Nonlocal Damage Model Research Articles

An interactive orthotropic non-local damage model for progressive failure analysis of composite laminates

In this study, the non-local continuum damage model is further developed for progressive failure analysis of laminated composites. Characterized by two internal length scales, an orthotropic non-local integral strategy is implemented to model the constrained matrix damage along fiber direction. The introduction of spatial averaging may avoid spurious strain localization and thus ensure the physically-meaningful energy dissipation during damage evolution process. Compared to standard damage descriptions, the pathological mesh sensitivity and mesh orientation bias are effectively alleviated, leading to objective and accurate solutions. Moreover, an interactive damage transfer scheme is proposed to capture the strong interaction between matrix cracking and interface delamination, so that faithful predictions on the complex failure mechanisms of composite laminates are guaranteed. Performance of the present model and its superiority over conventional methods are demonstrated through several numerical examples, including the typical tensile failure of notched and unnotched laminates.

Stress-based bi-directional evolutionary structural topology optimization considering nonlinear continuum damage

This paper proposes a new methodology for structural topology optimization that is capable of stress minimization design considering nonlinear continuum damage. A quasi-static non-local damage model is integrated into a linear finite element analysis for modeling the structural damage. The Bi-directional Evolutionary Structural Optimization (BESO) method is adopted to circumvent the singularity issue. To cope with large scale constraints, the maximal von Mises stress is measured by the global p-norm stress aggregation approach. The density filter BESO method is developed while the sensitivity expressions of proposed indices with respect to design variables are derived. Influences of varying damage threshold and p-norm parameter on final designs are investigated through 2D and 3D numerical tests. The effectiveness of the proposed method is further validated in comparison with the stiffness maximization design. It is revealed that the smaller value of the damage threshold results in higher strength while the damage in the majority of solid material increases. The results demonstrate that the proposed approach can achieve stress minimization design by simultaneously considering nonlinear continuum damage.

Nonlocal damage modelling for finite element simulations of ductile steel sheets under multiaxial loading

AbstractThe finite element simulation of forming processes or vehicle crash scenarios involves multiaxial loading conditions in general. Thereby, damage and failure in ductile steel sheets is triggered by void growth under hydrostatic tensile loading or by void elongation within a narrow band under shear dominated loading. These different damage mechanisms are considered by a stress state dependent continuum damage model. The damage initiation and failure are represented by a critical and a failure strain. Both are HOSFORD‐COULOMB type functions of the stress triaxiality and the LODE parameter and, thus, incorporate the stress state dependence of the damage mechanisms. Due to the pathological mesh sensitivity for continuum damage models, an enhancement towards nonlocal damage evolution is carried out. Thereby, two options are investigated and compared: the integral nonlocal formulation and the gradient enhanced approach. Finally, the proposed nonlocal ductile damage model is verified for a wide range of stress states by test data of various tensile and shear specimens.

ADAPT — A Diversely Applicable Parameter Identification Tool: Overview and full-field application examples

The free and open source software tool ADAPT (A Diversely Applicable Parameter Identification Tool) for the inverse parameter identification of constitutive material models is presented and applied in combination with Finite Element (FE) simulations. Different types of input data, namely integral data such as forces and field data such as displacement and strain fields, are considered within a multi-objective optimisation scheme. The effect of each of those data sets is investigated in detail for an elasto-plastic and a non-local ductile damage model applied to a dual-phase steel. To demonstrate the wide range of parameter identification applications, several examples are given. The behaviour of a soft polymer using a finite strain non-local visco-elastic damage model is predicted. It is shown that field data such as displacement fields are required to identify the material parameters of a non-local damage model. Differing from existing approaches, high resolution scanning electron microscopy (SEM) data of voids and optical strain fields are employed to identify the material parameters of a Gurson–Tvergaard–Needleman porous plasticity model. This strategy reduces the error in terms of the predicted void area fraction in the particular air bending example considered by approximately 500% compared to a parameter identification approach not using microstructural data. The applied identification scheme contributed to the achievement of a good qualitative agreement for the prediction of the void area fraction in the process chain of hot calibre rolling with subsequent forward rod extrusion of a case-hardened steel by using an uncoupled Lemaitre-type damage model with strain rate and temperature dependency.

Microstructural dimension involved in the damage of concrete-like materials: An examination by the lattice element method

With the lattice element method, it is required to introduce a length via, for example, a non-local approach in order to satisfy the objectivity of the mechanical response. In spite of this, the mesoscale structuring of inclusions within a matrix conveys the natural origin of the internal length for a fixed mesh. In other words, internal length is not explicitly provided to the model, but rather governed by the characteristics of the meso-structure itself. This study examines the influence that the meso-structure of quasi-brittle materials, like concretes, have on the width of the fracture process zone and thus the fracture energy. The size of the fracture process zone is assumed to correlate with a microstructural dimension of the quasi-brittle material. If a weakness is introduced by a notch, the involvement of the ligament size (a structural parameter) is also investigated. These analyses provide recommendations and warnings that could be beneficial when extracting, from material meso-structures, a required internal length for nonlocal damage models. Among the observations made, the study suggests that the property that best characterise a meso-structure length would be the spacing between inclusions rather than the size of the inclusions themselves. It is also shown that microstructural dimension and the width of the fracture process zone have comparable order of magnitude, and they trend similarly with respect to microstructural sizes such as the inclusion interdistances.

Gradient Nonlocal Enhanced Microprestress-Solidification Theory and Its Finite Element Implementation

Time-dependent responses of cracked concrete structures are complex, due to the intertwined effects between creep, shrinkage, and cracking. There still lacks an effective numerical model to accurately predict their nonlinear long-term deflections. To this end, a computational framework is constructed, of which the aforementioned intertwined effects are properly treated. The model inherits merits of gradient-enhanced damage (GED) model and microprestress-solidification (MPS) theory. By incorporating higher order deformation gradient, the proposed GED-MPS model circumvents damage localization and mesh-sensitive problems encountered in classical continuum damage theory. Moreover, the model reflects creep and shrinkage of concrete with respect to underlying moisture transport and heat transfer. Residing on the Kelvin chain model, rate-type creep formulation works fully compatible with the gradient nonlocal damage model. 1-D illustration of the model reveals that the model could regularize mesh-sensitivity of nonlinear concrete creep affected by cracking. Furthermore, the model depicts long-term deflections and cracking evolutions of simply-supported reinforced concrete beams in an agreed manner. It is noteworthy that the gradient nonlocal enhanced microprestress-solidification theory is implemented in the general finite element software Abaqus/Standard with the implicit solver, which renders the model suitable for large-scale creep-sensitive structures.

Modelling of the fully coupled chemo-mechanical degradation for cement-based materials

The coupling of chemically induced degradation and mechanical damage is fundamental to investigate the durability of load-bearing cement-based materials subject to aggressive environments. This study is devoted to the incorporation of a reactive transport model and non-local damage model to integrate the complete process of chemical degradation and appropriate propagation of mechanical damage in the framework of coupled chemo-mechanical degradation. The chemical degradation of cement-based materials subject to leaching solution is principally concluded as the dissolution of portlandite and decalcification of C–S–H, and the reduction of Young's modulus is evaluated based on the microstructure of cement-based materials by the Mori–Tanaka scheme. The displacement-based non-local damage model is incorporated to capture the post-peak behaviour of the leached mortar beam in three-point bending test. The significance of coupling chemical and mechanical damage for the degraded beam is demonstrated by comparing the fully coupled damage with outcomes of the non-coupled case.

A thermodynamically nonlocal damage model using a surface-residual-based nonlocal stress

ABSTRACT In this research, a surface-residual-based nonlocal stress was introduced into nonlocal damage theory to describe the long-range actions among microstructures that were excluded in the definition of Cauchy stress. By using the surface-residual-based nonlocal stress tensor, a thermodynamically consistent nonlocal integral damage model was established to simulate the strain localization behavior for elastic-brittle damage problems. In this model, both the strain and the damage were taken as nonlocal variables in the free energy function, and the integral-type damage constitutive relationships and the evolution equation were derived via thermodynamic laws in order to ensure the self-consistency within the thermodynamic framework. Based on the nonlocal damage formulations using a real nonlocal stress concept, we simulated the strain localization phenomenon in an elastic bar subjected to uniaxial tension. The results showed clear localizing and softening features of strain in the damage zone, and the boundary effects arising from the nonlocal surface residual were illuminated. Furthermore, the strain localization behaviors for different internal characteristic lengths were simulated, through which we found that the characteristic length was comparable to the size of the strain localization zone.

Tensile failure model of carbon fibre in unidirectionally reinforced epoxy composites with mean-field homogenisation

This paper presents an extension of the so-called incremental-secant mean-field homogenisation (MFH) formulation accounting for fibre bundle failure and matrix cracking in Unidirectional (UD) composites. First a model for fibre bundle failure is developed based on the strength failure probability of the carbon fibre described by a Weibull distribution. This fibre bundle failure model is then framed in a damage model of embedded bundles in a matrix by considering an exponential relation to describe the longitudinal stress build-up profile experimentally observed during failure of embedded fibre bundles. Cracking of the matrix in UD composites is accounted for through an anisotropic non-local damage model, which allows capturing the so-called 0° splits experimentally observed during the longitudinal tension of UD plies. A Mean Field Homogenisation (MFH) model is then extended to account for these damage models as component behaviours of the 2-phase composite material. A finite element multi-scale simulation of a notched laminate shows that the intra-laminar failure modes observed by an in situ experiment reported in the literature are well captured by the damage variables related to the matrix and fibre bundle failure processes. Inter-laminar failure is also captured by an extrinsic cohesive law introduced between the plies.

Mesoscale damage modelling of concrete by using image-based scaled boundary finite element method

As a thin layer surrounding aggregates, the interfacial transition zone (ITZ) has a significant influence on the mechanical performance of concrete, a large number of small elements is required to discretize the ITZ and its vicinity owing to its small thickness. In this contribution, the mesostructure of concrete is explicitly built by a novel method named improved random walking algorithm (IRWA), with which the grading, content, shape of aggregates and the thickness of ITZs, can be conveniently controlled by users, and high volume fraction of aggregate and remarkable efficiency can be achieved. In the framework of the scaled boundary finite element method (SBFEM), the image-based quadtree decomposition is employed to automatically generate the multi-level mesh for the concrete composites, with no trouble in dealing with hanging nodes or the dramatically changing element size in the vicinity of the material interface. The nonlocal damage model in integral format is extended to simulate progressive damage of concrete at mesoscale with good mesh independence. Four benchmarks are modelled to demonstrate the performance of the proposed approach, and the effects of model parameters on the structural responses are also discussed.

Local and non-local damage model with extended stress decomposition for concrete

In this work, a new damage model for concrete is proposed with an extension of the stress decomposition (limited to biaxial cases), to capture shear damage due to the opposite signed principal stresses. To extract the pure shear stress, the assumption is made that one component of the shear stress is a minimum absolute of the two principal stresses. The opposite signed principal stresses are decomposed into shear stress and uniaxial tensile/compressive stress. A local model is implemented in Abaqus UMAT and it is further extended to a non-local model by utilization of the gradient theory. The concept of three length scales (tension, compression, and shear) is kept the same as the recently proposed nonlocal damage model by the authors. The nonlocal model is implemented in the Abaqus UEL-UMAT subroutine with an eight-node quadrilateral user-defined element, having five degrees of freedom at corner nodes (displacement in X/Y direction and tensile/compressive and shear nonlocal equivalent strain) and two degrees of freedom at internal nodes. Some examples of a local model including uniaxial and biaxial loading are addressed. Also, five examples of mixed crack mode and mode-I cracking are presented to comprehensively show the performance of this model.

A nonlocal damage model for concrete with three length scales

In the presented work, a nonlocal gradient enhanced damage model for concrete is proposed with a stress decomposition, to account for shear induced damage. The nonlocal model is an extension of the recently proposed local plasticity damage model by the authors, which can handle directional dependency of damage, pure shear and biaxial damage, damage activation/deactivation and microcracks opening/closure. The gradient enhanced approach is utilized for the extension of the local model. Due to the distinct behavior of concrete in tension, compression and shear, three length scales (tension, compression and shear) are incorporated, depending on local damage variables. The model is implemented in Abaqus UEL-UMAT subroutine with eight node quadrilateral user defined element, having five degrees of freedom $$ \left( {{\text{u}}_{x} ,{\text{u}}_{y} ,\underline{\text{eq}}^{ + } ,\underline{\text{eq}}^{ - } ,\underline{\text{eq}}^{s} } \right) $$ at corner nodes and two degrees of freedom at internal nodes $$ \left( {{\text{u}}_{x} ,{\text{u}}_{y} } \right) $$ . Five examples of mixed crack mode and mode-I cracking are modeled to show the performance of the model.

Modelling stress‐state dependent nonlocal damage and failure of ductile metals

AbstractDamage and failure of ductile metals is highly influenced by the stress state. In fact, void growth takes place under hydrostatic tensile loading and void coalescence leads eventually to ductile failure. A completely different damage mechanism occurs for shear dominated loading that results in void elongation within a narrow shear band. In order to account for both failure mechanisms, a stress state dependent damage model is proposed. The model is based on a continuum damage mechanics approach, whereby nonlinear damage evolution is taken into account. The influence of the stress triaxiality and the LODE parameter on damage initiation and failure is considered by the Hosford‐Coulomb model. Pathological mesh sensitivity is prevented by an integral type nonlocal formulation. Finally, the proposed stress‐state dependent nonlocal damage model is verified by test data for the microalloyed high strength steel HX340LAD for a wide range of stress states.

Parallel Computing with the Thick Level Set Method

The thick level set (TLS) method has been proposed as a nonlocal damage model for the modeling of failure in solids being able to deal with crack initiation, branching, and merging. The nonlocality...

One-dimensional Eikonal Non-Local (ENL) damage models: Influence of the integration rule for computing interaction distances and indirect loading control on damage localization

The Eikonal Non-Local approach models damage dependent non-local interactions by considering that interaction distances between material points are computed from solving an eikonal equation with an isotropic/anisotropic damage dependent metric Riemann function. In the finite element context, such a formulation exhibits good regularization properties and naturally allows one to model strain localization when non-local interactions vanish. In an isotropic damage mechanics framework, the damage variable tends to unity on a single integration point, and no damage diffusion occurs. In a one-dimensional context, evaluating the interaction distance between two points comes into computing an integral where the integrated function depends on the damage field between them. Desmorat et al. [9] performed the integration using a trapezoidal rule (under the assumption that the damage field is constant between adjacent integration points). Conversely, Jirásek and Desmorat [19] assumed that the damage field is linear between nearby integration points. Mainly motivated by improved integration accuracy, this method leads to a more gradual damaging process and makes it easier to follow the structural response when damage localizes. In this contribution, we show that despite these advantages, such a choice induces parasite damage diffusion issues, making the use of a standard trapezoidal rule more relevant. Moreover, once this choice is made, the structural response during damage localization can be conveniently described using a path-following method based on controlling the non-local strain variation of the localizing element instead of the local strain as in the works cited above.

Polytopal composite finite elements for modeling concrete fracture based on nonlocal damage models

The paper presents an assumed strain formulation over polygonal meshes to accurately evaluate the strain fields in nonlocal damage models. An assume strained technique based on the Hu-Washizu variational principle is employed to generate a new strain approximation instead of direct derivation from the basis functions and the displacement fields. The underlying idea embedded in arbitrary finite polygons is named as Polytopal composite finite elements (PCFEM). The PCFEM is accordingly applied within the framework of the nonlocal model of continuum damage mechanics to enhance the description of damage behaviours in which highly localized deformations must be captured accurately. This application is helpful to reduce the mesh-sensitivity and elaborate the process-zone of damage models. Several numerical examples are designed for various cases of fracture to discuss and validate the computational capability of the present method through comparison with published numerical results and experimental data from the literature.

Stochastic nonlocal damage analysis by a machine learning approach

A machine learning aided stochastic nonlocal damage analysis framework is proposed for quasi-brittle materials. The uncertain system parameters, including the material properties and loading actions, have been incorporated and analysed within a unified safety assessment framework against various working conditions. A three-dimensional integral-type nonlocal damage model through finite element method (FEM) has been adopted. For the purpose of investigating the probabilistic damage analysis problems, a freshly established machine learning approach, namely the capped-extended-support vector regression method (C-X-SVR), is proposed to eliminate the influences of random outliers in the first step, then establish the relationship between the uncertain systemic inputs and structural responses. Such that the training robustness and computational adaptability of the proposed regression model can be reinforced. Moreover, the proposed approach is competent of efficiently predicting the statistical information (i.e., means, standard deviations, probability density functions and cumulative density functions) of structural behaviours under continuous information update of the uncertain working condition from mercurial environment. One real-life experimental validation and two numerical investigations are implemented to further verify the effectiveness and efficiency of the uncertainty quantification framework against probabilistic damage analysis.

A non-local damage model for the fatigue behaviour of metallic polycrystals

ABSTRACT The development of fatigue damage in metallic materials is a complex process influenced by both intrinsic (e.g. texture, defects) and extrinsic (e.g. loading mode, frequency) factors. To better understand this process, some efforts have been made to develop microstructure-sensitive models that consider the impact of microstructural heterogeneities on the formation of fatigue cracks. An important limitation of such models is their inability to describe the transition between the nucleation and growth stages in a consistent thermodynamic framework. To circumvent this limitation, a constitutive model, which is appropriate for the description of both nucleation and early growth, is proposed. For this purpose, non-local damage mechanics is used to construct a set of constitutive relations that explicitly considers the progressive stiffness reduction due to the formation of fatigue cracks. Specifically, a damage variable is attached to each slip system, which allows considering both the anisotropic aspect of fatigue damage and closure effects. The spatial gradients of the damage variables are treated as additional state variables to account for the increase of surface energy associated with the formation of fatigue cracks. For the evolution equations to be compatible with the second law of thermodynamics, a modified form of the entropy production inequality is adopted. For illustration purposes, the non-local damage model is implemented in a spectral solver. Some numerical examples are then presented. These examples allow discussing the ability of the proposed model to describe the impact of loading conditions and pre-existing defects on the fatigue behaviour of metallic polycrystals.

A phase field approach enhanced with a cohesive zone model for modeling delamination induced by matrix cracking

The progressive damage analysis of fiber-reinforced composite materials is a challenging task, especially when complicated cracking scenarios arise due to the onset and progression of several damage mechanisms. From a modeling point of view, a particularly complex failure scenario is the interaction between intralaminar and interlaminar cracks. This work proposes a novel framework accounting for this interaction through the coupling of a nonlocal damage model based on the phase field approach for the intralaminar failure with a cohesive zone model for the interlaminar one. The modular variational formalism of the method presented leads to a very compact and efficient numerical strategy, which endows the fulfillment of the thermodynamic consistency restrictions and provides a relatively simple basis for its finite element implementation due to the preclusion of complex crack tracking procedures with standard element architectures. After addressing its implementation in the context of the finite element method in a high performance computing environment, the capabilities of the proposed formulation are explored through a numerical investigation of a cross-ply laminate subjected to a 4-point bending configuration. The comparison of the numerical predictions against the experimental observations demonstrates the reliability of the proposed framework for capturing the delamination induced by matrix cracking failure scenario.

Mode I Non-Linear Fracture Model: Cases on Concrete and Fiber Reinforced Concrete

Model fraktur ragam I non-linier telah banyak digunakan untuk memperoleh faktor intensitastegangan KSIc dan perpindahan bukaan ujung retak CTODc sebagai kriteria fraktur untuk beton danbeton serat. Beberapa model fraktur ragam I non-linier terdahulu antara lain Model Retak Fiktif olehHillerborg, (1976), Model Pita Retak oleh Bazant (1983, 1986), Model Dua-Parameter oleh Jenq danShah (1986), Model Penjalaran Retak Mode I oleh Zhang dan Li (2005), dan Model Kerusakan Non-Lokal oleh Ferrara dan Prisco (2005). Tulisan ini mengimplementasikan model fraktur ragam I nonlinierpada 2 kasus. Kasus pertama diimplementasikan pad beton sedangkan kasus keduadiimplementasikan pada beton serat. Kedua kasus tersebut akan memperoleh nilai faktor intensitastegangan KSIc dan perpindahan bukaan ujung retak CTODc. Kasus 1 adalah kasus benda uji balokbeton bertakik model fraktur ragam I non-linier dan kasus 2 adalah beton serat tak hingga modelfraktur ragam I non-linier. Kasus 1 menghasilkan nilai faktor intensitas tegangan KSIc sebesar 15.078MPa mm-1/2 dan perpindahan bukaan ujung retak CTODc sebesar 0.023 mm. Kasus 1 menghasilkannilai faktor intensitas tegangan KSIc sebesar 3.917.10-4 MPa mm-1/2 dan perpindahan bukaan ujung retak CTODc sebesar –1.994.10-4 mm. Secara umum, keberadaan serat sangat mempengaruhi solusianalitis. Tulisan ini memperoleh kesimpulan sebagai berikut: (1) Model fraktur ragam I non-linierdapat digunakan untuk memperoleh faktor intensitas tegangan KSIc dan perpindahan bukaan ujungretak CTODc sebagai kriteria fraktur untuk beton dan beton serat, (2) Perilaku fraktur beton seratadalah spesifik dibandingkan beton karena adanya fenomena penjembatanan serat, (3) Dalamperhitungan hasil faktor intensitas tegangan KSIc dan perpindahan bukaan ujung retak CTODc akanberlebihan bila traksi serat diabaikan dan kurang bila Zona Proses Fraktur diabaikan, (4) Akan sangatbaik bila mengkombinasikan Kasus 1 dan Kasus 2 bersama-sama untuk memperoleh nilai faktorintensitas tegangan KSIc dan perpindahan bukaan ujung retak CTODc dengan memperhatikankeberadaan serat dalam komposit matriks berserat.