东南大学：《固体力学基础》课程教学课件（英文讲稿）02 Strain Measures

Strain Measuresmi@se.ed.cn

Strain Measures

Outline·Conceptof strain（应变的概念）·Deformation and displacement gradient（变形和位移梯度）·Cauchy-Green strain tensors （C-G应变）·Polar decomposition（极分解）·Jacobianof deformation（变形雅可比）·Differentmeasures of strain（应变度量）·Simpledeformations（简单变形）·Small strain theory（小应变理论）·Materialand spatialtimederivatives（材料时间导数和空间时间导数）·Stretch rate and spin rate（变形率和转动率）2

Outline • Concept of strain（应变的概念） • Deformation and displacement gradient（变形和位移梯度） • Cauchy-Green strain tensors（C-G应变） • Polar decomposition（极分解） • Jacobian of deformation（变形雅可比） • Different measures of strain （应变度量） • Simple deformations（简单变形） • Small strain theory（小应变理论） • Material and spatial time derivatives（材料时间导数和 空间时间导数） • Stretch rate and spin rate（变形率和转动率） 2

Concept of Strain·Elongation: S? Percentage of elongation: = S/ l?Stretch: a=l/l。=1+Vo83

Concept of Strain • Elongation: δ • Percentage of elongation: • Stretch: 3 0    / l      l l0 1

Different Measures of Strain? Engineering strain:S. True strain:Ss/l8 / l.-(8 / 1.)2lo+s1+8/1? Difference in length square:P-1 _ (1+1.)(1-10)_ (2l +8)(8)_ 8, 82212121§+2%? Difference in length square:-(1+1.)(1-1) _ (21。 +8)(8)2122122(%+)? Logarithmic strain:aIn==In(1+8/10)=8 //-(8/1)lo: All these measures are equivalent for small elongationand thus equivalent from an engineering point of view..How to generalize these to 3D?4

Different Measures of Strain • Engineering strain: • True strain: • Difference in length square: • Difference in length square: • Logarithmic strain: 4                     0 1 0 2 0 2 0 0 0 0 2 2 2 0 0 0 0 3 2 2 2 2 0 0 0 0 0 2 2 0 0 0 0 4 2 2 2 0 2 5 0 0 0 0 ; / / / ; 1 / 2 ; 2 2 2 2 2 ; 2 2 2 1 ln ln 1 / / / ; 2 l l e l l e l l l l l l l l l l l l e l l l l l l l l l l l l e l l l dl l e l l l l l                                                     • All these measures are equivalent for small elongation and thus equivalent from an engineering point of view. • How to generalize these to 3D?

Deformation Gradient Tensorü=j-x(displacementvector)=(,1)u(x)y, =y(,x,g,t)yOyidx, = F,dxjDeformeddyOriginalconfigurationOxjeiconfigurationQyi_Qu,(x,t).Deformation gradientatatou;x,=constayiSOxjoxjWe wish to find a measure of strain,a relative measure ofhow material points move with respect to each other, thatis independent of rigid body rotation5

Deformation Gradient Tensor • Deformation gradient: • We wish to find a measure of strain, a relative measure of how material points move with respect to each other, that is independent of rigid body rotation. 5

Displacement Vector1 Consider an arbitrary fiberdy=ndldx = mdl。? Define displacement vectoru(x+dx)j=j(x,x,x)=x+i(x,x,x)dx? Deformation gradientu(x)contains informationDeformedabout both stretch andOriginalconfigurationconfigurationrotation:2= Fm= andy = ndl = Fmdl.= Fm = ndl/dl..In order to separate stretch from rigid body rotationaconsider the dot product of two fibersaI6-: Since the dot product only depends on the relative angle6between the two vectors, rigid body rotation can be beeneffectivelyfiltered""out.6

Displacement Vector • Consider an arbitrary fiber d d d d d 0 0 y n l Fm l Fm n l l Fm n        • Define displacement vector • Deformation gradient contains information about both stretch and rotation: • In order to separate stretch from rigid body rotation, consider the dot product of two fibers. • Since the dot product only depends on the relative angle between the two vectors, rigid body rotation can be been effectively “filtered” out. 6

Cauchy-Green Strain Tensors.Consider the dot product of two differential segments inboth the undeformed and deformed configurationsdj·dy,=Fdx·Fdx,=d·FFdxddx,=F'djF'dy,=dFF"dy,=dFF'dy? Right Cauchy-Green strain tensordy2dx,ü1dx,djiC=F'F, C, =FhFxyLeft Cauchy-Green strain tensorB=FF',B,=FiFaae0h·Both are symmetricb7

Cauchy-Green Strain Tensors • Consider the dot product of two differential segments in both the undeformed and deformed configurations • Right Cauchy-Green strain tensor • Left Cauchy-Green strain tensor • Both are symmetric. 7

Physical/Geometric Interpretations of C: Principal values/directions. General matrix form00[CnCr3[CCi2Ci2 C22C230C=0CnC=，mlmlmm[Ci3C33C23100Cm2: Consider a fiber initially alongone of the base vectorsdx,djIXd,=d1o,=dnJd/2 = dji -dy = Fdx-Fdx, = dlioe ·Cd/1oe, = d/iCnd72=RC1do: Cu, is the stretch of a fiber initially aligned along e1.8

Physical/Geometric Interpretations of C • General matrix form 11 12 13 12 22 23 13 23 33 C C C C C C C C C C            • Principal values/directions • Consider a fiber initially along one of the base vectors • C11 is the stretch of a fiber initially aligned along e1 . 8

Physical/Geometric Interpretations of C2? Consider two initiallydydxperpendicular fibersedjdiXdx = dl1o, dx, = dl2oe,idj, =dl, n, djz =dl,n,dj1 dy2 =Fdx Fdx, = dx-Cdx2 = d/,dl, cos 012 =d/oe-Cd/20eC2cos 612CiC22? Ci2 is a measure of the angle between two fibersinitially aligned in the e, and e, directions9

Physical/Geometric Interpretations of C • Consider two initially perpendicular fibers • C12 is a measure of the angle between two fibers initially aligned in the e1 and e2 directions. 9

Physical/Geometric Interpretations of C. The right Cauchy-Green strain 2gives the information how a国dlaldl,small block of materialaltodl,d/20deforms.[CuCr3CC23C 22C12C=1C33C13C 23dl, = /C1d/1o , dl, = /C2 d/20 , dl, = /C33 d/30C23C13C2cose,cosOcose.CiC33C.C22CC3310

Physical/Geometric Interpretations of C • The right Cauchy-Green strain gives the information how a small block of material deforms. 11 12 13 12 22 23 13 23 33 C C C C C C C C C C            10