Singular relaxation moduli and smoothing in three-dimensional viscoelasticity. (English) Zbl 0672.73039
A semigroup setting is developed for linear viscoelasticity in three- dimensional space with tensor-valued relaxation modulus. A criterion on the relaxation kernel is given for differentiability and analyticity of solutions. Extension is also made to a simple problem in thermoviscoelasticity.
Reviewer: G.A.C.Graham
MSC:
| 74D99 | Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) |
| 45K05 | Integro-partial differential equations |
| 34G10 | Linear differential equations in abstract spaces |
| 74J10 | Bulk waves in solid mechanics |
Keywords:
smoothing properties of systems of linear integral and differential; equations; Hilbert spaces; semigroup setting; tensor-valued relaxation modulus; relaxation kernel; differentiability; analyticity of solutions; thermoviscoelasticityReferences:
| [1] | J. Achenbach and D. Reddy, Note on wave propagation in linearly viscoelastic media, Z. Angew. Math. Phys. 18 (1967), 141-144. · Zbl 0152.43303 |
| [2] | R. L. Bagley and P. J. Torvik, Fractional calculus, a different approach to viscoelastically damped structures, AIAA J. 21 (1983), 741-748. · Zbl 0514.73048 |
| [3] | M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (2) 1 (1971), 161-198. |
| [4] | Goong Chen and Ronald Grimmer, Semigroups and integral equations, J. Integral Equations 2 (1980), no. 2, 133 – 154. · Zbl 0449.45007 |
| [5] | R. M. Christensen, Theory of viscoelasticity. An introduction, 2nd ed., Academic Press, 1982. |
| [6] | Boa-Teh Chu, Stress waves in isotropic linear viscoelastic materials. I, J. Mécanique 1 (1962), 439 – 462. |
| [7] | Bernard D. Coleman and Morton E. Gurtin, Waves in materials with memory. II. On the growth and decay of one-dimensional acceleration waves, Arch. Rational Mech. Anal. 19 (1965), 239 – 265. · Zbl 0244.73017 · doi:10.1007/BF00250213 |
| [8] | Paul Germain and Bernard Nayroles , Applications of methods of functional analysis to problems in mechanics, Lecture Notes in Mathematics, vol. 503, Springer-Verlag, Berlin-New York, 1976. Joint Symposium, IUTAM/IMU, held in Marseille, September 1 – 6, 1975. |
| [9] | G. W. Desch and R. Grimmer, Propagation of singularities for integro-differential equations, J. Differential Equations 65 (1986), no. 3, 411 – 426. · Zbl 0611.45008 · doi:10.1016/0022-0396(86)90027-6 |
| [10] | W. Desch and R. C. Grimmer, Initial-boundary value problems for integro-differential equations, J. Integral Equations 10 (1985), no. 1-3, suppl., 73 – 97. Integro-differential evolution equations and applications (Trento, 1984). · Zbl 0596.45025 |
| [11] | W. Desch and R. Grimmer, Smoothing properties of linear Volterra integro-differential equations, SIAM J. Math. Anal. 20 (1989), no. 1, 116 – 132. · Zbl 0681.45008 · doi:10.1137/0520009 |
| [12] | W. Desch, R. C. Grimmer, and W. Schappacher, Propagation of singularities by solutions of second order integrodifferential equations (to appear). · Zbl 0675.45021 |
| [13] | W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels (in preparation). · Zbl 0673.45008 |
| [14] | J. D. Ferry, Viscoelastic properties of polymers, 2nd ed., Wiley, New York, 1970. |
| [15] | Y. C. Fung, A first course in continuum mechanics, 2nd ed., Prentice-Hall, Englewood Cliffs, N. J., 1977. |
| [16] | Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. · Zbl 0585.65077 |
| [17] | D. Gram, Mathematical models and waves in linear viscoelasticity, Wave Propagation in Viscoelastic Media , Res. Notes in Math., 52, Pitman, London, 1982, pp. 1-27. |
| [18] | R. C. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in Banach space, J. Differential Equations 50 (1983), no. 2, 234 – 259. · Zbl 0519.45011 · doi:10.1016/0022-0396(83)90076-1 |
| [19] | Kenneth B. Hannsgen and Robert L. Wheeler, Behavior of the solution of a Volterra equation as a parameter tends to infinity, J. Integral Equations 7 (1984), no. 3, 229 – 237. · Zbl 0552.45010 |
| [20] | Kenneth B. Hannsgen, Yuriko Renardy, and Robert L. Wheeler, Effectiveness and robustness with respect to time delays of boundary feedback stabilization in one-dimensional viscoelasticity, SIAM J. Control Optim. 26 (1988), no. 5, 1200 – 1234. · Zbl 0655.93056 · doi:10.1137/0326066 |
| [21] | W. J. Hrusa and M. Renardy, On wave propagation in linear viscoelasticity, Quart. Appl. Math. 43 (1985), no. 2, 237 – 254. · Zbl 0571.73026 |
| [22] | John Arthur Hudson, The excitation and propagation of elastic waves, Cambridge University Press, Cambridge-New York, 1980. Cambridge Monographs on Mechanics and Applied Mathematics. · Zbl 0435.35002 |
| [23] | J. Kazakia and R. S. Rivlin, Run-up and spin-up in a viscoelastic fluid I, Rheol. Acta 20 (1981), 111-127. · Zbl 0463.76002 |
| [24] | Richard K. Miller, Volterra integral equations in a Banach space, Funkcial. Ekvac. 18 (1975), no. 2, 163 – 193. · Zbl 0326.45007 |
| [25] | A. Narain and D. D. Joseph, Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid, Rheol. Acta 21 (1982), no. 3, 228 – 250. · Zbl 0498.76006 · doi:10.1007/BF01515712 |
| [26] | Amitabh Narain and Daniel D. Joseph, Classification of linear viscoelastic solids based on a failure criterion, J. Elasticity 14 (1984), no. 1, 19 – 26. · Zbl 0528.73038 · doi:10.1007/BF00041080 |
| [27] | A. Pazy, On the differentiability and compactness of semigroups of linear operators, J. Math. Mech. 17 (1968), 1131 – 1141. · Zbl 0162.45903 |
| [28] | -, Semigroups of linear operators and applications to linear partial differential equations, Springer, Berlin, 1983. |
| [29] | Jan Prüss, Positivity and regularity of hyperbolic Volterra equations in Banach spaces, Math. Ann. 279 (1987), no. 2, 317 – 344. · Zbl 0608.45007 · doi:10.1007/BF01461726 |
| [30] | -, Regularity and integrability of resolvents of linear Volterra equations, Proc. Conf. on Volterra Integral Equations in Banach Spaces and Applications, Trento, 1987 (to appear). |
| [31] | M. Renardy, Some remarks on the propagation and nonpropagation of discontinuities in linearly viscoelastic liquids, Rheol. Acta 21 (1982), no. 3, 251 – 254. · Zbl 0488.76002 · doi:10.1007/BF01515713 |
| [32] | Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. · Zbl 0719.73013 |
| [33] | Marshall Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal. 62 (1976), no. 4, 303 – 321. · Zbl 0362.76019 · doi:10.1007/BF00248268 |
| [34] | Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. · Zbl 0426.35003 |
| [35] | John N. Welch, On the construction of the Hilbert space \?_{2,\?}for an operator-valued measure \?, Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972) Academic Press, New York, 1973, pp. 387 – 397. |
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