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