# Mathematical Problems in Elasticity by Russo R. (ed.)

By Russo R. (ed.)

This quantity beneficial properties the result of the authors' investigations at the improvement and alertness of numerical-analytic tools for traditional nonlinear boundary price difficulties (BVPs). The equipment into consideration provide a chance to unravel the 2 vital difficulties of the BVP idea, particularly, to set up lifestyles theorems and to construct approximation strategies

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Z. angew. Math . Phys. (to appear). , J. Mech. Phys. Solids 18, 149-163 (1970) . , J. Ration. Mech. Anal. 2,329-337 (1953). 17 Hayes , M. , Z. angew. Math. Phys. 22 , 1173-1176 (1971). , Proc . R. Soc. Land. A370 , 417-429 (1980). 19 Boulanger, Ph. , Q. Jt Mech . appl. Math. 44 , 235-240 (1991). 20 Boulanger, Ph. , Arch. Rational Mech . Anal. 116, 199-222 (1991). 21 Boulanger, Ph. and Hayes , M. (submitted for publication) 47 DECAY ESTIMATES FOR BOUNDARY-VALUE PROBLEMS IN LINEAR AND NONLINEAR CONTINUUM MECHANICS C.

Thus any pair (n, a) of orthogonal directions in a principal plane of the basic deformation is a reciprocal pair . We now consider a reciprocal pair and take here n and a in the Xl x3-plane. Thus b = j (b is along the x2-axis), and n B- 1n = ),~a . Ba. (169) Here, we denote by v~( a ) the squared wave speed of the wave propagating along n and polarized along a , and by v~( n) the squared wave speed of the wave propagating along a and pola rized along n. From (46) and (47), we now obtain , using (169), (170) where V8 is the speed of propagation along the acous tic axes, given by (95).

Ta = n Ba = n . B- 1 a = O. Similarly, (167) does not apply when n or b is along a principal axis of the basic static deformation. We now consider pairs of ort hogonal directions which are reciprocal in the sense of Schouten 13 : a pair of orthogonal unit vectors (n, a) is said to be "reciprocal" when a wave propagating along n and polarized along a , and a wave propagating along a and polarized along n are both possible. For waves propagating along a direction n , the two polarization directions a and b must satisfy (45).