A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids

In order to simulate quasi-brittle failure in porous elastic solids, a continuum damage model has been developed within the framework of strain gradient elasticity. An essential ingredient of the continuum damage model is the local strain energy density for pure elastic response as a function of the void volume fraction, the local strains and the strain gradients, respectively. The model adopts Griffith’s approach, widely used in linear elastic fracture mechanics, for predicting the onset and the evolution of damage due to evolving micro-cracks. The effect of those micro-cracks on the local material stiffness is taken into account by defining an effective void volume fraction. Thermodynamic considerations are used to specify the evolution of the latter. The principal features of the model are demonstrated by means of a one-dimensional example. Key aspects are discussed using analytical results and numerical simulations. Contrary to other continuum damage models with similar objectives, the model proposed here includes the effect of the internal length parameter on the onset of damage evolution. Furthermore, it is able to account for boundary layer effects.     

Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials

A numerical method is developed to simulate complex two-dimensional crack propagation in quasi-brittle materials considering random heterogeneous fracture properties. Potential cracks are represented by pre-inserted cohesive elements with tension and shear softening constitutive laws modelled by spatially-varying Weibull random fields. Monte Carlo simulations of a concrete specimen under uni-axial tension were carried out with extensive investigation of the effects of important numerical algorithms and material properties on numerical efficiency and stability, crack propagation processes and load-carrying capacities. It was found that the homogeneous model led to incorrect crack patterns and load–displacement curves with strong mesh-dependence, whereas the heterogeneous model predicted realistic, complicated fracture processes and load-carrying capacity of little mesh-dependence. Increasing the variance of the tensile strength random fields with increased heterogeneity led to reduction in the mean peak load and increase in the standard deviation. The developed method provides a simple but effective tool for assessment of structural reliability and calculation of characteristic material strength for structural design.     

Interaction-based non-local damage model for failure in quasi-brittle materials

The purpose of this paper is to present a new macroscopic approach to describe the evolving non-local interactions which are produced at the mesoscale during damage and failure in quasi-brittle materials. A new-integral type non-local model is provided where the weight function is directly built from these interactions, and therefore takes into account their evolution during the material failure intrinsically.     

Component assembling model and its application to quasi-brittle damage

Potential energy can be approximated by “pair-functional” potentials which is composed of pair potentials and embedding energy. Pair potentials are grouped according to discrete directions of atomic bonds such that each group is represented by an orientational component. Meanwhile, another kind of component, the volumetric one is derived from embedding energy. Damage and fracture are the changing and breaking of atomic bonds at the most fundamental level and have been reflected by the changing of these components’ properties. Therefore, material is treated as a component assembly, and its constitutive equations are formed by means of assembling these two kinds of components’ response functions. This material model is referred to as the component assembling model. Theoretical analysis and numerical computing indicate that the proposed model has the capacity of reproducing some results satisfactorily, with the advantages of physical explicitness and intrinsic induced anisotropy, etc.     

Stochastic functional expansion in elasticity of heterogeneous solids

The Volterra-Wiener functional expansion is employed to the analysis of statistic properties for random heterogeneous solids. For simplicity, the technique is displayed on an elastic suspension of spheres. The basis function in the expansion is chosen as that corresponding to the so-called “perfect disorder” of spheres (PDS), recently introduced by the authors. An infinite hierarchy of equations for the kernels in the expansion is derived whose truncating after the nth equation is shown to yield results for the averaged statistical characteristics which are valid to order cⁿ_f, where c_f is the volume fraction of the spheres. The kernels for the first and the second approximations, n = 1, 2, are found and related to the displacement fields in an infinite elastic body containing, respectively, one and two spherical inhomogeneities. Within the frame of the so-called singular approximation the overall tensor of elastic moduli for a suspension of perfectly disordered spheres is shown to coincide to the order c²_f with a formula, earlier obtained by means of the method of the effective field.     

Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-brittle materials

The aim of this paper is to develop a thermodynamically consistent micromechanical concept for the damage analysis of viscoelastic and quasi-brittle materials. As kinematical damage variables a set of scalar-, vector-, and tensor-valued functions is chosen to describe isotropic and anisotropic damage. Since the process of material degradation is governed by physical mechanisms on levels with different length scale, the macro- and mesolevel, where on the mesolevel microdefects evolve due to microforces, we formulate in this paper the dynamical balance laws for macro- and microforces and the first and second law of thermodynamics for macro- and mesolevel.Assuming a general form of the constitutive equations for thermo-viscoelastic and quasi-brittle materials, it is shown that according to the restrictions imposed by the Clausius–Duhem inequality macro- and microforces consist of two parts, a non-dissipative and a dissipative part, where on the mesolevel the latter can be regarded as driving forces on moving microdefects. It is shown that the non-dissipative forces can be derived from a free energy potential and the dissipative forces from a dissipation pseudo-potential, if its existence can be assured.The micromechanical damage theory presented in this paper can be considered as a framework which enables the formulation of various weakly nonlocal and gradient, respectively, damage models. This is outlined in detail for isotropic and anisotropic damage.     

Hysteretic deteriorating model for quasi-brittle materials based on micromechanical damage approach

A damage model, which is based on the stochastic modeling of the microstructures, is developed for the quasi-brittle materials subjected to repeated loading. According to this model, the overall response of the material is represented with a series of micro-elements joined in parallel. A combined model is proposed for the micro-element considering the fracture as well as the hysteretic energy dissipation. To account for the progressive failure, the random fracture strains are assigned to the micro-elements. Therefore the overall parallel bundle is considered as a stationary random field. Then by averaging the microscopic random field, the overall loading, unloading and reloading curves are derived analytically. Two hysteretic rules are derived from the proposed model, and the overall hysteretic deteriorating behaviors could be well reproduced. To demonstrate the validity of the present model, the numerical results are shown against the stochastic simulated curves as well as the experimental data. The present model provides an alternative approach for the efficient modeling of the hysteretic deteriorating behaviors for the quasi-brittle materials.     

Characterization of heterogeneous solids via wave methods in computational microelasticity

Real solids are inherently heterogeneous bodies. While the resolution at which they are observed may be disparate from one material to the next, heterogeneities heavily affect the dynamic behavior of all microstructured solids. This work introduces a wave propagation simulation methodology, based on Mindlin's microelastic continuum theory, as a tool to dynamically characterize microstructured solids in a way that naturally accounts for their inherent heterogeneities. Wave motion represents a natural benchmark problem to appreciate the full benefits of the microelastic theory, as in high-frequency dynamic regimes do microstructural effects unequivocally elucidate themselves. Through a finite-element implementation of the microelastic continuum and the interpretation of the resulting computational multiscale wavefields, one can estimate the effect of microstructures upon the wave propagation modes, phase and group velocities. By accounting for microstructures without explicitly modeling them, the method allows reducing the computational time with respect to classical methods based on a direct numerical simulation of the heterogeneities. The numerical method put forth in this research implements the microelastic theory through a finite-element scheme with enriched super-elements featuring microstructural degrees of freedom, and implementing constitutive laws obtained by homogenizing the microstructure characteristics over material meso-domains. It is possible to envision the use of this modeling methodology in support of diverse applications, ranging from structural health monitoring in composite materials to the simulation of biological and geomaterials. From an intellectual point of view, this work offers a mathematical explanation of some of the discrepancies often observed between one-scale models and physical experiments by targeting the area of wave propagation, one area where these discrepancies are most pronounced.     

An isotropic unilateral damage model coupled with frictional sliding for quasi-brittle materials

In this paper, we present an original extension of an isotropic damage model for quasi-brittle materials and assess its predictive capabilities. The proposed model accounts not only for unilateral behavior related to the opening and closure of microcracks but also for inelastic strains reflecting the frictional sliding along closed microcracks. More importantly, owing to its careful mathematical formulation with a particular attention paid to the continuous differentiability of the underlined thermodynamic potential, the model ensures the continuity of the inelastic stress–strain response. First applications show that it is able to predict the asymmetric behavior and hysteretic response of microcracked materials such as concrete and some rocks     

Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials: A 3D study

In a recent publication (Yang et al., 2009. Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials. Int. J. Solids Struct. 46 (17) 3222–3234), we developed a finite element method capable of modelling complex two-dimensional (2D) crack propagation in quasi-brittle materials considering random heterogeneous fracture properties. The present study extends the method to model three-dimensional (3D) problems. First, 3D cohesive elements are inserted into the initial mesh of solid elements to model potential crack surfaces by a specially designed, flexible and efficient algorithm and corresponding computer program. The softening constitutive laws of the cohesive elements are modelled by spatially-varying 3D Weibull random fields. Monte Carlo simulations are then carried out to obtain statistical information of structural load-carrying capacity. A concrete cube under uniaxial tension was analysed as an example. It was found that as the 2D heterogeneous model, the 3D model predicted realistic, complicated fracture processes and load-carrying capacity of little mesh-dependence. Increasing heterogeneity in terms of the variance in the tensile strength random fields resulted in lower mean and higher standard deviation of peak loads. Due to constraint effects and larger areas of unsmooth, non-planar fracture surfaces, 3D modelling resulted in higher mean and lower standard deviation of peak loads than 2D modelling.     

On the modeling of thermo-mechanical phase transformations in solids

The present paper represents an effort to model coupled thero-mechanical effects in the mcroscopic response of solids that arise from the occurrence of phase transformations. A Helmholtz free energy potential is constructed to describe the response of the thermoelastic material to be considered. Apart from some considerations pertaining to properties of the hypothetical material, the analysis is carried out in the context of a simple problem, idealized from an experiment, in which an annular cylinder is deformed to a state of radially symmetric, finite anti-plane shear in the presence of differing inner and outer surface temperatures. After constructing all radially symmetric weak solutions involving at most a single surface of discontinuity of strain or temperature gradient, we determine the implications for quasi-static motions of the second law of thermodynamics. In particular, the results concerning creep as predicted by the present model are in qualitative agreement with the results of the motivating experiment.     

基于真实细观结构的准脆性材料三维建模及应用初探

采用连续切片的方法获取准脆性材料的表面图像,利用数字图像处理技术检测材料的细观结构并进行矢量化。通过一种简单的变换,将每一切片矢量化的细观结构转换成单层的三维结构,然后将这些切片连续的细观结构逐层叠加,形成整个试件的三维真实细观结构,并针对准脆性材料图像的特点,编制了能够批量处理数字图像并进行细观结构矢量化的程序,建立了与有限元三维网格模型之间的数据接口,模型数据可直接导入岩石三维破裂过程分析RFPA3D系统中,研究真实细观结构对准脆性材料破坏力学行为的影响。以颗粒材料为例,分析了在单轴受压情况下的三维空间裂纹的产生及扩展过程,计算结果显示颗粒分布与界面显著影响材料的破裂模式。     

Micro-cracks informed damage models for brittle solids

A class of micro-cracks informed damage models for describing the softening behavior of brittle solids is proposed, in which damage evolution is treated as a consequence of micro-crack propagation. The homogenized stress–strain relation in the cracked microscopic cell defines the degradation tensor, which can be obtained by the equivalence between the averaged strain energy of the microscopic cell and the strain energy density of the homogenized material. This energy equivalence relationship serves as an energy bridging vehicle between the damaged continuum and the cracked microstructure. Several damage evolution equations are obtained by this energy bridging method. The size effect of the micro-cracks informed damage law is characterized through the microscopic cell analysis, and the proper scaling of the characterized damage evolution functions to eliminate mesh dependency in the continuum solution is introduced.     

Modelling of quasi-brittle behaviour: a discrete approach coupling anisotropic damage growth and frictional sliding

The paper provides development of the model of anisotropic damage by microcracking proposed by Bargellini et al. 2006. This model is based on a discrete approach, which introduces a finite set of microcrack densities associated with fixed directions. This approach avoids inconveniences encountered when using a single second order tensor damage variable D (non uniqueness of the free energy) and strain decomposition into positive and negative parts (spurious dissipation at crack closure). Frictional sliding on closed microcracks is introduced as an additional dissipative mechanism; it is represented by a second order sliding variable in each damage direction. Corresponding sliding criteria and non-associated sliding evolution laws, formulated in the strain space for the model coherence, permit to account for hysteretic phenomena. Unilateral effect is taken into account; Young's and shear moduli are correctly restored at microcrack closure. The crucial requirements of continuity of the energy and of stress–strain response are ensured through relevant conditions on parameters and sliding variables values at opening-closure. The discrete approach, associated with some hypotheses concerning damage evolution, permits to couple damage and dissipative sliding. The pertinence of the proposed theory is illustrated by simulating first elastic properties at constant damage, then by considering a specific loading path involving both damage and friction evolutions.     

Multiscale cohesive failure modeling of heterogeneous adhesives

A novel multiscale cohesive approach that enables prediction of the macroscopic properties of heterogeneous thin layers is presented. The proposed multiscale model relies on the Hill's energy equivalence lemma, implemented in the computational homogenization scheme, to couple the micro- and macro-scales and allows to relate the homogenized cohesive law used to model the failure of the adhesive layer at the macro-scale to the complex damage evolution taking place at the micro-scale. A simple isotropic damage model is used to describe the failure processes at the micro-scale. We establish the upper and lower bounds on the multiscale model and solve several examples to demonstrate the ability of the method to extract physically based macroscopic properties.