Abstract
The fully dynamical motion of a phase boundary is examined for a specific class of elastic materials whose stress-strain relation in simple shear is nonmonotone. Previous work has shown that a preexisting stationary phase boundary in such a material can be set in motion by a finite amplitude shear pulse and that an infinity of solutions is possible according to the present theory. In this work, these solutions are examined in detail from the perspective of energy and dissipation. It is shown that there exists at most two solutions which involve no dissipation (corresponding to conservation of mechanical energy). It is also shown that there exists one solution that maximizes the mechanical energy dissipation rate. The total mechanical energy remaining in the dynamical fields after one such pulse-phase boundary encounter is shown to exceed the total methanical energy after either an energy minimal quasi-static motion or a maximally dissipative quasi-static motion.
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References
J.L. Ericksen, Equilibrium of bars,J. Elasticity 5 (1975) 191–201.
R.D. James, Co-existent phases in the one-dimensional static theory of elastic bars,Arch. Rational Mech. Anal. 72 (1979) 99–140.
R.D. James, Finite deformation by mechanical twinning,Arch. Rational Mech. Anal. 77 (1981) 143–176.
M.E. Gurtin, Two-phase deformations of elastic solids,Arch. Rational. Mech. Anal. 84 (1984) 1–29.
R.D. James, On the stability of phases,Int. J. Eng. Sci. 22 (1984) 1193–1197.
J. Carr, M. Gurtin and M. Slemrod, One dimensional structured phase transitions under prescribed loads,J. Elasticity 15 (1985) 133–142.
R.D. James, Displacive phase transformations in solids,J. Mech. Phys. Solids 34 (1986) 359–394.
J.K. Knowles and E. Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics,J. Elasticity 8 (1978) 329–379.
J.K. Knowles and E. Sternberg, Discontinuous deformation gradients near the tip of a crack in finite anti-phase shear: an example,J. Elasticity 10 (1980) 81–110.
J.K. Knowles and E. Sternberg, Anti-plane shear fields with discontinuous deformation gradients near the tip of a crack in finite-elastostatics,J. Elasticity 11 (1981) 129–164.
R. Abeyaratne, Discontinuous deformation gradients away from the tip of a crack in anti-plane shear,J. Elasticity 11 (1981) 373–393.
R. Abeyaratne, Discontinuous deformation gradients in the finite twisting of an incompressible elastic tube,J. Elasticity 11 (1981) 43–80.
N. Kikuchi and N. Triantafyllidis, On a certain class of elastic materials with non-elliptic energy densities,Quart. Appl. Math. 40 (1982) 241–248.
R.L. Fosdick and G. MacSithigh, Helical shear of an elastic, circular tube with a non-convex stored energy,Arch. Rational Mech. Anal. 84 (1983) 31–53.
R. Abeyaratne and J.K. Knowles, Non-elliptic elastic materials and the modeling of elastic-plastic behavior for finite deformation.J. Mech. Phys. Solids 35 (1987) 343–365.
R. Abeyaratne and J.K. Knowles, Non-elliptic elastic materials and the modeling of dissipative mechanical behavior: an example,J. Elasticity 18 (1987) 227–278.
R.D. James, The propagation of phase boundaries in elastic bars,Arch. Rational Mech. Anal. 73 (1980) 125–150.
T.J. Pence, On the emergence and propagation of a phase boundary in an elastic bar with a suddenly applied end load,J. Elasticity 16 (1986) 3–42.
T.J. Pence, Formulation and analysis of a functional equation describing a moving one-demensional elastic phase boundary,Quart. Appl. Math. 45 (1987) 293–304.
T.J. Pence, On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: reflection and transmission behavior,J. Elasticity 25 (1991) 31–79.
J.K. Knowles, On the dissipation associated with equilibrium shocks in finite elasticity,J. Elasticity 9 (1979) 131–158.
R.C. Abeyaratne, An admissibility condition for equilibrium shocks in finite elasticity,J. Elasticity 13 (1983) 175–184.
H. Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion—isothermal case,Arch. Rational Mech. Anal. 92 (1986) 247–263.
M.E. Gurtin, On phase transitions with bulk, interfacial, and boundary energy,Arch. Rational Mech. Anal. 96 (1986) 243–264.
S.A. Silling, Numerical studies of loss of ellipticity near singularities in an elastic material,J. Elasticity 19 (1988) 213–239.
S.A. Silling, Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid,J. Elasticity 19 (1988) 241–284.
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Pence, T.J. On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: energy and dissipation. J Elasticity 26, 95–146 (1991). https://doi.org/10.1007/BF00041218
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DOI: https://doi.org/10.1007/BF00041218