Ket qualitative features of solutions exhibiting strong discontinuities in rate-independent inelastic solids are identified and exploited in the design of a new class of finite element approximations. The analysis shows that the softening law must be re-interpreted in a distributional sense for the continuum solutions to make mathematical sense and provides a precise physical interpretation to the softening modulus. These results are verified by numerical simulations employing a regularized discontinuous finite element method which circumvent the strong mesh-dependence exhibited by conventional methods, without resorting to viscosity or introducing additional ad-hoc parameters. The analysis is extended to a new class of anisotropic rate-independent damage models for brittle materials.

An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids

A formulation and algorithmic treatment of static and dynamic plasticity at finite strains based on the multiplicative decomposition is presented which inherits all the features of the classical models of infinitesimal plasticity. The key computational implication is this: the closest-point-projection algorithm of any classical simple-surface or multi-surface model of infinitesimal plasticity carries over to the present finite deformation context without modification. In particular, the algorithmic elastoplastic tangent moduli of the infinitesimal theory remain unchanged. For the static problem, the proposed class of algorithms preserve exactly plastic volume changes if the yield criterion is pressure insensitive. For the dynamic problem, a class of time-stepping algorithms is presented which inherits exactly the conservation laws of total linear and angular momentum. The actual performance of the methodology is illustrated in a number of representative large scale static and dynamic simulations.

Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory

This paper presents a complete formulation of a model of coupled associative thermoplasticity at finite strains, addresses in detail the numerical analysis aspects involved in its finite element implementation, and assesses the performance of the proposed mechanical and finite element models in a comprehensive set of numerical simulations. On the thermomechanical side, novel aspects of the proposed model of thermoplasticity are (1) the explicit characterization of the plastic (configurational) entropy as an independent internal variable, (2) a thermomechanical extension of the principle of maximum dissipation consistent with the multiplicative decomposition of the deformation gradient, and (3) the exploitation of this extended principle in the formulation of an associative flow which characterizes the evolution of the plastic entropy in terms of the change of the flow criterion with respect to temperature. On the numerical analysis side, salient features of the proposed approach are (4) a new global product formula algorithm constructed via an operator split of the nonlinear initial value problem, which leads to a two-step solution procedure, (5) a unified class of local return mapping algorithms which preserves exactly the incompressibility constraint on the plastic flow and reduces to the classical radial return method for isothermal J 2 - flow theory, and (6) the formulation of a mixed finite element method in terms of the elastic entropy and the temperature field which circumvents well-known difficulties associated with the incompressibility constraint on the plastic flow. The exact linearization of both the product formula algorithm and an alternative simulataneous solution scheme for the coupled thermomechanical problem is given in two appendices.

Associated coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation

Rate-independent plasticity and viscoplasticity in which the boundary of the elastic domain is defined by an arbitrary number of yield surfaces intersecting in a non-smooth fashion are considered in detail. It is shown that the standard Kuhn-Tucker optimality conditions lead to the only computationally useful characterization of plastic loading. On the computational side, an unconditionally convergent return mapping algorithm is developed which places no restrictions (aside from convexity) on the functional forms of the yield condition, flow rule and hardening law. The proposed general purpose procedure is amenable to exact linearization leading to a closed-form expression of the so-called consistent (algorithmic) tangent moduli. For viscoplasticity, a closed-form algorithm is developed based on the rate-independent solution. The methodology is applied to structural elements in which the elastic domain possesses a non-smooth boundary. Numerical simulations are presented that illustrate the excellent performance of the algorithm.

Non‐smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms

Numerical analysis and simulation of plasticity

On plasticity with hardening

We prove an error estimate for an incremental finite element method for obtaining approximations to the stresses in an elastic-perfectly plastic body. We also comment on the limit load problem.

Claes Johnson

Papers

On plasticity with hardening

On finite element methods for plasticity problems