B. Schieck

Theory and numerical analysis of shells undergoing large elastic strains

Abstract A non-linear bending theory of rubber-like shells undergoing large elastic strains is proposed. The theory is based on a relaxed normality hypothesis and the incompressibility condition. Using series expansion in the normal direction and applying some estimation technique a consistent first approximation and a simplest approximation to the elastic strain energy of the shell are constructed. Lagrangian displacement shell equations are derived and the incremental shell deformation is considered. The numerical results presented for one- and two-dimensional large strain shell problems confirm the accuracy and efficiency of the proposed shell theory.

The appropriate corotational rate, exact formula for the plastic spin and constitutive model for finite elastoplasticity

Abstract The exact formulae for the plastic and the elastic spin referred to the deformed configuration are derived, where the plastic spin is a function of the plastic strain rate and the elastic spin a function of the elastic strain rate. With these exact formulae we determine the macroscopic substructure spin that allows us to define the appropriate corotational rate for finite elastoplasticity. Plastic, elastic and substructure spin are considered and simplified for various sub-classes of restricted elastic-plastic strains. It is shown that for the special cases of rigid-plasticity and hypoelasticity the proposed corotational rate is identical with the Green-Naghdi rate, while the ZarembaJaumann rate yields a good approximation for moderately large strains. To compare our exact plastic spin formula with the constitutive assumption for the plastic spin introduced by Dafalias and others, we simplify our result for small elastic-moderate plastic strains and introduce a simplest evolution law for kinematic hardening leading to the Dafalias formula and to an exact determination of its unknown coefficient. It is also shown that contrary to statements in the literature the plastic spin is not zero for vanishing kinematic hardening. For isotropic-elastic material with induced plastic flow undergoing isotropic and kinematic hardening constitutive and evolution laws are proposed. Elastic and plastic Lagrangean and Eulerian logarithmic strain measures are introduced and their material time derivatives and corotational rates, respectively, are considered. Finally, the elastic-plastic tangent operator is derived. The presented theory is implemented in a solution algorithm and numerically applied to the simple shear problem for finite elastic-finite plastic strains as well as for sub-classes of restricted strains. The results are compared with those of the literature and with those obtained by using other corotational rates.

Theory and analysis of shells undergoing finite elastic-plastic strains and rotations

SummaryIn this paper theory and analysis of shells undergoing finite elastic and finite plastic strains and rotations are presented. The shell kinematics are based on a relaxed normality hypothesis allowing transverse normal material fibers to be stretched and bended, whereas shear deformations are neglected. Lagrangean logarithmic membrane and logarithmic bending strain measures are introduced, and it is shown that they can be additively decomposed into purely elastic and purely plastic parts for superposed moderately large strains and unrestricted rotations. The logarithmic strain measures are used to formulate thermodynamic-based constitutive equations for isotropic elastic and plastic material behavior with isotropic and kinematic hardening induced by continuous plastic flow. To analyse path-dependent elastic-plastic shell deformations by iterative procedures the application of logarithmic strain measures allows to realize load steps with corresponding moderate strains and unrestricted rotations. The moderate strain restriction for superposed deformations can be assured by an appropriate update procedure. Formulae are given to determine exactly the rotational change of the reference configuration during the update. Finally, the principle of virtual work with corresponding elastic-plastic material tensor is formulated and it is shown that the weak form of the virtual work leads to the Lagrangean equilibrium equations and boundary conditions well-known from the nonlinear theory of elastic shells.

A shell finite element for large strain elastoplasticity with anisotropies—: Part I: Shell theory and variational principle

A shell model for finite elastic and finite plastic strains is derived taking into account initial and induced anisotropies. A corresponding eight-node C 0 shell element with three displacement and three director degrees-of-freedom at each node is developed, which combines the advantages of an isoparametric description of geometry and deformation with an effective plane stress formulation. The element accounts for isochoric or approximately isochoric deformation due to finite plastic strains. Because of the three displacement and three director degrees-of-freedom at each node, it is easily possible to link different parts of a composed irregular shell structure or to connect the derived shell element with solid (brick) elements. This paper presents the shell theory based on the kinematics of finite elastoplasticity proposed in Schieck and Stumpf (1995) and the special geometric concept of the derived shell model. The Lagrange multiplier method is applied to introduce into the virtual work principle the transverse normality constraint and the condition of isochoric deformation, where the Lagrange multipliers can be condensed inside the element procedure. Various assumed strain techniques designed to avoid the membrane locking are compared with known methods in the literature. According to the numerical experience so far the proposed shell finite element is free of locking effects and spurious modes. Part II presents the constitutive equations for finite elastic–plastic strains accounting for initial and induced anisotropies and the implementation of the model into the FE-code. A comprehensive set of numerical examples is provided, involving the tension of a plane specimen with necking and shear-band localization, the elastic–plastic response of a simply supported plate with a localization of the plastic bending strains in the four corner zones, the elastic–plastic deformation mode of the so-called Scordelis-Lo roof, and the elastic–plastic buckling of a cylindrical shell showing an essential influence of the anisotropic material behavior. The results illustrate the performance of the proposed shell finite element for a wide range of engineering applications.

Deformation analysis for finite elastic-plastic strains in a lagrangean-type description

Abstract A general concept is presented to analyse the deformation of structures undergoing arbitrarily large elastic and arbitrarily large plastic strains. Based on the multiplicative decomposition of the deformation gradient into elastic and plastic contributions the kinematics of two superposed finite, non-coaxial deformations are investigated. Lagrangean-type elastic and plastic stretch tensors are introduced and multiplicative decompositions of the total stretch into these elastic and plastic stretches are derived. It is shown that the result is independent of any decomposition of the total rotation into an elastic and a plastic rotation. For the second, superposed deformation the total Lagrangean logarithmic (Hencky) strain tensor with corresponding elastic and plastic logarithmic strains is defined. If in a large deformation analysis the first deformation is updated such that the second deformation is constrained to be moderately large, then the total logarithmic strain tensor of the second deformation can be additively decomposed into purely elastic and purely plastic parts. This enables an appropriate formulation of constitutive equations for isotropic hyperelastic material behavior with associated flow rule and evolution laws for combined isotropic-kinematic hardening. Work-conjugate to the elastic logarithmic strain tensor is a “back-rotated” Kirchhoff stress tensor. The rotational change of its reference configuration during the update is given explicitly. Finally the principle of virtual work with corresponding equilibrium equations and static and geometric boundary conditions is given. The virtual work functional is transformed to deliver the consistent tangent stiffness matrix as basis for a finite element solution algorithm.

A shell finite element for large strain elastoplasticity with anisotropies—: Part II: Constitutive equations and numerical applications

Abstract An eight-node C 0 shell element for finite elastic–plastic deformations with anisotropies is developed. It combines the advantages of an isoparametric description of geometry and deformation, the application of tensors in Cartesian components, and a real and effective plane stress description with three displacement and three director degrees-of-freedom at each node. In Part I of the paper the shell theory including the kinematics, the variational principle, the application of Lagrange multipliers with their condensation on the element level and a comparative study of various assumed strain techniques were presented and the results of convergence tests given. In this paper, we consider the constitutive equations for large elastic and large plastic strains accounting for initial and induced anisotropies and the corresponding thermodynamics. Then we investigate the return algorithm for finite strains and the implementation of the element procedure including stiffness matrix and residual force vector. Finally, we present the results of extended numerical applications and a comparison with FE solutions published in the literature, as far as such are available.

SHAKEDOWN AT FINITE ELASTO-PLASTIC STRAINS

For structures undergoing small elastic-plastic deformations (i. e. small rotations and small strains), shakedown or non-shakedown can be determined by applying the well-known statical shakedown theorem of Melan [1], which leads to a lower bound of the load factor. Dual to the Melan’s theorem Koiter [2] formulated a kinematical shakedown theorem yielding an upper bound of the load factor. While many publications deal with shakedown of structures at small deformations taking into account linear and non-linear hardening material behaviour (e. g. Stein et al. [3]) the search for generalisations of the shakedown theorem for large deformations was initiated by the paper of Maier [4]. Weichert [5, 6], Gros-Weege [7], Saczuk and Stumpf [8], Tritsch and Weichert [9] and Weichert and Hachemi [10] presented generalisations of the Melan’s theorem for structures undergoing large deformations with large plastic strains and moderate or large rotations. The underlying idea is to determine a moderately or finitely deformed configuration of the structure as new reference and then to investigate the shakedown behaviour for superposed small deformations. Typical applications for these methods are thin plate and shell structures. While the shakedown analysis for arbitrary superposed small deformation histories can be performed with optimisation technique, the reference configuration has to be determined by using an appropriate shell finite element for moderate or large deformations.