A NONLINEAR CONSTITUTIVE THEORY FOR HEAT CONDUCTION IN LAGRANGIAN DESCRIPTION BASED ON INTEGRITY

Year 2017, , 1615 - 1631, 04.10.2017

Karan Surana

https://doi.org/10.18186/journal-of-thermal-engineering.358150

Abstract

If the deforming matter is to be in thermodynamic equilibrium, then all constitutive theories, including

those for heat vector, must satisfy conservation and balance laws. It is well known that only the second law

of thermodynamics provides possible conditions or mechanisms for deriving constitutive theories, but the

constitutive theories so derived also must not violate other conservation and balance laws. In the work presented

here constitutive theories for heat vector in Lagrangian description are derived (i) strictly using the

conditions resulting from the entropy inequality and (ii) using theory of generators and invariants in conjunction

with the conditions resulting from the entropy inequality. Both theories are used in the energy equation

to construct a mathematical model in R1 that is utilized to present numerical studies using p-version least

squares finite element method based on residual functional in which the local approximations are considered

in higher order scalar product spaces that permit higher order global differentiability approximations.

The constitutive theory for heat vector resulting from the theory of generators and invariants contains up to

cubic powers of temperature gradients and is based on integrity, hence complete. The constitutive theory

in approach (i) is linear in temperature gradient, standard Fourier heat conduction law, and shown to be

subset of the constitutive theory for heat vector resulting from the theory of generators and invariants.

Keywords

Nonlinear Heat Conduction, Solid Continua, Lagrangian Description, Generators and Invariants, Entropy Inequality, Integrity, Temperature Gradient

References

• [1] C. O. Bennett and J. E. Myers. Momentum, Heat, and Mass Transfer. McGraw-Hill, 1962.
• [2] D. A. Anderson, J. C. Tannehill, and R. H. Pletcher. Computational Fluid Mechanics and Heat Transfer. CRC Press, 1984.
• [3] F. P. Incropera and D. P. DeWitt. Introduction to Heat Transfer. John Wiley & Sons, 1985.
• [4] K. S. Surana. Advanced Mechanics of Continua. CRC/Taylor and Francis, Boca Raton, FL, 2015.
• [5] J. N. Reddy. An Introduction to Continuum Mechanics. Cambridge University Press, 2013.
• [6] A. C. Eringen. Mechanics of Continua. Robert E. Krieger Publishing Co., 1980.
• [7] K. S. Surana and J. N. Reddy. The Finite Element Method for Boundary Value Problems: Mathematics and Computations. CRC/Taylor and Francis, 2016.
• [8] K. S. Surana and J. N. Reddy. The Finite Element Method for Initial Value Problems. CRC/Taylor and Francis, 2017 (In Preparation).
• [9] A. C. Eringen. Nonlinear Theory of Continuous Media. McGraw-Hill, 1962.
• [10] W. Prager. Strain Hardening under Combined Stresses. Journal of Applied Physics, 16:837–840, 1945.
• [11] M. Reiner. A Mathematical Theory of Dilatancy. American Journal of Mathematics, 67:350–362, 1945.
• [12] J. A. Todd. Ternary Quadratic Types. Philosophical Transactions of the Royal Society of London. Series A: Mathematical and Physical Sciences, 241:399–456, 1948.
• [13] R. S. Rivlin and J. L. Ericksen. Stress-Deformation Relations for Isotropic Materials. Journal of Rational Mechanics and Analysis, 4:323–425, 1955.
• [14] R. S. Rivlin. Further Remarks on the Stress-Deformation Relations for Isotropic Materials. Journal of Rational Mechanics and Analysis, 4:681–702, 1955.
• [15] C. C. Wang. On Representations for Isotropic Functions, Part I. Archive for Rational Mechanics and Analysis, 33:249, 1969.
• [16] C. C. Wang. On Representations for Isotropic Functions, Part II. Archive for Rational Mechanics and Analysis, 33:268, 1969.
• [17] C. C. Wang. A New Representation Theorem for Isotropic Functions, Part I and Part II. Archive for Rational Mechanics and Analysis, 36:166–223, 1970.
• [18] C. C. Wang. Corrigendum to ‘Representations for Isotropic Functions’. Archive for Rational Mechanics and Analysis, 43:392–395, 1971.
• [19] G. F. Smith. On a Fundamental Error in two Papers of C.C. Wang, ‘On Representations for Isotropic Functions, Part I and Part II’. Archive for Rational Mechanics and Analysis, 36:161–165, 1970.
• [20] G. F. Smith. On Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors and Vectors. International Journal of Engineering Science, 9:899–916, 1971.
• [21] A. J. M. Spencer and R. S. Rivlin. The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua. Archive for Rational Mechanics and Analysis, 2:309–336, 1959.
• [22] A. J. M. Spencer and R. S. Rivlin. Further Results in the Theory of Matrix Polynomials. Archive for Rational Mechanics and Analysis, 4:214–230, 1960.
• [23] A. J. M. Spencer. Theory of Invariants. Chapter 3 ‘Treatise on Continuum Physics, I’ Edited by A. C. Eringen, Academic Press, 1971.
• [24] J. P. Boehler. On Irreducible Representations for Isotropic Scalar Functions. Journal of Applied Mathematics and Mechanics / Zeitschrift f ¨ur Angewandte Mathematik und Mechanik, 57:323–327, 1977.
• [25] Q. S. Zheng. On the Representations for Isotropic Vector-Valued, Symmetric Tensor-Valued and Skew- Symmetric Tensor-Valued Functions. International Journal of Engineering Science, 31:1013–1024, 1993.
• [26] Q. S. Zheng. On Transversely Isotropic, Orthotropic and Relatively Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors, and Vectors. International Journal of Engineering Science, 31:1399–1453, 1993.