Sanjay Govindjee

Finite element implementation of incompressible, transversely isotropic hyperelasticity

This paper describes a three-dimensional constitutive model for biological soft tissues and its finite element implementation for fully incompressible material behavior. The necessary continuum mechanics background is presented, along with derivations of the stress and elasticity tensors for a transversely isotropic, hyperelastic material. A particular form of the strain energy for biological soft tissues is motivated and a finite element implementation of this model based on a three-field variational principle (deformation, pressure and dilation) is discussed. Numerical examples are presented that demonstrate the utility and effectiveness of this approach for incompressible, transversely isotropic materials.

A theory of finite viscoelasticity and numerical aspects

Most current models for finite deformation viscoelasticity are restricted to linear evolution laws for the viscous material behaviour. In this paper, we present a model for finite deformation viscoelasticity that utilizes a nonlinear evolution law, and thus is not restricted to states close to the thermodynamic equilibrium. We further show that upon appropriate linearization, we can recover several established models of finite linear viscoelasticity and linear viscoelasticity. The utilization of this model in a computational setting is also addressed and examples are presented to highlight the differences of the proposed model in relation with other available models.

On the use of continuum mechanics to estimate the properties of nanotubes

Abstract In experimental and theoretical investigations of the properties of nanostructures, the equations of continuum beam theory are often used to interpret the mechanical response of nanotubes. In particular, Bernoulli–Euler beam bending theory is being utilized to infer the Youngs Modulus. In this work, we examine the validity of such an approach using a simple elastic sheet model and show that at the nanotube scale the assumptions of continuum mechanics must be carefully respected in order to obtain reasonable results. Relations are derived for pure bending of nanotubes that show the explicit dependence of the “material properties” on system size when a continuum cross-section assumption is made. Two alternate approaches are proposed that provide a more reliable scheme for property extraction from experiments.

Mullins' effect and the strain amplitude dependence of the storage modulus

Abstract A micromechanically based continuum damage model for carbon black filled elastomers exhibiting Mullins effect is extended to incorporate viscous response within the framework of a theory of viscoelasticity which includes the classical BKZ model as a particular case and is not restricted to isotropy. The resulting model is shown to qualitatively predict the important effect of a strain amplitude dependent storage modulus even without the inclusion of healing effects. The proposed model for filled elastomers is shown to be well motivated from micromechanical considerations and suitable for large scale numerical simulations.

A multi-variant martensitic phase transformation model: formulation and numerical implementation

The development of models for shape memory alloys and other materials that undergo martensitic phase transformations has been moving towards a common generalized thermodynamic framework. Several promising models utilizing single martensitic variants and some with multiple variants have appeared recently. In this work we develop a model in a general multi-variant framework for single crystals that is based upon lattice correspondence variants and the use of dissipation arguments for the generation of specialized evolution equations. The evolution equations that appear are of a unique nature in that not only are the thermodynamic forces restricted in range but so are their kinematic conjugates. This unusual situation complicates the discrete time integration of the evolution equations. We show that the trial elastic state method that is popular in metal plasticity is inadequate in the present situation and needs to be replaced by a non-linear programming problem with a simple geometric interpretation. The developed integration methodology is robust and leads to symmetric tangent moduli. Example computations show the behavior of the model in the pseudoelastic range. Of particular interest is the fact that the model can predict the generation of habit plane-like variants solely from the lattice correspondence variants; this is demonstrated through a comparison to the experimental work of Shield [J. Mech. Phys. Solids 43 (1995) 869].

The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis

Abstract The construction of effective models for materials that undergo martensitic phase transformations requires usable and accurate functional representations for the free energy density. The general representation of this energy is known to be highly non-convex; it even lacks the property of quasi-convexity. A quasi-convex relaxation, however, does permit one to make certain estimates and powerful conclusions regarding phase transformation. The general expression for the relaxed free energy is however not known in the n-variant case. Analytic solutions are known only for up to 3 variants, whereas cases of practical interests involve 7–13 variants. In this study we examine the n-variant case utilizing relaxation theory and produce a seemingly obvious but very powerful observation regarding a lower bound to the quasi-convex relaxation that makes practical evolutionary computations possible. We also examine in detail the 4-variant case where we explicitly show the relation between three different forms of the free energy of mixing: upper bound by lamination, the Reus lower bound, and a lower estimate of the H -measure bound. A discussion of the bounds and their utility is provided; sample computations are presented for illustrative purposes.

Computational methods for inverse finite elastostatics

Abstract In the inverse motion problem in finite hyper-elasticity, the classical formulation relies on conservation laws based on Eshelbys energy-momentum tensor. This formulation is shown to be lacking in several regards for a particular class of inverse motion problems where the deformed configuration and Cauchy traction are given and the undeformed configuration must be calculated. It is shown that for finite element calculations a simple re-examination of the equilibrium equations provides a more suitable finite element formulation. This formulation is also shown to involve only minor changes to existing elements designed for forward motion calculations. Examples illustrating the method in simple and complex situations involving a Neo-Hookean material are presented.

Theoretical and Numerical Aspects in the Thermo-Viscoelastic Material Behaviour of Rubber-Like Polymers

Most current models for finite deformation thermo-viscoelasticity are restricted to linear evolution laws for the viscous behaviour and to thermorheologically simple materials. In this paper, we extend a model for finite deformation viscoelasticity that utilizes a nonlinear evolution law to include thermal effects. In particular, we present a thermodynamically consistent framework for the model and give a detailed form for then on-equilibrium Helmholtz free energy of the material in terms of the isothermal free energy function. The use of the model in a computational setting is addressed and it is shown that an efficient predictor-correct oralgorithm can be used to integrate the evolution equation of the proposed constitutive model. The integration algorithm makes crucial use of the exponential map as has been done previously in elastoplasticity. Numerical examples are presented to show some interesting features of the new model.

A Presentation and Comparison of Two Large Deformation Viscoelasticity Models

In this paper we present a theory of finite deformation viscoelasticity. The presentation is not restricted to small perturbations from the elastic equilibrium in contrast to many viscoelasticity theories. The fundamental hypothesis of our model is the multiplicative viscoelastic decomposition of Sidoroff (1974). This hypothesis is combined with the assumption of a viscoelastic potential to give a model that is formally similar to finite associative elasto-plasticity. Examples are given to compare the present proposal to an alternative formulation in the literature for the cases of uniaxial plane strain relaxation and creep.

A computational model for shape memory alloys

An investigation into the computational aspects of a multi-well mixture approach to shape memory modeling is undertaken with the goals of determining its qualitative behavior as well as its efficiency in a numerical setting. A basic rate dependent model for the transformation is first introduced, followed by a discussion of the steps taken to implement the constitution in discrete form. Numerical simulations demonstrate the quantitative and qualitative response of shape memory alloy structural systems to various thermal and mechanical cycles.