UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering,Department of Civil Engineering Mechanics and MaterialsFall 2002 Professor: S. GovindjeeUniqueness in linear elasticityOne important point to finding solutions in linear elasticity is that they areunique. This is a general feature of any type of linear problem with suitableboundary conditions. Thus if one finds a solution to a particular linear elasticboundary value problem then it is the solution to the problem. There aremany ways to prove this and we shall review here the classical method basedupon Clapeyron’s Theorem. This theorem states:Theorem 1 (Clapeyron’s Theorem) Consider a linear elastic body Ω ⊂R3with boundary ∂Ω. ThenZ∂Ωtiui+ZΩbiui= 2ZΩW ,where tiare the surface tractions, biis the body force, uiis the displacementfield, and W is the strain energy density.The proof of the theorem is rather straightforward. Simply go through thederivation of the weak from using the displacement field as a “test function”.Now Consider that the boundary ∂Ω = ∂ΩuS∂Ωt, where the traction anddisplacement parts of the boundary are mutually exclusive (∂ΩuT∂Ωt= ∅).On ∂Ωt, assume an imposed traction of¯ti. On ∂Ωu, assume an imposed dis-placement of ¯ui. To show uniqueness of the solution let us assume that thereare two different solutions to the problem and then show (by Clayperon’sTheorem) that their difference must be zero.Call the two solutions (u(1)i, ε(1)ij, σ(1)ij) and (u(2)i, ε(2)ij, σ(2)ij). Now define thedifference between the two solutions as:uDi= u(1)i− u(2)i(1)εDij= ε(1)ij− ε(2)ij(2)σDij= σ(1)ij− σ(2)ij. (3)Note that the difference functions are the solution to a problem with appliedbody force bi= 0, and applied boundary conditions¯ti= 0 on ∂Ωtand ¯ui= 01on ∂Ωu. If we apply Clapeyron’s Theorem to this difference problem thenwe have that:0 =ZΩεDijCijklεDkl. (4)If we assume that the integrand is a positive definite quadratic form, thenwe have thatεDij= 0 . (5)Note that this implies that uDi= 0 and that σDij= 0. Thus the differencebetween the two solutions is indeed zero and the solution is unique.Remark: The assumption that Cijklbe positive definite essentially says thatwe are assuming the material to be stable and it is perfectly compatible withour normal physical experience with stable materials.Remark: The assumption that ∂ΩuT∂Ωt= ∅ is essential for the existenceof solutions. Note that this notation implies that at a given point in a givencoordinate direction one can only impose a displacement or a traction.Remark: Similar results hold in rate form for viscoelasticity and