Constraints on the possible forms of material response, such as incompressibility or inextensibility, have long been used to simplify constitutive response models, and have resulted in substantial progress in fields such as fluid mechanics and the mechanics of composite materials. A method of imposing these constraints for thermoelastic materials is considered that follows steps that remove the need for assuming an additive term resulting from the constraint. In the process, three methods are considered for the separation of the constitutively prescribed part of the response from the part that is in reaction to the constraints. Both the case of single and multiple constraints are considered with extensive examples including special considerations for including effects associated with isotropic or anisotropic thermal expansion.
Spencer, A.J.M.Deformations of Fiber-Reinforced Materials, Clarendon/ Oxford University Press, London, 1972.
2.
Antman, S.S. and Marlow, R.S.Material constraints, lagrange multipliers, and compatibility. Applications to rod and shell theories, Archive for Rational Mechanics and Analysis, 116, 257—299 (1991).
3.
Negahban, M.A study of thermodynamic restrictions, constraint conditions, and material symmetry in fully strainspace theories of plasticity, International Journal of Plasticity, 11, 679—724(1995 ).
4.
Ericksen, J.L. and Rivlin, R.S.Large elastic deformations of homogeneous anisotropic materials, Journal of Rational Mechanics and Analysis , 3, 281—301 (1954).
5.
Carlson, D.E. and Tortorelli, D.A.On hyperelasticity with internal constraints , Journal of Elasticity, 42, 91—98(1996 ).
6.
Green, A.E. and Adkins, J.E.Large Elastic Deformations, Clarendon, Oxford, 1960 and 1970.
7.
Truesdell, C. and Noll, W.The Non-Linear Field Theories of Mechanics, Second Edition, Springer, Berlin, 1965/1992.
8.
Wang, C.-C.Mathematical Principles of Mechanics and Electromagnetism, Part A: Analytical and Continuum Mechanics, Plenum, New York, 1979.
9.
Beatty, M.F.Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues-with examples, Applied Mechanics Review, 40, 1699—1734 ( 1987).
10.
Cohen, H. and Wang, C.-C.On the response and symmetry of elastic materials with internal constraints, Archive for Rational Mechanics and Analysis, 99, 1—34 (1987 ).
11.
Casey, J.A treatment of internally constrained materials , Journal of Applied Mechanics, Transaction of ASME, 62, 542—543 (1995).
12.
Casey, J. and Carroll, M.M.A Treatment of internally constrained materials , Journal of Applied Mechanics, Transaction of ASME, 63, 240 (1996).
13.
Rivlin, R.S.The Fundamental Equations of Nonlinear Continuum Mechanics, Dynamics of Fluids and Plasmas, Academic, New York , 1966.
14.
Green, A.E. and Zerna, W.Theory of Elasticity, Second Edition, Oxford University Press, London, 1968.
15.
Green, A.E. , Naghdi, P.M. and Trapp, J.A.Thermodynamics of Continuum with Internal Constraints, International Journal of Engineering Science , 8, 891—908 (1970).
16.
Green, A.E. and Naghdi, P.M.A note on thermodynamics of constrained materials, Journal of Applied Mechanics, Transactions of ASME , 44, 787—789 (1977).
17.
Gurtin, M.E. and Podio-Guidugli , P.The thermodynamics of constrained materials, Archive for Rational Mechanics and Analysis, 51, 192—208 (1973).
18.
Batra, G.On Hamilton's principle for thermo-elastic fluids and solids, and internal constraints in thermo-elasticity, Archive for Rational Mechanics and Analysis, 99, 35—59 (1987).
19.
Casey, J. and Krishnaswamy, S.A characterization of internally constrained thermoelastic materials, Mathematics andMechanics of Solids , 3, 71—89 (1998).
20.
Martins, L.C. and Duda, F.P.Constrained elastic bodies and universal solutions in the class of plane deformations with uniform transverse stretch, Mathematics andMechanics of Solids, 3, 91—106 (1998).
21.
Klisch, S.M.Internally constrained mixtures of elastic continua, Mathematics andMechanics of Solids, 4, 481—498 (1999).
22.
Podio-Guidugli, P.A primer in elasticity, Journal of Elasticity,58, 1—104(2000).
23.
Beatty, M.F. and Hayes, M.A.Deformations of an elastic, internally constrained material. Part 1: Homogeneous deformations, Journal of Elasticity ,29, 1—84(1992).
24.
Pipkin, A.C.Constraints in linearly elastic materials, Journal of Elasticity, 6, 179—193(1976 ).
25.
Rostamian, R.Internal constraints in linear elasticity , Journal of Elasticity, 11, 11—31(1981 ).
26.
Podio-Guidugli, P. and Vianello, M.Constraint manifolds for isotropic solids , Archive for Rational Mechanics and Analysis, 105, 105—121 (1989).
27.
Saccomandi, G. and Beatty, M.F.Universal relations for fiber-reinforced elastic materials, Mathematics andMechanics of Solids, 7, 95—110 (2002).
