Variational principles in dissipative electro‐magneto‐mechanics: A framework for the macro‐modeling of functional materials

This paper presents a general framework for the macroscopic, continuum‐based formulation and numerical implementation of dissipative functional materials with electro‐magneto‐mechanical couplings based on incremental variational principles. We focus on quasi‐static problems, where mechanical inertia effects and time‐dependent electro‐magnetic couplings are a priori neglected and a time‐dependence enters the formulation only through a possible rate‐dependent dissipative material response. The underlying variational structure of non‐reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so‐called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum‐based modeling in macroscopic electro‐magneto‐mechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle‐point principles. Next, the implications on the formulation of boundary‐value problems are addressed, which appear in energy‐based formulations as minimization principles and in enthalpy‐based formulations in the form of saddle‐point principles. Furthermore, the material stability of dissipative electro‐magneto‐mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro‐magneto‐mechanics, an energy‐based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy‐based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials. Copyright © 2011 John Wiley & Sons, Ltd.     

B‐bar FEMs for anisotropic elasticity

Anisotropic elastic materials, such as the homogenized model of a fiber‐reinforced matrix, can display near rigidity under certain applied stress–the resulting strains are small compared with the strains that would occur for other stresses of comparable magnitude. The anisotropic material could be rigid under hydrostatic pressure if the material were incompressible, as in isotropic elasticity, but also for other stresses. Some commonly used finite elements are effective in dealing with incompressibility, but are ill‐equipped to handle materials that lock under non‐hydrostatic stress states (e.g., uniformly reduced serendipity and Q1/Q0 B‐bar hexahedra). The failure of the original B‐bar method is attributed to the assumption that the mode of deformation to be relieved is one of near incompressibility. The remedy proposed here is based on the spectral decomposition of the compliance matrix of the material. The spectrum can be interpreted to separate nearly‐rigid and flexible modes of stress and strain, which leads naturally to a generalized selective reduced integration. Furthermore, the spectral decomposition also enables a three‐field elasticity formulation that results in a B‐bar method that is effective for general anisotropic materials with an arbitrary nearly‐rigid mode of deformation.Copyright © 2014 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.     

Floating stress‐point integration meshfree method for large deformation analysis of elastic and elastoplastic materials

A new meshfree formulation of stress‐point integration, called the floating stress‐point integration meshfree method, is proposed for the large deformation analysis of elastic and elastoplastic materials. This method is a Galerkin meshfree method with an updated Lagrangian procedure and a quasi‐implicit time‐advancing scheme without any background cell for domain integration. Its new formulation is based on incremental equilibrium equations derived from the incremental virtual work equation, which is not generally used in meshfree formulations. Hence, this technique allows the temporal continuity of the mechanical equilibrium to be naturally achieved. The details of the new formulation and several examples of the large deformation analysis of elastic and elastoplastic materials are presented to show the validity and accuracy of the proposed method in comparison with those of the finite element method. Copyright © 2012 John Wiley & Sons, Ltd.     

Damage evolution with growing cyclic creep and life prediction of MDYB‐3 PMMA

The influence of cyclic creep accumulation rate on the damage evolution of MDYB‐3 polymethyl methacrylate (PMMA) was experimentally investigated. Fatigue tests were carried out at four stress levels by stress control mode. The steady cyclic creep accumulation stage was observed occupying a substantial proportion of all specimens fatigue processes. Cyclic creep strain growth speed and relaxed modulus degradation rate were deduced as two important indicators for describing the damage evolution characteristics. Linear evolution relations of cyclic creep strain and modulus degradation with cycle times were retrieved from different terms of hysteresis loops. A preliminary model was proposed to be able to estimate the damage extent at different cyclic stress levels. The life predictions by the proposed model were compared with the experiment results and the classical power S–N model, which were demonstrated as a good estimation for the fatigue life. It is feasible to make durability evaluations by the characteristics of steady cyclic creep for multiaxis directed PMMA material.     

Non‐local damage model based on displacement averaging

Continuum damage models describe the changes of material stiffness and strength, caused by the evolution of defects, in the framework of continuum mechanics. In many materials, a fast evolution of defects leads to stress–strain laws with softening, which creates serious mathematical and numerical problems. To regularize the model behaviour, various generalized continuum theories have been proposed. Integral‐type non‐local damage models are often based on weighted spatial averaging of a strain‐like quantity. This paper explores an alternative formulation with averaging of the displacement field. Damage is assumed to be driven by the symmetric gradient of the non‐local displacements. It is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium. Around physical boundaries of the analysed body, both formulations differ and the non‐local displacement model generates spurious damage in the boundary layers. The paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the non‐local weight function. Alternatively, an implicit gradient formulation could be used. Issues of algorithmic implementation, computational efficiency and smoothness of the resolved stress fields are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

A variational framework for fiber‐reinforced viscoelastic soft tissues

The mechanical properties of soft biological tissues vary depending on how the internal structure is organized. Classical examples of tissues are ligaments, tendons, skin, arteries, and annulus fibrous. The main element of such tissues is the fibers which are responsible for the tissue resistance and the main mechanical characteristic is their viscoelastic anisotropic behavior. The objective of this paper is to extend an existing model for isotropic viscoelastic materials in order to include anisotropy provided by fiber reinforcement. The incorporation of the fiber allows the mechanical behavior of these tissues to be simulated. The model is based on a variational framework in which its mechanical behavior is described by a free energy incremental potential whose local minimization provides the constraints for the internal variable updates for each load increment. The main advantage of this variational approach is the ability to represent different material models depending on the choice of suitable potential functions. Finally, the model is implemented in a finite‐element code in order to perform numerical tests to show the ability of the proposed model to represent fiber‐reinforced materials. The material parameters used in the tests were obtained through parameter identification using experimental data available in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐performance geometric nonlinear analysis with the unsymmetric 4‐node, 8‐DOF plane element US‐ATFQ4

A recent unsymmetric 4‐node, 8‐DOF plane element US‐ATFQ4, which exhibits excellent precision and distortion‐resistance for linear elastic problems, is extended to geometric nonlinear analysis. Since the original linear element US‐ATFQ4 contains the analytical solutions for plane pure bending, how to modify such formulae into incremental forms for nonlinear applications and design an appropriate updated algorithm become the key of the whole job. First, the analytical trial functions should be updated at each iterative step in the framework of updated Lagrangian formulation that takes the configuration at the beginning of an incremental step as the reference configuration during that step. Second, an appropriate stress update algorithm in which the Cauchy stresses are updated by the Hughes‐Winget method is adopted to estimate current stress fields. Numerical examples show that the new nonlinear element US‐ATFQ4 also possesses amazing performance for geometric nonlinear analysis, no matter whether regular or distorted meshes are used. It again demonstrates the advantages of the unsymmetric finite element method with analytical trial functions.     

Progressive damage modeling in fiber-reinforced materials

This paper presents an anisotropic damage model suitable for predicting failure and post-failure behavior in fiber-reinforced materials. In the model the plane stress formulation is used and the response of the undamaged material is assumed to be linearly elastic. The model is intended to predict behavior of elastic-brittle materials that show no significant plastic deformation before failure. Four different failure modes – fiber tension, fiber compression, matrix tension, and matrix compression – are considered and modeled separately. The onset of damage is predicted using Hashin’s initiation criteria [Hashin Z, Rotem A. A fatigue failure criterion for fiber-reinforced materials. J Compos Mater 1973;7:448; Hashin Z. Failure criteria for unidirectional fiber composites. J Appl Mech 1980;47:329–34] and the progression of damage is controlled by a new damage evolution law, which is easy to implement in a finite element code. The evolution law is based on fracture energy dissipation during the damage process and the increase in damage is controlled by equivalent displacements. The issues related to numerical implementation, such as mesh sensitivity and convergence in the softening regime, are also addressed.     

拉压加载下纤维增强铝合金层板疲劳裂纹扩展的压载荷效应与预测模型

基于增量塑性损伤理论与纤维增强金属层板疲劳裂纹扩展唯象方法, 推导出在拉-压循环加载下, 纤维增强金属层板疲劳裂纹扩展速率预测模型。并通过玻璃纤维增强铝合金层板在应力比R=-1,-2的疲劳裂纹扩展实验对预测模型进行验证。结果表明, 纤维增强铝合金层板疲劳裂纹扩展的压载荷效应分为两种情况: 在有效循环应力比R_C>0时, 表现为压载荷对铝合金层所承受残余拉应力的抵消作用; 当R_C<0时, 表现为压载荷抵消残余拉应力后, 对纤维增强铝合金层板金属层的塑性损伤, 对疲劳裂纹扩展存在促进作用。纤维铝合金层板疲劳裂纹扩展的压载荷效应不可忽略, 本文中得出的在拉-压循环加载下疲劳裂纹扩展速率预测模型与实验结果符合较好。     

A Gibbs‐energy‐barrier‐based computational micro‐sphere model for the simulation of martensitic phase‐transformations

We introduce a material model for the simulation of polycrystalline materials undergoing solid‐to‐solid phase‐transformations. As a basis, we present a scalar‐valued phase‐transformation model where a Helmholtz free energy function depending on volumetric and deviatoric strain measures is assigned to each phase. The analysis of the related overall Gibbs energy density allows for the calculation of energy barriers. With these quantities at hand, we use a statistical‐physics‐based approach to determine the resulting evolution of volume fractions. Though the model facilitates to take into account an arbitrary number of solid phases of the underlying material, we restrict this work to the simulation of phase‐transformations between an austenitic parent phase and a martensitic tension and compression phase. The scalar model is embedded into a computational micro‐sphere formulation in view of the simulation of three‐dimensional boundary value problems. The final modelling approach necessary for macroscopic simulations is accomplished by a finite element formulation, where the local material behaviour at each integration point is governed by the response of the micro‐sphere model.Copyright © 2014 John Wiley & Sons, Ltd.     

Anisotropic Gurson‐Tvergaard‐Needleman plasticity and damage model for finite element analysis of elastic‐plastic problems

Implementation and analysis of the anisotropic version of the Gurson‐Tvergaard‐Needleman (GTN) isotropic damage criterion are performed on the basis of Hill's quadratic anisotropic yield theory with the definition of an effective anisotropic coefficient to represent the elastic‐plastic behavior of ductile metals. This study aims to analyze the extension of the GTN model suitable for anisotropic porous metals and to investigate the GTN model extension. An anisotropic damage model is implemented using the user material subroutine in ABAQUS/standard finite element code. The implementation is verified and applied to simulate a uniaxial tensile test on a commercially produced aluminum sheet material for three‐dimensional and plane stress test cases. Spherical and ellipsoidal micro voids are considered in the matrix material, and their effects on the uniaxial stress‐strain response of the material are analyzed. Hill's quadratic anisotropic yield theory predicts substantially large damage evolution and a low stress‐strain curve compared with those predicted by the isotropic model. An approximate model for anisotropic materials is proposed to avoid increased damage evolution. In this approximate model, Hill's anisotropic constants are replaced with an effective anisotropy coefficient. All model‐generated stress‐strain predictions are compared with the experimental stress‐strain curve of AA6016‐T4 alloy.     

Multi‐scale modeling of surface effects in nano‐materials with temperature‐related Cauchy‐Born hypothesis via the modified boundary Cauchy‐Born model

In nano‐structures, the influence of surface effects on the properties of material is highly important because the ratio of surface to volume at the nano‐scale level is much higher than that of the macro‐scale level. In this paper, a novel temperature‐dependent multi‐scale model is presented based on the modified boundary Cauchy‐Born (MBCB) technique to model the surface, edge, and corner effects in nano‐scale materials. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo‐mechanical finite element model. The temperature‐related Cauchy‐Born hypothesis is implemented by using the Helmholtz free energy to evaluate the temperature effect in the atomistic level. The thermo‐mechanical multi‐scale model is applied to determine the temperature related characteristics at the nano‐scale level. The first and second derivatives of free energy density are computed using the first Piola‐Kirchhoff stress and tangential stiffness tensor at the macro‐scale level. The concept of MBCB is introduced to capture the surface, edge, and corner effects. The salient point of MBCB model is the definition of radial quadrature used at the surface, edge, and corner elements as an indicator of material behavior. The characteristics of quadrature are derived by interpolating the data from the atomic level laid in a circular support around the quadrature in a least‐square approach. Finally, numerical examples are modeled using the proposed computational algorithm, and the results are compared with the fully atomistic model to illustrate the performance of MBCB multi‐scale model in the thermo‐mechanical analysis of metallic nano‐scale devices. Copyright © 2013 John Wiley & Sons, Ltd.     

A quasi-incompressible and quasi-inextensible element formulation for transversely isotropic materials

The contribution presents a new finite element formulation for quasi-inextensible and quasi-incompressible finite hyperelastic behavior of transeversely isotropic materials and addresses its computational aspects. The material formulation is presented in purely Eulerian setting and based on the additive decomposition of the free energy function into isotropic and anisotropic parts, where the former is further decomposed into isochoric and volumetric parts. For the quasi-incompressible response, the Q1P0 element formulation is outlined briefly, where the pressure-type Lagrange multiplier and its conjugate enter the variational formulation as an extended set of variables. Using the similar argumentation, an extended Hu-Washizu–type mixed variational potential is introduced, where the volume averaged fiber stretch and fiber stress are additional field variables. Within this context, the resulting Euler-Lagrange equations and the element formulation resulting from the extended variational principle are derived. The numerical implementation exploits the underlying variational structure, leading to a canonical symmetric structure. The efficiency of the proposed approached is demonstrated through representative boundary value problems. The superiority of the proposed element formulation over the standard Q1 and Q1P0 element formulation is studied through convergence analyses. The proposed finite element formulation is modular and exhibits very robust performance for fiber reinforced elastomers in the inextensibility limit.     

Fatigue modeling of materials with complex microstructures

A new approach and method of the analysis of microstructure-lifetime relationships of materials with complex structures is presented. The micromechanical multiscale computational analysis of damage evolution in materials with complex hierarchical microstructures is combined with the phenomenological model of fatigue damage growth. As a result, the fatigue lifetime of materials with complex structures can be determined as a function of the parameters of their structures. As an example, the fatigue lifetimes of wood modeled as a cellular material with multilayered, fiber reinforced walls were determined for different parameters of wood microstructures. In so doing, 3D hierarchical finite element models of softwood, and a computational technique, including the repeating restart and model change procedures, have been employed to model the fatigue response of latewood.     

A generalized finite‐strain damage model for quasi‐incompressible hyperelasticity using hybrid formulation

A new generalized damage model for quasi‐incompressible hyperelasticity in a total Lagrangian finite‐strain framework is presented. A Kachanov‐like reduction factor (1 ? D) is applied on the deviatoric part of the hyperelastic constitutive model. Linear and exponential softening are defined as damage evolution laws, both describable in terms of only two material parameters. The model is formulated following continuum damage mechanics theory such that it can be particularized for any hyperelastic model based on the volumetric–isochoric split of the Helmholtz free energy. However, in the present work, it has been implemented in an in‐house finite element code for neo‐Hooke and Ogden hyperelasticity. The details of the hybrid formulation used are also described. A couple of three‐dimensional examples are presented to illustrate the main characteristics of the damage model. The results obtained reproduce a wide range of softening behaviors, highlighting the versatility of the formulation proposed. The damage formulation has been developed to be used in conjunction with mixing theory in order to model the behavior of fibered biological tissues. As an example, the markedly different behaviors of the fundamental components of the rectus sheath were reproduced using the damage model, obtaining excellent correlation with the experimental results from literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Damage zones based on Dugdale model for materials

In this paper, an infinite sheet with a crack is studied using continuum damage mechanics technique. The formulation is based on the hypothesis of incremental complementary energy equivalence model for damage evaluation. Damage distributions in the region of a macrocrack tip are calculated for an elastic-perfectly plastic material. The size of the damage zone is also derived via the Dugdale model with damage which considers the interactions between the macrocracks and microcracks. To assess the results, comparisons are made between proposed damage model, Dugdale plastic model and finite element solutions. Good agreement is observed.     

Multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks

Multiscale computational techniques play a major role in solving problems related to viscoelastic composites due to the complexities inherent to these materials. In this paper, a numerical procedure for multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks is proposed in which the (global scale) homogenized viscoelastic incremental constitutive equations have the same form as the local‐scale viscoelastic incremental constitutive equations, but the homogenized tangent constitutive tensor and the homogenized incremental history‐dependent stress tensor at the global scale depend on the amount of damage accumulated at the local scale. Furthermore, the developed technique allows the computation of the full anisotropic incremental constitutive tensor of viscoelastic solids containing evolving cracks (and other kinds of heterogeneities) by solving the micromechanical problem only once at each material point and each time step. The procedure is basically developed by relating the local‐scale displacement field to the global‐scale strain tensor and using first‐order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify the approach and demonstrate the model capabilities. Copyright © 2009 John Wiley & Sons, Ltd.