Modelling of strain softening materials based on equivalent damage force (original) (raw)
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ScienceDirect Modelling of strain softening materials based on equivalent damage force
The main aim of the work presented in this paper was treatment of damage and deformation localisation observed in the finite element method (FEM) analysis of strain softening materials combined with local constitutive models where damage is represented using continuum damage mechanics (CDM). The CDM/FEM approach typically suffers from a number of shortcomings, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). The approach proposed here is still based on the local constitutive model including damage, but introduces an alternative representation of damage effects in the system of linear momentum balance equations. The damage effects are included through equivalent damage force (EDF), which contributes to the right-hand side of the momentum balance equations. The main advantages of this approach are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening; numerical stability is preserved without a need for regularisation measures; and significantly reduced mesh dependency. In addition, the EDF approach can be used in combination with existing local CDM damage models and does not violate symmetry of the material stiffness tensor.
Theoretical and Computational Aspects of Non Local Damage Coupling with Elastic Behaviour
2013
Softening due to damage is the source of the strain localization in the materials as concrete. In such a situation, using the finite element analysis provides results that are directly depending to the spatio-temporal discretization. In this article, the adjustment of an isotropic elastic model coupled to non-local damage is presented in order to regularize the associated initial and boundary value problem. This formulation consists in delocalizing the damage variable “D”. Dispersion analysis and numerical simulations are used to compare the local and the non local models.
Stress-based nonlocal damage model
International Journal of Solids and Structures, 2011
Progressive microcracking in brittle or quasi-brittle materials, as described by damage models, presents a softening behavior that in turn requires the use of regularization methods in order to maintain objective results. Such regularization methods, which describe interactions between points, provide some general properties (including objectivity and the non-alteration of a uniform field) as well as drawbacks (damage initiation, free boundary). A modification of the nonlocal integral regularization method that takes the stress state into account is proposed in this contribution. The orientation and intensity of nonlocal interactions are modified in accordance with the stress state. The fundamental framework of the original nonlocal method has been retained, making it possible to maintain the method's advantages. The modification is introduced through the weight function, which in this modified version depends not only on the distance between two points (as for the original model) but also on the stress state at the remote point. The efficiency of this novel approach is illustrated using several examples. The proposed modification improves the numerical solution of problems observed in numerical simulations involving regularization techniques. Damage initiation and propagation in mode I as well as shear band formation are analyzed herein.
A nonlocal elastic damage theory: Mesh-insensitivity under strain softening
Computers & Structures, 1993
AIrstrati-A nonlocal damage theory was recently proposed by Valanis (J. Appl. Mech., ASME 58, 311-316, 1991). In the theory the basic conservation equations take a conventional local form and only the damage evolution law becomes nonlocal. In contrast to local theories, the resulting initial value problem of an elastic damaging solid retains its hyperbolicity even under strain softening. Thus, mesh sensitivity of the numerical solution observed in local damage theories is eliminated. The domain of damage localization, controlled by the heterogeneous microstructural interaction, converges to a Snite material region as the mesh size decreases to zero. This is in contrast to local models which predict an infinitely thin fracture domain of vanishing volume. Nonlocal fracture models based on the theory can be easily implemented into existing finite element wave codes by merely adding the constitutive subroutine for damage evolution. Convergence and mesh insensitivity are demonstrated by solving initial value problems using an updated Lagrangian finite element wave code.
Computer Methods in Applied Mechanics and Engineering
Integral Non-Local (INL) formulations are often used to regularize Continuum Damage computations, in the presence of stress softening for instance. The introduction of a characteristic/internal length allows for avoiding pathological mesh dependency. Some questions concerning the identification of the characteristic length, its possible evolution during damage process and the need for special treatments of non-locality operators near boundaries (e.g. edges, cracks) are however still open. A physical request is that material points separated by a crack (or an highly damaged zone) should not interact. Despite what is done in standard Integral Non-Local theories, this can be obtained by allowing non-local interactions to evolve depending on mechanical fields (e.g. damage, strain, stress). The Eikonal Non-Local (ENL) formulation provides a novel interpretation of damage dependent non-local interactions. Based on the Wentzel-Kramers-Brillouin (WKB) approximation for high-frequency wave propagation in a damaged medium, this formulation defines the interaction distances as the solution of a stationary damage dependent Eikonal equation. It allows for the modeling of non-local interactions which gradually vanish in damaged zones, thus ensuring a progressive transition from diffuse damage to fracture in a natural way. The numerical implementation and properties of this regularization technique are investigated and discussed. From a numerical viewpoint, a Fast Marching method is used to compute non-local interaction distances between Gauss integration points. Geodesic distances are then used to define the kernel of weighting function to be used in integral non-local averaging. Several numerical results of quasi-statics simulations of quasi-brittle fracture in isotropic media are presented.
Continuum damage modelling and some computational issues
Revue Française de Génie Civil, 2002
Continuum damage mechanics is a framework for describing the variations of the elastic properties of a material due to microstructural degradations. This paper presents the application of this theory to the modelling of concrete. Several constitutive relations are devised, including incremental, explicit, and non local damage models. A general framework for damage induced anisotropy is also presented. In the second part of this contribution, computational issues in damage mechanics related to iterative schemes and solution control in non linear computations are considered. The paper concludes with an example of 3D finite element computation of a reinforced concrete beam, as part of a benchmark initiated by Electricite de France.
Journal of the Serbian Society for Computational Mechanics, 2017
The main aim of this work is investigation of localization problem in strain softening materials and regularization techniques, which will reduce and possibly remove mesh dependency of the numerical results and balance the effects of heterogeneous microstructure on local continua while keeping the boundary value problem of softening (damaged) continua well-posed. Finite Element Method (FEM) and Smooth Particle Hydrodynamic (SPH) combined with a local continuum damage model (CDM) were used for analysis of a dynamic stress wave propagation problem, which was analytically solved in (Bažant and Belytschko 1985). The analytical solution was compared to the numerical results, obtained by using a stable, Total-Lagrange form of SPH (Vignjevic et al. 2006, Vignjevic et al. 2009), and two material models implemented in the FEM based on: 1) classic CDM; and 2) equivalent damage force. The numerical results demonstrate that the size of the damaged zone is controlled by element size in classic FEM and the smoothing length in the SPH, which suggests that the SPH method is inherently non-local method and that the smoothing length should be linked to the material characteristic length scale in solid mechanics simulations.
Non-local boundary integral formulation for softening damage
International Journal for Numerical Methods in Engineering, 2003
A strongly non-local boundary element method (BEM) for structures with strain-softening damage treated by an integral-type operator is developed. A plasticity model with yield limit degradation is implemented in a boundary element program using the initial-stress boundary element method with iterations in each load increment. Regularized integral representations and boundary integral equations are used to avoid the di culties associated with numerical computation of singular integrals. A numerical example is solved to verify the physical correctness and e ciency of the proposed formulation. The example consists of a softening strip perforated by a circular hole, subjected to tension. The strainsoftening damage is described by a plasticity model with a negative hardening parameter. The local formulation is shown to exhibit spurious sensitivity to cell mesh reÿnements, localization of softening damage into a band of single-cell width, and excessive dependence of energy dissipation on the cell size. By contrast, the results for the non-local theory are shown to be free of these physically incorrect features. Compared to the classical non-local ÿnite element approach, an additional advantage is that the internal cells need to be introduced only within the small zone (or band) in which the strain-softening damage tends to localize within the structure. Copyright ? 2003 John Wiley & Sons, Ltd.
A stress-return algorithm for nonlocal constitutive models of softening materials
International Journal for Numerical Methods in Engineering, 2009
We develop a fully implicit stress-return algorithm for a wide range of nonlocal inelastic models (of integral type) with coupling among integration points. Two examples of nonlocal models are presented, which combine plasticity with either damage or breakage. Using these models, we emphasize the need for a versatile integration scheme for nonlocal incremental constitutive equations. Furthermore, these models are used to demonstrate the importance of adequately selecting the way constitutive equations for softening materials are regularized. The stress update process requires the construction of a nonlocal constitutive matrix: we highlight and take advantage of its sparsity. Numerical solutions to relevant structural and geomechanical boundary value problems are used to demonstrate the performance of the proposed approach.