Nonlocal Damage Research Articles

Multiscale modeling of thermo-hydromechanical behavior of clayey rocks and application to geological disposal of radioactive waste

This work is devoted to numerical analysis of thermo-hydromechanical problem and cracking process in saturated porous media in the context of deep geological disposal of radioactive waste. The fundamental background of thermo-poro-elastoplasticity theory is first summarized. The emphasis is put on the effect of pore fluid pressure on plastic deformation. A micromechanics-based elastoplastic model is then presented for a class of clayey rocks considered as host rock. Based on linear and nonlinear homogenization techniques, the proposed model is able to systematically account for the influences of porosity and mineral composition on macroscopic elastic properties and plastic yield strength. The initial anisotropy and time-dependent deformation are also taken into account. The induced cracking process is described by using a non-local damage model. A specific hybrid formulation is proposed, able to conveniently capture tensile, shear and mixed cracks. In particular, the influences of pore pressure and confining stress on the shear cracking mechanism are taken into account. The proposed model is applied to investigating thermo-hydromechanical responses and induced damage evolution in laboratory tests at the sample scale. In the last part, an in-situ heating experiment is analyzed by using the proposed model. Numerical results are compared with experimental data and field measurements in terms of temperature variation, pore fluid pressure change and induced damaged zone.

A theoretical framework for multi-physics modeling of poro-visco-hyperelasticity-induced time-dependent fracture of blood clots

Fracture resistance of blood clots plays a crucial role in physiological hemostasis and pathological thromboembolism. Although recent experimental and computational studies uncovered the poro-viscoelastic property of blood clots and its connection to the time-dependent deformation behavior, the effect of these physical processes on clot fracture and the underlying fracture mechanisms are not well understood. This work aims to formulate a thermodynamically consistent, multi-physics theoretical framework for describing the time-dependent deformation and fracture of blood clots. This theory concurrently couples fluid transport through porous fibrin networks, non-linear visco-hyperelastic deformation of the solid skeleton, solid–fluid interactions, mechanical degradation of tissues, gradient enhancement of energy, and protein unfolding of fibrin molecules. The constitutive relations of tissue constituents and the governing equation of fluid transport are derived within the framework of porous media theory by extending non-linear continuum thermodynamics at large strains. A physics-based, compressible network model is developed for the fibrin network of blood clots to describe its mechanical response. The kinetic equations of the internal variables, introduced for describing the non-linear viscoelastic deformation, non-local damage driving force and protein unfolding, are formulated according to the thermodynamics principles by incorporating a non-equilibrium energy of fibrin networks, a gradient-enhanced energy, and a stretch-induced energy of fibrin molecules, respectively, into the total free energy density function. An energy-based damage model is developed to predict the damage and fracture of blood clots, and an evolving regularization parameter is proposed to limit the damage zone bandwidth. The proposed model is implemented into finite element code by writing subroutines and is experimentally validated using single-edge cracked clot specimens with different constituents. The fracture of blood clots subject to different loading conditions is simulated, and the mechanisms of clot fracture are systematically analyzed. Computational results show that this model can accurately capture the experimentally measured deformation and fracture. The viscoelasticity and fluid transport play essential roles in the fracture of blood clots under physiological loading.

An efficient nonlocal integral method based on the octree algorithm

The nonlocal integral method typically requires a very high computing cost to search neighborhood integration points for calculating the nonlocal variable, which limits its application in large-scale problems. This paper proposes an efficient nonlocal integral method based on the octree algorithm, in which the integration point information is stored in the tree data structure to accelerate the search task. Firstly, the fundamental principles and implementation details of using the octree algorithm to search neighborhood integration points are described in detail. Subsequently, a Mohr-Coulomb nonlocal damage plastic model is presented as the application object of the proposed method. The model is implemented further in the ABAQUS using the octree-based nonlocal method and the return mapping algorithm enhanced by a line search method. Finally, two typical boundary value problems are simulated to verify the effectiveness and to assess the computational efficiency of the proposed nonlocal method. For the given test environment, the octree algorithm can achieve up to 100 times speedup at the integration point level compared to the traversal algorithm, and the developed efficient nonlocal method can achieve up to 7.9 times speedup at the boundary value problem level compared to the original nonlocal method.

Topology optimization with a finite strain nonlocal damage model using the continuous adjoint method

This study presents a unified formulation of topology optimization with a finite strain nonlocal damage model using the continuous adjoint method. For the primal problem to describe the material response including deterioration, we consider the standard Neo–Hookean constitutive model and incorporate crack phase-field theory for brittle fracture within the finite strain framework. For the optimization problem, the objective function is set to accommodate multiple objectives by weighting each sub-function, and the continuous adjoint method is employed to derive the sensitivity. Thus, both the governing equations for primal and adjoint problems are written as strong forms and hold at any moment and at any location in the continuum body or on its boundary. Accordingly, the proposed formulation is independent of the requirements from numerical implementation, such as element type or discretization method. In addition, the reaction–diffusion equation is used to update the design variable in an optimizing process, by which the continuous distribution of the design variable, as well as material properties, are realized. After the basic performance of the proposed formulation is demonstrated with a simple numerical setup, two-material (matrix and inclusion materials) and single-material (material and null) topology optimizations are presented, and discussions are made.

An extended gradient damage model for anisotropic fracture

This paper combines energy decomposition and an extended gradient damage (EGD) model to develop an anisotropic fracture framework with decoupling of tensile and shear cohesive laws. By introducing the shear-normal decomposition in the energy form, the driving force of the damage variable is established within the framework of the EGD model, which is then capable of capturing the traction-separation of potential crack surfaces in both shear and normal directions. The intrinsic correspondence between the cohesive law and the damage evolution enables the accurate prediction of anisotropic fracture behavior in the mixed form of Mode I and Mode II. Furthermore, the proposed model also addresses the damage unloading issue, which still remains a challenge in classic phase field theory or non-local damage theory. A number of numerical examples are presented as validation. Some cutting-edge benchmarks, such as complex mixed-mode fracture and perfect shear fracture, are well reproduced.

Recent trends in computational damage models: An overview

This work presents an introduction to popular damage and fracture models along with the major breakthroughs and the current state of the art. The theoretical and computational aspects of each model are discussed. While the damage models incorporate the microstructural effects, the fracture models consider the damage mainly at the macro level. The damage models such as Continuum damage mechanics, lattice models, nonlocal damage models, and fracture models such as Extended Finite Element method, Phase field and Peridynamics are presented. Both the discrete and diffused crack approaches along with the numerical solution techniques are presented. A special emphasis on the recently developed Peridynamic models and the different formulations associated with Phase field models is made.

On the development of a constitutive model for steel subjected to fire and explosion

The behavior of steel under thermal and mechanical loading including high transient temperatures and strain rates (e.g., fire, creep or explosion) is extremely complex. To study this behavior, we propose a constitutive model accordance with the laws of thermodynamic, the principle of virtual power and experimental results. The thermodynamic framework considers the Clausius-Duhem inequality and maximum dissipation rate principle, as well as the theory of finite deformations, linear isotropic viscoelasticity, nonlinear isotropic and kinematic hardening, local and non-local viscoplasticity, local and non-local anisotropic viscodamage and parameters associated to material behavior dependent on loadings. The components of the thermodynamic conjugate forces are defined from the Helmholtz free energy function and the rate of energy dissipation, which are postulated considering the creep phenomenon, thermal-mechanical transient phenomenon, cyclic phenomenon and plastic wave propagation phenomenon. The balance of microforces is defined based on the principle of virtual power. A novel rule of non-associated thermo-viscoplastic flow of the Perzyna type and a novel rate dependent local and non-local damage evolution law are introduced from constitutive model. The verification of the model was carried out in two benchmark studies composed of fire tests. The predictions of the model have shown in according with the results of the experiments. A sensitivity study was also performed with a view to analysis the influence of the parameters in achieved results. From this study a simplified constitutive model is presented. Benchmark studies composed of explosion tests are part of future work.

An Improved Kernel Function in Nonlocal Damage Model with the Boundary Effect

An Improved Kernel Function in Nonlocal Damage Model with the Boundary Effect

A novel FFT framework with coupled non-local elastic-plastic damage model for the thermomechanical failure analysis of UD-CF/PEEK composites

A novel FFT-based computational framework for coupling non-local elastic-plastic damage model is proposed in this paper, which is employed to accurately simulate the transverse tensile behaviors of unidirectional (UD) CF/PEEK composite materials under different temperatures. To address the distinct material properties exhibited by different microscopic constituents within the composite, two approaches are employed: (i) an integral-type regularized traction-separation damage model is applied to the interphase debonding, and (ii) an implicit gradient regularization technique for Lemaitre-type damage (matrix cracking) of the PEEK. By coupling the non-local damage model capable of accurately characterizing different failure modes within composites with an FFT computational framework, accurate predictions of the transverse tensile performance of UD-CF/PEEK composites at different temperature conditions can be achieved. Furthermore, the influence of non-local feature parameters and interphase mechanical properties on the composite's mechanical behavior is thoroughly discussed. Comparisons with experimental results affirm the accurate prediction of the transverse tensile behavior of UD-CF/PEEK composites through the proposed computational framework. Ultimately, based on this framework, the transverse tensile modulus and strength of UD-CF/PEEK composites, considering different fiber volume fractions and temperatures, are predicted, successfully demonstrating the effectiveness and applicability of the proposed approach in forecasting the mechanical behavior of composites.

A thermodynamically consistent strain difference-based nonlocal damage mechanics approach for failure analysis of quasi-brittle materials

The nonlocal integral approaches are established methods known for their effectiveness in dealing with the challenges of ill-posed behaviour and mesh sensitivity during the strain softening process, especially within classical continuum theories. The present work extends the strain difference-based nonlocal approach to continuum damage mechanics (CDM) theories for predicting the failure behaviour of quasi-brittle materials. The strain difference approach is adopted for two reasons: (i) it maintains physical realism and indirectly incorporates nonlocal effects while computing both the damage variable and the damage energy release rate, and (ii) the constitutive model maintains a symmetry nature in constructing the nonlocal part of the stiffness matrix, reducing the computational cost in strain-softening problems. The state and damage evolution equations for the present nonlocal approach are formulated within the thermodynamic framework. Numerical examples involving 2D and 3D cases are solved: (i) to study the influence of nonlocal parameter ‘α’ on the damage distribution while capturing the post-peak behaviours, (ii) to demonstrate the effectiveness of the present methodology by obtaining mesh-independent results and accurately predicting damage propagation paths, and (iii) to compare the experimental results and numerical structural response for coarse mesh, highlighting both computational efficiency and reduction of mesh-bias dependency. The results obtained by the strain difference-based nonlocal damage model agree well with the experimental results in the literature.

Block preconditioning strategies for generalized continuum models with micropolar and nonlocal damage formulations

AbstractIn this work, preconditioning strategies are developed in the context of generalized continuum formulations used to regularize multifield models for simulating localized failure of quasi‐brittle materials. Specifically, a micropolar continuum extended by a nonlocal damage formulation is considered for regularizing both, shear dominated failure and tensile cracking. For such models, additional microrotation and nonlocal damage fields, and their interactions, increase the complexity and size of the arising linear systems. This increases the demand for specialized preconditioning strategies when iterative solvers are adopted. Herein, a block preconditioning strategy, employing algebraic multigrid methods (AMG) for approximating the application of sub‐block inverses, is developed and tested in three steps. First, a block preconditioner is introduced for linear systems resulting from micropolar models. For this case, a simple sparse Schur complement approximation, which is practical to compute, is proposed and analyzed. It is tested for three different discretizations. Second, the developed preconditioner is extended to reflect the additional nonlocal damage field. This extended three‐field preconditioner is tested on the simulation of a compression test on a sandstone sample. All numerical tests show an improved performance of the block preconditioning approach in comparison to a black‐box monolithic AMG approach. Finally, a problem‐adapted preconditioner setup strategy is proposed, which involves a reuse of the multigrid hierarchy during nonlinear iterations, and additionally accounts for the different stages occurring in the simulation of localized failure. The problem‐adapted preconditioning strategy has the potential to further reduce the total computation time.

Determination of nonlocal kernels in nonlocal models for solving strain localization problems

ABSTRACT The strain localization during the damage and failure of materials can be accurately predicted by nonlocal continuum theories. Yet, the results of theoretical predictions are directly affected by the nonlocal kernel formulation. In this study, the correlation between the nonlocal kernel formulation and the strain localization simulation results is revealed. First, based on the nonlocal damage theory, a novel free energy functional was proposed with strain and damage parameters as independent variables, and a constitutive relation considering nonlocality and elastic damage was derived within the framework of continuum thermodynamics. Then, one-dimensional governing equations were obtained through the dual-linearization method. Finally, strain localization predictions obtained with several nonlocal kernel formulations were presented, and the relationship between the nonlocal kernels and the solution was analyzed, thereby determining an appropriate nonlocal kernel function for the strain localization problem.

Nonlocal damage evaluation of a sigmoid-based damage model for fibrous biological soft tissues.

The comprehension and modeling of the mechanical behavior of soft biological tissues are essential due to their clinical applications. This knowledge is essential for predicting tissue responses accurately and enhancing our ability to compute the behavior of biological structures and bio-prosthetic devices under specific loading conditions. The current research is centered on modeling the initiation and progression of soft tissues damage, which typically exhibit intricate anisotropic and nonlinear elastic characteristics. For this purpose, the following study presents a comparative analysis of the computational performance of two distinct damage modeling techniques. The first technique employs a well-established damage model, based on a piece-wise exponential damage function as proposed by Calvo et al. (Int J Numer Methods Eng 69:2036-2057, 2007. https://doi.org/10.1002/nme.1825 ). The second approach adopts a sigmoid function, as proposed by López-Campos et al. (Comput Methods Biomech Biomed Eng 23(6):213-223. https://doi.org/10.1080/10255842.2019.1710742 ). The aim of this study is to verify the validity of the López-Campos sigmoid-based damage model to be used in finite element simulation, the implementation of which is unknown. For this proposal, both models were implemented within a commercial Finite Element software package, and their responses to local and non-local damage algorithms were assessed in depth through two standard benchmark tests: a plate with a hole and a ball burst. The results of this study indicate that, for a wide range of cases, such as in-plane stresses, out-plane stresses, stress concentration and contact, all over large displacement conditions, the López-Campos damage model shows a good response to non-local algorithms achieving mesh independence and convergence in all these cases. The results obtained are in line with those obtained for the Calvo's damage model, showing, in addition, larger deformations under in-plane stress and stress concentration conditions and a lower number of iterations under out-plane stress and contact conditions. Consequently, the López-Campos' damage model emerges as a valuable and useful tool in the field of mechanical damage research in biological systems.

Extreme load analysis of flexible wave energy converters utilising nonlocal continuum damage mechanics

In recent years, there has been a notable increase in interest towards Flexible Wave Energy Converters (FlexWECs). These flexible energy harvesters solve structural design challenges faced by rigid-body WECs by responding to external loading by changing shapes. Typically, the structures are made from rubber-like materials which pose few challenges from a material modelling point of view. Firstly, the material is in the finite strain regime requiring a hyperelastic modelling approach, but more critically the material response is expected to change during the operational lifetime. There is softening from both time-dependent viscoelasticity and micro-void growth caused by fatigue loading. The goal of this paper is to understand the latter mechanism and how it manifests within a membrane. To account for this damage accumulation, the gradient-enhanced nonlocal damage model is coupled to a hyperelastic Neo-Hookean constitutive law. The framework has been implemented in the commercial finite element software ABAQUS by exploiting its fully coupled thermo-mechanical formulation. A parametric study is performed on two FlexWEC archetypes: a submerged pressure differential and a floating bulge wave attenuator. The performance evaluation of these devices is carried out by analysing the evolution of the pressure–volume relation and pressure-stretch relation, respectively. The results show that the nonlocal aspects of damage in the pressure differential FlexWECs are small due to membrane action, but the saturation of damage does affect the pressure–volume function of each membrane. However, in the case of attenuator, the damage regularisation plays a crucial role in its behaviour due to the steep stress gradient from the crest of the wave. The outcomes from these analyses suggest FlexWEC design is advantageous from a fatigue loading perspective as it always reaches an equilibrium state which minimises the stress-differential, reducing the likelihood of localised crack growth.

Simulation of hydraulic fracture propagation based on implicit gradient non-local model

Fracture extension simulation is an important tool for studying hydraulic fracture characteristics. To solve the problem of mesh sensitivity of strain and fluid pressure during hydraulic crack propagation, the equivalent strain in the damage evolution equation was non-localized, and implicit gradient was performed. A hydraulic-mechanical coupling model based on implicit gradient non-local damage was established. Numerical simulation program was further developed. Comparison of simulation results with experimental data on ‘L’-brittle material verified the validity of the program. The fracture propagation process of hydraulic fracturing in a perforated well was simulated. On this basis, the influence of internal length scale parameters and mesh size of the non-local model on the calculation results was analyzed. The influence of mesh size on hydraulic fracture morphology and damage variables is reduced by introducing an implicit gradient nonlocal model, which effectively alleviates the localization effect in hydraulic-mechanical coupling damage simulation. This new model achieves the nonlocalization of the fluid field by introducing the relationship among porosity-permeability and deformation and damage, which avoids the mesh sensitivity of equivalent strain, damage, fluid pressure, and improves the numerical stability. It provides a new method for simulating hydraulic fracture propagation and studying the failure behavior of porous media materials.

On the gradient-enhanced damage model for a hyperelastic material

An approach for modeling damage in the material in a geometrically nonlinear setting is studied. We consider a gradient-enhanced damage approach where the damage variable is taken as an independent variable, and the fluxes conjugated to the damage gradient are computed. The concept of maximum dissipation is used to derive the global Kuhn–Tucker conditions. The total free energy function consists of two nonlocal terms corresponding to referential gradient of the nonlocal damage variable and a penalty term to bridge the local damage variable and the nonlocal damage variable. Based on the principle of minimum total potential energy, a coupled system of Euler–Lagrange equations is obtained and solved in weak form. These nonlinear system of equations is solved by a standard Newton–Raphson-type solution scheme. The regularization capabilities, as well as the accuracy and efficiency of the proposed damage models are demonstrated by some standard benchmark examples.

Comparison of local, gradient-enhanced and integral form of continuum damage approaches to strain localization in fiber reinforced composites

Structures often have inserts and notches that cause stress concentration and consequent reduction in the load bearing capacity. The literature on numerical implementation of continuum damage mechanics (CDM) has generally focused on using local theories of damage. However, their use to analyze deformations of notched laminates that are prone to strain localization can cause numerical convergence issues that can be alleviated by employing a non-local theory. There are a few studies using a non-local damage theory for fiber-reinforced polymeric composites (FRPCs) and even fewer for strain localization in FRPCs. The non-local theories employ an equivalent (or effective or von Mises) strain often used in metal plasticity while most theories for studying failure of FRPCs use individual strain/stress components. Here, we compare predictions from local and nonlocal CDM approaches that are effective in studying strain localization in FRPCs. For the local approach, we employ a rate-dependent evolution relation. For the non-local approach, we either compute damage at a point as the weighted sum of damage at its neighbors that introduces a length scale into the problem, or employ a gradient-enhanced approach that also introduces a length scale. It is shown that numerical predictions from both the rate-dependent and the non-local theory alleviate to different degrees sensitivity of results to the finite element mesh used and the strain localization issues. Of these, a local rate-type approach is more convenient and cost effective.

Simulation of the dynamic cracking of brittle materials using a nonlocal damage model with an effective strain rate effect

In this study, a nonlocal multiscale damage model that considers the effect of a strain rate is presented. Moreover, this model is combined with the scaled boundary finite element method (SBFEM) and quadtree mesh for dynamic crack propagation in quasi-brittle materials. The scaling centre of each SBFEM subdomain was selected as the material point, and the bond connecting two material points was referred to as the material bond. Microscopic damage was quantified based on the stretch rate of the material bond, and macroscale topological damage was calculated by taking the weighted average of the microscopic damage over the bonds within the influence domain. The bell distribution function is utilised as the weight function in the calculation of the nonlocal weight function. To account for the impact of the strain rate during dynamic loading, an effective rate R is introduced to enhance multiscale damage by modifying the damage threshold and bond softening coefficient throughout the analysis process. By analysing the energetic degradation function of the phase field, which establishes a connection between the macroscale topological damage and energy-based damage, a nonlocal multiscale damage model was incorporated into the framework of the SBFEM. Quadtree mesh discretization was employed to achieve rapid and high-quality multilevel mesh generation by effectively utilising the hanging nodes. Three typical dynamic cracking examples were simulated using the proposed model, and the results were compared with the relevant experimental results and references. The results show that the method is suitable for accurate dynamic cracking simulation in quasi-brittle materials. Moreover, the results indicate that the proposed method can spontaneously simulate dynamic crack bifurcation without introducing additional crack bifurcation criteria. For a compact tensile test of concrete under different loading rates, the method proposed in this study can accurately simulate the corresponding failure modes. The results also show that without considering the effect of the strain rate, the damage develops rapidly in an unconstrained state, which is inconsistent with the actual case.

Differential geometry-based thermodynamics derivation of isotropic and anisotropic eikonal non-local gradient (ENLG) damage models using a micromorphic media framework

The Eikonal non-local damage (ENL) approach models damage as a space-deforming phenomenon that affects the interaction distance between material points. As damage increases, interactions between points decrease, ultimately resulting in no interaction. In the integral version of such an approach, non-local interaction distances between material points are computed by solving a stationary Hamilton–Jacobi equation with a damage-dependent Riemannian metric. In the implicit gradient version of ENL models, the Riemannian metric figures in the Helmholtz equation to be solved for computing the non-local field controlling damage evolution. However, one of the main criticisms of such formulation is the lack of thermodynamics basis in its derivation. This paper presents a thermodynamics derivation of the Eikonal implicit gradient formulation based on differential geometry concepts to overcome this issue. A free-energy potential is defined considering the non-local strain as a morphological descriptor belonging to the abstract differentiable manifold (where the Riemannian metric is defined). Following a micromorphic media framework, the balance equations of the model are obtained. It is shown that the stress tensor and thermodynamic force associated with damage are the sum of standard and additional non-local contributions. It is also shown that the resulting energy dissipation is always positive, thus verifying the Clausius–Duhem inequality. After presenting all development considering second-order anisotropic continuum damage, the isotropic formulation is obtained as a particular case. Finally, a two-dimensional numerical implementation of an isotropic implicit Eikonal non-local gradient damage model is illustrated. Test cases are simulated to show the relocalization features of the considered formulation and its natural capability of naturally representing damage-to-fracture transition for high damage levels.

Development of a fully automatic damage simulation framework for quasi-brittle materials

This paper proposes a fully automatic analysis framework for static damage analysis of quasi-brittle materials exploiting adaptive mesh refinement approaches as well as adaptive load stepping techniques. The damage process is modelled using an integral-type nonlocal damage model which alleviates the mesh dependence issue. The scaled boundary finite element method and quadtree meshing algorithm are employed in the adaptive analysis. The quadtree algorithm is simple for adaptively remeshing and the mesh refinement is limited to local regions around the damage zones. The elements of the quadtree mesh are modelled by the scaled boundary finite element method, naturally resolving the hanging nodes encountered by the standard finite elements. Furthermore, the regular element shapes and small number of unique element patterns permit a significant reduction of computational cost on processing the element formulations, and an easy implementation of data transfer between two successive meshes. In the last step, the adaptive meshing strategy is combined with an automatic load stepping scheme to create a fully automatic analysis framework. By means of various numerical examples the performance of the proposed method predicting the damage evolution in structures is analysed in terms of its accuracy and efficiency.