Compressive failure of microcracked porous brittle solids

Constitutive equations for porous, brittle solids are developed based on the damage mechanics of elastic materials containing cavities and microcracks. For homogeneous deformation modes, microcrack growth from pores causes changes in the average elastic compliance of the material. Failure criteria in terms of bifurcations of the constitutive paths are established by examining the properties of the evolving tangent stiffness tensor. Limit points as well as localized shear band failure modes are addressed. The influence of moderate levels of lateral stresses is studied for biaxial stress states.     

Incremental viscoelastic formulation using generalized variables for thin structures: relaxation differential approach

This paper presents a new incremental formulation in the time domain for linear, non-ageing viscoelastic materials undergoing mechanical deformation. The transformation of the viscoelastic continuum problem from the integral to the differential form is achieved. The formulation is derived from linear differential equations based on a discrete spectrum representation for the relaxation tensor using generalized variables and applied to thin structures. This leads to incremental constitutive formulations using the finite difference integration. Thus, the difficulty of retaining the strain history in computer solutions is avoided. A complete general formulation of linear viscoelastic strain analysis is developed in terms of increments of generalized stresses and strains. An illustrative example is included to demonstrate the method.     

A multiscale theoretical model for fluid flow in cellular biological media

An integrated methodology is developed for the theoretical analysis of momentum transfer in cellular biological media, such as biofilms and tissues. First, the method of local spatial averaging via a weight function is used to establish the equations that describe momentum transfer at the cellular biological medium scale, starting with a continuum-based formulation of the process at finer spatial scales. The constitutive behavior of each constituent phase is postulated at the polymer- or cell-scale and, through the averaging procedure, appropriate constitutive relations are developed for the upscaled stress tensors and the fluid–structure interaction forces. Further, closure problems are developed for the theoretical calculation of the effective material properties that appear in the constitutive relations. The developed closure problem for the static hydraulic permeability tensor is solved using a finite element method in the context of a periodic spherocylinder-in-cell model, which accounts for salient geometric features of microbial aggregates and biofilms at the cell-scale. The degree of structural anisotropy resulting from the shape, orientation, and spatial arrangement of biological cells (from stack formation to nematic alignment), is examined and shown to affect strongly the permeability tensor. Very good agreement is observed with results from previous theoretical studies for sphere packings and experimental data for the hydraulic permeability of mycelial cakes.     

Edge stabilization and consistent tying of constituents at Neumann boundaries in multi‐constituent mixture models

A mixture‐theory‐based model for multi‐constituent solids is presented where each constituent is governed by its own balance laws and constitutive equations. Interactive forces between constituents that emanate from maximization of entropy production inequality provide the coupling between constituent‐specific balance laws and constitutive models. The deformation of multi‐constituent mixtures at the Neumann boundaries requires imposing inter‐constituent coupling constraints such that the constituents deform in a self‐consistent fashion. A set of boundary conditions is presented that accounts for the non‐zero applied tractions, and a variationally consistent method is developed to enforce inter‐constituent constraints at Neumann boundaries in the finite deformation context. The new method finds roots in a local multiscale decomposition of the deformation map at the Neumann boundary. Locally satisfying the Lagrange multiplier field and subsequent modeling of the fine scales via edge bubble functions result in closed‐form expressions for a generalized penalty tensor and a weighted numerical flux that are free from tunable parameters. The key novelty is that the consistently derived constituent coupling parameters evolve with material and geometric nonlinearity, thereby resulting in optimal enforcement of inter‐constituent constraints. Various benchmark problems are presented to validate the method and show its range of application. Copyright © 2016 John Wiley & Sons, Ltd.     

Incremental constitutive formulation for time dependent materials: creep integral approach

This paper deals with the development of a mathematical approach for the solution of linear, non-ageing viscoelastic materials undergoing mechanical deformation. The formulation is derived from integral approach based on a discrete spectrum representation for the creep tensor. Finite difference integration is used to discretize the integral operators. The resulting constitutive model contains an internal state variable which represents the influence of the whole past history of stress and strain. Thus the difficulty of retaining the strain history in computer solutions is avoided. A complete general formulation of linear viscoelastic stress-strain analysis is developed in terms of increments of stresses and strains. Numerical simulations are included in order to validate the incremental constitutive equations.     

A study on the S‐version FEM for a dynamic damage model

A mesh superposition technique is developed based on the superimposed version FEM (s‐version FEM) for dynamic analysis of damage to structures. In the proposed technique, the concerned area is discretized as a rough global mesh and is overlaid into a fine local mesh of the proposed s‐version FEM. To describe the small‐scale stress wave, the governing equations for the internal part and boundary of the local region are proposed. The damage evolving constitutive model is substituted into these governing equations to describe the distributions of damage on a small scale. L‐shape domain problem, modal, time history, and damage examples are given for the validations. The results show that the proposed s‐version FEM can refine a global mesh and accurately describe the damage, stress, and deformation of a concerned area of structure on the small scale. The proposed s‐version FEM can be applied to not only a dynamic problem but also a damage analysis of structures.     

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

This study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.     

Tensor representations and solutions of constitutive equations for viscoelastic fluids

An approach is developed using tensor representations to assess and characterize both the transient behavior and equilibrium states of viscoelastic fluid constitutive equations in viscometric flows. The methodology is based on the replacement of the differential constitutive equation for the deviatoric part of the viscoelastic stress tensor by an equivalent and more tractable set of differential equations for the characteristic scalar invariants. In the case of planar flows, this equivalence leads to an explicit, closed-form analytic solution for the time evolution of the extra-stress tensor that is formally expressed as a second-order fluid relation, with time-dependent coefficients. As a validation of the approach, an analysis of the transient and equilibrium system characteristics of fluid flows described by the corotational Jeffreys model and general Oldroyd-type constitutive equations is presented.     

On time integration of viscoplastic constitutive models suitable for creep

Creep of critical components such as electrical solder connections may occur over long periods of time. Efficient numerical simulations of such problems generally require the use of quasi‐static formulations with conjugate‐gradient techniques for solving the large number of algebraic equations. Implicit in the approach is the need to solve the constitutive equation several times for large time steps and for loading directions that may have no resemblance to the actual solution. Therefore, an unconditionally stable and efficient algorithm for solving the constitutive equation is essential for the overall efficiency of the solution procedure. Unfortunately, constitutive equations suitable for simulating the materials of interest are notoriously difficult to solve numerically and most existing algorithms have a stability limit on the time step which may be several orders of magnitude smaller than the desired time step. Here an algorithm is proposed which is a combination of the use of a trapezoidal rule and an iterative Newton–Raphson method for solving implicitly the non‐linear equations. The key to the success of the proposed approach is to always use an initial guess based on the steady‐state solution to the constitutive equation. A representative viscoplastic constitutive equation is used as a model for illustrating the approach. The algorithm is developed and typical numerical results are provided to substantiate the claim that stability has been achieved. Copyright © 2001 John Wiley & Sons, Ltd.     

Micromechanically Established Constitutive Equations for Multiphase Materials with Viscoelastic–Viscoplastic Phases

A model for viscoelastic–viscoplastic solids is incorporated in a micromechanical analysis of composites with periodic microstructures in order to establish closed-form coupled constitutive relations for viscoelastic–viscoplastic multiphase materials. This is achieved by employing the homogenization technique for the establishment of concentration tensors that relate the local elastic and inelastic fields to the externally applied loading. The resulting constitutive equations are sufficiently general such that viscoelastic, viscoplastic and perfectly elastic phases are obtained as special cases by a proper selection of the material parameters the phase. Results show that the viscoelastic and viscoplastic mechanisms have significant effect on the global stress-strain, relaxation and creep behavior of the composite, and that its response is strongly rate-dependent in the reversible and irreversible regimes.     

On the representation and implicit integration of general isotropic elastoplasticity based on a set of mutually orthogonal unit basis tensors

A simple and compact representation framework and the corresponding efficient numerical integration algorithm are developed for constitutive equations of isotropic elastoplasticity. Central to this work is the utilization of a set of mutually orthogonal unit tensor bases and the corresponding invariants. The set of bases can be regarded equivalently as a local cylindrical coordinate system in the three‐dimensional coaxial tensor subspace, namely, the principal space. The base tensors are given in the global coordinate system. Similar to the principal space approach, the proposed method reduces the problem dimension from six to three. In contrast to the conventional approach, the transformation procedure between the principal space and the general space is avoided and explicit computation of the principal axes is bypassed. With the proposed technique, the matrices, which need to be inverted during iteration, take a simple form for the great majority of constitutive equations in use. The tangent operator consistent with the proposed algorithm can be decomposed into the direct sum of two linear maps over the coaxial tensor subspace and the subspace orthogonal to it. Consequently, its closed form is derived in an extremely simple manner. Finally, numerical examples demonstrate the high quality performances of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Viscoelastic incremental formulation using creep and relaxation differential approaches

A new incremental formulation in the time domain for linear, non-ageing viscoelastic materials undergoing mechanical deformation is presented in this work. The formulation is derived from linear differential equations based on a discrete spectrum representation for the creep and relaxation tensors. The incremental constitutive equations are then obtained by finite difference integration. Thus the difficulty of retaining the stress and strain history in computer solutions is avoided. A complete general formulation of linear viscoelastic stress analysis is developed in terms of increments of strains and stresses in order to establish the constitutive stress–strain relationship. The presented method is validated using numerical simulations and reliable results are obtained.     

Modeling failure of heterogeneous viscoelastic solids under dynamic/impact loading due to multiple evolving cracks using a two-way coupled multiscale model

This paper presents a model for predicting damage evolution in heterogeneous viscoelastic solids under dynamic/impact loading. Some theoretical developments associated with the model have been previously reported. These are reviewed briefly, with the main focus of this paper on new developments and applications. A two-way coupled multiscale approach is employed and damage is considered in the form of multiple cracks evolving in the local (micro) scale. The objective of such a model is to develop the ability to consider energy dissipation due to both bulk dissipation and the development of multiple cracks occurring on multiple length and time scales. While predictions of these events may seem extraordinarily costly and complex, there are multiple structural applications where effective models would save considerable expense. In some applications, such as protective devices, viscoelastic materials may be preferred because of the considerable amount of energy dissipated in the bulk as well as in the fracture process. In such applications, experimentally based design methodologies are extremely costly, therefore suggesting the need for improved models. In this paper, the authors focus on the application of the newly developed multiscale model to the solution of some example problems involving dynamic and impact loading of viscoelastic heterogeneous materials with growing cracks at the local scale.     

Computational modeling of cardiac electrophysiology: A novel finite element approach

The key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element‐based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell‐specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non‐oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non‐linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar‐valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two‐dimensional patient‐specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Consistent micro–macro transitions at large objective strains in curvilinear convective coordinates

This paper presents a computational homogenization scheme that is of particular interest for problems formulated in curvilinear coordinates. The main goal of this contribution is to generalize the computational homogenization scheme to a formulation of micro–macro transitions in curvilinear convective coordinates, where different physical spaces are considered at the homogenized macro‐continuum and at the locally attached representative micro‐structures. The deformation and the coordinate system of the micro‐structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro‐continuum. For the consistent formulation of micro–macro transitions, the operations scale‐up and scale‐down are introduced, considering the rotated representation of tensor variables at the different physical reference frames of micro‐ and macro‐structure. The second goal of this paper is to use objective strain measures like the Green–Lagrange strain tensor for the solution of boundary value problems on the micro‐ and macro‐scale by providing the required transformations for the work‐conjugate stress, strain and tangent tensors into variables admissible for the considered micro–macro transitions and satisfying the averaging theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi‐static analysis of three‐dimensional crack propagation in brittle and quasi‐brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity‐regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack‐band’ version of XFEM. The evolving discontinuity surface is discretized through a C⁰ surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large‐scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.     

Computational structural and material stability analysis in finite electro‐elasto‐statics of electro‐active materials

Dielectric materials like electro‐active polymers (EAPs) exhibit coupled electro‐mechanical behavior at large strains. They respond by a deformation to an applied electrical field and are used in advanced industrial environments as sensors and actuators, for example, in robotics, biomimetics and smart structures. In field‐activated or electronic EAPs, the electric activation is driven by Coulomb‐type electrostatic forces, resulting in Maxwell stresses. These materials are able to provide finite actuation strains, which can even be improved by optimizing their composite microstructure. However, EAPs suffer from different types of instabilities. This concerns global structural instabilities, such as buckling and wrinkling of EAP devices, as well as local material instabilities, such as limit‐points and bifurcation‐points in the constitutive response, which induce snap‐through and fine scale localization of local states. In this work, we outline variational‐based definitions for structural and material stability, and design algorithms for accompanying stability checks in typical finite element computations. The formulation starts from stability criteria for a canonical energy minimization principle of electro‐elasto‐statics, and then shifts them over to representations related to an enthalpy‐based saddle point principle that is considered as the most convenient setting for numerical implementation. Here, global structural stability is analyzed based on a perturbation of the total electro‐mechanical energy, and related to statements of positive definiteness of incremental finite element tangent arrays. We base the local material stability on an incremental quasi‐convexity condition of the electro‐mechanical energy, inducing the positive definiteness of both the incremental electro‐mechanical moduli as well as a generalized acoustic tensor. It is shown that the incremental arrays to be analyzed in the stability criteria appear within the enthalpy‐based setting in a distinct diagonal form, with pure mechanical and pure electrical partitions. Applications of accompanying stability analyses in finite element computations are demonstrated by means of representative model problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite element analysis of viscoelastic composite structures based on a micromechanical material model

The stress and creep analysis of structures made of micro-heterogeneous composite materials is treated as a two-scale problem, defined as a mechanical investigation on different length scales. Reinforced composites show by definition a heterogeneous texture on the microlevel, determined by the constitutive behaviour of the matrix material and the embedded fibres as well as the characteristics of the bonding properties in the interphase. All these heterogeneities are neglected by the finite element analysis of structural elements on the macroscale, since a ficticious and homogeneous continuum with averaged properties is assumed. Therefore, the constitutive equations of the substitute material should well reflect the mechanical behaviour of the existing micro-heterogeneous composite in an average sense.The paper at hand starts with the brief outline of a micromechanical model, named generalized method of cells (GMC), which provides the macrostress responses due to macrostrain processes as well as the homogenised constitutive tensor of the substitute material. The macroscopic stresses and strains are obtained as volume averages of the corresponding microfields within a representative volume element. The effective material tensor constitutes the mapping between the macro-strains and the macro-stresses. The cells method is used for the homogenisation of the unidirectionally reinforced single layers of laminates made of viscoelastic resins and flexibly embedded elastic fibres. The algorithm for the homogenisation of the constitutive properties runs simultaneously to the finite element analysis at each point of numerical integration and provides the macro-stresses and the homogenised constitutive properties. The validity of the proposed two-scale simulation is investigated by solving boundary value problems and comparing the numerical results for the structures to the experimental data of creep and relaxation tests or analytical solutions.     

A new nonlinear constitutive theory for conducting magnetothermoelastic solids

A new nonlinear theory of constitutive equations for electrically and thermally conducting magnetothermoelastic (MTE) solids is developed. In the theory, the electric current and heat flux vectors are also considered to be independent variables in the argument of each constitutive function. It is shown that the modified Helmholtz free energy (MH FE) density, which is a thermodynamical potential for the specific entropy, the magnetization and the stress tensor, does no longer appear as a function of the temperature, the magnetic field and the strain tensor, but it also depends upon the electric current and heat flux vectors. Furthermore, referring to the mentioned constitutive equations, the Gibbs equation is also generalized. In order to expose the constitutive theory developed here, an appropriate polynomial expression of the MH FE density for the anisotropic materials is proposed, and, exploiting the method of the theory of invariants, its exact expression is also determined. With the use of these two expressions, a set of rather general nonlinear constitutive equations, which governs a lot of magnetoelastothermo-electrical (MET-E) effects, is then obtained explicitly. It is interesting to notice that each of the constitutive equations mentioned above has a pseudo (ir) reversible part in vicinities of the new equilibrium state, namely the thermo-electrical equilibrium (T-EE) state. According to the deductive scheme, the generalized constitutive equations and the Gibbs equation in the present work are finally discussed for special materials, and/or vanishing some of the fields. The resulting expressions are, as they should be, in full mutual agreement with the established theories on the same subject.