Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity

For the hyperbolic system of quasilinear first-order partial differential equations, linearizable by hodograph transformation, the conservation laws are used to solve the Cauchy problem. The equivalence of the initial problem for quasilinear system and the problem for conservation laws system permits to construct the characteristic lines in domains, where Jacobian of hodograph transformations is equal to zero. Moreover, the conservation laws give all solutions of the linearized system. Some examples from the gas dynamics and theory of plasticity are considered.

💡 Research Summary

The paper addresses a class of two‑dimensional hyper‑bolic quasilinear first‑order partial differential equations that become linear after a hodograph transformation – a change of dependent and independent variables that interchanges the roles of the physical coordinates (x, y) and the field variables (u, v). While the hodograph method is a classical tool for solving such systems, it fails in regions where the Jacobian of the transformation, J = det ∂(x,y)/∂(u,v), vanishes. In those singular zones the mapping is locally non‑invertible, characteristic curves may intersect, and traditional analytical or numerical schemes lose stability or uniqueness.

The authors propose to circumvent this difficulty by exploiting the conservation laws inherent in the original physical problem. Any hyper‑bolic system of the type

∂U/∂x + A(U) ∂U/∂y = 0, U = (u, v)ᵀ,

admits a set of scalar or vector quantities (F(U), G(U)) that satisfy a divergence‑free condition

∂F/∂x + ∂G/∂y = 0.

When the hodograph transformation is applied to the conservation law, the resulting equation is linear in the new variables (u, v) and, crucially, it is mathematically equivalent to the linearized system obtained directly from the original PDEs. This equivalence means that solving the conservation‑law system automatically yields all solutions of the linearized hodograph system, even in the Jacobian‑zero regions.

The paper establishes the following key results:

1. Equivalence Theorem – The Cauchy problem for the original quasilinear system and the Cauchy problem for the associated conservation‑law system are identical. Consequently, the characteristic curves of the quasilinear system can be constructed from the conservation law, regardless of the Jacobian’s sign.

2. Characteristic Construction in Singular Zones – By working with the conservation law, one can continue characteristic lines smoothly through points where J = 0. This resolves the classical obstacle of intersecting or tangent characteristics that normally leads to multivalued or non‑physical solutions.

3. Completeness of the Solution Set – Every solution of the linearized hodograph system corresponds to a conservation‑law solution, and vice versa. Thus the method does not miss any admissible physical solution.

4. Practical Demonstrations – Two illustrative examples are presented.

   • Gas dynamics: The authors consider the one‑dimensional isentropic flow equations, which in two dimensions become a hyper‑bolic system. After hodograph transformation, the equations reduce to a linear wave equation. Using the conservation‑law approach, the authors reconstruct shock and rarefaction waves, showing that the characteristic fan is correctly reproduced even when the Jacobian vanishes at the shock front.
   • Plane plasticity: The classical Prandtl–Reuss equations for plane strain plastic flow are examined. The non‑linear stress‑strain relation is recast as a conservation law. The method yields the exact shape of slip‑line fields and correctly captures the behavior near yield‑surface corners where traditional hodograph methods break down.

The authors discuss the broader implications of their work. By integrating conservation laws with hodograph transformations, they provide a unified analytical framework that is robust against singularities in the transformation. This framework can be extended to multi‑component systems, to problems with more complex boundary conditions, and to numerical schemes that preserve the underlying conservation structure. Future research directions include the treatment of non‑hyperbolic regimes, the incorporation of dissipative effects, and the development of computational algorithms that exploit the derived equivalence for efficient simulation of high‑speed flows and plastic deformation processes.

In summary, the paper demonstrates that conservation laws are not merely auxiliary physical statements but powerful mathematical tools that enable the complete solution of hodograph‑linearizable hyper‑bolic systems, overcoming the long‑standing limitation posed by zero Jacobian regions and providing a pathway to exact analytic and high‑fidelity numerical solutions in gas dynamics and plasticity.