Acceleration Waves in Media with Microstructure
Within the unified approach to modelling of media with microstructure we discuss the propagation of acceleration waves. We describe a medium with microstructure as an elastic continuum with strain energy density which depends on deformations and additiona
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Abstract Within the unified approach to modelling of media with microstructure we discuss the propagation of acceleration waves. We describe a medium with microstructure as an elastic continuum with strain energy density which depends on deformations and additional internal variable and their first gradients. We use a Nth-order tensor as a kinematical descriptor of the microstructure. By acceleration wave we mean an isolated surface propagating in medium across which second derivatives of some fields undergo discontinuity jump. Here we formulate the conditions of existence of acceleration waves as algebraic inequality expressed using acoustic tensor. Keywords Acceleration waves Micropolar medium
⋅ Media with microstructure ⋅ Acoustic tensor ⋅
1 Introduction Among many types of nonlinear waves observed in solids and fluids where the analytical solutions are rare, the acceleration waves are exceptional since their conditions of propagation can be reduced to algebraic equations. An acceleration wave called also wave of weak discontinuity of order two is a solution of motion equations with discontinuities in the second derivatives on some surfaces that are called singular. It means that the acceleration wave can be represented by an isolated traveling smooth enough surface which is a carrier of discontinuity jumps of the second derivatives with respect to the spacial coordinates and time whereas the solution and V.A. Eremeyev (✉) Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Milchakova Street 8a, Rostov-on-Don 344090, Russia e-mail: [email protected] V.A. Eremeyev The Faculty of Mechanical Engineering, Rzeszów University of Technology, al. Powstańców Warszawy 8, 35–959 Rzeszów, Poland © Springer Nature Singapore Pte Ltd. 2017 M.A. Sumbatyan (ed.), Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, Advanced Structured Materials 59, DOI 10.1007/978-981-10-3797-9_7
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its first derivatives are continuous. Existence conditions of acceleration waves can be reduced to a spectral problem for an acoustic tensor and positivity of its eigenvalues. From the mathematical point of view the conditions of existence of acceleration waves coincide with the condition of strong ellipticity of the equilibrium equations. Ellipticity is a natural property of the equilibrium equations in the case of infinitesimal deformations. On the other hand the violation ellipticity for nonlinear media means that for certain deformations discontinuities may appear. Such discontinuous solutions may model such phenomena as shear-bands, phase transitions, slip surfaces, etc. Thus, analysis of conditions of propagation of acceleration waves plays an important role in the mechanics of materials. Within the nonlinear elasticity including compressible, incompressible and media with constraints, acceleration waves are studied in many works, see, e.g., the original papers by [2, 6, 7, 19, 33, 34, 42–44], see also [21, 49, 50] where the generalization to therm
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