The generalized partial correspondence principle in linear viscoelasticity

Authors:
G. A. C. Graham and J. M. Golden

Journal:
Quart. Appl. Math. **46** (1988), 527-538

MSC:
Primary 73F15

DOI:
https://doi.org/10.1090/qam/963588

Erratum:
Quart. Appl. Math. **49** (1991), 397.

MathSciNet review:
963588

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A result, referred to as the generalized partial correspondence principle, is proved for noninertial viscoelastic boundary value problems. This states that if , are the complementary regions of the boundary of a viscoelastic medium with a unique Poisson's ratio, on which displacements and stresses are specified, respectively, then if for all , the stresses satisfying this mixed boundary value problem at time are the stresses for the elastic problem with the same boundary regions and the same specified stresses while known functions take the place of the specified displacements. It is noteworthy that , , is not required to be monotonic increasing.

**[1]**R. M. Christensen,*Theory of viscoelasticity*, Academic Press, New York, 1971**[2]**G. A. C. Graham,*On the solution of mixed boundary value problems that involve time-dependent boundary regions for viscoelastic bodies with temperature dependent properties*, Arch. Mech. Stos.**5**, 771-785 (1967)**[3]**G. A. C. Graham,*The correspondence principle of linear viscoelasticity theory for mixed boundary value problems involving time-dependent boundary regions*, Quart Appl. Math.**26**, 167-174 (1968)**[4]**T. C. T. Ting,*A mixed boundary value problem in viscoelasticity with time-dependent boundary regions*, Proc. of the Eleventh Midwestern Mechanics Conference, H. J. Weiss, D. F. Young, W. F. Riley, and T. R. Rogge (Editors), Iowa University Press, 591-599, 1969**[5]**G. A. C. Graham and G. C. W. Sabin,*The correspondence principle of linear viscoelasticity for problems that involve time-dependent regions*, Internat J. Engng. Sci.**11**, 123-140 (1973) MR**0455723****[6]**G. A. C. Graham,*The solution of mixed boundary value problems that involve time-dependent boundary regions for viscoelastic materials with one relaxation function*, Acta Mech.**8**, 188-204 (1969)**[7]**J. M. Golden and G. A. C. Graham,*Boundary value problems in linear viscoelasticity*, Springer-Verlag, Berlin, 1988 MR**958684****[8]**M. E. Gurtin and E. Sternberg,*On the linear theory of viscoelasticity*, Arch. Rat. Mech. Anal.**11**, 291-356 (1962) MR**0147047****[9]**J. M. Golden,*Causality and viscoelastic boundary value problems*, Internat J. Engng. Sci.**24**, 1141-1149 (1986) MR**852153****[10]**A. E. Green and W. Zerna,*Theoretical elasticity*, Clarendon Press, Oxford, 1968 MR**0245245****[11]**G. A. C. Graham and G. C. W. Sabin,*The opening and closing of a growing crack in a linear viscoelastic body that is subject to alternating tensile and compressive loads*, Internat J. Fracture**14**, 639-649 (1978) MR**600045****[12]**J. M. Golden and G. A. C. Graham,*Energy balance criteria for viscoelastic fracture*, submitted for publication MR**1074956****[13]**T. C. T. Ting,*Contact problems in the linear theory of viscoelasticity*, J. Appl. Mech.**35**, 248-254 (1968)**[14]**G. A. C. Graham and F. M. Williams,*Boundary value problems for time-dependent regions in aging viscoelasticity*, Utilitas Math.**2**, 291-303 (1972)**[15]**I. N. Sneddon,*The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile*, Internat J. Engng. Sci.**3**, 47-57 (1965) MR**0183169****[16]**J. M. Golden and G. A. C. Graham,*The steady-state plane normal viscoelastic contact problem*, Internat J. Engng. Sci.**25**, 227-291 (1987)**[17]**G. A. C. Graham and J. M. Golden,*The three-dimensional steady-state viscoelastic indentation problem*, Internat J. Engng. Sci.**26**, 121-126 (1988)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73F15

Retrieve articles in all journals with MSC: 73F15

Additional Information

DOI:
https://doi.org/10.1090/qam/963588

Article copyright:
© Copyright 1988
American Mathematical Society