Betti's Reciprocal Relationships for the Displacements of an Elastic Infinite Space Bounded Internally by a Rigid Inclusion

A Reciprocal Theorem for Finite Deformations in Incompressible Bodies

“…( 11). Accordingly, we have identically recovered the auxiliary boundary value problem ( 7)- (11). But, since x * is a solution of the auxiliary boundary value problem, x = x * identically satisfies (3)-( 6) and our proposition is true.…”

“…where we identify that the constant hydrostatic pressure vanishes from the equation of motion (16), and that the traction conditions ( 18) and (19), are identical to the applied tractions, t * , and t * i , of the auxiliary boundary value problem ( 7)- (11), as defined in eq. ( 2), and eq.…”

“…The idea that one can experimentally observe a body deforming under a given set of boundary conditions, and then directly infer its response due to an alternative set, without solving the boundary value problem, has proven to be extremely useful, and has become one of the most classical results in elasticity. Other than its mathematical elegance [7], it has been particularly useful in contact mechanics, to interpret indentation measurements conducted with different indentor shapes [8,9]; it has enabled solution of various inclusion problems [10,11,12]; and extends to dynamics [13,14], as well as various additional fields, such as acoustics [15,16], optics [17], and fluids [18,19,20]. The appeal of using the reciprocal theorem to understand the response at finite deformations, which can be exceedingly more difficult to capture experimentally, or to analytically resolve, is clear; even if for a limited class of boundary value problems.…”