Integral-type Nonlocal Model Research Articles

Peridynamic modeling of delayed fracture in electrodes during lithiation

Material failure caused by diffusion-induced stress is one of the main obstacles to fulfill the potential of electrodes in lithium-ion batteries. The conventional numerical methods find difficulty in simulating material singularities such as cracks. In contrast, the peridynamic theory is an integral-type nonlocal model that allows fracture to be treated as a natural part of the solution process. Here we have developed a concurrently-coupled and multi-physical model considering lithium-ion diffusion, mechanical deformation, and crack growth based on peridynamics to simulate delayed fracture in electrodes under mechanical and chemical loads. The ability and the advantage of the model are demonstrated in the following simulations: (1) In the process of lithiation, lithium ions tend to accumulate around the crack tip due to the concentration of the stress field. The lithium-ion concentration in turn reduces the tensile stress and the bond stretch. Further, the lithium reaction causes embrittlement of the host materials. (2) The crack propagates in a “wait and go” fashion of delayed fracture in which the steady state of “wait” experiences the transport and supply of lithium ions at the crack tip and the rapid crack propagation of “go” depends on the lithium embrittlement of the host materials. (3) The model with double-sided pre-crack reveals that the cracks can deflect or even intersect due to the interference of the stress fields around the crack tips in the intermittent cracking process. The work lays a foundation for corrosive fracture and provides a useful tool for the future design of lithium-ion batteries.

Nonlocal dynamic damage modelling of quasi-brittle composites using the scaled boundary finite element method

In this contribution, a nonlocal approach is proposed to model the dynamic damage of quasi-brittle composites within the framework of the scaled boundary finite element method (SBFEM). The image-based quadtree decomposition is used to automatically achieve multi-level mesh of a domain with geometric as well as material discontinuities. By enforcing a 2:1 balanced condition, only limited types of S-domains are used. Owning to the salient capability of the SBFEM in constructing polygons with arbitrary edges and nodes, there is no extra burden on dealing with hanging nodes between neighbouring S-domains in different sizes. Furthermore, the strain modes and the original stiffness matrix of each S-domain and the weighting matrix for nonlocal formulation are all pre-computed initially and extracted in subsequent time steps. Consequently, the computational efficiency of the proposed approach is considerably accelerated. The mesh dependence is well eliminated by using an integral-type nonlocal model. Several benchmarks are simulated to demonstrate the efficiency, robustness and variability of the proposed approach for macroscale and mesoscale problems. It is found that the proposed method can model multiple crack propagation and crack branching as well. Compared with the case under static load, the failure mode of quasi-brittle materials under dynamic load depends not only on the strength of each component but also on the propagation of stress wave.

Nonlocal damage modelling by the scaled boundary finite element method

Summary The progressive damage of structures is fruitfully simulated by using the semi-analytical scaled boundary finite element method (SBFEM). The integral-type nonlocal model combined with the isotropic damage model is extended to eliminate the mesh sensitivity concerning the strain localization. An automatic and efficient quadtree mesh generation algorithm is employed to refine the localized damage process zone (DPZ) and reduce the number of degrees of freedom (DOFs). Owing to the salient advantage of the SBFEM in using arbitrary polygonal subdomains, side-effects associated with hanging nodes can be eliminated. Furthermore, the computational effort of strain/stress field and damage variables can be considerably saved in the framework of the SBFEM. Four numerical benchmarks with regular-shaped domain and a porous plate with irregular holes are simulated to demonstrate the effectiveness and robustness of the proposed approach.

Three-dimensional damage analysis by the scaled boundary finite element method

A novel and effective approach within the framework of the scaled boundary finite element method (SBFEM) is proposed for the damage analysis of structures in three dimensions. The integral-type nonlocal model is extended to SBFEM to eliminate the mesh sensitivity concerning the strain localization. In order to reduce the number of degrees of freedoms (DOFs), an automatic mesh generation algorithm using octree decomposition is employed to refine the localized damage process zone (DPZ), but no extra effort is required to deal with hanging nodes existing between adjacent subdomains with different sizes. A double-notched tension beam is simulated with two different meshes to illustrate the mesh-independence. Three benchmarks are modelled to further verify the effectiveness and robustness of the proposed approach. It is shown that the proposed computational approach is capable of accurately capturing the damage evolution under complicated boundary conditions, and the results agree well with the experimental observations and prior numerical simulations reported in the literatures.

Analysis of size effect on strength of quasi-brittle materials using integral-type nonlocal models

The influence of the averaging operator of nonlocal continuum damage models near specimen boundaries on the size effect on strength of quasibrittle materials is investigated. Two phenomenological approaches, namely standard rescaling and distance-based models, are considered. The numerical results are compared to data from three-point bending tests of notched and unnotched beams recently reported in the literature. It is shown that both approaches can reproduce the experiments well for one type of geometry with one set of input parameters. However, only the distance-based model provides a good agreement for both unnotched and notched beams with the same set of parameters.

Size and Boundary Effects During Failure in Quasi-brittle Materials: Experimental and Numerical Investigations

The degradation of quasi-brittle materials encompasses micro-crack propagation, interaction and coalescence in order to form a macro-crack. These phenomena are located progressively within the so-called Fracture Process Zone (FPZ). The shape and growth of the FPZ, and its interaction with boundaries lead to typical phenomena such as size effects, boundary effects and shielding effects. Classical failure constitutive models involve strain softening due to progressive cracking and a regularization technique for avoiding spurious strain and damage localization. Different approaches have been promoted in the literature such as integral-type non-local models, gradient damage formulations, cohesive cracks models or strong discontinuity approaches. Such macroscale failure models have been applied on a wide range of problems, including the description of damage and failure in strain softening quasi-brittle materials, softening plasticity, creep or composite degradation. An important element of validation of failure models is that they should be able to capture size and boundary effects for various geometries. However, numerical predictions of size effect on different geometries or the description of boundary effects are quite rare in the literature because experimental data on different specimen geometries and on the same material are not available for comparison. If experiments involving size effect are numerous in the literature, they are restricted to a specific geometry and barely consider structures made of the same material, with different geometries. Most of the time, the notch-to-depth ratio tends to zero without reaching zero and unnotched specimens are studied separately, with different materials compared with size effect tests on notched specimens. This paper aims at presenting new experimental and numerical investigations of failure for geometrically similar notched and unnotched concrete specimens made of the same mix. Different geometries (four depth and three notch sizes) have been considered to obtain results involving size and boundary effects at the same time. A mesomodel is used to study the FPZ evolution upon damage depending on the geometry and boundary conditions. A very good agreement with the experimental results is obtained. An analysis of the correlations involved during the fracture process at the mesoscale is performed and a good agreement with acoustic emissions data is revealed.

Numerical simulation of shear band localization in geotechnical materials based on a nonlocal plasticity model

A new nonlocal plasticity model, which is based on the integral-type nonlocal model and the cubic representative volumetric element (RVE), is proposed to simulate shear band localization in geotechnical materials such as soils and rocks. An algorithm is developed to solve the resulting nonlinear system of equations. In this algorithm, the nonlocal averaging of plastic strain over the RVE is evaluated using C^0 elements instead of using C^1 elements to solve the second-order gradient of plastic strains. To obtain the average plastic strain, a set of special elements, called the nonlocal elements, are constucted to approximate the RVE. The updating of average stresses of the local element is based on the nonlocal plastic strain of the corresponding nonlocal elements. Numerical examples show that mesh-independent results can be achieved using the proposed model and the algorithm, and the thickness of the shear band is insensitive to the mesh refinement.

Multiscale Modeling of Concrete

In this paper, a mesoscale model of concrete is presented, which considers particles, matrix material and the interfacial transition zone (ITZ) as separate constituents. Particles are represented as ellipsoides, generated according to a prescribed grading curve and placed randomly into the specimen. Algorithms are proposed to generate realistic particle configurations efficiently. The nonlinear behavior is simulated with a cohesive interface model for the ITZ. For the matrix material, different damage/plasticity models are investigated. The simulation of localization requires to regularize the solution, which is performed by using integral type nonlocal models with strain or displacement averaging. Due to the complexity of a mesoscale model for a realistic structure, a multiscale method to couple the homogeneous macroscale with the heterogeneous mesoscale model in a concurrent embedded approach is proposed. This allows an adaptive transition from a full macroscale model to a multiscale model, where only the relevant parts are resolved on a finer scale. Special emphasis is placed on the investigation of different coupling schemes between the different scales, such as the mortar method and the arlequin method, and a discussion of their advantages and disadvantages within the current context. The applicability of the proposed methodology is illustrated for a variety of examples in tension and compression.

Meso-scale approach to modelling the fracture process zone of concrete subjected to uniaxial tension

A meso-scale analysis is performed to determine the fracture process zone of concrete subjected to uniaxial tension. The meso-structure of concrete is idealised as stiff aggregates embedded in a soft matrix and separated by weak interfaces. The mechanical response of the matrix, the inclusions and the interface between the matrix and the inclusions is modelled by a discrete lattice approach. The inelastic response of the lattice elements is described by a damage approach, which corresponds to a continuous reduction of the stiffness of the springs. The fracture process in uniaxial tension is approximated by an analysis of a two-dimensional cell with periodic boundary conditions. The spatial distribution of dissipated energy density at the meso-scale of concrete is determined. The size and shape of the deterministic FPZ is obtained as the average of random meso-scale analyses. Additionally, periodicity of the discretisation is prescribed to avoid influences of the boundaries of the periodic cell on fracture patterns. The results of these analyses are then used to calibrate an integral-type nonlocal model.

Numerical simulations of void linkage in model materials using a nonlocal ductile damage approximation

Experiments on the growth and linkage of 10 μm diameter holes laser drilled in high precision patterns into Al-plates were modelled with finite elements. The simulations used geometries identical to those of the experiments and incorporated ductile damage by element removal under the control of a ductile damage indicator based on the micromechanical studies of Rice and Tracey. A regularization of the problem was achieved through an integral-type nonlocal model based on the smoothing of the rate of a damage indicator D over a characteristic length L. The simulation does not predict the experimentally observed damage acceleration either in the case where no damage is included or when only a local damage model is used. However, the full three-dimensional simulations based on the nonlocal damage methodology do predict both the failure path and the failure strain at void linkage for almost all configurations studied. For the cases considered the critical parameter controlling the local deformations at void linkage was found to be the ratio between hole diameter and hole spacing.