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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.

Unlike common engineering materials, living matter can autonomously respond to environmental changes to become stronger, weaker, larger, or smaller within months, weeks, days, or even seconds as a result of a continuous microstructural renewal and adaption. The importance of mechanics in developmental biology is well established, where biomechanical forces and constraints are the bridge that connects genetic and molecular-level events to tissue-level deformations that shape the developing embryo. Moreover, feedback from the mechanical environment affects gene expression and differentiation. Therefore, mechanical forces and constraints are not only a proximal cause of biological growth and morphogenesis, but also play a regulatory role. Regarding the simulation of these processes, however, it is important to recognize that mathematical models can never include all of the complexities inherent in biological systems. Rather, they are ad hoc constructions designed to help understand some aspect of the system. In this sense, continuum mechanics, accompanied by growth theories, provide the theoretical framework for the computational modeling of biological growth and morphogenesis. In this article, we proposed an advanced continuum theory by integrating with the growth theory, proposed by Rodriguez et al. [1]. The shape change of a load-free tissue during growth is described by a mapping analogous to the deformation gradient tensor. This mapping is decomposed into a transformation to a local zero-stress grown state and an accompanying deformation that ensures the compatibility of the total growth deformation. Beyond autonomous growth, the geometric constraints confine the tissue developments, which are usually irreversible processes. Here, we propose to introduce plastic deformation to simulate the irreversible development and adopt return mapping algorithm for the numerical simulation. It is noticed that most biological growth and morphogenesis behaviors involve large deformation. Hence, we constructed a large strain theory of plasticity which is based on the Lagrangian description of balance laws and constitutive relations and incorporates the fundamental features of the aforementioned growth theory. To verify our theory and the corresponding code, several sample problems will be shown and discussed. REFERENCE [1] Rodriguez, E.K., Hoger, A., McCulloch, A.D., 1994. Stress-dependent finite growth in soft elastic tissue. J Biomechanics. 27, 455–567.

Coarse-grained molecular dynamics (CG-MD) is a multiscale method for concurrently coupling atomic subdomain and continuum subdomain in a nano/micro material system. In this study, we first present the theoretical framework of multiscale material modeling. Based on that, we derived two numerical algorithms: force-based coarse-grained molecular dynamics (FB-CG-MD) and stiffness-based coarse-grained molecular dynamics (SB-CG-MD). In the first case, we investigate the effect of mesh sizes on accuracy through the bending of a magnesium oxide (MgO) bar. The interatomic interaction of MgO is described by Coulomb–Buckingham potential. The result obtained by classical MD simulation is considered as the standard solution. From CG-MD simulations, we compare the results of different mesh sizes with the standard solution and the error tells the accuracy of both FB-CG-MD and SB-CG-MD. In the second case, we study the heat conduction problem of an MgO specimen, which is subdivided into two subdomains and each subdomain can be either atomic subdomain or continuum subdomain. One subdomain is controlled at a desired temperature through the use of Upgraded Nosé–Hoover Thermostat, which eliminates unphysical phenomena due to reference frame translation and/or rotation when one uses the original Nos é–Hoover thermostat. The other subdomain is free from temperature control. In the CG-MD model, the temperatures of continuum subdomain and atomic subdomain are determined by nodal velocities and atomic velocities, respectively. Our results show that the thermal energy transfers successfully from atomic subdomain to atomic subdomain, as well as from continuum subdomain to continuum subdomain. It is shown that both -FB-CG-MD and SB-CG-MD reduce significantly the number of degrees of freedom in the material system and provide reliable results and improve numerical efficiency.

When Molecular Dynamics (MD) was originally conceived, the trajectories of atoms are determined by numerically solving the Newton’s equations for a system under equilibrium condition. However, to study the multifunctionalities of advanced nanomaterial, such as piezoelectricity or thermoelectricity, the nonequilibrium MD should be considered and further developed to incorporate the multiphysics phenomena, i.e., thermo-mechanical-electromagnetic coupling effects. For the temperature simulation, the revolutionary Nosé–Hoover dynamics, modified Newtonian dynamics so as to reproduce canonical and isobaric–isothermal ensemble equilibrium systems. However, there is an increasing interest in conducting MD simulation for a non-equilibrium system whose temperature varies spatially and temporally during the simulation with the imposition of a temperature gradient. Clearly, this is a heat conduction problem and requires non-equilibrium MD with suitable algorithmic thermostat for local temperature regulation. Inspired by Nosé–Hoover thermostat, this study reformulates the feedback force caused by the temperature control, aiming at (i) controlling the temperature locally at several distinct spots and (ii) eliminating the rigid-body translation and rotation which are irrationally introduced into the system due to the temperature force. This reformulation will generate accurate and rigorous trajectories of atoms and thus the heat conduction can be simulated and performed successfully at nanoscale. Correspondingly, the definition of temperature is modified; the expression of Hamiltonian is upgraded. To demonstrate the capability and feasibility of this new algorithm, we studied heat conduction phenomena in a beam-like and a ring-like finite size specimen by using our in-house developed computer code. The results from the reformulated Nosé–Hoover thermostat show the correct (reasonable and logical) temperature distributions across the specimens after the steady state arrives for long time duration until the steady state arrives. Yet, the results from the original Nosé–Hoover thermostat cannot reach yield the steady state solution. In addition, it reaches the conclusion that the heat conduction at nanoscale exhibits the same feature of Fourier’s law at macroscopic scale if the temperature is averaged over a sufficiently large time interval and spatial region. By following this conclusion, we can proceed to calculate the thermal conductivity of MgO material system, which fits the experimental results very well. To incorporate the electromagnetic effect, the Lorentz force is rigorously derived based on the Maxwell’s equations at atomic scale. This allows us to unveil the resonance phenomena at nanoscale, which resemble the relationship between the frequency and response in classical mechanical vibration. Moreover, the coupling effects, including the temperature variation induced by electromagnetic field, and the electromagnetic quantities induced by mechanical input, will be discussed in detail.