Strain History and Polar Decomposition

Abstract

The effect of two consecutive strains (only two states enter into the calculation of a strain, the states before and after, independently of the actual strain path) can be calculated by premultiplying the transformation matrix of the first strain (its stretch tensor) with that of the second. Unless the two strains are coaxial (their principal directions coincide), however, the resulting cumulative transformation matrix represents not only a strain but also a rigid-body rotation; in that case the matrix is asymmetric. The method of polar decomposition allows one to interpret the combined transformation as if it had come about either by a strain followed by a rotation (right polar decomposition) or by a rotation followed by a strain (left polar decomposition). Let 𝔸 and 𝔹 be two stretch tensors, or transformation matrices, representing each a strain without rotation; and let the strain 𝔹 follow the strain 𝔸. Then the combined transformation matrix 𝔽 is: . . . 𝔹𝔸 = 𝔽 = ℝ𝕌= 𝕍ℝ, (8.1) . . . where 𝔽 results from premultiplication of the earlier stretch 𝔸 with the later 𝔹, where ℝ𝕌 is the “right” and 𝕍ℝ the “left” decomposition of 𝔽, where 𝕌 and 𝕍 are two distinct stretch tensors, and where ℝ is the transformation matrix for a rotation (elements of rotation matrices are indicated by the symbol aij elsewhere in this book). 𝔽 is asymmetric and ℝ differs from the identity matrix (δij) except when 𝔸 and 𝔹 are coaxial. 𝕌 and 𝕍 have the same principal stretches and differ by orientation only. In Problems 120 to 122, false approaches in the search for an appropriate decomposition of an asymmetric transformation were recognized by yielding impossible values for a rotation. Application of eq. (8.1) makes such a trial-and-error approach unnecessary.