Hadamard’s conditions of compatibility from Cesaro’s line-integral representation

Abstract A given symmetric and compatible linear transformation field D determines, through Cesaro representation theorem, a vector field v whose symmetric gradient ∇sv equals D, up to an additive vector field vh satisfying ∇ s v h = 0 . If D represents an infinitesimal strain field, then vh is an infinitesimal rigid-body displacement. If D represents a stretching field, then vh is a rigid-body velocity which characterizes a rate of translation and rotation of the whole (fluid or solid) body. If D is compatible in regions separated by a smooth singular surface across which D suffers a jump, then in our main Theorem 2.3 we show that the associated vector field v must satisfy the Hadamard rank 1 compatibility condition across the surface, i.e., the jump ⟦ ∇v ⟧ is rank 1 of the form d⊗n, where n is a unit normal to the surface, if and only if ⟦ ∇v ⟧ is constant across the singular surface. Importantly, no a priori stated restriction on the skewsymmetric part of ∇v is needed. We, then, record two corollaries that characterize the possible jump behavior at a corner line where two smooth singular surfaces meet, discussing how our results apply in two example elementary flow problems in classical fluid dynamics.