The Tensor Equation AX + XA = G

Abstract

Approved REPOT DCUMETATON PGE0MBNo 0704-0188Publc reporting ourden for this collection of information is estimated to average i hour per resopor.5e. inctuong thre time for reviewing instructions. searching exinsting data sources. Approved for public release; distribution is unlimited. ABSTRACT (Maximum 200 words)We study the second-order tensor equation AX + IA = G for symmetric, positive-definite A and arbitrary G.Motivated by applications in the continuum mechanics literature, we also exUamine some special cams where G depends on A and another tensor H. For arbitrary dimensions we establish relations between the solutions X for various forms of G. These results, together with Rivlin's identities for tensor polynomials in two variables, we applied in two and three dimensions to obtain explicit formulas for X in direct (component-free) notation. The results include new formulas as well as new derivations of previously known formulas. An application to the kinatics of rigid motions; is considered.

The Tensor Equation AX+XA=Phi (A, H) with Applications to Kinematics of Continua

“…1 Then a is an eigenvalue of A, and it is not hard to show that a is independent of b, so that A 2 = aA. If P := IA then p2 = P. Hence, A = aP for some projection 1 2 The identities (3.30)--(3.32) were derived in Scheidler (23] by the method.s u&ed here but under the assumption that A is symmetric. A major special case of (3.31) was obtained independently and by a different method by Chen and Wheeler [20].…”

“…Direct formulas for one work-conjugate stress tensor in terms of another, or for a work-conjugate stress tensor in terms of the Cauchy or first Piola-Kirchhoff stress tensors (cf. Guo and Man [22]): Here A = U or V, and D(A,L1) has some of the forms listed in (1.13) [23]) and pseudo-rigid bodies (cf. Cohen and Muncaster [24]): Here the symmetric tensor A is either the current or the referential Euler tensor.…”

“…They derived the analog of the tensor equation (1.12) for na and solved this equation to obtain the analog of (6.33)6 for 12R; they also, Ac.ed the corresponding result for f2. 23 A variety of additi,,v 1 formulas for V and U can be obtained by substituting the formulas (6.30) for Q into (6.23), and by substituting the analog of (6.30) for OR into (6.24). In partic~ular, the formulas for V and U which follow from (6.30)3, (6.23)1, and (6.24), ar.e due to Guo rS], and the formula for V which follows from (6.30), and (6.23), was obtained by Chen and Wheeler [20].…”

“…By substituting the analog of (6.33) for f2R -Wil into (6.38), we recover the formulas (6.10) and (6.11). 23 Equations similar to (1.12) arise in the kinematics of elastic-plastic deformations based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts; cf. Nernat- …”

The tensor equation AX+XA=?(A,H), with applications to kinematics of continua

“…Direct formulas for one work-conjugate stress tensor in terms of another, or for a work-conjugate stress tensor in terms of the Cauchy or first Piola-Kirchhoff stress tensors (cf. Guo and Man [22]): Here A = U or V, and ¢ ( A , H) has some of the forms listed in (1.13) [23]) and pseudo-rigid bodies (cf. Cohen and Muncaster [24]): Here the symmetric tensor A is either the current or the referential Euler tensor.…”