The purpose of this paper is to review and contrast the Lagrangian and Eulerian formulations of inelasticity as they apply to metal plasticity and growth of biological tissues. In contrast with the ...Lagrangian formulation of inelasticity, the Eulerian formulation is unaffected by arbitrary choices of the reference configuration, an intermediate configuration, a total deformation measure and an inelastic deformation measure. Although the Eulerian formulation for growth of biological tissues includes a rate of mass supply and can be used to understand the mechanics of growth, it does not yet model essential mechanobiological processes that control growth. Much research is needed before this theory can help design medical treatments for growth related disease. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.
The objective of this classroom note is to propose a generalized model for a compressible elastically isotropic material with elastic-inelastic response. The model is generalized for an exponential ...Fung-type strain energy with a sum of new higher order elastic distortional deformation invariants that can model elastic-inelastic distortional deformation. In contrast with an Ogden-type model, the coefficients of the proposed higher order invariants do not affect the small deformation response. Moreover, there is no need to determine eigenvalues and eigenvectors of the elastic distortional deformation tensor. Examples show that the equations can model preconditioning of biological tissues as well as elastic instability.
A thermomechanical large deformation Eulerian formulation of a multi-phase soft tissue model with homeostasis and phase transformation has been developed. The model has been simplified by considering ...a single velocity field and a single temperature field with evolution equations for volume fractions of the phases which are not driven by diffusion. Specific constitutive equations are developed for an exponential form of the Helmholtz free energy. An example of spherical expansion of a three-phase material is considered in the absence of thermal effects. This example considers phase transformations from healthy tissue to a composite of healthy, diseased and necrotic tissues. Although much work is needed to model the mechanobiological aspects of the disease progress, this simplified model indicates that the state of the tissue at the onset of its non-zero volume fraction (i.e. when it begins to influence stress) can significantly effect the resulting stress distribution.
In the constrained Cosserat theory of a rod with a rigid cross-section, the balance of angular momentum is satisfied by a restriction on the constitutive equations that requires a second order tensor ...to be symmetric. The kinematics of the rod are determined by satisfying the balances of linear and director momentum and kinematic constraints. In contrast, the Antman model for a special Cosserat theory of rods proposes constitutive equations directly for the force and mechanical moment applied to the rod and the kinematics are determined by the balances of linear and angular momentum. These two models differ by their treatment of angular momentum. This note poses and answers the question: Are the solutions of these two models identical for the same strain energy?
Recently, an Eulerian formulation based on evolving microstructural vectors was used (Lee and Rubin, 2020) to model anisotropic inelastic deformation rate in sheet metals. In contrast to Lagrangian ...formulations, the Eulerian formulation is insensitive to arbitrariness of reference and intermediate configurations. Here, additional constitutive equations are introduced to model asymmetric inelastic behaviors, such as the Bauchinger effect, permanent softening and the strength-differential effect within the context of the Eulerian formulation. The resulting model is calibrated to produce good agreement with several material data sets. In addition, example data sets are used to demonstrate advantages of the new model, especially for severe anisotropic response and large deformations.
•An Eulerian formulation based on microstructural vectors is used for sheet metal.•Microstructural vectors determine elastic deformations and orthotropic axes.•Strength-differential Effect modeled using the Lode angle and a hardening parameter.•Baushinger effect and permanent softening in cyclic loading modeled using a directional hardening tensor.•The constitutive equations are validated using five material data sets.
A thermomechanical generalized Cosserat membrane has no resistance to bending but it allows for deformations through the membrane’s thickness. This theory is generalized to allow for growth due to ...mass supply and is constrained to eliminate shear deformations but retain thickness changes. An Eulerian formulation of isothermal elastic-viscoplastic constitutive equations is developed using evolution equations for elastic dilatation and elastic distortional deformations. These evolution equations include time-dependent inelastic effects of homeostasis which cause a tendency for the elastic deformation measures to approach their homeostatic values. An important feature of the Eulerian evolution equations is that they depend only on the current state of the membrane and therefore are appropriate for modeling growth. Examples of zero-stress growth and constrained growth are considered.
This paper develops an approximate stress distribution in a conical heap of jammed dry granular material loaded by gravity. An Eulerian formulation of elastic-inelastic response is used to explain ...why the residual stresses in the heap can be approximated by the current state of stress in the material. The proposed normalized stress components are functions of the normalized radial and vertical coordinates and are parameterized by only the angle of repose. It is shown that the vertical stress distribution applied to the base of the heap compares well with experiments using a rain procedure for sand deposition.
An analytical expression for the strain energy of a constrained extensible Cosserat elastica is developed for general planar shapes and deformations of the rod. This strain energy function naturally ...couples tangential stretch and reference and current curvatures of the centroidal curve. The model considers a rigid rectangular cross-section of the rod which remains normal to the centroidal curve. The constitutive equations for the tangential force, shear force and bending moment are consistent with a restriction based on the balance of angular momentum that requires a stress-like tensor to be symmetric in a similar manner to the symmetry of the Cauchy stress in a three-dimensional continuum. Examples show that coupling of tangential stretch and reference and current curvatures of the centroidal curve in the new strain energy function can significantly influence predictions of tangential force, shear force and bending moments.
This paper presents a constitutive equation for anisotropic inelastic deformation rate applied to sheet metal forming based on recently developed invariants for nonlinear orthotropic ...elastic–inelastic material response. These invariants are based on Eulerian evolution equations for microstructural vectors that characterize directions of anisotropy and elastic deformations which cause stress. These microstructural vectors are used here to model directions of orthotropy for sheet metal forming. This Part I of a two-part paper modifies an anisotropic yield function for sheet metal analysis based on stress components determined by elastic deformations of microstructural vectors. Specific expressions are proposed for the direction of inelastic distortional deformation rate, which automatically satisfies the rate of material dissipation inequality, and for the direction of inelastic spin. The material parameters are calibrated for the aluminum alloy AA6022-T43, steel MP980 sheets and cold rolled low carbon steel sheets. The results of stress anisotropy and R-value distributions are shown to be in good agreement with the experimental data. Part II describes the implementation of this model into a finite element program and presents a comparison of Eulerian and Lagrangian formulations.
•Anisotropic inelastic sheet metal behavior is modeled using microstructural vectors.•Microstructural vectors determine elastic deformations and axes for orthotropic inelastic response.•Hyperelastic constitutive equation for Cauchy stress.•Validation with two materials shows that this model can capture anisotropies of R-value and yield stress.•Inelastic spin can shift R-value curves.
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