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

Number Theory

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).

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