Nonlocal Damage Variable Research Articles

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.

Simulation of hydrogen embrittlement of steel using mixed nonlocal finite elements

A new simulation strategy to model hydrogen embrittlement based on a multi-field finite element using displacements, pressure, volume variation, nonlocal damage variables and lattice hydrogen concentration as unknowns is proposed. The material is described using a modified GTN model, which includes the description of hydrogen enhanced decohesion (HEDE). The finite element problem is solved using a fully implicit formulation. Numerical problems related to volumetric locking are solved using a mixed pressure/volume variation formulation. The use of the mixed formulation allows a straightforward evaluation of the pressure gradient, which drives hydrogen diffusion. Mesh size dependence is solved using an implicit gradient nonlocal formulation. Two variables are used to represent damage: the plastic volume variation and the accumulated plastic strain, controlling nucleation. The model is used to simulate an existing experimental database (Moro et al., 2010; Briottet et al., 2012) including tensile, fracture toughness and pressurized disk tests. The model, after adjusting the various coefficients, can represent the main experimental findings: the effect of deformation rate on the failure of tensile specimens, the transition from surface to internal fracture with increasing deformation rate, the sharp toughness drop under hydrogen (CT specimen), the fracture location and the effect of the pressurization rate in the case of the tests on disks.

Advances in computational dynamics for inelastic continua with anisotropic material behavior: Formulation and numerical implementation of inelastic ductile behavior of spruce wood

A new material model for wood is proposed and analyzed in this contribution. It is focused on anisotropic ductile failure at large strains and high strain-rates. The anisotropy of the ductile failure is represented by a multi-surface yield criterion coupled to anisotropic damage, which represents the anisotropic material behavior of wood due to its fibre orientation. The rate-dependency of the approach is modelled by visco-plasticity for large strain-rates in order to enable its application to the simulation of experiments at high loading velocities, e.g. impact and drop tests. To account for high strain-rates requires transient simulations of the nonlinear dynamic response of the structure. Moreover, a novel time integration method based on a three point backward Euler scheme is introduced. The localization of strain and mesh dependent results arising from simulations with softening formulations are overcome by gradient enhancement of the damage formulation. To this end, three nonlocal damage variables are introduced as additional degrees of freedom, which results in a coupled system of partial differential equations for displacement and damage variables. The material model is applied to the simulation of experiments on wood in a Split Hopkinson Pressure Bar experiment and in a drop weight test.

An FFT-based spectral solver for interface decohesion modelling using a gradient damage approach

This work presents a fast Fourier transform (FFT) based method that can be used to model interface decohesion. The inability of an FFT solver to deal with sharp interfaces discards the use of conventional cohesive zones to model the interfacial mechanical behaviour within this framework. This limitation is overcome by approximating sharp interfaces (e.g. grain/phase boundaries) with an interphase. Within the interphase, the background plastic constitutive behaviour is inherited from the respective adjacent grains. The anisotropic kinematics of the decohesion process is modelled using a damage deformation gradient that is constructed by mapping the opening strains (in normal and tangential modes) to the associated projection tensors. The degradation (damage) of the interfacial opening resistances is modelled via a dimensionless nonlocal damage variable that prevents localisation of damage within the interphase. This nonlocal variable results from the solution of a gradient damage based regularisation equation within the interphase subdomain. The damage field is constrained to the interphase region by applying a relatively large penalisation on the damage gradients inside the interphase. The extent of nonlocality ensures that the damage is largely uniform in the direction perpendicular to the interphase, thus making its thickness the theoretical lengthscale for dissipation. To achieve model predictions that are objective with respect to the interphase thickness, scaling relations of the model parameters are proposed. The numerical performance is shown for a uniform opening case and then for a propagating crack. Finally, an application to an artificial polycrystal is shown.

On the implementation of finite deformation gradient-enhanced damage models

We introduce a comprehensive framework for the efficient implementation of finite deformation gradient-regularised damage formulations in existing finite element codes. The numerical implementation is established within a thermo-mechanically fully coupled finite element formulation, where the heat equation solution capabilities are utilised for the damage regularisation. The variationally consistent, gradient-extended and geometrically non-linear damage formulation is based on an overall free energy function, where the standard local free energy contribution is additively extended by two non-local terms. The first additional term basically contains the referential gradient of the non-local damage variable. Secondly, a penalty term is added to couple the local damage variable—the evolution of which is governed by an ordinary differential equation—and the non-local damage field variable that is governed by an additional balance equation of elliptic type.

A meshless approach to non-local damage modelling of concrete

A non-linear continuum damage mechanics model for concrete constructions is analysed using a radial point interpolation meshless method (RPIM). The fundamental mathematical relations and the material model are fully characterized. The 2D plane stress RPIM formulation is extended to a rate-independent standard (local) damage model considering both tension and compression static states. Additionally, in this work, the local damage formulation is modified considering a non-local constitutive damage criterion with regard to a Helmholtz free energy potential. Here, the internal variational fields, such as local and non-local damage variables, are determined by a return-mapping damage algorithm. Due to the non-linear nature of the phenomenon, a displacement controlled Newton-Raphson iterative approach is adopted to attain the non-linear damage solution. In the end, the performance of the proposed non-local damage model is evaluated using an experimental test of a notched-three-point bending beam available in the literature. The obtained solution shows that the meshless methods are capable to effectively analyse concrete structures assuming a non-linear non-local continuum damage model.

A computational framework for modelling damage-induced softening in fibre-reinforced materials – Application to balloon angioplasty

A computational framework for modelling damage-induced softening in fibre-reinforced materials is presented. The main aspect of this framework is the proposed non-local gradient-enhanced continuum damage formulation. At the material level, the elastic constitutive behaviour is defined by a hyperelastic functional including a volumetric and an isochoric contribution. The isochoric contribution is subdivided into three contributions associated to three different phases i=0,1,2. Phase 0 is represented by an incompressible neo-Hookean material, whereas phases 1 and 2 are represented by an exponential format that accounts for the stretching along two preferred anisotropy directions, i.e. two fibre families. Furthermore, a 1−di–type damage function, is introduced to reproduce the loss of stiffness in each phase i. Following the ideas discussed in (Dimitrijević and Hackl, 2008; Waffenschmidt et al. 2014) and references cited therein, the model is built around the enhancement of the local free energy function by means of terms that contain the referential gradients of the non-local damage variables ϕi. The inclusion of these terms ensures an implicit regularisation of the finite element implementation. A finite element implementation of the non-local gradient-enhanced continuum damage model is presented. To this end we develop an 8-noded Q1Q1P0 hexahedral element following a variational approach, in order to efficiently model the quasi-incompressible behaviour of the hyperelastic material. This element is implemented in Abaqus by means of a user subroutine UEL. Three boundary value problems are studied: an anisotropic plate with a hole, a balloon angioplasty and a full-3D artery-like tube. These computational experiments serve to illustrate the main capabilities of the proposed model.

A unilateral nonlocal tensile damage model for masonry structures

In the present paper, a constitutive nonlocal damage model is proposed for the non-linear incremental finite element analysis of masonry structures. The mechanical model is based on the assumptions of linear elasticity under compression and softening behaviour under tension, described by the adoption of a unique strain-driven nonlocal damage variable. Specifically, non-locality of the integral type is introduced in order to prevent spurious strain localization. It can be noted that the unilateral nature of the model is suitable to contemplate both diffused macro-cracks induced by the tensile damage process and the stiffness recovery in the transition from tension to compression, considering the anisotropy induced by the damage process as well. This is performed by realizing a decomposition of the strain tensor in its positive and negative components, and accounting for stiffness degradation only along tensile direction. The assumption of a linear elastic behaviour in compression is motivated by the fact that the main interest of the model is represented by investigating the response of masonry structures under service loads, condition in which very low compressive states are usually predominant. Consequently, the number of constitutive parameters is more limited with respect to other models that include a damage criterion also in compression. Finally, the validation of the proposed damage model is carried out with reference to a plane problem, in order to check the capability of the model to treat damage in an anisotropic way as well as the almost null dependence of the results on the discretization.

A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials

A non-local gradient-based damage formulation within a geometrically non-linear setting is presented. The hyperelastic constitutive response at local material point level is governed by a strain energy which is additively composed of an isotropic matrix and of an anisotropic fibre-reinforced material, respectively. The inelastic constitutive response is governed by a scalar [1–d]-type damage formulation, where only the anisotropic elastic part is assumed to be affected by the damage. Following the concept in Dimitrijević and Hackl [28], the local free energy function is enhanced by a gradient-term. This term essentially contains the gradient of the non-local damage variable which, itself, is introduced as an additional independent variable. In order to guarantee the equivalence between the local and non-local damage variable, a penalisation term is incorporated within the free energy function. Based on the principle of minimum total potential energy, a coupled system of Euler–Lagrange equations, i.e., the balance of linear momentum and the balance of the non-local damage field, is obtained and solved in weak form. The resulting coupled, highly non-linear system of equations is symmetric and can conveniently be solved by a standard incremental-iterative Newton–Raphson-type solution scheme. Several three-dimensional displacement- and force-driven boundary value problems—partially motivated by biomechanical application—highlight the mesh-objective characteristics and constitutive properties of the model and illustratively underline the capabilities of the formulation proposed.

Efficient numerical integration of an elastic–plastic damage law within a mixed velocity–pressure formulation

This study focuses on numerical integration of constitutive laws in numerical modeling of cold materials processing that involves large plastic strain together with ductile damage. A mixed velocity–pressure formulation is used to handle the incompressibility of plastic deformation. A Lemaitre damage model where dissipative phenomena are coupled is considered. Numerical aspects of the constitutive equations are addressed in detail. Three integration algorithms with different levels of coupling of damage with elastic–plastic behavior are presented and discussed in terms of accuracy and computational cost. The implicit gradient formulation with a non-local damage variable is used to regularize the localization phenomenon and thus to ensure the objectivity of numerical results for damage prediction problems. A tensile test on a plane plate specimen, where damage and plastic strain tend to localize in well-known shear bands, successfully shows both the objectivity and effectiveness of the developed approach.

A nonlocal damage model for mild steel under inelastic cyclic straining

A material model for mild construction steel is developed in order to analyse damage in 3D-steel structures under inelastic cyclic straining as it may occur during earthquakes. The constitutive equations consider viscoplastic material behavior, isotropic and kinematic hardening, softening, and isotropic ductile damage. For compression the evolution equation for ductile damage is disabled and the impact of damage on the material stiffness is reduced. The phenomenon of strain localization and its mesh-independent modelling is taken into account by a nonlocal implicit gradient enhanced extension provided by an additional partial differential equation for the nonlocal damage variable. The global structural stability is evaluated by a global damage indicator.

Nonlocal damage‐viscoplastic model for high temperature creep of single crystal superalloys

AbstractA new nonlocal damage‐viscoplastic model for high temperature creep of single crystal superalloys is developed. It is based on the variational formulation consisting of free energy, plastic and damage dissipation potentials. Evolution equations for plastic strain and damage variables are derived from the minimum principle for dissipation potentials [1]. The model is capable of describing different stages of creep in a unified way. The evolution of dislocation densities of gamma and gamma prime phases in superalloys incorporates plastic deformation. It results in the time‐dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage variables. Herein the nonlocal one is considered as numerical treatment to remove mesh‐dependence. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A new mesh-independent Rousselier's damage model: Finite element implementation and experimental verification

Numerical analyses based on local damage models are often found to depend on the spatial discretisation (i.e., mesh size of the numerical method used). The growth of damage tends to localize in the smallest band that can be captured by the spatial discretisation. As a consequence, increasingly finer discretisation grids can lead to crack initiation earlier in the loading history and to faster crack growth. The reason behind this non-physical behaviour is the loss of ellipticity of the set of partial differential equations, which govern the rate of deformation locally at a certain level of accumulated damage. Displacement discontinuities and damage rate singularities can be avoided by adding nonlocality to the damage model. The enhanced continuum description which is thus obtained results in smooth damage fields, in which the localization of damage is limited to the length scale introduced by the averaging. In this work, a new nonlocal form of Rousselier's damage model has been developed by introducing an additional partial differential equation (diffusion type) for the nonlocal damage variable in terms of the ductile void volume fraction. The diffusion equation has been discretised along with the stress equilibrium equation of the mechanical continuum using finite element (FE) method. The nonlocal damage variable has been used as an additional degree of freedom in the FE model. Several example problems have been solved to demonstrate the mesh-independent nature of the new nonlocal formulation.

Gradient-enhanced Coupled Plasticity-anisotropic Damage Model for Concrete Fracture: Computational Aspects and Applications

It is widely studied that classical continuum damage theory for concrete fracture exhibits an extreme sensitivity to the spatial discretization in the finite element simulations. This sensitivity is caused by the fact that the mathematical description becomes ill-posed at a certain level of accumulated damage. A well-posed problem can be recovered by using a gradient-enhanced damage model in which a material length scale is introduced as a localization limiter. In this work, a nonlocal gradient-enhanced fully coupled plastic-damage constitutive model for plain concrete is developed. Anisotropic damage with a plasticity yield criterion and a damage criterion are introduced to be able to adequately describe the plastic and damage behavior of concrete. In order to account for different effects under tensile and compressive loadings, nonlocal damage variables that account for the progressive degradation of mechanical properties under stress states of prevailing tension and compression and two internal length scales, one for tension and the other for compression, are introduced as localization limiters. Therefore, two nonlocal damage criteria are used: one for compression and a second for tension such that the total stress is decomposed into tensile and compressive components. In order to solve the time step problem, a decoupled elastic predictor and plastic corrector steps are performed first in the effective configuration where damage is absent, and then a nonlocal damage corrector step is applied in order to update the final stress state. The algorithmic treatment of both tension and compression is presented in a unified way. A simple procedure to calculate the gradient of the tensile/compressive damage variables is described which can be used directly without the need of intensive numerical modifications of an existing finite element code. The effectiveness of the proposed local model has been demonstrated in both uniaxial and biaxial tension and compression problems and compared with experimental data. Numerical results obtained with the proposed nonlocal model are compared with experimental results concerning bending of three-point notched and four-point notched concrete beams. As the mesh is refined, convergence of numerical results is observed both in terms of damage patterns and of the global response.

Local and Global Damages of Quasi-Brittle Material in Uniaxial Compression Based on Gradient-Dependent Plasticity

Theoretical relations among local and nonlocal damage variables in localized band as well as global damage variable according to the measured stress-strain curve of quasi-brittle material subjected to shear and tensile failures in uniaxial compression and in three-point bending were presented. The nolocal damage variable depends on the local counterpart and its second spatial gradient based on nonlocal theory. Analytical solution for the local damage variable was derived by substitution of the nonlocal damage variable for averaged damage variable in shear band and by using boundary condition as well as by resorting to the assumption that the actual thickness of shear band corresponds to the maximum local damage variable. It is found that local damage variable depends on the internal length parameter, coordinate, flow shear stress, shear strength, shear elastic and softening moduli. Assuming that the shear localization is initiated just at the peak strength, and that afterwards the strain-softening behavior of specimen occurs. The relation among shear strength, uniaxial compressive strength, flow shear and compressive stresses was established. The global damage variable was defined according to the measured stress-strain curve whether the post-peak response is snap-back or snap-through, which depends on the uniaxial compressive strength and flow compressive stress. A relation among local, nonlocal, and global damage variables in uniaxial compression was proposed analytically, and then a relation applicable to three-point bending was directly presented. The latter relation can be simplified for uniaxial tension condition if tensile stress is assumed to be uniform. Two examples were presented to investigate the two- and three-dimensional distributions of local damage variable in shear and tensile localized bands. The present analytical solutions for the distributed damage in shear and tensile localized bands qualitatively agree with the previous numerical predictions.

A gradient model for finite strain elastoplasticity coupled with damage

This paper describes the formulation of an implicit gradient damage model for finite strain elastoplasticity problems including strain softening. The strain softening behavior is modeled through a variant of Lemaitre's damage evolution law. The resulting constitutive equations are intimately coupled with the finite element formulation, in contrast with standard local material models. A 3 D finite element including enhanced strains is used with this material model and coupling peculiarities are fully described. The proposed formulation results in an element which possesses spatial position variables, nonlocal damage variables and also enhanced strain variables. Emphasis is put on the exact consistent linearization of the arising discretized equations. A numerical set of examples comparing the results of local and the gradient formulations relative to the mesh size influence is presented and some examples comparing results from other authors are also presented, illustrating the capabilities of the present proposal.