Consistent Tangent Stiffness Matrix Research Articles

A formulation of the MITC4 shell element for finite strain elasto-plastic analysis

A finite strain elasto-plastic formulation is developed for the MITC4 shell element. The developed formulation is based on Lee's multiplicative decomposition of the deformation gradient and on the hyperelastic expression of Von Mises flow rule expressed in terms of Hencky's strain tensor. The formulation incorporates thickness stretching degrees of freedom that are condensed at the element level by imposing the classical ‘in-layer’ plane stress condition. A symmetric and consistent tangent stiffness matrix is also developed.

Quasi-static implicit and transient explicit analyses of sheet-metal forming using a C0 three-node shell element

A three-node isoparametric shell element well suited for extensive non-linear analyses is developed and used for the simulation of sheet-metal forming processes. Two computational strategies based on this element are presented: a quasi-static implicit one based on an updated lagrangien formulation with a consistent tangent stiffness matrix and a transient explicit one where the diagonal mass matrix is used to compute the nodal acceleration vector. The anisotropic elastic-plastic constitutive equations are treated using a rotated frame subjected to follow the anisotropy axes. Two numerical examples are presented and the two computational strategies are compared.

On the thermodynamic framework of generalized coupled thermoelastic-viscoplastic-damage modeling

A complete potential based framework using internal state variables is put forth for the derivation of reversible and irreversible constitutive equations. In this framework, the existence of the total (integrated) form of either the (Helmholtz) free energy or the (Gibbs) complementary free energy are assumed a priori. Two options for describing the flow and evolutionary equations are described, wherein option one (the fully coupled form) is shown to be over restrictive while the second option (the decoupled form) provides significant flexibility. As a consequence of the decoupled form, a new operator, i.e., the Compliance operator, is defined which provides a link between the assumed Gibb's and complementary dissipation potential and ensures a number of desirable numerical features, for example the symmetry of the resulting consistent tangent stiffness matrix. An important conclusion reached, is that although many theories in the literature do not conform to the general potential framework outlined, it is still possible in some cases, by slight modifications of the used forms, to restore the complete potential structure.

Design sensitivity analysis for rate-independent elastoplasticity

A new incremental, direct differentiation method for design sensitivity analysis of structures with rate-independent elastoplastic behavior is presented. We formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions. The sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation. Combined with the inherent advantages of implicit integration strategies, this represents a significant improvement over previous sensitivity formulations for history-dependent materials. First-order sensitivity expressions involving the complete set of design variables, including shape design variables, are derived for a generic response functional. The reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step. No iterations are needed in the sensitivity computations. A numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem. Explicit sensitivities from the new method are confirmed by finite difference estimates.

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

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.

Finite rotation analysis and consistent linearization using projectors

A systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis. This procedure is a generalization of the work presented in Computers and Structures, Volume 30, pp. 257–267. The basis of our approach is the element-independent co-rotational algorithm, where the element rigid body motion (translations and rotations) is separated from the deformational part of its total motion. The variation of this co-rotational relation results in a projector matrix, with the property that a consistent internal force vector is invariant under its action. The consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector. This results in an unsymmetric tangent matrix whose anti-symmetric part depends on the out-of-balance force vector. In this paper we prove that using the symmetric part of the tangent matrix, the Newton iteration retains its quadratic rate of convergence. This approach has been used to solve a number of large rotation test example problems. The results demonstrate that it is possible to analyze structures undergoing large rotations within a general co-rotational framework, using simple and economical finite elements. The resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre- and post-processors to the element routines.

An investigation of a finite rotation four node assumed strain shell element

AbstractThe paper presents a shell formulation based on the ‘degenerated solid approach’. The theory employs covariant strains and performs explicit integration through the shell thickness. The rigid body motion is exactly represented. The consistent tangent stiffness matrix is evaluated for the four node quadrilateral. It is shown, in the final part, that this type of element, which distinguishes itself by a very simple and easily understandable theory, gives good answers for linear as well as non‐linear applications.

A consistent tangent stiffness matrix for three‐dimensional non‐linear contact analysis

AbstractA consistent tangent stiffness matrix for the analysis of non‐linear contact problems is presented. The associated element has three or four nodes and establishes contact between three‐dimensional structures like solids and shells. It accounts for the non‐linear kinematics of large deformation analysis and guarantees a quadratic convergence rate. Two formulations, the penalty method and the Lagrange multiplier method, are investigated.

Consistent linearization in elasto‐plastic shell analysis

The present paper is directed towards elasto‐plastic large deformation analysis of thin shells based on the concept of degenerated solids. The main aspect of the paper is the derivation of an efficient computational strategy placing emphasis on consistent elasto‐plastic tangent moduli and stress integration with the radial return method under the restriction of ‘zero normal stress condition’ in thickness direction. The advantageous performance of the standard Newton iteration using a consistent tangent stiffness matrix is compared to the classical scheme with an iteration matrix based on the infinitesimal elasto‐plastic constitutive tensor. Several numerical examples also demonstrate the effectiveness of the standard Newton iteration with respect to modified and quasi‐Newton methods like BFGS and others.