Coupled 1-D stress and thermal waves in temperature dependent nonlinear elastic and viscoelastic media*

Abstract

A general analysis is formulated for the closed loop coupled thermal and displacement viscoelastic 1-D wave problem. The proper inclusion of the highly temperature sensitive viscoelastic material properties renders the problem nonlinear, even though the displacements and material properties are considered to obey linear relations. In the present article. the previous analysis is enlarged and reformulated by (a) the inclusion of nonlinear elastic and viscoelastic constitutive relations as formulated in Hilton, (b) the addition of thermal waves to the displacement waves, and by (c) temperature dependent material density and viscoelastic moduli and compliances. The wave problem studied here is of significant importance in modeling, material characterization, determination of instantaneous moduli, nonlinear analytical solution protocols and the nonlinear interaction of temperature, material properties, and wave motions. Analytical and numerical solution protocols are presented and evaluated.

Keywords:

• Coupled 1-D waves
• longitudinal displacement and thermal waves
• material temperature sensitivity
• nonlinear coupled variable coefficient integral PDEs
• nonlinear elasticity and viscoelasticity
• nonlinear moduli

Acknowledgement

Support for HHH by the Aerospace Engineering Department of the College of Engineering and by the Computing and Data Sciences Division (CDS) of the National Center for Supercomputing Applications (NCSA) at the University of Illinois at Urbana-Champaign (UIUC) is gratefully acknowledged. MHE, HSE and MK thank the University of Science and Technology (UST) and the Natural Sciences Publishing Co. (NSP) for financially supporting their internship. Special thanks are also due to Professor A. J. Hildebrand of the UIUC Mathematics Department who supplied invaluable help on a number of difficult LATEX problems.

Notes

1 The dimensions of R are [[work/(${}^{o}K\otimes$ mole)]]

2 Also known as Williams-Landel-Ferry or WLF (shift) functions [116–118]. For finite linear viscoelasticity TSM see [119].

3 Typical values for the constants are ${\tilde{C}}_1=15$ and ${\tilde{C}}_2={50}^{°}$C.

4 The explicit dependence of the moduli on x, t and $t\prime$ is governed by whether or not the material is nonhomogeneous separately from any nonlinearities.

5 More often than not, these coefficients are independent of $\left[x,t,T(x,t)\right]$ or (x, t).

6 All functions are dependent on $(x_1,t)$ except for ${\rho }_0,u_2(x_1,x_2,t)$ and $E_{\mathit{\text{ijkl}}}(x_1,t,t\prime )$ in time integrals.