Uniqueness of the Elastostatic Stress and Strain Field

15 Because the equations of linear elasticity are linear equations, the principles of superposition may be used to obtain additional solutions from those established. Hence, given two sets of solution T(1, ui1^, and Tj\ ui2), then Tij — Tj — Tj\ and ui — u[2) — ui1 with bi — b\2) — bi1 — 0 must also be a solution.

16 Hence for this "difference" solution, Eq. 9.18 would yield / tiuidT — 2 u*di but the left hand

J r Jo side is zero because ti — t^ — ti1 — 0 on T„, and ui — u\2 — nip — 0 on Tj, thus / u*dl — 0.

v or

17 But u* is positive-definite and continuous, thus the integral can vanish if and only if u* =0 everywhere, and this is only possible if Eij = 0 everywhere so that

hence, there can not be two different stress and strain fields corresponding to the same externally imposed body forces and boundary conditions1 and satisfying the linearized elastostatic Eqs 9.1, 9.14 and 9.3.

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