By Robin John Knops, Lawrence Edward Payne (auth.)
The classical outcome for area of expertise in elasticity conception is because of Kirchhoff. It states that the traditional combined boundary worth challenge for a homogeneous isotropic linear elastic fabric in equilibrium and occupying a bounded 3-dimensional zone of area possesses at so much one resolution within the classical feel, supplied the Lame and shear moduli, A and J1 respectively, obey the inequalities (3 A + 2 J1) > zero and J1>O. In linear elastodynamics the analogous consequence, because of Neumann, is that the initial-mixed boundary worth challenge possesses at so much one answer supplied the elastic moduli fulfill a similar set of inequalities as in Kirchhoffs theorem. most traditional textbooks at the linear concept of elasticity point out simply those classical standards for specialty and overlook altogether the ample literature which has seemed because the unique guides of Kirchhoff. To treatment this deficiency it sort of feels applicable to aim a coherent description ofthe a number of contributions made to the research of specialty in elasticity idea within the desire that such an exposition will supply a handy entry to the literature whereas while indicating what growth has been made and what difficulties nonetheless look forward to answer. clearly, the continued assertion of recent effects thwarts any try and offer a whole overview. except linear elasticity thought itself, there are a number of different components the place elastic distinctiveness is significant.
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Extra resources for Uniqueness Theorems in Linear Elasticity
Example text
This observation is repeatedly used in what follows. 12) but whose displacement vanished on 9 Q=xi +a 2 x~ +b 2 x~ -1 =0. 20) u;=QP; where P; are polynomials of non-negative degree. u 2 (a 2 +b 2 )<0. 21). u = 0, A=l= 0, by observing that any solenoidal displacement satisfying zero data is a solution to the problem. 2) become (12 u. ), _ _)_=0 (lx i (lx j , which clearly are satisfied by imposing the constraint (lUi =0, or equivalently: . t h e permutatIOn . /. (lX ib· f' IS sym b 0 an '1'. an ar Itrary vector unctIOn (11/1.
4 Mixed Boundary Value Problems 51 For regions w,hose boundaries extend to infinity, Mindlin, referring to work by Neuber, has produced examples in which the traction boundary value problem fails to have a unique solution for values of (J in either of the intervals - 00 < (J ~ -1 or 1 ~ (J < 00. 4 Mixed Boundary Value Problems For homogeneous isotropic elastic bodies occupying bounded three-dimensional regions, Kirchhoffs theorem guarantees uniqueness in all the standard boundary value problems including those of mixed type.
Here we have used integration by parts and the fact that Vi vanishes on oB. 10) is non-negative and moreover aC" kl SrP - 2 -a'J Xj Vi Vk dx~ -k1 SrP - B S rP B Xl 2 Cijkj Vi V k B d X- SrP - 1 Vi Vi dx~ -k1 a SrP - 2 Vi Vi dx, B 2 C ~j ~j ilkm Vi V k Ul U m d X= > 3 C1 Srp - B 2 Vi Vi d X, B for computable constant k1 independent of p. 7) we obtain [(p+ 1) C1 -k1 a] Sr P - 2 Vi Vi dx~O. B By choosing p sufficiently large, the term in square brackets can be made positive which implies that Vi vanish identically in B.