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