Progressive damage in brittle heterogeneous materials produces at the macroscopic level strain softening from which theoretical difficulties arise (e.g. ill-posedness and multiple bifurcation points). This characteristics behavior favours spurious strain localization in numerical analyses and calls for the implementation of localization limiters, for instance nonlocal damage constitutive relations. The issue of possible (stable or unstable) equilibrium paths, multiple localization zones, and of the detection of bifurcation points has, however, never been addressed in the context of nonlocal constitutive laws. We extend here the eigenmode analysis and perturbation method proposed by De Borst to the study of the bifurcation and post-bifurcation response of discrete nonlocal strain softening solids. Numerical applications on beams show that bifurcation and instability may occur in the post-peak regime. As opposed to the case of local constitutive relations, the number of possible solutions at a bifurcation point is restricted due to the constraint introduced by the localization limiter and these solutions are shown to be mesh independent.
Nonlocal damage models are typically used to model failure of quasi-brittle materials. Due to brittleness, the choice of a particular model or set of parameters can have a crucial influence on the structural response. To assess this influence, it is essential to keep finite element discretization errors under control. If not, the effect of these errors on the result of a computation could be erroneously interpreted from a constitutive viewpoint. To ensure the quality of the FE solution, an adaptive strategy based on error estimation is proposed here. It is based on the combination of a residual-type error estimator and quadrilateral h-remeshing. Another important consequence of brittleness is that it leads to structural responses of the snap-through or snap-back type. This requires the use of arc-length control, with a definition of the arc parameter that accounts for the localized nature of quasi-brittle failure. All these aspects are discussed for two particular nonlocal damage models (Mazars and modified von Mises) and for two tests: the Brazilian tensile splitting test and the single-edge notched beam test. For the latter test, the capability of the Mazars model to capture the curved crack pattern observed in experiments – a topic of debate in the literature – is confirmed.
This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method (XFEM). The paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a FEM framework. The upper bound property requires the recovered solution to be equilibrated and continuous. The proposed technique consists of using a recovery technique, especially adapted to the XFEM framework that yields equilibrium at a local level (patch by patch). Then a postprocess
based on the partition of unity concept is used to obtain continuity. The result is a very accurate but only nearly-statically admissible recovered stress field, with small equilibrium defaults introduced by the
postprocess. Sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults, as demonstrated by the numerical tests.
An adaptive strategy for nonlinear finite-element analysis, based on the combination of error estimation and h-remeshing, is presented. Its two main ingredients are a residual-type error estimator and an unstructured quadrilateral mesh generator. The error estimator is based on simple local computations over the elements and the so-called patches. In contrast to other residual estimators, no flux splitting is required. The adaptive strategy is illustrated by means of a complex nonlinear problem: the failure analysis of a single-edge notched beam. The quasi-brittle response of concrete is modelled by means of a nonlocal damage model.
We present an efficient and reliable approach for the numerical modelling of failure with nonlocal damage models. The two major numerical challenges––the strongly nonlinear, highly localized and parameter-dependent structural response of quasi-brittle materials, and the interaction between nonadjacent finite elements associated to nonlocality––are addressed in detail. Reliability of the numerical results is ensured by an h-adaptive strategy based on error estimation. We use a residual-type error estimator for nonlinear FE analysis based on local computations, which, at the same time, accounts for the nonlocality of the damage model. Efficiency is achieved by a proper combination of load-stepping control technique and iterative solver for the nonlinear equilibrium equations. A major issue is the computation of the consistent tangent matrix, which is nontrivial due to nonlocal interaction between Gauss points. With computational efficiency in mind, we also present a new nonlocal damage model based on the nonlocal average of displacements. For this new model, the consistent tangent matrix is considerably simpler to compute than for current models. The various ideas discussed in the paper are illustrated by means of three application examples: the uniaxial tension test, the three-point bending test and the single-edge notched beam test.
A new non-local damage model is presented. Non-locality (of integral or gradient type) is incorporated into the model by means of non-local displacements. This contrasts with existing damage models, where a non-local strain or strain-related state variable is used. The new model is very attractive from a computational viewpoint, especially regarding the computation of the consistent tangent matrix needed to achieve quadratic convergence in Newton iterations. At the same time, its physical response is very similar to that of the standard models, including its regularization capabilities. All these aspects are discussed in detail and illustrated by means of numerical examples.
A well-controlled and minimal experimental scheme for dynamic fracture along weak planes is specifically designed for the validation of large-scale simulations using cohesive finite elements. The role of the experiments in the integrated approach is two-fold. On the one hand, careful measurements provide accurate boundary conditions and material parameters for a complete setup of the simulations without free parameters. On the other hand, quantitative performance metrics are provided by the experiments, which are compared a posteriori with the results of the simulations. A modified Hopkinson bar setup in association with notch-face loading is used to obtain controlled loading of the fracture specimens. An inverse problem of cohesive zone modeling is performed to obtain accurate mode-I cohesive zone laws from experimentally measured deformation fields. The speckle interferometry technique is employed to obtain the experimentally measured deformation field. Dynamic photoelasticity in conjunction with high-speed photography is used to capture experimental records of crack propagation. The comparison shows that both the experiments and the numerical simulations result in very similar crack initiation times and produce crack tip velocities which differ by less than 6%. The results also confirm that the detailed shape of the non-linear cohesive zone law has no significant influence on the numerical results.
The conceptual simplicity and the ability of cohesive finite element models to describe complex fracture phenomena makes them often the approach of choice to study dynamic fracture. These models have proven to reproduce some experimental features, but to this point, no systematic study has validated their predictive ability; the difficulty in producing a sufficiently complete experimental record, and the intensive computational requirements needed to obtain converged simulations are possible causes. Here, we present a systematic integrated numerical–experimental approach to the verification and validation (V&V) of simulations of dynamic fracture along weak planes. We describe the intertwined computational and the experimental sides of the work, present the V&V results, and extract general conclusions about this kind of integrative approach.