A new damage model for microcrack-weakened brittle solids

In the present paper, a micromechanically based damage model for microcrack-weakened solids is developed. The concept of the domain of microcrack growth (DMG) is defined and used to describe the damage state and the anisotropic properties of brittle materials. After choosing an appropriate fracture criterion of microcrack, we obtain the analytical expression of DMG under a monotonically increasing proportional plane stress. Under a complex loading path, the evolution equation of DMG and the overall effective compliance tensor of damaged materials are given. The project supported by National Natural Science Foundation of China     

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.     

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.     

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。     

An interacting micro-crack damage model for failure of brittle materials under compression

A model is developed for brittle failure under compressive loading with an explicit accounting of micro-crack interactions. The model incorporates a pre-existing flaw distribution in the material. The macroscopic inelastic deformation is assumed to be due to the nucleation and growth of tensile “wing” micro-cracks associated with frictional sliding on these flaws. Interactions among the cracks are modeled by means of a crack-matrix-effective-medium approach in which each crack experiences a stress field different from that acting on isolated cracks. This yields an effective stress intensity factor at the crack tips which is utilized in the formulation of the crack growth dynamics. Load-induced damage in the material is defined in terms of a scalar crack density parameter, the evolution of which is a function of the existing flaw distribution and the crack growth dynamics. This methodology is applied for the case of uniaxial compression under constant strain rate loading. The model provides a natural prediction of a peak stress (defined as the compressive strength of the material) and also of a transition strain rate, beyond which the compressive strength increases dramatically with the imposed strain rate. The influences of the crack growth dynamics, the initial flaw distribution, and the imposed strain rate on the constitutive response and the damage evolution are studied. It is shown that different characteristics of the flaw distribution are dominant at different imposed strain rates: at low rates the spread of the distribution is critical, while at high strain rates the total flaw density is critical.     

Simulation of damage evolution in ductile metals undergoing dynamic loading conditions

A set of constitutive equations for large rate-dependent elastic-plastic-damage materials at elevated temperatures is presented to be able to analyze adiabatic high strain rate deformation processes for a wide range of stress triaxialities. The model is based on the concepts of continuum damage mechanics. Since the material macroscopic thermo-mechanical response under large strain and high strain rate deformation loading is governed by different physical mechanisms, a multi-dissipative approach is proposed. It incorporates thermo-mechanical coupling effects as well as internal dissipative mechanisms through rate-dependent constitutive relations with a set of internal variables. In addition, the effect of stress triaxiality on the onset and evolution of plastic flow, damage and failure is discussed.Furthermore, the algorithm for numerical integration of the coupled constitutive rate equations is presented. It relies on operator split methodology resulting in an inelastic predictor-elastic corrector technique. The explicit finite element program LS-DYNA augmented by an user-defined material subroutine is used to approximate boundary-value problems under dynamic loading conditions. Numerical simulations of dynamic experiments with different specimens are performed and good correlation of numerical results and published experimental data is achieved. Based on numerical studies modified specimens geometries are proposed to be able to detect complex damage and failure mechanisms in Hopkinson-Bar experiments.     

Constitutive and damage model for 63Sn37Pb solder under uniaxial and torsional cyclic loading

A constitutive model with Ohno–Wang kinematic hardening rule is developed and employed to simulate the isothermal cyclic behavior of Sn–Pb solder under uniaxial and torsional loading. An implicit constitutive integration scheme is presented for inelastic flow of solder. Then a modified low cycle fatigue life prediction model is put forward in which the sum of maximum shear strain range and normal strain range based on the critical plane concept is adopted to replace the uniaxial strain range used by Stolkarts et al. [Stolkarts, V., Keer, L.M., Fine, M.E., 1999. Damage evolution governed by microcrack nucleation with application to the fatigue of 63Sn–37Pb solder. J. Mech. Phys. Solids 47, 2451–2468]. Comparison of the experimental results and simulation verifies that the stress strain hysteresis loops and peak stress decline curve of solder can be reasonably modeled over a wide range of loading conditions with implement of damage coupled constitutive model, and the lifetime estimations of 63Sn37Pb solder based on the assumption of microcrack nucleation governed damage is effective to provide a conservative prediction.     

Fractal effect and anisotropic constitutive model for concrete

An elasto-anisotropic damage constitutive model for concrete is developed in this work. Disregarding the coupling between the isotropic and the anisotropic damage, the isotropic damage variables are defined as functions of the microcrack fractal dimension, and the anisotropic parts are expressed by the lengths of cracks in concrete which various in different directions. The Helmholtz free energy is decomposed into the elastic deforming, damage and irreversible deforming components, with the last component used to replace the plastic deformation. Therefore the damage constitutive formulas for concrete are derived based on continuum damage mechanics. Evolution laws for both isotropic and anisotropic damage variables are derived, in which the anisotropic parts are obtained by modifying an empirical model. The critical fracture stress and the fracture toughness are investigated for materials with a single fractal crack based on the fractal geometry and the Griffith fracture criterion. Numerical computation is conducted for concrete under the uniaxial and the biaxial compression. The results indicate that the material stiffness degradation can be well addressed when the anisotropic damage is incorporated; the irreversible deformation is greatly related to the behavior of the descending branch beyond the peak load. The validation of the presented model is proofed by comparing results with the experimental data. This model provides an approach to link the macro properties of a material with its micro-structure change.     

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。     

考虑微缺陷形状的细片层状珠光体钢的损伤本构模型

基于非经典塑性理论和连续介质损伤力学,利用在一个特殊坐标系下基于椭球形孔洞模型得到的可考虑孔洞形状变化混合强化材料的损伤演化率得到了铁素体相的损伤本构方程,通过混合物理论利用铁素体和渗碳体相各自本构关系并考虑其几何特征得到了珠光体团的损伤本构模型。进而采用Hill自洽方法,得到了珠光体材料的宏观损伤本构描述,发展了相应的数值方法与程序。讨论了孔洞形状对材料损伤的影响,并对典型珠光体双相材料BS11在非对称循环加载史下的弹塑性响应特性进行了分析,得到了与实验较为一致的结果。     

单轴拉伸条件下脆性岩石微裂纹损伤模型研究

利用断裂力学、损伤力学和均匀化原理,对脆性岩石单轴拉伸条件下的力学特性进行分析,建立了脆性岩石的微裂纹损伤本构模型.首先对岩石内部微裂纹的统计分布规律进行分析,给出了理论分析过程中微裂纹分布的假设条件,在此基础上,参考已有研究成果,得到含细长微裂纹脆性岩石有效弹性参数的计算公式.然后,对岩石内部单一微裂纹进行断裂力学和损伤力学分析,得到了扩展裂纹尖端的应力强度因子计算公式,在一定微裂纹断裂扩展准则和断裂扩展速率的假设基础上,利用积分原理,得到了岩石整体的损伤变量和损伤演化方程,由此建立单轴拉伸条件下脆性岩石的微裂纹损伤本构模型.最后,通过一花岗岩的单轴拉伸试验结果对微裂纹损伤本构模型进行了验证.     

动力损伤后的脆性岩石静力蠕变断裂模型研究

应力波动力扰动下脆性岩石的静力蠕变特性,对深部地下工程围岩变形的评价有重要的实践意义.动力载荷作用导致的局部细观裂纹损伤严重影响脆性岩石蠕变力学行为.基于细观裂纹扩展与应力关系模型、动力扰动损伤演化函数、静动力载荷演化路径函数与黏弹性本构模型,提出一种应力波动力扰动下脆性岩石蠕变断裂特性的宏细观力学模型.其中动力损伤通过控制岩石内部细观裂纹数量变化实现.模型描述了应力波动力扰动下岩石的应变时间演化曲线,解释了岩石动力扰动下蠕变失效特性.研究了不同应力波幅值及周期影响下的脆性岩石应变-时间关系曲线,并通过试验结果验证了模型的合理性.讨论了动力损伤变化形式,突变发生时刻,突变量的大小对岩石蠕变失效特性的影响.分析了应力波幅值、周期对岩石动态动力损伤效应以及蠕变失效特性的影响.主要研究结果:动力损伤的变化值越大,岩石蠕变失效发生时间越短.冲击载荷扰动期间,动力损伤发生的时刻及增加的形式,对动力扰动后的岩石应变及蠕变破坏时间影响很小.动力损伤变化量随应力波幅值增加、周期减小而加速增大.应力波幅值越大、周期越小,岩石发生蠕变失效时间越短.     

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．     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

砌体复合剪压受力特性与数值分析方法

 从砌体细观结构出发，提出了用于砌体复合剪压受力过程模拟的细观力学 模型. 该模型假定砌体是由砂浆、块体及两者之间的粘结带组成的三相复合材料，以此为基 础对砌体三相组成的非均匀性予以表征. 提出了基于弹性损伤力学的本构关系，以最大拉应 力准则和摩尔库仑准则作为损伤发生的阈值，并借助复合材料破坏过程分析(MFPA)程序进 行受力分析. 结果表明，所采用的数值模型和模拟方法能够较好地模拟砌体复合剪压受力破 坏过程，为砌体开裂过程的机理分析提供了一种新的数值方法.     

Anisotropic density growth of bone—A computational micro-sphere approach

Bones are able to adapt their local density when exposed to mechanical loading. Such growth processes result in densification of the bone in regions of high loading levels and in resorption of the material in regions of low loading levels. This evolution and optimisation process generates heterogeneous distributions of bone density accompanied by pronounced anisotropic mechanical properties. While several constitutive models reported in the literature assume the growth process to be purely isotropic, only few studies focus on the modelling and simulation of anisotropic functional adaptation we can observe in vivo. Some of these few computational models for anisotropic growth characterise the evolution of anisotropy by analogy to anisotropic continuum damage mechanics while others include anisotropic growth but assume isotropic elastic properties.The objective of this work is to generalise a well-established framework of energy-driven isotropic functional adaptation to anisotropic microstructural growth and density evolution. We adopt the so-called micro-sphere concept, which proves to be extremely versatile and flexible to extend sophisticated one-dimensional constitutive relations to the three-dimensional case. In this work we apply this framework to the modelling and simulation of anisotropic functional adaptation by means of a directional density distribution, which evolves in time and in response to the mechanical loading condition. Several numerical studies highlight the characteristics and properties of the anisotropic growth model we establish. The formulation is embedded into an iterative finite element algorithm to solve complex boundary value problems. In particular, we consider the finite-element-simulation of a subject-specific proximal tibia bone and a comparison to experimental measurements. The proposed model is able to appropriately represent the heterogeneous bone density distribution. As an advantage over several other computational growth models proposed in the literature, a pronounced local anisotropy evolution is identified and illustrated by means of orientation-distribution-type density plots.     

Thermomechanical constitutive equations for the dynamic response of ceramics

The behavior and failure of brittle materials is significantly influenced by the existence of inhomogeneities such as pores and cracks. The proposed constitutive equations model the coupled micro-mechanical response of these inhomogeneities through evolution equations for scalar measures of porosity, and a “density” function of randomly oriented penny-shaped cracks. A specific form for the Helmholtz free energy is proposed which incorporates the known Mie–Grüneisen constitutive equation for the nonporous solid. The resulting thermomechanical constitutive equations are valid for large deformations and the elastic response is hyperelastic in the sense that the stress is related to a derivative of the Helmholtz free energy. These equations allow for the simulation of the following physical phenomena exhibited by brittle materials: (1) high compressive strength compared with much lower tensile strength; (2) inelastic deformation due to growth and nucleation of cracks and pores instead of due to dislocation dynamics associated with metal plasticity; and (3) loss of integrity (degradation of elastic moduli) due to damage accumulation. The main features of the model are demonstrated by examples of cyclic loading in homogeneous deformation and by a simulation of a dynamic plate-impact experiment on AD85 ceramic. The theoretical predictions of the model are in excellent agreement with the dynamic experimental data.     

ANALYSIS OF THE LOCALIZATION OF DAMAGE AND THE COMPLETE STRESS-STRAIN RELATION FOR MESOSCOPIC HETEROGENEOUS ROCK UNDER UNIAXIAL TENSILE LOADING

The mechanical behavior of rock under uniaxial tensile loading is different from that of rock under compressive loads. A micromechanics-based model was proposed for mesoscopic heterogeneous brittle rock undergoing irreversible changes of their microscopic structures due to microcrack growth. The complete stress-strain relation including linear elasticity, nonlinear hardening, rapid stress drop and strain softening was obtained. The influence of all microcracks with different sizes and orientations were introduced into the constitutive relation by using the probability density function describing the distribution of orientations and the probability density function describing the distribution of sizes. The influence of Weibull distribution describing the distribution of orientations and Rayleigh function describing the distribution of sizes on the constitutive relation were researched. Theoretical predictions have shown to be consistent with the experimental results.