弹性力学基础-中英

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1、2021/3/101 Elastic-plastic Mechanics of Materials Ming-an CHEN（陈明安）（陈明安）School of Materials Science and Engineering Central South University2021/3/102教材与主要参考书教材与主要参考书弹性力学弹性力学（上册，第三版）（上册，第三版）徐芝纶徐芝纶 编编高等教育出版社高等教育出版社弹性理论弹性理论铁木辛柯铁木辛柯 （Timoshenko）编编科学出版社科学出版社弹性力学弹性力学吴家龙吴家龙 编编同济大学出版社同济大学出版社弹性力学学习方法及解题指导弹性

2、力学学习方法及解题指导王俊民王俊民 编编同济大学出版社同济大学出版社弹性与塑性力学弹性与塑性力学（例题与习题）（例题与习题）徐秉业徐秉业 编编机械工业出版社机械工业出版社弹性与塑性力学基础弹性与塑性力学基础哈尔滨工业大学出版社哈尔滨工业大学出版社王仲仁王仲仁 编编2021/3/103 Typical tensile specimen Typical tensile test machinegauge length(portion of sample with reduced cross section)Chapt.1 Introduction1.1 Elasticity and plastic

3、ity（弹性与塑性）（弹性与塑性）1.gauge length=2021/3/1042021/3/105Fbonds stretchreturn to initialElastic means reversible!It is reversible and time independent.The deformation vanishes instantaneously as soon as forces are removed.2021/3/106Plastic means permanent!Flinear elasticlinear elasticplasticPlastic defor

4、mation-it is irreversible or permanent.2021/3/107EepOABCDEepselastic regionyield strength 屈服应力屈服应力屈服强度屈服强度plastic regionultimate tensile strength 抗拉强度抗拉强度breaking strengthStrain hardening加工(应变)硬化Flow stress流动应力Strain hardening exponent(n)加工硬化指数加工硬化指数Bauschinger effectStrain-rate sensitivity(m)2021/3

5、/1082003 Brooks/Cole,a division of Thomson Learning,Inc.Thomson Learning is a trademark used herein under license.2021/3/1092021/3/1010杨氏（弹性）模量杨氏（弹性）模量 E2021/3/10110.280.61Magnesium,AluminumPlatinumSilver,GoldTantalumZinc,TiSteel,NiMolybdenumG raphiteSi crystalGlass-sodaConcreteSi nitrideAl oxidePCW

6、ood(grain)AFRE(fibers)*CFRE*GFRE*Glass fibers onlyCarbon fibers onlyAramid fibers onlyEpoxy only0.40.8246102040608010 020060080010 001200400TinCu alloysTungstenSi carbideDiamondPTF EHDP ELDPEPPPolyesterPSPETC FRE(fibers)*G FRE(fibers)*G FRE(|fibers)*A FRE(|fibers)*C FRE(|fibers)*MetalsAlloysGraphite

7、CeramicsSemicondPolymersComposites/fibersE(GPa)109 PaBased on data in Table B2,Callister 6e.Composite data based onreinforced epoxy with 60 vol%of alignedcarbon(CFRE),aramid(AFRE),orglass(GFRE)fibers.2021/3/1012 Plastic tensile strain at failure:Another ductility measure:%AR Ao AfAox100 Note:%AR and

8、%EL are often comparable.-Reason:crystal slip does not change material volume.-%AR%EL possible if internal voids form in neck.%EL LfLoLox1004.Ductile and brittle materials（韧性与脆性材料）2021/3/1013 Energy to break a unit volume of material Approximate by the area under the stress-strain curve.smaller toug

9、hness-unreinforced polymersEngineering tensile strain,Engineering tensile stress,smaller toughness(ceramics)larger toughness(metals,PMCs)2021/3/1014 Low-carbon(mild)steel is different from most other metals in that there is a sudden small drop of load at the yield point followed by an extension at c

10、onstant stress.The lower load is usually referred to as the yield point for mild steel.2021/3/10152021/3/10162021/3/10172021/3/10182021/3/10192021/3/10202021/3/10212021/3/10222021/3/10232021/3/10242021/3/1025The actual point of yield is often difficult to identify.A number of techniques are used to

11、locatey.The tangent method(or knee method)locates the yield strength at the intersection of the elastic slope and the initial portion of the plastic region(not reliably).The preferred method is the percentage offset method where yield strength is obtained by drawing a line parallel to the initial el

12、astic region data at 0.2%strain(0.002)offset.Where this line intersects the stress-strain curve then becomes known as the 0.2%yield strength.5.Determination of yield strength屈服强度屈服强度2021/3/1026 For most metals,loading beyond the yield point causes a permanent deformation.When a material is loaded to

13、 point B and then unloaded,it returns to a zero stress state along a line parallel to the initial elastic region but directly from B.The strain remaining in the material at point D is known as the plastic deformation.On reloading from D there is a departure from linearity at C,slightly below B,and t

14、he stress-strain curve becomes the same as the original stress-strain curve(at E).Note that the point of departure from linearity on the reload curve(C)is slightly higher than for the first loading curve.6.Unloading and reloading（卸载与再加载）（卸载与再加载）2021/3/1027s7.Idealizations of stress-strain curves 应力应

15、变曲线简化应力应变曲线简化sElastic-plasticsIdeal elastic-plasticsLinear stress-strainss2021/3/1028sIdeal rigid-plastics Rigid-plasticss2021/3/10291.2 Research objects and contents2021/3/10301.DifferencesMech.Mater.-（1）Research objectsBarStruc.Mech.-Structure of bars,truss structureElastic mech.-Elastic objects:1

16、D,2D and 3D2021/3/10312021/3/1032 On January 18,2005 the first Airbus A380 was revealed and presented to the public.F-22,the 4th generation flight,USA air force2021/3/1033Manufacturing processes that make use of cold working as well as hot working.Common metalworking methods轧制、挤压、轧制、挤压、锻造、冲压、锻造、冲压、拉

17、拔等拉拔等2021/3/1034（2）Research approachesMech.Mater.-Struc.Mech.-Elastic mech.-借助于直观和实验现象作一些假定，如借助于直观和实验现象作一些假定，如平面假设等，然后由静力学、几何关系、平面假设等，然后由静力学、几何关系、物理方程三方面进行分析。物理方程三方面进行分析。与材力类同。与材力类同。仅由静力平衡、几何方程、物理方程三仅由静力平衡、几何方程、物理方程三方面分析，方面分析，放弃了材力中的大部分假定放弃了材力中的大部分假定。2021/3/1035Example：Beam in bending弹性力学结果弹性力学结果材料力

18、学结果材料力学结果当当 l h 时，两者误差很小时，两者误差很小如：变截面杆受拉伸如：变截面杆受拉伸 弹性力学以微元体为研弹性力学以微元体为研究对象，建立方程求解，得究对象，建立方程求解，得到弹性体变形的一般规律。到弹性体变形的一般规律。所得结果更符合实际。所得结果更符合实际。2021/3/10362.Contents of researchStresses,strain and deformation,displacements for bars structure under loads including thermal load.Stresses,strain and deformat

19、ion,mechanical properties,failure criterion for bars under loads including thermal load.Strengthen,rigidity and stability of loaded bars.Mech.Mater.-Struc.Mech.Elastic mech.Strengthen,rigidity and stability of loaded bars structure.Stresses,strain and deformation,displacements for elastic objects un

20、der loads including thermal load.Strengthen,rigidity and stability of loaded elastic objects.（任务）解决杆件的强度、刚度、稳定性问题。（任务）解决杆件的强度、刚度、稳定性问题。2021/3/10371.连续性假定连续性假定整个物体的体积都被组成物体的介质充满，不留下任何空隙。整个物体的体积都被组成物体的介质充满，不留下任何空隙。该假定在研究物体的该假定在研究物体的宏观力学特性宏观力学特性时，与工程实际吻时，与工程实际吻合较好；研究物体的合较好；研究物体的微观力学性质微观力学性质时不适用。时不适用。作用

21、：作用：使得使得、u 等量表示成坐标的连续函数。等量表示成坐标的连续函数。),(zyx),(zyxuu),(zyxxx保证保证ssQslim0中极限的存在。中极限的存在。1.3 Assumptions in elastic-plastic mechanics of materials2021/3/10382.线弹性假定线弹性假定 假定物体完全服从虎克（假定物体完全服从虎克（Hooke）定律，）定律，应力与应变间应力与应变间成线性比例关系成线性比例关系（正负号变化也相同）。（正负号变化也相同）。比例常数比例常数 弹性常数（弹性常数（E、）脆性材料脆性材料 一直到破坏前，都可近似为线弹性的；一直到

22、破坏前，都可近似为线弹性的；塑性材料塑性材料 比例阶段，可视为线弹性的。比例阶段，可视为线弹性的。3.均匀性假定均匀性假定作用：作用：可使求解方程线性化可使求解方程线性化 假定整个物体是由同一种材料组成假定整个物体是由同一种材料组成 的，各部分材料性的，各部分材料性质相同。质相同。作用：作用：弹性常数（弹性常数（E、）不随位置坐标而变化；不随位置坐标而变化；取微元体分析的结果可应用于整个物体。取微元体分析的结果可应用于整个物体。Linear Elasticity Metals are composed of grains of material having directional and b

23、oundary characteristics.However,these grains are usually microscopic and when a larger portion of the material is considered,these random variations average out to produce a macroscopically uniform material.Macroscopic uniformity=homogenous.2021/3/10394.各向同性假定各向同性假定 假定物体内一点的假定物体内一点的弹性性质弹性性质在所有在所有各个方

24、向都相同各个方向都相同。作用：作用：弹性常数（弹性常数（E、）不随坐标方向而变化；不随坐标方向而变化；金属金属 上述假定符合较好；上述假定符合较好；木材、岩石木材、岩石 上述假定不符合，称为上述假定不符合，称为各向异性材料各向异性材料；符合上述符合上述4个假定个假定的物体，称为的物体，称为理想弹性体理想弹性体。5.小变形假定小变形假定 假定位移和形变是微小的，即物体受力后物体内各点假定位移和形变是微小的，即物体受力后物体内各点位移远远小于物体原来的尺寸。位移远远小于物体原来的尺寸。1,1作用：作用：建立方程时，可略去高阶微量；建立方程时，可略去高阶微量；可用变形前的尺寸代替变形后的尺寸。可用变

25、形前的尺寸代替变形后的尺寸。使求解的方程使求解的方程线性化线性化。2021/3/10402003 Brooks/Cole,a division of Thomson Learning,Inc.Thomson Learning is a trademark used herein under license.Anisotropic behavior in a rolled aluminum-lithium sheet material used in aerospace applications.The sketch relates the position of tensile bars to

26、 the mechanical properties that are obtained2021/3/1041弹性力学是固体力学的一个分支，研弹性力学是固体力学的一个分支，研究弹性体由于外力作用或温度改变等原因究弹性体由于外力作用或温度改变等原因而发生的应力、形变和位移。而发生的应力、形变和位移。本课程较为完整的表现了力学问题的数本课程较为完整的表现了力学问题的数学建模过程，建立了弹性力学的基本方程和学建模过程，建立了弹性力学的基本方程和边值条件，并对一些问题进行了求解。弹性边值条件，并对一些问题进行了求解。弹性力学基本方程的建立为进一步的数值方法奠力学基本方程的建立为进一步的数值方法奠定了基

27、础。定了基础。2021/3/1042Chapt.2 Stress Analysis（应力分析）（应力分析）2.1 Force and stress 2.1.1 Force（力）（力）Structural mechanics-describing the relations between external forces,internal forces(gravity,centrifugal,magnetic attractions,etc.)and deformation of structural materials.Forces are vector quantities,thus hav

28、ing direction and magnitude.They have special names depending upon their relationship to a reference plane:a)Compressive forces act normal and into the plane;b)Tensile forces act normal and out of the plane;c)Shear forces act parallel to the plane.Pairs of oppositely directed forces produce twisting

29、 effects called moments.内力内力(1)物体内部分子或原子间的相互作用力物体内部分子或原子间的相互作用力;(2)由于外力作用引起的相互作用力由于外力作用引起的相互作用力.2021/3/1043Rectangular Coordinates 2021/3/1044The system of particles in the Figure is said to be in equilibrium if every one of its constitutive particles is in equilibrium.Consequently,the first conditi

30、on for equilibrium:the vector sum of all the forces is zero.where rn extends from point 0 to an arbitrary point on the line of action of force Fn.If the surface and body forces are in balance,the body is in static equilibrium.The second condition for equilibrium:the total moment of all the external

31、forces about an arbitrary point 0 must be zero.An Isolated System of Particles Showing External and Internal ForcesFor an object to be at rest(ie static eq),net force and net moment must be zero.2021/3/10452021/3/10462.1.2 Stress（应力）（应力）Examination of Internal Forces of a Body in EquilibriumIn the F

32、igure,an arbitrary plane,aa,divides a body in equilibrium into regions I and II.Since the force acting upon the entire body is in equilibrium,the forces acting on part I alone must be in equilibrium.In general,the equilibrium of part I will require the presence of forces acting on plane aa.These int

33、ernal forces applied to part I by part II are distributed continuously over the cut surface,but,in general,will vary over the surface in both direction and intensity.2021/3/1047AFSAlim0ASnStress has units:N/m2Stress is the term used to define the intensity and direction of the internal forces acting

34、 at a particular point on a given plane.The stress acting at a point on a specific plane is a vector.Its direction is the limiting direction of force dF as area dA approaches zero.It is customary to resolve the stress vector into two components whose scalar magnitudes are:normal stress component:act

35、ing perpendicular to the plane shear stress component:acting in the plane.由外力引起的在由外力引起的在 P点的某一面上内力分布集度点的某一面上内力分布集度应力分量应力分量应力的法向分量应力的法向分量 正应力正应力应力的切向分量应力的切向分量 剪应力剪应力应力关于坐标应力关于坐标连续分布的连续分布的:),(zyx),(zyx2021/3/1048 Shear Stress（剪应力）（剪应力）Shear stresses can also be generated by applied shear loads.Conside

36、r two equal and opposite shear forces V acting on a rectangular block as shown.2021/3/10492021/3/10502.2 State of Stress（应力状态）（应力状态）2.2.1 Stress components in Rectangular coordinatesRight-handed System of Rectangular Coordinates The selection of different cutting planes through point 0 would,in gene

37、ral,result in stresses differing in both direction and magnitude.Stress is thus a second-order tensor quantity,because not only are magnitude and direction involved but also the orientation of the plane on which the stress acts is involved.NOTE:A complete description of the magnitudes and directions

38、 of stresses on all possible planes through point 0 constitutes the state of stress at point 0.2021/3/1051xzxyx,yzyxy,zyzxz,Note:Three stress components at a given point on the plane through the point.应力符号的意义：应力符号的意义：第第1个下标表示应力所在面的法线个下标表示应力所在面的法线方向；第方向；第2个下标表示应力的方向个下标表示应力的方向.2021/3/1052The direction

39、 of a positive normal or shear stress is determined by the direction of the normal on the face upon which the normal or shear stress is applied.If the normal of the face is in the positive direction(as defined by local Cartesian coordinates)then a positive normal or shear stress is also in the posit

40、ive direction(two positives make a positive).If the normal is in the negative direction then a positive normal or shear stress must also be in the negative direction.Positive and negative of normal and shear stressNormal stresses are positive for tensile stress and negative for compressive stress.20

41、21/3/1053Now we consider the stresses components on the three mutually perpendicular planes as faces of a cube of infinitesimal size(element)which surround the given point.2021/3/10542021/3/1055A positive component of stress acts on a positive face in a positive coordinate direction or on a negative

42、 face in a negative coordinate direction.2021/3/1056Thus an applied shear stress is automatically accompanied by a shear stress of equal intensity at right angles(and causing an opposite turning moment)to the original shear stress.These are called complementary shear stresses.2021/3/1057xyxyxyxyxyyx

43、xyyxxyIn mechanics of materials and for drawing of Mohrs circle:(Right-hand rule)与材力中剪应力与材力中剪应力正负号正负号规定的区别：规定的区别：规定使得规定使得单元体顺时的剪应力单元体顺时的剪应力为正，反之为负。为正，反之为负。2021/3/1058 The state of stress at a point can normally be determined by computing the stresses acting on certain conveniently oriented planes pa

44、ssing through the point of interest.Stresses acting on any other planes can then be determined by means of simple,standardized analytical or graphical methods.If so we can use the stresses,acting on these conveniently oriented planes passing through the point,for representing the stress state of the

45、 given point,and that the stress state at this point is known.The selection of different cutting planes through a given point would,in general,result in stresses differing in both direction and magnitude.A complete description of the magnitudes and directions of stresses on all possible planes throu

46、gh the given point constitutes the state of stress at the given point.Problem:The stress components,on which of and how much different planes,can be used for representing the stress state of the given point?2.2.2 State of stress at a point（点的应力状态）（点的应力状态）2021/3/10592021/3/1060 State of uniaxial stre

47、ss:单向应力状态单向应力状态The stress normal to the cross-sectional surface:2021/3/1061Stresses:Forces:2021/3/10622.General Stress Systems in 2-Dimensions （双向应力状态）（双向应力状态）The stresses on the element ABCD in a component subjected to combined 2D loading(assuming no through thickness stresses,i.e.plane stress)are

48、schematically shown in the Figure.The reference system of coordinate axes are as shown also.What is the stress state on a chosen plane of interest?2021/3/10632021/3/1064The stresses x andxy are obtained by resolution of forces in the respective directions.2021/3/1065 yxyyxxyyxxyx cossinsincos cossin

49、sincos y yx xyxxyx-y x y y y x xx xyy y xypxpxp-yp2021/3/10663.General Stress Systems in 3-Dimensions （三向应力状态）（三向应力状态）cos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnnSxSySzcos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,

50、cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,

51、cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnn SDecompose S into Sx,Sy

52、 and Sz,then S2=Sx2+Sy2+Sz22021/3/1067These results clearly illustrate how the values for the normal and shear stress components of a force distributed over a plane inside of an object depends upon how you look at the point inside the object in the sense that the values of the shear and normal stres

53、ses at a point within a continuum depend upon the orientation of the plane you have chosen to view.n=Sxl+Sym+Szn =xl2+ym2+zn2+2(xylm+yzmn+zxnl)Sn n n l=cos(n,x),m=cos(n,y)and n=cos(n,z)n2=S2-n22021/3/1068The above mentioned showes that if we know the 9 stresses components on the three mutually perpe

54、ndicular planes as faces of a cube of infinitesimal size(element)which surround the given point we can determine the sterss components acting on any plane through the point.So these 9 stresses components can be used to represent the stress state of a point.2021/3/1069一点应力状态可表示为：一点应力状态可表示为：xyzyz xy z

55、x 2021/3/10702.3 Stress tensor and principal stresses （应力张量与主应力）（应力张量与主应力）2.3.1 Stress tensor（应力张量）（应力张量）Tensor is the generalised term for a vector.Its full mathematical definition is:A mathematical entity specifiable by a set of components with respect to a system of co-ordinates and such that the

56、 transformation that has to be applied to the components to obtain components with respect to a new system of co-ordinates is related in a certain way to the transformation that has to be applied to the system of coordinates.The components of a vector change when the co-ordinate system is rotated.Ho

57、wever,the vector still has the same magnitude and direction as it did before the co-ordinate system was rotated.Second rank tensors(e.g.stress,inertia)see their components change when a co-ordinate system is rotated and unlike vectors the magnitude and orientation of the tensor may also change.2021/

58、3/1071xyzOxyxxzyxyyzzzyzxyxyyzzzyzxzzyzxyzyyxxzxyx yxxyzyyzxzzx Theorem of conjugate shearing stressesTherefore only 6 independent stress componentsStress tensor is symmetric.2021/3/10722021/3/1073Could you write out the stress tensors corresponding to the following figures？2021/3/1074xyz504030ABC20

59、21/3/1075Could you show the stress tensor in a corresponding element？060060000030ij 2021/3/1076740410006ij 2021/3/1077三向应力状态下的应力变换三向应力状态下的应力变换2021/3/10782021/3/10792021/3/10802021/3/10812.3.2 Principal stresses（主应力）（主应力）2021/3/10822021/3/1083 一点处一般有三个主平面，互相垂直。一点处一般有三个主平面，互相垂直。主主平面平面切应力等于零的平面。切应力等于零的

60、平面。主应力主应力主平面上的正应力。一点处一般有三个主应力。主平面上的正应力。一点处一般有三个主应力。2021/3/1084 三向应力状态三向应力状态cos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnnSxSySzcos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyzn

61、nnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyzn

62、nnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyznnnnnnnnn SDecompose S into Sx,Sy and Sz,then S2=Sx2+Sy2+Sz22021/3/1085假设该斜微

63、分面即为待求的主平面，面上0，正应力全应力S。全应力S在3个坐标轴上的投影为：xyzssllssmmssnn；()0()0()0 xyxzxxyyzyxzyzzlmnlmnlmn以l、m、n为未知数的齐次线性方程组，其解就是应力主轴的方向。显然lmn0是一组解，但l2m2n21，故应求其非0解。2021/3/1086032213III0)()(321)()0 xyxzxxyyzyxzyzz（2021/3/1087321zyzyxzxyxI.3133221222zxyzxyxzzyyxxxzzxzzzyyzyyyxxyxI23211zyxIStress invariants(应力张量不变量应力张

64、量不变量)第一、第二、第三应力不变量第一、第二、第三应力不变量2021/3/1088xyz 2 1 3123000000 xxyxzijyxyyzzxzyz2021/3/10892021/3/10902021/3/1091用主应力表示的各种应力状态的图示：用主应力表示的各种应力状态的图示：2021/3/10922021/3/10932021/3/10942021/3/10952021/3/10962021/3/10972021/3/10982021/3/10992021/3/101002021/3/101012.3.3 Principal shear stresses（主剪应力）主剪应力）剪应

65、力取极值的平面上的剪应力剪应力取极值的平面上的剪应力-主剪应力；主剪应力；主剪应力所在的平面主剪应力所在的平面-主剪应力平面；主剪应力平面；主剪应力平面的法线方向主剪应力平面的法线方向-主剪应力方向。主剪应力方向。n=Sxl+Sym+Szn =xl2+ym2+zn2+2(xylm+yzmn+zxnl)n2=S2-n2Sn n ncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyzt

66、xyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos,cos,cos,xxyxzxyxyyzyzxzyzztxyztxyztxyz nnnnnnnnncos,cos,cos,cos,cos,cos,cos