Stretch, Strain, and Rotation

Abstract

Stretch and Rotation Tensors. Strain Consider the (pointwise) polar decomposition of the deformation gradient F into a rotation R and positive-definite symmetric tensors U and V ; using terminology motivated in §7.3, we refer to U as the right stretch tensor and to V as the left stretch tensor . The tensors U and V , which have the explicit representations are useful in theoretical discussions but are often problematic to apply because of the square root. For that reason, we introduce the right and left Cauchy–Green (deformation) tensors C and B defined by Then, by (7.1), For future reference,we list the properties of the stretch and Cauchy–Green tensors: A tensor useful in applications is the Green–St. Venant strain tensor Note that E vanishes when F is a rotation, for then F ⊤ F = 1 . This property of E is often adopted as one necessary for a tensor to qualify as a meaningful measure of strain. Next, by (M1) and (M2) on page 65 and (7.3) and (7.6), (M3) U, C , and E map material vectors to material vectors ; (M4) V and B map spatial vectors to spatial vectors ; (M5) R maps material vectors to spatial vectors . To verify (M3), consider F ⊤ F : (M1) implies that F maps a material vector f R to a spatial vector Ff R and, by (M2), this spatial vector is mapped back to a material vector. Thus C = F ⊤ F maps material vectors to material vectors and since UU = C , with U symmetric, U must also map material vectors to material vectors.