Physical/Geometric Interpretations of C2·In principal coordinates, theabove geometric团interpretations suggest[0000C=场00Cm=m = m,lmlmm This indicates that the fibers along the principaldirections will remain perpendicular to each other afterdeformation.11
Physical/Geometric Interpretations of C • In principal coordinates, the above geometric interpretations suggest • This indicates that the fibers along the principal directions will remain perpendicular to each other after deformation. 11
Lagrangian vs. Eulerian Descriptions. The above formulation of strain has been focused on makingpredictions about the deformed configuration (called Eulerian)based on know information in the undeformed (called Lagrangian)configuration.: Alternatively, we could reverse the direction of analysis. We couldstart from the deformed configuration and try to predict theundeformed configurationLe0nardEuler(1707-1783)JosephLouisLagrange(1736-1813)12
Lagrangian vs. Eulerian Descriptions • The above formulation of strain has been focused on making predictions about the deformed configuration (called Eulerian) based on know information in the undeformed (called Lagrangian) configuration. • Alternatively, we could reverse the direction of analysis. We could start from the deformed configuration and try to predict the undeformed configuration. Joseph Louis Lagrange (1736-1813) Leonard Euler (1707-1783) 12
Physical/Geometric Interpretations of Bdji = dle. Consider a fiber aligned ir2 + dx, = dlomdirection after deformatiodx,djix元dj = Fdx = dx = F-'djdx,·dx = d/= F-ldji·F-ldji = d·(FF)'dji = dji·B-'dyd = d/e B-dlé =→Bl =: The stretch that has happened to a fiber aligned in the ejdirection after deformation is given by Brl13
Physical/Geometric Interpretations of B • Consider a fiber aligned in e1 direction after deformation • The stretch that has happened to a fiber aligned in the e1 direction after deformation is given by 1 13
Physical/Geometric Interpretations of B: The off-diagonal term has the following interpretationBi2dx, -dx, = dl1od/20 cos α12 = dl,d/,e ·B'e, = cos α12Bi"B!. The original angle of two fibers that have becomealigned in the e, and ez directions after deformation isgiven by Bi[BT]Bt[B1Bi2B12Br3Bi1B-1BrBi. In matrix form: B=B21B2B237BgiB,2B[B31 Bs2 B,]00[M]In principal coordinates.00Λwe may define the right U = C =00Mmstretch tensor14
Physical/Geometric Interpretations of B • The off-diagonal term has the following interpretation • The original angle of two fibers that have become aligned in the e1 and e2 directions after deformation is given by • In matrix form: • In principal coordinates, we may define the right stretch tensor 14