The Physics of Rubber Elasticity (J. Meixner)

An explicit solution for implicit time stepping in multiplicative finite strain viscoelasticity

This paper introduces a fully implicit partitioned coupling scheme for problems of thermoelasticity at finite strains utilizing the p-version of the finite element method. The mechanical and the thermal fields are partitioned into symmetric subproblems where algorithmic decoupling has been obtained by means of an isothermal operator-split. Numerical relaxation methods have been implemented to accelerate the convergence of the algorithm. Such methods are well-known from coupled fluid-structure interaction problems leading to highly efficient algorithms. Having studied the influence of three different strategies: polynomial prediction methods, numerical relaxation with constant relaxation coefficients, its dynamic variant with a residual based relaxation coefficient and a variant of a reduced order model - quasi-Newton method, we present several numerical simulations of quasi-static problems investigating the performance of accelerated coupling schemes.

Accelerated staggered coupling schemes for problems of thermoelasticity at finite strains

The purposes of this article are to present new aspects of modeling multi-physically coupled fields, focusing particularly on the partitioned treatment of electro-thermo-mechanical problems. Coupled problems of this kind occur in many industrial applications, such as micro-electrical devices, field-assisted sintering processes or electrical fuses. In this paper, we restrict ourselves to the case of nonlinear thermo-elasticity at finite strains and a heat source resulting from the electrical field. Plasticity effects are not taken into consideration. The objective is to ascertain the coupling of the algorithm relating to the fields involved individually and to demonstrate a global partitioned solution strategy. We also introduce several methods that increase algorithmic stability and accelerate the iterative coupling process. This article aims to present an efficient partitioned coupling strategy for different coupling levels between the fields. To this end, we study the proposed algorithm with the help of several numerical examples ranging from linear to highly nonlinear problems, involving substantial geometric changes and finite strains.

A partitioned solution approach for electro-thermo-mechanical problems

In view of code verification of finite element implementations and for material parameter identification purposes it is of interest to make use of stress algorithms developed for three-dimensional finite element computations. In the case of homogeneous deformations various boundary-conditions for given displacements or stresses are possible and define a sub-problem of three-dimensional stress---strain states, which are either one-, two- or three-dimensional. Examples are uniaxial tension/compression, plane stress conditions or biaxial tensile problems. Caused by the fact that the stress algorithms are strain-driven, the constraints of zero stresses in a specific direction lead for elastic and inelastic constitutive models to a particular system of differential-algebraic equations. How to treat such stress algorithms and how to solve the resulting system of differential-algebraic equations, which are developed for finite element programs, for specific stress and displacement boundary conditions is discussed in this article. Additionally, it is worked out that the consistent tangent operator is required in the same manner as in 3D-FE computations. The second topic treats the extension of the entire procedure for material parameter identification procedure applied to test data for different materials such as steel, rubber material and powder. In this respect, uniaxial tensile, biaxial tensile tests, and laterally constrained loading paths are exemplarily investigated. These investigations and the proposed procedure are applied for small and finite strain problems. In this investigation measure of the quality of identification is discussed as well.

Homogeneous stress---strain states computed by 3D-stress algorithms of FE-codes: application to material parameter identification

Rosenbrock-type methods applied to finite element computations within finite strain viscoelasticity

On the one hand, displacement controlled processes are frequently applied in Computational Mechanics using the finite-element method, on the other hand, there are some shortcomings regarding the study and the presentation of this particular case. In this article, we focus our theoretical considerations on quasi-static problems with constitutive equations of evolutionary type. In this case, it is currently known that the solution procedure of global and local iterations within the “Newton-Raphson method” is related to the method of lines, where one arrives at a system of differential-algebraic equations after the spatial discretization by means of the finite element method. This could be solved by means of the Backward-Euler method or more appropriately using time-adaptive, stiffly accurate, diagonally implicit Runge-Kutta methods in combination with the Multilevel-Newton method. In this respect the theoretical basis of currently applied implicit non-linear finite element analyses is extended to displacement controlled processes. In this article, the calculation of the reaction forces using the method of Lagrange multipliers and the penalty method are of special interest in view of the new DAE-approach. Accordingly, an important objective lies in a consistent notation so that the dependence of known and unknown variables as well as the transition from local to global level becomes obvious. These investigations are of utmost importance for micro-macro homogenization techniques in FE$^2$ approaches and the development of new global time and space-adaptive integration methods.

Displacement control in time-adaptive non-linear finite-element analysis

The method of vertical lines in the case of quasi-static solid mechanics applying constitutive models of evolutionary-type yields after the spatial discretization by means of finite elements a system of differential-algebraic equations. It is of substantial interest how fast, accurate, and stable such computations can be carried out. Moreover, the questions are how simple the implementation can be done and how susceptible a procedure is to programming errors. In this article, this is investigated for half-explicit Runge–Kutta methods that are applied to small and finite strain viscoelasticity. The advantage of the method is given by a non-iterative scheme on element level. Additionally, it turns out that for models where linear elasticity is one ingredient in the constitutive model, the method leads to only one required LU-decomposition at the beginning of the entire computation, and in each time step, only one back-substitution step has to be carried out. This outperforms current finite element computations. Order investigations of various integration schemes and the automatic step-size behavior are studied. This new proposal is compared to a classical Backward-Euler implementation, high-order stiffly accurate diagonally implicit Runge–Kutta, and recently proposed Rosenbrock-type methods. Advantages and disadvantages of the applied schemes are discussed.

Comparison of diagonal-implicit, linear-implicit and half-explicit Runge–Kutta methods in non-linear finite element analyses

In this essay, a constitutive model for nearly incompressible elastic behavior is extended to the case to thermal effects. First, the use is made of the multiplicative decomposition of the deformation gradient into a thermal and a mechanical part. The thermal part is purely volumetric. Additionally, the mechanical part is multiplicatively decomposed into a volume-preserving and a volume-changing part so that the final stress state shows the influences of the temperature-dependence. The proposed model is carefully studied in view of the thermo-mechanical coupling effects. Second, the model is implemented into a time-adaptive finite element formulation based on higher-order Rosenbrock-type methods, which is a completely iteration-free procedure so that really fast computations are available. The article concludes with a three-dimensional numerical simulation of a representative elastomeric tensile specimen.

https://www.ptmts.org.pl/jtam/index.php/jtam/article/download/v50n1p3/97

Ahmad-Wahadj Hamkar

Papers

Rosenbrock-type methods applied to finite element computations within finite strain viscoelasticity

Displacement control in time-adaptive non-linear finite-element analysis

Comparison of diagonal-implicit, linear-implicit and half-explicit Runge–Kutta methods in non-linear finite element analyses

Theoretical and numerical aspects in weak-compressible finite strain thermo-elasticity

A stiffly accurate Rosenbrock-type method of order 2 applied to FE-analyses in finite strain viscoelasticity