损伤力学—第3章 - 第7节

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损伤力学ppt课件第三章几何损伤理论

1 1 无损时， e ( ) : E : 2 1 ~ 1 T 1 1~ 受损伤后， e ( , D) : E : : M ( D) : E : M ( D) : 2 2 1 ~ 1 : E : 2 ~ 1 T 1
11 11
1
F
o 当 90 时，令 11
f
( 9011 ) f
，有
f
~ ( 9011 ) f 2 F
由此可确定任意角度下的 11 的计算值，将之与实验结果比较，如二者吻合很好，则可证明定义的损伤张量可用。
第四节等价性原理
三维应变等价性原理：Lemaitre & Chaboche
有效弹性张量
* 1 (E ) * 1 E ( D) M ( D) : E
能量等价性原理：Sidoroff 无耦合的各向异性损伤和应变等价性假设不相容受损材料的性能可以用无损材料的余弹性能表示，只要把其中的应力换成有效应力即可。例如：弹性
E ( D) M ( D) : E : M ( D) 1 T , 1 ~ E ( D) M ( D) : E : M ( D)
损伤张量可以定义为表观面积和实际受载截面积的差与表观面积之比，即：
* dx G dx
* dy G dy
1 T * 1 D I K (G ) [dA dA ] (dA)
S A DD D
对损伤张量反对称部分引起的面积变化分析：
第二节有效应力张量
定义为Cauchy应力张量, T 为作用在PQR上的面力
则:
TdA (v dA)

损伤力学PPT课件

损伤准则与损伤演化
一、损伤力学的定义
Damage Mechanics Continuum Damage Mechanics (CDM) 损伤力学研究材料在损伤阶段的力学行为及相应的边值问题。它系统地讨论微观缺陷对材料的机械性能、结构的应力分布的影响以及缺陷的演化规律。主要用于分析结构破坏的整个过程，即微裂纹的演化、宏观裂纹的形成直至结构的破坏。
在这些点处只在一些平面上会产生穿晶微开裂。
失效的循环数很高， NR>10000
复合材料拉伸断口
损伤的宏观测量
直接测量间接测量
剩余寿命密度电阻率疲劳极限弹性模量塑性特征声速变化粘塑性特征
损伤变量和结构寿命预报
损伤演变依赖于：延性失效或疲劳失效中的应力蠕变、腐蚀或辐照过程中的应力疲劳损伤时载荷循环周数
拉伸试样在拉断前产生银纹化现象，银纹方向与应力方向垂直
损伤的分类
宏观（变形状态）： ➢ 弹性损伤 ➢ 弹塑性损伤 ➢ 蠕变损伤 ➢ 疲劳损伤
微观（损伤形式）： ➢ 微裂纹损伤（micro-crack） ➢ 微孔洞损伤（micro-void） ➢ 剪切带损伤（shear bond） ➢ 界面（interface）
损伤力学与断裂力学的关系
损伤力学分析材料从变形到破坏，损伤逐渐积累的整个过程；断裂力学分析裂纹扩展的过程。
微裂纹孕育萌生扩展汇合
剪切带
形成
快速扩展
微孔洞
形核
长大汇合
损伤力学
脆断
宏观裂纹
分岔驻止
启裂
扩展
韧断
失稳
疲劳
断裂力学
损伤力学的应用
物理性能
断裂过程（脆、韧）

超经典损伤力学讲义

（1）奇异损伤
奇异损伤主要是指在岩体工程范围内所含有的一条或若干条较大的断裂带，且内部常有充填物。对于这类工程岩体，虽然断裂带数目较少，但其力学性质和没有断裂带部位的岩体相比差异较为明显，所以断裂带的力学性质，对工程岩体具有决定性的作用，应重点分析。著名的Goodman单元，界面接触单元（COJB）等都是这方面的代表性方法。
1. 岩石材料损伤的微细观表现
一般岩石材料的组织结构为：
晶体+晶间质微层理（沉积造成）泥质颗粒+胶结质钙质硅质微劈理（造岩造成）微节理（受力造成）
岩石组织
+孔隙（水）+微裂纹
Sprunt和Brace（1974）详细研究了不同岩石中的自然孔隙和微裂纹，注意到孔隙率可由两种方式定义，即总的和表观的。表观孔隙率是相连于岩石外表面的孔隙和微裂纹互相连通的体积的量度；总的孔隙率是所有孔隙和微裂纹所占体积的量度，它既包含与外表面相连的孔隙和微裂纹，也包含与外表面不相连的孔隙和微裂纹。岩石中自然存在的孔隙和微裂纹的形状和尺寸如下图所示。
Chelmsford花岗闪长Granodiorite016
0.29
0.9
0.8
1.0
1.1
1500
2000
150
360
扫描电镜技术也充分证明了岩石是一种自然损伤材料，其自然损伤大致有下面几种：（1）孔隙在沉积岩中最为常见，在颗粒支撑、接触式胶结物连接或颗粒连接等胶结类的碎屑岩（如砂岩）中，孔隙的体积含量相当高。砂岩孔隙的类型主要是粒间孔隙，分布比较均匀，孔隙的大小与沉积颗粒的尺寸和分选性有关，形态取决于颗粒形态和颗粒粒径的级配。石灰岩中的孔隙含量低于砂岩，形态还与胶结物有关。孔隙是一种典型的三维细观损伤。（2）颗粒边界及界面裂纹沉积岩中，颗粒与颗粒之间或颗粒与各种胶结物之间的结合一般都比较薄弱；或者结晶质岩浆岩中矿物颗粒之间或者结晶界面，以及变质岩中重结晶矿物之间的结合相对较弱。所以岩石中的颗粒边界成为重要的初始细观损伤。在颗粒边界常形成界面裂纹、或者结晶界面裂纹的细观损伤的形状受碎屑颗粒、矿物颗粒或者晶体颗粒的外形所控制。

损伤力学资料

Effect of manufacturing defects on mechanical properties and failure features of 3D orthogonal woven C/CcompositesAi Shigang a ,Fang Daining b ,⇑,He Rujie b ,1,Pei Yongmao ba Institute of Engineering Mechanics,Beijing Jiaotong University,Beijing 100044,PR ChinabState Key Laboratory of Turbulence and Complex System,College of Engineering,Peking University,Beijing 100871,PR Chinaa r t i c l e i n f o Article history:Received 8September 2014Received in revised form 1November 2014Accepted 3November 2014Available online 10November 2014Keywords:A.Carbon-carbon composites (CCCs)B.DefectsC.Damage mechanics C.Numerical analysisD.Non-destructive testinga b s t r a c tFor high performance 3D orthogonal textile Carbon/Carbon (C/C)composites,a key issue is the manufac-turing defects,such as micro-cracks and voids.Defects can be substantial perturbations of the ideal archi-tecture of the materials which trigger the failure mechanisms and compromise strength.This study presents comprehensive investigations,including experimental mechanical tests,micron-resolution computed tomography (l CT)detection and ﬁnite element modeling of the defects in the C/C composite.Virtual C/C specimens with void defects were constructed based on l CT data and a new progressive dam-age model for the composite was proposed.According to the numerical approach,effects of voids on mechanical performance of the C/C composite were investigated.Failure predictions of the C/C virtual specimens under different void fraction and location were presented.Numerical simulation results showed that voids in ﬁber yarns had the greatest inﬂuences on performance of the C/C composite,espe-cially on tensile strength.Ó2014Elsevier Ltd.All rights reserved.1.IntroductionCarbon ﬁber reinforced carbon composites (C/C)have high ther-mal stability,thermal shock resistance,strength and stiffness in non-oxidizing atmosphere.Due to its superior speciﬁc strength and toughness,C/C composites can be considered as favourite materials for highly demanding thermostructural lightweight applications e.g.in aerospace and nuclear industry [1–6].Nowa-days C/C components are leading candidates for applications under extreme conditions.C/C composites are produced by chemical vapor inﬁltration (CVI)of a textile ﬁber preform.After the CVI pro-cess and high temperature heart-treatments,generally,manufac-turing defects exist inner the materials.In particular,porosity/voids and micro-cracks are typical defects in C/C composites,and seriously affect the performance of the composites [7–9].So,it is mandatory to account for the effects of defects and their evolution,even in the early stages of the design process.With the increasing use of C/C composites as advanced structural materials,the deter-mination of damage criticality and structural reliability of compos-ites has become an important issue in recent years.Defects–mechanical property relationships of ﬁber reinforced composites have always been of interest to scientists addressing the composite performance.In Gowayed et al.’s work [10],defects in an as-manufactured oxide/oxide and two non-oxide (SiC/SiNC and MI SiC/SiC)ceramic matrix composites were categorized and their volume fraction quantiﬁed using optical imaging and image analysis.Aslan and Sahin [11]investigated the effects of delamin-ations size on the critical buckling load and compressive failure load of E-glass/epoxy composite laminates with multiple large del-aminations by experiments and numerical simulations.In Masoud et al.’s work [12]effects of manufacturing and installation defects on mechanical performance of polymer matrix composites appear-ing in civil infrastructure and aerospace applications were studied.Damage onset and propagation were studied used time-dependent nonlinear regression of the strain ﬁeld.In Refs.[13–17],the ﬁnite element method (FEM)was followed by various authors to study the delamination problems.FEM is preferred than analytical solu-tions because it can handle various laminate conﬁgurations and boundary conditions.In recent decades,high-ﬁdelity X-ray micro-computed tomog-raphy (l CT)technology has been used to characterize defects and reconstruct meso-structure of textile composites [18].In Cox et al.’s work [19–21],three-dimensional images of textile com-posites were captured by X-ray l CT on a synchrotron beamline.Based on a modiﬁed Markov Chain algorithm and the l CT data,/10.1016/positesb.2014.11.0031359-8368/Ó2014Elsevier Ltd.All rights reserved.⇑Corresponding author.E-mail addresses:sgai@ (F.Daining),rujh@ (H.Rujie).1Co-corresponding author.a computationally efﬁcient method has been demonstrated for generating virtual textile specimens.In Fard et al.’s work [22],manufacturing defects in stitch-bonded biaxial carbon/epoxy composites were studied through nondestructive testing (NDT)and the mechanical performance of the composite structures was investigated using strain mapping technique.In Desplentere et al.’s work [23],X-ray l CT was used to characterize the micro-structural variation of four different 3D warp-interlaced fabrics.And the inﬂuence of the variability of the fabric internal geometry on the mechanical properties of the composites was estimated.In Guillaume et al.’s work [24]effects of porosity defects on the interlaminar tensile (ILT)fatigue behavior of car-bon/epoxy tape composites were studied.In that work,CT mea-surements of porosity defects present in specimens were integrated into ﬁnite element stress analysis to capture the effects of defects on the ILT fatigue behavior.In Thomas et al.’s work [25]X-ray microtomography technology was adopted to measure the dimensions and orientation of the critical defects in short-ﬁber reinforced composites.Generally,geometry reconstruction based on l CT data is a huge and complex work,sometimes,virtual specimens explored through this approach are difﬁcult to use for numerical analysis.For 3D fabric composites,because of the 2.Material and experimentsMaterial studied in this article is C/C 3-D orthogonal woven ceramic composite (fabricated by National Key Laboratory of Ther-mostructure Composite Materials,Northwestern Polytechnical University,China)in which T300carbon ﬁber (Nippon Toray,Japan)tows rigidiﬁed by carbon matrix.The C/C composite was prepared using chemical vapor inﬁltration (CVI)method.T300car-bon ﬁber was used as reinforcement of the C/C composites with the ﬁber volume fraction was 56.5%.The ﬁber preforms,as shown in Fig.1a,were inﬁltrated with carbon matrix using multiple cycles of inﬁltration and heat treatment at 1373K,0.03MPa (the thick-ness of the ﬁber preforms is about 5mm).With increasing cycles,a matrix with near full density can be asymptotically approached,generally,it was about 10cycles (1200h).The C/C specimens are illustrated in Fig.1b (the thickness of the tensile specimen is 5.0mm).However,from the l CT images of the C/C materials,it was found that manufacturing defects such as voids and micro-cracks appeared inner the composites.It is because of the special material preparation process.The manufacturing defects are illus-trated in Fig.1c.Uniaxial tensile experiments were carried out under a Shima-Fig.1.C/C 3-D orthogonal woven composite.Fig.2.Stress–strain curve of the C/C composite under uniaxial tension.114 A.Shigang et al./Composites:Part B 71(2015)113–121In the tensile experiments,ﬁve specimens in total were tested and the tensile strengths were217.3,185.1,219.8,176.5and 187.3MPa correspondingly.The average value of the tensile strength was197.2MPa and the dispersion of the experimental results was less than11.5%.Other more,the fracture behaviors of theﬁve specimens were similar with the failure locations almost all located in the middle of the specimens.From the experiments, deformation of the C/C3-D orthogonal composite under uniaxial tension comprises with three stages:linear elastic stage,damage initiation/evolution stage and the material fracture stage.In the ﬁrst stage the stress–strain curve increased linearly and in the sec-ond stage the stress–strain curve increased nonlinearly.In the frac-ture stage the stress–strain curve rapidly declined.3.Numerical programmer3.1.3Dﬁnite element modelFiber tows in the3-D orthogonal architecturesﬁt together snugly in the woven pattern by a system of periodic motions, and approximately in the same cross-sectional geometry.In this study,cross-sections of the warpﬁber yarns and weftﬁber yarns wereﬁtted as rectangle.The cross-sections of the z-binder tows wereﬁtted as circular.Geometric parameters of theﬁber yarn cross-sections were recorded.For the warp yarns and weft yarns the side lengths of the cross-section rectangle were0.786mm and0.340mm.For the z-binderﬁber yarns the diameter of the cross-section circular was0.790mm.The smallest repeatable rep-resentative volume element(RVE)of the textile architecture was constructed and shown in Fig.3.The lengths of the RVE model in X and Y direction both were1.96mm and the height of the RVE model in Z direction was0.76mm.To reveal the internal defects in theﬁnite element model,l CT technology was used to investigate the meso-structures of the fore,three local coordinates were constructed to identify the mate-rial directions.Then,an interface zone with a constant thickness 0.01mm was generated based on the geometrical model of the ﬁber yarns,as shown in Fig.3d.Finally,a solid block model with the same size of the composite specimen was constructed.Boolean operation were carried out among the solid block,interfaces and theﬁber yarns to generate the geometrical model of the carbon matrix,which is shown in Fig.3b.A whole RVE model of the com-posite is illustrated in Fig.3a.A Monte Carlo algorithm was adopted to choose elements one-by-one randomly as‘‘void defects’’until the volume fraction of the voids satisﬁed the threshold values in the three zones respectively. For the C/C composite studied in this paper,the void fractions of theﬁber yarns,matrix and the interfaces are0.51%,0.47%and 1.94%respectively.It must be noted out that those elements which identiﬁed as‘‘void defects’’were not moved away from the FE model,but the stiffness was degenerated by10eÀ6times in the simulation process.The void defects in the three zones are high-light as‘‘red’’,as shown in Fig.3.3.2.Progressive damage modelThe failure criterion proposed here is a strain-based continuum damage formulation with different failure criteria applied for matrix andﬁber yarns.A gradual degradation of the material prop-erties is assumed.This gradual degradation is controlled by the individual fracture energies of matrix andﬁber yarns,respectively. Theﬁber yarn is in the X(1)–Y(2)–Z(3)Cartesian coordinate sys-tem,and the X direction corresponds to theﬁber longitudinal direction.For theﬁber yarns,two different modes of failure are considered:ﬁber failure in longitudinal direction and matrix fail-ure in transverse direction.The damage mechanism consists of two ingredients:the damage initiation criteria and the damage evolution law.orthogonal textile C/C composite,(a)RVE model,(b)carbon matrix,(c)ﬁber yarns and(d)ﬁber yarns-matrixinterpretation of the references to colour in thisﬁgure legend,the reader is referred to the web version of thisA.Shigang et al./Composites:Part B71(2015)113–121115failure strains inﬁber direction in tension and compression,F f;tX and F f;cXare the tensile and compressive strength of theﬁberyarns in X direction,respectively.Once the above criterion is sat-isﬁed,theﬁber damage variable,f Xf,evolves according to the fol-lowing equation law:d X f ¼1Àe f;t11f XfeÀC11e f;t11f X fÀe f;t11ðÞL c=G fðÞð2Þwhere L c is the characteristic length associated with the material point.For matrix failure the following failure criterion is used:f Y m ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃe f;t22e f;c22ðe22Þ2þe f;t22Àe f;t222e f;c22B@1C A e22þe f;t22e f;s12!2ðe12Þ2>e f;t22v uu uu tð3Þf Z m ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃe f;t3333ðe33Þ2þe f;t33Àe f;t33233B@1C A e33þe f;t3313!2ðe13Þ2>e f;t33v uu uu tð4Þwhere e f;t22;e f;t33;e f;c22and e f;c33are the failure strains perpendicular to the ﬁber direction in tension and compression,respectively.The failure strain for shear are e f;s13and e f;s12.Failure occurs when f Y m exceeds its threshold value e f;t22or f Z m exceeds its threshold value e f;t33.The evolu-tion law of the matrix damage variable,d m,is:d Ym¼1Àe f;t22f YmeÀC22e f;t22f Y mÀe f;t22ðÞL c=G mðÞð5Þd Zm¼1Àe f;t33fmeÀC33e f;t33f Z mÀe f;t33ðÞL c=G mðÞð6ÞAs damage progressing,the effective elasticity matrix isreduced as functions of the three damage variables f Xf,d Ymand d Zm, as follows:3.2.2.Failure criterion for matrixDamage in theﬁber is initiated when the following criterion is reached:where e f;t and e f;c are the failure strains in tension and compression respectively and e f,t=r f,t/C11,e f,c=r f,c/C11.Once the above criterionis satisﬁed,theﬁber damage variable,f XðY=ZÞm,evolves according to the equation:d XðY=ZÞm¼1Àef;tfmeÀC11e f;t f XðY=ZÞmÀe f;tL c=G mð9ÞThe modulus matrix of the matrix will be reduced according to:In user subroutine UMAT the stresses are updated according to the following equation:C f d ¼1Àd XfC111Àd Xf1Àd YmC121Àd Xf1Àd ZmC130001Àd YmC221Àd Ym1Àd ZmC230001Àd ZmC330001Àd Xf1Àd YmC4400Symmetric1Àd Xf1Àd ZmC5501Àd Ym1Àd ZmC66ð7Þf XðY=ZÞm ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃef;tef;cðe11ð22=33ÞÞ2þe f;tÀðe f;tÞ2ef;c!e11ð22=33Þþef;tef;s2ðe12ð23=13ÞÞ2þef;tef;s2ðe13ð12=23ÞÞ2v uu t>e f;tð8ÞC m d ¼1Àd XmC111Àd Xm1Àd YmC121Àd Xm1Àd ZmC130001Àd YmC221Àd Ym1Àd ZmC230001Àd ZmC330001Àd Xm1Àd YmC4400Symmetric1Àd Xm1Àd ZmC5501Àd Ym1Àd ZmC66ð10Þ116 A.Shigang et al./Composites:Part B71(2015)113–121r ¼C d :eTo improve convergence,a technique based ization (Duvaut–Lions regularization [27])of is implemented in the user subroutine.In this age variables are ‘‘normalized’’via the _d t ;X ðY =Z Þf ðm Þ¼1gd X ðY =Z Þf ðm ÞÀd t ;X ðY =Z Þf ðm Þwhere d X f and d X ðY =Z Þm are the ﬁber and matrix culated according to the damage evolution d t ;X f and d t ;X ðY =Z Þm are the ‘‘normalized’’the real calculations of the damaged elasticity bian matrix,and g is the viscosity parameter.and d t ;X ðY =Z Þm can be calculated according to the d t ;X ðY =Z Þf ðm Þt 0þD t¼D t t 0þt d X ðY =Z Þf ðm Þ t 0þD t þg g þt dt ;X ðY =f ðm ÞTherefore,for the ﬁber yarns and matrix,can be further formulated as Eqs.(14)and (15)correspondingly@r e ðf Þ¼C f d þ@C f d@d f:e !@d X f @f f Á@f Xfe!þ@C f d @d Y m :e !@d Y m @f Y m Á@f Y m @e !þ@C f d @d Z m :e !@d Z m @f Zm Á@f Z m@e !ð14Þ@r @e ðm Þ¼C m d þ@C md @d m :e !@d X m @f m Á@f X m@e!þ@C md @d m :e !@d Y m @f m Á@f Y m @e !þ@C m d @d m :e !@d Z m @f m Á@f Z m @e!ð15Þ3.3.Material parameters3D orthogonal C/C composites are composed by T300ﬁber yarns and carbon matrix.The ﬁber yarns can be regarded as unidi-rectional ﬁber-reinforced C/C composites and are assumed to be one transversely isotropic entity in each local material coordinate system.The mechanical properties of the ﬁber yarns can be calcu-lated using the properties of the component materials (ﬁbers and matrix):E 1¼e E f 11þð1Àe ÞE mE 2¼E 3¼E m1Àﬃﬃe p 1ÀE m =E f 22ðÞG 12¼G 13¼G m 1Àﬃp 1ÀG m =G f 12ðÞG 23¼G m1Àﬃp 1ÀG m =G f23ðÞl 12¼l 13¼e l f 12þð1Àe Þl m l 23¼E 222G 23À19>>>>>>>>>>>>=>>>>>>>>>>>>;ð16Þwhere e is the yarn pack factor,for the C/C composite studied in this paper,e =0.81.E f 11,E f 22are the Young’s elastic modulus of the ﬁberin the principal axis 1and 2,respectively.Axis 1is the longitudinal direction of the ﬁber yarns.G f 12,G f 23are the shear modulus of the ﬁber in the 1–2and 2–3plane,respectively.l f 12is the primary Pois-son’s ratio of the ﬁber,E m ,l m and G m represent the Young’s elastic modulus,Poisson’s ratio and shear modulus of the matrix,respec-tively.Materials parameters are listed in Table 1.It should be noted that the mechanical parameters of the carbon matrix and the T300ﬁbers changed after the CVI process.In particular,strength of the ﬁber will had a greater decline.The tensile and com-pressive strength of the T300ﬁber yarns were tested with the values listed in Table 1.The elasticity modular of the carbon matrix was tested by a nanoindentor system,which developed by Fang’s research team from Peking University [28].In the carbon matrix modular tests,the experiments repeated 20times for statistical averaging.The val-ues in the 20measurements were 7.18,9.77,8.58,10.01,11.92,5.30,9.98,8.14,8.63,7.31,6.19,10.69,11.10,13.45,9.15,9.13,11.27,9.20,10.14and 10.06GPa;average value was 9.36GPa.Mate-rial parameters of the ﬁber/matrix interface are not very clear so far,in this study the Young’s elastic modulus and Poisson’s ratio of the inter-faces were assumed as same as the carbon matrix.G f is one of the key parameters which control the failure pro-gress of the ﬁber yarns,however,different values were recom-mended in reported articles.In this study,inﬂuences of G f on the mechanical properties of the C/C composite were investigated ﬁrstly.Based on the values reported in Refs.[29,30],ﬁve virtual specimens with different G f values (0.5,2.0,6.0,10.0,14.0)were constructed and numerical tested.Simulation results were com-pared with the experimental result,as illustrated in Fig.4.It was found that G f has inﬂuences on tensile strength and fracture strain of the C/C composite.When G f were 0.5,2.0,6.0,10.0and 14.0,ten-sile strengths of the specimens were 200.5,205.1,205.3,214.2and 214.8MPa.When G f were 0.5,2.0,6.0,the failure strains were 0.36%,0.43%and 0.57%correspondingly.When G f is bigger than 10.0,failure strains of the C/C specimens were bigger than 1.0%.So,by the simulation results,in the present study the value of G f was set to 6.0.Table 1Materials parameters.E 11(GPa)E 22(GPa)ˆ12G 12(GPa)G 23(GPa)F t (MPa)F c (MPa)S (MPa)G f (m )(N/mm)gT300ﬁber 230400.262414.389075650 6.00.001C matrix 9.360.338210050 1.00.001Interface9.360.3382100501.00.001Fig.4.Stress–strain curves of the C/C virtual specimens under different G f .4.Simulation results and discussionThe anisotropic damage model of theﬁber yarns and the isotro-damage model of the matrix and the interface were carriedmaterial constitutive equations by User subroutine UMAT ABAQUS nonlinearﬁnite element codes.Static uniaxial tensile sim-ulations were carried out.In order to keep forces continuity and displacements compatibility of the opposite faces of the unit cell, periodic boundary conditions were imposed in the simulation. Because the opposite faces of the unit cell have the same geomet-rical features,the nodes on the faces were controlled in the same position to form the corresponding nodes in the process of meshing.The periodic BCs were imposed on the corresponding nodes by FORTRAN pre-compiler code,detailed in Ref.[26].The RVE model subjected to a constant displacement load in Y direction and the loading strain is1%.4.1.Effects of the void defectsIn order to investigate the void defects on the mechanical prop-erties and failure behaviors of the C/C composite,two RVE models of the C/C composite were numerical simulated.In one RVE model (FE_D),theﬁber yarns,interface and matrix all had void defects with the void fractions are0.51%,1.94%and0.47%respectively. For the other RVE model(FE_Intact),no defect inside.The stress–strain curves of the C/C composite in the simulations and experi-ment are illustrated in Fig.5.By the experimental results,the elas-ticity modular of this C/C composite was58.4GPa.By the numerical simulations,for the intact model,the elasticity modular was56.6GPa;for the‘defected’model the modular was56.3GPa. In the view of modular,the simulation error of the two models were3.1%and3.6%compared with the experimental results.The difference between the two FE models was only0.53%,so,void defects have relatively limited effects on the elastic modular of the C/C composite.The uniaxial tensile strength of the C/C compos-ite was197.2MPa by the experiments.In the simulations,the ten-sile strengths were231.4MPa and205.3MPa corresponding to the intact model and the‘defected’model.It was about17.3%and4.1% difference compared with experimental results.It is clear that,theFig.5.Stress–strain curves of the C/C composite under uniaxial tension.Fig.6.Damage evolution inﬁber yarns,(a)RVE model with voids defects,(b)intact model.Part B71(2015)113–121yarns are corresponding to the three pictures‘o’,‘p’and‘q’in Fig.6. For the RVE model with defects,it was found that damages were ﬁrstly generated besides the defects.During the loading process, damages were growing in several sections in theﬁber yarns.How-ever,for the intact model,damages were almost generated in one section in theﬁber yarns.Damage evolution in carbon matrix and the interface zone are illustrated in Figs.7and8.From the simulation results,in all of the three zones,damages wereﬁrstly generated in the‘defected’RVE model.For the‘defected’model,when e=0.27%damages appeared in the interface zone,while for the intact model the strain was0.33%.In the matrix zone,the strains in the two models were0.31%and0.33%,respectively,when damages appeared.In ﬁber yarns,the strains when damages appear for the two models were0.37%and0.44%.So,because of the internal defects,in load-ing progress damages will generate early inner the material.Fail-ure strain of the materials which with defects is comparatively small when compared with the materials without defects.4.2.Inﬂuence of void locationBy the l CT images,it is clear that voids and micro cracks exist in ﬁber yarns,carbon matrix and the interface zones.By statistical analysis for those defects,fraction of the voids in those three zones was calculated.To study the inﬂuence of void location on the mechanical properties of the C/C materials,threeﬁnite elementFig.7.Damage evolution in carbon matrix,(a)model with voids defects,(b)intact model.Fig.8.Damage evolution in interface,(a)model with void defects,(b)intact model.models were constructed and numerically analyzed.In the three RVE models,one model has defects only in theﬁber yarns(FE_DF) and another model has the defects only in carbon matrix(FE_DM), while the other one has defects only in the interface zone(FE_DI). Simulation results were compared with the experimental results. Stress–strain curves in the numerical simulations and experiments are illustrated in Fig.9.Tensile strength calculated by the simulations were208.5MPa, 229.7MPa and230.1MPa,corresponding to the threeﬁnite ele-ment models:FE_DF,FE_DI and FE_DM.By the simulation results of theﬁnite element models FE_D and FE_Intact,as mentioned in above section,the tensile strengths were205.3MPa and 231.4MPa.It can give the conclusion that,defects inﬁber yarns has the biggest effects on the mechanical properties of the C/C composite.If theﬁber yarns are perfect and defects only exist in carbon matrix and interfaces,void defects have limited inﬂuences on the mechanical properties of the C/C composites under the cur-rent void volume fractions.4.3.Inﬂuence of void volume fractionBy the statistical analysis in Section3.1,volume fractions of voids in theﬁber yarns,matrix and the interface zones are0.51%, 0.47%and1.94%.Under this defect fraction,as calculated in Section,tensile strength of the C/C composite declined12.7%compared with the material which contains no defects.So,it is important and meaningful that if we can make sure about the mechanical behav-iors of C/C composites when we exactly know the void defect frac-tion.If so,it will be helpful for the performance evaluation of C/C composites and structures.To investigate the inﬂuence of the void defect fraction on the mechanical performance of the C/C composite,ﬁve RVE models were constructed and the defect fractions of theﬁber yarns were 0.25%,0.5%,1.0%,2.0%and4.0%.In this study,void defect was assumed only exist inﬁber yarns.Because,as calculated in Section 4.2,voids in carbon matrix and interfaces zone had very little effects on the mechanical properties of the C/C composite.Uniaxial tension simulations were carried out and the stress–strain curves of theﬁve C/C virtual specimens are illustrated in Fig.10.From the simulation results,it is clear that as the defect fraction increased tensile strength of the C/C composite decreased.For the intact FE model,the tensile strength was231.4MPa.For theﬁve FE models with voids defects,the tensile strengths were214.8MPa, 206.6MPa,197.1MPa,182.3MPa and152.8MPa.For the FE model under the defect density0.25%,tensile strength decreased7.2% compared with the intact model.So,if there exist defects inner the C/C materials,even if the volume fraction of the defects was small,it will has obvious effects on the mechanical performance of the composite,especially on the tensile strength.When the defect density was4.0%,tensile strength of the C/C virtual speci-men declined33.9%compared with the intact specimen.5.ConclusionUniaxial tensile properties and meso-structure of the3D orthogonal C/C composite were studied by experimental approaches.Manufacturing defects inner the C/C composite were investigated though a micron-resolution computed tomography (l CT)approach.From the l CT photos of the3-D orthogonal car-bon/carbon composite,it was found that voids and microcracks are two classic type of manufacture defects inner the C/C materials. Base on the statistical analysis of the l CT data,ﬁnite element mod-els of the C/C composite were constructed.According to a new pro-gressive damage model,failure behaviors and mechanical properties of the C/C composites were studied by ABAQUS code. Effects of the void defects on the mechanical performances of the C/C material were numerically investigated.From the numerical simulation results,manufacturing defects such as voids have great effects on the mechanical performance of the carbon/carbon com-posite,especially on the tensile strength.With0.51%void volume fraction,tensile strength of the carbon/carbon composite has 13.2%declines compared with the intact material.When void defects exist inﬁber yarns,even if the volume fraction of the defects is small it still will has great inﬂuence on tensile strength of the C/C composite.However,the defects which exist in carbon matrix and interface have limited effects on the mechanical prop-erties of the C/C materials.So,keep the continuity and improve the density of the carbonﬁber yarns in C/C composite manufacture process is the key to improve the mechanical properties of the C/ C composites.AcknowledgementsFinancial support from the National Natural Science Founda-tions of China(Nos.11202007,11232001,11402132)and the Foundation of Beijing Jiaotong University(KCRC14002536)are gratefully acknowledged.Fig.9.Stress–strain curves of the C/C composite in experiment and simulations.10.Stress–strain curves of the virtual specimens with different void defectfraction.Part B71(2015)113–121。

第六章-连续损伤力学

第六章连续损伤力学
第一节弹脆性损伤理论第二节粘脆性(蠕变)损伤理论第三节弹塑性损伤理论第四节疲劳损伤理论
第一节弹脆性损伤理论
1）弹性各向同性损伤模型对于等温和线弹性情况下的弹性各向同性损伤材
料，由于塑性变形很小、温度梯度为零，因此耗散不等式变为： •
R 0
其中损伤扩展力R的含义是表征材料提供产生新的弹脆性损伤的能力，数量上等于损伤扩展所耗散的能量密度。因此， R也可称为损伤能量释放率密度。
f
是相应于恒应力
k
的脆断时间，由式
（9）决定。对上式求和，并考虑初始条件（ t=0时，
ψ=1）和破坏条件（ t=t*f时，ψs=0），则有：
s tk 1
t k 1 k f
多级载荷下的断裂时间为：
t f
s
tk
k 1
（2）非均匀损伤场
如果弹性固体受应力场是均匀的，如等截面的受拉杆，其损伤从理论上说也是均匀的。加载过程中，损伤场将均匀增强，直到发生瞬时破坏。
伤度取最大值。在 r ri 处，断裂起始条件为t=tf，ψ
（ri）=0 或ω（ri）=1 ，或。将此条件代入上式，得
脆性断裂起始时间：
t fi
n 1
A
n max
1
应当指出，在断裂潜伏阶段（ 0 t t f）i ， r 0
或 r 1。
例1 等矩形截面梁受纯弯曲（小变形情况）设断裂潜伏阶段，应力场不随时间变化，即：
弹性损伤下，Helmholtz自由能密度函数可表示为
f , W e , 1 1 : :
2
（1）
式中，ω是各向同性标量损伤变量；ε是二阶应变张量；E是四阶弹性系数张量。
由应力等效性假设有： 1 :

损伤力学基础知识损伤理论的研究内容和意义机械设备和工程结构中

损伤力学基础知识一、损伤理论的研究内容和意义机械设备和工程结构中的构件，从毛坯制造到加工成形的过程中，不可避免地会使构件的内部或表面产生微小的缺陷（如小于l mm的裂纹或空隙等）。

在一定的外部因素（载荷、温度变化以及腐蚀介质等）作用下，这些缺陷会不断扩展和合并，形成宏观裂纹。

裂纹继续扩展后，最终可能导致构件或结构的断裂破坏。

微缺陷的存在与扩展也是使构件的强度、刚度、韧性下降或剩余寿命降低的原因。

这些导致材料和结构力学性能劣化的微观结构的变化称为损伤。

损伤理论研究材料或构件从原生缺陷到形成宏观裂纹直至断裂的全过程，也就是通常指的微裂纹的萌生、扩展或演变、体积元的破裂、宏观裂纹形成、裂纹的稳定扩展和失稳扩展的全过程。

损伤理论，主要是在连续介质力学和热力学的基础上，用固体力学的方法，研究材料或构件宏观力学性能的演变直至破坏的全过程，从而形成了固体力学中一个新的分支——损伤力学。

长期以来，人们对材料和构件宏观力学性能的劣化直至破坏过程的机理、本构关系、力学模型和计算方法都非常重视，并且用各种理论和方法进行了研究。

材料学家和物理学家从微观的角度研究微缺陷产生和扩展的机理，但是所得结果不易与宏观力学量相联系。

力学工作者则着眼于宏观分析，其中最常用的是断裂力学的理论和方法。

裂断力学主要研究裂纹尖端附近的应力场和应变场、能量释放率等，以建立宏观裂纹起裂、裂纹的稳定扩展和失稳扩展的判据。

但是断裂力学无法分析宏观裂纹出现前材料中的微缺陷或微裂纹的形成及其发展对材料力学性能的影响，而且许多微缺陷的存在并不都会简化为宏观裂纹，这是断裂力学的局限性。

经典的固体力学理论虽然完备地描述了无损材料的力学性能(弹性、粘弹性、塑性、粘塑性等)，然而，材料或构件的工作过程就是不断损伤的过程，用无损材料的本构关系描绘受损材料的力学性能显然是不合理的。

损伤理论旨在建立受损材料的本构关系、解释材料的破坏机理、建立损伤的演变方程、计算构件的损伤程度，从而达到预估其剩余寿命的目的。

纸页结构与性能(纸页的损伤力学理论及其特性)

进入二十世纪90年代，损伤力学研究的重点是损伤的宏细微观理论，其特征是引入多层次的缺陷几何结构，在材料的宏观体元中引入细观或微观的缺陷结构，在材料细观结构的演化与宏观力学响应之间建立联系，从而对材料的本构行为进行宏、细、微观相结合的研究。
二、纸页的结构、组成及其损伤力学原理
纸张是由纤维、填料、胶料等不同化学组成和物理性状的组分构成、三维各向异性的非均匀固体材料。
从研究角度，如果以产生损伤的加载过程来区分，材料的后天性损伤可分为以下几种：
（1) 延性、塑性损伤：微孔洞和微裂纹的形成和扩展，使
材料产生大塑性应变，最后导致塑性断裂。
与这类损伤相伴发生的是不可恢复的塑性变形。这类损伤的表现形式主要是微空洞、微裂纹的萌生、成长和聚合。主要发生于塑性材料。
(2) 蠕变损伤：在长期载荷作用或高温环境下，伴随着蠕变变形会发生蠕变损伤，其宏观表现形式为微裂纹、微空洞，蠕变损伤的扩展导致材料的耐久性下降。蠕变损伤使蠕变变形增加，最后导致断裂。（3)疲劳损伤：在循环载荷作用下，材料性能逐渐劣化。在每一步载荷循环中的损伤累积起来，将导致材料的寿命减少，导致疲劳破坏。有趣的是纸页开始受到第一轮低应力循环载荷后其抗张强度反而会有一定程度的提高。？!
损伤力学主要研究材料在受力过程产生的损伤及相应出现的各种材料结构变化和相应的某些力学参数和热力学性能的变化。
以纸页为例，其在作为印刷材料或包装材料使用过程中，均会受到力的作用。当纸页受力较小，低于纸页的破坏强度时，纸页不会受到破坏，但会受到不同程度的损伤。这些损伤将会导致纸页强度的下降。
事实上：作为一种网络结构型纤维材料，纸页中存在着大量的空隙。纸页中纤维之间氢键结合点及其分布也是不均匀的。
在宏观的角度，人们更多注意的是材料宏观上的表现以及由此造成的材料的力学性能劣化。

第七章_细观损伤力学

2）细观损伤力学的概念细观损伤力学，是从材料的细观结构出发，对不同的细观损伤机制加以区分，通过对细观结构变化的物理与力学过程的研究来了解材料的破坏，并通过体积平均化的方法从细观分析结果导出材料的宏观性质。细观损伤力学主要是从美国发展起来；常与材料的力学行为和变形过程相联系。起初，连续损伤力学和细观损伤力学是相互独立发展，直到80年代中后期，这两个损伤力学分支才被力学家和材料学家在不同程度上加以认可。实际上，这两种理论在工程应用、理论分析等方面可相互补充。
（2）刚性楔的体胞单元：有限体积的圆柱体中的圆柱形孔洞，有限体积的球体中的球形孔洞。
通过该模型，研究微孔洞损伤下，孔洞萌生的临界应变；孔洞体积的变化规律及材料的塑性变形行为等损伤规律。
详细版社，1997”
假设在单位体积的材料中有完全随机分布的N个椭圆形微裂纹，微裂纹的存在使得材料在有效弹性模量变为 E 和 G 。
自洽方法估计损伤材料有效模量的基本思想是：把每个微裂纹置于具有自洽等效模量的基体材料中，分析单个微裂纹的变形及其引起的模量变化，然后对所有微裂纹取总体平均，建立含有效模量的方程，求解得到材料的有效力学性质。
为了描述韧性材料细观损伤的机制及其演化过程，须建立适当的模型来描述材料的细观结构。Gurson摈弃无限大基体的假设，提出有限大基体含微孔洞的体胞模型。这种模型更加接近于真实的材料细观结构，为损伤的描述（如作为损伤变量的孔洞体积百分比）及宏观体积膨胀塑性理论的建立奠定了基础。
Gurson给出了4种微孔洞的体胞模型：（1）全塑性体胞单元：有限体积的圆柱体中的圆柱形孔洞，有限体积的球体中的球形孔洞。
6）细观损伤机制材料的细观损伤机制有多种，比较典型的有微孔洞、微裂纹、微滑移带、银纹、晶界滑移等。

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而损伤力学的方法则是在可用范围内比较详尽地分析裂纹尖端附近的应力场及扩展过程，利用恰当的局部损伤断裂准则，处理裂纹尖端的断裂行为。因此这种方法常常被称为断裂问题的局部方法。由于损伤力学中的本构关
土木工程专业教学质量工作汇报
系描述了材料逐渐劣化的过程，不需要再人为地引入材料断裂条件，裂纹扩展的路径就是构件中已经完全损伤的所有质点的集合，从而非常自然地刻划了裂纹的逐渐发展过程。
土木工程专业教学质量工作汇报
损伤不仅影响粘塑性应变率，而且影响弹性应变，也就说有限元
中的刚度矩阵是随时间不断变化的。第三，损伤导致材料的软化
行为，即对于宏观单调加载情况，裂纹尖端可以出现非单调的材
料变形，这可以引起解的不唯一（分叉）、数值结果不稳定、变形局部化以及网格敏感性等一系列问题。
为了避免损伤的局部效应，可以采用的几种方法有：（1）根据细观缺陷的统计结果，用一个特征尺寸来限定有限单元的最小尺寸；（2）在非局部的连续介质理论的框架上，引入应力和应变的高阶梯度；（3）采用局部限制手段；（4）用非局部的方法定义损伤演化律。例如，在任意一点x处的损伤演化律可以表达为
变形和损伤全耦合的局部方法是很有吸引力的一种方法，因为它能够更恰当更完整地预测裂纹尖端的变形、损伤和断裂行为。但是在实施这种方法时，往往遇到一些具体的问题。首先，由于损伤的引入，控制裂纹尖端场的微分方程更加复杂，对其进行求解有更多的困难，对于蠕变损伤的情况尤其明显。由于解析解一般难以得到，断裂的局部方法经常借助于有限元来实现。有限元计算中经常用到的单元消去技术，一旦某个单元满足了损失断裂准则，则人为消去该单元，使其不再承受应力。另一个技术是在耦合损伤的有限元程序中，采用自适应的时间步长。第二，用连续损伤力学方法描述裂纹扩展时需要全耦合的本构关系和损伤演化方程，例如在蠕变裂纹扩展问题中，
3.7 考虑塑性损伤的断裂问题
3.7.1 韧性损伤材料断裂的局部方法如上节所述，损伤力学和断裂力学处理裂纹问题的基
本方法有明显不同。断裂力学方法主要是寻求裂纹扩展和断裂的过程、规律与总体的载荷参数（如K,J,C*）的关系。在相当广泛的范围内，尤其是对于二维的弹性裂纹问题，比例加载条件下的小范围屈服裂纹问题和等幅应力作用下的循环加载断裂问题，断裂力学方法是非常有效的，具有良好的精度，并且得到了工程上的广泛应用，然而在另外一些情况下，如对于非比例加载条件下的裂纹扩展问题、与时间相关的裂纹扩展问题、裂纹尖端微孔洞损伤比较明显的问题，断裂力学方法会遇到一些难以克服的困难。
土木工程专业教学质量工作汇报
第一个算例是圆柱形试件, 试件的形状及网格划分如图 3.32( a) 所示, 采用均匀应变率的单向拉伸加载由于粘塑性变形, 应力在试件中部越来越集中, 图 3.32( b) 给出了有限元预测的载荷-位移曲线及损伤演化曲线。计算的结果表明, 如果采用局部意义上的损伤定义 ( d * = 0) , 则出现明显的网格敏感性, 即构件的寿命与网格划分的大小直接相关, 而且在计算过程中的最后几秒内, 应力分布发生混沌变化。而如果采用非局部意义上的损伤定义( 这里取d * = 100μ m) , 则上述局部化效应可以避免, 得到比较稳定的计算结果。此外, 图 3.33 对照了损伤和变形全耦合的方法和解耦的方法得到的损伤演化曲线, 解耦方法预测的构件寿命往往是偏于保守的。
.
vp ij
3 2
.
p
sij
e
（3.7.10）
（3.7.11）
对于铬镍铁合金 IN CON EL718, 实验测定了本构关系中的各个材料常数, 在 Norton 方程中, K = 1786, n = 18; 在 Lemait re 方程中, K = 2432, n = 20, m = 16.75, 其它参数为 r = 14, k = 21.6, A = 2177, α= 0.15, β= 0。
ij
e ij
vp ij
（3.7.8）
利用有效应力的定义
# ij
ij
/(1)
和应变等效假设，各向同性的弹
性本构关系为
土木工程专业教学质量工作汇报
e ij
E
1 (1
)
[(1
)
sij
1 2
3
kk ij ]
（3.7.9）
对于没有应变强化和有应变强化的情况，分别利用幂次蠕变律（即
Norton方程，得到粘塑性流动律表示为
式中1是最大主应力， m 是静水应力， e 是Mises等效应力，，
是材料常数。式（3.7.5）和式（3.7.6）都考虑了拉伸和压缩时
损伤演化的不同，对于式（3.7.5），在单压时
.
0
；对于式
（3.7.6），则有
.
.
comp (1 2 )r tens
（3.3.7）
材料的应变可以分解为弹性应变和粘塑性应变，即
.
(x)
1 *d
.
(x, )( )d
d
（3.71）
式中 d 是x点附近的一个小体元，是体元内任意一点，. 是局部意
义上的损伤演化律，. 是非局部意义上的损伤演化律。（x,）是人为
土木工程专业教学质量工作汇报
取定的一个函数，如
（x,
）
exp[

2 (x, d *2
)
]
（3.7.2）
式中d* 是一个特征长度，d(x,)是x和之间的距离。*d 定义为
第二个算例是弹性 -粘塑性裂纹的扩展问题 , 图3. 34是网格的划分，图3.35给出了用全耦合方法得到的裂纹尖端前方延
土木工程专业教学质量工作汇报
图3.32 单拉试件
土木工程专业教学质量工作汇报
长线上最大主应力和最大主应变的分布随时间的变化。在加载的初始时刻, 裂纹尖端附近有很强的应力集中, 由于蠕变损伤的逐渐演化, 裂尖附近应力分布逐渐平滑, 这种应力和应变场的变化以及裂纹的逐渐扩展过程是经典的断裂力学方法难以描述的。
*d
(x, )d
d
(3.7.3)
3.7.2 弹性-粘塑性材料断裂的局部方法各向同性的蠕变损伤演化通过宏观变量在0和1间的变化来
定量描述，损伤演化方程表示为
.
[
（）]r
(1
)k
A
（3.7.4）
式中是依赖于应力不变量的函数，如
(
)
1
(1 ) e
（3.7.5）
土木工程专业教学质量工作汇报
或
( ) 1 3 m (1 ) e （3.7.6）