Flexural behavior of strain-softening solids

The flexural behavior of a beam is investigated in an attempt to establish a correlation between the tensile and bending properties of strain-softening solids. Given the complete uniaxial stress—strain relations, including the post-peak tension-softening portion, it is possible to predict the flexural behavior in moment—curvature and load—deflection relations. The results indicate that strain-softening gives rise to enhanced bending strength in agreement with experimental data. Conversely, given the bending responses together with the softening characteristics the complete tensile behavior can be determined. Since bending experiments are easier to perform than uniaxial tensile tests, this well-defined correlation provides a feasible means to obtain the entire tensile behavior of strain-softening solids such as concrete, rocks and ceramics.     

A FRACTURE-ENERGY-BASED ELASTO-SOFTENING-PLASTIC CONSTITUTIVE MODEL FOR JOINTS OF GEOMATERIALS

On the basis of plasticity and fracture mechanics for quasi-brittle materials , this article presented a constitutive model for gradual softening behavior of joints of geomateri-als . Corresponding numerical tests are carried out at the local level. Characteristics of the model proposed are 1) plastic softening and dilatancy behavior are directly related to the fracture process of joint, and much less material and model parameters are required compared with those proposed by references ; 2) the process of decohesion coupled with friction-al sliding at both micro-scale and macro-scale is described.     

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.     

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.     

Discrete fracture modeling of asphalt concrete

A heterogeneous fracture approach is presented for modeling asphalt concrete that is composed of solid inclusions and a viscous matrix, and is subjected to mode-I loading in the fracture test configuration. A heterogeneous fracture model, based on the discrete element method (DEM), is developed to investigate various fracture toughening mechanisms of asphalt materials using a high-resolution image processing technique. An energy-based bilinear cohesive zone model is used to model the crack initiation and propagation of materials, and is implemented as a user-defined model within the discrete element method. Experimental fracture tests are performed to investigate various fracture behavior of asphalt concrete and obtain material input parameters for numerical models. Also, bulk material properties are necessary for each material phase for heterogeneous numerical models; these properties are determined by uniaxial complex modulus tests and indirect tensile strength tests. The main objective of this study is to integrate the experimental tests and numerical models in order to better understand the fracture mechanisms of asphaltic heterogeneous materials. Experimental results and numerical simulations are compared at different test conditions with excellent agreement. The heterogeneous DEM fracture modeling approach has the potential capability to understand various crack mechanisms of quasi-brittle materials.     

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.     

A statistical model of microcracking of concrete under uniaxial compression

At increasing external load, numerous microcracks propagate in discrete and successive stages within a body of concrete material according to the hierarchy of their tensile fracture strengths. Each microcrack propagation is conditional upon the statistical encounter of its associated fracture criterion. This paper shows the development of a statistical model for the progressive microcrack growth process within a body of concrete material at monotonic uniaxial loading in compression to ultimate failure. This model is formulated by using the Weibull's statistical theory of the strength of materials. The body of heterogeneous concrete material is simulated as a continuum comprising a large population of microscopic “weakest-link” isoenergy elements, each of which contains a unit-volume of representative micro-structural material which is linearly elastic, homogeneous and isotropic. The statistical modelling is derived from the stochastic evaluation of the tensile micro-fracture probabilities of these isoenergy elements at increasing global uniaxial compressive strains.     

Tensile failure of two-dimensional quasi-brittle foams

Stress redistribution caused by damage onset and the subsequent local softening plays an important role in determining the ultimate tensile strength of a cellular structure. The formation of damage process zones with struts dissipating a finite amount of fracture energy will require the macroscopic stress to be increased in order to continue structural damage. The goal of this paper is to investigate the influence of the fracture energy of the solid on the tensile fracture strength and the strain to fracture in quasi-brittle two-dimensional foams using a microstructural model. We analyze the mesoscopic damage and failure mechanisms in uniaxial tension. Relative density, strut cross-sectional profile, solid’s fracture strain, and fracture energy are varied systematically. The effect of the specific fracture energy on the peak behavior has been shown to be captured by the ratio of the fracture energy to the stored elastic energy. We have also explored the net section strength variation in the presence of a central crack at two different fracture energies. Comparison is made between two structurally identical quasi-brittle and ductile strain hardening foams to identify the differences in the damage mechanisms.     

Fracture analysis of concrete using singular fractal functions with lattice beam network and confirmation with acoustic emission study

In the present study singular fractal functions (SFF) were used to generate stress-strain plots for quasi-brittle material like concrete and cement mortar and subsequently stress-strain plot of cement mortar obtained using SFF was used for modeling fracture process in concrete. The fracture surface of concrete is rough and irregular. The fracture surface of concrete is affected by the concrete’s microstructure that is influenced by water cement ratio, grade of cement and type of aggregate [1], [2], [3] and [4]. Also the macrostructural properties such as the size and shape of the specimen, the initial notch length and the rate of loading contribute to the shape of the fracture surface of concrete. It is known that concrete is a heterogeneous and quasi-brittle material containing micro-defects and its mechanical properties strongly relate to the presence of micro-pores and micro-cracks in concrete [1], [2], [3] and [4]. The damage in concrete is believed to be mainly due to initiation and development of micro-defects with irregularity and fractal characteristics. However, repeated observations at various magnifications also reveal a variety of additional structures that fall between the ‘micro’ and the ‘macro’ and have not yet been described satisfactorily in a systematic manner [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [15], [16] and [17]. The concept of singular fractal functions by Mosolov was used to generate stress-strain plot of cement concrete, cement mortar and subsequently the stress-strain plot of cement mortar was used in two-dimensional lattice model [28]. A two-dimensional lattice model was used to study concrete fracture by considering softening of matrix (cement mortar). The results obtained from simulations with lattice model show softening behavior of concrete and fairly agrees with the experimental results. The number of fractured elements are compared with the acoustic emission (AE) hits. The trend in the cumulative fractured beam elements in the lattice fracture simulation reasonably reflected the trend in the recorded AE measurements. In other words, the pattern in which AE hits were distributed around the notch has the same trend as that of the fractured elements around the notch which is in support of lattice model.     

Compression-shear failure of cementitious materials with oblique weak layers

Compressive/shear failure and strain-softening behavior of a bi-material system consisting of two different mortar compositions are studied. The bulk part of the bi-material specimen was made from the stronger mortar and was cast first, and then an oblique weak layer made from the weaker mortar was introduced in the middle of the specimen. By controlling the weak layer angle, thickness and strength, the compressive/shear failure characteristics and Mode-II shear strain-softening behavior have been determined. A bi-linear strain-softening model is proposed to consider both the Mode-II shear strain-softening behavior and the influence of friction due to compression. A linear softening law for the first part of the bi-linear model is sufficient to describe the softening curve after the peak load, but the second linear ‘softening’ relation is required to explain the influence of friction on the load and displacement curve. With the bi-linear model the Mode-II fracture energy G^f-ll can be separated from the frictional energy dissipation. It is also found that two different frictional coefficients exist if a load and displacement curve has distinct softening and pure frictional regions.     

拉伸载荷下混凝土体分形内聚模型

在综合考虑混凝土试件损伤面分形分布及损伤面分形演化基础上,论文提出了体分形内聚模型,用于描述混凝土试件在准静态拉伸载荷作用下的破坏行为.数值计算结果与现有实验数据吻合较好.此外,采用该模型分析了混凝土骨料级配对材料软化性质的影响.结果表明,在拉伸载荷作用下,混凝土骨料级配越均匀,材料的软化特征越明显.     

Nonlinear analysis of beam–column joints using softened truss model

The aim of this study is to expand the application of the nonlinear softened truss model for membrane elements on beam–column joints. The softened truss model employs three equations for equilibrium, three for compatibility and four equations for the constitutive laws of materials. The constitutive equations for both the concrete and steel are based on the actually observed stress–strain relationships. The model has three important attributes. The first is the nonlinear association of stress and strain. The second, and conceivably more noteworthy, is the softening of concrete in compression due to tensile strains in the perpendicular direction. The third is that the influence of the concrete tensile stresses between cracks on the average stress–strain relationship for reinforcing steel and the influence of orthogonal tensile stresses on the compression stress–strain relationship for concrete can be considered in the model. For beam–column joints, one of the most important factors influencing the behaviour is certainly the bond conditions of the beam bars. In this study, the softened truss model is expanded to take into account the influence of this important factor into account. In the revised version of the model, full strain compatibility does not exist between the steel reinforcement and the surrounding concrete and thus the factors influencing the bond-slip between concrete and reinforcement is adequately considered. The improved softened truss model is applied on 51 exterior beam–column joint tests. It is apparent from the results that the revised model gives very accurate predictions of the shear strength of joints and is an improvement on the existing version of the model proposed by Hsu.     

岩土介质的分形孔隙和分形粒子

岩土介质是晶粒状材料,存在大量的孔隙。这些孔隙的存在严重地影响岩土介质的力学、物理和化学性能。大量的研究表明,岩土介质的孔隙几何从原子尺度到晶粒尺寸范围内均表现出分形特征。目前分形几何已被广泛应用来研究岩土的孔隙率,输运特性和渗透性等等。本文从5个方面来讨论岩土介质的分形孔隙和分形粒子:①分形几何简介;②孔隙介质的分形模型;③岩土介质的分形孔隙特性和它们的分形量测方法;④岩土的分形粒子;⑤水土保持估计中的分形毛细管模型。      

A “quasi-elastic” affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites

The derivation of the overall behaviour of nonlinear viscoelastic (or rate-dependent elastoplastic) heterogeneous materials requires a linearisation of the constitutive equations around uniform per phase stress (or strain) histories. The resulting Linear Comparison Material (LCM) has to be linear thermoviscoelastic to fully retain the viscoelastic nature of phase interactions. Instead of the exact treatment of this LCM (i.e., correspondence principle and inverse Laplace transforms) as proposed by the “classical” affine formulation, an approximate treatment is proposed here. First considering Maxwellian behaviour, comparisons for a single phase as well as for two-phase materials (with “parallel” and disordered morphologies) show that the “direct inversion method” of Laplace transforms, initially proposed by Schapery (1962), has to be adapted to fit correctly exact responses to creep loading while a more general method is proposed for other loading paths. When applied to nonlinear viscoelastic heterogeneous materials, this approximate inversion method gives rise to a new formulation which is consistent with the classical affine one for the steady-state regimes. In the transient regime, it leads to a significantly more efficient numerical resolution, the LCM associated to the step by step procedure being no more thermoviscoelastic but thermoelastic. Various comparisons for nonlinear viscoelastic polycrystals responses to creep as well as relaxation loadings show that this “quasi-elastic” formulation yields results very close to classical affine ones, even for high contrasts.     

轻质泡沫混凝土的劈裂破坏力学行为分析

利用平台巴西圆盘加载方式和钢质压条加载方式,对两种厚度为25mm和50mm、不同密度的轻质泡沫混凝土(400～1000kg/m3)进行巴西圆盘劈裂试验,研究密度和厚度对泡沫混凝土裂纹宽度、劈裂强度、断裂韧度、断裂能的影响规律。结果表明,在橡胶垫平台巴西圆盘和钢质压条加载方式下,其劈裂断裂特征大致分为四个阶段:线性弹性段、非线性弹性段、起裂阶段、失稳阶段。同样加载率下最大裂纹宽度随着泡沫混凝土密度增加逐渐减小,劈裂拉伸强度、断裂韧度、断裂能呈幂函数形式增加。借鉴Reinhardt非线性软化曲线,对不同密度泡沫混凝土的应力软化关系进行曲线拟合,建立基于拉伸强度、断裂韧度等控制参数的应力-裂纹宽度关系三段式模型。基于试验结果,对理想多孔材料细观力学预测模型进行修正,获得泡沫混凝土孔隙率与拉伸强度的半经验公式。     

An enriched damage-frictional cohesive-zone model incorporating stress multi-axiality

The paper presents an interphase cohesive zone model (CZM) incorporating stress multi-axiality devised to capture, by simplified micro-modeling, the influence of the in-plane strain and stress state in the mechanical response of the CZM. Moreover, the model is able to account for the Poisson-related effect in the interphase, which can play an important role in the modeling of heterogeneous masonry elements. From the constitutive point of view, the proposed formulation couples damage and friction by addressing a smooth transition from a quasi-brittle response to a residual frictional behavior described by a Coulomb law with unilateral contact. As in-plane stresses are accounted for, damage activation and evolution are governed by a Drucker–Prager law with linear softening. A predictor-corrector procedure based on a backward Euler scheme is detailed for integrating the nonlinear evolutive problem together with the related tangent operator which consistently linearizes the algorithmic strategy. This framework is embedded into a kinematically-enriched finite element interphase formulation incorporating stress multi-axiality. The modeling features of the resulting numerical tool are tested both at the local level, for the typical interphase point, and in meso-structural tests consisting of brick-mortar triplets, investigating the capability of the proposed model and numerical procedure to simulate the brick-mortar decohesion mechanism during confined slip tests.     

Bounds for the effective properties of heterogeneous plates

This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variational principle is used with a self-adjoint Green-operator with traction-free boundary conditions proposed by the authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory are considered: first, in-plane invariant polarization fields are used to recover the “first-order” bounds proposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneous plate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, “second-order bounds” for n-phase plates whose constituents are statistically homogeneous in the in-plane directions are obtained. The results related to a two-phase material made of elastic isotropic materials are shown. The “second-order” bounds for the plate elastic properties are compared with the plate properties of homogeneous plates made of materials having an elasticity tensor computed from “second-order” Hashin–Shtrikman bounds in an infinite domain.     

Mechanical behavior of non-crimp fabric PEEK/C thermoplastic composites

The thermoplastic resin Poly-Ether-Ether-Ketone (PEEK) was used to develop four new NCF composite materials. They refer to two different principal concepts, while each concept was investigated for two different material modifications. The tensile and compression behavior of the newly developed NCF materials was experimentally investigated. For comparison, same tests were also performed on APC-2/AS4 reference material. Prior to the mechanical tests, the quality of the produced laminates was evaluated by means of non destructive investigation (C-Scan tests) and optical microscopy analyses to obtain defects such as delaminations, porosities, micro-cracks etc. The results of the mechanical tests were exploited to obtain the “optimal” NCF fabrication process; the mechanical properties of the material solution considered to be “optimal” compare well to the respective properties of the reference material thus providing evidence for improved cost efficiency by the production of thermoplastic composite components.     

Continuum Dyson's equation and defect Green's function in a heterogeneous anisotropic solid

A continuum Dyson's equation and a defect Green's function (GF) in a heterogeneous, anisotropic and linearly elastic solid under homogeneous boundary conditions have been introduced. The continuum Dyson's equation relates the point-force Green's responses of two systems of identical geometry and boundary conditions but of different media. Given the GF of either system (i.e., a reference), the GF of the other (i.e., a defect system with “defect” change of materials property relative to the reference) can be obtained by solving the Dyson's equation. The defect GF is applied to solve the eigenstrain problem of a heterogeneous solid. In particular, the problem of slightly inhomogeneous inclusions is examined in detail. Based on the Dyson's equation, approximate schemes are proposed to efficiently evaluate the elastic field. Numerical results are reported for inhomogeneous inclusions in a semi-infinite substrate with a traction-free surface to demonstrate the validity of the present formulation.     

A simple model of the mechanical behavior of ceramic-like materials

This paper presents a macroscopic mechanical theory for ceramic-like materials undergoing isothermal deformations. The proposed model describes an elastic brittle material which is damageable only under tensile loading. The damage lowers the elastic stiffness in traction simulating hence the softening and the fracture (zero stillness) of the material. The basic idea is to consider the continuum as a mixture of two phases—a linear elastic phase and a masonry phase (which shows a linear elastic behavior under compression but cannot hold tractive loads at all). The damage is then related to the volume fraction β of the clastic constituent. The constitutive relations are derived from macroscopic thermodynamics with the volume fraction β and its gradient β taken as state variables.