# Linear Theories of Elasticity and Thermoelasticity: Linear by Morton E. Gurtin (auth.), Professor Dr. C. Truesdell (eds.) By Morton E. Gurtin (auth.), Professor Dr. C. Truesdell (eds.)

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Extra info for Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells

Example text

0 or 1p(~. , o) 1p; (vii) if t0 =oo and qJ,1p possess Laplace transforms, then so also does 91*11'· and lf'*1p=cp1p on D X ['1]0 , oo) for some 1Jo ;;::;;o. Proof. The results (i)-(iii), (v), and (vii) are well known,1 while (vi) follows directly upon differentiating cp*1J'· To establish (iv) let 91 = 91(~. •), 1p =1p{~. •), assume that 1p does not vanish identically on [0, t 0), and let ~ =inf {tE [0, t0): 1p(t)=l= 0}. Since 1p is continuous, there exists a number t2 with ~ < t2 < t0 such that 1p > 0 on (t1 , t 2) or 1p < 0 on (t1 , t2).

E13 =E2a =Eaa =0. It is a simple matter to verify that a plane displacement field is rigid if and only if u"'(x) =a"' +we"' 11 x11 where x = (x1 , x2) and a"' and w are constants. We call a two-dimensional field u"' of this form a plane rigid displacement and a complex function of the form ti(Z) =tZt +ia2 -iwz, z=x1 +ix2 , a compleoz: rigid displacement. Clearly, a complex function u is a complex rigid displacement if and only if its real and imaginary parts ~ and u 2 are the components of a plane rigid displacement.

The values of M are four-tensors). We define the divergences of u and M by the relations: div(4) u =div u +,U. (div(4)M) ·G=div(4)(MTo) for every oEi"'(4l. For future use, we now record the following (1) Identity. Let M be a smooth four-tensor field on an open set D in cf(4l, and let (a) be the space-time partition of M. Then Proof. Since it follows that div(4l M = (div M +m, div m + ~). MT a= (JUT a +mcx:, m · a+Acx:), div(4)(MT a) =div(MT a) +ex: divm+a · m+cx:i =(div M which implies the desired identity.