figura 1: Campo de deformación de un medio continuo
El gradiente de deformación se define como Failed to parse (MathML with SVG or PNG fallback (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}=D \boldsymbol{\varphi }}
y se adoptará la hipótesis de que admite una descomposición multiplicativa en tres partes, Failed to parse (MathML with SVG or PNG fallback (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}^{m}, \boldsymbol{F}^{s}}
y Failed to parse (MathML with SVG or PNG fallback (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}^{t}}
, tal que:
Failed to parse (MathML with SVG or PNG fallback (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}=\boldsymbol{F}^{m} \boldsymbol{F}^{s} \boldsymbol{F}^{t} \quad }
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \quad \boldsymbol{F}^{s}=\lambda ^{s} \mathbf{1} ; \boldsymbol{F}^{t}=\lambda ^{t} \mathbf{1} ; \quad \lambda ^{s}>0 \quad \lambda ^{t}>0}
.
En esta expresión Failed to parse (MathML with SVG or PNG fallback (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}^{s}}
está asociado a la deformación del material debida a las moléculas absorbidas por la estructura base, Failed to parse (MathML with SVG or PNG fallback (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 ^{s}}
representa el estiramiento correspondiente, Failed to parse (MathML with SVG or PNG fallback (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}^{t}}
está relacionado con la deformación del material debido al efecto térmico, Failed to parse (MathML with SVG or PNG fallback (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 ^{t}}
representa su correspondiente estiramiento y Failed to parse (MathML with SVG or PNG fallback (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}^{m}}
está asociado a la deformación mecánica.
Esta descomposición multiplicativa del tensor gradiente de deformación se le atribuye a Flory [31] y su uso está muy extendido [32,33]. Existen diversas situaciones en las que también se utiliza
una descomposición multiplicativa del tensor gradiente de deformación, tales como termoelasticidad, elastoplasticidad y biomecánica.
El jacobiano de la transformación se define como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J=\operatorname{det}(\boldsymbol{F})} , continuando con la notación indicada, se tiene que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J^{m}=\operatorname{det}\left(\boldsymbol{F}^{m}\right), J^{s}=\operatorname{det}\left(\boldsymbol{F}^{s}\right)}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J^{t}=\operatorname{det}\left(\boldsymbol{F}^{t}\right)}
. Consecuentemente, a la vista de la descomposición multiplicativa del gradiente de deformación planteada en (4), el jacobiano de la transformación puede escribirse:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J=J^{m} J^{s} J^{t} \quad }
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \quad J^{s}=\left(\lambda ^{s}\right)^{3}, J^{t}=\left(\lambda ^{t}\right)^{3}}
.
Si se supone que el volumen de un mol de moléculas de soluto, que se denotará como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } , es constante y que además todo el cambio por unidad de volumen de referencia se debe al cambio de concentración de las moléculas de soluto en la estructura base, entonces:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J^{s}=1+\Omega g \quad \Rightarrow \quad \dot{J}^{s}=\Omega \dot{g}} ,
siendo Failed to parse (MathML with SVG or PNG fallback (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}
la concentración del soluto por unidad de volumen de referencia de la estructural huésped en seco. A la vista de la relación entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J^{s}}
y Failed to parse (MathML with SVG or PNG fallback (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 ^{s}}
planteada en (5), la primera ecuación de (6) puede expresarse también como
Failed to parse (MathML with SVG or PNG fallback (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 ^{s}=(1+\Omega g)^{\frac{1}{3}}} .
A continuación se acometerá la descripción matemática de las leyes fundamentales de la física que rigen el comportamiento de un medio continuo: las ecuaciones de balance.
2.1 Forma local de las ecuaciones de balance
La ecuación de balance de masa, que determina la evolución de la concentración de soluto en la estructura base, junto con la ecuación de la cantidad de movimiento y con la ecuación de balance de energía establecen el sistema local de ecuaciones diferenciales en derivadas parciales que, conjuntamente con la restricción impuesta por el segundo principio de la termodinámica, gobiernan el problema acoplado termo-difusivo-mecánico. Estos principios son válidos siempre y para cualquier material.
Para un cuerpo arbitrario con configuración de referencia Failed to parse (MathML with SVG or PNG fallback (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{B}_{{ref }}}
y contorno Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma }
, la forma local (material) de las tres ecuaciones de balance puede escribirse como [34,35]:
donde Failed to parse (MathML with SVG or PNG fallback (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{P}}
representa el primer tensor de Piola Kirchhoff, Failed to parse (MathML with SVG or PNG fallback (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{B}}
la descripción material del campo vectorial de fuerzas másicas por unidad de masa, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _{0}}
la densidad, Failed to parse (MathML with SVG or PNG fallback (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{V}}
la velocidad material Failed to parse (MathML with SVG or PNG fallback (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{V}=\frac{d}{d t} \boldsymbol{\varphi }, \dot{\boldsymbol{V}}=\frac{d}{d t} \boldsymbol{V}}
es la aceleración material, Failed to parse (MathML with SVG or PNG fallback (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}
es la energía interna por unidad de volumen, Failed to parse (MathML with SVG or PNG fallback (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{Q}}
el vector flujo de calor y Failed to parse (MathML with SVG or PNG fallback (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}
es el calor generado por las fuentes internas por unidad de volumen en la configuración de referencia y por unidad de tiempo, Failed to parse (MathML with SVG or PNG fallback (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{h}}
es el flujo másico, Failed to parse (MathML with SVG or PNG fallback (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 }
el potencial químico y Failed to parse (MathML with SVG or PNG fallback (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}
la concentración. Los símbolos GRAD ( Failed to parse (MathML with SVG or PNG fallback (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 }
) y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \operatorname{DIV}(\cdot )}
denotan, respectivamente, el gradiente y la divergencia materiales.
Las ecuaciones correspondientes al balance de la cantidad de movimiento, balance de masa y balance de la energía están complementadas por el segundo principio de la termodinámica cuya
expresión material es:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{s} \geq \frac{Q}{\Theta }-\operatorname{DIV}\left(\frac{\boldsymbol{Q}}{\Theta }\right)} .
Multiplicando ambos términos de la ecuación (9) por Failed to parse (MathML with SVG or PNG fallback (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 }
y definiendo Failed to parse (MathML with SVG or PNG fallback (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}=-\frac{1}{\Theta } \operatorname{GRAD}(\Theta )}
, se llega a la expresión
Failed to parse (MathML with SVG or PNG fallback (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{Q} \cdot \boldsymbol{G}-\boldsymbol{h} \cdot \operatorname{GRAD}(\mu ) \geq 0} .
Las ecuaciones (8) definen un problema inicial de valores de contorno con las siguientes condiciones de frontera:
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y con las siguientes condiciones iniciales:
Failed to parse (MathML with SVG or PNG fallback (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{\varphi }=\boldsymbol{\varphi }_{0}, \quad \boldsymbol{V}=\boldsymbol{V}_{0}, \quad \Theta =\Theta _{0}, \quad \mu =\mu _{0}, \quad }
para Failed to parse (MathML with SVG or PNG fallback (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}
,
siendo Failed to parse (MathML with SVG or PNG fallback (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}}
la normal unitaria hacia el exterior de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma }
. Se ha utilizado la notación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\bar{\cdot })}
y Failed to parse (MathML with SVG or PNG fallback (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 )_{0}}
para valores conocidos de las funciones en sus respectivos dominios y el contorno Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma }
se ha descompuesto como:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma =\overline{\Gamma _{\varphi } \cup \Gamma _{t}}=\overline{\Gamma _{\Theta } \cup \Gamma _{q}}=\overline{\Gamma _{\mu } \cup \Gamma _{h}}}
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \quad \Gamma _{\varphi } \cap \Gamma _{t}=\Gamma _{\Theta } \cap \Gamma _{q}=\Gamma _{\mu } \cap \Gamma _{h}=\emptyset }
.
Nótese que la variable fundamental del problema de transporte de masa es el potencial electroquímico Failed to parse (MathML with SVG or PNG fallback (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 }
y no la concentración Failed to parse (MathML with SVG or PNG fallback (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}
, que es su variable conjugada. Esto está motivado porque físicamente no se puede imponer la concentración en la frontera; tan sólo el potencial electroquímico. Sin embargo, sí que se puede imponer el flujo de masa en el contorno y no existen dispositivos para imponer el flujo de potencial electroquímico. Ver, por ejemplo [36].
En virtud del postulado II de la termodinámica se considerará la existencia de una función, denominada entropía Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s} , de los parámetros extensivos, definida para todos los estados de equilibrio y que tiene la propiedad siguiente: los valores que toman los parámetros extensivos, en ausencia de ligaduras internas, son aquellos que maximizan la entropía respecto al conjunto de los estado de equilibrio ligados. Además se supondrá que el estado termodinámico estará unívocamente determinado por el tensor gradiente de deformación Failed to parse (MathML with SVG or PNG fallback (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}} , la entropía Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s}
y la concentración Failed to parse (MathML with SVG or PNG fallback (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}
- la energía interna, que es una función de estado termodinámico, podrá escribirse como Failed to parse (MathML with SVG or PNG fallback (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=E(\boldsymbol{F}, s, g)}
. Mediante la transformada de Legendre pueden obtenerse otros potenciales, definiéndose la energía libre Failed to parse (MathML with SVG or PNG fallback (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 }
como:
Failed to parse (MathML with SVG or PNG fallback (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 (\boldsymbol{F}, \Theta , g)=E(\boldsymbol{F}, s, g)-s \Theta } ,
y el potencial gran canónico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathfrak{G}}
:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathfrak{G}(\boldsymbol{F}, \Theta , \mu )=\Psi (\boldsymbol{F}, \Theta , g)-g \mu } .
Teniendo en cuenta que la entropía Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s}
puede definirse como:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s=-\frac{\partial \mathfrak{G}}{\partial \Theta }} ,
aplicando la regla de la cadena en el primer término de la segunda ecuación de (8) se sigue que:
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o equivalentemente:
Failed to parse (MathML with SVG or PNG fallback (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 \dot{s}=Q-\operatorname{DIV}(\boldsymbol{Q})-\boldsymbol{h} \operatorname{GRAD}(\mu )} .
Previamente a establecer la formulación débil del problema, resulta necesario caracterizar dos tipos de funciones. En primer lugar se tienen las funciones de prueba, que son aquellas que pertenecen al espacio de funciones de cuadrado integrable cuya derivada también es de cuadrado integrable (espacio de Hilbert Failed to parse (MathML with SVG or PNG fallback (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}}
) y cumplen la primera de las condiciones de contorno establecidas en (11). Así se tiene que los espacios de funciones de prueba para los tres campos del problema son:
Failed to parse (MathML with SVG or PNG fallback (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{S}_{1}=\left\{\varphi \mid \boldsymbol{\varphi } \in \left[H^{1}(\mathcal{B})\right]^{3}, \boldsymbol{\varphi }=\bar{\varphi }\right.}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left.\Gamma _{\varphi }\right\}}
,
Failed to parse (MathML with SVG or PNG fallback (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{S}_{2}=\left\{\Theta \mid \Theta \in H^{1}(\mathcal{B}), \Theta =\bar{\Theta }\right.}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left.\Gamma _{\Theta }\right\}}
,
Failed to parse (MathML with SVG or PNG fallback (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{S}_{3}=\left\{\mu \mid \mu \in H^{1}(\mathcal{B}), \mu =\bar{\mu }\right.}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left.\Gamma _{\mu }\right\}}
,
y las funciones de ponderación:
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Denominando Failed to parse (MathML with SVG or PNG fallback (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{S}}
el segundo tensor de Piola-Kirchhoff, la formulación débil correspondiente a la ecuación del balance de la cantidad de movimiento, primera ecuación de (8), puede escribirse como:
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En esta ecuación, y a partir de ahora, se ha utilizado la notación Failed to parse (MathML with SVG or PNG fallback (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 , \cdot \rangle _{\Xi }}
para indicar el producto Failed to parse (MathML with SVG or PNG fallback (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_{2}}
de dos campos vectoriales o tensoriales sobre un conjunto Failed to parse (MathML with SVG or PNG fallback (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 }
. Cuando no se indica ningún conjunto se sobreentenderá que el producto escalar se extiende a toda la configuración de referencia. Más adelante se considerará el caso en el que las fuerzas inerciales son despreciables frente al resto. En esta situación la forma débil de la ecuación del balance de cantidad de movimiento se simplifica a:
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Respecto a la ecuación correspondiente al balance de la energía, segunda ecuación de (8), multiplicando dicha ecuación por una función de ponderación Failed to parse (MathML with SVG or PNG fallback (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 \Theta \in \mathcal{W}_{2}} , integrando y aplicando el teorema de Gauss, se tiene que:
Failed to parse (MathML with SVG or PNG fallback (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 \Theta \dot{s}, \delta \Theta \rangle -\langle \boldsymbol{Q}, \operatorname{GRAD}(\delta \Theta )\rangle +\langle \boldsymbol{h} \cdot \operatorname{GRAD}(\mu ), \delta \Theta \rangle =\langle Q, \delta \Theta \rangle -\langle \boldsymbol{Q} \cdot \boldsymbol{N}, \delta \Theta \rangle _{\Gamma _{q}}} ,
que es la formulación débil del balance de la energía en coordenadas materiales. En cuanto al balance de masa, tercera ecuación de (8), multiplicando dicha ecuación por una función de ponderación Failed to parse (MathML with SVG or PNG fallback (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 \mu \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 \mathcal{W}_{3}}
e integrando por partes se llega a:
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Á. Ortiz-Toranzo and I. Romero,
Utilizando la relación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \operatorname{DIV}(\boldsymbol{h} \delta \mu )=\operatorname{DIV}(\boldsymbol{h}) \delta \mu +\boldsymbol{h} \operatorname{GRAD}(\delta \mu )}
y aplicando el teorema de la divergencia, se obtiene:
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Obteniéndose así la forma débil del balance de masa.
2.2 Ecuaciones constitutivas del material
En el apartado precedente, las leyes de balance junto con la descripción cinemática habitual de los medios continuos [35] permiten establecer las ecuaciones del equilibrio del problema termo-difusivomecánico planteado. Sin embargo, es necesario definir una regla de comportamiento del material que permita vincular la cinética del problema con el equilibrio del cuerpo, a fin de poder resolver las ecuaciones de gobierno establecidas. Esta regla de comportamiento del material es lo que se denomina habitualmente modelo constitutivo.
Los modelos constitutivos elásticos más ampliamente estudiados en Mecánica de Medios continuos son los modelos hiperelásticos también llamados modelos elásticos de Green. Estos modelos son aquellos en los que se supone la existencia de una función de energía libre de Helmholtz Failed to parse (MathML with SVG or PNG fallback (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 }
definida por unidad de volumen, de forma que el primer tensor de tensiones de Piola-Kirchhoff se puede calcular simplemente derivando la función de energía libre de Helmholtz. Para el modelado del comportamiento del material termo-difusivo-mecánico, se plantea extender el modelo hiperelástico de forma que también exista un funcional, la energía libre de Helmholtz, cuyas derivadas den lugar a los campos conjugados.
Para facilitar el modelado de los procesos termo-difusivo-mecánicos y sus acoplamientos, supondremos que la energía libre estará construida por una expresión aditiva de una parte que establece la relación de acoplamiento térmico-difusivo ( Failed to parse (MathML with SVG or PNG fallback (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 ^{t-d}}
) y otra parte correspondiente al acoplamiento termomecánico ( Failed to parse (MathML with SVG or PNG fallback (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 ^{t-m}}
), otra difusivo-mecánico ( Failed to parse (MathML with SVG or PNG fallback (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 ^{d-m}}
) además de las contribuciones puramente mecánica, térmica y difusiva ( Failed to parse (MathML with SVG or PNG fallback (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 ^{m}, \Psi ^{t}, \Psi ^{d}}
). Según esta hipótesis simplificadora, la energía libre tendría el siguiente aspecto:
Failed to parse (MathML with SVG or PNG fallback (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 (\boldsymbol{F}, \Theta , g)=\Psi ^{m}(\boldsymbol{F})+\Psi ^{t}(\Theta )+\Psi ^{d}(\mu )+\Psi ^{t-d}(\Theta , g)+\Psi ^{t-m}(\boldsymbol{F}, \Theta )+\Psi ^{d-m}(\boldsymbol{F}, g)} .
Considerando en este caso que el material es hiperelástico, la energía libre de Helmholtz se expresaría Failed to parse (MathML with SVG or PNG fallback (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 =\Psi (\boldsymbol{F}(\boldsymbol{X}, t), \Theta , g, \boldsymbol{X})} , y el primer tensor de tensiones de Piola-Kirchhoff, se escribiría:
Failed to parse (MathML with SVG or PNG fallback (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{P}=\frac{\partial \Psi (\boldsymbol{F}, \Theta , g, \boldsymbol{X})}{\partial \boldsymbol{F}}} .
La definición de un material hiperelástico dada por la ecuación (27) no tiene en cuenta el principio de objetividad del modelo constitutivo, que establece que las relaciones constitutivas deben de ser válidas para cualquier observador. Denominando Failed to parse (MathML with SVG or PNG fallback (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}}
el tensor derecho de Cauchy-Green y Failed to parse (MathML with SVG or PNG fallback (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}}
el tensor derecho de Cauchy-Green, una forma alternativa de definición de material hiperelástico, que incluye intrínsecamente el principio de objetividad, es:
Failed to parse (MathML with SVG or PNG fallback (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{S}=2 \frac{\partial \Psi (\boldsymbol{C}, \Theta , g, \boldsymbol{X})}{\partial \boldsymbol{C}}} ,
La ecuación constitutiva correspondiente a la conducción de calor establece que el flujo de calor depende linealmente del gradiente térmico:
Failed to parse (MathML with SVG or PNG fallback (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{Q}=-\boldsymbol{K} \operatorname=\Theta} ,
donde Failed to parse (MathML with SVG or PNG fallback (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}}
es el tensor material de conductividad térmica, que en el caso, como se va a suponer, de un medio isótropo puede escribirse como Failed to parse (MathML with SVG or PNG fallback (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=K \mathbf{1}}
, siendo Failed to parse (MathML with SVG or PNG fallback (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}
la constante de conductividad térmica.
En cuanto a la ecuación constitutiva correspondiente al flujo difusivo, vendrá dada por la Ley de Fick, que en este contexto establece que el flujo del soluto depende linealmente del gradiente del potencial químico ([34], ec. 66.23):
Failed to parse (MathML with SVG or PNG fallback (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{h}=-\boldsymbol{M} \operatorname=\mu} ,
siendo Failed to parse (MathML with SVG or PNG fallback (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{M}}
el tensor de movilidad. Considerando medio isotrópico, en tensor de movilidad puede escribirse Failed to parse (MathML with SVG or PNG fallback (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{M}=m \boldsymbol{I}}
siendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m}
el coeficiente de movilidad (escalar) y la ecuación anterior puede expresarse de manera más sencilla:
Failed to parse (MathML with SVG or PNG fallback (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{h}=-m \operatorname=\mu} ,
donde se supondrá que el coeficiente de movilidad dependerá de la temperatura según la expresión:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m=\frac{D g}{R \Theta J}} ,
siendo Failed to parse (MathML with SVG or PNG fallback (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}
el coeficiente de difusión, Failed to parse (MathML with SVG or PNG fallback (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}
es la constante universal de los gases ideales y Failed to parse (MathML with SVG or PNG fallback (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 }
la temperatura absoluta.
El potencial químico podrá calcularse como derivada de primer orden del potencial (26), mediante la expresión:
Failed to parse (MathML with SVG or PNG fallback (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 =\frac{\partial \Psi }{\partial g}} .
En cuanto a la capacidad térmica:
Failed to parse (MathML with SVG or PNG fallback (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=-\Theta \frac{\partial ^{2} \Psi }{\partial \Theta ^{2}}} .
Teniendo en cuenta que Failed to parse (MathML with SVG or PNG fallback (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 }
es positiva, la positividad estricta de la capacidad térmica para todo valor de Failed to parse (MathML with SVG or PNG fallback (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 }
es equivalente a expresar que la función entropía es estrictamente creciente con la temperatura Failed to parse (MathML with SVG or PNG fallback (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 }
y , asimismo, equivale a indicar que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial \Psi ^{2}}{\partial \Theta ^{2}}<0}
para todo valor de Failed to parse (MathML with SVG or PNG fallback (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 }
, esto es, que Failed to parse (MathML with SVG or PNG fallback (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 }
es estrictamente cóncava considerando la 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 \Theta }
.
En lo que respecta a las derivadas segundas del potencial termodinámico correspondiente a la energía libre de Helmholtz, se define el tensor de acoplamiento termo-mecánico, Failed to parse (MathML with SVG or PNG fallback (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{M}} , y su correspondiente parte simétrica, Failed to parse (MathML with SVG or PNG fallback (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{M}^{s}}
:
Failed to parse (MathML with SVG or PNG fallback (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{M}=\Theta \frac{\partial ^{2} \Psi }{\partial \boldsymbol{F} \partial \Theta }, \quad \boldsymbol{M}^{s}=\Theta \frac{\partial ^{2} \Psi }{\partial \Theta \partial \boldsymbol{C}}} .
Para el cálculo de Failed to parse (MathML with SVG or PNG fallback (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{M}}
se tendrá en cuenta que:
Failed to parse (MathML with SVG or PNG fallback (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{M}^{s}=\Theta \frac{\partial ^{2} \Psi }{\partial \Theta \partial C}=\frac{\Theta }{2} \frac{\partial \boldsymbol{S}}{\partial \Theta }=\frac{\Theta }{2} \frac{\partial \left(\boldsymbol{F}^{-1} \boldsymbol{P}\right)}{\partial \Theta }=\frac{\Theta }{2} \boldsymbol{F}^{-1} \frac{\partial ^{2} \Psi }{\partial \Theta \partial \boldsymbol{F}}=\frac{1}{2} \boldsymbol{F}^{-1} \boldsymbol{M}} ,
consecuentemente
Failed to parse (MathML with SVG or PNG fallback (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{M}=2 \boldsymbol{F} \boldsymbol{M}^{s}} .
Se definirá un segundo tensor de acoplamiento, difusivo-mecánico, que vendrá dado por:
Failed to parse (MathML with SVG or PNG fallback (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{L}=\frac{\partial ^{2} \Psi }{\partial \boldsymbol{F} \partial \mu }=\frac{\partial \boldsymbol{P}}{\partial \mu }} .
En este caso, al existir tres campos acoplados, surge un nuevo acoplamiento, en este caso un escalar: el coeficiente de acoplamiento difusivo-térmico:
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3 Discretización de las ecuaciones y linealización
A continuación, se presentará la formulación del problema algebraico discreto asociado a las ecuaciones que gobiernan el problema de sólidos con grandes deformaciones y acoplamiento termo-difusivo-mecánico y que resulta de la discretización espacial y temporal de dichas ecuaciones.
Para ello, se partirá de la formulación débil del problema expresado mediante las ecuaciones (22), (23) y (25). En primer lugar se dividirá el dominio en el que está definido el problema, Failed to parse (MathML with SVG or PNG fallback (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 } , en Failed to parse (MathML with SVG or PNG fallback (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}
elementos o subdominios Failed to parse (MathML with SVG or PNG fallback (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 ^{(h)}}
y se reemplazan los espacios de dimensión infinita Failed to parse (MathML with SVG or PNG fallback (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{S}_{1}, \mathcal{S}_{2}}
y Failed to parse (MathML with SVG or PNG fallback (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{S}_{3}}
a los cuales pertenecen las soluciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi , \Theta }
y Failed to parse (MathML with SVG or PNG fallback (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 }
, por los espacios de dimensión finita Failed to parse (MathML with SVG or PNG fallback (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{S}_{1}^{h}, \mathcal{S}_{2}^{h}}
y Failed to parse (MathML with SVG or PNG fallback (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{S}_{3}^{h}}
, tales que Failed to parse (MathML with SVG or PNG fallback (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{S}_{1}^{h} \subset \mathcal{S}_{1}, \mathcal{S}_{2}^{h} \subset \mathcal{S}_{2}}
y Failed to parse (MathML with SVG or PNG fallback (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{S}_{3}^{h} \subset \mathcal{S}_{3}}
a los cuales pertenecen las funciones aproximadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi ^{h}, \Theta ^{h}}
y Failed to parse (MathML with SVG or PNG fallback (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 ^{h}}
, respectivamente. Asimismo, los espacios de dimensión infinita Failed to parse (MathML with SVG or PNG fallback (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}_{1}, \mathcal{W}_{2}}
y Failed to parse (MathML with SVG or PNG fallback (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}_{3}}
, a los cuales pertenecen las funciones de ponderación Failed to parse (MathML with SVG or PNG fallback (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 \varphi } , Failed to parse (MathML with SVG or PNG fallback (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 \Theta }
y Failed to parse (MathML with SVG or PNG fallback (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 \mu }
, se reemplazan por los espacios de dimensión finita Failed to parse (MathML with SVG or PNG fallback (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}_{1}^{h}, \mathcal{W}_{2}^{h}}
y Failed to parse (MathML with SVG or PNG fallback (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}_{3}^{h}}
, tales que Failed to parse (MathML with SVG or PNG fallback (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}_{1}^{h} \subset \mathcal{W}_{1}, \mathcal{W}_{2}^{h} \subset }
Failed to parse (MathML with SVG or PNG fallback (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}_{2}}
y Failed to parse (MathML with SVG or PNG fallback (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}_{3}^{h} \subset \mathcal{W}_{3}}
, a los cuales pertenecen las funciones aproximadas Failed to parse (MathML with SVG or PNG fallback (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 \varphi ^{h}, \delta \Theta ^{h}}
y Failed to parse (MathML with SVG or PNG fallback (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 \mu ^{h}}
, respectivamente.
Suponiendo que cada miembro de Failed to parse (MathML with SVG or PNG fallback (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{S}_{1}^{h}, \mathcal{S}_{2}^{h}}
y Failed to parse (MathML with SVG or PNG fallback (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{S}_{3}^{h}}
puede expresarse de la forma:
Failed to parse (MathML with SVG or PNG fallback (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{\varphi }^{h}=\boldsymbol{\varphi }_{v}^{h}+\boldsymbol{\varphi }_{g}^{h}, \quad \Theta ^{h}=\Theta _{v}^{h}+\Theta _{g}^{h}, \mu ^{h}=\mu _{v}^{h}+\mu _{g}^{h}} ,
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{v}^{h} \in \mathcal{W}_{1}^{h}, \Theta _{v}^{h} \in \mathcal{W}_{2}^{h}}
y Failed to parse (MathML with SVG or PNG fallback (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 _{v}^{h} \in \mathcal{W}_{3}^{h}}
satisfacen las condiciones de contorno Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi _{g}^{h}=\boldsymbol{\varphi }^{h}, \Theta _{g}^{h}=\Theta ^{h} \mathrm{y}}
Failed to parse (MathML with SVG or PNG fallback (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 _{g}^{h}=\mu ^{h}}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _{\varphi }, \Gamma _{\Theta } \mathrm{y} \Gamma _{\mu }}
, respectivamente.
Cualquiera de las funciones de ponderación Failed to parse (MathML with SVG or PNG fallback (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 \varphi ^{h}, \delta \Theta ^{h}}
y Failed to parse (MathML with SVG or PNG fallback (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 \mu ^{h}}
, al pertenecer a espacios lineales de dimensión finita, se pueden escribir como combinación de Failed to parse (MathML with SVG or PNG fallback (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}
funciones de forma, Failed to parse (MathML with SVG or PNG fallback (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_{i} \operatorname{con} i=1,2, \ldots , n}
, linealmente independientes. La expresión discretizada correspondiente al balance de la cantidad de movimiento se obtiene sustituyendo en su formulación débil Failed to parse (MathML with SVG or PNG fallback (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{\varphi }}
por su aproximación Failed to parse (MathML with SVG or PNG fallback (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{\varphi }^{h}}
, expresada como combinación lineal de las funciones de forma Failed to parse (MathML with SVG or PNG fallback (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_{i}} , quedaría:
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Del mismo modo, sustituyendo Failed to parse (MathML with SVG or PNG fallback (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 \Theta }
por Failed to parse (MathML with SVG or PNG fallback (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 \Theta ^{h}=\sum _{a=1}^{N_{n d}} N^{a}(\boldsymbol{X}) \delta \Theta ^{a}}
en la ecuación correspondiente a la formulación débil del balance de la energía, se obtiene la ecuación discretizada del balance de la energía:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\langle \Theta \dot{s}, \sum _{a} N^{a} \delta \Theta ^{a}\right\rangle -\left\langle \boldsymbol{Q}, \sum _{a} \nabla N^{a} \delta \Theta ^{a}\right\rangle +\left\langle \boldsymbol{h} \cdot \operatorname{GRAD}(\mu ), \sum _{a} N^{a} \delta \Theta ^{a}\right\rangle }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle =\left\langle Q, \sum _{a} N^{a} \delta \Theta ^{a}\right\rangle -\left\langle \boldsymbol{Q} \cdot \boldsymbol{N}, \sum _{a} N^{a} \delta \Theta ^{a}\right\rangle _{\Gamma _{q}}}
.
De forma similar, a partir de la formulación débil del balance de masa, sustituyendo Failed to parse (MathML with SVG or PNG fallback (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 \mu }
por su aproximación correspondiente Failed to parse (MathML with SVG or PNG fallback (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 \mu ^{h}}
expresada como combinación lineal de las funciones de forma, esto
es, Failed to parse (MathML with SVG or PNG fallback (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 \mu ^{h}=\sum _{a=1}^{N_{n d}} N^{a}(\boldsymbol{X}) \delta \mu ^{a}} , se obtiene la formulación discretizada de la expresión correspondiente al balance de masa:
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Se ha realizado la discretización espacial de las ecuaciones de evolución por aplicación del método de elementos finitos, obteniendo así las ecuaciones (41), (42) y (43) procedentes de la aplicación del balance de la cantidad de movimiento, balance de la energía y principio de conservación de la masa respectivamente. Sin embargo, en las ecuaciones (42) y (43) aparecen las derivadas temporales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{\Theta }, \dot{\boldsymbol{F}}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{g}}
, por lo que dichas ecuaciones tambien deben ser discretizadas con respecto al tiempo. Para ello se realizará una partición del intervalo de integración Failed to parse (MathML with SVG or PNG fallback (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, T]}
en Failed to parse (MathML with SVG or PNG fallback (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}
subintervalos del tipo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[t_{n}, t_{n+1}\right]}
, con Failed to parse (MathML with SVG or PNG fallback (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=t_{0}<t_{1}<\ldots t_{N}=T}
tal que Failed to parse (MathML with SVG or PNG fallback (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 t=t_{n+1}-t_{n}}
y se utilizará el método implícito de Euler, según el cual:
Failed to parse (MathML with SVG or PNG fallback (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}_{n+1} \approx \boldsymbol{F}_{n}+\Delta t \cdot \dot{\boldsymbol{F}}_{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/":): {\textstyle s_{n+1} \approx s_{n}+\Delta t \cdot \dot{s}_{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/":): {\textstyle g_{n+1} \approx g_{n}+\Delta t \cdot \dot{g}_{n+1}} ,
donde los subíndices Failed to parse (MathML with SVG or PNG fallback (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}
y Failed to parse (MathML with SVG or PNG fallback (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}
denotan el instante de la evaluación, ya sea Failed to parse (MathML with SVG or PNG fallback (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_{n} \mathrm{o} t_{n+1}}
. Según lo anterior, sustituyendo los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{\Theta }, \dot{\boldsymbol{F}}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{g}}
, de las expresiones (44), en la ecuación (42) quedaría:
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Sustituyendo el valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{g}} , de la última expresión de (44), en la ecuación (43) puede escribirse:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left\langle \frac{g_{n+1}-g_{n}}{\Delta t}, \sum _{a} N^{a} \delta \mu ^{a}\right\rangle -\left\langle \boldsymbol{h}, \sum _{a} \nabla N^{a} \delta \mu ^{a}\right\rangle =-\left\langle \boldsymbol{h} \cdot \boldsymbol{N}, \sum _{a} N^{a} \delta \mu ^{a}\right\rangle _{\Gamma _{h}}} .
Las ecuaciones de conservación-balance expuestas, constituyen un sistema de ecuaciones no lineal que deben ser previamente linealizadas en aras de acometer su resolución numérica por las técnicas habituales de resolución de sistemas de ecuaciones lineales basadas en el método de Newton Raphson. Los detalles del proceso de linealización pueden encontrarse en el Apéndice A.
4 Aplicaciones
El objetivo de esta sección es ilustrar las posibilidades que ofrece el modelo planteado de cara a resolver diferentes problemas acoplados, destacando además la generalidad del método propuesto que es capaz de resolver problemas no lineales complejos que involucran tres campos, o únicamente uno o dos campos, bloqueando la variación de las variables correspondientes mediante la imposición de las correspondientes condiciones de contorno. En todos los ejemplos se ha empleado una discretización espacial con elementos hexaédricos trilineales con una formulación de deformación supuesta (la formulación "B-barra") para los términos de la energía mecánica.
4.1 Ejemplo: geles elastoméricos con dos campos acoplados
Como primer ejemplo, para ilustrar la confluencia del modelo de tres campos a un modelo de dos campos, se resolverá un problema de tensión-difusión acoplado, imponiendo en cada punto el valor de
la temperatura. Se trata de los problemas resueltos mediante modelos de dos campos tensión-difusión tradicionales [32], variacionales [11], y sobre los que hay estudios experimentales [37].
En particular, se considerará un tubo cilíndrico de gel, con diámetro exterior Failed to parse (MathML with SVG or PNG fallback (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} , espesor Failed to parse (MathML with SVG or PNG fallback (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}
y longitud Failed to parse (MathML with SVG or PNG fallback (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}
al que se le sujeta una de sus bases mientras se inyecta fluido por la cara opuesta mediante la imposición de un potencial químico, utilizando el modelo de tres campos presentado bloqueando las variaciones térmicas. La ecuación del potencia químico utilizada para establecer la condición de contorno propuesta es:
Failed to parse (MathML with SVG or PNG fallback (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 (t)=\mu ^{0}+\mu _{0} \exp \left(-t / t_{d}\right)\quad \operatorname{con} t_{d}=60} .
Para el modelado del comportamiento del material termo-difusivo-mecánico, se va a plantear un modelo constitutivo hiperelástico, de forma que el problema termo-difusivo-mecánico puede describirse a partir un funcional, energía libre de Helmholtz, dada por la expresión:
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En la expresión anterior Failed to parse (MathML with SVG or PNG fallback (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 _{0}}
es el potencial químico de referencia, Failed to parse (MathML with SVG or PNG fallback (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}
es la concentración de fluido en moles de fluido por unidad de volumen de estructura base seca correspondiente al fenómeno difusivo, Failed to parse (MathML with SVG or PNG fallback (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}
es la constante universal de los gases ideales, Failed to parse (MathML with SVG or PNG fallback (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 }
es la temperatura absoluta, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }
denota el volumen de un mol de moléculas de fluido, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi }
es un parámetro adimensional que establece una medida de la entalpía de la mezcla conocido como parámetro de Flory-Huggins, Failed to parse (MathML with SVG or PNG fallback (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(\theta )=N_{R} k_{B} \theta }
es el módulo de cizalladura, que se considerará dependiente de la temperatura, de la constante de Bolzman ( Failed to parse (MathML with SVG or PNG fallback (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_{B}}
) y del número de cadenas de polímero por volumen de referencia, Failed to parse (MathML with SVG or PNG fallback (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}
es el módulo de rigidez volumétrica de la estructura base, Failed to parse (MathML with SVG or PNG fallback (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 }
y Failed to parse (MathML with SVG or PNG fallback (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 }
son los coeficientes de expansión térmica y química respectivamente, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\lambda }}
representa el alargamiento efectivo, definido como [34]:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\lambda } \stackrel{{ def }}{=} \sqrt{\frac{1}{3}\left(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}\right)}=\sqrt{\frac{1}{3} \operatorname{tr} \boldsymbol{B}}=\sqrt{\frac{1}{3} \operatorname{tr} \boldsymbol{C}}} ,
donde Failed to parse (MathML with SVG or PNG fallback (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 _{i} i=1,2,3} , son los estiramientos principales, y se ha tenido en cuenta que Failed to parse (MathML with SVG or PNG fallback (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{B}=\boldsymbol{F} \boldsymbol{F}^{T}}
(tensor izquierdo de Cauchy-Green), y la propiedad conmutativa de la traza del producto.
El material utilizado se trata de un gel y sus propiedades son las mismas que [32], ampliadas con el coeficiente de expansión térmica y el coeficiente de expansión debida a la difusión. Las propiedades del material se resumen en la tabla Failed to parse (MathML with SVG or PNG fallback (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[10,32]} .
Tabla 1: Propiedades material para los ejemplos de la sección 4 relativos a geles
| Propiedad | Valor |
| Failed to parse (MathML with SVG or PNG fallback (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 ^{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 0 \mathrm{~J} / \mathrm{mol}} |
| Failed to parse (MathML with SVG or PNG fallback (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} | 100 MPa |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10^{-4} \mathrm{~m}^{3} / \mathrm{mol}} |
| 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/":): {\textstyle 5 \cdot 10^{-9} \mathrm{~m}^{2} / \mathrm{s}} |
| Failed to parse (MathML with SVG or PNG fallback (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_{r}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2.43 / \mathrm{m}^{3}}
|
(Continued)
Tabla 1 (continued)
| Propiedad | Valor |
| Failed to parse (MathML with SVG or PNG fallback (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} | 1 MPa (a la temp. de ref.) |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{L}} | 0.1 (adimensional) |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{H}} | 0.7 (adimensional) |
| Failed to parse (MathML with SVG or PNG fallback (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 } | 5.0 K |
| Failed to parse (MathML with SVG or PNG fallback (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_{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 10^{7} \mathrm{~J} /\left(\mathrm{m}^{3} \mathrm{~K}\right)} |
| Failed to parse (MathML with SVG or PNG fallback (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} | Failed to parse (MathML with SVG or PNG fallback (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 \mathrm{~W} /(\mathrm{m} \mathrm{K})} |
| Failed to parse (MathML with SVG or PNG fallback (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 } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10^{-5} \mathrm{~K}^{-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 \beta } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 6 \cdot 10^{-5} \mathrm{~mm}^{3} / \mathrm{mol}}
|
Se supondrá una dependencia térmica del parámetro de Flory-Huggins dada por la siguiente expresión:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi (\Theta )=\frac{1}{2}\left(\chi _{L}+\chi _{H}\right)-\frac{1}{2}\left(\chi _{L}-\chi _{H}\right)\tanh \left(\frac{\Theta{-\Theta}_{T}}{\Delta }\right)} ,
donde Failed to parse (MathML with SVG or PNG fallback (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 _{T}}
es la temperatura de transición del gel, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{L}}
es el valor del parámetro de Flory-Huggins a temperaturas inferiores a la temperatura de transición del gel y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{H}}
es el valor de dicho parámetro a temperaturas superiores y Failed to parse (MathML with SVG or PNG fallback (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 }
es la anchura de la zona de transición de temperatura entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{L}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{H}}
.
A partir de la energía libre de Helmholtz propuesta, en primer lugar, puede obtenerse el segundo tensor de Piola-Kirchhoff:
Failed to parse (MathML with SVG or PNG fallback (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{S}=2 \frac{\partial \Psi }{\partial \boldsymbol{C}}=G\left(\boldsymbol{I}-\boldsymbol{C}^{-1}\right)+K J^{s} \ln \boldsymbol{J}^{e} \boldsymbol{C}^{-1}-3 \alpha K\left(\theta{-\theta}_{0}\right)\boldsymbol{C}^{-1}-3 \beta K\left(g-g_{0}\right)\boldsymbol{C}^{-1}} ,
y a partir de éste, el primer tensor de Piola-Kirchhoff y el tensor de tensiones de Cauchy pueden obtenerse mediante las conocidas relaciones Failed to parse (MathML with SVG or PNG fallback (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{P}=\boldsymbol{F} \boldsymbol{S}}
y Failed to parse (MathML with SVG or PNG fallback (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 }=J^{-1} \boldsymbol{F} \boldsymbol{S} \boldsymbol{F}^{T}}
, quedando:
Failed to parse (MathML with SVG or PNG fallback (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{P}=G\left(\boldsymbol{F}-\boldsymbol{F}^{T}\right)+\boldsymbol{J}^{s} K \mathbf{J}^{\mathrm{e}} \boldsymbol{F}^{-T}-3 \alpha K\left(\theta{-\theta}_{0}\right)\boldsymbol{F}^{-T}-3 \beta K\left(g-g_{0}\right)\boldsymbol{F}^{-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 \boldsymbol{\sigma }=J^{-1}\left[G(\boldsymbol{B}-\boldsymbol{I})+K J^{s} \ln J^{e} \boldsymbol{I}-3 \alpha K\left(\theta{-\theta}_{0}\right)\boldsymbol{I}-3 \beta K\left(g-g_{0}\right)\boldsymbol{I}\right]} .
El potencial químico se obtiene por derivación respecto a la concentración de la expresión de energía libre de Helmholtz propuesta:
Failed to parse (MathML with SVG or PNG fallback (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 =\frac{\partial \Psi }{\partial g}=\mu _{0}+R \Theta \left[\ln (1-\phi )+\phi{+\chi}\phi ^{2}\right]+\Omega \frac{1}{2} K\left(\ln J^{e}\right)^{2}-K \Omega \ln J^{e}-3 \beta K \ln J} ,
siendo:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \phi =\frac{1}{1+\Omega g}} .
En lo que respecta a las derivadas segundas del potencial termodinámico correspondiente a la energía libre de Helmholtz, la parte simétrica del tensor de acoplamiento termo-mecánico, puede obtenerse como:
Failed to parse (MathML with SVG or PNG fallback (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{M}^{s}=\Theta \frac{\partial ^{2} \Psi }{\partial \Theta \partial \boldsymbol{C}}=\frac{\Theta }{2} \frac{1}{2} \frac{\partial \boldsymbol{S}}{\partial \Theta }=\frac{\Theta }{2} \frac{\partial }{\partial \Theta }\left[G\left(\boldsymbol{I}-\boldsymbol{C}^{-1}\right)+K J^{s} \ln J^{e} \boldsymbol{C}^{-1}\right]=}
|
El tensor de acoplamiento termo-mecánico, Failed to parse (MathML with SVG or PNG fallback (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{M}} , se obtendrá mediante la ecuación (37) a partir de su parte simétrica (56).
En cuanto al tensor material de acoplamiento difusivo-mecánico, definido en la ecuación (38), y el coeficiente de acoplamiento difusivo-térmico, definido en la ecuación (39), se calcularán de forma numérica en la implementación computacional del modelo.
La figura 2 y la figura 3 muestran, respectivamente, los valores de la concentración y la tensión de von Mises. En todos los casos el fluido se inyecta en el fluido desde el exterior por la cara contraria a su base, a la que se han impedido los desplazamientos. Los tubos con geometrías 1 y 2 se pandean como resultado del hinchamiento presentado, 8 y 6 pétalos. El tubo con la geometría 3 es más grueso y no se pandea. Todo ello coincidiendo con los resultados experimentales [37] y una simulación de referencia [11].
Figura 2: De arriba a abajo, evolución de la concentración para Failed to parse (MathML with SVG or PNG fallback (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=0,300,600,900 \mathrm{~s}} . De izquierda a derecha, geometrías 1,2 y 3 de la tabla 2 .
Figura 3: De arriba a abajo, evolución de la tensión de von Mises Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mathrm{Pa}]}
para Failed to parse (MathML with SVG or PNG fallback (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=0,300,600,900 \mathrm{~s}}
. De izquierda a derecha, geometrías 1,2 y 3 de la tabla 2
4.2 Ejemplo: geles elastoméricos con tres campos acoplados
Para ilustrar el acoplamiento triple de los campos mecánico, térmico y difusivo se presenta un segundo ejemplo de los tres campos fuertemente acoplados describiéndose la respuesta del sólido como se indica en la sección 2.
En este ejemplo se ha utilizado una corona cilíndrica con las dimensiones correspondientes a la geometría 1 indicada en la tabla 2, las propiedades del material siguen siendo las indicadas en la tabla 1. En cuanto a las condiciones de contorno, se ha retenido el desplazamiento de los puntos de la superficie exterior impidiendo la deformación de la misma pero permitiendo los desplazamientos de los puntos interiores y se ha inyectado fluido por una de las bases mediante la imposición del potencial químico indicado en la expresión (47).
En la figura 4 y la figura 5 se muestran la temperatura y la tensión de von Mises en varios instantes de la simulación. En la figura 5 se ha practicado una sección radial en la corona cilíndrica para poder ver los puntos interiores y un factor de escala que permita apreciar su desplazamiento.
Tabla 2: Dimensiones tubo cilíndrico de diámetro exterior Failed to parse (MathML with SVG or PNG fallback (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} , altura Failed to parse (MathML with SVG or PNG fallback (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}
y espesor Failed to parse (MathML with SVG or PNG fallback (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}
| Geometría | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{D}(\mathrm{mm})} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{t}(\mathrm{mm})} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{h}(\mathrm{mm})} |
| 1 | 4.636 | 0.206 | 0.6 |
| 2 | 3.191 | 0.118 | 0.6 |
| 3 | 3.191 | 0.355 | 0.6 |
|
|
Figura 4: Evolución de la temperatura Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mathrm{K}]}
para Failed to parse (MathML with SVG or PNG fallback (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=0,100,200,300 \mathrm{~s}}
.png)
Figura 5: Evolución de la tensión de von Mises [Pa] para Failed to parse (MathML with SVG or PNG fallback (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=0,100,200,300 \mathrm{~s}} . Se ha aplicado un factor de escala para apreciar la deformación de los puntos que no están situados en la superficie exterior
4.3 Ejemplo: con tres campos acoplados aplicado a baterías de ion litio
En aras de ilustrar la flexibilidad del modelo presentado para resolver problemas que involucran deformación mecánica, transmisión de calor y transporte de masa, con todos los campos acoplados, se planteará un tercer ejemplo, tomando como base el ejemplo del artículo [38] referente a baterías de ion litio, extendiéndolo para incorporar el campo térmico. Se analizará un cuarto de corona cilíndrica, cuya geometría y condiciones de contorno se muestran en la figura 6. Como condición de contorno se mantendrá un potencial prefijado en la superficie exterior utilizando una expresión similar a la empleada en [39]:
Failed to parse (MathML with SVG or PNG fallback (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 (t)=\mu ^{0}+\left(\mu ^{0}-\mu ^{f}\right)\frac{t}{T}} ,
de forma que el potencial químico aumenta linealmente desde un valor Failed to parse (MathML with SVG or PNG fallback (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 ^{0}}
hasta un valor Failed to parse (MathML with SVG or PNG fallback (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 ^{f}}
, siendo Failed to parse (MathML with SVG or PNG fallback (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 ^{0}=-5.800 \mathrm{~kJ} / \mathrm{mol}, \mu ^{f}=5.800 \mathrm{~kJ} / \mathrm{mol}}
y Failed to parse (MathML with SVG or PNG fallback (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=30}
minutos.
Figura 6: Cuarto de corona cilíndrica: geometría y discretización
Para el modelado del comportamiento del material termo-difusivo-mecánico, se utilizará la siguiente energía libre de Helmholtz, construida como una expresión aditiva de una parte que establece la relación de acoplamiento térmico-difusivo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(\Psi ^{t-d}\right)} , otra parte correspondiente al acoplamiento termo-mecánico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(\Psi ^{t-m}\right)}
y otra relativa al acoplamiento difusivo-mecánico ( Failed to parse (MathML with SVG or PNG fallback (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 ^{d-m}}
), además de las partes puramente térmica, mecánica y difusiva ( Failed to parse (MathML with SVG or PNG fallback (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 ^{t}, \Psi ^{m}, \Psi ^{d}}
, respectivamente).
|
La expresión anterior es similar a la planteada en la expresión (48), con algunas salvedades. Por un lado, se ha considerado que el módulo elasticidad transversal no tiene dependencia de la temperatura para el rango térmico previsto en el problema en cuestión. Además, el término de acoplamiento térmico difusivo se ha visto modificado, desapareciendo el término que involucraba el parámetro de Flory-Huggins en el los ejemplos de geles, quedando para dicho acoplamiento una expresión similar a la utilizada en [11].
La expresión correspondientes al potencial químico quedaría:
Failed to parse (MathML with SVG or PNG fallback (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 =\frac{\partial \Psi }{\partial g}=\mu _{0}+R \Theta [\ln (1-\phi )+\phi ]+\Omega \frac{1}{2} K\left(\ln J^{e}\right)^{2}-K \Omega \ln J^{e}-3 \beta K \ln J} .
Manteniéndose invariables respecto a los ejemplos anteriores las expresiones analíticas correspondientes a los tensores de Piola-Kirchhoff y tensor de tensiones de Cauchy ((51)-(53)) y a las
correspondientes a las derivadas segundas del la energía libre de Helmholtz. Las propiedades del material del material del electrodo pueden verse en la tabla 3 [38,40,41].
Tabla 3: Propiedades material
| Propiedad | Valor |
| Failed to parse (MathML with SVG or PNG fallback (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 ^{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 0 \mathrm{~J} / \mathrm{mol}} |
| Failed to parse (MathML with SVG or PNG fallback (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} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 6.94 \cdot 10^{4} \mathrm{MPa}} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3.5 \cdot 10^{-6} \mathrm{~m}^{3} / \mathrm{mol}} |
| 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/":): {\textstyle 10^{-14} \mathrm{~m}^{2} / \mathrm{s}} |
| Failed to parse (MathML with SVG or PNG fallback (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} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3.97 \cdot 10^{4} \mathrm{MPa}} |
| Failed to parse (MathML with SVG or PNG fallback (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_{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 0.5 \cdot 10^{7} \mathrm{~J} /\left(\mathrm{m}^{3} \mathrm{~K}\right)} |
| Failed to parse (MathML with SVG or PNG fallback (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} | Failed to parse (MathML with SVG or PNG fallback (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 \mathrm{~W} /(\mathrm{m} \mathrm{K})} |
| Failed to parse (MathML with SVG or PNG fallback (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 } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10^{-5} \mathrm{~K}^{-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 \beta } | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 6 \cdot 10^{-5} \mathrm{~mm}^{3} / \mathrm{mol}}
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En la figura 7 puede verse la evolución de la deformación y de la temperatura. En la figura 8 puede verse la tensión de von Mises obteniéndose valores máximos de la tensión de von Mises similares a los presentados en diversos estudios [42-44].
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Figura 7: Evolución de la temperatura Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mathrm{K}]}
para Failed to parse (MathML with SVG or PNG fallback (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=0,300,900,1000 \mathrm{~s}}
Figura 8: Evolución de la tensión de von Mises [Pa] para Failed to parse (MathML with SVG or PNG fallback (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=0,300,900,1000 \mathrm{~s}}
5 Resumen y conclusiones
En este artículo se ha presentado una discretización por el método de los elementos finitos del problema de tres campos térmico-difusivo-mecánico. El problema incluye la transmisión de calor, el transporte de masa y el equilibrio mecánico, y todos los acoplamientos posibles han sido tenidos en cuenta. Además, la formulación es adecuada para grandes deformaciones.
A. Ortiz-Toranzo and I. Romero Discretización por elementos finitos del problema termo-difusivo-mecánico con grandes deformaciones, Rev. int. métodos numér. cálc. diseño ing. (2024). Vol.40, (4), 47
El modelo de tres campos se ha presentando dentro del marco de la termo-mecánica de medios continuos y se ha mostrado cómo las ecuaciones constitutivas derivan de una energía libre, confiriendo al planteamiento una gran generalidad. Después, el artículo describe la formulación por el método de los elementos finitos de una discretización espacial y temporal de los tres problemas acoplados. Como novedad adicional frente a otros trabajos de la literatura, se describe con detalle la implementación de las ecuaciones, incluyendo todos los térmicos de la linealización necesaria para su resolución por el método de Newton-Raphson.
Se ha empleado la formulación obtenida para simular problemas de geles elastoméricos y un ejemplo relacionado con baterías de ion litio. En este último ejemplo se consideran los procesos de difusión de iones en el interior del ánodo que lo deforman, provocando variaciones térmicas y del campo de tensiones.
El artículo proporciona los fundamentos teóricos para la descripción de los problemas termo-difusivo-mecánicos de medios continuos y su aproximación numérica. Este marco, junto con todos los detalles que se proporcionan para la implementación de sus ecuaciones, serán de gran utilidad para los investigadores que deseen desarrollar formulaciones de elementos finitos para este tipo problemas, unificando y complementando la información disponible en la literatura.
Un aspecto relevante que no se trata con profundidad en este trabajo, por limitar su extensión y sus objetivos, es la posibilidad de emplear métodos de resolución de ecuaciones no lineales distintos al de Newton-Raphson (cuasi-Newton, BFGS, gradiente conjugado no lineal, etc.). Los desarrollos de este trabajo y en particular los detalles del Apéndice Apéndice A pueden servir como guía para la formulación de dichos métodos más eficientes.
Agradecimientos/Acknowledgement: No aplica.
Fuentes de financiación/Funding Statement: I.R. agradece la financiación parcial de los proyectos TED2021-130255B-C32 y PID2021-128812OB-I00 del Ministerio de Ciencia e Innovación de España.
Aportaciones de los autores/Author Contributions: Ángel Ortiz-Toranzo: software, redacción del manuscrito, análisis, investigación. Ignacio Romero: conceptualización, metodología, revisión y edición del manuscrito, supervisión. Todos los autores revisaron los resultados y aprobaron la versión final del manuscrito.
Disponibilidad de datos y materiales/Availability of Data and Materials: Los autores proporcionarán información y datos sobre las simulaciones a aquellas personas que los soliciten por correo electrónico.
Consideraciones éticas/Ethics Approval: No aplican.
Conflictos de interés/Conflicts of Interest: Los autores declaran que no tienen intereses financieros que pudieran haber influido en el trabajo reportado en este artículo.
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Apéndice A. Linealización ecuaciones
La linealización de los términos de la ecuación (41), correspondiente al balance de la cantidad de movimiento, quedaría:
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Teniendo en cuenta la definición del tensor de acoplamiento difusivo-mecánico Failed to parse (MathML with SVG or PNG fallback (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{L}}
(véase (38)), y del tensor de acoplamiento termo-mecánico Failed to parse (MathML with SVG or PNG fallback (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{M}}
(según la expresión (35)), la linealización de la expresión (60) puede expresarse:
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La linealización de la ecuación (42) correspondiente al balance de la energía resulta en los siguientes bloques:
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \operatorname{Lin}\left[\left\langle -\boldsymbol{Q}, \sum _{a} \nabla N^{a} \delta \Theta ^{a}\right\rangle \right]=\left\langle \Delta \left(\boldsymbol{F}^{-1} K_{0} \boldsymbol{F}^{-T}\right)\mathrm{GRAD} \Theta , \sum _{a} \nabla N^{a} \delta \Theta ^{a}\right\rangle =}
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \operatorname{Lin}\left[\left\langle \boldsymbol{h} \cdot \operatorname{GRAD}(\mu ),\left(\sum _{a} N^{a} \delta \Theta ^{a}\right)\right\rangle \right]=}
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En cuanto a la linealización de los términos de la ecuación correspondiente a la conservación de masa (43), los términos resultantes son:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \operatorname=\underbrace{\left\langle\dot{g},\left(\sum_{a} N^{a} \delta \mu^{a}\right)\right\rangle}_{1}-\underbrace{\left\langle\boldsymbol{h},\left(\sum_{a} \nabla N^{a} \delta \mu^{a}\right)\right\rangle}_{2}} .
La linealización del término 1 de la expresión anterior puede escribirse como
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En cuanto al primer término del miembro de la derecha de la expresión (66), se tiene:
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En cuanto al segundo término del miembro de la derecha de la expresión (66), se tiene:
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En cuanto al tercer término del miembro de la derecha de la expresión (66), se tiene:
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La linealización del término 2 de la expresión (65) puede escribirse:
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Las ecuaciones linealizadas correspondientes al balance de la cantidad de movimiento, balance de la energía y balance de masa, pueden escribirse matricialmente como un sistema de ecuaciones Failed to parse (MathML with SVG or PNG fallback (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{K} \boldsymbol{d}=\boldsymbol{F}} , siendo Failed to parse (MathML with SVG or PNG fallback (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{K}}
la matriz tangente, Failed to parse (MathML with SVG or PNG fallback (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{d}}
el vector de desplazamientos y Failed to parse (MathML with SVG or PNG fallback (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}}
el vector de cargas, esto es:
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siendo:
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