Kaspar Willam

TRIAXIAL FAILURE CRITERION FOR CONCRETE AND ITS GENERALIZATION

The authors present a failure criterion that describes the triaxial strength of concrete in terms of three independent stress invariants. Its geometric representation in principal stress space is convex and smooth and is characterized by two parabolic meridians and a deviatoric section that changes from triangular to circular shape with increasing confinement. The three-parameter description is calibrated from elementary strength data of uniaxial compression and uniaxial tension, as well as equibiaxial compression experiments. The failure criterion is verified with different biaxial and triaxial strength data on plain concrete. Finally, the failure criterion is generalized to a format that includes the standard strength hypotheses of Huber-Mises, Drucker-Prager, Rankine, Mohr-Coulomb, and Leon as special cases.

Recent developments in the finite element analysis of prestressed concrete reactor vessels

Abstract This paper describes recent developments in the nonlinear deformation and ultimate load analysis of prestressed concrete reactor vessels using finite elements. First, a number of finite element models are called into attention for the idealization of composite structures such as reinforced and prestressed concrete components. Then different inelastic constitutive models are proposed for the behaviour of concrete in the pre- and post-failure regime. Subsequently various numerical techniques are examined for the solution of nonlinear problems, especially with regard to their distortion of the constitutive model. In conclusion these modelling techniques are applied to the analysis of four typical examples, the nonlinear deformation analysis of a concrete specimen subjected to biaxial compression, the crack analysis of a thick-walled concrete cylinder, the overload analysis of the THTR 1 : 5 scale model, and the ultimate load analysis of a concrete top closure model.

Computational aspects of welding stress analysis

Abstract The residual stresses and deformations due to arc-welding are determined in a rectangular steel plate. The thermomechanical response behaviour is computed in two steps (i), the heat flow analysis of the entire plate due to the moving electrode, and (ii) the thermoelastic-viscoplastic response analysis of the midsection for the transient temperature history at this section. The staggered solution strategy leads to a fully integrated approach of the thermal and mechanical solution steps within each time increment providing for weak thermomechanical coupling. The main difficulties arise due to the transient character of the heating-cooling cycle which spans the entire temperature regime from room temperature up to melting and vice versa. Clearly, highly localized material phase changes and associated stress redistributions complicate the time history analysis in addition to volume changes and latent heat effects during recristallization. The incremental solution of the thermoelastic-viscoplastic process is illustrated with the arc-welding of a rectangular steel plate for which limited experimental data are also available.

A unified theory of elastic degradation and damage based on a loading surface

A number of new models with stiffness degradation have been proposd in recent literature in the small strain regime. However, most of these works represent specific formulations, each using its own terminology, notation and assumptions, and relatively little effort has been spent so far towards achieving a common theoretical framework similar for instance to the theory of elastoplasticity. Moreover, most of the existing damage models are presented with intensive recourse to abstract thermodynamics concepts, and they combine stiffness degradation with plasticity, which (though being ultimately necessary to represent the actual material behavior) makes it much more difficult to isolate, analyse and understand the properties of the formulation for elastic stiffness degradation. As a contribution in this field, this paper presents a unifying theoretical framework to describe a class of models for elastic stiffness degradation based on the concept of loading surface. The derivation includes two consecutive steps: first, the constitutive framework for elastic-degrading models with evolution laws which are expressed directly in terms of the secant stiffness (or compliance) tensor, and second the elastic-damage models, in which the scant stiffness (or compliance) is assumed to depend on a reduced set of damage variables with clearer physical meaning and simpler evolution laws. Whenever possible, terminology is borrowed from the classical formulation of elastoplasticity, and thermodynamic concepts are introduced only as needed. Both stress-based and strain-based developments are compared throughout the paper, and the concept of associativity is reanalysed and generalized within the new unified framework of elastic degradation. The most significant scalar damage models found in the literature are reinterpreted in the context of this unified theory. Finally, a general expression is obtained for the tangential stiffness operator of associated scalar models (stress- and strain-based) of the (1 —D) type, that includes all the models considered as particular cases. More general damage formulations [scalar non-(1 — D), vectorial, tensorial] are reviewed and discussed systematically in a sequel paper.

Fundamental Issues of Smeared Crack Models

For numerical simulation of fracture in concrete and rock the “smeared crack approach” is receiving increasing attention. On one hand renewed attempts in terms of the fixed and rotating crack models resort to fracture mechanics in order to refine the traditional orthotropic crack formulation. Along this approach the original concept involving Mode I type cracking is being broadened to include mixed mode fracture interpretation of the shear retention factor, if the crack memory is fully retained. On the other hand, fracture energy-based plasticity models are advocated by the authors, as well as other investigators, which describe the degradation of strength due to tensile cracking and decohesion in shear in terms of isotropic and anisotropic strain-softening concepts.

Higher order methods for transient diffusion analysis

Abstract In this paper the finite element approach is examined for transient heat conduction and moisture migration problems. In particular, several finite element in time algorithms are developed based on Hermitian expansions of increasing order which are then compared with Pade approximations and alternative Norsett formulas. The individual time operators are investigated with regard to 1. a)accuracy of the incremental algorithm, 2. b)stability and implications of the integration of stiff systems, 3. c)computational implementation of higher order operators. After a thorough assessment of each family an incremental step-by-step algorithm is presented in which the solution of the linear expansion is corrected for higher order accuracy. To this end an iterative improvement technique is developed which does not destroy the sparse structure of the initial system matrices or increase their order. The theoretical exposition is concluded with numerical examples to compare the solution effort and the accuracy of the different operators.

Localized failure analysis in elastoplastic Cosserat continua

This paper summarizes our experience with localized failure analysis in micropolar Cosserat continua. The main intent is to examine the regularization properties of discontinuous bifurcation problems when higher grade micropolar materials that exhibit an internal length scale are introduced. Thereby, the key feature is the endowment of micropolar continua with rotational drilling degrees of freedom in addition to the traditional description of motion in terms of displacements. After a brief review of elastoplastic Cosserat continua, we examine recent findings of localization analysis of discontinuous bifurcation [1]. To this end we generalize the strength concept of the Mohr envelope condition to non-symmetric stress states to analyze the localization condition of discontinuous failure [2]. The Mohr representation of stress leads to a geometrical condition which is used to determine critical hardening/softening moduli and localization directions that characterize the particular mode of failure. This geometric interpretation of the localization condition lends itself to a systematic study of regularizing weak discontinuities within elastoplastic Cosserat continua. For illustration of these theoretical findings, numerical failure studies are carried out on a representative volume element made of J2-type elastoplastic Cosserat material. The computational failure simulations demonstrate the non-local character of Cosserat continua that depends on an internal length scale and on the degree of non-symmetry in the stress and strain states. The computational results also illustrate the conversion of the underlying failure mode from mixed mode to mode I type failure of decohesion and separation.

Spurious energy dissipation/generation in stiffness recovery models for elastic degradation and damage

Abstract A number of constitutive models have been proposed in recent years for elastic degradation and damage, many of which include procedures for the recovery of stiffness upon closure of tensile raicrocracks. Most of these recovery procedures are based on the decomposition of stress or strain into positive (tensile) and negative (compressive) components, which are incorporated in the elastic formulation taking recourse to fourth-order positive and negative projection operators. Due to the non-dissipative nature of microcrack closure-reopening for a certain fixed state of degradation, the recovery formulation should possess a well-defined energy potential along the line of hyperelasticity, which conserves energy upon closed-loop load histories. This condition seems to have escaped the rapidly expanding literature on damage mechanics, i.e. closure formulations have not been verified in this regards. In the paper, the (lack of) energy conservation is examined in terms of the spurious dissipation rate, which is developed for a relatively general class of recovery models. They include the positive-negative projection operators and the bimodular formulations with different stiffnesses for tension and compression. It is shown that under proportional loading in strain or stress, all these formulations are energy conservative. Under non-proportional loading, however, they are only conservative in conjunction with isotropic degradation, and they exhibit spurious dissipation-generation when anisotropic degradation is considered and the load history involves rotation of principal directions.

Micropolar elastoplasticity and its role in localization

Abstract In this article we focus on micropolar elastoplasticity and the conditions for the formation of spatial discontinuities. After a brief review of micropolar kinematics and balance equations, an augmented localization tensor is developed for elastoplastic constitutive behavior that describes discontinuous bifurcation at the constitutive level. In this context an auxiliary condition is encountered that restricts the jump of the stress rate to remain symmetric across discontinuity surfaces of second order. The expanded localization conditions are studied with two constitutive models — micropolar Rankine and micropolar von Mises J 2 - flow theories of elatoplasticity — for which definite statements are derived with regard to their regularization properties.

Localization within the Framework of Micropolar Elasto-Plasticity

Recently increasing attention is paid to continuum models which describe failure and post-peak behaviour in terms of localized deformations in failure bands of finite width. The classical Boltzmann description of local continua signals loss of ellipticity as soon as localization occurs causing strong mesh size dependence. To remedy this pathological behaviour an internal length scale was introduced by Bazant, Belytschko and Chang [1] via non-local concepts, or alternatively, by higher order gradient approaches involving an internal length scale, as proposed by Coleman and Hodgdon [2], Triantafyllidis and Aifantis [3]. In contrast, the micropolar theories, as advocated recently by Muhlhaus [4], Muhlhaus and Vardoulakis [5] and de Borst [6] possess an internal length scale to start with, which tends to regularize the impending loss of ellipticity near localization. For wave propagation problems Belytschko and Lasry [7] demonstrated that non-local and higher order gradient approaches exhibit a stable response to short wave length inputs and an unstable response to long wave lengths.