Conjugate stresses of the Seth±Hill strain tensors K. Farahani a , R. Naghdabadi b, * a Department of Civil Engineering, Sharif University of Technology, Tehran, Iran b Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Received 30 November 1998; in revised form 18 July 1999 Abstract In this paper, conjugate stresses of the Seth±Hill strain measures are treated by means of the de®nition of energy conjugacy and Hill's principal axis method. This approach results in relations between components of two dierent conjugate stress tensors. The results are valid for distinct as well as coalescent principal stretches. Illustrative examples are solved to demonstrate the simplicity and usefulness of the derived relations between conjugate stress tensors. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Conjugacy; Conjugate stress; Seth±Hill strain 1. Introduction The concept of energy conjugacy for stress and strain measures was ®rst introduced by Hill (1968). This concept plays an important role when writing the internal power of a deforming body. Also the virtual work, as the weak form of equilibrium equations, can be established in terms of a stress measure and the variation of its conjugate strain, as a basis for linear and nonlinear analysis of solids and structures. A stress measure T is said to be conjugate to a strain measure E if T: Ç E represents power per unit reference volume, _ w: That is _ w  IIIs:D  T: Ç E 1 where s and D are the Cauchy stress and stretching tensors, respectively, and III  detU is the third invariant of the right stretch tensor U. By the spectral decomposition theorem U can be recast as International Journal of Solids and Structures 37 (2000) 5247±5255 0020-7683/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S0020-7683(99)00209-7 www.elsevier.com/locate/ijsolstr * Corresponding author. Fax.: +98-21-6000021. E-mail address: naghdabd@mech.sharif.ac.ir (R. Naghdabadi).