Spurious Mesh Research Articles

Strain gradient plasticity modeling and finite element simulation of Lüders band formation and propagation

An analytical solution of the problem of the propagation of a Luders band in an isotropic strain gradient plasticity medium is provided based on a softening–hardening constitutive law. A detailed description is given of the plastic strain distribution in the finite size band front. The solution is shown to be harmonic in the band front and exponential in the band tail. Particular attention is paid to the conditions to be applied at the interface between both regions. This solution is then used to validate finite element simulations of the Luders band formation and propagation in a plate in tension. The approach is shown to suppress the spurious mesh dependence exhibited by conventional finite element simulations of the Luders behavior and to provide a finite width band front in agreement with the experimental observations from strain field measurements.

Numerical method to control high levels of damage growth using an implicit finite element solver applied to notched cross-ply laminates

Numerical modelling of composites is becoming ever more important to commercial aircraft manufacturers for design-phase cost reductions and improved lead times. The implementation of a user-defined numerical model for damage prediction in composites is described. All modelling is conducted in ABAQUS/Standard (an implicit solver) and the models demonstrate stable performance with significant levels of damage. This stability is often difficult to achieve when modelling damage with implicit solvers. The favourable performance is due to the damage variable evaluation, which uses information from the previous iteration of an increment to drive damage development during a given load step. This “pseudo-current” damage evaluation is complex to implement, however, the stable performance and the improved time-step size dependence justifies this approach. The method is applied to analyse cross-ply open-hole tension specimens having similar layups but demonstrating different damage growth and varying failure loads. The analysis includes in-plane damage, delamination and splitting mechanisms and a nonlocal regularisation scheme to mitigate spurious mesh sensitivity. The delamination and splitting mechanisms are essential for differentiating between the various specimen types. The damage evolution method provides stability to the implicit solver allowing significant levels of damage to develop within the laminates, giving a valuable prediction of laminate behaviour.

Assessment and comparison of non-local integral models for ductile damage

Over the past years, the non-local method has established itself as an effective remedy to the well-known pathological mesh dependency that inherently affects softening media. The non-local method incorporates an intrinsic length into the traditional continuum theory and therefore the size of the localising zone is resolved, attenuating the unwanted effects of spurious mesh dependency. However, despite many contributions that have successfully employed the non-local theory, it is still not clear how exactly should non-locality be formulated in the general sense. Aiming to answer the question of which non-local formulations effectively lead to mesh-insensitive results, we select in this article several constitutive variables to be non-local quantities by taking both Lemaitre and Gurson–Tvergaard–Needleman models as the base for the non-local enhancement. The resulting non-local constitutive models are employed in the numerical simulation of various specimens which are subjected to different values of stress triaxiality and third invariant at the fracture zone. The goal is to find which models present the best performance in the task of providing mesh-insensitive solutions for different stress states. The results show that strain-softening mesh dependency is stronger in plane strain than in the axisymmetric case. It is also found that the variables that regularise the solution in the axisymmetric case do not necessarily eliminate mesh sensitivity in the other cases. Furthermore, the results indicate that damage should be the preferred non-local variable in the case of implicit damage models. This result is in sharp contrast with the case of explicit damage models, for which it has already been shown in the literature that damage is a bad candidate for non-local variable.

A structural modelling framework for prediction of damage development and failure of composite laminates

This article presents the latest developments of a constitutive modelling framework, CODAM (COmposite DAmage Model), for predicting the non-linear in-plane response of composite laminates using continuum damage mechanics. The methodology is best suited for non-linear structural analysis of large-scale laminated composites whose boundaries do not interfere/interact with the damage zone that develops and grows within the structure. The new development presented here, CODAM2, addresses the deficiencies in both the numerical and material objectivity of the original version of CODAM. While the previous CODAM formulation was essentially a local smeared crack model that was augmented with crack band scaling to overcome one aspect of the numerical objectivity, namely the mesh-sensitivity, CODAM2 introduces a non-local regularisation scheme to alleviate both the spurious mesh dependency and mesh orientation problems that plague all local strain-softening models. Two of the 13 test cases, provided in the third-world wide failure exercise, which were related to the in-plane tensile and compressive loading of open hole specimens, were used in order to demonstrate the effectiveness of CODAM2 in predicting the damage development and the corresponding overall response in such structural loading configurations.

Identification of a strain-aging model accounting for Lüders behavior in a C-Mn steel

Strain-aging constitutive models are suitable to simulate the formation and propagation of Lüders bands in complex specimens and components. The identification of the corresponding material parameters is difficult because the strain localization phenomena associated with the Lüders behavior must be taken into account. The spurious mesh dependence of standard finite element simulations of Lüders band propagation is illustrated in the present work and removed using a strain gradient plasticity model. Furthermore the gradient approach introduces a characteristic size corresponding to the finite dimension of the Lüders band front as observed from strain field measurements. The parameters are identified from the experimental measurement of the peak stress and plastic strain carried by the band for a C-Mn steel over a temperature range from −150°C to room temperature and for several strain rates. The validity of the model is tested on 3D simulations of the Lüders band propagation in a strain gradient plasticity medium.

Three‐dimensional dynamic rupture simulation with a high‐order discontinuous Galerkin method on unstructured tetrahedral meshes

Accurate and efficient numerical methods to simulate dynamic earthquake rupture and wave propagation in complex media and complex fault geometries are needed to address fundamental questions in earthquake dynamics, to integrate seismic and geodetic data into emerging approaches for dynamic source inversion, and to generate realistic physics‐based earthquake scenarios for hazard assessment. Modeling of spontaneous earthquake rupture and seismic wave propagation by a high‐order discontinuous Galerkin (DG) method combined with an arbitrarily high‐order derivatives (ADER) time integration method was introduced in two dimensions by de la Puente et al. (2009). The ADER‐DG method enables high accuracy in space and time and discretization by unstructured meshes. Here we extend this method to three‐dimensional dynamic rupture problems. The high geometrical flexibility provided by the usage of tetrahedral elements and the lack of spurious mesh reflections in the ADER‐DG method allows the refinement of the mesh close to the fault to model the rupture dynamics adequately while concentrating computational resources only where needed. Moreover, ADER‐DG does not generate spurious high‐frequency perturbations on the fault and hence does not require artificial Kelvin‐Voigt damping. We verify our three‐dimensional implementation by comparing results of the SCEC TPV3 test problem with two well‐established numerical methods, finite differences, and spectral boundary integral. Furthermore, a convergence study is presented to demonstrate the systematic consistency of the method. To illustrate the capabilities of the high‐order accurate ADER‐DG scheme on unstructured meshes, we simulate an earthquake scenario, inspired by the 1992 Landers earthquake, that includes curved faults, fault branches, and surface topography.

Microplane Model for Fracturing Damage of Triaxially Braided Fiber-Polymer Composites

AbstractA material model for the fracturing behavior for braided composites is developed and implemented in a material subroutine for use in the commercial explicit finite element code ABAQUS. The subroutine is based on the microplane model in which the constitutive behavior is defined not in terms of stress and strain tensors and their invariants but in terms of stress and strain vectors in the material mesostructure called the “microplanes.” This is a semi-multiscale model, which captures the interactions between inelastic phenomena such as cracking, splitting, and frictional slipping occurring on planes of various orientations though not the interactions at a distance. To avoid spurious mesh sensitivity due to softening, the crack band model is adopted. Its band width, related to the material characteristic length, serves as the localization limiter. It is shown that the model can realistically predict the orthotropic elastic constants and the strength limits. More importantly, the present model can also fit the tests of size effect on the strength of notched specimens and the post-peak behavior, which have been conducted for this purpose. When used in the ABAQUS software, the model gives a realistic picture of the axial crushing of a braided tube by a divergent plug.

An isogeometric analysis approach to gradient damage models

AbstractContinuum damage formulations are commonly used for the simulation of diffuse fracture processes. Implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models. The C0‐continuity of traditional finite elements has hindered the study of higher order gradient damage approximations. In this contribution we use isogeometric finite elements, which allow for the construction of higher order continuous basis functions on complex domains. We study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations. Copyright © 2011 John Wiley & Sons, Ltd.

A global arbitrary Lagrangian–Eulerian method for stratified Richtmyer–Meshkov instability

Richtmyer–Meshkov (RM) instability arises when a material interface is accelerated impulsively by shock waves. In this work, an arbitrary Lagrangian–Eulerian method, global ALE method, was proposed for the simulation of stratified RM instability. In the global ALE method, an Eulerian diffusion interface model was implemented based on mass fraction function. Thus all the meshes can be remeshed arbitrarily no matter whether they are material interface or not. Some benchmark problems, such as shock tube problem with different specific ratio, RM instability with small initial perturbation, were computed with the global ALE method, and the numerical results agree well with exact solution or theoretical model. Also, we proposed some stratified RM instability model problems with two or more material interfaces in planar, cylindrical and spherical geometries. Then the stratified RM instabilities were simulated with global ALE method. The interface evolution process was studied and compared in different geometry cases based on simulation results. To overcome the spurious mesh distortion, a sub-zonal Riemann solver method was proposed in appendix part of the paper based on the analysis of the error source of 2D Lagrangian computation due to non-uniform multi-dimensional mesh.

Delay-active damage versus non-local enhancement for anisotropic damage dynamics computations with alternated loading

Abstract Anisotropic damage thermodynamics framework allows to model the concrete-like materials behavior and in particular their dissymmetric tension/compression response. To deal with dynamics applications such as impact, it is furthermore necessary to take into account the strain rate effect observed experimentally. This is done in the present work by means of anisotropic visco-damage, by introducing a material strain rate effect in the cases of positive hydrostatic stresses only. The proposed delay-damage law assumes no viscous effect in compression as the consideration of inertia effects proves sufficient to model the apparent material strength increase. High-rate dynamics applications imply to deal with wave propagation and reflection which can generate alternated loading in the impacted structure. In order to do so, the key concept of active damage is defined and introduced within both the damage criterion and the delay-damage evolution law. At the structural level, strain localization often leads to spurious mesh dependency. Three-dimensional Finite Element computations of dynamic tensile tests by spalling are presented, with visco-damage and either without or with non-local enhancement. Delay-damage, as introduced, regularizes the solution in fast dynamics. The location of the macro-crack initiated is found influenced by non-local regularization. The strain rate range in which each enhancement, delay-damage or non-local enhancement, has a regularizing effect is studied.

Computational Modeling of Damage Development in Composite Laminates Subjected to Transverse Dynamic Loading

This paper presents a robust computational model for the response of composite laminates to high intensity transverse dynamic loading emanating from local impact by a projectile and distributed pressure pulse due to a blast. Delaminations are modeled using a cohesive type tie-break interface introduced between sublaminates while intralaminar damage mechanisms within the sublaminates are captured in a smeared manner using a strain-softening plastic-damage model. In the latter case, a nonlocal regularization scheme is used to address the spurious mesh dependency and mesh-orientation problems that occur with all local strain-softening type constitutive models. The results for the predicted damage patterns using the nonlocal approach are encouraging and qualitatively agree with the experimental observations. The predictive performance of the proposed numerical model is assessed through comparisons with available instrumented impact test results on a class of carbon-fiber reinforced polymer (CFRP) composite laminates. Force-time histories and other derived cross-plots such as the force versus projectile displacement and progression of projectile energy loss as a function of time are compared with available experimental results to demonstrate the efficacy of the model in capturing the details of the dynamic response. Another case study involving the blast loading of CFRP composite laminates is used to further highlight the capability of the proposed model in simulating the global structural response of composite laminates subjected to distributed pressure pulses.

Size effect and localization in J2 plasticity

This paper studies the phenomenon of structural size effect and strain localization in J2 plasticity. Size effect is here understood as the change in the response of a given structure when the spatial dimensions are scaled up or down while the geometry and other relevant properties of the structure are preserved. The work exploits the advantages of the mixed displacement–pressure formulation in incompressible or quasi-incompressible situations. Elasto-J2-plastic constitutive behaviour with regularized softening is considered. Stability issues are discussed to ensure existence and uniqueness of the solution of the corresponding discrete finite element formulation. Numerical examples show that the formulation derived is able to solve a wide range of structural scales, including real life engineering applications. The results obtained do not suffer from spurious mesh dependence. Furthermore, the formulation includes the classical theories of perfect plasticity and linear fracture mechanics as limit cases.

Loss of uniqueness of numerical solutions of the borehole problem modelled with enhanced media

It is well known that, initial boundary value problems involving constitutive equations modeling the degradation of the strength of materials are not well posed, which renders computations questionable. To overcome this issue it is necessary to enhance the models by incorporating some internal length. It has been shown that such an enhancement restores the objectivity of the computation as spurious mesh dependency is avoided. However, at least for simple problems (e.g. one dimensional ones), it has been proven that uniqueness of the underlying mathematical problem itself is not restored. Moreover numerical modeling of element tests yields several solutions. This paper demonstrates that several numerical solutions can be obtained also for less simple problems, namely the borehole problems. Even when a defect is introduced in the computed problems, different numerical solutions are found. Contrary to the one dimensional problem there is no proof that this loss of uniqueness comes from the underlying mathematical problem. It is our opinion that this is an inherent property of initial boundary value problems where, broadly speaking, strong degradation of the mechanical properties is modeled. In any case, it is necessary to be aware of this issue. For problems involving constitutive equation modeling strength degradation, it is important to try to find other solutions than the one obtained by using routinely a numerical code. The failure patterns of the different solutions found are however similar to experimental observations. This possible loss of uniqueness can then be seen as a counterpart of the difficulties encountered when attempting to reproduce experiments. This is crucial when dealing with geomaterials.

A non-local finite element based on volumetric strain gradient: Application to ductile fracture

The aim of this work is to propose a finite element formulation adapted to ductile fracture simulation using Continuum Damage Mechanics and unravelling two main encountered difficulties. First, as ductile damage represents voids nucleation and growth, constitutive behaviors constrain the plastic volumetric strain through damage evolution laws, leading to volumetric locking. A specific formulation is consequently needed. A three-field formulation is defined, in which volume change is treated as a new unknown; its relation to the displacement field is weakly enforced by mean of a Lagrange multiplier. Then, strain and damage localisation occurs due to softening, and leads to spurious mesh dependent solutions. A coupling between neighbouring material points is thus introduced through a volume change gradient term added in the three field formulation. First results show that this formulation permits to control localisation and to unravel mesh dependency.

A simplified second gradient model for dilatant materials: Theory and numerical implementation

This paper deals with the development of a new second gradient model, its numerical implementation and its validation. In order to remedy to the spurious mesh dependency of the post localized computation enhanced models incorporating some internal length are necessary. These models are very time consuming. In this paper we present a simplified theory within the framework of constrained micromorphic models involving only the micro volumetric strain. Provided the use of an additional penalty term in the numerical treatment, this model is quite efficient to regularize problems modelling behaviors exhibiting plastic volumetric strain such as the ones of geomaterials. More over this model is notably less time consuming than the more general ones.

Numerical simulation of ductile fracture with the Rousselier constitutive law

This contribution aims at developing robust and reliable tools dedicated to predictive numerical simulation of ductile fracture. Description of damage mechanisms relies here on the Rousselier constitutive law. To achieve robustness and objectivity, several points need then to be addressed: reduction of the set of internal variables so as to recast the model into the framework of generalised standard materials; proper expression of the normal plastic flow rule at the vertex of the yield surface; finite strain formulation based on a multiplicative split of the deformation gradient; fully implicit integration of the constitutive equations leading to a single scalar equation that admits a unique root; use of mixed finite elements to avoid volumetric locking; strain gradient-based nonlocal model to control strain localisation and remedy spurious mesh dependency. The capabilities of the resulting numerical formulation are demonstrated through simulating the cup–cone fracture of a notched tensile specimen.

An elastoplastic damage model for semi-brittle rocks

A new elastoplastic damage model, based on experimental evidences and using a thermodynamic framework, is proposed for rock materials under a large range of stresses. The elastic and plastic properties are affected by induced damage, which leads to material softening behaviour. After formulation of the model and determination of its parameters, triaxial compression tests on sandstone are simulated to show the model's consistency. In order to describe progressive failure process due to onset of strain localization, the elastoplastic model is extended by including a non-local damage formulation. Three representative cases are studied. The numerical results verify the well-known fact that the elastoplastic non-local damage model avoids spurious mesh sensitivity and provides a reliable description of the growth of the damage zone and the failure process.

Multiscale Modelling of Softening Materials

A hierarchical multiscale procedure for softening materials is proposed in this paper. A standard multiscale model has been analysed with respect to macro-level mesh dependence and meso-level cell size dependence. In order to eliminate spurious macro-level mesh dependence and meso-level cell size dependence a coupled-volume approach has been proposed. A discussion on the various interacting length scales in the model is included.

Computational Techniques and Shear Band Development for Cylindrical and Spherical Penetrometers in Strain-Softening Clay

This paper addresses numerical simulation of deep penetration of full-flow penetrometers in strain-softening, rate-dependent, cohesive soil, and the observed phenomenon of periodic shear bands. The analysis was conducted using a large deformation finite element approach, modifying the simple elastic–perfectly plastic Tresca soil model to allow strain-softening, with strain-rate dependency being incorporated in order to avoid spurious mesh dependency. Parametric analyses were carried out varying the strain-softening parameters (hence the relative brittleness of the soil), the rigidity index of the soil, and the strain-rate parameter. Increased brittleness of the soil led to reduction in the penetration resistance, but also to increasingly significant oscillations in the resistance–penetration responses. The oscillation was found to result from periodic shear bands evolving cyclically ahead of the advancing cylindrical and spherical penetrometers. Analyses with different values of rigidity index confirmed further that the periodic shear bands were a real material phenomenon, rather than due to errors in numerical simulation. Similar phenomena have been observed for continuous flow problems in granular materials. However, rising strain-rate dependency tended to suppress the oscillations.

Nonlocal microplane model with strain-softening yield limits

The paper deals with the problem of nonlocal generalization of constitutive models such as microplane model M4 for concrete, in which the yield limits, called stress–strain boundaries, are softening functions of the total strain. Such constitutive models call for a different nonlocal generalization than those for continuum damage mechanics, in which the total strain is reversible, or for plasticity, in which there is no memory of the initial state. In the proposed nonlocal formulation, the softening yield limit is a function of the spatially averaged nonlocal strains rather than the local strains, while the elastic strains are local. It is demonstrated analytically as well numerically that, with the proposed nonlocal model, the tensile stress across the strain localization band at very large strain does soften to zero and the cracking band retains a finite width even at very large tensile strain across the band only if one adopts an “over-nonlocal” generalization of the type proposed by Vermeer and Brinkgreve [In: Chambon, R., Desrues, J., Vardoulakis, I. (Eds.), Localisation and Bifurcation Theory for Soils and Rocks, Balkema, Rotterdam, 1994, p. 89] (and also used by Planas et al. [Basic issue of nonlocal models: uniaxial modeling, Tecnical Report 96-jp03, Departamento de Ciencia de Materiales, Universidad Politecnica de Madrid, Madrid, Spain, 1996], and by Strömberg and Ristinmaa [Comput. Meth. Appl. Mech. Eng. 136 (1996) 127]). Numerical finite element studies document the avoidance of spurious mesh sensitivity and mesh orientation bias, and demonstrate objectivity and size effect.