1、2 Deformation: Displacements and StrainsWe begin development of the basic field equations of elasticity theory by first investigating thekinematics of material deformation. As a result of applied loadings, elastic solids will changeshape or deform, and these deformations can be quantified by knowing
2、 the displacements ofmaterial points in the body. The continuum hypothesis establishes a displacement field at allpoints within the elastic solid. Using appropriate geometry, particular measures of deformationcan be constructed leading to the development of the strain tensor. As expected, the strain
3、components are related to the displacement field. The purpose of this chapter is to introduce thebasic definitions of displacement and strain, establish relations between these two fieldquantities, and finally investigate requirements to ensure single-valued, continuous displace-ment fields. As appr
4、opriate for linear elasticity, these kinematical results are developed underthe conditions of small deformation theory. Developments in this chapter lead to two funda-mental sets of field equations: the strain-displacement relations and the compatibility equa-tions. Further field equation developmen
5、t, including internal force and stress distribution,equilibrium and elastic constitutive behavior, occurs in subsequent chapters.2.1 General DeformationsUnder the application of external loading, elastic solids deform. A simple two-dimensionalcantilever beam example is shown in Figure 2-1. The undef
6、ormed configuration is taken withthe rectangular beam in the vertical position, and the end loading displaces material points tothe deformed shape as shown. As is typical in most problems, the deformation varies frompoint to point and is thus said to be nonhomogenous. A superimposed square mesh is s
7、hown inthe two configurations, and this indicates how elements within the material deform locally. It isapparent that elements within the mesh undergo extensional and shearing deformation. Anelastic solid is said to be deformed or strained when the relative displacements between pointsin the body ar
8、e changed. This is in contrast to rigid-body motion where the distance betweenpoints remains the same.In order to quantify deformation, consider the general example shown in Figure 2-2. In theundeformedconfiguration,weidentifytwoneighboringmaterialpointsPoandPconnectedwiththe relative position vecto
9、r r as shown. Through a general deformation, these points are mappedto locations P0oand P0in the deformed configuration. For finite or large deformation theory, the27Sadd / Elasticity Final Proof 3.7.2004 2:59pm page 27undeformed and deformed configurations can be significantly different, and a dist
10、inctionbetween these two configurations must be maintained leading to Lagrangian and Euleriandescriptions; see, for example, Malvern (1969) or Chandrasekharaiah and Debnath (1994).However, since we are developing linear elasticity, which uses only small deformation theory,the distinction between und
11、eformed and deformed configurations can be dropped.Using Cartesian coordinates, define the displacement vectors of points Poand P to be uoandu, respectively. Since P and Poare neighboring points, we can use a Taylor series expansionaround point Poto express the components of u asu uouxrxuyryuzrzv vo
12、vxrxvyryvzrzw wowxrxwyrywzrz(2:1:1)(Undeformed) (Deformed)FIGURE 2-1 Two-dimensional deformation example.PPPoPorr(Undeformed) (Deformed)FIGURE 2-2 General deformation between two neighboring points.Sadd / Elasticity Final Proof 3.7.2004 2:59pm page 2828 FOUNDATIONS AND ELEMENTARY APPLICATIONSNote th
13、at the higher-order terms of the expansion have been dropped since the components of rare small. The change in the relative position vector r can be written asDr r0C0r uC0uo(2:1:2)and using (2.1.1) givesDrxuxrxuyryuzrzDryvxrxvyryvzrzDrzwxrxwyrywzrz(2:1:3)or in index notationDri ui,jrj(2:1:4)The tens
14、or ui,jis called the displacement gradient tensor, and may be written out asui,juxuyuzvxvyvzwxwywz2666666437777775(2:1:5)From relation (1.2.10), this tensor can be decomposed into symmetric and antisymmetricparts asui,j eij!ij(2:1:6)whereeij12(ui,juj,i)!ij12(ui,jC0uj,i)(2:1:7)The tensor eijis called
15、 the strain tensor, while !ijis referred to as the rotation tensor. Relations(2.1.4) and (2.1.6) thus imply that for small deformation theory, the change in the relativeposition vector between neighboring points can be expressed in terms of a sum of strain androtation components. Combining relations
16、 (2.1.2), (2.1.4), and (2.1.6), and choosing ri dxi,we can also write the general result in the formui uoieijdxj!ijdxj(2:1:8)Because we are considering a general displacement field, these results include both straindeformation and rigid-body motion. Recall from Exercise 1-14 that a dual vector !ican
17、Sadd / Elasticity Final Proof 3.7.2004 2:59pm page 29Deformation: Displacements and Strains 29be associated with the rotation tensor such that !iC01=2eijk!jk. Using this definition, it isfound that!1 !3212u3x2C0u2x3C18C19!2 !1312u1x3C0u3x1C18C19!3 !2112u2x1C0u1x2C18C19(2:1:9)which can be expressed c
18、ollectively in vector format asv (1=2)(rC2u). As is shown in thenext section, these components represent rigid-body rotation of material elements aboutthe coordinate axes. These general results indicate that the strain deformation is related to thestrain tensor eij, which in turn is a related to the
19、 displacement gradients. We next pursue a moregeometric approach and determine specific connections between the strain tensor componentsand geometric deformation of material elements.2.2 Geometric Construction of Small Deformation TheoryAlthough the previous section developed general relations for s
20、mall deformation theory, wenow wish to establish a more geometrical interpretation of these results. Typically, elasticityvariables and equations are field quantities defined at each point in the material continuum.However, particular field equations are often developed by first investigating the be
21、havior ofinfinitesimal elements (with coordinate boundaries), and then a limiting process is invoked thatallows the element to shrink to a point. Thus, consider the common deformational behavior ofa rectangular element as shown in Figure 2-3. The usual types of motion include rigid-bodyrotation and
22、extensional and shearing deformations as illustrated. Rigid-body motion does notcontribute to the strain field, and thus also does not affect the stresses. We therefore focus ourstudy primarily on the extensional and shearing deformation.Figure 2-4 illustrates the two-dimensional deformation of a re
23、ctangular element withoriginal dimensions dx by dy. After deformation, the element takes a rhombus form asshown in the dotted outline. The displacements of various corner reference points are indicated(Rigid Body Rotation)(Undeformed Element)(Horizontal Extension) (Vertical Extension) (Shearing Defo
24、rmation)FIGURE 2-3 Typical deformations of a rectangular element.Sadd / Elasticity Final Proof 3.7.2004 2:59pm page 3030 FOUNDATIONS AND ELEMENTARY APPLICATIONSin the figure. Reference point A is taken at location (x,y), and the displacement components ofthis point are thus u(x,y) and v(x,y). The co
25、rresponding displacements of point B areu(xdx,y) and v(xdx,y), and the displacements of the other corner points are defined inan analogous manner. According to small deformation theory, u(xdx,y) C25 u(x,y)(u=x)dx, with similar expansions for all other terms.The normal or extensional strain component
26、 in a direction n is defined as the change inlength per unit length of fibers oriented in the n-direction. Normal strain is positive if fibersincrease in length and negative if the fiber is shortened. In Figure 2-4, the normal strain in the xdirection can thus be defined byexA0B0C0ABABFrom the geome
27、try in Figure 2-4,A0B0dxuxdxC18C192vxdxC18C192s12uxuxC18C192vxC18C192dxsC25 1uxC18C19dxwhere, consistent with small deformation theory, we have dropped the higher-order terms.Using these results and the fact that AB dx, the normal strain in the x-direction reduces toexux(2:2:1)In similar fashion, th
28、e normal strain in the y-direction becomeseyvy(2:2:2)A second type of strain is shearing deformation, which involves angles changes (see Figure2-3). Shear strain is defined as the change in angle between two originally orthogonalu(x,y)u(x+dx,y)v(x,y)v(x,y+dy)dxdyA BC DABCDdydxvxxbyuy FIGURE 2-4 Two-
29、dimensional geometric strain deformation.Sadd / Elasticity Final Proof 3.7.2004 2:59pm page 31Deformation: Displacements and Strains 31directions in the continuum material. This definition is actually referred to as the engineeringshear strain. Theory of elasticity applications generally use a tenso
30、r formalism that requires ashear strain definition corresponding to one-half the angle change between orthogonal axes;see previous relation (2:1:7)1. Measured in radians, shear strain is positive if the right anglebetween the positive directions of the two axes decreases. Thus, the sign of the shear
31、 straindepends on the coordinate system. In Figure 2-4, the engineering shear strain with respect tothe x- and y-directions can be defined asgxyp2C0C0A0B0 abFor small deformations, a C25 tana and b C25 tanb, and the shear strain can then be expressed asgxyvxdxdxuxdxuydydyvydyuyvx(2:2:3)where we have
32、 again neglected higher-order terms in the displacement gradients. Note thateach derivative term is positive if lines AB and AC rotate inward as shown in the figure. Bysimple interchange of x and y and u and v, it is apparent that gxy gyx.By considering similar behaviors in the y-z and x-z planes, t
33、hese results can be easilyextended to the general three-dimensional case, giving the results:exux, eyvy, ezwzgxyuyvx, gyzvzwy, gzxwxuz(2:2:4)Thus, we define three normal and three shearing strain components leading to a total of sixindependent components that completely describe small deformation th
34、eory. This set ofequations is normally referred to as the strain-displacement relations. However, these resultsare written in terms of the engineering strain components, and tensorial elasticity theoryprefers to use the strain tensor eijdefined by (2:1:7)1. This represents only a minor changebecause
35、 the normal strains are identical and shearing strains differ by a factor of one-half; forexample, e11 ex exand e12 exy 1=2gxy, and so forth.Therefore, using the strain tensor eij, the strain-displacement relations can be expressed incomponent form asexux,eyvy, ezwzexy12uyvxC18C19, eyz12vzwyC18C19,
36、ezx12wxuzC18C19(2:2:5)Using the more compact tensor notation, these relations are written aseij12(ui,juj,i)(2:2:6)Sadd / Elasticity Final Proof 3.7.2004 2:59pm page 3232 FOUNDATIONS AND ELEMENTARY APPLICATIONSwhile in direct vector/matrix notation as the form reads:e 12ru(ru)TC2C3(2:2:7)where e is t
37、he strain matrix and ru is the displacement gradient matrix and (ru)Tis itstranspose.The strain is a symmetric second-order tensor (eij eji) and is commonly written in matrixformat:e e exexyexzexyeyeyzexzeyzez2435(2:2:8)Before we conclude this geometric presentation, consider the rigid-body rotation
38、 of our two-dimensional element in the x-y plane, as shown in Figure 2-5. If the element is rotated througha small rigid-body angular displacement about the z-axis, using the bottom element edge, therotation angle is determined as v=x, while using the left edge, the angle is given byC0u=y.These two
39、expressions are of course the same; that is, v=x C0u=y and note that thiswould imply exy 0. The rotation can then be expressed as !z (v=x)C0(u=y)=2,which matches with the expression given earlier in (2:1:9)3. The other components of rotationfollow in an analogous manner.Relations for the constant ro
40、tation !zcan be integrated to give the result:u* uoC0!zyv* vo!zx(2:2:9)where uoand voare arbitrary constant translations in the x- and y-directions. This resultthen specifies the general form of the displacement field for two-dimensional rigid-bodymotion. We can easily verify that the displacement f
41、ield given by (2.2.9) yields zero strain.xdxdyyuyvxFIGURE 2-5 Two-dimensional rigid-body rotation.Sadd / Elasticity Final Proof 3.7.2004 2:59pm page 33Deformation: Displacements and Strains 33For the three-dimensional case, the most general form of rigid-body displacement can beexpressed asu* uoC0!z
42、y!yzv* voC0!xz!zxw* woC0!yx!xy(2:2:10)As shown later, integrating the strain-displacement relations to determine the displacementfield produces arbitrary constants and functions of integration, which are equivalent to rigid-body motion terms of the form given by (2.2.9) or (2.2.10). Thus, it is impo
43、rtant to recognizesuch terms because we normally want to drop them from the analysis since they do notcontribute to the strain or stress fields.2.3 Strain TransformationBecause the strains are components of a second-order tensor, the transformation theorydiscussed in Section 1.5 can be applied. Tran
44、sformation relation (1:5:1)3is applicable forsecond-order tensors, and applying this to the strain givese0ij QipQjqepq(2:3:1)where the rotation matrix Qij cos(x0i, xj). Thus, given the strain in one coordinate system,we can determine the new components in any other rotated system. For the general th
45、ree-dimensional case, define the rotation matrix asQijl1m1n1l2m2n2l3m3n32435(2:3:2)Usingthisnotationalscheme,thespecifictransformationrelationsfromequation(2.3.1)becomee0x exl21eym21ezn212(exyl1m1eyzm1n1ezxn1l1)e0y exl22eym22ezn222(exyl2m2eyzm2n2ezxn2l2)e0z exl23eym23ezn232(exyl3m3eyzm3n3ezxn3l3)e0x
46、y exl1l2eym1m2ezn1n2exy(l1m2m1l2)eyz(m1n2n1m2)ezx(n1l2l1n2)e0yz exl2l3eym2m3ezn2n3exy(l2m3m2l3)eyz(m2n3n2m3)ezx(n2l3l2n3)e0zx exl3l1eym3m1ezn3n1exy(l3m1m3l1)eyz(m3n1n3m1)ezx(n3l1l3n1)(2:3:3)For the two-dimensional case shown in Figure 2-6, the transformation matrix can be ex-pressed asQijcosy siny 0
47、C0siny cosy 00012435(2:3:4)Sadd / Elasticity Final Proof 3.7.2004 2:59pm page 3434 FOUNDATIONS AND ELEMENTARY APPLICATIONSUnder this transformation, the in-plane strain components transform according toe0x excos2yeysin2y2exysinycosye0y exsin2yeycos2yC02exysinycosye0xyC0exsinycosyeysinycosyexy(cos2yC
48、0sin2y)(2:3:5)which is commonly rewritten in terms of the double angle:e0xexey2exC0ey2cos2yexysin2ye0yexey2C0exC0ey2cos2yC0exysin2ye0xyeyC0ex2sin2yexycos2y(2:3:6)Transformation relations (2.3.6) can be directly applied to establish transformations betweenCartesian and polar coordinate systems (see Exercise 2-6). Additional applications of theseresults can be found when dealing with experimental strain gage measurement systems. Forex
