A geometrically nonlinear (3,2) reduced degree-of-freedom unified zigzag laminated beam element for large deformation analysis

A geometrically nonlinear (3,2) unified zigzag beam element is developed with a reduced number of degree-of-freedom for the large deformation analysis. The main merit of the beam element model is the Kirchhoff and Cauchy shear stress solution for large deformation and large strain analysis is more accurate. The geometrically nonlinearity is considered in the calculation of the zigzag coefficients. Thus, the results of shear Cauchy stress are matching well with solid element analysis in case of the beam with aspect ratio greater than 20 under large deformation. The zigzag coefficients are derived explicitly. The Green strain and the second Piola Kirchhoff stress are used. The second Piola Kirchhoff shear stress is continuous at the interface between adjacent layers priori. The bottom surface second Piola Kirchhoff shear stress condition is used to determine the zigzag coefficient and the top surface second Piola Kirchhoff shear stress condition is used to reduce one degree-of-freedom. The nonlinear finite element equations are derived. In the numerical tests, several benchmark problems with large deformation are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for beam with aspect ratio greater than 20. The second Piola Kirchhoff and Cauchy shear stress accuracy is also good. A convergence study is also presented.     

非线性弹性大变形问题的率型解法

本文分别采用Ｊａｕｍａｎｎ应力率、Ｔｒｕｅｓｄｅｌｌ应力率和Ｇｒｅｅｎ－Ｎａｇｈｄｉ应力率导出了非线性各向同性弹性体的率型本构表达形式，通过对Ｍｏｏｎｅｙ－Ｒｉｖｌｉｎ材料的简单剪切大变形分析表明，三种率型的本构关系均与全量本构关系相等价。文中还给出了相应的率型变分原理，并采应Ｒｉｔｚ法，数值求解了受单轴拉伸的矩形橡皮薄膜的大变形问题．     

形状记忆合金相变过程三维大变形有限元模拟

形状记忆合金（SMA）一直被作为智能材料开发,并被用于阻尼器、促动器和智能传感器元件．形状记忆合金（SMA）的一项重要特性,是它具有恢复在机械加卸载周期下产生的大变形而不表现出永久变形的能力．该文旨在介绍一种由应力产生的相变且可以描述马氏体和奥氏体之间的超弹性滞回环现象本构方程．形状记忆合金的马氏体系数假设为应力偏张量的函数,因此形状记忆合金在相变过程中锁定体积．本构模型是在大变形有限元的基础上执行的,采用了现时构型Lagrange大变形算法．为了方便地使用Cauchy应力和线性应变本构关系,使用了与旋转无关的Jaumann应力增率计算应力．数值分析结果表明,相变引起的超弹性滞回环可以有效地通过该文提出的本构方程和大变形有限元模拟．     

Cyclic martensitic phase transformation of monocrystals at finite strains

Based on Bain's principle, a C¹-continuous thermo-mechanical micro-macro constitutive model for martensitic phase transformation (PT) of monocrystals at finite strains and hyperelastic free energy function is used. It is represented by a unified non-convex Lagrangian variational functional. The convexification problem is solved here by generalizing the explicit form of the lower Reuss bound for small strains given in [1] to finite strains. Abaqus is used for implementation of 3D finite elements in space, via UMAT-interface which requires Jaumann rate of Kirchhoff stress tensor. Deterministic validation of the model is presented by comparing verified numerical results with experimental data for Cu₈₂Al₁₄Ni₄ [6] for quasiplastic PT. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

简单剪切有限塑性内时理论分析

在有限塑性内时理论中引入Jaumann率、广义Jaumann率、扶率及Wu率,并以此分析了简单剪切大变形问题．结果验证了简单剪切变形中,采用次弹性或内时刚塑性材料的Jaumann率客观模型,随单调递增的剪切变形剪切应力和法向应力都会出现振荡现象．这说明振荡现象的出现不取决于弹塑性模型,而与选取不同的客观率有很大的关系．同时指出在简单剪切大变形时,法向应力并不为零．     

Numerical calculation of the tangent stiffness matrix in materials modeling

The Finite Element Method in the field of materials modeling is closely connected to the tangent stiffness matrix of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness matrix can be derived analytically only with great effort, if at all. Numerical methods are therefore widely used for its calculation. We present our method to calculate the tangent stiffness matrix for the logarithmic strain measure. The approach is compared with other commonly used procedures. A significant increase in accuracy can be achieved with the proposed method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational implementation of stress integration in FE analysis of elasto-plastic large deformation problems

In this paper, we compare different numerical implementation algorithms for the rate type constitutive equation and present an integration scheme based on the physical meaning of the stress. Numerical implementation of various schemes is investigated in conjunction with the return mapping algorithm and the conditions to maintain plastic consistency. Jaumann and Truesdell rates are taken as the objective stress rates in the constitutive equation. An alternative numerical treatment for rate of deformation tensor D_ij is presented and is shown to maintain incremental objectivity. Numerical examples included a single element under rigid body rotation, a necking bifurcation of a bar in tension and a punch indentation process. It is shown that the use of Truesdell stress rate with specific numerical integration procedure gives more accurate results than other procedures presented.     

Computation of energy dissipation in visco-elastic materials at finite deformation

In this contribution, the derivation of the energy dissipation rate in generalized visco-elastic material models with internal stress-type variables and linear evolution equations is outlined. The approximated dissipation rate is computed from a positive quadratic form of the nonlinear non-equilibrium stresses and the inverse of the consistent material tangent tensor. The presented method is used to compute the energy dissipation of visco-elastic rubber material in a large scale application of a steady state rolling tire structure. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polyconvex potentials,invertible deformations,and thermodynamically consistent formulation of the nonlinear elasticity equations

It is shown that the nonstationary finite-deformation thermoelasticity equations in Lagrangian and Eulerian coordinates can be written in a thermodynamically consistent Godunov canonical form satisfying the Friedrichs hyperbolicity conditions, provided that the elastic potential is a convex function of entropy and of the minors of the elastic deformation Jacobian matrix. In other words, the elastic potential is assumed to be polyconvex in the sense of Ball. It is well known that Ball’s approach to proving the existence and invertibility of stationary elastic deformations assumes that the elastic potential essentially depends on the second-order minors of the Jacobian matrix (i.e., on the cofactor matrix). However, elastic potentials constructed as approximations of rheological laws for actual materials generally do not satisfy this requirement. Instead, they may depend, for example, only on the first-order minors (i.e., the matrix elements) and on the Jacobian determinant. A method for constructing and regularizing polyconvex elastic potentials is proposed that does not require an explicit dependence on the cofactor matrix. It guarantees that the elastic deformations are quasiisometries and preserves the Lame constants of the elastic material.     

Finite strain inelasticity for isotropy,a simple and efficient finite element formulation

We consider a formulation of associative isotropic J₂‐elastoplasticity at finite inelastic strains and aspects of its numerical implementation. The essential ingredients include the multiplicative decomposition of the deformation gradient in elastic and inelastic parts, the definition of a convex elastic domain in stress space and a material representation of the constitutive equations for general non‐Cartesian coordinate charts. On the numerical side we propose a stress update algorithm for elasto‐plastic response, including isotropic hardening. The finite element formulation is based on assumed strain and enhanced strain variational principles, for a complete outline see [3]. Remarkably the formulation is very similar to the case of infinitesimal plasticity: (i) The scheme of linear return mapping algorithm takes the form of standard return mapping of the infinitesimal theory for the case of isotropic elastic response. (ii) The algorithmic elastoplastic moduli have a similar structure as in the linear case. Together with an exact fulfillment of plastic incompressibility by means of a simple correction one achieves an advantageously efficient finite element formulation. Its performance is documented by a numerical example.     

Influence of the nonlinear matrix material behaviour on the effective properties of textile-reinforced thermoplastics

For a consistent lightweight design the consideration of the nonlinear macroscopic material behaviour of composites, which is amongst others driven by damage and strain-rate effects on the mesoscale, is required. Therefore, a modelling approach using numerical homogenization techniques is applied to predict the effective nonlinear material behaviour of the composite based on the finite element simulation of a representative volume element (RVE). In this RVE suitable constitutive relations account for the material behaviour of each constituents. While the reinforcing glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law is applied to represent the strain-rate dependent, inelastic deformation of the matrix material. In order to analyse the influence of the nonlinear matrix material behaviour on the global mechanical response of the composite, effective stress-strain-curves are computed for different load cases and compared to experimental observations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Incremental self-consistent approach for the estimation of nonlinear material behavior of metal matrix composites

The application of homogenization methods to compute the macroscopic material response of metal matrix composites is a possibility to save memory and computation time in comparison to full field simulations. This paper deals with a method to extend the self-consistent scheme from linear elasticity theory to nonlinear problems. The idea is to approximate the nonlinear problem by an incrementally linear one. Since time discretization of the deformation process implies a certain linearization, we use the algorithmic consistent tangent operator of the composite for defining the linear comparison material in each time step. This is in contrast to classical incremental self-consistent approaches which use continuum tangent or secant operators. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of the effective nonlinear material behaviour of composites using X-FEM - Analysis of the matrix material behaviour

For a consequent lightweight design the consideration of the nonlinear macroscopic material behaviour of composites, which is amongst others driven by damage and strain–rate effects on the mesoscale, is required. Therefore, the modelling approach using numerical homogenization techniques based on the simulation of representative volume elements which are modelled by the extended finite element method (X–FEM) is currently extended to nonlinear material behaviour. While the glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law accounts for strain–rate dependence and inelastic deformation of the matrix material. This paper describes the process of finding suitable constitutive relations for the polymeric matrix material Polypropylene in the small–strain regime. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modified Newton’s method for systems of nonlinear equations with singular Jacobian

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.     

Consistent macroscopic tangent computation for FFT-based computational homogenization at finite strains

The present work addresses the efficient computation of effective properties of periodic microstructures by the use of Fast Fourier Transforms. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of an RVE, the computation of the associated moduli is not straight-forward. The contribution of the present paper is thus the derivation and implementation of an algorithmically consistent macroscopic tangent operator that comprises the effective properties of the RVE. In contrast to finite-difference based approaches, an exact solution for the macroscopic tangent is derived by means of the classical Lippmann-Schwinger equation. The problem then reduces to the solution of a system of linear equations even for nonlinear material behaviour. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stress recovery procedure for solving geometrically non-linear problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric Piola–Kirchhoff stress tensor. Vectors of the forces reduced to the mesh points are constructed using the displacements at the mesh points found by solving this equation and for the known stiffness matrices of the elements. On the other hand, these forces at the mesh points are defined in terms of unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The values of the Piola–Kirchhoff stress tensor of the first kind at the mesh points are determined using the values found for the distributed forces on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations for the initial configuration. The linearized representation of this tensor enables all the derivatives of the increment in the strain vector with respect to the coordinates to be found without invoking the operation of differentiation. The particular features of the use of the stress recovery procedure are demonstrated for a plane problem in the non-linear theory of elasticity.     

Error estimates for mixed finite element discretization of the Richards equation

We present a numerical scheme based on the mixed finite element method (MFEM) for the Richards equation, a nonlinear, degenerate parabolic equation. Due to the degeneracy, the solution of the equation has low regularity and therefore only lower order finite elements are recommended. We review the main posibilities for proving the convergence of the scheme. Especially for the case without using the Kirchhoff transformation a new result is given. We also briefly discuss how to solve the nonlinear fully discrete problems appearing at each time step and refer to papers where the convergence of these methods is rigurously studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the thermodynamics of fluids defined by implicit constitutive relations

In this paper, we develop a thermodynamically consistent theory for describing the response of nonlinear viscous fluids whose constitutive equations are of the form f (T, D) = 0. We show that such constitutive equations which include classical constitutive equations wherein the stress is expressed explicitly in terms of the kinematical quantities, provide a rich class of physically meaningful fluid response functions which allows us to describe a wider range of material behavior, including that of a general class of incompressible fluids, incompressible fluids with pressure dependent viscosity, and Bingham (or pseudoplastic) materials.     

Asymmetric effects of polymers in finite thermoviscoplasticity

Glassy polymers such as polycarbonate exhibit different behaviours in different loading scenarios, such as tension and compression. For the simulation of these asymmetric effects we present a framework for thermoviscoplastic modelling of polymers at large strains. To this end a flow rule is postulated within a thermodynamic consistent framework in a mixed variant formulation which is decomposed into a sum of weighted stress mode related quantities. The different stress modes are chosen such that they are accessible to individual examination in the laboratory, where tension and compression are typical examples. The characterisation of the stress modes is obtained in the octahedral plane of the deviatoric stress space in terms of the Lode angle, such that stress mode dependent scalar weighting functions can be constructed. Furthermore the numerical implementation of the resulting set of constitutive equations is used in the finite element program ABAQUS to simulate the laser transmission welding process. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)