Singular relaxation moduli and smoothing in three-dimensional viscoelasticity

Authors:
Wolfgang Desch and Ronald Grimmer

Journal:
Trans. Amer. Math. Soc. **314** (1989), 381-404

MSC:
Primary 73F15; Secondary 45K05, 45N05, 47D05, 47G05

DOI:
https://doi.org/10.1090/S0002-9947-1989-0939803-3

MathSciNet review:
939803

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop a semigroup setting for linear viscoelasticity in three-dimensional space with tensor-valued 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.

**[1]**J. Achenbach and D. Reddy,*Note on wave propagation in linearly viscoelastic media*, Z. Angew. Math. Phys.**18**(1967), 141-144.**[2]**R. L. Bagley and P. J. Torvik,*Fractional calculus, a different approach to viscoelastically damped structures*, AIAA J.**21**(1983), 741-748.**[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. MR**572484****[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. MR**0149753****[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. MR**0195336**, https://doi.org/10.1007/BF00250213**[8]**Paul Germain and Bernard Nayroles (eds.),*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. MR**0521351****[9]**G. W. Desch and R. Grimmer,*Propagation of singularities for integro-differential equations*, J. Differential Equations**65**(1986), no. 3, 411–426. MR**865070**, https://doi.org/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). MR**831236****[11]**W. Desch and R. Grimmer,*Smoothing properties of linear Volterra integro-differential equations*, SIAM J. Math. Anal.**20**(1989), no. 1, 116–132. MR**977492**, https://doi.org/10.1137/0520009**[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., 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. MR**851383****[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. 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. MR**719448**, https://doi.org/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. MR**770149****[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. MR**957661**, https://doi.org/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. MR**793532**, https://doi.org/10.1090/S0033-569X-1985-0793532-0**[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. MR**572263****[23]**J. Kazakia and R. S. Rivlin,*Run-up and spin-up in a viscoelastic fluid*I, Rheol. Acta**20**(1981), 111-127.**[24]**Richard K. Miller,*Volterra integral equations in a Banach space*, Funkcial. Ekvac.**18**(1975), no. 2, 163–193. MR**0410312****[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. MR**669373**, https://doi.org/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. MR**739116**, https://doi.org/10.1007/BF00041080**[27]**A. Pazy,*On the differentiability and compactness of semigroups of linear operators*, J. Math. Mech.**17**(1968), 1131–1141. MR**0231242****[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. MR**919509**, https://doi.org/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. MR**669374**, https://doi.org/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. MR**919738****[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. MR**0416245**, https://doi.org/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. MR**603444****[35]**John N. Welch,*On the construction of the Hilbert space 𝐿_{2,𝑀}for an operator-valued measure 𝑀*, Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972) Academic Press, New York, 1973, pp. 387–397. MR**0342999**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
73F15,
45K05,
45N05,
47D05,
47G05

Retrieve articles in all journals with MSC: 73F15, 45K05, 45N05, 47D05, 47G05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0939803-3

Article copyright:
© Copyright 1989
American Mathematical Society