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The aim of this paper is to present and analyze a new direct method forsolving the linear elasticity inverse problem. Given measurements of somedisplacement fields inside a medium, we show that a stable reconstruction ofelastic parameters is possible, even for discontinuous parameters and withoutboundary information. We provide a general approach based on the weakdefinition of the stiffness-to-force operator which conduces to see the problemas a linear system. We prove that in the case of shear modulus reconstruction,we have an $L^2$-stability with only one measurement under minimal smoothnessassumptions. This stability result is obtained though the proof that the linearoperator to invert has closed range. We then describe a direct discretizationwhich provides stable reconstructions of both isotropic and anisotropicstiffness tensors.

A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only