A finite element displacement formulation for gradient elastoplasticity (original) (raw)
Related papers
A Finite Element approach with patch projection for strain gradient plasticity formulations
Several strain gradient plasticity formulations have been suggested in the literature to account for inherent size effects on length scales of microns and submicrons. The necessity of strain gradient related terms render the simulation with strain gradient plasticity formulation computationally very expensive because quadratic shape functions or mixed approaches in displacements and strains are usually applied. Approaches using linear shape functions have also been suggested which are, however, limited to regular meshes with equidistanced Finite Element nodes. As a result the majority of the simulations in the literature deal with plane problems at small strains. For the solution of general three dimensional problems at large strains an approach has to be found which has to be computationally affordable and robust.
Finite elements for materials with strain gradient effects
International Journal for Numerical Methods in Engineering, 1999
A finite element implementation is reported of the Fleck-Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin-Mindlin framework and deals with first-order strain gradients and the associated work-conjugate higher-order stresses. In conventional displacement-based approaches, the interpolation of displacement requires C-continuity in order to ensure convergence of the finite element procedure for higher-order theories. Mixed-type finite elements are developed herein for the Fleck-Hutchinson theory; these elements use standard C-continuous shape functions and can achieve the same convergence as C elements. These C elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated.
The current trend towards miniaturization in the microelectronics industry has pushed for the development of theories intended to explain the behavior of materials at small scales. In the particular case of metals, a class of available non-classical continuum mechanics theories has been recently employed in order to explain the wide range of observed behavior at the micron scale. The practical use of the proposed theories remains limited due to issues in its numerical implementation. First, in displacement-based finite element formulations the need appears for higher orders of continuity in the interpolation shape functions in order to maintain the convergence rate upon mesh refinement. This limitation places strong restrictions in the geometries of the available elements. Second, the available inelastic constitutive models for small scale applications have been cast into deformation theory formulations limiting the set of problems to those exhibiting proportional loading only. In this article two contributions are made for the particular case of a Cosserat couple stress continuum. First it describes a numerical scheme based on a penalty function/reduced integration approach that allows for the proper treatment of the higher order terms present in Cosserat like theories. This scheme results in 1 PhD in computational mechanics, jgomezc1@eafit.edu.co, associate professor, applied mechanics group, EAFIT University, Medellín-Colombia.
A gradient model for finite strain elastoplasticity coupled with damage
Finite Elements in Analysis and Design, 2003
This paper describes the formulation of an implicit gradient damage model for ÿnite strain elastoplasticity problems including strain softening. The strain softening behavior is modeled through a variant of Lemaitre's damage evolution law. The resulting constitutive equations are intimately coupled with the ÿnite element formulation, in contrast with standard local material models. A 3D ÿnite element including enhanced strains is used with this material model and coupling peculiarities are fully described. The proposed formulation results in an element which possesses spatial position variables, nonlocal damage variables and also enhanced strain variables. Emphasis is put on the exact consistent linearization of the arising discretized equations.
European Journal of Mechanics - A/Solids, 2019
In this work, a higher-order irrotational strain gradient plasticity theory is studied in the small strain regime. A detailed numerical study is based on the problem of simple shear of a non-homogeneous block comprising an elastic-plastic material with a stiff elastic inclusion. Combinations of micro-hard and micro-free boundary conditions are used. The strengthening and hardening behaviour is explored in relation to the dissipative and energetic length scales. There is a strong dependence on length scale with the imposition of micro-hard boundary conditions. For micro-free conditions there is marked dependence on dissipative length scale of initial yield, though the differences are small in the post-yield regime. In the case of hardening behaviour, the variation with respect to energetic length scale is negligible. A further phenomenon studied numerically relates to the global nature of the yield function for the dissipative problem; this function is given as the least upper bound of a function of plastic strain increment, and cannot be determined analytically. The accuracy of an upper-bound approximation to the yield function is explored, and found to be reasonably sharp in its prediction of initial yield.
A strain space gradient plasticity theory for finite strain
Computer Methods in Applied Mechanics and Engineering, 2004
In this paper, an extension to the finite deformation regime of the infinitesimal theories of strain gradient plasticity discussed in a paper by R. Chambon, D. Caillerie and T. Matsushima [Int. J. Solids Struct. 38 (2001) 8503-8527] is presented which extends and generalizes the previous works of R. Chambon, D. Caillerie and C. Tamagnini [C.R. Acad. Sci. 329 (S erie IIb) (2001) 797-802] and C. Tamagnini, R. Chambon and D. Caillerie [C.R. Acad. Sci. 329 (S erie IIb) (2001) [735][736][737][738][739]. Central to the proposed theory are the kinematic assumptions concerning the decomposition of the assumed measures of strain and hyperstrain into an elastic and a plastic part. Following modern treatments of finite deformation plasticity, a multiplicative decomposition of the deformation gradient is postulated, while an additive decomposition is adopted for the second deformation gradient, as in the paper by R. Chambon, D. Caillerie and C. [C.R. Acad. Sci. 329 (S erie IIb) (2001) 797-802]. The elastic constitutive equations for Kirchhoff stress and double stress tensors are obtained by assuming the existence of a suitable free energy function in the spatial description. The requirements of (i) invariance for rigid body motions superimposed upon the intermediate configuration; and, (ii) spatial covariance, see the book by J.E. Marsden and T.J.R. Hughes [Mathematical Foundations of Elasticity, Dover Publications Inc. , New York, 1994], provide the corresponding material version of the hyperelastic constitutive equations. A fully covariant formulation of the evolution equations for the plastic strain and hyperstrain tensors, as well as for the internal variables is obtained by a straightforward application of the principle of maximum dissipation, after introducing a suitable yield condition in strain space. The resulting strain-space theory of second gradient plasticity can be considered an extension of the finite deformation plasticity theory proposed by J.C. Simo [Comput. Methods Appl. Mech. Engrg. 66 (1988) 199-219] to elastoplastic media with microstructure. As an example of application, a single-mechanism, isotropic hardening second gradient model for cohesive-frictional materials is proposed in which the cohesive component of the shear strength is assumed to increase with the magnitude of elastic hyperstrains, as advocated by, e.g. N.metals, based on both experimental observations and micromechanical considerations. Due to the internal length scales provided by the microstructure, the model is ideally suited for the analysis of failure problems in which strain localization into shear band occurs.
Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour
International Journal of Plasticity, 2003
An improved gradient-enhanced approach for softening elasto-plasticity is proposed, which in essence is fully nonlocal, i.e. an equivalent integral nonlocal format exists. The method utilises a nonlocal field variable in its constitutive framework, but in contrast to the integral models computes this nonlocal field with a gradient formulation. This formulation is considered 'implicit' in the sense that it strictly incorporates the higher-order gradients of the local field variable indirectly, unlike the common (explicit) gradient approaches. Furthermore, this implicit gradient formulation constitutes an additional partial differential equation (PDE) of the Helmholtz type, which is solved in a coupled fashion with the standard equilibrium condition. Such an approach is particularly advantageous since it combines the long-range interactions of an integral (nonlocal) model with the computational efficiency of a gradient formulation. Although these implicit gradient approaches have been successfully applied within damage mechanics, e.g. for quasi-brittle materials, the first attempts were deficient for plasticity. On the basis of a thorough comparison of the gradient-enhancements for plasticity and damage this paper rephrases the problem, which leads to a formulation that overcomes most reported problems. The two-dimensional finite element implementation for geometrically linear plain strain problems is presented. One-and two-dimensional numerical examples demonstrate the ability of this method to numerically model irreversible deformations, accompanied by the intense localisation of deformation and softening up to complete failure.
A reformulation of strain gradient plasticity
Journal of the Mechanics and Physics of Solids, 2001
A class of phenomenological strain gradient plasticity theories is formulated to accommodate more than one material length parameter. The objective is a generalization of the classical J2 ow theory of plasticity to account for strain gradient e ects that emerge in deformation phenomena at the micron scale. A special case involves a single length parameter and is of similar form to that proposed by Aifantis and co-workers. Distinct computational advantages are associated with this class of theories that make them attractive for applications requiring the generation of numerical solutions. The higher-order nature of the theories is emphasized, involving both higher-order stresses and additional boundary conditions. Competing members in the class of theories will be examined in light of experimental data on wire torsion, sheet bending, indentation and other micron scale plasticity phenomena. The data strongly suggest that at least two distinct material length parameters must be introduced in any phenomenological gradient plasticity theory, one parameter characterizing problems for which stretch gradients are dominant and the other relevant to problems when rotation gradients (or shearing gradients) are controlling. Flow and deformation theory versions of the theory are highlighted that can accommodate multiple length parameters. Examination of several basic problems reveals that the new formulations predict quantitatively similar plastic behavior to the theory proposed earlier by the present authors. The new formulations improve on the earlier theory in the manner in which elastic and plastic strains are decomposed and in the representation of behavior in the elastic range.
Computer Methods in Applied Mechanics and Engineering, 1984
Finite element models for elasto-plastic incremental analysis are derived from a three-field variational principle. The Newton-Raphson method is applied to solve the nonlinear system of equations which is obtained from the stationarity condition of this principle. The iterative schemes are discussed in detail for pure displacement and for pure equilibrium models from which iterative schemes for hybrid models folfow directly. In the displacement model, the compatibility of the strains and the plasticity criterium are satisfied during the whole iterative process, while the equilibrium of the stresses is restored only in the mean after convergence. In the equilibrium model, the plasticity criterium and the compatibility of the strains are verified in the mean during the iterative process; when convergence is achieved, the stresses are locally in equilibrium with the applied external loads. In both cases, a tangential stiffness matrix can be constructed, even for perfectly plastic materials and it allowsone to obtain always very good convergence properties. Examples are shown for plane stress and axisymmetric cases.