#### Description

In continuum mechanics, the number of balance laws in Lagrangian description can be condensed into two: balance of linear momentum and conservation of energy. In principle, one may perform a finite element analysis of any material provided that we have the constitutive equations of stress tensor, heat flux, and internal energy density. During the process of formulating a constitutive theory, if internal variables are introduced, then correspondingly for each internal variable a governing equation has to be derived. In this study, we formulate a set of generalized constitutive relations, including strain-based return mapping formula, for thermo-visco-elastic-plastic solid. In our formulation, the second order Piola–Kirchhoff stress tensor is decomposed into two parts: the reversible part and dissipative part. Based on Onsager’s postulate, as usual, the dissipative stress tensor is linearly proportional to Lagrangian strain rate tensor and the heat flux vector is linearly proportional to temperature gradients – a generalized Fourier’s law. The reversible part of the stress tensor has three parts: initial stress, thermal stress, and elastic stress. The initial stress may exist in the muscle when activated; therefore it is named as the active stress in living biological tissue. It may also be the stress because of biological growth [1]. Here the elastic stress tensor depends on the elastic strain tensor that equals to the total strain tensor minus the plastic strain, which is obtained through the return mapping algorithm. From the energy equation, it is found that the dissipation comes from the inner products between (i) the dissipative stress tensor and the total strain rate tensor and (ii) the reversible part of the stress tensor and the plastic strain rate tensor. Corresponding to this constitutive theory, we developed a generalized finite element computer program for large strain thermomechanically coupled material system. In this conference, the numerical results of two sample problems, (i) temperature elevation during the process of dynamic crack propagation and (ii) the role of eye muscle in focusing, will be presented and discussed. REFERENCE [1] Rodriguez, E.K., Hoger, A., McCulloch, A.D. Stress-dependent finite growth in soft elastic tissue. J. Biomechanics. , 1994, 27, 455–567.

#### Recommended Citation

Li, J.
(2014).
Finite element analysis of thermomechanical coupling.
In A. Bajaj, P. Zavattieri, M. Koslowski, & T. Siegmund (Eds.).
*
Proceedings of the Society of Engineering Science 51st Annual Technical Meeting, October 1-3, 2014
*,
West Lafayette: Purdue University Libraries Scholarly Publishing Services, 2014.
https://docs.lib.purdue.edu/ses2014/mss/gmss/1

Finite element analysis of thermomechanical coupling

In continuum mechanics, the number of balance laws in Lagrangian description can be condensed into two: balance of linear momentum and conservation of energy. In principle, one may perform a finite element analysis of any material provided that we have the constitutive equations of stress tensor, heat flux, and internal energy density. During the process of formulating a constitutive theory, if internal variables are introduced, then correspondingly for each internal variable a governing equation has to be derived. In this study, we formulate a set of generalized constitutive relations, including strain-based return mapping formula, for thermo-visco-elastic-plastic solid. In our formulation, the second order Piola–Kirchhoff stress tensor is decomposed into two parts: the reversible part and dissipative part. Based on Onsager’s postulate, as usual, the dissipative stress tensor is linearly proportional to Lagrangian strain rate tensor and the heat flux vector is linearly proportional to temperature gradients – a generalized Fourier’s law. The reversible part of the stress tensor has three parts: initial stress, thermal stress, and elastic stress. The initial stress may exist in the muscle when activated; therefore it is named as the active stress in living biological tissue. It may also be the stress because of biological growth [1]. Here the elastic stress tensor depends on the elastic strain tensor that equals to the total strain tensor minus the plastic strain, which is obtained through the return mapping algorithm. From the energy equation, it is found that the dissipation comes from the inner products between (i) the dissipative stress tensor and the total strain rate tensor and (ii) the reversible part of the stress tensor and the plastic strain rate tensor. Corresponding to this constitutive theory, we developed a generalized finite element computer program for large strain thermomechanically coupled material system. In this conference, the numerical results of two sample problems, (i) temperature elevation during the process of dynamic crack propagation and (ii) the role of eye muscle in focusing, will be presented and discussed. REFERENCE [1] Rodriguez, E.K., Hoger, A., McCulloch, A.D. Stress-dependent finite growth in soft elastic tissue. J. Biomechanics. , 1994, 27, 455–567.