Differential Equations and Nonlinear Mechanics pp 285-291 | Cite as

# On the Modelling of Dissipative Processes

## Abstract

A basic premise in the classical theory of elasticity (see Truesdell and Noll [1]) is that the body has a unique “natural configuration” modulo rigid motion, the “natural configuration” being usually understood as the stress free state. Such a premise is not generally true for most if not all materials, there being numerous other “configurations” in which they can exist naturally. Not all such states may be accessed by a body in a specific process, but that is not to say that such natural configurations do not exist. For instance, a virgin specimen of metal (if such a specimen is possible for after all such a specimen is produced via a complicated manufacturing process), that is in a stress free state, could be subject to sufficiently small deformations by the application of sufficiently small loads, which on the removal of the loads could return to its original stress free state (or “natural configuration”). However, the same body when subjected to a sufficiently large homogenous deformation would not return to its original stress free configuration on the removal of the load, as it might have undergone “plastic” deformation. A piece of steel, for instance, can undergo classical slip, or twin, or undergo solid to solid phase change on the application of loads, or it could melt due to an increase in temperature. At the end of any one of these above mentioned processes, the piece of steel would be associated with a different natural configuration from its initial natural configuration. In order to model the response of materials undergoing such processes, it is imperative to explicitly take cognizance of the fact that different natural configurations are accessed during such processes.

### Keywords

Entropy Crystallization## Preview

Unable to display preview. Download preview PDF.

### References

- [1]Truesdell, C. and Noll, W., The non-linear field theories of mechanics, 2nd edition, Springer-Verlag, Berlin 1992.Google Scholar
- [2]Rajagopal, K.R. and Srinivasa, A.R., Mechanics of the inelastic behavior of materials: Part I — Theoretical underpinnings, International Journal of Plasticity, 14, 1998, 945–967.MATHCrossRefGoogle Scholar
- [3]Rajagopal, K.R. and Srinivasa, A.R., Mechanics of the inelastic behavior of materials: Part II — Inelastic response, International Journal of Plasticity, 14, 1998, 969–995.MATHCrossRefGoogle Scholar
- [4]Rajagopal, K.R. and Srinivasa, A.R., On the inelastic behavior of solids: Part I — Twinning, International Journal of Plasticity, 11, 1995, 653–678.MATHCrossRefGoogle Scholar
- [5]Rajagopal, K.R. and Srinivasa, A.K., Inelastic behavior of materials: Part ll — Energetic associated with discontinuous twinning, International Journal of Plasticity, 13, 1997, 1–35.MATHCrossRefGoogle Scholar
- [6]Srinivasa, A.R., Rajagopal, K.R., and Armstrong, R., A phenomenological model of twinning based on dual reference structures, Acta Materialia, 46, 1998, 1235–1248.CrossRefGoogle Scholar
- [7]Lapczyk, E., Rajagopal, K.R., and Srinivasa, A.R., Deformation twinning during impact — Numerical calculations using a constitutive theory based on multiple natural configurations, Computational Mechanics, 21, 1998, 20–27.MATHCrossRefGoogle Scholar
- [8]Rajagopal, K.R. and Srinivasa, A.R., On the thermodynamics of shape memory wires, ZAMP, 50, 1999, 459–494.MathSciNetMATHCrossRefGoogle Scholar
- [9]Rajagopal, K.R. and Wineman, A.S., A constitutive equation for non-linear solids which undergo deformation induced microstructural changes, International Journal of Plasticity, 8, 1992, 385–395.MATHCrossRefGoogle Scholar
- [10]Wineman, A.S. and Rajagopal, K.R., On a constitutive theory for materials undergoing microstructural changes, Archives of Mechanics, 42, 1990, 53–75.MathSciNetMATHGoogle Scholar
- [11]Rao, I.J. and Rajagopal, K.R., Phenomenological modelling of polymer crystallization using the notion of multiple natural configurations, Interfaces and Free Boundaries, 2, 2000, 73–94.MathSciNetMATHCrossRefGoogle Scholar
- [11]Rao, I.J. and Rajagopal, K.R., Phenomenological modelling of polymer crystallization using the notion of multiple natural configurations, Interfaces and Free Boundaries, 2, 2000, 73–94.MathSciNetMATHCrossRefGoogle Scholar
- [13]Rajagopal, K.R. and Srinivasa, A.R., A thermodynamic framework for rate type fluid models, 88, 2000, 207–227.MATHGoogle Scholar
- [14]Rajagopal, K.R., On the constitutive modeling of materials, to appear.Google Scholar
- [15]Truesdell, C., A
*First Course in Rational Continuum Mechanics*, Academic Press, 19.Google Scholar - [16]Noll, W., On the foundations of mechanics of continuous media, Carnegie Institute of Technology, Department of Mathematics, Report 7, 1957.Google Scholar
- [17]Noll, W., A new mathematical theory of simple materials, Archive for Rational Mechanics and Analysis, 48, 1972, 243–293.MathSciNetCrossRefGoogle Scholar