]damage model for brittle elastic solids with unequal tensile and compressive strengths

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Birtel-paper(损伤因子定义)

Formulas for the resistances of the tensile and the compressive shear struts of beams under biaxial shear forces have been developed by the coauthor (Mark, 2004; Mark 2005). They are valid for beams with rectangular cross sections, typical side aligned stirrups, normal strength concrete and arbitrary distributions of the longitudinal reinforcement. Moreover, there is no limitation for an
Abstract: A spatial finite element model of reinforced concrete (RC) beams with rectangular cross sections, typical side aligned stirrups and distributed or edge concentrated longitudinal reinforcement is presented. It is parameterised in its properties of geometry, material, discretisation and loads in biaxial directions. The concrete volume is discretisised into 8 or 20node solid elements. Truss elements discretely model each single reinforcement bar. They are coupled to the concrete elements using the ″embedded modelling″ technique. The ″concrete damage plasticity″ model of ABAQUS is used to describe the nonlinear material behaviour of concrete. Suitable material functions and material parameters are derived and verified to experimental data of (cyclic) uniaxial, biaxial or triaxial stress tests. Energy criteria and internal length parameters ensure almost mesh independent results of the simulations. An elasto-plastic material model with a gradually rising plastic branch is adopted for the reinforcing steel. The parametric model is verified to experimental data of uniaxial shear tests taken from the literature. Afterwards, it is used to establish a data base of biaxial shear resistances to check developed biaxial shear design formulas that base on simple strut and tie models. More than 100 simulations guarantee an extended and reliable verification that experiments – almost none of them are available in the literature – are not able to give. Moreover, the arrangement of the stirrups is optimised in dependence upon the distribution of the longitudinal reinforcement to minimise reinforcement amounts and increase bearing capacities.

力学专业英语考试重点整理

学而不思则惘，思而不学则殆力学专业英语考试重点整理一、单词：英译汉、汉译英Centroid 质心，形心Elasticity 弹力，弹性Linear 线性的，直线的Prismatic 棱镜的，棱柱形的Strain 应变Stress 应力Tension 张力，拉力，拉紧Alloy 合金Aluminium 铝Ductile 易延展的，韧性的Failure 失败，破坏，失效Lateral 侧面的，横向的Necking 颈缩Couple 力偶Cylinder 圆筒，圆筒状物Inertia 惯性，惯量，惰性Shaft 轴，杆状物Torsion 扭矩Cantilever 悬臂梁，伸臂Neutral 中性的Statics 静力学，静止状态，静态Symmetry 对称Transverse 横向的，横断的Astronomer 天文学家Galaxy 天河，银河，星系Planet 行星Collinear 共线的，在同一直线上的Dimension 尺寸，大小，维数Equate 使相等，等同Parameter 参量，参数Visualize 想象，形象化，使看得见Acceleration 加速度Dynamics 动力学Stationary 不动的，稳定的，定常的Vector 矢量Velocity 速度Angular 角的，角度的Coordinate 坐标Radian 弧度Shaft 轴Symbol 符号Conservation 守恒，保存Differentiation 微分Integration 积分，集成，一体化Interval 间隔，间隙，空隙Linear 直线的，线性的，一次的Moment 力矩，瞬间，片刻Vector 矢量Velocity 速度Derivative 导数，派生的Frequency 频率Friction 摩擦Magnitude 大小，量级Power 力量，乘方，幂Viscous 黏性的Continuum 连续体，连续统Rectangular 矩形的，成直角的Resultant 合成的，合力，合量Torque 扭转力，扭矩Dilatation 膨胀，扩张Distortion 扭曲，变形Isotropic 各向同性的Tensor 张量Coordinate 坐标Crack 裂缝Curvature 曲率Ellipse 椭圆Formula 公式Function 函数，功能Buckle 屈曲，皱曲，弄弯，翘曲Deflection 挠曲，偏向Wrinkle 皱纹，皱褶，起皱Factor 因数，系数Flexural 弯曲的，挠曲的Notch 缺口，凹槽，刻痕Vibrate 振动（v）Vibration 振动（n）Detector 发现者，侦察器，探测器，检波器Vacuum 真空，空间，真空的，产生真空的Other than 除了二、句子：英译汉1、The concepts of stress and strain can be illustrated in an elementary way byconsidering the extension of a prismatic bar. As shown in Fig. 1, a prismatic bar is one that has constant cross section throughout its length and a straight axis. In this illustration the bar is assumed to be loaded at its ends by axial forces P that produce a uniform stretching, or tension, of the bar.翻译：应力和应变的概念可以通过考虑一个棱柱形杆的拉伸这样一个简单的方式来说明。

An Elastoplastic Damage Constitutive Model for Concrete

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LIU Jun et al. / China Ocean Eng., 27(2), 2013, 169 − 182
account. As well known, the irreversible deformation in concrete is closely related to microcracks. The growth and coalescence of microcracks lead to the degradation of stiffness and strength, while interfaces deterioration of microcracks results in incomplete closure after unloading. Therefore, these two theories should be combined, as have been researched by many scholars in various ways. In their research, plasticity can be described in effective stress space (weak coupling) (Lee and Fenves, 1998; Jason et al., 2006), or nominal stress space (strong coupling) (Lubliner et al., 1989; Salari et al., 2004; Kratzig and Polling, 2004), while damage can be described based on isotropic (scalar) (Mazars and Pijaudier-Cabot, 1989; Hassler-Combe and Hartig, 2008) or anisotropic (tensor) (Skrzypek and Ganczarski,1998; Meschke et al., 1998; Voyiadjis et al., 2008) idealization. In order to enhance the numerical efficiency and reflect the nonlinear feature of concrete as well as possible, in the presented research, plasticity is formulated in effective stress space, and isotropic hardening has been taken into account. Besides that, two scalar indicators of tensile damage and compressive damage are introduced to establish strict constitutive model that ensure thermodynamic consistency in describing the different features of concrete under tension and compression. Fully implicit backward-Euler integration algorithm characterized by unconditional stability is employed in the numerical computation. The proposed model is examined firstly by using uniaxial, biaxial and triaxial loading cases, monotonically as well as cyclically. And then, damage process of a double-edge-notched specimen subjected to loading combination of shear and tension is simulated to further verify the model. The tensor product and contractions in this paper are operated by using notational convention as

土木工程专业英语课后题答案

英译汉Lesson One Sandcrete is a yellow-white building material made from Portland cement and sand in a ratio of circa 1:8.It is the main building material for walls of single-storey buildings (such as houses and schools)in countries such as Ghana and Nigeria. Measured strengths fo commercially available Sandcrete blocks in Nigeria were found to be between 0.5 and 1 N/mm2,which is well below the 3.5 N/mm2that is legally required there. This may be due to the need of the manufacturers to keep the price low,and since the main cost-factor is the Portland cement, they reduce that, which results in a block that starts behaving more like loose sand.翻译：sandcrete是黄白色建筑材料制成的波特兰水泥和沙子的比例大约1:8.it是主要的建筑墙体材料的单层建筑（如房屋和学校）在加纳和尼日利亚等国家。

测量优势的商用sandcrete块在尼日利亚被发现之间的0.5和1牛顿/毫米，这是远远低于3.5牛顿/毫米，是法律所要求的有。

这可能是由于需要的厂家保持价格低，因为主要cost-factor是波特兰水泥，他们减少，从而在一块，开始表现得更像松砂。

An elasto-plastic damage model for reinforced concrete with minimum number of material parameters

An elasto-plastic damage model for reinforced concrete with minimum number of material parametersWilfried B.Kr€a tzig a,Rainer P€o lling b,*a Institute for Statics and Dynamics,Ruhr-University Bochum,44780Bochum,Germanyb Building Contractors of Mesenbrock,Welter Straße4,48249D€u lmen,GermanyReceived1November2002;accepted4March2004Available online20April2004AbstractBased on a fully3-D elasto-plastic damage theory,the material behavior of all reinforced concrete components––concrete,reinforcement and bond––is,for biaxial loading,realistically modelled,including cyclic action.Thereby emphasis is layed on concrete in tension and compression.The presented model contains a minimum number of material parameters.It further enables to map exact uniaxial stress–strain curves as proposed by modern codes of practise,like the EC2.All material parameters of the model can be readily interpreted and determined by few standard experiments,or approximated from concrete compression strength.Finally,the concrete model is veriﬁed by numerical simulation of experiments.Ó2004Elsevier Ltd.All rights reserved.Keywords:Concrete;Reinforced concrete;Material damage;Plasticity;Crushing energy;Material model;Softening1.IntroductionIn structural engineering,reinforced concrete is considered as a2-phase composite of uniaxial steel and multi-dimensional concrete components.Hardened concrete itself,predominantly a mixture of aggregates and a cementitious matrix,exhibits a rather complicated deformation behavior,mainly because of initiation and growth of micro-cracking in the matrix.Constitutive models for concrete,the central concern of this treatise,exist on very diﬀerent quality levels.Even the most advanced structural codes of practice[1]oﬀer mainly elastic behavior,at most some monotonic non-linear stress–strain curves without any rheo-mechanical speciﬁcation.On the other(research)extreme,weﬁnd highly sophisticated constitutive attempts with few relations to engineering design practice.But for com-puter simulation of advanced structural problems,like failure analyses or life-cycle assessments,constitutive models are required,improved compared to those from codes of practice and more exact,nevertheless of lowest possible complication.Severe deﬁciencies of code mod-els are their uniaxial formulation and the lack of infor-mation for cyclic processes.If one attempts to set up a constitutive model for cyclic behavior of concrete as pictured in Fig.6or7,one observesﬁrst the residual strains after complete unloading.These generally are interpreted as‘plastic’deformations,although in concrete there will be cer-tainly no plastic slip like in metal plasticity.The second observation is the stiﬀness degradation along the cycles which requires a‘damage’component in the model. Obviously in constitutive modeling of concrete,an elasto-plastic component has to be combined with an elasto-damaging one.Hence the key aim here is a3-dimensional elasto-plastic description of mechanical responses of reinforced*Corresponding author.Tel.:+49-2548-9193-883;fax:+49-2548-1085.E-mail address:rainer.poelling@(R.P€o lling).0045-7949/$-see front matterÓ2004Elsevier Ltd.All rights reserved.doi:10.1016/pstruc.2004.03.002Computers and Structures82(2004)1201–1215/locate/compstrucconcrete,supplemented by a damage component.Be-cause of few cyclic experiments for concrete and the problem of reproductivity of results,a minimum number of well-deﬁned,experimentally identiﬁable material parameters shall control the model[2].Classical elasto-plastic material theories[3,4]are standard in engineering,they need no explanation here. They are‘stress-based’since their yield conditions and yield potentials are formulated in stress-space.‘Strain-based’plasticity theories,with plastic stresses as internal thermodynamic variables and consequently yield con-ditions in strain-space,have their origins in the works [5,6].Remarkably early,Dougill[7]proposed a‘frac-turing theory’for concrete under compression,in our terminology a‘strain-based’damage concept.Opposite to plasticity,its basic idea required that inelastic defor-mations result merely in stiﬀness degradations,thus all unloading paths cross through the origin of the stress–strain diagram,and no residual strains remain after unloading.In a re-work Dougill and Rida[8]deﬁned‘micro-fracturing’materials,equipped with stiﬀness degrada-tion from micro-crack evolution,linear-elastic behavior during un-/reloading,and without strain/stress residuals after complete unloading.Hence they laid the basis of ‘strain-based’continuum damage theories[9,10],in which the material stiﬀness tensor itself acts as internal damage variable.If instead of this theﬂexibility(com-pliance)tensor is applied,‘stress-based’continuum damage theories can be established[11,12].In order to properly model the constitutive behavior of concrete in compression or tension,both constitu-ents––plasticity as well as continuum damage theory––have to be combined,both either in‘stress-based’or ‘strain-based’alternatives.Suchﬁrst coupling work was due to Ba z ant and Kim[13].Their‘stress-based’plas-ticity combined with‘strain-based’damage theory led to an‘elasto-plastic fracturing’concept with a total of26 material parameters.Coupling concepts of both con-stituents in stress-space are due to[14],a work speciﬁed by[15]to isotropic damage of concrete under com-pression.[16]therein uniﬁed both limit surfaces for yielding and fracturing to one.Han and Chen[17]ﬁrst combined both constituents in complete tensorial manifold in strain space.In order to reduce the number of material constants,they united yield and damage surface,an idea repeated in[10]. Further,[18]oﬀered a scalar coupling of plasticity and damage theories introducing again a uniﬁed yield/dam-age condition with the key argument that micro-crack-ing in the cementitious matrix is the single source for both inelastic phenomena.Finally we mention the survey over all basic condi-tions of coupling alternatives in[19].We stress,that there only elasto-plastic-damage concepts are listed with a sound basis in continuum mechanics.Empirical models of reinforced concrete which all combine the mentioned constituents are legion[2];a prominent concept leading to excellent results is that one of Darwin and Pecknold[20].All mentioned constitutive models call,in principle, for the ability to describe material failure in compression as well as in tension,obviously both strongly localized phenomena[21].To avoid mesh sensitivity of a partic-ular FE-solution,such localized fracture processes re-quire special treatment[22].The intended constitutive law of this work is aimed to the analysis of complete structures,such that the classical‘smeared crack’con-cept shall be maintained[23].Such concept avoids the immense numerical eﬀort of modeling hundreds of single cracks,presently to be mastered only by parallel com-puting techniques,in view of the physical irreproduc-ibility of crack-patterns in concrete structures,even under laboratory conditions.So in order to avoid modelling of localization bands or surfaces as weak or strong discontinuities,or to avoid enhancements by non-local kinematics,the use of crushing respectively fracture energy is chosen here[2]. Advantageously,such regularization requires only the determination of2additional material constants, namely the crushing and the fracture energy.Consequently,in this paper a‘stress-based’elasto-plastic damage model for reinforced concrete will be derived.It will be demonstrated,that not only the fun-damental ideas of micro-fracturing and damage theories are equal,but that both concepts lead to identical material descriptions[18],if the material stiﬀness tensor itself is used as internal thermodynamic variable.It hence is obvious that both phenomena––damage by tension cracks as well as by compressive micro-crack-ing––follow a uniﬁed elasto-plastic damage theory.The ﬁnal constitutive law will possess a minimum number of material parameters,an important aspect for applica-tion.For this reason,the model is adapted without exact description of the response under highly triaxial com-pression.2.Theoretical basis of the elasto-plastic damage theoryThe fundamental framework of the presented con-cept is the‘stress-based’elasto-plastic damage theory. An excellent summary of this theory with the compli-ance tensor as internal variable has been given by Govindjee et al.[12]in complete analogy to plasticity theory.Recently a scalar combination of both damage and plasticity theory has been performed by[16].We start with a brief summary of the underlying concept.As in[12]our starting point is the assumption of a Helmholtz-free energy W and the deﬁnition of internal thermo-dynamical variables.To consider both,plastic deformation and stiﬀness reduction,we introduce as1202W.B.Kr€a tzig,R.P€o lling/Computers and Structures82(2004)1201–1215internal thermo-dynamical variables the plastic strain e plas in ordinary plasticity theory and the change D da of the compliance tensor like in [12].Then the Helmholtz-free energy reads:q 0W ðe ;e pl ;D da ;q Þ¼1ðe Àe pl Þ:ðD 0þD da|ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ}DÞÀ1:ðe Àe plÞþq 0W inðq Þ;ð1Þwhere r and e denote the stress and strain tensors,respectively.D 0abbreviates the initial compliance ten-sor,and q as further internal variable describes the hardening/softening behavior of the material.W in ðq Þcontains the free energy associated with progressive degradation and plastic deformations.Finally,with q 0as initial density we identify the thermodynamically associated variables Àr ¼q 0o W =o e pl ;Àa ðq Þ¼q 0o W =o q ;ð2ÞÀ12r r ¼q 0o W =o D da :ð3ÞApplying the C LAUSIUS –D UHEM inequality q 0_W6r :_e ,we arrive on one hand at the compliance rela-tione ¼ðD 0þD da Þ:r þe pl !r ¼ðD 0þD da ÞÀ1:ðe Àe pl Þ;ð4Þand on the other hand at the inequality of dissipation P dis ¼12r :_D da :r þr :_e pl þa _q P 0:ð5ÞAfter introducing the elastic region E in the space of the associated variables (stresses):E :¼fðr ;a Þj U ðr r ;a Þ60g ;ð6Þwe deduce from the principle of maximum inelastic dissipation the following normality rules:_epl :o r o ðr r Þþ12_D da ¼o U o ðr r ÞÁ_k ;ð7Þ_a¼Ào U Á_k :ð8ÞAt this point we have to separate the inelastic strains into those of plastic and damaged origin.Because of few available cyclic experiments for later parameter ﬁtting,the use of just one scalar b ,as proposed in [16]for separation of plastic from damaged parts,is adopted.Thus we assume _Dda ¼2_kb o U o ðr r Þand_epl ¼_k ð1Àb Þo U o r:ð9ÞObviously with (34),the normality rule (7)is fulﬁlled.With the consistency condition _U¼0one ﬁnally derives the tangential stiﬀness relation and the evolution equa-tions for the internal variables as summarized in Table 1.3.Uniﬁed elasto-plastic damage model for reinforced concrete3.1.Concrete under compression3.1.1.Yield/damage potentialA yield/damage potential of Drucker–Prager-type has been selected for description of concrete in com-pression.Such type of potential can be handled rela-tively easy,it enables a suﬃcient modelling quality at least for uniaxial and biaxial proportional loading.This potential reads U c ðr ;a c Þ¼11ﬃﬃ3p Àl l I 1ÂþﬃﬃﬃﬃJ 2p ÃÀa c ðq c Þ;ð10Þwith I 1and J 2as ﬁrst invariant of r and as secondinvariant of its deviator s ,respectively.The parameter l herein controls the inﬂuence of the hydrostatic stress on damage or yield.Further,the term left of the brackets guarantees that during plastic/damage loading a c always corresponds to the negative uniaxial compression stress.Diﬀerentiation of this potential with respect to r deliversTable 1Tangential stiﬀness relation and evolution equations of internal variables of the stress-based elasto-plastic damage theoryElastic loading/unloading _k ¼0Plastic-damaging loading k>0,U ¼0Tangential stiﬀness relation:_r ¼D À1:_e _r ¼ D À1ÀD À1:o U o r o Uo r:D À1o U :D :o U Ào U a ;q o U!:_e Equations of evolution of internal variables:_D da ¼0_D da ¼2b o U Á_k or ð_D da :r Þ¼b o U _k _e pl ¼0_e pl ¼ð1Àb Þo U o rÁ_k _q¼0_q¼o U o aÁ_k with _k ¼o Uo r:D À1o U:D :o U Ào Ua ;q o U:_eW.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–12151203o U c o r ¼11ﬃﬃ3p Àl l o I 1o r þ12ﬃﬃﬃﬃJ 2p o J 2o r¼11ﬃﬃ3p Àl l I þs þs T4ﬃﬃﬃﬃJ 2p :ð11Þ3.1.2.Modiﬁcation of the internal variable,yield anddamage ruleWe now introduce a new internal variable q Ãc ,replacing the original one q c :_q Ãc ¼À2b w ðr Þ_q c ¼2b w ðr Þ_k c with w ðr Þ¼21ﬃﬃ3p Àl l1þ1ﬃﬃﬃ2p:ð12ÞThen a c in (10)shall be only a function of q Ãc .Due toSection 3.1.4this transformation allows an analytical solution for the hardening/softening function a c ðq Ãc Þfrom a given uniaxial stress–strain curve [2],one of our goals.The rate of this variable a c reads_a c ¼d a c Ãc _q Ãc ¼d a c Ãc 2b _k c :ð13ÞFurthermore we assume,that the parameter b ,which separates plastic from damage parts,shall also be a function of q Ãc .The equations of evolution of the plastic strain _epl and the strain _eda ¼_D da :r can directly be determined from the potential (10).With respect to (11)and (12)we gain:_e pl ¼ð1Àb ðq Ãc ÞÞo U c o r _k c ð14Þ¼w ðr Þ121b ðq Ãc Þ À111ﬃﬃ3p Àl l I þ14ﬃﬃﬃﬃJ 2p s Àþs TÁ _q Ãc ð15Þand_e da ¼_D da :r ¼b ðq ÃcÞo U c o r _k c ð16Þ¼b ðq Ãc Þ11ﬃﬃ3p Àl l I þ14ﬃﬃﬃﬃJ 2p s Àþs T Á_k c :ð17ÞAssuming now an isotropic damage evolution due to compression,we ﬁnd the following compliance evolu-tion law,the correctness of which can be veriﬁed by an double scalar contraction with r :_Dda ;c ¼b ðq Ãc Þ1ﬃﬃ3p Àl l 1 À16ﬃﬃﬃﬃJ 2p I I þ14ﬃﬃﬃﬃJ 2p ðI þI Þ!Á_k c ð18Þ¼w ðr Þ1211ﬃﬃ3p Àl l I 1À16ﬃﬃﬃﬃJ 2pI I þ14ﬃﬃﬃﬃJ 2p ðI þI Þ!Á_q Ãc :ð19ÞThe tensor D da ;c describes the compliance evolution and can be represented by two scalars in the isotropic case.Consequently,we replace the fourth-order tensor D da ;cin the set of internal variables by D da ;c s1and D da ;cs2as fol-lows:D da ;c ¼D da ;c s1I I þD da ;c s2ðI þI Þ:ð20ÞBy comparison with the coeﬃcients in (18)and (20),onearrives at evolution equations for D da ;c s1and D da ;cs2:_D da ;c s1¼b ðq Ãc Þ1ﬃﬃ3p Àl l 1 À16ﬃﬃﬃﬃJ 2p _k c ;ð21Þ_D da ;c s2¼b ðq ÃcÞ1ﬃﬃ3p Àl14ﬃﬃﬃﬃJ 2p_k c :ð22Þ3.1.3.Tangential stiﬀness relationBecause of all above assumptions the tangential stiﬀness relation slightly diﬀers from the standard form in Table 1,and one arrives with q Ãc at:_r ¼D À1"ÀD À1:o U c o r o U c o r:D À1o U c o r :D À1:o U c o r þd a c d q ÃcÃc w ðr Þ#:_e ð23Þin the case of plastic-damaging loading.3.1.4.Concept of determination of the hardening/softeningfunctionWe now consider uniaxial loading with respect to an orthogonal cartesian frame,where the unit base vector i 1coincides with the loading direction.Then the compli-ance relation (4)becomese ¼1E cþD da ;c !r þe pl with e ¼e 11;e pl ¼e pl 11;r ¼r 11;1E c¼D 1111;D da ;c ¼D da ;c 1111ð24Þand the evolution equation (19)for the compliance boilsdown to_D da ;c ¼w ðr Þ1ﬃﬃ3p Àl l r "À16ﬃﬃ3p r !þ14ﬃﬃ3p r Á2#Á_q Ãc ¼_q Ãc :ð25ÞBy integration of this relation one arrives at theimportant statement D da ;c ¼q Ãc ;ð26Þinterpreting the internal variable q Ãc as the uniaxial change of the compliance due to damage.Furthermore we assume the split of inelastic strains into plastic and damaging parts by the scalar parameter b ,as illustrated in Fig.1.From all previous transformations follows for the plastic strain e pl :1204W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–1215e pl¼b e pl h þD da ;cÁ rðe Þi!e pl ¼b1ÀbD da ;c Á rðe Þ¼b1Àb q ÃcÁ r ðe Þ:ð27ÞWe then obtain by substituting (26)and the latter identity into (24)e ¼1c þ1q Ãc ! r ðe Þ;ð28Þwhich equation solves with given function rðe Þfor e as e ¼e ðq Ãc Þ,yielding together with the stress–strain instruction ﬁnally to a c ðq Ãc Þ¼ r ðe ðq Ãc ÞÞ:ð29ÞFurthermore we need the diﬀerential quotient d a c =d q Ãcand ﬁndd a c d q Ãc ¼Àd r de d e d q Ãc ¼a c d r d e ð1Àb Þ1À1E c þq Ãc 1Àb dr d eh i ;ð30Þwith d e =d q Ãc gained by total diﬀerentiation of Eq.(28).Finally we need the explicit form of b ðq Ãc Þ.Starting point is the evolution equation (15),which reduces under uniaxial loading conditions to_e pl¼1Ãc À1r Á_q Ãc ;ð31Þfrom which we ﬁnd directly b ðq Ãc Þ¼11þo e =o q cc c :ð32ÞFor further transformation we can conclude from (27)o e pl o q Ãc ¼b 1Àb a c ðq Ãc Þþq Ãcd a c d q Ãc !;ð33Þleading ﬁnally to b ðq Ãc Þ¼11þb 1þq Ãcc Ãcd a cÃchi :ð34Þ3.1.5.Mapping to uniaxial stress–strain curveFor convenience the uniaxial stress–strain curve of concrete,assumed here as given function,shall be sub-divided into three parts as illustrated in Fig.2.Region 1:Elastic.Below the initial yield/damage stress f c y we assume linear-elastic behavior,deﬁned by Young’s modulus E c and Poisson’s ratio m c .The initial yield stress f c y is taken as one third of the failure strength f c ,such that we obtain the initial conditiona c ðq Ãc ¼0Þ¼f c y ¼13f c :ð35ÞRegion 2:Hardening.In this region the stress grows until failure strength f c ,consequently the tangential stiﬀness decreases from initial stiﬀness to zero (hori-zontal tangent).An analytical function of this behavior,which ﬁts well with experiments is found in [2].Eq.(2.1-18)therein readsr2ðe Þ¼E ci e cþe c21ÀE ci e c fcÀ2e e cf c ;ð36Þwith f c and e c as failure stress and accompanying strain,respectively.The modulus E ci ;originally denotes the initial modulus of elasticity.But in order to guarantee the stress–strain curve to cross the point ðÀe c y ;Àf c y Þ,we deﬁne E ci ;with respect to (35)as secant modulus[2]Fig.1.Deﬁnition of material parameter b.Fig.2.Assumed uniaxial stress–strain curve of concrete undercompression.W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–12151205E ci¼12E cf ce c2Àf ce cþ32E c:ð37ÞRegion3:Softening.After exceeding the compression strain e c,localization of damage occurs in this softening region.A suitable concept to overcome possible ill-po-sedness there,originally derived for tension softening in smeared crack idealization,is the use of fracture energy: Then the softening branch depends on the fracture en-ergy,a material parameter,and on the characteristic length l eq[24].Use of fracture energy as material parameter instead of a softening function is generally accepted for tension cracks.The transfer of this concept to softening under compression has beenﬁrst proposed by Feenstra[25] and accepted since that,i.e.[26].To distinguish fracture energy under tension from that one under compression, the latter will be denoted as‘crushing energy’.But one should notice,that because of nonlinear hardening also a diﬀuse crushing energy g cu;exists[27],not included in the presently described localized one.Thus one sets thevolume speciﬁc localized crushing energy gÃcl (compareFig.2)equal to G cl=l eq,where G cl is the material parameter‘crushing energy’and l eq the characteristic length of the respective FE integration point.Clearly,l eq depends on type,quadrature rule and form of the ele-ment[28].Consequently,the following function has been cho-sen for the softening branch of region3:rðeÞ¼À12þc c f c e ccþc c eþc ccÁe2with c c>0ð38Þin which c c is the only free parameter controlling thearea under the stress–strain curve,corresponding to gÃcl .Note,that this area isﬁnite.Determination delivers the following relation to the localized crushing energy G cl,as explained in detail in[19]:c c¼p2f c e c2G cleq À1f c e cð1ÀbÞþb f cch i2:ð39ÞRemark.If the term in square brackets becomes nega-tive,the above equation renders invalid,since this would describe a‘snap-back’-behavior in the considered material point.To avoid this,one should elude elements leading tol eq6G clf c e cð1ÀbÞþb f cE c:ð40ÞFor determination of a cðqÃc Þ(29)from re-branches,seeAppendix A.3.2.Concrete under tensionFor description of concrete behavior in tension we apply the stress-based continuum damage theory again. But for simplicity,all inelastic deformations shall be due to pure damaging,i.e.b¼1.Also,opposite to com-pression,we exclude nonlinear hardening before soft-ening.Thus the initial damage surface and the failure surface are identical.3.2.1.Damage potentialFor tension failure,we use the widely accepted damage potential of Rankine type,a criterium in good accordance with biaxial experiments[29].To allow for softening in diﬀerent directions to model cracks in dif-ferent directions at least a kinematic softening rule is required.Hence introducing the back-stress a t of the stress state r as internal variable,the damage potential readsU tð1Þðr;a tÞ¼nð1ÞÀf ct60;ð41Þabbreviating with nð1Þtheﬁrst eigenvalue of n¼rÀa t.The derivatives of this potential with respect to r and a t areo U to r¼Mð1Þnando U to a t¼ÀMð1Þn;ð42Þwhere Mð1Þndenotes the eigenvalue basis of theﬁrst eigenvalue of n.3.2.2.Modiﬁcation of internal variable,kinematic soften-ing and damage ruleIf one follows strictly the thermodynamic concept of the continuum damage theory,one has to introduce asecond-order tensor qt,thermodynamically conjugate to a t.Furthermore one has toﬁnd an evolution law as normality condition for this tensor and a tensor-valued function a tðq tÞas softening rule.The evolution law then reads_a t¼o a t=o q t:_q t.Instead of such complexity determination,we replace the fourth-order tensoro a t=o q t by a scalar function Zða1t nÞ,with a1t n¼a t:Mð1Þnas normal component of the back-stress in crack-direction.Introduction of an explicit conjugate variable q1t ncan be omitted,because no hardening regime has been as-sumed,and the function q1t nða1t nÞtherefore is bijective. These simpliﬁcations lead to the softening rule,illus-trated in Fig.3:_a t¼Zða1t nÞo U to a t_ktð1Þ¼À_k tð1ÞZða1t nÞMð1Þnð43Þcorresponding to Prager’s hardening law.(43)guaran-tees crack formation in each principal stress direction, independent of other existing cracks.Now the normality rule can be derived from the potential U tð1Þ(41)reading1206W.B.Kr€a tzig,R.P€o lling/Computers and Structures82(2004)1201–1215_e da ;t :¼_D da ;t :r ¼_k t ð1Þo U t ð1Þ¼_k t ð1ÞM ð1Þn :ð44ÞThe anisotropic damage rule:_Dda ;t ¼1n_k t ð1ÞM ð1Þn M ð1Þnwithr 1n¼M ð1Þn:r ð45Þhas been taken over from [12].This rule guarantees the accomplishment of (44),and preserves symmetry of the compliance tensor.3.2.3.Consideration of further cracksAs well known,a second crack in 2D-plane and a third one in 3D-space can appear,each orthogonal to the existing ones.Therefore two more damage potentials are introduced analogously to the original one (41):U t ð2Þðr ;a t Þ¼n ð2ÞÀf ct ;60;ð46ÞU t ð3Þðr ;a t Þ¼n ð3ÞÀf ct ;60:ð47ÞThe complete theory thus develops to a non-smoothmulti-surface continuum damage theory.According to Koiter’s rule [30],we then obtain _at ¼ÀX 3i ¼1_k t ði ÞZ ða i t nÞM ði Þn ;ð48Þ_Dda ;t ¼X 3i ¼11r i n _k t ði ÞM ði Þn M ði Þn with r i n ¼M ði Þn :r :ð49ÞLoading conditions have to been introduced for each crack direction separately:_kt ði ÞP 0;U t ði Þ60;_kt ði ÞÁU t ði Þ¼0:ð50Þ3.2.4.Closure of cracksCrack closures results in instantaneous local re-stiﬀ-ening of the concrete,where the cracks remain as ‘pas-sive’ones.Following an idea of Ortiz [11]we thusdisregard in the compliance tensor D the constituent D da ;t ,which represents all cracks,but maintain D da ;ta of all ‘active’cracks:D ¼D 0þD da ;c þD da ;ta :ð51ÞKraj c inovi c [31]obtained the following relation between D da ;ta and D da ;t ,assembling only those active cracks in principal directions,which belong to positive principal stresses:D da ;ta ¼P þ:D da ;t :P þ;ð52Þwhere P þdenotes a fourth-order projection tensor.According to [31],one possible form of P þwith H ðÞas Heaviside-function is:P þ¼X 3i ¼1H r ði ÞÀÁM ði Þr M ði Þr :ð53Þ3.2.5.Tangential stiﬀness relationWith all these assumptions the tangential stiﬀness relation will expectedly slightly diﬀer from that one in Section 2.For its evaluation,we determine the consis-tency parameters by use of the consistency condition _Ut ði Þ¼0,the damage rule (49)and the kinematic soft-ening rule (48),ﬁnally arriving at:_r ¼D À1"ÀX 3i ¼1H _k t ði Þ D À1:M ði Þn M ði Þn :D À1M ði Þn :D À1:M ði ÞnÀZ a i t n ÀÁ#:_e :ð54Þ3.2.6.Assumed uniaxial stress–strain curve of concreteunder tensionIn order to relate our theory to experiments,the softening function Z a it n ÀÁwill be determined directly from a given uniaxial curve analogously to the proce-dure in Section 3.1.5.Due to Fig.4we hence distinguish:Region 1:Elastic region.Below the tension strength,the initial damage stress f ct ,linear-elastic behavior isassumed.Fig.3.Illustration of the assumed kinematic softening rule.W.B.Kr €a tzig,R.P €o lling /Computers and Structures 82(2004)1201–12151207。

Multiscale damage modelling for composite materials- theory and computational framework

Jacob Fish∗; † and Qing Yu
Departments of Civil; Mechanical and Aeronautical Engineering; Rensselaer Polytechnic Institute; Troy; NY 12180; U.S.A.
SUMMARY A non-local multiscale continuum damage model is developed for brittle composite materials. A triplescale asymptotic analysis is generalized to account for the damage phenomena occurring at micro-, meso- and macroscales. A closed-form expressions relating microscopic, mesoscopic and overall strains and damage is derived. The damage evolution is stated on the smallest scale of interest and non-local weighted phase average ÿelds over micro- and mesophases are introduced to alleviate the spurious mesh dependence. Numerical simulation conducted on a composite beam made of Blackglas= Nextel 2-D weave is compared with the test data. Copyright ? 2001 John Wiley & Sons, Ltd.

Using the Brittle Cracking Material Model in AbaqusExplicit

Using the Brittle Cracking Material Model in Abaqus/ExplicitThe brittle cracking constitutive model in Abaqus/Explicit is applicable to all brittle materials, not just concrete. This model can be expected to yield more accurate and realistic results than the tensile failure criterion (*TENSILE FAILURE), which is also applicable to brittle materials. In addition, the brittle cracking model allows more detailed modeling of post-cracking response than the simple tensile failure model.Capability summaryThe brittle cracking model in Abaqus/Explicit:•Is applicable to brittle materials whose compressive and pre-cracking tensile behavior can be represented through linear elasticity•Does not track individual macro-cracks, but captures the effect of cracking and damage on stresses and stiffness through constitutive calculations•Accounts for crack-induced anisotropy•Is conceptually compatible with the ideas of the Abaqus progressive damage framework (i.e.“damage initiation” and “damage evolution”), even though the user interface is different •Can be viewed as complementary to the ductile damage capability in Abaqus/Explicit•Accomplishes the same end-result effect as the Abaqus/Standard XFEM approach (i.e. through jumps in displacements) though the two modeling approaches are quite differentCrack initiation• A simple Rankine criterion used:•Crack initiates when the maximum principal stress exceeds the material tensile strength •Multiple cracks (as many as the number of direct stress components) can initiate at a point, in orthogonal directions•The crack orientations are governed by the principal stress directions •Post-cracking tensile (Mode I) and shear (Mode II) behavior can be independently controlled •Conceptually similar to “damage initiation & evolution” (i.e. material stiffness degradation is specified with a post-initiation “damage evolution” behavior)Post-cracking behavior•Mode I (i.e. normal, tensile) response specified by maximum strain, maximum displacement or Mode I fracture energy•Strain-based definition can introduce mesh sensitivity (due to the same arguments as for ductile damage)•Not recommended except for reinforced concrete•Displacement- or energy-based specification of post-cracking Mode I response is recommended (one can be derived from the other )•Element characteristic length may be used to convert strain-type data to displacement (or energy-based) definition (again, this is similar to ductile damage modeling)•Mode II (shear) response is specified by providing shear stiffness as a function of opening strain across the crack•Direct tabular entries (fraction of initial stiffness vs. crack opening strain) or analytical form (power law) can be usedElement deletion option•Elements may be removed from the model once the failure strain (or displacement) is reached •Allows specification of number of cracks at a material point that should fail before complete failure of the material occurs•Useful in overcoming possible excessive element distortion issues when the material point fails in tension•Exercise caution when using element removal option, since it precludes availability of the element to resist compression and hence may not be realistic for all brittle damage applications! Required input parametersFigure 1 shows a schematic of the material parameters and properties that are applicable when using the brittle cracking model:Figure 1: Required input parameters for the Abaqus/Explicit brittle cracking modelExample 1 - Crack propagation in a plate with a holeThis is the model in Abaqus Benchmark problem 1.19.2 (Release 6.9); there, it is solved in Abaqus/Standard using XFEM. The half-symmetric model is shown in Figure 2. Equal and opposite displacements are prescribed at the two ends of the plate.Fracture parameters used:-Max principal stress: 22.0E6 Pa-Fracture toughness (Modes I & II): 2.87E3 N/m (BK-type mode-dependence is used for damage evolution in the XFEM version of the model)-Linear shear stiffness degradation over a crack opening strain of 0.001 used in the brittle cracking model, along with element deletion (using “brittle failure” option)Figure 2: Half-symmetry model of a plate with a holeFigure 3 presents the predicted load-displacement histories from both the brittle cracking and XFEM approaches – a good agreement is observed.Figure 3: Comparison of load-displacement responses from brittle cracking and XFEM approaches The following points should be noted:•CPE4R elements with enhanced hourglass control were used in the Explicit analysis (XFEM model used fully integrated element)-Default hourglass control predicts lower peak load, with a somewhat larger deviation from the XFEM result•No significant sensitivity to load amplitude type (smooth step vs. ramp vs. constant velocity) observed in the quasi-static, Explicit result (all results not shown here)•Lack of crack orientation output and crack status output to the output database (ODB) prevents a direct comparison of brittle cracking result with the analytical applied stress vs. variation of crack length•Rough comparison using element status can still be made (not presented here)Example 2 – Brazilian Fracture TestIn the so-called Brazilian test for brittle fracture, a cylinder is compressed between two rigid platens ( Figure 4).Figure 4: Cylinder compressed between two rigid platensThe peak load is given by the theoretical expression:F max= π R f toWhere R = radius of the cylinder and f t0 = tensile strength of the material.Fracture parameters & dimensions used-Radius of cylinder: 150 mm-Max principal stress: 2.0 MPa-Fracture toughness: 0.055 N/mm-Linear shear stiffness degradation over a crack opening strain of ~0.008 was used in the brittle cracking model, along with element deletion (with the “brittle failure” option).Note that the actual damage evolution data and specification of element removal are not strictly necessary here, since we are mainly interested in predicting the peak load and comparing with the theoretical value. The use of element deletion is also optional, except to visualize the blunt cracks resulting from deleted elements (an approximate workaround for not being able to visualize the actual crack orientations and crack status in Abaqus/Viewer).Figure 5: Comparison of predicted peak load with theoretical valueFigure 5 shows that the brittle cracking model predicts a peak load that is consistent with the theoretically computed value. Note that enhanced hourglass control was used in this simulation.Limitations•Crack visualization is not possible the with the brittle cracking model, except through element deletion which, if used, can be misleading•Compressive and pre-cracking tensile behavior is limited to linear elasticity•No damage is possible in compression. Note however, that there are other models in Abaqus/Explicit, such as concrete damaged plasticity, that can represent brittle damage inelasto-plastic materials。

机械制造专业英语课后翻译标准答案

机械制造专业英语课后翻译标准答案应力与应变第一单元That branch of scientific analysis which motions, times and forces is called mechanicsand is made up of two parts, statics and dynamics.研究位移、时间和力运动乘力是科学分析法的一个分歧，被称作力学，力学由两大部静力学和动力学。

分组成，For example, if the force operating on a sleeve bearing becomes too high, it will squeeze outthe oil film and cause metal-to-metal contact, overheating and rapid failure of the bearing.例如：如果止推轴承上的作用力过大的话，会挤出油膜，引起金属和金属之间的相互接触，轴承将过热而迅速失效。

and of place application, direction, concept Our intuitive of force includes such ideas asmagnitude, and these are called the characteristics of a force.。

力的直观概念包括力的作用点、大小、方向，这些被称为力的三要素All bodies are either elastic or plastic and will be deformed if acted upon by forces. When thedeformation of such bodies is small, they are frequently assumed to be rigid, i.e., incapableof deformation, in order to simplify the analysis.的，如果受到力的作用就产生变形。

A plasticity and anisotropic damage model for plain concrete

A plasticity and anisotropic damage model forplain concreteUmit Cicekli,George Z.Voyiadjis *,Rashid K.Abu Al-RubDepartment of Civil and Environmental Engineering,Louisiana State University,CEBA 3508-B,Baton Rouge,LA 70803,USAReceived 23April 2006;received in ﬁnal revised form 29October 2006Available online 15March 2007AbstractA plastic-damage constitutive model for plain concrete is developed in this work.Anisotropic damage with a plasticity yield criterion and a damage criterion are introduced to be able to ade-quately describe the plastic and damage behavior of concrete.Moreover,in order to account for dif-ferent eﬀects under tensile and compressive loadings,two damage criteria are used:one for compression and a second for tension such that the total stress is decomposed into tensile and com-pressive components.Stiﬀness recovery caused by crack opening/closing is also incorporated.The strain equivalence hypothesis is used in deriving the constitutive equations such that the strains in the eﬀective (undamaged)and damaged conﬁgurations are set equal.This leads to a decoupled algo-rithm for the eﬀective stress computation and the damage evolution.It is also shown that the pro-posed constitutive relations comply with the laws of thermodynamics.A detailed numerical algorithm is coded using the user subroutine UMAT and then implemented in the advanced ﬁnite element program ABAQUS.The numerical simulations are shown for uniaxial and biaxial tension and compression.The results show very good correlation with the experimental data.Ó2007Elsevier Ltd.All rights reserved.Keywords:Damage mechanics;Isotropic hardening;Anisotropic damage0749-6419/$-see front matter Ó2007Elsevier Ltd.All rights reserved.doi:10.1016/j.ijplas.2007.03.006*Corresponding author.Tel.:+12255788668;fax:+12255789176.E-mail addresses:voyiadjis@ (G.Z.Voyiadjis),rabual1@ (R.K.AbuAl-Rub).International Journal of Plasticity 23(2007)1874–1900U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001875 1.IntroductionConcrete is a widely used material in numerous civil engineering structures.Due to its ability to be cast on site it allows to be used in diﬀerent shapes in structures:arc,ellipsoid, etc.This increases the demand for use of concrete in structures.Therefore,it is crucial to understand the mechanical behavior of concrete under diﬀerent loadings such as compres-sion and tension,for uniaxial,biaxial,and triaxial loadings.Moreover,challenges in designing complex concrete structures have prompted the structural engineer to acquire a sound understanding of the mechanical behavior of concrete.One of the most important characteristics of concrete is its low tensile strength,particularly at low-conﬁning pres-sures,which results in tensile cracking at a very low stress compared with compressive stresses.The tensile cracking reduces the stiﬀness of concrete structural components. Therefore,the use of continuum damage mechanics is necessary to accurately model the degradation in the mechanical properties of concrete.However,the concrete material undergoes also some irreversible(plastic)deformations during unloading such that the continuum damage theories cannot be used alone,particularly at high-conﬁning pressures. Therefore,the nonlinear material behavior of concrete can be attributed to two distinct material mechanical processes:damage(micro-cracks,micro-cavities,nucleation and coa-lescence,decohesions,grain boundary cracks,and cleavage in regions of high stress con-centration)and plasticity,which its mechanism in concrete is not completely understood up-to-date.These two degradation phenomena may be described best by theories of con-tinuum damage mechanics and plasticity.Therefore,a model that accounts for both plas-ticity and damage is necessary.In this work,a coupled plastic-damage model is thus formulated.Plasticity theories have been used successfully in modeling the behavior of metals where the dominant mode of internal rearrangement is the slip process.Although the mathemat-ical theory of plasticity is thoroughly established,its potential usefulness for representing a wide variety of material behavior has not been yet fully explored.There are many research-ers who have used plasticity alone to characterize the concrete behavior(e.g.Chen and Chen,1975;William and Warnke,1975;Bazant,1978;Dragon and Mroz,1979;Schreyer, 1983;Chen and Buyukozturk,1985;Onate et al.,1988;Voyiadjis and Abu-Lebdeh,1994; Karabinis and Kiousis,1994;Este and Willam,1994;Menetrey and Willam,1995;Grassl et al.,2002).The main characteristic of these models is a plasticity yield surface that includes pressure sensitivity,path sensitivity,non-associativeﬂow rule,and work or strain hardening.However,these works failed to address the degradation of the material stiﬀness due to micro-cracking.On the other hand,others have used the continuum damage theory alone to model the material nonlinear behavior such that the mechanical eﬀect of the pro-gressive micro-cracking and strain softening are represented by a set of internal state vari-ables which act on the elastic behavior(i.e.decrease of the stiﬀness)at the macroscopic level (e.g.Loland,1980;Ortiz and Popov,1982;Krajcinovic,1983,1985;Resende and Martin, 1984;Simo and Ju,1987a,b;Mazars and Pijaudier-Cabot,1989;Lubarda et al.,1994). However,there are several facets of concrete behavior(e.g.irreversible deformations, inelastic volumetric expansion in compression,and crack opening/closure eﬀects)that can-not be represented by this method,just as plasticity,by itself,is insuﬃcient.Since both micro-cracking and irreversible deformations are contributing to the nonlinear response of concrete,a constitutive model should address equally the two physically distinct modes of irreversible changes and should satisfy the basic postulates of thermodynamics.1876U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900 Combinations of plasticity and damage are usually based on isotropic hardening com-bined with either isotropic(scalar)or anisotropic(tensor)damage.Isotropic damage is widely used due to its simplicity such that diﬀerent types of combinations with plasticity models have been proposed in the literature.One type of combination relies on stress-based plasticity formulated in the eﬀective(undamaged)space(e.g.Yazdani and Schreyer,1990; Lee and Fenves,1998;Gatuingt and Pijaudier-Cabot,2002;Jason et al.,2004;Wu et al., 2006),where the eﬀective stress is deﬁned as the average micro-scale stress acting on the undamaged material between micro-defects.Another type is based on stress-based plastic-ity in the nominal(damaged)stress space(e.g.Bazant and Kim,1979;Ortiz,1985;Lubliner et al.,1989;Imran and Pantazopoulu,2001;Ananiev and Ozbolt,2004;Kratzig and Poll-ing,2004;Menzel et al.,2005;Bru¨nig and Ricci,2005),where the nominal stress is deﬁned as the macro-scale stress acting on both damaged and undamaged material.However,it is shown by Abu Al-Rub and Voyiadjis(2004)and Voyiadjis et al.(2003,2004)that coupled plastic-damage models formulated in the eﬀective space are numerically more stable and attractive.On the other hand,for better characterization of the concrete damage behavior, anisotropic damage eﬀects,i.e.diﬀerent micro-cracking in diﬀerent directions,should be characterized.However,anisotropic damage in concrete is complex and a combination with plasticity and the application to structural analysis is straightforward(e.g.Yazdani and Schreyer,1990;Abu-Lebdeh and Voyiadjis,1993;Voyiadjis and Kattan,1999;Carol et al.,2001;Hansen et al.,2001),and,therefore,it has been avoided by many authors.Consequently,with inspiration from all the previous works,a coupled anisotropic dam-age and plasticity constitutive model that can be used to predict the concrete distinct behavior in tension and compression is formulated here within the basic principles of ther-modynamics.The proposed model includes important aspects of the concrete nonlinear behavior.The model considers diﬀerent responses of concrete under tension and compres-sion,the eﬀect of stiﬀness degradation,and the stiﬀness recovery due to crack closure dur-ing cyclic loading.The yield criterion that has been proposed by Lubliner et al.(1989)and later modiﬁed by Lee and Fenves(1998)is adopted.Pertinent computational aspects con-cerning the algorithmic aspects and numerical implementation of the proposed constitu-tive model in the well-knownﬁnite element code ABAQUS(2003)are presented.Some numerical applications of the model to experimental tests of concrete specimens under dif-ferent uniaxial and biaxial tension and compression loadings are provided to validate and demonstrate the capability of the proposed model.2.Modeling anisotropic damage in concreteIn the current literature,damage in materials can be represented in many forms such as speciﬁc void and crack surfaces,speciﬁc crack and void volumes,the spacing between cracks or voids,scalar representation of damage,and general tensorial representation of damage.Generally,the physical interpretation of the damage variable is introduced as the speciﬁc damaged surface area(Kachonov,1958),where two cases are considered:iso-tropic(scalar)damage and anisotropic(tensor)damage density of micro-cracks and micro-voids.However,for accurate interpretation of damage in concrete,one should con-sider the anisotropic damage case.This is attributed to the evolution of micro-cracks in concrete whereas damage in metals can be satisfactorily represented by a scalar damage variable(isotropic damage)for evolution of voids.Therefore,for more reliable represen-tation of concrete damage anisotropic damage is considered in this study.The eﬀective(undamaged)conﬁguration is used in this study in formulating the damage constitutive equations.That is,the damaged material is modeled using the constitutive laws of the eﬀective undamaged material in which the Cauchy stress tensor,r ij,can be replaced by the eﬀective stress tensor, r ij(Cordebois and Sidoroﬀ,1979;Murakami and Ohno,1981;Voyiadjis and Kattan,1999):r ij¼M ijkl r klð1Þwhere M ijkl is the fourth-order damage eﬀect tensor that is used to make the stress tensor symmetrical.There are diﬀerent deﬁnitions for the tensor M ijkl that could be used to sym-metrize r ij(see Voyiadjis and Park,1997;Voyiadjis and Kattan,1999).In this work the deﬁnition that is presented by Abu Al-Rub and Voyiadjis(2003)is adopted:M ijkl¼2½ðd ijÀu ijÞd klþd ijðd klÀu klÞ À1ð2Þwhere d ij is the Kronecker delta and u ij is the second-order damage tensor whose evolution will be deﬁned later and it takes into consideration diﬀerent evolution of damage in diﬀer-ent directions.In the subsequence of this paper,the superimposed dash designates a var-iable in the undamaged conﬁguration.The transformation from the eﬀective(undamaged)conﬁguration to the damaged one can be done by utilizing either the strain equivalence or strain energy equivalence hypoth-eses(see Voyiadjis and Kattan,1999).However,in this work the strain equivalence hypothesis is adopted for simplicity,which basically states that the strains in the damaged conﬁguration and the strains in the undamaged(eﬀective)conﬁguration are equal.There-fore,the total strain tensor e ij is set equal to the corresponding eﬀective tensor e ij(i.e.e ij¼ e ijÞ,which can be decomposed into an elastic strain e eij (= e eijÞand a plastic straine p ij(= e p ijÞsuch that:e ij¼e eij þe p ij¼ e eijþ e p ij¼ e ijð3ÞIt is noteworthy that the physical nature of plastic(irreversible)deformations in con-crete is not well-founded until now.Whereas the physical nature of plastic strain in metals is well-understood and can be attributed to the generation and motion of dislocations along slip planes.Therefore,in metals any additional permanent strains due to micro-cracking and void growth can be classiﬁed as a damage strain.These damage strains are shown by Abu Al-Rub and Voyiadjis(2003)and Voyiadjis et al.(2003,2004)to be minimal in metals and can be simply neglected.Therefore,the plastic strain in Eq.(3) incorporates all types of irreversible deformations whether they are due to tensile micro-cracking,breaking of internal bonds during shear loading,and/or compressive con-solidation during the collapse of the micro-porous structure of the cement matrix.In the current work,it is assumed that plasticity is due to damage evolution such that damage occurs before any plastic deformations.However,this assumption needs to be validated by conducting microscopic experimental characterization of concrete damage.Using the generalized Hook’s law,the eﬀective stress is given as follows: r ij¼E ijkl e eklð4Þwhere E ijkl is the fourth-order undamaged elastic stiﬀness tensor.For isotropic linear-elas-tic materials,E ijkl is given byE ijkl¼2GI dijkl þKI ijklð5ÞU.Cicekli et al./International Journal of Plasticity23(2007)1874–19001877where I dijkl ¼I ijklÀ13d ij d kl is the deviatoric part of the fourth-order identity tensorI ijkl¼12ðd ik d jlþd il d jkÞ,and G¼E=2ð1þmÞand K¼E=3ð1À2mÞare the eﬀective shearand bulk moduli,respectively,with E being the Young’s modulus and m is the Poisson’s ratio which are obtained from the stress–strain diagram in the eﬀective conﬁguration.Similarly,in the damaged conﬁguration the stress–strain relationship in Eq.(4)can be expressed by:r ij¼E ijkl e eklð6Þsuch that one can express the elastic strain from Eqs.(4)and(5)by the following relation:e e ij ¼EÀ1ijklr kl¼EÀ1ijklr klð7Þwhere EÀ1ijkl is the inverse(or compliance tensor)of the fourth-order damaged elastic tensorE ijkl,which are a function of the damage variable u ij.By substituting Eq.(1)into Eq.(7),one can express the damaged elasticity tensor E ijkl in terms of the corresponding undamaged elasticity tensor E ijkl by the following relation:E ijkl¼MÀ1ijmnE mnklð8ÞMoreover,combining Eqs.(3)and(7),the total strain e ij can be written in the following form:e ij¼EÀ1ijkl r klþe p ij¼EÀ1ijklr klþe p ijð9ÞBy taking the time derivative of Eq.(3),the rate of the total strain,_e ij,can be written as _e ij¼_e eijþ_e p ijð10Þwhere_e eij and_e p ij are the rate of the elastic and plastic strain tensors,respectively.Analogous to Eq.(9),one can write the following relation in the eﬀective conﬁguration:_e ij¼EÀ1ijkl _ rklþ_e p ijð11ÞHowever,since E ijkl is a function of u ij,a similar relation as Eq.(11)cannot be used. Therefore,by taking the time derivative of Eq.(9),one can write_e ij in the damaged con-ﬁguration as follows:_e ij¼EÀ1ijkl _r klþ_EÀ1ijklr klþ_e p ijð12ÞConcrete has distinct behavior in tension and compression.Therefore,in order to ade-quately characterize the damage in concrete due to tensile,compressive,and/or cyclic loadings the Cauchy stress tensor(nominal or eﬀective)is decomposed into a positive and negative parts using the spectral decomposition technique(e.g.Simo and Ju, 1987a,b;Krajcinovic,1996).Hereafter,the superscripts‘‘+”and‘‘À”designate,respec-tively,tensile and compressive entities.Therefore,r ij and r ij can be decomposed as follows:r ij¼rþij þrÀij; r ij¼ rþijþ rÀijð13Þwhere rþij is the tension part and rÀijis the compression part of the stress state.The stress tensors rþij and rÀijcan be related to r ij byrþkl ¼Pþklpqr pqð14ÞrÀkl ¼½I klpqÀPþijpqr pq¼PÀklpqr pqð15Þ1878U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900such that Pþijkl þPÀijkl¼I ijkl.The fourth-order projection tensors Pþijkland PÀijklare deﬁnedas follows:Pþijpq ¼X3k¼1Hð^rðkÞÞnðkÞi nðkÞj nðkÞpnðkÞq;PÀklpq¼I klpqÀPþijpqð16Þwhere Hð^ rðkÞÞdenotes the Heaviside step function computed at k th principal stress^rðkÞof r ij and nðkÞi is the k th corresponding unit principal direction.In the subsequent develop-ment,the superscript hat designates a principal value.Based on the decomposition in Eq.(13),one can assume that the expression in Eq.(1) to be valid for both tension and compression,however,with decoupled damage evolution in tension and compression such that:rþij ¼Mþijklrþkl; rÀij¼MÀijklrÀklð17Þwhere Mþijkl is the tensile damage eﬀect tensor and MÀijklis the corresponding compressivedamage eﬀect tensor which can be expressed using Eq.(2)in a decoupled form as a func-tion of the tensile and compressive damage variables,uþij and uÀij,respectively,as follows:Mþijkl ¼2½ðd ijÀuþijÞd klþd ijðd klÀuþklÞ À1;MÀijkl¼2½ðd ijÀuÀijÞd klþd ijðd klÀuÀklÞ À1ð18ÞNow,by substituting Eq.(17)into Eq.(13)2,one can express the eﬀective stress tensor as the decomposition of the fourth-order damage eﬀect tensor for tension and compression such that:r ij¼Mþijkl rþklþMÀijklrÀklð19ÞBy substituting Eqs.(14)and(15)into Eq.(19)and comparing the result with Eq.(1), one can obtain the following relation for the damage eﬀect tensor such that:M ijpq¼Mþijkl PþklpqþMÀijklPÀklpqð20ÞUsing Eq.(16)2,the above equation can be rewritten as follows:M ijpq¼Mþijkl ÀMÀijklPþklpq þMÀijpqð21ÞOne should notice the following:M ijkl¼Mþijkl þMÀijklð22Þoru ij¼uþij þuÀijð23ÞIt is also noteworthy that the relation in Eq.(21)enhances a coupling between tensileand compressive damage through the fourth-order projection tensor Pþijkl .Moreover,forisotropic damage,Eq.(20)can be written as follows:M ijkl¼Pþijkl1ÀuþþPÀijkl1ÀuÀð24ÞIt can be concluded from the above expression that by adopting the decomposition of the scalar damage variable u into a positive u+part and a negative uÀpart still enhances adamage anisotropy through the spectral decomposition tensors Pþijkl and PÀijkl.However,this anisotropy is weak as compared to the anisotropic damage eﬀect tensor presented in Eq.(21).U.Cicekli et al./International Journal of Plasticity23(2007)1874–190018793.Elasto-plastic-damage modelIn this section,the concrete plasticity yield criterion of Lubliner et al.(1989)which was later modiﬁed by Lee and Fenves(1998)is adopted for both monotonic and cyclic load-ings.The phenomenological concrete model of Lubliner et al.(1989)and Lee and Fenves (1998)is formulated based on isotropic(scalar)stiﬀness degradation.Moreover,this model adopts one loading surface that couples plasticity to isotropic damage through the eﬀective plastic strain.However,in this work the model of Lee and Fenves(1998)is extended for anisotropic damage and by adopting three loading surfaces:one for plastic-ity,one for tensile damage,and one for compressive damage.The plasticity and the com-pressive damage loading surfaces are more dominate in case of shear loading and compressive crushing(i.e.modes II and III cracking)whereas the tensile damage loading surface is dominant in case of mode I cracking.The presentation in the following sections can be used for either isotropic or anisotropic damage since the second-order damage tensor u ij degenerates to the scalar damage vari-able in case of uniaxial loading.3.1.Uniaxial loadingIn the uniaxial loading,the elastic stiﬀness degradation variables are assumed asincreasing functions of the equivalent plastic strains eþeq and eÀeqwith eþeqbeing the tensileequivalent plastic strain and eÀeq being the compressive equivalent plastic strain.It shouldbe noted that the material behavior is controlled by both plasticity and damage so that, one cannot be considered without the other(see Fig.1).For uniaxial tensile and compressive loading, rþij and rÀijare given as(Lee and Fenves,1998)rþ¼ð1ÀuþÞE eþe¼ð1ÀuþÞEðeþÀeþpÞð25ÞrÀ¼ð1ÀuÀÞE eÀe¼ð1ÀuÀÞEðeÀÀeÀpÞð26ÞThe rate of the equivalent(eﬀective)plastic strains in compression and tension,eÀep and eþep,are,respectively,given as follows in case of uniaxial loading:1880U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900_eþeq ¼_e p11;_eÀeq¼À_e p11ð27Þsuch thateÀeq ¼Z t_eÀeqd t;eþeq¼Z t_eþeqd tð28ÞPropagation of cracks under uniaxial loading is in the transverse direction to the stress direction.Therefore,the nucleation and propagation of cracks cause a reduction of the capacity of the load-carrying area,which causes an increase in the eﬀective stress.This has little eﬀect during compressive loading since cracks run parallel to the loading direc-tion.However,under a large compressive stress which causes crushing of the material,the eﬀective load-carrying area is also considerably reduced.This explains the distinct behav-ior of concrete in tension and compression as shown in Fig.2.It can be noted from Fig.2that during unloading from any point on the strain soften-ing path(i.e.post peak behavior)of the stress–strain curve,the material response seems to be weakened since the elastic stiﬀness of the material is degraded due to damage evolution. Furthermore,it can be noticed from Fig.2a and b that the degradation of the elastic stiﬀ-ness of the material is much diﬀerent in tension than in compression,which is more obvi-ous as the plastic strain increases.Therefore,for uniaxial loading,the damage variable can be presented by two independent damage variables u+and uÀ.Moreover,it can be noted that for tensile loading,damage and plasticity are initiated when the equivalent appliedstress reaches the uniaxial tensile strength fþ0as shown in Fig.2a whereas under compres-sive loading,damage is initiated earlier than plasticity.Once the equivalent applied stressreaches fÀ0(i.e.when nonlinear behavior starts)damage is initiated,whereas plasticityoccurs once fÀu is reached.Therefore,generally fþ¼fþufor tensile loading,but this isnot true for compressive loading(i.e.fÀ0¼fÀuÞ.However,one may obtain fÀ%fÀuin caseof ultra-high strength concrete.3.2.Multiaxial loadingThe evolution equations for the hardening variables are extended now to multiaxial loadings.The eﬀective plastic strain for multiaxial loading is given as follows(Lubliner et al.,1989;Lee and Fenves,1998):U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001881_e þeq ¼r ð^ r ij Þ^_e p maxð29Þ_e Àeq ¼Àð1Àr ð^ r ij ÞÞ^_e p min ð30Þwhere ^_e p max and ^_e p min are the maximum and minimum principal values of the plastic strain tensor _e p ij such that ^_e p 1>^_e p 2>^_e p 3where ^_e p max ¼^_e p 1and ^_e p min ¼^_ep 3.Eqs.(29)and (30)can be written in tensor format as follows:_j p i ¼H ij ^_e p jð31Þor equivalently _e þeq 0_e Àeq8><>:9>=>;¼H þ0000000H À264375^_e p 1^_e p 2^_e p 38><>:9>=>;ð32ÞwhereH þ¼r ð^ rij Þð33ÞH À¼Àð1Àr ð^ r ij ÞÞð34ÞThe dimensionless parameter r ð^ rij Þis a weight factor depending on principal stresses and is deﬁned as follows (Lubliner et al.,1989):r ð^ r ij Þ¼P 3k¼1h ^ r k i P k ¼1j ^ r kj ð35Þwhere h i is the Macauley bracket,and presented as h x i ¼1ðj x j þx Þ,k ¼1;2;3.Note that r ð^ rij Þ¼r ð^r ij Þ.Moreover,depending on the value of r ð^r ij Þ,–in case of uniaxial tension ^ r k P 0and r ð^ r ij Þ¼1,–in case of uniaxial compression ^ rk 60and r ð^ r ij Þ¼03.3.Cyclic loadingIt is more diﬃcult to address the concrete damage behavior under cyclic loading;i.e.transition from tension to compression or vise versa such that one would expect that under cyclic loading crack opening and closure may occur and,therefore,it is a challenging task to address such situations especially for anisotropic damage evolution.Experimentally,it is shown that under cyclic loading the material goes through some recovery of the elastic stiﬀness as the load changes sign during the loading process.This eﬀect becomes more sig-niﬁcant particularly when the load changes sign during the transition from tension to com-pression such that some tensile cracks tend to close and as a result elastic stiﬀness recovery occurs during compressive loading.However,in case of transition from compression to tension one may thus expect that smaller stiﬀness recovery or even no recovery at all may occur.This could be attributed to the fast opening of the pre-existing cracks that had formed during the previous tensile loading.These re-opened cracks along with the new cracks formed during the compression will cause further reduction of the elastic stiﬀness that the body had during the ﬁrst transition from tension to compression.The1882U.Cicekli et al./International Journal of Plasticity 23(2007)1874–1900consideration of stiﬀness recovery eﬀect due to crack opening/closing is therefore impor-tant in deﬁning the concrete behavior under cyclic loading.Eq.(21)does not incorporate the elastic stiﬀness recovery phenomenon as well as it does not incorporate any coupling between tensile damage and compressive damage and,therefore,the formulation of Lee and Fenves(1998)for cyclic loading is extended here for the anisotropic damage case.Lee and Fenves(1998)deﬁned the following isotropic damage relation that couples both tension and compression eﬀects as well as the elastic stiﬀness recovery during transi-tion from tension to compression loading such that:u¼1Àð1Às uþÞð1ÀuÀÞð36Þwhere sð06s61Þis a function of stress state and is deﬁned as follows: sð^ r ijÞ¼s0þð1Às0Þrð^ r ijÞð37Þwhere06s061is a constant.Any value between zero and one results in partial recovery of the elastic stiﬀness.Based on Eqs.(36)and(37):(a)when all principal stresses are positive then r=1and s=1such that Eq.(36)becomesu¼1Àð1ÀuþÞð1ÀuÀÞð38Þwhich implies no stiﬀness recovery during the transition from compression to tension since s is absent.(b)when all principal stresses are negative then r=0and s¼s0such that Eq.(36)becomesu¼1Àð1Às0uþÞð1ÀuÀÞð39Þwhich implies full elastic stiﬀness recovery when s0¼0and no recovery when s0¼1.In the following two approaches are proposed for extending the Lee and Fenves(1998)model to the anisotropic damage case.Theﬁrst approach is by multiplying uþij in Eq.(18)1by the stiﬀness recovery factor s:Mþijkl ¼2½ðd ijÀs uþijÞd klþd ijðd klÀs uþklÞ À1ð40Þsuch that the above expression replaces Mþijkl in Eq.(21)to give the total damage eﬀecttensor.Another approach to enhance coupling between tensile damage and compressive dam-age as well as in order to incorporate the elastic stiﬀness recovery during cyclic loading for the anisotropic damage case is by rewriting Eq.(36)in a tensor format as follows:u ij¼d ijÀðd ikÀs uþik Þðd jkÀuÀjkÞð41Þwhich can be substituted back into Eq.(2)to get theﬁnal form of the damage eﬀect tensor, which is shown next.It is noteworthy that in case of full elastic stiﬀness recovery(i.e.s=0),Eq.(41)reducesto u ij¼uÀij and in case of no stiﬀness recovery(i.e.s=1),Eq.(41)takes the form ofu ij ¼uÀijþuþikÀuþikuÀjksuch that both uþijand uÀijare coupled.This means that duringthe transition from tension to compression some cracks are closed or partially closed which could result in partial recovery of the material stiﬀness(i.e.s>0)in the absence U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001883。

混凝土热开裂数值描述方法1(英文)

Thermal induced stress and associated crackingin cement-based composite at elevated temperatures––Part I:Thermal cracking around single inclusionY.F.Fu a ,Y.L.Wonga,*,C.A.Tang b ,C.S.PoonaaDepartment of Civil and Structural Engineering,The Hong Kong Polytechnic University,Hong Kong,ChinabLab of Numerical Test of Material Failure,Northeastern University,Shenyang 110006,ChinaAccepted 25April 2003AbstractThis paper presents the development and veriﬁcation of 2-D mesoscopic thermoelastic damage model used to numerically quantify the thermal stresses and crack development of a cement-based composite subjected to elevated temperatures.The program is then used to study the thermal fracture behavior of a cement-based matrix with a single inclusion.The results show that the mechanisms of thermal damage and fracture of the composite depend on (i)the diﬀerence between the coeﬃcients of thermal ex-pansion (CTE)of the inclusion and the cement-based matrix,(ii)the strengths of materials,and (iii)the heterogeneity of materials at meso-scale.The thermal cracking is an evolution process from diﬀused damage,nucleation,and ﬁnally linkage of cracks.If the CTE of the inclusion is greater than that of the matrix,radial cracks will form in the matrix.On the other hand,inclusion cracks and tangential cracks at the interface between inclusion and matrix will form if the CTE of the inclusion is smaller than that of the matrix.Ó2003Elsevier Ltd.All rights reserved.Keywords:Thermal stress;Thermal induced cracking;Heterogeneity;Numerical simulation1.IntroductionThermal cracking induced by thermal mismatch has been one of the problems in a cement-based composite material under elevated temperatures.For a multi-phase material,the eigenstrains deriving from the heteroge-neous deformations among phase components inevita-bly cause cracking in the composite,even though it is under a uniform temperature ﬁeld.Experimental results [1]have shown that this type of cracking signiﬁcantly reduces the strength and elastic modulus of a cement-based composite.However,the entire thermal cracking process (initiation,propagation and linkage of cracks)and the associated stress distributions under elevated temperatures are diﬃcult to quantify experimentally,mainly because of the limitation of equipment and the complex structure of a composite material.In order to understand the failure mechanism of a composite material due to thermal eﬀects,many math-ematical models have been proposed [1–3].In these models,the driving stresses for the crack initiation and propagation are the heterogeneous eigenstresses,which develop in and around the restraining inclusion.These eigenstresses might be caused by thermal expansion,shrinkage [4,5],initial strains and misﬁt strains.Timo-shenko and Goodier in 1970[6]proposed a closed-form solution for the axisymmetric problem of a circular in-clusion concentrically embedded in the circular disc of another phase material with diﬀerent thermal and mechanical properties.Hsueh et al.[7]transformed a composite with a microstructure of square-array,hexagon-array,brick-array grains,as well as the actual microstructure of random-array grains into a simple composite-circle analytical model.The residual thermal stresses were predicted reasonably well using the pro-posed linear elastic solutions except for the model mi-crostructure of brick-array grains.A modiﬁed version of Timoshenko and Goodier Õs solution incorporating the longitudinal strain proposed by Gentry and Husain*Corresponding author.Tel.:+852-2766-6009;fax:+852-2334-6389.E-mail address:ceylwong@.hk (Y.L.Wong).0958-9465/$-see front matter Ó2003Elsevier Ltd.All rights reserved.doi:10.1016/S0958-9465(03)00086-6Cement &Concrete Composites 26(2004)99–111[2]was also used to study the diﬀerential pressure de-veloped in the interface between concrete and a com-posite rod.As for a40°C temperature increase,the concrete was modeled with a linear-elastic and nonlin-ear tension-softening material model using aﬁnite ele-ment approach.The calculated results showed that the large spacing of the rods and the thick concrete cover were helpful to reduce the tensile stress in concrete as well as the potential for thermally induced cracking. Based on a fracture mechanics model,Timoshenko and GoodierÕs solution was adopted by Dela and Stang[3] to calculate the crack growth with time in a high-shrinkage cement paste with a single aggregate disc.The experimentally measured stresses in the selected circular aggregate were employed to predict the stresses dis-tributed in cement paste and the crack growth at a crack tip close to the aggregate in terms of a given stress in-tensity factor.Although the above-mentioned models deepen the understanding on thermal stress and cracking,essen-tially,none of them can simulate the entire thermal cracking process from crack initiation to propagation. HsuehÕs and RussellÕs models can determine the stress distribution around a single inclusion in the composite before crack initiates.DelaÕs model was suitable to cal-culate the critical stress value when an existing crack starts to grow.The stress distribution represented by this model would be invalid as soon as the crack is ex-tended.A fracture mechanics model is able to study the growth of existing single crack,but it is not suitable to explain the initiation and coalescence of cracks.More importantly,the phase materials of a cement-based composite are often heterogeneous so that the eﬀect of change in microstructure(mesostructure)on the mac-roscopic behavior is diﬃcult to be studied by using an analytical model.Consequently,a numerical method appears to be an eﬀective tool to model cracking processes.Substantial progress[8,9]has been achieved in numerical simulation of failure occurring in a cement-based composite at ambient temperatures.However,a satisfactory model to simulate the cracking processes caused by the thermal induced stresses in a heated cement-based composite is still not available.The aim of this paper is to propose and verify a mesoscopic thermoelastic damage(MTED)model,that can numerically simulate the formation,extension and coalescence of cracks in a cement-based composite ma-terial(cement-based matrix+aggregate inclusion), caused by the thermal mismatch of the matrix and the inclusion under uniform temperature variations and free boundary conditions.Numerical studies of the eﬀects of the thermal mismatch between the matrix and a single circular inclusion on the stress distribution and crack development are also presented.2.Numerical modelIn the MTED model,phase materials of a composite are considered to be heterogeneous following the Wei-bull distribution.Tensile and shear cracking at meso-scale occur if the stress in the composite subjected to high temperatures satisﬁed with the failure criteria of Coulomb–Mohr with tensioncutoﬀ. 100Y.F.Fu et al./Cement&Concrete Composites26(2004)99–1112.1.Material modelFor a cement-based composite material,the phase materials are cement mortar matrix and aggregate in-clusions.Although the composite material is regarded as an isotropic elastic-brittle solid at a macroscopic scale, while the individual grains in the matrix and inclusions are distinguished at microscopic or mesoscopic scales [8].The eﬀect of heterogeneity on the stress distribution has been studied[10],and much of the behavior ob-served at a macro-level can be explained in terms of the material structure at a meso-level.As a result,the matrix and the inclusions are considered as disorder solids in a meso-scale in this study.To account for the heterogeneity of the matrix and inclusions,their statistical distributions of properties (elastic modulus,compressive strength and Poisson ratio)are assumed to follow the Weibull distribution:uðh;bÞ¼hb0Ább0hÀ1ÁeÀðb=b0Þhð1aÞwhere uðh;bÞis the distribution density of parameter b which is a material property(such as strength,elasticity and Poisson ratio)of a representative volume element (RVE)in the mesh divisions,and b0is the mean value of the material property under consideration.h is the ho-mogeneity index of the RVE which represents the degree of homogeneity.The statistical distribution function Uðh;bÞis expressed by Eq.(1b)after integrating Eq.(1a):Uðh;bÞ¼1ÀeÀðb=b0Þhð1bÞThe randomness of the mechanical properties of RVE can be simulated using the distribution function with given parameters h and b0,i.e.Uðh;b0Þ.The relationship of distribution density of RVE strength and homoge-neity index is shown in Fig.1.With increasing h,the material is more homogeneous or vice versa.For in-stance,we consider a material with a mean strength of 200MPa.If the material has a homogeneity indexðhÞof 30,the distribution probabilities Uð30;200Þwill be close to zero and unity for the strengths of RVE less than160 and210MPa,respectively as shown in Fig.1b.In an-other case,if it has a homogeneity indexðhÞof1.1,the corresponding distribution probabilities Uð1:1;200Þwill become0.54and0.64for the strengths of RVE less than 160and210MPa,respectively.From these strength distributions,it is evident that increasing heterogeneity of a material will increase both the diﬀerence in me-chanical properties among the RVEs,and the popula-tion of the RVE with lower strengths.The strength,elastic modulus and Poisson ratio are randomly allocated to each RVE so to account for the inherent variability in phase materials,using the Monte-Carlo method.A more detailed introduction and ex-planation to the material model were reported in our previous publications[11–13].The thermal properties (CTE)of the phase materials are assumed to be uniform and location-invariant,and only depend on the indi-vidual phase.2.2.Mesoscopic thermoelastic damage(MTED)modelIt has been known that the thermal damages of a heated concrete is a complex problem.There are a number of aﬀecting factors,such as thermal mismatch, temperature gradient,degradation of mechanical prop-erties of cementitious materials due to chemical de-composition,and pore water pressure,that cause such damages.However,the focus of this paper is on the damage caused by diﬀerential thermal strains as aresultof diﬀerent CTEs of the phase materials(matrix and inclusion).Studies[8,14]show that the macroscopic fracture of materials is always related to the initiation and propagation of cracks at a meso-scale.Hence,it is assumed that the damage of a cement-based composite is due to the cracking caused by thermal induced stresses at a meso-scale.The bond between the matrix and the inclusion is considered to be perfect.In fact,the pro-posed model can be further modiﬁed to incorporate the eﬀects of temperature gradients and temperature-de-pendent mechanical properties,pending on the avail-ability of experimental data to quantify the associated simulation functions,details of which are under inves-tigation by the authors of this paper.In the numerical modeling,each phase material is discretized into many RVEs with a suitable charac-teristic length.In general,the precision of computa-tional results will increase with decreasing RVE size,at the expense of longer computational time.The RVE has the same size as the meshedﬁnite element in this paper.It is also assumed that the stress–strain rela-tionship of a RVE is linearly elastic till its peak-strength is reached,and thereafter follows an abrupt drop to its residual strength.Cracking is treated as a smeared phenomenon.That is,a crack is not consid-ered as a discrete displacement jump,but rather changing the properties of the RVE according to a continuum law,such as damage mechanics.Although this modeling approach might appear to be crude, however,the complex failure phenomenon(such as compressive and tensile failure)and the nonlinear be-havior in a macro-scale have been proved to be suc-cessfully simulated using the material heterogeneity [11].The behavior laws of the RVE are implemented by introducing a MTED variable D into a constitutive relationship.Based on the above-mentioned ideas and the damage mechanics[21],the general form of an eﬀective stress for a given state of damage for a RVE can be expressed as follows:r¼ð1ÀDÞÁE0Áe rð2Þwhere r is the eﬀective stress,D is the damage variable, E0is the elastic modulus at a reference/undamaged condition(such as at reference temperature),and e r is the strain.Under a uniform temperatureﬁeld,the damage is induced both by diﬀerential thermal strains and by the temperature increment D T,the general ex-pression of damage variable is D¼Dðe r;D TÞ.Let D m and D T denote the damages by the thermal strain and temperature increment,respectively.They can be ex-pressed in terms of the stiﬀness degradation as follows: D m¼1ÀEðe rÞE0ð3ÞD T¼1ÀEðD TÞE0ð4Þwhere Eðe rÞand EðD TÞare the elastic modulus corre-sponding to a given thermal strain e r and the elastic modulus at temperature increment of D T,respectively. If they are independent,the damage variable Dðe r;D TÞcan be expressed as follow:Dðe r;D TÞ¼1Àð1ÀD mÞÁð1ÀD TÞDðe r;D TÞ¼1ÀEðe rÞE0ÁEðD TÞE0ð5ÞSince the temperature-dependent properties are not considered,the damage D T is equal to zero.According to the description of the damage process of a material by Mazars[14]and Yu[15],the thermal induced damage before and after the peak-strength can be determined by the thermal strain and the temperature increment D T102Y.F.Fu et al./Cement&Concrete Composites26(2004)99–111through a separation function,respectively.Fig.2shows a general constitutive relationship of a RVE under thermal loading.At a temperature increment of D T ,the initial thermal strain e thermal is equal to a ÁD T ,and the damage at any given thermal strain can be calculated from Eq.(6)D ðe ;D T Þ¼0;e thermal 6e 6e r 01Àn ðe r 0Àa ÁD T Þðe Àa ÁD T Þ;e P e r 0under compression1;e P e r 0under tension8<:ð6Þwhere D ðe r ;D T Þrepresents the thermal damage with respect to the thermal strains.e r 0is the strain at peak-strength;n ð¼S r =S Þis the coeﬃcient of residual strength for a RVE,S and S r are the peak-strength and residual strength,respectively.Under compression,n is less than 1but greater than 0.Under tension,n is equal to 0.When the strain e becomes smaller than or equal to e r 0,the RVE is undamaged and intact,and D ¼0.When the strain e is larger than e r 0,and under a compressive state,the RVE is damaged,i.e.D >0,and damage variable shall be calculated by the residual strength.Under a tensile state,the RVE is fully damaged and does not sustain any load,and D ¼1.The behavior for a given state of thermal induced damage can be represented by r ¼½1ÀD ðe r ;D T Þ ÁE 0Áðe r Àe thermal Þð7ÞHence,substituting Eq.(6)into Eq.(7),a mesoscopic nonlocal damage model,which can describe the com-plete thermal induced damage process,is expressed as:r ¼E 0Áðe Àa ÁD T Þ;e thermal 6e 6e r 0n ÁE 0Áðe r 0Àa ÁD T Þ;e P e r 0under compression0;e P e r 0under tension8<:ð8ÞIn order to simulate the thermal damage induced by thermal tensile or compressive stresses,a failure crite-rion,which can consider the eﬀects of both tension and compression,is necessary.In this study,the Mohr–Coulomb criterion with tension cutoﬀ[16]is chosen as the criterion of cracking:r 1À1þSin h r 2P S c if r 1P S c 1À1þSin h Á1ÀÁor r 26ÀS t if r 16S c 1À1þSin h 1ÀSin h Á1k ÀÁ8<:ð9Þwhere S c and S t are the uniaxial compressive strengthand tensile strength respectively,S t ¼Àk ÁS c ,and k is the ratio of tensile strength to compressive strength.h is the friction angle of the material.All these parameters can be obtained experimentally.r 1and r 2are the maximum and minimum principal stresses respectively.A compressive stress is positive,and a tensile stress is negative.Finally,a ﬁnite element program T-MFPA,incor-porating the above-mentioned MTED model and failurecriteria,was developed based on the Material Failure Process Analysis (MFPA)program [11,12],using a four-node isoperimetric element.2.3.Numerical specimensNumerical tests of ﬁve specimens (one circular spec-imen and four square specimens)using the T-MFPA program are reported in following sections.Let a i de-note the CTE of the inclusion and a m be the CTE of the matrix.The specimens were analyzed under a plain stress condition without external loading.Specimen no.1is a circular specimen comprising two diﬀerent homogeneous phase materials (matrix and in-clusion,see Fig.3a).It is numerically heated under a uniform temperature ﬁeld of 50°C,and free boundary conditions.The numerical thermal stresses determined from the proposed program are compared with those derived from the classical theory of thermoelasticity,from which the validity of the MTED model in an elastic and undamaged state can be justiﬁed.The me-chanical and geometrical properties of the phase mate-rials are listed in Table 1.In this case,the CTE of the inclusion is greater than that of the matrix.A homoge-neity index h ¼300is chosen so that the phase materials are basically homogeneous in nature.The numerical results are shown in the following section.Y.F.Fu et al./Cement &Concrete Composites 26(2004)99–111103In order to determine the eﬀects of material hetero-geneity,material strength,and CTE on the stress de-velopment and the process of thermal cracking around a single inclusion,four square specimens (Specimens no.2to no.5,see Fig.4)with diﬀerent thermal and me-chanical properties (see Table 2)are numerically stud-ied.Basically,the specimens can be classiﬁed into two groups.In Group 1(Specimens no.2and no.3),the CTE of the matrix is smaller than that of the inclusion.In Group 2(Specimens no.4and no.5),the CTE of matrix is larger than that of the inclusion.Within a group,the only variable is the mean strength of the in-clusion.The four specimens have the same homogeneity index h equal to 3,representing a high degree of heter-ogeneity.They are subjected to a uniform temperature increment from 20to 620°C at an incremental step of 10°C.3.Model validationFig.3b shows the comparison of the thermal stresses around the single inclusion of Specimen no.1calculated from the T-MFPA program,and from the analytical solutions (Eqs.(10)and (11))derived from the classical theory of thermo-elasticity [6,17].It is evident that under an elastic and undamaged state,an excellent agreement between the stresses ob-tained from the two diﬀerent approaches has been ob-tained.4.Thermal cracking history of square specimens Fig.5shows the eﬀect of thermal mismatch on the thermal induced damages and fracture processes of Specimen no.2(Group 1)and Specimen no.4(Group 2)due to increasing temperatures.Fig.6illustrates the inﬂuence of the mean strength of the inclusions on the crack development in each group.Detailed descriptions of crack formation of the specimens are shown below.4.1.Thermal cracking in composite of a i >a mIn the case of Specimen no.2,since the a i (CTE)of the inclusion is greater than that of the matrix ða m Þ,the incompatibility of thermal deformation at the interface between the matrix and the inclusion leads to the stress concentration around the inclusion (see Fig.5a(a)).The inclusion is under a statistically hydrostatic compres-sion,and the matrix is under a combination of com-pression and tension.When the temperature reaches 200°C,a few broken elements randomly occur (due to heterogeneity)in the high stress zone around the inclu-sion.With increasing temperatures,the number of the diﬀused damaged elements increases.The damaged ele-ments exist in both the high stress zone and in the low stress zone,but most of them are located near the for-Table 1Material properties of circular Specimen no.1ParameterValue Matrix Inclusion Heterogeneity index (h )300300Mean elastic modulus (MPa)60,000100,000Mean compressive strength (MPa)3060Poisson ratio0.250.20Coeﬃcient of thermal expansion (/°C) 1.0E )5 1.1E )5Temperature increment (°C)1010Tension cutoﬀ0.10.1Frictional angle (°)3030Diameter (mm)10020Number of elements31,4001256Fig.4.Numerical square specimen with single inclusion.Table 2Material properties of square Specimens no.2to no.5ParameterValue Matrix Inclusion Heterogeneity index (h )33Mean elastic modulus (MPa)60,000100,000Mean compressive strength (MPa)Specimen no.2200300Specimen no.3150Specimen no.4300Specimen no.5150Poisson ratio0.250.20Coeﬃcient of thermal expansion (/°C)Specimen no.2 1.0E )51.1E )5Specimen no.3 1.1E )5Specimen no.40.9E )5Specimen no.50.9E )5Temperature increment (°C)1010Tension cutoﬀ0.10.1Frictional angle (°)3030Dimension (mm)100Â100U 30Number of elements200Â2001412104Y.F.Fu et al./Cement &Concrete Composites 26(2004)99–111merly broken elements in the high stress zone (see Fig.5a(c)).When the temperature increases to 430°C,a macro-crack is formed ﬁrstly at the top-left area around the inclusion.At the same time,only a few of cracks nucleate far away from the high stress zone around the inclusion (see Fig.5a(d)).As the temperature further increases,the broken elements around the inclusion nucleate into several discontinuous macro-cracks (see Fig.5a(e)and (f)),and simultaneously corresponding tensile stress zones are formed at the tips of these cracks.Bridges are formed between the cracks due to the fact that many small cracks simultaneously grow at diﬀerent locations caused by the heterogeneity.This phenomenon is also described by Van Mier [19].As the temperature rises to 570°C,all these macro-cracks further propagate under the tensile stresses at their tips,followed by the occurrence of dispersed damaged elements in the frac-ture process zone.During the heating process,the macro-cracks are formed in the way that the discontin-uous cracks continue to grow and bridges are formed.The shapes of these cracks are irregular,rough and bi-furcate (see Fig.5a(e)–(h)).The macro-cracks formed along the radial direction around the inclusion can be called ‘‘radial cracks’’,which were also evident intheFig.5.(a)Thermal cracking of cement-based composite of Specimen no.2(inclusion diameter ¼30mm and a i ¼1:1Â10À6/°C).(b)Thermal cracking of cement-based composite of Specimen no.4(inclusion diameter ¼30mm and a i ¼0:9Â10À6/°C).Y.F.Fu et al./Cement &Concrete Composites 26(2004)99–111105experiments reported by Zhou et al.[18]and Golter-mann [5].It is also noted that when the main macro-cracks begin to propagate,the pace of minor crack develop-ment is slow down (see Fig.5a(g)and (h)).Fig.6a shows the thermal fracture process of the companion Specimen no.3with a lower mean strength ðr i3Þof the inclusion than that ðr i2Þin Specimen no.2.It is evident that the variation of the mean inclusion strength does not aﬀect the patterns of thermal damage initiation and propagation.4.2.Thermal cracking in composite of a i <a mIn the case of Specimen no.4,since the a i (CTE)of the inclusion is smaller than that of the matrix ða m Þ,azone of stress concentration also occurs around the in-clusion.The inclusion is stressed under tension and the matrix is under a combination of tension and com-pression (see Fig.5b(a)).When the temperature reaches 160°C,a few of the damaged elements distribute dis-orderly inside the inclusion.With increasing tempera-tures,the number of broken elements grows,and a few of them occur in the stress concentration zone outside the inclusion (see Fig.5b(c)).When the temperature increases to 400°C,the broken elements at the interface between the matrix and the inclusion nucleate and form several small discontinuous cracks.As the temperature becomes further higher,the discontinuous cracks at the interface propagate gradually and coalesce with the stress transferring from the inclusion and the matrix nearby the inclusion to the tips of the cracks (seeFig.Fig.6.(a)Thermal cracking processes of specimens in Group 1.(b)Thermal cracking processes of specimens in Group 2.106Y.F.Fu et al./Cement &Concrete Composites 26(2004)99–1115b(e)–(g)).Eventually,after the temperature has reached 620°C,most of all the elements around the interface between the matrix and the inclusion are broken and a nearly close circular macro-crack is formed at the in-terface.The high stress distributing inside the inclusion is transferred into the crack tips.This kind of crack is called‘‘tangential crack’’,which is also observed in the experiments by Zhou et al.[18]and Goltermann[5].Fig.6b demonstrates the thermal fracture process of the companion Specimen no.5with a lower mean strengthðr i5Þinclusion than thatðr i4Þin Specimen no.4. Although the thermal damage initiation and crack propagation of the two specimens are similar,the number of the broken elements and the kinds of cracks at each temperature level are diﬀerent.At a lower tem-perature,more elements in the inclusion of Specimen no. 5are damaged than those in Specimen no.4(see Fig. 6b(a)and(a0)).When the temperature reaches360°C, the macro-cracks pass through partly or wholly the in-clusion,and high stresses previously distributed around the inclusion are transferred into the tips of these cracks (as shown in Fig.6b(b0)).With increasing temperatures, these discontinuous cracks nucleate and coalesce with the redistributing stressﬁeld(as shown in Fig.6b(c0)and (d0)).This kind of crack occurred inside the inclusion is called‘‘inclusion crack’’.5.Thermal stressﬁelds of square specimens5.1.Eﬀect of thermal mismatchAlthough the four specimens are subjected to uniform temperature changes,local stress concentration occurs around the inclusion due to the thermal mismatch be-tween the matrix and the inclusion.When the CTE of the inclusion is greater than that of the matrix,the inclusion in Specimen no.2is stressed under a state of statistically hydrostatic compression due to the restriction from the matrix,and the matrix is under a general bi-axial state of stresses(tensile/com-pressive and shear stresses)due to the outward expan-sion from the inclusion.The distribution of maximum and minimum principal stresses and the maximum shear stress along the mid-section of Specimen no.2can be shown in Fig.7a(a).Although the maximum and mini-mum principal stresses in the inclusion are high,the maximum shear stress is much smaller so that few ele-ments with lower strength in this area reach their failure strength.The absence of tensile stresses in the inclusion delays the attainment of the Mohr–Coulomb with ten-sion cutoﬀfailure criterion.Unless the inclusion is ab-normally weak in compression,the strength of inclusion has no eﬀect on damage initiation(see Fig.6a).As a result,most of the diﬀused damages distribute in the high stress zone of the matrix around the inclusion for Group1specimens.With increasing temperatures,these broken elements nucleate and form several discontinu-ous cracks due to the stress redistribution at the crack tips.Since the minimum principal stress is nearly per-pendicular to the radial direction of the inclusion and is in tension,these cracks are developed in the manner of radial cracks in the matrix.When the CTE of the inclusion is smaller than that of the matrix,the inclusion in Specimen no.4is stressed under bi-axial tension,and the matrix remains in a state of compressive/tensile and shear stresses(see Fig.7b). Since a bi-axial tension leads to early attainment of the failure criterion,it is not surprised that the initiation of damage takes place only in the inclusion of Group2 specimens.In such a case,the strength of the inclusion has considerable eﬀects on the crack formation.That is, a weaker inclusion will have damage initiated at a lower temperature and grow more rapidly at high tempera-tures(see Fig.6b).The minimum principal stress in the matrix is parallel to the radius direction of the inclusion and is in tension,so that the main cracks propagate in the manner of tangential cracks at the matrix–inclusion interfacial region.5.2.Eﬀect of heterogeneity at meso-scaleThe thermal stressﬁelds are shown in Figs.4–6.The bright color indicates the higher stress,and vice versa.It is found that the points with diﬀerent scale colors exist in a same zone.It means that there are existing points subjected to diﬀerent stresses due to the heterogeneity at meso-scale in such zone,where the stressﬁeld is statis-tically uniform at macro-scale.The ratio of the local stress to the local strength is a very important parameter which can be used to decide whether or not an element fails.The eﬀect of the heterogeneity at meso-scale can be reﬂected by the stressﬂuctuation shown in Fig.7.In comparison with the results from Fig.3,the curves of stress distribution along the mid-section E–E in Speci-mens no.2to no.5before crack initiating are charac-terized by an irregular variation of stress values(see Fig. 7a and b).Such a strong thermal stressﬂuctuation in a heterogeneous composite,which can be quantitatively identiﬁed in our numerical study,is diﬃcult to be de-termined by experiments.Taking into account of the material heterogeneity, the failure of a material is dependent both on the in-duced stress level and on the strength itself.An element subjected to high stress may not break due to the fact that this element has higher strength;whereas an ele-ment subjected to low stress may break because of its low strength.These kinds of failure are deﬁnitely dif-ferent,since their released energies are diﬀerent.Con-sequently,some RVE can still remain un-fractured in a zone of high stress,if these elements have higherY.F.Fu et al./Cement&Concrete Composites26(2004)99–111107。

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SOLIDS WITH STRENGTHS
Tempe,
and S. MASTILOVIC State University,
Abstract-The paper presents a rate-type constitutive analysis of damage, applicable to brittle materials whose elastic properties degrade during a deformation process. Different tensile and compressive material responses are modeled incorporating positive and negative projections of the stress or strain tensors. Proposed evolution laws for the rate of compliance tensors are consistent with some of the prominent features of brittle material response. A new structure of the damage surface is introduced for a more accurate account of the effects of the hydrostatic states of stress on the overall response. Derived rate constitutive equations provide the explicit representation of the tangent compliance tensor. The proposed model is applied to uniaxial tension and compression to illustrate nonlinear relationships between stress and longitudinal, lateral, and volumetric strains. The proposed model is compared with some of the existing theories.
681
682
V. A. LUBARDA etal
and strain tensors are obtained by using the positive stress and strain projection operators P,’ and
P;‘. When applied to stress c and strain c tensors, these operators remove their negative eigenvalue
components G+=P;:b, L’=P::c. (1)
In (I), the symbol (:) denotes the trace product of the fourth-order projection operator and the second-order tensor. If ny) and ny) (c( = 1,2, 3) are the three orthogonal principal directions of the stress and strain tensors, respectively, the corresponding positive projection operators are defined by
Engineering Fra~~rureMechrmic.~ Vol. 49. No. 5. pp. 6XlH97. 1994
Pergamon
0013-7944(94)00183-9
Copyright ‘c 1994 Elsevier Science Ltd
Printedin Great Britain.All rights reserved
1. INTRODUCTION
DEGRADATION of elastic properties reflecting accumulating damage in brittle materials is primarily a consequence of the evolution of internal microcrack structure. Depending on the material microstructure and the current state of stress and strain and their rates, instantaneous material response and further evolution of damage may involve activation of different microcrack mechanisms. The tensile hoop stress generated at the surface of relatively large pores represents a preferential crack nucleation mechanism in very porous rocks. In low porosity, compact rocks frictional sliding of crack surfaces can destabilize original shear cracks, causing them to kink and develop wing cracks. Other mechanisms of microcracking are also possible. Some of these mechanisms were implemented in the failure analysis of brittle solids by Ashby and Sammis [I], and others. Progressive degradation of mechanical properties is an inherent feature of brittle material behavior. Several analytical models were developed to estimate the effective elastic properties of a solid weakened by a given distribution of microcracks or other defects. A comprehensive review of these models can be found in a recent treatise by Nemat-Nasser and Hori [2]. An important issue in the formulation of continuum damage models is related to the appropriate choice of the mathematical form for the damage variable. This has been recently studied by Lubarda and Krajcinovic [3], who considered several frequently encountered two and three dimensional distributions of microcracks. Their analysis demonstrated the shortcomings of the scalar and secondorder tensor damage variables, and the accuracy gained by using the fourth-order tensor variable. The mode and stability of crack growth and, therefore, behavior of brittle material strongly depends on the sign and magnitude of applied stresses. For instance? the response of brittle material subjected to a compressive loading is strongly dependent on the magnitude of lateral confinement. An unconfined specimen fails by axial splitting, attributed to unstable growth of a single crack, at relatively small microcrack density. As the confinement is increased, axial splitting is suppressed and at large levels of confinement homogeneous microcracking prevails throughout the sample, resulting in a quasi-ductile overall response (Horii and Nemat-Nasser [4]). The analysis presented in this paper addresses the important issue of modeling different tensile and compressive responses of brittle materials. The concept of positive and negative projections of stress and strain tensors is used to account approximately for two basic, tensile and compressive, damage evolution modes. This idea was independently introduced by Ladeveze and Lemaitre [5], Ortiz [6] and Mazars [7], and was subsequently utilized by Simo and Ju [8], Yazdani and Schreyer [9], Ju [IO], Stevens and Liu [l 11, Yazdani [12], and others. The positive parts of the stress