Singular relaxation moduli and smoothing in threedimensional viscoelasticity
Authors:
Wolfgang Desch and Ronald Grimmer
Journal:
Trans. Amer. Math. Soc. 314 (1989), 381404
MSC:
Primary 73F15; Secondary 45K05, 45N05, 47D05, 47G05
MathSciNet review:
939803
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Abstract: We develop a semigroup setting for linear viscoelasticity in threedimensional space with tensorvalued relaxation modulus and give a criterion on the relaxation kernel for differentiability and analyticity of the solutions. The method is also extended to a simple problem in thermoviscoelasticity.
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𝐿_{2,𝑀}for an operatorvalued measure 𝑀,
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Utah, 1972), Academic Press, New York, 1973, pp. 387–397. MR 0342999
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 [1]
 J. Achenbach and D. Reddy, Note on wave propagation in linearly viscoelastic media, Z. Angew. Math. Phys. 18 (1967), 141144.
 [2]
 R. L. Bagley and P. J. Torvik, Fractional calculus, a different approach to viscoelastically damped structures, AIAA J. 21 (1983), 741748.
 [3]
 M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (2) 1 (1971), 161198.
 [4]
 G. Chen and R. C. Grimmer, Semigroups and integral equations, J. Integral Equations 2 (1980), 133154. MR 572484 (81f:45026)
 [5]
 R. M. Christensen, Theory of viscoelasticity. An introduction, 2nd ed., Academic Press, 1982.
 [6]
 B. T. Chu, Stress waves in isotropic linear viscoelastic materials, J. Mécanique 1 (1962), 439462. MR 0149753 (26:7238)
 [7]
 B. D. Coleman and M. E. Gurtin, Waves in materials with memory II. On the growth and decay of onedimensional acceleration waves, Arch. Rational Mech. Anal. 19 (1965), 239265. MR 0195336 (33:3538)
 [8]
 C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuum mechanics, IUTAM/IMU Sympos. on Applications of Methods of Functional Analysis to Problems in Mechanics (P. Germain and P. Nayroles, eds.), Lecture Notes in Math., vol. 503, Springer, Berlin, 1976. MR 0521351 (58:25196)
 [9]
 W. Desch and R. C. Grimmer, Propagation of singularities for integrodifferential equations, J. Differential Equations 65 (1986), 411426. MR 865070 (88b:45013)
 [10]
 , Initialboundary value problems for integrodifferential equations, J. Integral Equations 10 (1985), 7397. MR 831236 (87f:45025)
 [11]
 , Smoothing properties of linear Volterra integrodifferential equations, SIAM J. Math. Anal. 20 (1989), 116132. MR 977492 (89m:45014)
 [12]
 W. Desch, R. C. Grimmer, and W. Schappacher, Propagation of singularities by solutions of second order integrodifferential equations (to appear).
 [13]
 W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels (in preparation).
 [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., PrenticeHall, Englewood Cliffs, N. J., 1977.
 [16]
 V. Girault and P. A. Raviart, Finite element methods for NavierStokes equations. Theory and algorithms, Springer, Berlin, 1986. MR 851383 (88b:65129)
 [17]
 D. Gram, Mathematical models and waves in linear viscoelasticity, Wave Propagation in Viscoelastic Media (F. Mainardi, ed.), Res. Notes in Math., 52, Pitman, London, 1982, pp. 127.
 [18]
 R. C. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in Banach space, J. Differential Equations 50 (1983), 234259. MR 719448 (85k:45023)
 [19]
 K. Hannsgen and R. L. Wheeler, Behavior of the solution of a Volterra equation as a parameter tends to infinity, J. Integral Equations 7 (1984), 229237. MR 770149 (86b:45004)
 [20]
 K. Hannsgen, Y. Renardy, and R. L. Wheeler, Effectiveness and robustness with respect to time delays of boundary feedback stabilization in onedimensional viscoelasticity, SIAM J. Control Optim. 26 (1988), 12001234. MR 957661 (89k:93165)
 [21]
 W. J. Hrusa and M. Renardy, On wave propagation in linear viscoelasticity, Quart. Appl. Math. 43 (1985), 237254. MR 793532 (86j:45022)
 [22]
 J. A. Hudson, The excitation and propagation of elastic waves, Cambridge Univ. Press, London, 1980. MR 572263 (81h:73001)
 [23]
 J. Kazakia and R. S. Rivlin, Runup and spinup in a viscoelastic fluid I, Rheol. Acta 20 (1981), 111127.
 [24]
 R. K. Miller, Volterra integral equations in a Banach space, Funkcial. Ekvac. 18 (1975), 163194. MR 0410312 (53:14062)
 [25]
 A. Narain and D. D. Joseph, Linearlized dynamics for step jumps in velocity and displacement of shearing flows of a simple fluid, Rheol. Acta 21 (1982), 228250. MR 669373 (83j:76006)
 [26]
 , Classification of linear viscoelastic solids based on a failure criterion, J. Elasticity 14 (1984), 1926. MR 739116 (85f:73048)
 [27]
 A. Pazy, On the differentiability and compactness of semigroups of linear operators, J. Math. Mech. 17 (1968), 11311141. MR 0231242 (37:6797)
 [28]
 , Semigroups of linear operators and applications to linear partial differential equations, Springer, Berlin, 1983.
 [29]
 J. Prüss, Positivity and regularity of hyperbolic Volterra equations in Banach spaces, Math. Ann. 279 (1987), 317344. MR 919509 (89h:45004)
 [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), 251254. MR 669374 (83j:76007)
 [32]
 M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical problems in viscoelasticity, Longman, 1987. MR 919738 (89b:35134)
 [33]
 M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal. 62 (1976), 303322. MR 0416245 (54:4320)
 [34]
 R. Temam, NavierStokes equations. Theory and numerical analysis, rev. ed., NorthHolland, Amsterdam, 1979. MR 603444 (82b:35133)
 [35]
 J. N. Welch, On the construction of the Hilbert space for an operator valued measure , Vector and Operator Valued Measures and Applications (D. H. Tucker and H. B. Maynard, eds.), Academic Press, New York, 1973. MR 0342999 (49:7743)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198909398033
PII:
S 00029947(1989)09398033
Article copyright:
© Copyright 1989 American Mathematical Society
