Fundamental issues in finite element analyses of localization of deformation
TL;DR: In this article, three different approaches are scrutinized which may be used to remedy these two intimately related deficiencies of the classical theory, namely (i) the addition of higher-order deformation gradients, (ii) the use of micropolar continuum models, and (iii) the adding of rate dependence.
Abstract: Classical continuum models, i.e. continuum models that do not incorporate an internal length scale, suffer from excessive mesh dependence when strain‐softening models are used in numerical analyses and cannot reproduce the size effect commonly observed in quasi‐brittle failure. In this contribution three different approaches will be scrutinized which may be used to remedy these two intimately related deficiencies of the classical theory, namely (i) the addition of higher‐order deformation gradients, (ii) the use of micropolar continuum models, and (iii) the addition of rate dependence. By means of a number of numerical simulations it will be investigated under which conditions these enriched continuum theories permit localization of deformation without losing ellipticity for static problems and hyperbolicity for dynamic problems. For the latter class of problems the crucial role of dispersion in wave propagation in strain‐softening media will also be highlighted.
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TL;DR: In this paper, a new plastic-damage model for concrete subjected to cyclic loading is developed using the concepts of fracture-energy-based damage and stiffness degradation in continuum damage mechanics.
Abstract: A new plastic-damage model for concrete subjected to cyclic loading is developed using the concepts of fracture-energy-based damage and stiffness degradation in continuum damage mechanics. Two damage variables, one for tensile damage and the other for compressive damage, and a yield function with multiple-hardening variables are introduced to account for different damage states. The uniaxial strength functions are factored into two parts, corresponding to the effective stress and the degradation of elastic stiffness. The constitutive relations for elastoplastic responses are decoupled from the degradation damage response, which provides advantages in the numerical implementation. In the present model, the strength function for the effective stress is used to control the evolution of the yield surface, so that calibration with experimental results is convenient. A simple and thermodynamically consistent scalar degradation model is introduced to simulate the effect of damage on elastic stiffness and its recovery during crack opening and closing. The performance of the plastic-damage model is demonstrated with several numerical examples of simulating monotonically and cyclically loaded concrete specimens.
2,825 citations
TL;DR: In this article, a consistent numerical solution procedure of the governing partial differential equations is presented, which is shown to be capable of properly simulating localization phenomena, and the introduction of higher-order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models.
Abstract: SUMMARY Conventional continuum damage descriptions of material degeneration suffer from loss of well-posedness beyond a certain level of accumulated damage. As a consequence, numerical solutions are obtained which are unacceptable from a physical point of view. The introduction of higher-order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models. A consistent numerical solution procedure of the governing partial differential equations is presented, which is shown to be capable of properly simulating localization phenomena.
1,207 citations
TL;DR: In this paper, a model which allows the introduction of displacements jumps to conventional finite elements is developed, where the path of the discontinuity is completely independent of the mesh structure.
Abstract: A model which allows the introduction of displacements jumps to conventional finite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so-called ‘embedded discontinuity’ models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid finite element that can be used and displacement jumps are continuous across element boundaries. Using finite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi-brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes. Copyright © 2001 John Wiley & Sons, Ltd.
914 citations
TL;DR: In this paper, a unifying thermomechanical framework is presented that reconciles several classes of gradient elastoviscoplasticity and damage models proposed in the literature during the last 40 years.
Abstract: A unifying thermomechanical framework is presented that reconciles several classes of gradient elastoviscoplasticity and damage models proposed in the literature during the last 40 years . It is based on the introduction of the micromorphic counterpart ϕχ of a selected state or internal variable ϕ in a standard constitutive model. In addition to the classical balance of momentum equation, a balance of micromorphic momentum is derived that involves generalized stress tensors. The corresponding additional boundary conditions are also deduced from the procedure. The power of generalized forces is assumed to contribute to the energy balance equation. The free energy density function is then chosen to depend on a relative generalized strain, typically ϕ- ϕχ , and the microstrain gradient ∇ ϕχ . When applied to the deformation gradient itself, ϕ≡ F , the method yields the micromorphic theory of Eringen and Mindlin together with its extension to finite deformation elastoviscoplasticity by Forest and Sievert. If...
504 citations
TL;DR: An overview of the results obtained with lattice models of the fracture, highlighting the relations with statistical physics theories and more conventional fracture mechanics approaches is presented.
Abstract: Disorder and long-range interactions are two of the key components that make material failure an interesting playfield for the application of statistical mechanics. The cornerstone in this respect has been lattice models of the fracture in which a network of elastic beams, bonds, or electrical fuses with random failure thresholds are subject to an increasing external load. These models describe on a qualitative level the failure processes of real, brittle, or quasi-brittle materials. This has been particularly important in solving the classical engineering problems of material strength: the size dependence of maximum stress and its sample-to-sample statistical fluctuations. At the same time, lattice models pose many new fundamental questions in statistical physics, such as the relation between fracture and phase transitions. Experimental results point out to the existence of an intriguing crackling noise in the acoustic emission and of self-affine fractals in the crack surface morphology. Recent advances ...
464 citations
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TL;DR: In this paper, the authors investigated the hypothesis that localization of deformation into a shear band may be considered a result of an instability in the constitutive description of homogeneous deformation.
Abstract: T his paper investigates the hypothesis that localization of deformation into a shear band may be considered a result of an instability in the constitutive description of homogeneous deformation. General conditions for a bifurcation, corresponding to the localization of deformation into a planar band, are derived. Although the analysis is general and applications to other localization phenomena are noted, the constitutive relations which are examined in application of the criterion for localization are intended to model the behavior of brittle rock masses under compressive principal stresses. These relations are strongly pressure-sensitive since inelasticity results from frictional sliding on an array of fissures; the resulting inelastic response is dilatant, owing to uplift in sliding at asperities and to local tensile cracking from fissure tips. The appropriate constitutive descriptions involve non-normality of plastic strain increments to the yield hyper-surface. Also, it is argued that the subsequent yield surfaces will develop a vertex-like structure. Both of these features are shown to be destabilizing and to strongly influence the resulting predictions for localization by comparison to predictions based on classical plasticity idealizations, involving normality and smooth yield surfaces. These results seem widely applicable to discussions of the inception of rupture as a constitutive instability.
2,411 citations
01 Jan 1909
TL;DR: Cosserat and Hermann as mentioned in this paper discussed the kinematical and dynamical theories of the flexible line, the flexible surface, and the deformable three-dimensional medium in great detail.
Abstract: THE authors, who are well known by their writings on general elastic theory, here reprint in separate form an appendix contributed by them to M. Chwolson's “Traité de Physique.” The kinematical and dynamical theories of the flexible line, the flexible surface, and the deformable three-dimensional medium are discussed in turn in great detail. The dynamical standpoint adopted is that of the principle of action, which forms, in the authors' opinion, the only satisfactory basis for the “deductive” exposition of the subject. In each, case the most general form of the function representing the “action” is sought which is consistent with the necessary invariantive relations. This procedure is, of course, not altogether new, and an expert, turning over the pages, will recognise much that in one form or another is familiar to him. The treatment is necessarily somewhat abstract, and is i mathematically very elaborate, Cartesian, methods being followed throughout. To many readers the long train of general investigations, unrelieved by a single application, may prove deterrent; but the authors at all events claim that their procedure has never before been carried out so resolutely and completely, and may justly pride themselves on the mathematical elegance of their work. Apart from its other qualities, the treatise has a distinct value as a book of reference, and furnishes a whole arsenal of formulæ which may save trouble to future writers.Théorie des Corps déformables.By E. Cosserat F. Cosserat. Pp. vi + 226. (Paris: A. Hermann et Fils, 1909.) Price 6 francs.
1,783 citations
TL;DR: In this paper, the authors proposed a nonlocal damage theory, which is based on the nonlocal treatment of damage from the local treatment of elastic behavior, and the only required modification is to replace the usual local damage energy release rate with its spatial average over the representative volume of the material whose size is a characteristic of a material.
Abstract: In the usual local finite element analysis, strain softening causes spurious mesh sensitivity and incorrect convergence when the element is refined to vanishing size. In a previous continuum formulation, these incorrect features were overcome by the imbricate nonlocal continuum, which, however, introduced some unnecessary computational complications due to the fact that all response was treated as nonlocal. The key idea of the present nonlocal damage theory is to subject to nonlocal treatment only those variables that control strain softening, and to treat the elastic part of the strain as local. The continuum damage mechanics formulation, convenient for separating the nonlocal treatment of damage from the local treatment of elastic behavior, is adopted in the present work. The only required modification is to replace the usual local damage energy release rate with its spatial average over the representative volume of the material whose size is a characteristic of the material. Avoidance of spurious mesh ...
1,672 citations
TL;DR: A sufficient condition for uniqueness of the boundary-value problem set by given velocities on a part of the surface of a body and given nominal traction-rates on the remainder is established in this paper.
Abstract: A sufficient condition is established for uniqueness of the boundary-value problem set by given velocities on a part of the surface of a body and given nominal traction-rates on the remainder. No restriction is placed on changes in geometry either in the boundary-value problem itself or in the postulated material properties, which are a generalization of those conventionally assumed for workhardening elastic-plastic solids. The solution, when it is unique, is characterized by an extremum principle. Stability under dead loading is also examined, and a criterion is proposed. This has a formal resemblance to the uniqueness criterion, but differs from it in a significant respect when the body is partly plastic, so that stability and uniqueness need not be parallel properties. Finally, the present, theorems are compared with those previously proved for rigid-plastic solids ( Hill 1957a, b. d).
1,432 citations
1,286 citations