The formation of an anisotropic elastic medium on the compaction front of a stream of particles☆
Section snippets
Statement of the problem
A homogeneous stream of non-interacting particles impinges on a plane boundary at a certain angle to it. We take the direction orthogonal to the boundary and, therefore, to the compaction front formed as the x3 = x axis of a Cartesian coordinate system (Fig. 1). The x1 and x2 axes lie in the plane of the front. It is assumed that all tthe flow parameters depend only on the x coordinate and the time t, i.e., the flow is assumed to be one-dimensional in the form of plane waves.
In the region ahead
Motion of the medium behind the compaction front
The elastic medium formed behind the compaction front is assumed to be incompressible, weakly non-linear and weakly anisotropic. Its velocity in Lagrangian coordinates in the direction of the xα axis is defined in terms of the displacement vector component wα as υα = ∂wα/∂t. The elastic properties of the medium are given by its elastic potential Φ in the form of a function of the components of the shear strain uα = ∂wα/∂x. Assuming that uα ≪ 1, we can represent the elastic potential of such a medium
Relations on the compaction front following from the conservation laws
The mass conservation law (1.1) for V = const and the medium is incompressible, as was stated above, enables us to eliminate the longitudinal motions (along the x axis) of the medium from the discussion only and to consider the motion only in planes parallel to the front.
The law of conservation of momentum for the tangential components when ρ = 1 giveswhere and are the stresses and velocities ahead of the front, about which it is known that and due to the absence of
Evolutionarity of a compaction front
For the existence of a compaction front to be possible, the condition for its evolutionarity must be satisfied. In the mathematical literature it is called the correctness condition, and it ensures a unique solution of the problem of the interaction of a discontinuity with small perturbations which are orientated in the same direction as the discontinuity. The evolutionarity condition states7, 8 that just as many boundary conditions should be set on the front as there are small perturbations
Structure of a compaction front
The structure of a discontinuity (a compaction front in the present case) refers to a continuous solution of the equations of motion in the narrow transitional layer (which represents a discontinuity in a small-scale treatment of it), which describes the continuous variation of the flow parameters from the values ahead of the front to the values behind the front. The solution in this region is described by differential equations supplemented by dissipative terms and thus by a model that is more
Types of compaction fronts
The solutions of the front structure problem can be represented in the form of integral curves of a system of ordinary differential equations, which can be reduced using Eqs (5.1) and (2.1) to the form
The treatment will be performed in u1, u2, W phase space, i.e., in the u1, u2 phase planes for various values of W from the range 0 < W < ∞. Note that system (6.1) is identical to the system of equations used in Refs 5 and 9 to study the structure of shock waves in the elastic medium under
Isoclines of the equations of the structure and their properties
The singular points of system (6.1) in the u1, u2 plane lie on the intersection of the lines Lα(u1, u2) = 0. These same lines are isoclines for the integral curves of system (6.1). The integral curves intersect the line L1 = 0 in the direction parallel to the u2 axis (at points of intersection where du1/dξ = 0), and they intersect the line L2 = 0 in the direction parallel to the u1 axis (where du2/dξ = 0). The signs of the functions L1 and L2 specify directions corresponding to an increase in ξ along the
Sets of states behind a compaction front (an analogue of a shock adiabat) for an elastic medium with κ > 0
The goal is to determine the set of final points of the structure for all possible discontinuities with a specified initial state.
We will successively examine ranges (7.3), into which the W axis of the u1, u2, W phase space is divided at the values and .
In the first range in (7.3) the isocline picture in the u1, u2 plane at small positive values of Qα is presented in Fig. 3. The isoclines L1 are depicted by dashed lines, and the isoclines L2 are depicted by solid lines. In this case
Jouguet lines and surfaces. The shock adiabat
The Jouguet points in a W = const plane lie where the isoclines Lα come into contact. Using Eqs (7.1), we can conclude that the set of Jouguet points in u1, u2, W phase space is described by the equation
The function J(u1, u2, W) represents a figure consisting of two surfaces, one of which contains the Jouguet points of the slow waves, while the other contains the Jouguet points of the fast waves. Figure 5 shows the intersection of the surface J = 0 with the u2, W meridional plane, where u1 = 0. In a
The self-similar wave problem in a half-space
The self-similar wave problem in a half-space, which is often called the piston problem, refers here to the problem in which the medium formed behind a compaction front is adjacent to a plane, i.e., a piston, which moves with a specified constant velocity relative to the homogeneous medium ahead of the front. It is assumed here that all three components of the piston velocity are identical to the corresponding velocity components of the elastic medium adjacent to it. The solution of the piston
Conclusion
For non-linear elastic media with a stiffness that decreases as the strains increase (κ > 0) under the condition of small strains, the structure of the fronts for the formation of an elastic medium from a stream of free particles was investigated under the condition that the structure of the front is described by the Kelvin–Voigt equations. The shock adiabat, i.e., the set of states behind all possible fronts moving towards a specified stream of particles, was constructed under the assumption
Acknowledgement
This research was financed by the Russian Foundation for Basic Research (14-01-00049, 15-01-00361).
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Structures of non-classical discontinuities in solutions of hyperbolic systems of equations
2022, Russian Mathematical SurveysSimple One-Dimensional Waves in an Incompressible Anisotropic Elastoplastic Medium with Hardening
2020, Proceedings of the Steklov Institute of MathematicsFormation fronts of a nonlinear elastic medium from a medium without shear stresses
2017, Moscow University Mechanics Bulletin
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Prikl. Mat. Mekh. Vol. 79, No. 6, pp. 739–755, 2015.