Nonlocal Damage Model Research Articles

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.

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.

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.

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 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.

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.

A non-local damage model-based FFT framework for elastic-plastic failure analysis of UD fiber-reinforced polymer composites

A novel computational framework combining fast Fourier transform (FFT) with the non-local damage model is proposed to simulate the nonlinear mechanical behaviors of unidirectional composites. Integral-type non-local damage model and gradient-regularized non-local damage model are integrated to analyze brittle materials and ductile materials, respectively. Under transverse tensile load, the failure of matrix in the composites is characterized by non-local maximum principal stress damage model or the elastic-plastic damage model obeying the parabolic yield criterion, respectively. It is verified that predicted stiffness, strength, and ultimate crack morphologies obtained by the proposed model are in good agreement with traditional finite element method (FEM) and phase field method (PFM) in the literature. The effects of microstructure features on the mechanical behaviors and the suitability of multiple crack propagation are further studied effectively and accurately.

Fatigue crack growth: Validation of the Kmax-ΔK approach using the GTN damage model

Considering cyclic plastic deformation as the single damage mechanism behind the fatigue crack growth (FCG), the effect of the mean stress is null in the absence of crack closure. In this study, an additional damage mechanism was introduced, predicting the effects of the growth, nucleation and coalescence of micro-voids. Using a non-local damage model, the reverse plastic zone was identified as the fatigue process zone, allowing to establish a relation to calculate its size. Considering aluminium alloy 2024-T351, the predictions are in close agreement with the experimental da/dN-ΔK curves, for distinct stress ratios. The results highlight that the effect of Kmax on the FCG rate is small in the absence of crack closure. On the other hand, modelling the contact between the crack flanks, the effect of Kmax on the FCG rate is amplified, i.e. the ΔK-Kmax plots for a given da/dN presented a Kmax dominance zone. The developed model suggests a three-parameter approach: ΔKeff -ΔK-Kmax.

An implicit stress update algorithm for the plastic nonlocal damage model of concrete

The numerical implementation of advanced plastic damage models has been challenging due to the diverse computational difficulties caused by material nonlinearity. In this study, firstly, a plastic damage model is developed by a nonsmoothed Mohr–Coulomb yield criterion. The nonorthogonal flow rule without plastic potential function is employed to describe the dilatancy behavior of concrete. The stiffness degradation is considered through two tensile and compressive damage variables. Subsequently, an implicit stress update algorithm incorporating several advanced numerical methods is proposed to tackle difficulties arising in the numerical calculation of the plastic damage model. In the model implementation, the Mohr–Coulomb yield function with corners is replaced by a single smooth expression to ensure the stability of the numerical calculation. A line search method is used to solve nonlinear systems of integral stress equations to obtain better convergence than Newton method. The complex step derivative approximation is used to evaluate the consistent tangent operator to provide quadratic convergence speed of the global equilibrium iterations, thus avoiding tedious analytical derivative operations. The integral nonlocal method is also used to regularize the finite element solution. At the material point scale, the ability of the proposed model to describe the complex mechanical behavior of concrete is verified by utilizing nine sets of monotonic and cyclic loading and unloading tests. Finally, the algorithm and model’s performance is well tested by the failure analysis of five numerical examples. The code of this work will be freely available at https://github.com/zhouxin615/PD_model.

A new implicit gradient damage model based on energy limiter for brittle fracture: Theory and numerical investigation

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

A unified non-local damage model for hydraulic fracture in porous media

A unified non-local damage model for hydraulic fracture in porous media

A 2D/3D implicit gradient-enhanced nonlocal meso-scale damage model for deformation and fracturing of brittle materials

An implicit gradient-enhanced nonlocal meso‑scale damage model is developed to investigate deformation and fracturing of brittle materials. In the model, an implicit gradient integration scheme is adopted to deal with mesh dependency and strain localization. The mechanical parameters are assigned randomly using a Weibull statistical distribution to reflect the material heterogeneity at the meso‑scale. A linear elastic damage constitutive law with residual strength is used to describe the stress-strain relationship at the mesoscale and the Mohr-Coulomb criterion as well as the tensile stress criterion are used to evaluate the damage of the elements in tension or shear. The local equivalent strain is replaced by a nonlocal description employing the non-local implicit gradient model. Benchmark tests are conducted to document the potential of the model in simulating crack initiation and propagation processes in brittle materials. The benchmark tests also demonstrate that the proposed non-local damage model can effectively achieve strain localization and eliminate mesh dependency to some extent.

Experimental and numerical study of tensile behavior of carbon nanotube reinforced glass-epoxy composite: The multiscale approach

In this paper, a multiscale approach is presented for strength and tensile modulus prediction of carbon nanotube (CNT) reinforced glass-epoxy composite. At the micro-scale, the three-dimensional representative volume element is obtained by randomly orienting the carbon nanotube in the resin according to the weight percentage. The interphase of the nanotube and resin is modeled using a cohesive element method. The behavior of the resin is accounted with a quasi-brittle non-local damage model. At the Mesoscale, the representative volume element is estimated by using the weight percentage of glass fiber and resin generalized with nanotubes. Finite element analyses were conducted to estimate the effective mechanical properties through numerical homogenization. The effect of different percentages of carbon nanotubes on tensile behavior was investigated experimentally and numerically. The results show that adding carbon nanotubes improves the tensile modulus by about 25% and the tensile strength by about 53%. In the numerical study, the strength of the composite is predicted with good accuracy by modeling the damage of resin, glass fibers, and fracture toughness mechanisms of carbon nanotubes such as bridging and pull-out of carbon nanotubes. A good agreement was observed between the simulation results and the experimental results at different percentages of carbon nanotubes.

Identification of Constitutive Parameters for the Non-Local Damage Model of Soft Biological Tissues

Identification of Constitutive Parameters for the Non-Local Damage Model of Soft Biological Tissues

On a computational stress-based non-local damage model for quasi-brittle composites

Numerical models for the stress and strain analysis of quasi-brittle composites, such as cement-based ones, supplied by various stiffening particles, under mechanical, thermal, etc. loads should handle i) the creation of certain micro-damaged zones, antecedent to ii) the initiation and propagation of a system of macroscopic cracks. Whereas ii) can be analysed using some extended, generalized or similar finite element technique (XFEM, GFEM, etc. algorithms), i) must rely on a smeared crack formulation, whose regularization properties are derived from a non-local stress evaluation. This paper studies mathematical and computational properties of such model, based on the Eringen's approach, useful in many applications in civil engineering.

Non-local continuum damage model for poro-viscoelastic porous media

We present a novel poro-damage-viscoelastic model for predicting the failure response of fluid-saturated porous geomaterials. The Generalized Maxwell model is introduced for the representation of the viscoelastic behavior of the solid skeleton, which is achieved by a standard Prony-series type expansion. Damage regularization is obtained by an non-local integral-type formulation and damage behavior is described by Mazars model with the modified von Mises-type equivalent strain measure. The poromechanics parameters (Biot’s coefficient, Biot’s modulus) are functions of damage, and the fluid flow obeys Darcy’s seepage law in the entire domain, while the permeability is assumed to be anisotropic and strain dependent. The coupled system is discretized in time using a backward Euler scheme. The non-linear system is linearized using a Newton Raphson scheme and solved monolithically every time step. A consistent Jacobian matrix and residual vector are derived analytically. Several numerical examples are studied in order to investigate the performance of the proposed approach, including (i) a column undergoing hysteresis from cyclic loading, stress relaxation, creep and variable strain rate loading tests and (ii) fluid-driven fracturing in a 2D poro-viscoelastic domain. The numerical time-dependent results exhibit mesh insensitivity for all field variables, and confirm the feasibility and applicability of the proposed non-local damage model for simulating hydraulic fracture. • A non-local damage model for poro-viscoelastic saturated porous media is proposed. • Von-Mises strain measure is used to drive the non-local integral type damage growth. • Damage-response includes nonlinear Biot’s coefficients and anisotropic permeability. • A mixed FEM solution is developed, including a monolithic iterative solution. • Analytically derived Jacobian is used to drive the non-linear solution of equations. • Benchmark problems include poro-viscoelastic column and hydraulic fracture problems. • Objective mesh-independent and oscillation free numerical results are presented. • The model has the potential to represent time-dependent damage in several rock types.

A method for introducing mode II dissipation energy into mixed fracture analyses

AbstractThis paper presents a methodology to consider the mode II fracture energy density in those analyses in which the phenomenon of mixed fracture occurs. The method consists of using a fracture energy density calculated as the weighted average of the fracture energy densities in mode I and mode II, the weighting factors being the relative importance of the mode I and II deformations with respect to the total deformation. This methodology has been implemented in local damage models using the smeared crack approximation but can be implemented in more complex models such as nonlocal damage models. Since there is no specific experimental test to measure the mode II fracture energy density, this method allows this parameter to be measured indirectly by adjusting its value so that the numerical equilibrium curve matches the experimental results.

Numerical insights on quasi-static behaviors of UHPC-NC composite members by a phase-field approach enhanced with a cohesive-frictional interface model

The risk of cracking/debonding of UHPC-NC composite members, either for repairing/strengthening or newly-built structures, is an issue of great concern by the engineering community. Distinct failure modes are continuously reported in the literature and among them, the complex interactions between matrix (UHPC and NC) cracking and interfacial debonding play a vital role. From modeling aspects, the progressive damage analysis of UHPC-NC composite members is a rather challenging task, especially when complicated cracking scenarios arise. To address this shortfall, this work proposes a novel numerical framework for UHPC-NC members by combining: (i) a nonlocal damage model based on the phase-field approach for matrix cracking of quasi-brittle materials, and (ii) a coupled cohesive-friction model for debonding of UHPC-NC interface. Variational formalization of the method provides a simple basis for the finite element implementation and complex cracking tracking procedures are circumvented. The strengths of the model are explored through numerical investigations of several quasi-static behaviors of UHPC-NC composite members. Their damage evolutions and failure patterns are accurately captured by the model. For the sake of promoting relevant research, the source codes are disclosed to potential users.