Int. J. Non-Linen? Mechanics. Vol. 27. No. 4. pp. 63-38. 1992 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Printed in Great Britain. 002&7462/92 s5.00 + .oo Pergamon PIUS Ltd zyxwvutsrqpo THE SYMMETRY GROUP OF SECOND-GRADE MA TERIA LS MAREK EL~ANOWSKI Department of Mathematical Sciences, Portland State University, Portland, Oregon, U.S.A. and MARCELO EPSTEIN Department of Mechanical Engineering, The University of Calgary, Calgary, Alberta, Canada (Received 6 February 1991; zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ in revisedfirm 12 April 1991) Abstract-The mathematically correct form of the material symmetry group in the second-grade elasticity is obtained. Its possible ramifications for the theory of continuous distributions of disclinations are explored. INTRODUCTION A material of the second grade is, by definition, sensitive in its response to the first and second gradients of the deformation. Since, when evaluated at a point, the values of those gradients are independent of each other, it would appear that, as far as characterizing the symmetries of a material point, one should be concerned only with a response function of two completely independent arguments, namely, a second- and a third-order tensors. To demonstrate that this is not the case is the main concern of this paper. We show that the fact that the value of the aforementioned third-order tensor is always the result of evaluating the second derivative of a global deformation-or, in other words, the fact that the arguments of the response function of a point are actually equivalence classes of global configura- tions-has a crucial effect on the law of composition of the tensors involved which, in turn, determines what kind of an object the symmetry group of the material should be. Our interest in this particular question has arisen from the need to incorporate continu- ous distributions of disclinations within the rigorous framework of continuum mechanics. This problem will be, only briefly, addressed at the end of this paper, but is the subject of a forthcoming work [l]. SYMMETRY GROUP A second-grade elastic material [2, 33 is characterized by a response function which depends upon the second as well as the first derivatives of the deformation. Thus, if X’ (I = 1,2,3) denote (Cartesian) coordinates in a reference configuration and xi (i = 1,2,3) are their counterparts in an arbitrary space configuration, a deformation being given by functions xi = x’(X’) (1) its first and second gradients, F and VF, have the components F: = Xfl (2) and VFjJ = xfll (3) respectively, and a constitutive response-such as the strain energy density-relative to the Contributed by K. R. Rajagopal. 635