Kenneth Runesson

Properties of Discontinuous Bifurcations Solutions in Elasto-Plasticity

Explicit expressions for the spectral properties of the bifurcation problem involving discontinuities for general elastic-plastic materials are presented. It then follows that the classical value of the critical hardening modulus derived by Rice (1976. Proc. 14th IUTAM Congr., Delft, The Netherlands, pp. 207–220, North Holland, Amsterdam.) is, in fact, the only possible one. Furthermore, from the spectral analysis it follows in a very straightforward fashion that bifurcation displaying elastic unloading on one side of the singular surface can never precede bifurcation with plastic loading on both sides of this surface. Explicit analytical results for the critical bifurcation directions and the corresponding hardening modulus are derived for non-associated volumetric flow rules while the devialoric portion is associated. The considered yield and potential functions may depend on all three stress invariants and may involve mixed isotropic and kinematic hardening. The result obtained by Rudnicki and Rice (1975, J. Mech. Phys. Solids 23, 371–394) for a Drucker-Pruger material appear as a special case. Other criteria that are investigated arc those of Coulomb and Rankine. (Less)

Discontinuous bifurcations of elastic-plastic solutions at plane stress and plane strain

Abstract Conditions for discontinuous bifurcations of the incremental fields in elastic-platic materials subjected to the condition of either plane stress or plane strain are derived and explicit expressions for the critical hardening modulus and the corresponding bifurcation directions are obtained for a quite general class of plasticity models. The only restriction is that the gradients of the yield function and plastic potential, that defines the nonassociated flow rule, have the same principal directions and that two of these directions are located in the plane of interest. Drucker-Pragers and Mohr-Coulombs yield criteria are taken as typical for the behavior of pressure-dependent materials such as concrete and granular materials. For the latter criterion, results for plane strain have previously been obtained only for the very particular case when the intermediate principal stress is directed out-of-plane. These results are confirmed in this paper as a part of the investigation of the complete behavior.

A thermodynamically consistent theory of gradient-regularized plasticity coupled to damage

A thermodynamically consistent theory of gradient-regularized plasticity with coupling to isotropic damage is presented. If the internal variables are gradient-enhanced, this leads to a nonconventional definition of the corresponding dissipative stresses (with a separate definition associated with external boundaries of the body). The main purpose is to use this theory for describing the successive development of the localization zone that is pertinent to the softening regime due to excessive damage close to failure. Localization characteristics of the linearized deformation field are assessed for the simple homogeneous tension bar as well as for the general stress state. In particular, it is predicted that the localization zone narrows with accumulation of damage. This prediction is indeed verified by a numerical solution in the case of uniaxial stress.

A note on nonassociated plastic flow rules

Abstract The analytical properties of the constitutive equations in plasticity with a nonassociated flow rule are investigated. Under the assumption of small deformations the directional stiffness (and compliance) rule is considered and the relevant spectral properties of the tangent stiffness tensor are assessed. It is shown that the directional stiffness may be larger than the elastic. It may also be negative in the case of a formally perfectly plastic material and so the nonassociative flow rule represents (spurious) softening in terms of an associated flow rule. The issue of uniqueness at finite strains is briefly addressed, whereby use is made of the tangent stiffness tensor relating the velocity gradient to the first Piola-Kirchhoff stress rate. The relevant spectral properties, which generalise those from the small deformation case, are found explicit. A sufficient condition for uniqueness is given in terms of a critical (upper bound) value of the hardening modulus.

Embedded localization band in undrained soil based on regularized strong discontinuity theory and FE-analysis

Abstract The paper presents a novel approach to the analysis of a developing localization zone in undrained soil considered as a mixture of a solid skeleton and fluid-filled pores, where the solid phase is considered as elastic-plastic. The basic feature is the concept of regularized displacement and pore pressure discontinuities, which are assumed to occur across an internal discontinuity surface. In this respect the present developments are extensions of those suggested by Simo et al. (1993), Comput. Mech. 12 , 277–296, and, more recently, by Larsson and Runesson (1995), J. Engng Mech. ASCE (in press). A finite element formulation is proposed on the basis of a mixed variational formulation in the spirit of the “enhanced strain” concept by Simo and Rifai (1990), Int. J. Numer. Meth. Engng 29 , 1595–1638. In this fashion, the localization zone is embedded into a “base” element for the ordinary analysis of the (non-localized) mixture problem.

Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws

Abstract A physically sound three-dimensional anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a finite set of the pertinent internal variables is presented. It is shown that the internal variables can be expressed in terms of the strain as convolution integrals with kernels of Mittag–Leffler function type. A time integration scheme, based on the Generalized Midpoint rule together with the Grunwald algorithm for numerical fractional differentiation, for integration of the constitutive response is developed. The predictive capability of the viscoelastic model for describing creep, relaxation and damped dynamic responses is investigated both analytically and numerically. The algorithm and the present general linear viscoelastic model are implemented into the general purpose finite element code Abaqus. The algorithm is then used together with an explicit difference scheme for integration of structural responses. In numerical examples, the quasi-static and damped responses of a viscoelastic ballast material that is subjected to loads simulating the overrolling of a train are investigated.

Integration in computational plasticity

Abstract Temporal integration of the constitutive relations is an important topic in computational plasticity. A brief review of recent developments is given in this paper. The Generalized Midpoint Rule and the Generalized Closest Point Projection Method are emphasized in particular. Algorithmic issues are discussed, especially with respect to the treatment of non-smooth yield surfaces. Numerical analyses support the conclusion that midpoint integration is preferred for small and moderate strain increments, while the Backward Euler rule (Closest Point Projection Method) is best suited when large strain increments can be anticipated, e.g. in zones of localized deformation and in large deformation plasticity.

On the localization properties of multiplicative hyperelasto-plastic continua with strong discontinuities

Abstract The objective of this work is to examine the large strain localization properties of hyperelasto-plastic materials which are based on the multiplicative decomposition of the deformation gradient. Thereby, the case of strong discontinuities is investigated. To this end, first an explicit expression for the spatial tangent operator is given, taking into account anisotropic as well as nonassociated material behaviour. Then the structure of a regularized discontinuous velocity gradient is elaborated and discussed in detail. Based on these two results, the localization condition is derived with special emphasis on the loading conditions inside and outside an anticipated localization band. Thereby, the intriguingly simple structure of the tangent operator, which resembles the structure of the geometrically linear theory, is extensively exploited. This similarity carries over to the general representation for the critical hardening modulus which is exemplified for isotropic materials. As a result, analytical solutions are available under the assumption of small elastic strains, which is justified for metals. Finally, examples are given for the special case of the associated von Mises flow rule. To this end, the critical localization direction and the critical hardening modulus are investigated with respect to the amount of finite elastic strain within different modes of homogeneous elasto-plastic deformations.

Finite element simulation of granular material flow in plane silos with complicated geometry

A model with few material parameters is proposed for finite element simulation of the transient flow of cohesionless granular materials in silos. The constitutive model used is essentially a fluid model and, consequently, it is best established in the Eulerian format. The model allows the simulation of initial stress transients at the beginning of the discharging process. After long time, when the solution approaches steady state, the flow pattern that is developed represents either a mass or funnel flow situation. Numerical examples of transient stress fields are given, and the flow patterns in advanced silo geometries are studied. The numerical results are in good agreement with analytical solutions and experiments known from the literature.

Modeling of polycrystals with gradient crystal plasticity: A comparison of strategies

This paper treats the computational modeling of size dependence in microstructure models of metals. Different gradient crystal plasticity strategies are analyzed and compared. For the numerical implementation, a dual-mixed finite element formulation which is suitable for parallelization is suggested. The paper ends with a representative numerical example for polycrystals.