Gradient-enhanced Damage Model Research Articles

A computationally efficient gradient-enhanced healing model for soft biological tissues.

Soft biological tissues, such as arterial tissue, have the ability to grow and remodel in response to damage. Computational method plays a critical role in understanding the underlying mechanisms of tissue damage and healing. However, the existing healing model often requires huge computation time and it is inconvenient to implement finite element simulation. In this paper, we propose a computationally efficient gradient-enhanced healing model that combines the advantages of the gradient-enhanced damage model, the homeostatic-driven turnover remodeling model, and the damage-induced growth model. In the proposed model, the evolution of healing-related parameters can be solved explicitly. Additionally, an adaptive time increment method is used to further reduce computation time. The proposed model can be easily implemented in Abaqus, requiring only a user subroutine UMAT. The effectiveness of proposed model is verified through a semi-analytical example, and the influence of the variables in the proposed model is investigated using uniaxial tension and open-hole plate tests. Finally, the long-term development of aneurysms is simulated to demonstrate the potential applications of the proposed model in real biomechanical problems.

Analytical solution of a gradient-enhanced damage model for quasi-brittle failure

This paper presents an approach for analytically solving gradient-enhanced damage (GED) models. The proposed approach provides rigorous proofs for essential phenomena observed in the traditional GED model, including damage widening, characteristic length sensitivity, and stress-locking. The derived cohesive law is a useful technique for accurately determining the material parameters of the GED model, significantly reducing the sensitivity to characteristic length. Furthermore, this study presents an isotropic damage model that considers both tensile and shear failures and can capture complex crack paths. Finally, a series of numerical examples is shown to demonstrate the efficacy of the suggested method.

Finite element modelling of refractories fracture process zone with gradient enhanced damage models

This study investigates the numerical simulation of fracture behaviour in quasi-brittle materials like magnesia spinel refractories using the Gradient-Enhanced Damage (GED) model. It focuses on the complex modelling of these materials non-linear responses and compares conventional and variant GED models through a wedge splitting test. The results demonstrate that all GED models show a good fit to experimental data. However, the conventional GED model falls short in accurately depicting the fracture process zone. In contrast, the localizing GED model more accurately represents the fracture process zone, limiting spurious damage distribution, but requires finer meshing, elevating computational demands. The stress-based variant reduces spurious damage but is less effective comparatively. The study also assesses the role of heterogeneous strength distribution in replicating realistic crack patterns as observed in experiments.

A user material approach for the solution of multi-field problems in Abaqus: Theoretical foundations, gradient-enhanced damage mechanics and thermo-mechanical coupling

The solution of multi-field problems and the numerical implementation by means of the finite element method constitute a sophisticated part of the characterisation of industrial processes. A comprehensive implementation framework for such a system of coupled field equations into a non-linear large strain finite element formulation is provided. The procedure is derived for a micromorphic approach in a thermo-mechanical setting. Although the provided framework contributes to a particular three-field problem it is not limited to a specific application or a specific number of coupled field equations from a conceptual point of view. The solution of the considered system of equations is separated into two coupled domains, with the balance of linear momentum and a balance equation of heat equation-type being solved on each of them. Since both, the balance of micromorphic momentum and the heat balance equation, are partial differential equations of Laplace-type, the resulting two-instance problem can be solved in the framework of commercial finite element software, such as Abaqus, based on a thermo-mechanical user material. To assess the framework for a particular constitutive model, a gradient-enhanced damage model in a thermo-mechanical setting is applied and representative simulation results are discussed. The Abaqus framework is made available as an open-source code on GitHub (https://github.com/InstituteOfMechanics/Thermomechanical_Gradient_Enhanced_Damage_UMAT).

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.

A dynamic description of the smoothing gradient damage model for quasi-brittle failure

Quasi-static simulations are of limited interest because cracks, if they are not severely constrained, propagate dynamically. When natural disasters such as earthquakes or explosions happen, structures made of quasi-brittle or brittle materials can suffer from failures activated by, for instance, loading at a high rate. Dynamic fractures, especially dynamic crack branching, are often observed during those events. We present in this paper, for the first time, a dynamic description of the smoothing gradient-enhanced damage model towards the simulation of quasi-brittle failure localization under time-dependent loading conditions. We introduce two efficient rate-dependent damage laws and various equivalent strain formulations to analyze the complicated stress states and inertia effects of dynamic regime, enhancing the capability of the adopted approach in modeling dynamic fracture and branching. The study is carried out using low-order finite elements, and the merits of the developed approach are examined through our numerical experiments, including mixed-mode fracture and dynamic crack branching simulations.

A Gradient-Enhanced Damage Model with a New Equivalent Strain Based on the Menétrey–Willam Function

A Gradient-Enhanced Damage Model with a New Equivalent Strain Based on the Menétrey–Willam Function

A gradient-enhanced damage model for anisotropic brittle fracture with interfacial damage in polycrystalline materials

This article presents a nonlocal gradient-enhanced damage model that uses direction-dependent damage evolution and interfacial damage to predict transgranular and intergranular cracks in polycrystalline materials at the microstructural level. The distinct grains within the polycrystalline morphology are modeled as anisotropic linear elastic domains with random spatial orientation and cubic symmetries. Transgranular micro-cracks are assumed to occur along specific preferential cleavage planes within each randomly oriented crystal and are described using a bulk damage variable. For intergranular fracture, a smeared description of interface decohesion is incorporated through an interface damage variable which depends on the modified interface kinematics based on a cohesive law that uses a smoothed displacement jump approximation. The coupled system of equations in the proposed computational framework is decoupled using an operator-split methodology to ensure a robust and straightforward computational implementation. Several numerical examples are presented, and simulations are performed on single crystal, bicrystals, and polycrystalline domains to demonstrate the capabilities and validation of the proposed model.

Stochastic failure analysis of proximal femur using an isogeometric analysis based nonlocal gradient-enhanced damage model

Background and objectiveMedical imaging-based finite element methods are more accurate tools for fracture risk prediction than the traditional aBMD based methods. However, these methods have drawbacks like geometric errors, high computational cost, mesh-dependent results, etc. In this article, the authors have proposed an isogeometric analysis-based nonlocal gradient-enhanced damage model to overcome some of these issues. Moreover, there are uncertainties in the values of input parameters for such analysis due to various measurement errors. Hence, stochastic analysis is performed to quantify the effect of these parametric uncertainties on the fracture behavior of the proximal femur. MethodsComputed Tomography images of a patient are used to create a 2D proximal femur model with a heterogeneous description of material properties. A numerical model based on gradient-enhanced nonlocal continuum damage mechanics is used for fracture analysis of proximal femur to overcome the issues related to mesh dependency in traditional continuum damage mechanics models. Further, a multipatch isogeometric solver is developed to solve the governing equations. Monte Carlo simulations are used to understand the effect of parametric uncertainties on the fracture behavior of the proximal femur. ResultsThe developed numerical framework is used to solve the fracture problem of proximal femur under single leg stance loading conditions. The obtained results are validated by comparing the load-displacement response and the crack path with that given in the literature. Stochastic analysis is performed by considering a ±5% variation in the elastic modulus, damage initiation strain, and the neck-shaft angle values. ConclusionThe proposed numerical framework can correctly predict the damage initiation and propagation in a proximal femur. The results reveal that the heterogeneous nature of material properties of bone plays a significant role in determining the fracture characteristics of the proximal femur. Further, the results of the stochastic analysis reveal that the parametric uncertainties in the neck-shaft angle have a much more significant influence on the results of the analysis than the parametric uncertainties in the elastic modulus and damage initiation strain.

A direction-dependent smoothing gradient damage model for anisotropic brittle fracture

We present a novel smeared gradient-enhanced damage model which takes into account direction-dependent damage evolution in two-dimensional brittle materials (e.g., uniaxial fiber-reinforced composites and polycrystalline materials). The recently developed smoothing gradient damage model for isotropic localized failure analysis is revisited by introducing a new damage evolution equation towards anisotropic fracture. To this end, the evolving anisotropic gradient interaction parameter existed in the damage evolution law is redefined by associating with a second-order structural tensor, aiming to possibly drive the directional fracture properties. This structural tensor restrains the crack growth in the direction of predefined fracture plane. In this circumstance, all cleavage or family of cleavage in a grain (for instance in polycrystalline materials) are assumed to be described by a single cleavage plane (or fiber plane) where the resistance of materials is considered to be weakest. Consequently, the directional dependency of cleavage fracture and transgranular fracture are captured precisely. This new definition utilized solely a scalar damage variable is in contrast to most other existing gradient damage models which usually employ multiple damage variables or higher-order gradient terms. Here, we also introduce an effective staggered algorithm aiming to reduce the computational cost, and its desirable features are demonstrated in the numerical experiments for transversely isotropic materials. The predicted crack paths in unidirectional fiber-reinforced composites and polycrystalline materials are in good agreement with reference results derived from other numerical methods.

Failure zone homogenization for modeling damage- and debonding-induced softening in composites including gradient-extended damage at finite strains

Within multiscale modeling strategies for composite materials, standard Hill averaging is widely used for homogenization of repeating unit cells (RUCs). However, Hill’s homogenization approach reaches its limitations at the presence of strain softening, because in the standard averaging sense the representativeness of the considered micro-scale volume is lost. Hence, in order to overcome these limitations, we propose a new failure zone averaging scheme at finite strains, which is also applicable in the softening regime. First, starting from Hill’s assumption of equal virtual work densities on different scales, a novel homogenization approach is derived analytically. Then, the analytical results are verified by numerical simulations evaluating different RUC sizes and accounting for varying geometric realizations in a statistical manner. It is shown that the proposed homogenization scheme yields size-independent stress–strain relations even after localized damage has occurred leading to strain softening within the RUCs. In addition, the material model for the epoxy bulk is valid for the description of damage progression at large deformations using a gradient-enhanced damage model. It turns out that the micromorphic power density – related to the gradient enhancement – of the failure zone gives significant values during the formation of localization while the overall work density is small. Thus, it can be concluded that for accurate failure zone homogenization, the contribution of gradient terms has to be considered. Further, cohesive zone elements are incorporated to consider the decohesion between fibers and matrix. As a result, the stress–strain curves show a lower peak stress and a higher scatter in the stress response compared to samples with perfect interfaces. In contrast, the work density plots show a higher dissipation for the RUC with damaging interfaces.

A note on mixed-mode fracture by the smoothing gradient damage model

ABSTRACT The objective of this technical note is to illustrate twofold. The first issue is devoted to the ability of the recently developed smoothing gradient-enhanced damage model (SGDM), which is associated with the modified von Mises equivalent strain in modelling mixed-mode fracture problems, in particular the common Nooru–Mohamed's test. We show that the developed approach works pretty well for this test, yielding appropriate structural damage response and crack paths for all three specimen sizes, i.e. and . The second issue is, in a similar manner, to discuss the less accuracy of another damage model, the localising gradient damage scheme recently studied by Shedbale et al. Int. J. Mech Sci 2021, 199:106410, in reproducing crack paths and structural responses for the same mixed-mode problem.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

Application of a geometrically nonlinear elastoplastic gradient-enhanced damage model with incremental potential to composite microstructures

The prediction of inelastic processes like plastic deformations and cracks within the microstructure of modern man-made materials by realistic, yet simple and efficient continuum models remains a major task in material modelling. For this purpose, gradient-extended standard dissipative solids represent one of the most promising model classes, which is also formulated and applied in this work to investigate microscopic failure mechanisms in three exemplary three-dimensional composite microstructures. The model combines geometrically nonlinear isotropic elastoplasticity with an isotropic damage model with gradient-extension. For the numerical treatment, a variational constitutive update algorithm based on the exponential map is applied. The model is used to provide insight into the microscopic failure of a brittle woven composite material, a particle-reinforced plastic and a carbon fiber reinforced composite. The influence of different microstructural and material parameters on the overall failure behavior is characterized. Adaptive meshing is used to enable a refined numerical resolution of the cracked regions.

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.

An IGA based nonlocal gradient-enhanced damage model for failure analysis of cortical bone

The fracture behavior of a cortical bone is significantly affected by its hierarchal structure and a high degree of material anisotropy. Recently, continuum damage models have proven to be an effective tool to characterize such kind of material behavior. However, these models suffer from drawbacks such as mesh dependency, spurious damage growth, and incorrect prediction of damage initiation during the initial stage of loading process. To this end, the objective of this work is to develop a nonlocal gradient-enhanced damage model in an isogeometric setting to predict the fracture behavior of a cortical bone. The numerical framework is formulated considering an additional diffusion equation in conjunction with the standard equilibrium equation. The material is assumed to be transversely isotropic to incorporate the anisotropy in damage evolution. The versatility of the proposed framework is tested by solving problems for different modes of fracture. For numerical simulations, samples are taken from different quadrants of bovine cortical bone, and the effect of anisotropy on the damage characteristics of the bone are investigated.

A priorierror estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model

In this paper we investigatea priorierror estimates for the space-time Galerkin finite element discretization of a quasilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of two quasilinear elliptic PDEs which have to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. Numerical experiments are added to illustrate the proven rates of convergence.

Incorporation of localizing gradient-enhanced damage model into layered membrane elements for reinforced concrete structures subject to in-plane loading

Implementation of the localizing gradient-enhanced concrete model and the Menegotto-Pinto steel model into eight-node quadrilateral layered membrane elements for reinforcement concrete (RC) members subject to in-plane quasi-static loading is presented and evaluated. This RC element circumvents the pathological mesh-sensitivity of continuum damage models and the spurious damage growth of standard gradient damage models. Drilling DOFs are adopted, which improve the smoothness of strain distributions and aid in numerical convergence. The layered cross-section consists of fully bonded/smeared steel reinforcement layers and concrete layers. Different properties over the thickness can be easily assigned to each layer to account for heterogeneity. Several benchmark tests are simulated to demonstrate the versatility and effectiveness of the RC element. It is highlighted that all analyses are performed with the fully implicit incremental-iterative Newton-Raphson algorithm. Consequently, the RC element is suitable for modeling quasi-static behaviors of RC members, especially those coupled with other physical processes (creep, shrinkage, corrosion, etc.).

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.

Simulating 3D crack propagation with XFEM to investigate failure mechanism in high strength concrete

AbstractHigh strength concrete compared to normal strength concrete has a denser micro‐structure, i.e. tight packaging or bonding between glued aggregates and cement paste, which increases strength and durability. The disproportionate ratio in ingredients of concrete, water to cement ratio and binding agents to form cement paste are subjected to self‐desiccation during the hydration process which leads to shrinkage, that has potential to form micro‐cracks and delamination within the ITZ (Interfacial Transition Zone), which ultimately causes deterioration in health of the structure. In order to investigate this phenomenon, a computational model to simulate the formation and propagation of cracks within the micro‐structure is developed. The geometrical model for the computation and simulation of the micro‐structure is extracted by means of computer tomography (CT) scans of concrete samples. The eXtended Finite Element Method (XFEM) in combination with level set techniques is employed to simulate discrete cracks and their advancement within the micro‐structure. The criterion for the crack propagation is obtained from a gradient enhanced damage model and once it is fulfilled, the crack geometry is updated using an advance algorithm for level sets.