Thermoviscoelasticity

A theory of thermoviscoelasticity that includes the temperature dependence of the relaxation or retardation functions is necessarily nonlinear, and consequently the elastic-viscoelastic correspondence principle is not applicable. Nevertheless, a linear theory of thermoviscoelasticity can be developed in the framework of rational thermodynamics with further constitutive assumptions (Ref. 5, Chap. 3 see also Ref. 10). 

However, for thermorheologically simple materials, that is, for those materials for which the time-temperature superposition principle holds, the mechanical properties data can be shifted parallel to the time or frequency axis. This fact suggests an additional hypothesis that can be very useful in solving some specific thermoviscoelastic problems. According to this hypothesis, the net effect of temperature in the response must be equivalent to a variation in the rates of creep or relaxation of the material. Thus for T Tq the process occurs at a higher rate than at Tq. 

In general, the effective time will be related to t according to the expression 

When a is dependent on T — Tq [that is, a Williams-Landel-Ferry (WLF) dependence] or on T (Arrhenius dependence), then is a linear function of time. 

To formalize the main hypothesis, the nonisothermal functional given by Eq. (16.1) is modified by a new isothermic functional with a modified time scale to account for the temperature history. Now, according to the basic hypothesis of linear theory, the specific form of the stress-strain relationship can be written as 

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For a sinusoidal steady excitation and small deflections, the elastic and viscoelastic solutions are formally similar, as the separation of variables methodology outlined above suggests. Thus, in this case, the viscoelastic response is dependent on only the specific material properties of the sample under study. Moreover, on the basis of one of the hypotheses mentioned above, the thermoviscoelastic problem can be reduced to a thermoelastic one. Therefore, in the present context only the elastic solution of the problem will be discussed. 

Zheng, R. Kennedy, P. Phan-Thien, N. Fan, X.J. Thermoviscoelastic simulation of thermally and pressure induced stresses in injection molding. J. Non-Newtonian Fluid Mech. 1999, 84, 159-190. 

Use was made of a piezoresistive strain gauge array to measure the stress distribution on the surface of the die. A beam bending apparatus was used to study the importance of the thermoviscoelastic properties of the molding compound. The strain gauge allowed for the study of the effects of thermal shock testing. 

O Connell, P. A., and McKenna, G. B., Rheological measnrements of the thermoviscoelastic response of nltrathin polymer films. Science, 307, 1760-1763 (2005). 

The above-mentioned thermomechanical models only consider the elastic behavior of materials. Boyd et al. [13] reported on compression creep rapture tests performed on unidirectional laminates of E-glass/vinylester composites subjected to a combined compressive load and one-sided heating. Models were developed to describe the thermoviscoelasticity of the material as a function of time and temperature. In their work, the temperature-dependent mechanical properties were determined by fitting the Ramberg-Osgood equations and the temperature profiles were estimated by a transient 2D thermal analysis in ANSYS 9.0. 

Nguyen, T.D., Qi, H., Castro, F., and Long, K.N. (2008) A thermoviscoelastic model for amorphous shape memory polymers incorporating stmctural and stress relaxation. Journal of the Mechanics and Physics of Solids, 56, 2792-2814. 

Based on this understanding, a mechanism based constitutive model incorporating the nonlinear structural relaxation model into the continuum finite-deformation thermoviscoelastic framework was developed as follows. The aim of this effort was to estabUsh a quantitative understanding of the shape memory behavior of the thermally responsive thermoset SMP programmed at temperamres below Tg. To simplify the formulation, several basic assumptions were made in this study ... 

Despite this large volume of results, there are almost no papers on the thermoviscoelastic behavior of PLCs and their blends with EPs under conditions of creep or stress relaxation. At the same time, it is known that the time dependence of the mechanical behavior of these materials is significant, i.e. they are distinctly viscoelastic. Thus creep and stress relaxation studies can give important information for better understanding of the peculiarities of this type of polymer material and for the application conditions of PLCs as engineering materials. 

A major limitation of the model in the formulation of [71] is the prediction of stress and strain in dependency of temperature for only small unidirectional deformations of about 10%. As principal extension to large finite strains, the same authors published an improved 3-D, thermoviscoelastic approach to a phenomenological temperature dependence of the viscosity [87]. It allowed successful reanalysis of the experimental data of [71]. 

Bushko WC, Stokes VK (1995) Solidification of thermoviscoelastic melts part 1. formulation of model problem. Polym Eng Sci 35 351-364... 

Fan Z, Friedl C, Costa FS, Jin X (2010d) Three-dimensional anisotropic thermoviscoelastic simulation of residual stresses in injection moldings of thermoplastics. Proc 9th World Congr Comput Mech, Sydney (Abstract, p 69)... 

Kabanemi KK, Crochet Ml (1992) Thermoviscoelastic calculation of residual stresses and residual shapes of injection molded parts. Int Polym Process 7 60-70 Kaelble DH (1985) Computer-aided design of polymers and composites. Marcel Dekker, New York... 

R. A. Schapery, A Theory of Nonlinear Thermoviscoelasticity Based on Irreversible Thermodynamics Proc. 5th U. S. National Congress of Applied Mechanics, ASME, New York, 1966, pp. 511-530. 

D. S. McWilliams, Study of the Effect of Thermal History on the Structural Relaxation and Thermoviscoelasticity of Amorphous Polymers, Ph.D. Dissertation, School of Chemical Engineering, Purdue University, Lafayette, Ind., 1996. 

McCrum [213, 214] recently suggested that the above approach is subject to large errors and based on an irrational premiss. He proposed a new method of thermoviscoelasticity . Smith and Mark [215] have demonstrated McCrum s analysis to be flawed and have shown that the classical thermoelasticity approach is soundly based on theory. Indeed, there is excellent agreement between thermoelastic and viscometric results for poly(l-pentene) [206, 211], polyethylene [151, 154, 155, 211, 216], poly(dimethyl siloxane) [205, 211, 217], poly(ethylene oxide) [211, 218], poly (isobutylene) [207, 211, 216, 219] and poly(H-butyl methacrylate) [220, 221] (Table 6). 

This chapter has been used to illustrate the method discussed in Chap. 2, which may be used to solve thermoviscoelastic boundary value problems involving temperature fields simultaneously varying with position and time. The method relies upon the solution of integral equations in terms of Neumann series expansions. 

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