1、材料成形力学材料成形力学（双语教学）（双语教学）2 Stress Analysis2.1 Specification of stress at a point2.1.1 Internal forces (in.f.)（内力）（内力）一点的应力状态一点的应力状态No external forcesInternal forces existExternal forces act.Mutual position of molecular changeDistances between them changeAdditional internal forces The additional force
2、s are what we are interested in and are called internal forces2.1.2 External forces (En.f.)（外力）（外力）External forcesAct on every particlesAct on contact surface Gravitational Magnetic Normal pressureFriction forcesIn the direction opposite to the moving direction of the body Convert:In.f.Ex.f.Section
3、methodBody forcesContact forces2.1.3 Stress at a point in a continuous body (连续体内一点的应力连续体内一点的应力)System of ex.f.A bodyIn equilibrium（外力系）（外力系）（处于平衡状态）（处于平衡状态）Body inter-sectioned by a section plane through pass point PUse section method:FPart A.Part B.(remove)FF:Resultant exerted by B on A,A in equil
4、ibriumF:Resultant exerted by A on B(remain)in equilibriumAround point P on section planeisolateElemental areaAResultant exerted by A on isAFAverage stress on is AFA（上的平均应力为 ）AFAABPF1F2F3F4F5F6F7F8APBF5F6F7F8Fwhen 0A0rAFdFAdAlim(contract around point P)(Force/(lngth)2)Intensity of internal force at t
5、he point P on the section plane,in.f./per unit area.r:Stress at the point P on the section plane For another section plane passing through the point P,we have another stress at the same point.In general,0r0rrThe stress state at a point can be considered defined if the stress on any section plane pas
6、sing through the point have been determined.（如果过一点任意截面上的应力已知，则可以认为过该点的应力状态便确定了。）（如果过一点任意截面上的应力已知，则可以认为过该点的应力状态便确定了。）2.1.4 Stress components (应力分量应力分量)Fr,and also,needs not be normal to the section plane.Fresolve（分解）sFNF:Normal to the reference plane:tangential to the reference planeNFsFFAPNormal str
7、ess(正应力正应力)NNA0FdFAdAlim(Normal to the section plane)Positive:elongationNegative:compressionShear stress(剪应力剪应力)ssA0FdFAdAlim(tangential to the section plane)Change in shape（改变物体的形状）（改变物体的形状）Nine components and stress tensor(九个分量和应力张量九个分量和应力张量)Coordinate system:OxyzTake an infinite small element fro
8、m the body around point P.Six section planes parallel to the coordinate planesParallel-piped(平行六面体平行六面体)Three orthogonal planes:xoy,yoz and zoxOn the face parallel to the plane xoy(normal direction is oz)Normal stress:zzz(In the oz direction)Shear stress:zresolvezxzyAlong ox directionAlong oy direct
9、ionDouble subscript notationFirst subscript:The direction of the normal to the plane on which the stress acts.second subscript:The sense of the stress.zyxOzzxzyxxyxzyyxyzyozzoxxoyoxoyozyyxxzzxyyxxzyzzyzxplanenormal directionSense of stressox oy oz Another specified coordinate systemox y z Another co
10、mponent systemy y x x z z x y y x x z y z z y z x They can be transformedCoordinate systeminfinitecomponent systeminfiniteStress tensorDetermine the stress state at the pointExpression of stress tensor(matrix of tensor)应力张量的表示方法（张量的矩阵形式）应力张量的表示方法（张量的矩阵形式）xxxyxzyxyyyzzxzyzzxxyxzyxyyzzxzyzxxxyxzyxyyyz
11、zxzyzz(,)iji jx y zRowAct on the same plane,but in the different direction Columm Act on the different plane,but in the same direction 1.The nine stress components constitute a unity which is not separable.九个应力分量构成一个不可分隔的统一体九个应力分量构成一个不可分隔的统一体2.The stress components depend on the choice of the coordi
12、nate system.应力分量取决于坐标系的选择应力分量取决于坐标系的选择3.The stress components can be transformed when referring to different coordinate system.不同的坐标系之间应力分量可以不同的坐标系之间应力分量可以相互转换相互转换4.The stress components can constitute stress invariants independent of the choice of the coordinate system.应力分量应力分量可以构成应力不变量，该应力不变量与坐标系的
13、选择无关可以构成应力不变量，该应力不变量与坐标系的选择无关5.It is a symmetric tensor.应力分量是对称张量应力分量是对称张量Properties of stress tensor 应力张量的性质应力张量的性质2.2 Differential equation of equilibrium in the neighbour-hood of a pointxzyxzyABCDEFGO2.2.1 Force equilibrium (力平衡力平衡)xxyxzyyxyzzzxzyTake an element using section method.Analyses the
14、stresses on the face of the element.Stress components are the continuous function of the Cartesian coordinate.0,ijijx y z（力平衡微分方程）（力平衡微分方程）,Aijijxx yy zzAijExpand into Taylors series:,ijijijijijxx yy zzx y zxyzxyzNeglect the terms in higher powers of xyzand,ijijijijijxx yy zzx y zxyzxyzAt the point
15、A(O A).,xyzincremet:,ijijBijijijxx yy zx y zxyxyAt the point B(O B).,xyincremet:xzyxzyABCDEFGOAt the point G(O G).xincremet:,ijGijijijxx y zx y zxxxzyxzyABCDEFGOxxyxzyyxyzzzxzyxxxxxyxyxxxzxzxxyyyyyxyxyyy zy zyyzzzzzxzxzzzyzyzzxzyxzyABCDEFGO0X Consider that the stress on each face are uniform(infinit
16、esimal)000XYZThe element is in equilibriumxxxxxy zx zxzxzx yz 0yxxxzxxyzxzyxzyABCDEFGOxxyxzyyxyzzzxzyxxxxxyxyxxxzxzxxyyyyyxyxyyy zy zyyzzzzzxzxzzzyzyzzxxy z yxyxyx zy yxx z yxx y 00yxxxzxxyz0 xyyyzyxyz0yzxzzzxyz0ijixBy means of tensor notation:j:is free subscript,it appears only one time in one term
17、 and is the same in all terms,it is replaced by x,y and z cyclicly in different eqs.i:is dummy subscript,it appears twice in one term and it is considered as the sum of three terms,in which the sub is replaced by x,y and z cyclicly.0ijix0 xjyjzjxyzxxx,jx0yxxxzxxyz,jy0 xyyyzyxyz,jz0yzxzzzxyziixxyyzzi
18、ij jSldummy,;jx y ziix xiy yiz zSlll,ixxxx xxy yxz zSlllyyx xyy yyz zSlll,iyzzx xzy yzz zSlll,iz2.2.2 Couple equilibrium (力矩平衡方程力矩平衡方程)Consider the moment about the axes passing through point P and parallel to Ox,Oy and Oz Neglect body forces and inertia forces.Resultant couple about the three axes:
19、000 xyzMMM0 xM yzyzyyzyzyzzAfter neglecting the quantities of the fourth order and simplifyingyzzyx z 2yyz2yx y 2zzy2z0 xzyxzyABCDEFGOzyyxyxyyzyzyzzyzx z x y Similarly:0yM xzzx0zM xyyxTherefore:ijjiEquality of shear stress（切应力互等）（切应力互等）From above analysis we got conclusion that the stress tensor is
20、a symmetric tensor.Therefore,nine components become six components.所以，应力张量是对称张量，所以，应力张量是对称张量，9个分量简化为个分量简化为6个分量。个分量。2.3 Three-dimensional stress analysis(important feature of tensor)2.3.1 Resultant stress on an oblique plane inclined to three Cartesian axesTake an element using the method of sections
21、Intersected byThree Cartesian planeTetrahedron OABC(四面体四面体)xzyOABCxxzxyyyzyxzzxzyRSnSsSzSxSySlmnRSzSySxS222RxyzSSSSAn oblique plane ABC（三维应力分析）（三维应力分析）（与三个坐标轴相倾斜的斜面上的和应力）（与三个坐标轴相倾斜的斜面上的和应力）Determine according to on three Cartesian planes:RSijplaneareaoxoyozOBClxxxyxzOACmyxyyyzOABnzxzyzzABC1SxSySzOAB
22、Cin equilibrium000XYZxxxyxzxSlmnFrom first column of the table yxyyyzySlmnFrom second column of the tablezxzyzzzSlmnFrom third column of the tablenmlllllzyxiiijjlS jxj xyj yzj zxjyjzjSllllmnfrom sub.j=x xxxyxzxSlmn j=y yxyyyzySlmnj=z zxzyzzzSlmn222RxyzSSSSijilis knownis knownSj is known SRTherefore,
23、if the stresses on three orthogonal Cartesian planes are known,the stresses on any oblique plane can be determined.ij can be transformed ji 2.3.2 Normal stresses on the oblique plane（斜面上的正应力）（斜面上的正应力）SRON.prozON.proyON.proxON.proRSSSSnxxxyxzyxyyyzzxzyzzSlmn llmn mlmn n2222xyzxyyzzxlmnlmmnnl Sn norma
24、l component,is coincide with ON Ss shear component resolve2.3.3 shear stresses on the oblique plane（斜面上的剪应力）（斜面上的剪应力）222snRSSS 22sRnSSS2.3.4 Stresses boundary conditions（应力边界条件）（应力边界条件）Relations between the distribution load on the body surface and stresses within the body at the same boundary point
25、 are the stress boundary condition.（所谓应力边界条件是指物体表面上的应力分布与同一边界处物体内部应力（所谓应力边界条件是指物体表面上的应力分布与同一边界处物体内部应力之间的关系。）之间的关系。）Body surface:inclined,normal ON(direction cosines:l,m,n)Distribution load p px(ox),py(oy),pz(oz)jij iplnmlpzxyxxxxnmlpzyyyxyynmlpzzyzxzz0yxxxzxxyz0 xyyyzyxyz0yzxzzzxyz0ijixBy means of t
26、ensor notation:（力平衡微分方程）（力平衡微分方程）ijjiEquality of shear stress（切应力互等）（切应力互等）iijjlSjxj xyj yzj zxjyjzjSllllmn222RxyzSSSSSummary of Last Class（斜面上的正应力）（斜面上的正应力）nxxxyxzyxyyyzzxzyzzSlmn llmn mlmn n2222xyzxyyzzxlmnlmmnnl（斜面上的剪应力）（斜面上的剪应力）222snRSSS22sRnSSSjij iplnmlpzxyxxxxnmlpzyyyxyynmlpzzyzxzz（应力边界条件）（应力
27、边界条件）2.3.5 Principal Stresses（主应力）（主应力）已知条件已知条件：ij on three Cartesian plane.An oblique plane(l,m,n)SR,Sn,SS,on the oblique plane.If l,m,n changeSR,Sn,SS,changeSelect l,m,n so as to take SR=SnSs=0Definition:The oblique plane on which SS=0 is a principal plane.The stress Sn=SR acting on the principal
28、plane is a principal stress.The direction of the principal stress is a principal direction.剪应力剪应力SS=0 的平面称为的平面称为主平面主平面，主平面上的正应力称为，主平面上的正应力称为主应力主应力，主应力的方，主应力的方向称为向称为主方向主方向。The problem is how to determine the principal stress(magnitude and directions)问题是如何确定主应力（大小和方向）问题是如何确定主应力（大小和方向）Suppose that the
29、direction cosines of the normal to the principal plane are l,m,n.and Sn=SR on the principal plane.Resultant vector component vector jn jnij iSS lSllSSnxnSmSSnynSSnzstress on oblique plane stress on three Cartesian plane:nmlSzxyxxxxnmlSzyyyxyynmlSzzyzxzz0jnijijlSi=ji=j01ij kronecker delta 0nmlSzxyxnx
30、x0nmSlzynyyxy0nSmlnzzyzxzFor the equation to have a nontrivial solution for l,m,n.the determinant of the coefficients must vanish:0 xxnyxzxijijnxyyynzyxzyxzznSSSSAfter expanding the determinant and rearranging the terms the coefficients matrix is as follows:.32222nnxxyyzznxxyyyyzzzzxxxyyzzxSSS 22220
31、 xxyyzzxyyz zxyzxxzxyyxyzz Trivial solution（无效解）（无效解）is l=m=n=0,violate（违背）（违背）the relation 2221lmn为了使得以为了使得以 l、m、n 为变量的方程有非零解，其系数矩阵必须为零。为变量的方程有非零解，其系数矩阵必须为零。Which can be written as:WhereThe cubic equation（三次方程）（三次方程）has three real roots which are the three principal stresses,1,2 and 3 acting on thr
32、ee orthogonal plane.321230nnnSJ SJ SJ1xxyyzzJ2222xxyyyyzzzzxxxyyzzxJ 22232xxyyzzxyyzzxyzxxzxyyxyzzJ 三次方程有三个实数根，这三个根分别为作用在三个相互垂直的坐标面上的三次方程有三个实数根，这三个根分别为作用在三个相互垂直的坐标面上的主应力主应力 1、2 和和 3。Substituting ij into two of the equations(ij-ijSn)li=0 and adding the condition l2+m2+n2=1 ,the three set of direction
33、 cosines l1,m1,n1;l2,m2,n2;l3,m3,n3;can be found.They define the three principal directions that are perpendicular to each other.Conclusion（结论）（结论）:ij 321J,J,Jcubic equation321,321,in,m,liii (Independent of the choice of the coordinate system)roots of the equationA given stress state and coordinate
34、system OXYZ 将将 ij 代入代入(ij-ijSn)li=0 的两个方程中，并考虑到的两个方程中，并考虑到 l2+m2+n2=1 ,则可以得到三则可以得到三组方向余弦组方向余弦 l1,m1,n1;l2,m2,n2;l3,m3,n3;这三组方向余弦便确定了三个相互垂直这三组方向余弦便确定了三个相互垂直的主方向的主方向.2.3.6 Stress invariants（应力不变量）（应力不变量）At a point of a body(given stress state)ij 321J,J,J 032213JSJSJSnnn321,ji 321J,J,J032213JSJSJSnnnTh
35、erefore 11JJ22JJ33JJzzyyxxzzyyxxJ1 First(linear)invariant（第一不变量）（第一不变量）2222zxyzxyxxzzzzyyyyxxJ222xzzyyxxxzzzzyyyyxxSecond(quadratic)invariantFor a given stress state 1,2,3 are unique,independent of the choice of coordinate systemzz2xyyy2zxxx2yzzxyzxyzzyyxx32Jzzyxyyxzxxzyxzzyyxzzyyxx2222third(cubic)i
36、nvariant（三次不变量）（三次不变量）3211J1332212J3213J ij:nine components(ij=ji)six components(principal surface)three componentsAlgebraically 321When the direction of the coordinate axes coincide with the principal direction 123000000ij2.3.7 Some stress states in terms of principal stresses Uniaxial stress state
37、（单向应力状态）（单向应力状态）,(+,0,0)uniaxial tension（单向拉伸）（单向拉伸）01032(0,0,-)uniaxial compression（单向压缩）（单向压缩）02103Plane stress state(biaxial stress state)（平面应力状态）（平面应力状态）010203(+,+,0)biaxial tension 010203tension in one direction,compression in another direction.(+,0,-)010203(0,-,-)biaxial compression Triaxial s
38、tress state(三向应力状态三向应力状态)010203321(+,+,+)(-,-,-)(+,-,-)(+,+,-)Spherical stress state0103203201Drawing of a round barExtrusion of a round bar132321Cylindrical stress stateor0 triaxial uniform tension0 triaxial uniform compression (hydrostatic stress state).3212.3.8 principal shear stress and maximum
39、shear stress(主剪应力和最大剪应力主剪应力和最大剪应力)Relation between principal stress and principal shear stressThe principal directions the coordinate axes Coincide YOZ:1(xx=1,xy=xz=0)XOZ:2(yy=2,yx=yz=0)YOX:3(zz=3,zx=zy=0)321000000lSx1mSy2nSz32232222212222nmlSSSSzyxR232221nmlnSmSlSSzyxn(A)2232221223222221222nmlnmlSS
40、SnRs(B)jiijSl（主应力和主剪应力之间的关系）（主应力和主剪应力之间的关系）If(l,m,n)change,then Ss changeSelect l,m,n Ss extremum(极大，极大，极小极小)Two of l,m,n are independentThe problem of finding extremum value of Ss with restriction condition2221mlnSs is the function of l and m02lSs02mSs 02131232231mll02132232231mlm(C)1nl=m=0,m=n=0,1
41、ll=n=0,1mSs attains its minimum value on the three principal planes Ss=01nS0sS2nS0sS0sS3nRSS(A)411ncoslcos 21mcos From(C)we can get(1 3)(1-2l2)=022l22n22m 22n 411ncosmcos 21lcos 22l 411lcosmcos 21ncos321,is principal shear stresses 21312(B)31212(B)12312(B)m=00l l=00m n=00m 22mmax1312 is maximum shea
42、r stress 1-2l2=0Exercise Problem1 The stress state is shown as in Fig.1,Please determine the resultant stress SR,the components of the resultant,Sx,Sy,and Sz,the normal stress Sn and shear stress Ss on the oblique plane,when the three direction cosines of the oblique plane are xyz1051055513lmnFig.1E
43、xercise Problem2 The four stress tensor are known as Ta,Tb,Tc and Td,please determine whether they belong to the same stress state or not?300002000010aT300001550515cT200002000020bT255052500010dTPrincipal StressesDefinition:The oblique plane on which SS=0 is a principal plane.The stress Sn=SR acting
44、on the principal plane is a principal stress.The direction of the principal stress is a principal direction.Summary of Last Class0jnijijlS0 xxnyxzxijijnxyyynzyxzyxzznSSSSThe cubic equation（三次方程）（三次方程）has three real roots which are the three principal stresses,1,2 and 3 acting on three orthogonal pla
45、ne.321230nnnSJ SJ SJ1xxyyzzJ2222xxyyyyzzzzxxxyyzzxJ 22232xxyyzzxyyzzxyzxxzxyyxyzzJ Stress invariants First(linear)invariantSecond(quadratic)invariantthird(cubic)invariant 将将 ij 代入代入(ij-ijSn)li=0 的两个方程中，并考虑到的两个方程中，并考虑到 l2+m2+n2=1 ,则可以得则可以得到三组方向余弦到三组方向余弦 l1,m1,n1;l2,m2,n2;l3,m3,n3;这三组方向余弦便确定了三个相互这三组方向
46、余弦便确定了三个相互垂直的主方向垂直的主方向.3211J1332212J3213JWhen the direction of the coordinate axes coincide with the principal direction 123000000ijAlgebraically 321Some stress states in terms of principal stresses Uniaxial stress state（单向应力状态）（单向应力状态）Plane stress state(biaxial stress state)（平面应力状态）（平面应力状态）Triaxial
47、 stress state(三向应力状态三向应力状态)213123121212312max1312is maximum shear stress principal shear stress and maximum shear stress3 The stress tensors of the body is known as ij,Please determine the principal stress 1,2 and 3,the direction cosine of the principal plane,and the principal shear stress.101010101
48、ijExercise Problem4 The plane stress state is shown as in Fig.3,Please verify that:22cos2sincossinnxxyy22sincoscossinnxyxy11cos2sin222nxyxyxy1sin2cos22nxyxyorxyxyyxnnFig.3Exercise Problem5 在在oxy平面内按图平面内按图4所示的方向贴应变片，测出的应力为所示的方向贴应变片，测出的应力为,xyn 试求试求?xy?nxynnxy45oFig.4Exercise Problem2.3.9 Spherical and
49、 deviator stresses(球应力和偏差应力球应力和偏差应力)Deformation Volumetric component:elastic,volumetric change(hydrostatic component)Distortional component:change in geometric of a body,(elastic or plastic).ijvolumetric component(elastic)Spherical stressDeviator stressdistortional component(elastic or plastic)Spher
50、ical stress and spherical stress tensor（球应力和球应力张量）（球应力和球应力张量）Spherical stress state Spherical stress state is a uniform triaxial(tensile or compressive)stress state（球应力状态是均匀的三向应力状态（拉伸或压缩）（球应力状态是均匀的三向应力状态（拉伸或压缩）Features:1)shear stress is absent on any arbitrary plane2)No any distortional component of
