Shock wave propagation in elastic-plastic media

Conditions for the existence of discontinuity surfaces of irreversible strains in elastoplastic media

“…As such a condition we use the extension of the Mises maximum principle to the dissipative process on Σ(t): as the irreversible strains on the discontinuity surface vary from the value e p+ ij to the value e p− ij , the stresses vary so that the product (σ + ij + σ − ij )[e p ij ] takes the maximum value for all possible stresses that satisfy the conditions f (s) (σ + ij , τ + ) k and f (s) (σ − ij , τ − ) = k. This condition, which is a natural extension of the Mises maximum principle to shock-wave processes ([e p ij ] = 0) is the classical maximum principle when σ + ij tends to the 880 value σ − ij and [e p ij ] becomesė p ij = ε p ij . It has been shown [4] that the extremality condition of the stress-strain states on the discontinuity surface, which is similar to the extension of the maximum principle given above, implies invariance of the principal axes of the stress tensor across the discontinuity surface Σ(t). This result underlies all conclusions [4] on the existence conditions and propagation mechanisms of strong-discontinuity surfaces.…”

“…It has been shown [4] that the extremality condition of the stress-strain states on the discontinuity surface, which is similar to the extension of the maximum principle given above, implies invariance of the principal axes of the stress tensor across the discontinuity surface Σ(t). This result underlies all conclusions [4] on the existence conditions and propagation mechanisms of strong-discontinuity surfaces. Later, the invariance of the principal directions across a strong discontinuity surface was inferred using the extreme conditions of shock-wave transition formulated differently [5][6][7][8]14].…”

“…Eliminating plastic compressibility (q = 0) from these relation, we obtain the well-known value c 2 = (3λ + 2μ)/(3ρ) [4,7]. If the stress state ahead of the discontinuity surface and in the transition layer correspond to the edge of the pyramid formed by the intersection of its face and base, conditions (3.1) and (3.2) should be supplemented by the requirement σ = −ϕ(τ ).…”

“…For q = 0, these relation lead to the well-known values c 2 = (λ + 2μ/3)/ρ [4,14]. As above, the necessary condition for the existence of such discontinuities is the collinearity of the normal ν to one of the principal axes of the stress tensor.…”

“…Existence conditions and propagation mechanism of discontinuity surfaces of irreversible strains (dissipative shock waves) are necessary elements in the formulation of boundary-value problems of irreversible deformation dynamics and are a subject of continuous interest [1][2][3][4][5][6][7][8]. In model relations it is difficult to use rough loading surfaces with irreversible changes in volumetric strains taken into account.…”