Pathological Mesh Dependence Research Articles

A mixed nonlocal finite element model for thermo‐poro‐elasto‐plastic simulation of porous media with multiphase fluid flow

SummaryA new mixed nonlocal finite element framework is developed for nonlinear thermo‐poro‐elasto‐plastic simulation of porous media with multiphase pore fluid flow and thermal coupling. The solid‐fluid interaction is accounted for using the mixture theory of Biot based on the volume fractions concept. Different sources of nolinearities arising from the multiphase fluid flow effects, advective‐diffusive heat transfer, inelastic deformation, fluid flux injection induced mechanical tractions, solid skeleton deformation permeability dependence, and temperature dependent viscosity are included in developing a robust numerical solver for the targeted coupled multiphysics problem. To address the effect of microstructure in inelastic localized deformation behaviour with dilational softening, a nonlocal plasticity model is proposed based on a characteristic length scale which rectifies the non‐physical pathological mesh dependence problem encountered in conventional plasticity. The accuracy and strength of the developed model is shown with comparing the obtained numerical results of a benchmark bilateral compression test with existing published data in the literature. To show the versatility and robustness of the developed computational framework in modelling the geomechanics of real‐case engineering practices, large scale thermo‐hydro‐mechanical (THM) subsurface stimulation processes with applications in enhanced oil recovery (EOR) are effectively simulated and the targeted enhanced recovery and performances are demonstrated. The current formulation does not include phase transformation modelling capability, and therefore, the developed models may not be applicable for simulation of the engineering processes that involve phase change behaviour (e.g., steam injection).

A material model for thick, rate‐dependent adhesives with delocalized softening

AbstractThis contribution is dedicated to the development and implementation of a material model to describe the deformation behaviour of assembly adhesives. These assembly adhesives are usually used with thicknesses of several millimetres up to centimetres in order to compensate for large relative displacements of joining parts. These large displacements can be resisted by assembly adhesives, since they have a low shear modulus and thus a high strain until failure above their glass transition temperature. As a basis, the isothermal and isotropic hyperelasticity theory is used to describe the elastic deformation behaviour up to the postcritical range by using an isochoric–volumetric decomposition of the deformation gradient. The Arruda–Boyce and Ogden models are employed here for the isochoric‐elastic and volumetric‐elastic description. The modelling of the rate‐dependence concerning shear and bulk viscosity succeeds by extending the hyperelasticity theory to the theory of finite viscoelasticity with the generalized Maxwell model in each case. In order to model the postcritical deformation behaviour, an isotropic damage model is developed based on the energy limiting method and used to define an isochoric and a volumetric damage variable, so that the mapping of softening due to change of shape, as well as cavitation, succeeds. Furthermore, the damage variables are delocalized to prevent pathological mesh dependence by using the integral regularization method. Finally, the material parameters are determined and validated by means of measurement data due to tests at the assembly adhesive Betaforce 2850L. The results demonstrate a high prediction quality.

Integral-based non-local approach to ductile damage and mixed-mode fracture

The present study focuses on the application of an integral-based approach for constructing non-local ductile damage models. The incorporation of non-locality into the damage accumulation rule enhances the modeling framework and yields simulation results that are physically reasonable, avoiding the issue of pathological mesh-dependence. To achieve a precise depiction of damage accumulation and fracture under mixed-mode I/II loading, novel delocalization kernels are proposed. These kernels explicitly consider the heterogeneity of stresses and strains in the fracture process zone; the new kernels are categorized into stress-based and strain-based families. Furthermore, rigorous receiver-based normalization procedures and physically meaningful source-based normalizations are explored for the developed kernels. The outcome of this study is the modeling tool, suitable for end-to-end simulations of damage accumulation and fracture. Using a well-calibrated material model, the practical applicability of the new approach is demonstrated by performing finite element analysis on fracture tests conducted on compact tension–shear specimens. In contrast to the conventional delocalization kernels, the newly introduced families of kernels offer an additional fitting parameter, valuable in accurately describing the structural behavior under mixed-mode loading conditions. In particular, the proposed approach enables control over the shape of the predicted KIc–KIIc diagram.

Numerical implementation of tension–compression asymmetry in a gradient‐extended two‐surface damage–plasticity model

AbstractBased on the spectral decomposition of the strain tensor, the tension‐compression asymmetry (TCA) in a gradient‐extended two‐surface damage‐plasticity model is considered in this study. More specifically, the gradient extension is applied in the current model as a suitable damage regularization technique to prevent pathological mesh dependence issues. Furthermore, plasticity and damage are treated as distinct but strongly coupled dissipative mechanisms by using two different types of damage and yield criteria as well as the related loading/unloading conditions, respectively. Finally, a specific material parameter is adopted to control the degree of TCA which results in a unique damage evolution in compression.

A gradient-based non-local GTN model: Explicit finite element simulation of ductile damage and fracture

Non-local models have over the years been established as an effective approach to solve the pathological mesh dependency problem observed in finite element simulation of strain-softening materials. This paper presents the formulation, implementation, and application of a gradient-based non-local Gurson–Tvergaard–Needleman (GTN) model for explicit finite element analysis. The porosity is taken as the non-local variable where the increment in porosity is averaged over the volume using an implicit gradient model. The gradient model is implemented in Abaqus/Explicit by utilising the coupled thermal–mechanical solver, which proves to be both a simple and computationally efficient approach. Due to the use of an explicit integration scheme, a transient term is introduced to the partial differential equation of the gradient formulation. The non-local GTN model is compared to the local counterpart for increasing mesh refinements using a plane strain shear band specimen, a plane strain tension specimen, and a plane strain compact tension specimen. The proposed approach can remedy the pathological mesh dependency problem. For the plane strain tension specimen, it is shown that the non-local GTN model will preserve the fracture mode (slant versus cup–cup fracture mode) when the mesh is refined. The non-local GTN model is also able to predict the same fracture mode as observed in ductile tearing experiments. However, non-local averaging can over-smooth the fields and exclude the slant fracture mode from occurring if the material length Lc is too large.

A multiscale finite element method for soil-rock mixture

Soil-rock mixture is a complex multi-phase composite geotechnical material, and its strength is determined by the physical properties of constituent multi-phase materials and their coupling mechanical response between different phases of materials. Based on the Eshelby-Mori-Tanaka equivalent inclusion average stress principle, a theoretical model of multi-scale coupled shear strength of soil-rock mixture considering the interaction effect of rock block and soil is established, and the rotational freedom reflecting the microscopic motion details of rock block is introduced. Moreover, a multi-scale coupled constitutive relationship of soil-rock mixture is derived and compiled into a multi-scale finite element program. Based on the large-scale direct shear test of soil-rock mixture, the model parameters of the multi-scale finite element method are determined, and then the multi-scale finite element program is used to simulate and predict the cross-scale deformation process of the soil-rock mixture slope. The results show that the multi-scale finite element method can effectively describe the influence of the mechanism of the micro motion characteristics of the soil-rock mixture on the macro mechanical response, and can effectively overcome the pathological mesh-dependency of the classical finite element method; the rotation displacement of the rock block is mainly concentrated within the shear zone of the slope. The maximum rotational displacement of rock blocks inside the soil-rock mixture slope is 40.7°, and the rotational displacement of rock blocks outside the shear zone is about 0°. The physical mechanism of the cross scale evolution of the shear band of the soil-rock mixture slope is that: the rotation of the rock blocks weakens the strain transmission ability between the rock block and the matrix soil, thus forming the concentration and development of the plastic strain, and finally leading to the penetration of the shear bands of the slope and the overall sliding failure.

Nonlocal FEM modeling of piles as beam elements embedded within 3D continuum

Finite element analysis of piled rafts and other foundation systems, in which concrete columns/piles are used to strengthen weak subsoil, is a commonly accepted tool in modern geotechnical engineering. It enables one to analyze all important aspects of soil–structure interaction (SSI), including nonlinear stress–strain constitutive relations for soils, strong displacement discontinuity between rigid inclusions (piles/barettes/columns) and subsoil, but also coupling soil deformation with groundwater flow. In practice, the number of designed inclusions is usually large. Therefore, the standard approach in which the subsoil, and rigid inclusion as well, are discretized using continuum finite elements, may yield huge computational models, which are not acceptable from the practical point of view. To remedy this serious drawback a nonlocal method of modeling rigid inclusions as beam elements embedded within the 3D continuum is proposed and verified. Moreover, it removes another pathological mesh dependency effect when modeling beam–shell connection.

A finite deformation multiplicative plasticity model with non–local hardening for bonded geomaterials

The paper presents a finite deformation, isotropic hardening, non–associative elastic–plastic constitutive model (FD_Milan model) for describing the mechanical behavior of a wide range of bonded natural geomaterials such as stiff overconsonsolidated clays, porous soft rocks or bio–improved soils. The formulation of the model is based on the multiplicative split of the deformation gradient and on the assumption of hyperelastic behavior. To deal with the occurrence of strain localization, typically observed in this class of geomaterials, the model has been equipped with a non–local version of the hardening laws. This approach is capable of regularizing the pathological mesh dependence occurring in the post–localization regime when adopting classical plasticity models.In view of its application to practical geotechnical problems characterized by large displacements and deformations within a hydro-mechanical coupled environment, the model has been implemented in the recently developed Particle Finite Element code G–PFEM for geomechanics applications.To demonstrate the effectiveness of the numerical implementation, a series of numerical simulations has been performed considering two representative boundary value problems: the modeling of shear localization in plane strain biaxial tests and the simulation of CPTu tests in a saturated porous soil. The results of biaxial test simulations have highlighted the role of the characteristic length in controlling the thickness of the localized zone and the effect of the confining pressure in determining the pattern of shear band formation. An interesting feature emerging from the partially drained CPTu simulation results is the progressive formation of persistent shear bands, which originate from the cone tip and propagate outwards along the entire penetration depth.

Investigation of Instabilities in Granular Media and Their Numerical Simulation

The inception of instabilities in sand across different “length-scales,” viz., continuum, discrete, and laboratory element tests, has been highlighted in this study. With instability onset, the material behavior no longer remains “single element” in the sense of continuum mechanics, and due care is required while calibrating different constitutive relationships for use in various numerical simulations. A laboratory “element” test can instead be viewed as a boundary value problem at the instability onset. The post-instability response of a soil specimen represents the “system-response” with the evolution of inhomogeneities being primarily influenced by the boundary conditions among various other factors. Instability onset in drained and undrained biaxial tests has been explored by adopting the bifurcation framework with the aid of a generalized nonassociative elastoplastic material model. For the locally drained globally undrained scenario within a continuum numerical framework, instability onset is found to be influenced by the refinement of mesh discretization. Signatures of rate-dependent localized behavior of sand specimens are also commented on from numerical simulations of drained biaxial tests. To address the “pathological mesh dependence” of classical continuum modeling, we resort to a micromechanical discrete granular framework that considers the actual particle morphology. The numerical predictions match reasonably well with the flexible boundary plane strain experiments. The inherent grain fabric arrangement is found to be the triggering mechanism behind the localized strain accumulation at multiple zones. The influence of loading boundary conditions and various signatures of local nonuniformities is also explored with the aid of image analysis in plane strain and triaxial tests.

Analysis of cone penetration in brittle liquefiable soils

The paper presents a comprehensive set of simulations of the cone penetration test with pore pressure measurements (CPTu test) in brittle, potentially liquefiable, soils using advanced numerical techniques. The large displacements and large deformations involved in the cone penetration problem are tackled using the Particle Finite Element Method, especially adapted for the analysis of coupled geotechnical problems (G-PFEM). A non-local regularization technique is adopted to prevent the pathological mesh dependence associated with continuum analysis of softening materials. Brittle soil behaviour is described using the CASM constitutive model. Undrained CPTu tests are simulated for a range of soils with different degree of brittleness. Effects of permeability, partial drainage and cone roughness are also explored. The results of the analyses are assessed in terms of the interpretation and classification procedures usually applied in practice. In addition, advantage is taken of the availability of a full numerical solution to explore aspects of soil response during penetration that are not possible when only global CPTu observations are available. The full set of numerical analyses have been able to capture the essential features of the problem and to provide useful insights into the effects of brittleness on the mechanisms and results of the CPTu test.

Numerical analysis of hip fracture due to a sideways fall

The primary purpose of this paper is to outline a methodology for evaluating the likelihood of cortical bone fracture in the proximal femur in the event of a sideways fall. The approach includes conducting finite element (FE) analysis in which the cortical bone is treated as an anisotropic material, and the admissibility of the stress field is validated both in tension and compression regime. In assessing the onset of fracture, two methodologies are used, namely the Critical Plane approach and the Microstructure Tensor approach. The former is employed in the tension regime, while the latter governs the conditions at failure in compression. The propagation of localized damage is modeled using a constitutive law with embedded discontinuity (CLED). In this approach, the localized deformation is described by a homogenization procedure in which the average properties of cortical tissue intercepted by a macrocrack are established. The key material properties governing the conditions at failure are specified from a series of independent material tests conducted on cortical bone samples tested at different orientations relative to the loading direction. The numerical analysis deals with simulations of experiments involving the sideways fall, and the results are compared with the experimental data. This includes both the evolution of fracture pattern and the local load-displacement characteristics. The proposed approach is numerically efficient, and the results do not display a pathological mesh-dependency. Also, in contrast to the XFEM approach, the analysis does not require any extra degrees of freedom.

A phase field framework for dynamic adiabatic shear banding

We present a general variational formulation of phase field framework for dynamic adiabatic shear banding (ASBing). The ASB surface energy is derived from the Griffith’s regularized ASB surface function, and the concept of shear banding energy density that characterized by the energy jump contour integral. We reveal that the ASB phase field formulation reduces to a standard strain-gradient plasticity model by energy minimization and ASB canonical analysis, which indicates that strain-gradient regularization for problems involving strain softening has been incorporated reasonably. The phase-field and plasticity coupled constitutive relations are developed within an atomistic potentials consistent hyperelastic-plasticity framework. The degraded volumetric strain energy governs the elastic responses, while the degraded deviatoric strain energy, as well as the ASB surface energy, accounts for the combined effects of shear localization, plasticity and fracture, which directly relates energy dissipation to the evolution of dynamic ASBs. Numerical simulations demonstrate the ability of this predictive ASB phase field framework in capturing discontinuous surfaces, and the removal of pathological mesh-dependence.

A micromorphic approach for gradient-enhanced anisotropic ductile damage

This paper deals with the numerically effective modeling of anisotropic material degradation caused by ductile damage. Although standard local anisotropic damage models are relatively well-developed nowadays, their regularization which is required in order to eliminate their mathematical ill-posedness is far from being straightforward. It bears emphasis that this regularization is not only required from a mathematical point of view, since the aforementioned ill-posedness is known to be the source for the pathological mesh dependence as far as the finite element method is concerned. Within this paper, a general local framework for capturing anisotropic material degradation caused by ductile damage is extended to a non-local model by means of a gradient-enhancement. However, in order to achieve a numerically effective implementation, the gradient-enhancement is not implemented in a direct manner, but by means of a micromorphic approximation. By doing so, the implementation of the underlying local model is almost unaffected. Particularly the inequalities resulting from the yield function can be restricted to the local integration point level. Since a naive micromorphic implementation turns out to be unsuitable for regularizing the underlying local model, a novel adaption of the yield function is proposed. It is shown that the resulting single-surface-model is indeed able to capture anisotropic material degradation caused by ductile damage and, furthermore, that the finite element implementation is mesh objective.

Numerical investigation of wear processes by a gradient‐enhanced damage‐plasticity model

AbstractThe prediction of failure mechanism in structures are always an important topic in the field of computational mechanics. Finite element computations of an inelastic material involving softening behavior (e.g. softening plasticity or damage) can suffer from strongly mesh‐dependent results. Therefore, such continuum models should be equipped with a regularization (localization limiter) strategy to overcome the above‐mentioned problem.In this study, we present a framework for gradient‐enhancement for coupled damage‐plasticity material model derived by means of Hamilton's principle for non‐conservative continua. This model is applied for the numerical investigation of wear processes as they occur, e.g. in the case of mechanized tunneling. These investigations require a fine resolution of the involved constituents (cut sheet and abrasive particles in the soil). Consequently, a numerical strategy for the damage‐plasticity model is demanded that allows for time‐efficient simulations.In this paper, we present a first step to the mentioned ultimate goal. To this end, a numerical framework for gradient‐enhanced damage‐plasticity coupling is proposed that is based on a combination of the finite element method with strategies from meshless methods. We demonstrate that this framework keeps the computational effort limited and for each load step close to the purely elastic problems. Several numerical examples prove the elimination of the pathological mesh dependency of the results. Furthermore, first results to the simulation of wear in tunneling machines are presented.

A stable mesh-independent approach for numerical modelling of structured soils at large strains

We describe the large strain implementation of an elasto-plastic model for structured soils into G-PFEM, a code developed for geotechnical simulations using the Particle Finite Element Method. The constitutive model is appropriate for naturally structured clays, cement-improved soils and soft rocks. Structure may result in brittle behavior even in contractive paths; as a result, localized failure modes are expected in most applications. To avoid the pathological mesh-dependence that may accompany strain localization, a nonlocal reformulation of the model is employed. The resulting constitutive model is incorporated into a numerical code by means of a local explicit stress integration technique. To ensure computability this is hosted within a more general Implicit-Explicit integration scheme (IMPLEX). The good performance of these techniques is illustrated by means of element tests and boundary value problems.

A Constitutive Model for Locally Drained Shear Bands in Globally Undrained Dense Sand

When saturated granular materials which are dilative in nature are subjected to the undrained deformation, their strength increases due to the generation of negative excess pore pressure. This phenomenon is known as dilative hardening and can be witnessed in saturated dense sand or rocks during very fast loading. However, experimental evidence of undrained biaxial compression tests of dense sand shows a limit to this dilative hardening due to the formation of shear bands. There is no consensus in the literature about the mechanism which triggers these shear bands in the dense dilative sand under isochoric constraint. The possible theoretical reasoning is the local drainage inside the specimen under the globally undrained condition, which is challenging to be monitored experimentally. Hence, both incept of localisation and post-bifurcation of the saturated undrained dense sand demand further numerical investigation. Pathological mesh dependency hinders the ability of the finite element method to represent the localisation without advanced regularisation methods. This paper attempt to provide a macroscopic constitutive behaviour of the undrained deformation of the saturated dense sand in the presence of a locally drained shear band. Discontinuation of dilatant hardening due to partial drainage between the shear band and the adjacent material is integrated into the constitutive model without changing governing equilibrium equations. Initially, a classical bifurcation analysis is conducted to detect the inception and inclination of the shear band based on the underlying drained deformation. Then a post-bifurcation analysis is carried out assuming an embedded drained or partially drained shear band at gauss points which satisfy bifurcation criterion. The smeared shear band approach is utilised to homogenise the constitutive relationship. It is observed that the dilatant hardening in the saturated undrained dense sand is reduced considerably due to the formation of shear bands.

Microscale analysis of heterogeneous ductile materials with nonlocal damage models of integral type

A computational scheme for the analysis of damage localization in heterogeneous ductile materials in the case of non-separated scales is presented. The consistent linearization of nonlocal damage models of integral type at finite strains is addressed and the influence of nonlocal interactions on the homogenized material response is investigated. The constituents and phases of the material at the microstructural representative volume element (RVE) level are modeled with a nonlocal elasto-plastic isotropic damage model. The numerical integration of the constitutive equations within a nonlinear homogenization problem is described in detail. The scheme is applied to the simulation of ductile damage and the influence of the nonlocal averaging procedure, which can be evaluated at different configurations and include non-local interactions between different phases, is analysed. The quadratic rate of convergence of the Newton-Raphson iterative procedure is demonstrated and the capability to alleviate the pathological mesh dependence is illustrated through microstructural examples.

On the localisation of damage under pure bending using a nonlocal approach

To reproduce the behaviour of quasibrittle materials mathematically, constitutive laws with softening are needed, which often leads to development of damage localisation bands. If a standard local formulation is used, this approach leads to a pathological mesh dependence, which can be eliminated by making use of alternative nonlocal formulations. The aim of this paper is to assess the localisation properties of damage models under pure bending using different nonlocal formulations; to permit a partially analytical treatment, the idealised case of pure bending is studied. Under these conditions, the localisation process starts at the tensile face of the beam, which belongs to the boundary of the domain on which the problem is solved. Consequently, localisation patterns are affected by the boundary treatment as well as other parameters, such as the characteristic length that defines the area contributing to the nonlocal averaging. This paper presents an analytical study of the onset of localisation of different nonlocal formulations for a beam under pure bending. In addition to it, the subsequent evolution of the localised solution is explored by numerical simulations, analysing the localisation bands spacing, the dissipated energy profile along the fracture plane and the Moment-ϕL diagrams, with ϕL being a parameter that represents the rotation that drives the loading process (ϕ stands for the relative rotation angle of the cross section and L for the beam length). An analysis of damage localisation on longer beams where damage localises in several areas is also carried out and, finally, the damage localisation due to shrinkage is studied as a more realistic example of the problem addressed here.

From diffuse damage to strain localization from an Eikonal Non-Local (ENL) Continuum Damage model with evolving internal length

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.

Non-Local Ductile Damage Formulations for Sheet Bulk Metal Forming

A ductile damage model for sheet bulk metal forming processes and its efficient and accurate treatment in the context of the Finite Element Method is presented. The damage is introduced as a non-local field to overcome pathological mesh dependency. Since standard elements tend to show volumetric locking in the bulk forming process a mixed formulation is implemented in the commercial software simufact.forming to obtain better results.