Quasi-micromechanical constitutive theory for brittle damaged materials under tension

One of fundamental but difficult problems in damage mechanics is the formulation of the effective constitutive relation of microcrack-weakened brittle or quasi-brittle materials under complex loading, especially when microcrack interaction is taken into account. The combination of phenomenological and micromechanical damage mechanics is a promising approach to constructing an applicable damage model with a firm physical foundation. In this paper, a quasi-micromechanical model is presented for simulating the constitutive response of microcrack-weakened materials under complex loading. The microcracking damage is characterized in terms of the orientation domain of microcrack growth (DMG) as well as a scalar microcrack density parameter. The DMG describes the complex damage and its evolution associated with microcrack growth, while the scalar microcrack density factor defining the isotropic magnitude of damage yields an easy calculation of the effects of microcrack interaction on effective elastic moduli. Project supported by the National Natural Science Foundation of China (19891180).     

MICROMECHANICAL MODELLING OF STRAIN SOFTENING IN MICROCRACK-WEAKENED QUASI-BRITTLE MATERIALS

By using the concept of domain of microcrack growth(DMG),themicromechanisms of damage in quasi-brittle materials subjected to triaxial either tensileor compressive loading are investigated and the complete strew-strain relation includingfour stages is obtained from micromechanical analysis.The regime of pre-peaknonlinear hardening corresponds to the distributed damage,i.e.the stable propagationof microcracks.After the attainment of the ultimate strength of load-bearing capacity,some microcracks experience the second unstable growth and the distributed damage istransmitted to the localization of damage.These analyses improve our understanding ofthe hardening and softening behaviors of quasi-brittle materials.     

带微裂纹物体的有效断裂韧性

按照等效介质的思想，引进有效表面能密度的概念，建立了带微裂纹物体有效断裂韧性的公式．具体计算了微裂纹群分别平行和垂直于宏观裂纹两种情况的减韧比．表明微裂纹群在产生应力屏蔽（或反屏蔽）效应的同时，也降低了材料的有效断裂韧性，减小了对宏观裂纹的扩展阻力．     

一种基于材料细观特征量的PBX拉伸强度理论模型

为给塑性黏结炸药(PBX)的力学强度设计提供支撑、探索材料细观特征量与材料强度之间的定量规律,应用微裂纹扩展区理论,将PBX炸药的单轴拉伸过程中力学响应特征的变化归结为扩展裂纹取向角度的增加,将扩展裂纹最大取向角与拉伸强度相关联,构建了基于材料细观特征量的拉伸强度理论模型,并采用不同温度的单轴拉伸实验验证了该理论模型的有效性。研究表明:该拉伸强度理论模型可以实现对PBX炸药拉伸强度与炸药微裂纹密度、颗粒/黏结剂界面性能以及颗粒/黏结剂体系的表观杨氏模量、泊松比等细观特征量之间关系的定量描述。     

A micromechanics-based strain gradient damage model for fracture prediction of brittle materials – Part II: Damage modeling and numerical simulations

In this paper, we established a strain-gradient damage model based on microcrack analysis for brittle materials. In order to construct a damage-evolution law including the strain-gradient effect, we proposed a resistance curve for microcrack growth before damage localization. By introducing this resistance curve into the strain-gradient constitutive law established in the first part of this work (Li, 2011), we obtained an energy potential that is capable to describe the evolution of damage during the loading. This damage model was furthermore implemented into a finite element code. By using this numerical tool, we carried out detailed numerical simulations on different specimens in order to assess the fracture process in brittle materials. The numerical results were compared with previous experimental results. From these studies, we can conclude that the strain gradient plays an important role in predicting fractures due to singular or non-singular stress concentrations and in assessing the size effect observed in experimental studies. Moreover, the self-regularization characteristic of the present damage model makes the numerical simulations insensitive to finite-element meshing. We believe that it can be utilized in fracture predictions for brittle or quasi-brittle materials in engineering applications.     

Fractal dimension evolution of microcrack net in disordered materials

Fractal geometry is used to evaluate the degree of disorder of crack size distribution in brittle damaging materials. The fractal dimension of the 2D microcrack net turns out to increase from one to two during the loading process and microcrack propagation. This means that the material becomes more disordered with the damage evolution. The longer cracks, in fact, propagate more rapidly than the shorter and, at the same time, the crack size distribution increases its statistical dispersion. Some numerical examples, related to different initial microcrack densities and size distributions, are illustrated with the computer simulation of the system evolution.     

计算微裂纹损伤材料有效模量的一种简单方法

给出了一种基于Taylor模型的有效介质方法。用以计算微裂纹相互作用对有效本构关系的影响,该方法假设每一个微裂纹位于一种有效介质之中,该有效介质的弹性模量由不考虑微裂纹相互作用的Taylor模型计算、和自洽方法相比,这种方法计算简单,而且结果更准确。     

A heuristic approach to microcracking and fracture for ceramics with statistical consideration

Microcracking damage and toughening are examined for ceramics. These effects have been found to depend on the material microstructure and macrocrack growth. Isotropic damage, attributed to random distribution of microcrack location, length and orientation can be associated with a disordered microstructure and a non-uniform residual stress field. When the applied stress is the main cause of cracking, the microcrack distribution is no longer random such as a system of quasi-parallel cracks. To highlight the effect of crack interaction, discrete models are advanced where damage is simulated by a distribution of microcracks. The dilute concentration assumption is invoked to simplify the analysis.The two-dimensional discrete model is based on a phenomenological approach that is statistical in character. Interactions of microcracks and with a macrocrack are considered by means of a boundary element technique (A. Brencich, A. Carpinteri, Int. J. Fracture 76 (1996) 373–389; A. Brencich, A. Carpinteri, Eng. Fract. Mech. 59 (1998) 797–814) where both isotropic and anisotropic damage could be treated. Comparisons with other results are made to show that the model can be applied to analyse the fracture behaviour of different materials.     

A micromechanics-based thermodynamic formulation of isotropic damage with unilateral and friction effects

This paper is devoted to micromechanical modeling of isotropic damage in brittle materials. The damaged materials will be considered as heterogeneous media composed of solid matrix weakened by isotropically distributed microcracks. The original contribution of the present work is to provide a proper micromechanical thermodynamic formulation for damage-friction modeling in brittle materials with the help of Eshelby’s solution to matrix-inclusion problems. The elastic and plastic strain energy involving unilateral effects will be fully determined. The condition of microcrack opening–closure transition will be determined in both strain-based and stress-based forms. The effect of spatial distribution of microcracks will also be taken into account. Further, the damage evolution law is formulated in a sound thermodynamic framework and inherently coupled with frictional sliding. As a first phase of validation, the proposed micromechanical model is finally applied to reproduce basic mechanical responses of ordinary concrete in compression tests.     

一种固体推进剂破坏的细观实验研究

固体推进剂是一种颗粒填充高能聚合物,其破坏时所表现出的一些行为和金属的破坏相比,有明显不同之处。本文利用扫描电镜,给出了受载情况下固体推进剂试件的损伤发展以及裂纹扩展过程。结果表明,损伤的发展过程也就是微裂纹区扩展及微裂纹密度加大的过程,而裂纹扩展过程中的慢速扩展和亚临界扩展的最大区别则是其裂尖损伤区范围不同,同时也表明,裂纹尖端和边缘的微裂纹损伤区尺寸已不容忽略。     

Probabilistic character of the microcrack growth in brittle layered materials

Characteristics of microcrack initiation, multiplication and saturation in layered materials are discussed. A probabilistic-analytical method, the ‘characteristic curve method (CCM)’ is developed to correlate the initial defects and the microcrack evolution under static and cyclic loadings. The ‘equivalent applied loading’ and the ‘equivalent crack density’ concepts are introduced to describe different microcrack multiplication features in different layered materials. Microcrack multiplication processes in many layered materials with brittle matrices subjected to static and cyclic loadings can be easily predicted.     

内压对多场诱发表面扩散下微裂纹演化的影响

根据表面扩散控制下物质迁移机制,建立力、电、热和内压共同作用下的有限单元法,详细讨论了多场诱发表面扩散下金属材料内部微裂纹的演化过程。结果表明：对于形态比为 的微裂纹,存在一临界电场值 。当 时,微裂纹逐渐圆柱化;当 时,微裂纹分节为上下或左右两个较小的微裂纹。随着内压的增大,微裂纹圆柱化时间增长,而分节时间显著减小,且内压有助于微裂纹分节。对于 的微裂纹,当 、 和 一定时存在一临界内压值 。当 时,随着电场和内压的增大会率先在裂纹的上下两端发生裂纹扩展。对于 的微裂纹,当 MPa时,内压对临界值 的影响较小。     

动载荷作用下微裂纹的增韧分析

在实验研究的基础上，将细观统计力学应用于材料的动态增韧分析中，探索性的研究了在动态载荷作用下微裂纹的增韧机理，并进行了相应的有限元分析计算，得出在动载荷作用下，微裂纹增韧主要表现在微裂纹的快速成核。如果要提高材料在动态情况下的增韧幅值，必须提高增韧颗粒的特征尺寸。     

Finite element analysis on evolution process for damage microcrack healing

Based on the thermal kinetic and mass conservation, a series of controlling equations for the finite element are derived and related programs are developed to simulate the damage microcrack healing process controlled by surface diffusion. Two kinds of typical models for microcrack splitting are proposed, i.e., the grain boundary energy existing on the crack surface and residual stresses applying on the crack surface. And the conditions of microcrack splitting in the two models are given as a function of the microcrack aspect ratio. The microcrack with traction-free surfaces will directly evolve into a spheroid. The project supported by the National Natural Science Foundation of China (19972053 and 59889101), and the National Outstanding Young Scientist Fund of China (59925104)     

Uniqueness and minimum theorems for a multifield model of brittle solids

Some minimum theorems potentially useful to construct numerical schemes related to quasi-static evolution of damage in brittle elastic solids are proposed. The approach is that of multifield theories, with a second-order damage tensor describing the microcrack density. The use of damage entropy flux and damage pseudo-potential are both investigated.     

Dynamic damage constitutive relation of mesoscopic heterogenous brittle rock under rotation of principal stress axes

A micro-mechanics-based model is developed to investigate microcrack damage mechanism of four stages of brittle rock under rotation of the principal stress axes. They consist of linear elastic, non-linear hardening, rapid stress drop and strain softening. The frictional sliding crack model is applied to analyze microcracks nucleation, propagation and coalescence. The strain energy density factor approach is applied to determine the critical condition of microcrack nucleation, propagation and coalescence. The inelastic strain increments are formulated within the framework of thermodynamics with internal variables. Rotation of principal stress axes affect the dynamic damage constitutive relationship and the failure strength of brittle rock.     

A material model for cementitious composite materials with an exterior point Eshelby microcrack initiation criterion

A micromechanical model for cementitious composite materials is described in which microcrack initiation, in the interfacial transition zone between aggregate particles and cement matrix, is governed by an exterior-point Eshelby solution. The model assumes a two-phase elastic composite, derived from an Eshelby solution and the Mori–Tanaka homogenization method, to which circular microcracks are added. A multi-component rough crack contact model is employed to simulate normal and shear behaviour of rough microcrack surfaces. The development of the microcrack initiation criterion and the rules adopted for microcrack evolution are a particular focus of the paper. Finally, it is shown, on the basis of several numerical simulations, that the model captures key characteristics of the behaviour of cementitious composites such as concrete.     

Localization of deformation and stress–strain relation for mesoscopic heterogeneous brittle rock materials under unloading

Stress redistribution induced by excavation of underground engineering and slope engineering results in the unloading zone in parts of surrounding rock masses. The mechanical behaviors of crack-weakened rock masses under unloading are different from those of crack-weakened rock masses under loading. A micromechanics-based model has been proposed for brittle rock material undergoing irreversible changes of their microscopic structures due to microcrack growth when axial stress is held constant while lateral confinement is reduced. The basic idea of the present model is to classify the constitution relation of rock material into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The closed-form explicit expression for the complete stress–strain relation of rock materials containing cracks under unloading is obtained. The results show that the complete stress–strain relation and the strength of rock materials under unloading depend on the crack spacing, the fracture toughness of rock materials, orientation of the cracks, the crack half-length and the crack density parameter.     

Microcrack interaction brittle rock subjected to uniaxial tensile loads

A micromechanics-based model is proposed to describe unstable damage evolution in microcrack-weakened brittle rock material. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress–strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to be consistent with the experimental results.     

Constitutive relationship of brittle rock subjected to dynamic uniaxial tensile loads with microcrack interaction effects

A micromechanical model is proposed to describe both stable and unstable damage evolution in microcrack-weakened brittle rock material subjected to dynamic uniaxial tensile loads. The basic idea of the present model is to classify the constitution relationship of rock material subjected to dynamic uniaxial tensile loads into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress-strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to consistent with the experimental results.