Framework for non-coherent interface models at finite displacement jumps and finite strains

This paper deals with a novel constitutive framework suitable for non-coherent interfaces, such as cracks, undergoing large deformations in a geometrically exact setting. For this type of interface, the displacement field shows a jump across the interface. Within the engineering community, so-called cohesive zone models are frequently applied in order to describe non-coherent interfaces. However, for existing models to comply with the restrictions imposed by (a) thermodynamical consistency (e.g., the second law of thermodynamics), (b) balance equations (in particular, balance of angular momentum) and (c) material frame indifference, these models are essentially fiber models, i.e. models where the traction vector is collinear with the displacement jump. This constraints the ability to model shear and, in addition, anisotropic effects are excluded. A novel, extended constitutive framework which is consistent with the above mentioned fundamental physical principles is elaborated in this paper. In addition to the classical tractions associated with a cohesive zone model, the main idea is to consider additional tractions related to membrane-like forces and out-of-plane shear forces acting within the interface. For zero displacement jump, i.e. coherent interfaces, this framework degenerates to existing formulations presented in the literature. For hyperelasticity, the Helmholtz energy of the proposed novel framework depends on the displacement jump as well as on the tangent vectors of the interface with respect to the current configuration – or equivalently – the Helmholtz energy depends on the displacement jump and the surface deformation gradient. It turns out that by defining the Helmholtz energy in terms of the invariants of these variables, all above-mentioned fundamental physical principles are automatically fulfilled. Extensions of the novel framework necessary for material degradation (damage) and plasticity are also covered.     

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.     

A 3D finite volume scheme for solving the updated Lagrangian form of hyperelasticity

The finite volume discretization of nonlinear elasticity equations seems to be a promising alternative to the traditional finite element discretization as mentioned by Lee et al. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell‐centered finite volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency, which are imposed by mean of the Coleman–Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free energy with respect to the left Cauchy–Green tensor. Moreover, the materials being isotropic, the free‐energy is function of the left Cauchy–Green tensor invariants, which enable the use of the neo‐Hookean model. The hyperelasticity system is discretized using the cell‐centered Lagrangian scheme from the work of Maire et al. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. Copyright © 2016 John Wiley & Sons, Ltd.     

Visco-electro-elastic models of fiber-distributed active tissues

We present a constitutive model for stochastically distributed fiber reinforced visco-active tissues, where the behavior of the reinforcement depends on the relative orientation of the electric field. Following our previous works, for the passive behaviors we adopt a second order approximation of the strain energy density associated to the parameters of the fiber distribution. Consistently, we also assume that the active behavior accounts for the stochastic distribution of the fibers. The ensuing mechanical quantities result to be dependent on two average structure tensors. We introduce an extended Helmholtz free energy density characterized by the inclusion of a directional active potential, dependent on a stochastic anisotropic permittivity tensor. The permittivity tensor is expanded in Taylor series up to the second order, allowing to obtain an approximated active potential with the same structure of the passive Helmholtz free energy density. In particular, the explicit expression of active stress and stiffness are dependent on the two average structure tensors that characterize the passive response. Anisotropy follows from the fiber distribution and inherits its stochastic nature through statistics parameters. The active fiber distributed model is extended here to viscous materials by including the contribution of a dual dissipation potential in the variational formulation of the constitutive updates. Additionally, we present a computational example of application of the electro-viscous-mechanical material model by simulating peristaltic contractions on a portion of human intestine.     

一类铁电模型的参变量变分原理及其应用

将参变量变分原理引入铁电问题。对一类借用了经典弹塑性理论中的概念和方法的多轴铁电模型建立基于Helmholtz自由能的参变量变分原理,可以有效处理传统变分原理中由非关联流动法则或屈服面不考虑材料系数变化所引起的切线模量非对称困难。相应于参变量变分原理,引入参数二次规划算法,可获得具有可靠数值稳定性的一套铁电算法。将该算法应用于一个具体的铁电模型,数值计算结果表明本文方法的有效性。     

On the Comparison of Two Strategies to Formulate Orthotropic Hyperelasticity

The main goal of this work is to clarify the relation between two strategies to formulate constitutive equations for orthotropic materials at large strains. On the one hand, the classical approach is based on the incorporation of structural tensors into the free energy function via an enriched set of invariants. On the other hand, a fictitious isotropic configuration is introduced which renders an anisotropic, undeformed reference configuration via an appropriate linear tangent map. This formulation results in a reduced (with respect to the more general setting based on structural tensors) but nevertheless physically motivated set of invariants which are related to the invariants defined by structural tensors. As a main conceptual advantage standard isotropic constitutive equations can be applied and moreover, due to the reduced set of physically motivated invariants, the numerical treatment within a finite element setting becomes manageable.     

Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

This paper deals with the non-linear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the non-linear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time-dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to the third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the non-linear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed. The model is constructed by the introduction of a reference configuration which is deduced from the non-linear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.     

On configurational compatibility and multiscale energy momentum tensors

In this work the continuum theory of defects has been revised through the development of kinematic defect potentials. These defect potentials and their corresponding variational principles provide a basis for constructing a new class of conservation laws associated with the compatibility conditions of continua. These conservation laws represent configurational compatibility conditions which are independent of the constitutive behavior of the continuum. They lead to the development of a new concept termed configurational compatibility, dual to the concept of configurational force. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by the integral is defect-free. It is shown that the Peach-Koehler force can be recovered as one of these invariant integrals. Based on the proposed defect potentials and their corresponding defect energies, two-field multiscale mixed variational principles can be employed to construct multiscale energy momentum tensors. An application is outlined in the form of a mode III elasto-plastic crack problem for which the new configurational quantities are calculated.     

Use of isotropic thermoelasticity laws in finite deformation viscoplasticity models

We consider general finite deformation thermoviscoplasticity models and focus attention to the thermoelasticity laws involved in the constitutive theory. The goal is to show how such constitutive relations having forms as simple as possible may be defined with respect to both the plastic intermediate configuration and the actual configuration. In particular, using two families of strain and associated stress tensors, we arrive at two different thermoelasticity laws formulated with respect to the actual configuration. These are extensions to nonisothermal deformations of corresponding hyperelasticity laws introduced previously by Simo [1]. Received March 24, 1998     

Electrically polarized mixtures of porous bodies

In this paper we propose a mathematical model of a mixture between a porous material and a polarized fluid in the mechanics of complex bodies. We use a variational approach to derive the global and microstructural balance laws for each phase of the mixture and for the mixture as a whole. Such balances differ from those generally proposed in theory of mixtures. We consider an open system and we account for external sources. Moreover, via the Coleman and Noll procedure we point out that the transport equation for the matter in a polarized mixture due to electrical field is governed by the Nernst–Planck equation.     

Generalized heat-transport equations: parabolic and hyperbolic models

We derive two different generalized heat-transport equations: the most general one, of the first order in time and second order in space, encompasses some well-known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman–Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.     

On a regular convex solver potential for an elastic-damage constitutive model: a theoretical analysis

This work is related with the proposition of a so-called regular or convex solver potential to be used in numerical simulations involving a certain class of constitutive elastic-damage models. All the mathematical aspects involved are based on convex analysis, which is employed aiming a consistent variational formulation of the potential and its conjugate one. It is shown that the constitutive relations for the class of damage models here considered can be derived from the solver potentials by means of sub-differentials sets. The optimality conditions of the resulting minimisation problem represent in particular a linear complementarity problem. Finally, a simple example is present in order to illustrate the possible integration errors that can be generated when finite step analysis is performed.     

Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains

This work describes the finite element implementation of a generalised strain gradient and rate-dependent crystallographic formulation for finite strains and general anisothermal conditions based on a multiplicative decomposition of the deformation gradient. The implementation involved the development of both a novel finite element formulation to determine the spatial slip rate gradients at each material point, and an implicit numerical integration scheme at the constitutive level to update the stresses and solution dependent variables. The time-integration procedure uses a Newton–Raphson scheme with a single level of iteration to solve the incremental non-linear equations associated with the non-local constitutive formulation. Closed-form solutions for the relevant fourth-order Jacobian tensors are given. The proposed numerical scheme is formulated in a general form and hence should be applicable to most existing crystallographic models. The crystallographic formulation is then used to investigate the effect of the morphology and volume fraction of the reinforcing phase of a two-phase single crystal on its macroscopic behaviour.     

A unified potential-based cohesive model of mixed-mode fracture

A generalized potential-based constitutive model for mixed-mode cohesive fracture is presented in conjunction with physical parameters such as fracture energy, cohesive strength and shape of cohesive interactions. It characterizes different fracture energies in each fracture mode, and can be applied to various material failure behavior (e.g. quasi-brittle). The unified potential leads to both intrinsic (with initial slope indicators to control elastic behavior) and extrinsic cohesive zone models. Path dependence of work-of-separation is investigated with respect to proportional and non-proportional paths—this investigation demonstrates consistency of the cohesive constitutive model. The potential-based model is verified by simulating a mixed-mode bending test. The actual potential is named PPR (Park-Paulino-Roesler), after the first initials of the authors’ last names.     

Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors

In large strain elasticity the existence of minimizers is guaranteed if the variational functional to be minimized is sequentially weakly lower semicontinuous (s.w.l.s.) and coercive. Therefore, polyconvex functions which are always s.w.l.s. are usually considered. For isotropic as well as for transversely isotropic and orthotropic materials constitutive functions that are polyconvex already exist. The main goal of this contribution is to provide a new method for the construction of polyconvex hyperelastic models for more general anisotropy classes. The fundamental idea is the introduction of positive definite second-order structural tensors G=HH^T encoding the anisotropies of the underlying crystal. These tensors can be viewed as a push-forward of a cartesian metric of a fictitious reference configuration to the real reference configuration. Here the driving transformations H in the push-forward operation are mappings of the cartesian base vectors of the fictitious configuration onto crystallographic motivated base vectors. Restrictions of this approach are based on the polyconvexity condition as well as on the usage of second-order structural tensors and pointed out in detail.     

The finite element method of viscoelastic large deformation plane problem with Kirchhoff stress tensors-Green strain tensors constitutive relation

Based upon the updated Lagrangian approach, the principle of virtual work denoted by the updated Kirchhoff stress increment tensors and the updated Green strain increment tensors and the integral constitutive relation expressed by Kirchhoff stress tensors and Green strain tensors are used and the viscoelastic large deformation incremental variational equation is derived. By means of the 8-nodes isoparametric finite element the program of two-dimensional problem is written. Good agreement is found among the results obtained from this paper and other literatures.     

On the calculation of predeformation-dependent dynamic modulus tensors in finite nonlinear viscoelasticity

To simulate the frequency-dependent behaviour of nonlinear viscoelastic structures under loadings which consist of a finite predeformation in combination with a superimposed harmonic deformation with small amplitude, frequency-domain formulations of the constitutive models are needed. For this purpose, a recently developed approach of finite viscoelasticity is considered and the corresponding dynamic modulus tensors are derived. The constitutive equations are geometrically linearized in the neighbourhood of the predeformation and evaluated in the frequency-domain. This procedure is applicable to arbitrary constitutive models and can be used to derive their frequency-domain formulations for finite element implementations as proposed by Morman and Nagtegaal [Morman, K.N., Nagtegaal, J.C., 1983. Finite element analysis of sinusoidal small-amplitude vibrations in deformed viscoelastic solids. International Journal for Numerical Methods in Engineering, 19, 1079–1103].     

Thermomechanical material modelling based on a hybrid free energy density depending on pressure,isochoric deformation and temperature

In order to represent temperature-dependent mechanical material properties in a thermomechanical consistent manner it is common practice to start with the definition of a model for the specific Helmholtz free energy. Its canonical independent variables are the Green strain tensor and the temperature. But to represent calorimetric material properties under isobaric conditions, for example the exothermal behaviour of a curing process or the dependence of the specific heat on the temperature history, the temperature and the pressure should be taken as independent variables. Thus, in the field of calorimetry the Gibbs free energy is usually used as thermodynamic potential whereas in continuum mechanics the Helmholtz free energy is normally applied. In order to simplify the representation of calorimetric phenomena in continuum mechanics a hybrid free energy density is introduced. Its canonical independent variables are the isochoric Green strain tensor, the pressure and the temperature. It is related to the Helmholtz free energy density by a Legendre transformation. In combination with the additive split of the stress power into the sum of isochoric and volumetric terms this approach leads to thermomechanical consistent constitutive models for large deformations. The article closes with applications of this approach to finite thermoelasticity, curing adhesives and the glass transition.