本文考慮一種同時具有異向性(與方向有關)、黏彈性(與時間有關)和熱流變簡單(溫度相關)的材料。為了對這種材料進行暫態熱應力分析，本研究將許多基礎理論結合在一起，例如(1)異向性彈性的複變數史磋公式，(2)彈性-黏彈性的對應原理或黏彈性的時間步進法，(3)熱流變簡單材料的時間-溫度疊加原理。將這些理論結合，我們可以使用異向性彈性求解方法來解決暫態異向性熱黏彈性於均勻溫度場的複雜問題。為了提供準確且有效率的計算方法，我們採用邊界元素法，並針對此類問題提出邊界元素法-對應原理和邊界元素法-時間步進法。最終結果顯示出，兩者都是準確的，而邊界元素法-時間步進法的效率比邊界元素法-對應原理較好。

We consider a material which is anisotropic (direction-dependent), viscoelastic (time-dependent), and thermorheologically simple (temperature-dependent). To work out the transient thermal stress analysis for this kind of materials, in this study several fundamental theories are combined together such as (1) the complex variable Stroh formalism for anisotropic elasticity, (2) the elastic-viscoelastic correspondence principle (EVCP) or the time-stepping method (TSM) for viscoelasticity, (3) the time-temperature superposition principle for thermorheologically simple materials. The combination of these theories gives us a simple idea that we may use the available solution tool for anisotropic elasticity to solve the complicated problems of transient anisotropic thermo-viscoelasticity under uniform temperature fields. Thus, to provide an accurate and efficient computational tool, we select the boundary element method (BEM), and propose BEM-EVCP and BEM-TSM for the present problems. The final results show that both of them are accurate, and BEM-TSM is much more efficient than BEM-EVCP.

[1] Ferry JD. Viscoelastic properties of polymers. Wiley; 1980.
[2] Christensen RM. Theory of viscoelasticity: an introduction. Elsevier; 1982.
[3] Haddad YM. Viscoelasticity of engineering materials. Springer; 1995.
[4] Brinson HF, Brinson LC. Polymer engineering science and viscoelasticity: An introduction. Springer; 2008.
[5] Lakes RS. Viscoelastic materials. Cambridge University Press; 2009.
[6] Leaderman H. Elastic and creep properties of filamentous materials and other high polymers. The Textile Foundation; 1943.
[7] Ferry JD. Mechanical properties of substances of high molecular weight VI. Dispersion in concentrated polymer solutions and its dependence on temperature and concentration. J Am Chem Soc 1950;69:3746-3752.
[8] Freudenthal AM. Effect of Rheological Behavior on Thermal Stresses. J Appl Physi 1954;25:1110-1117.
[9] Hilton HH. Thermal stresses in thick walled cylinders exhibiting temperature dependent viscoelastic properties of the Kelvin type. Proceedings of the Second U.S. National Congress on Applied Mechanics 1954;547-553.
[10] Biot MA. Linear thermodynamics and the mechanics of solids. Proceedings of the Third U.S. National Congress on Applied Mechanics. American Society of Mechanical Engineers; 1958.
[11] Hunter SC. Tentative Equations for the propagation of stress, strain and temperature fields in viscoelastic solids. J Mech Phys Solids 1961;9:39-51.
[12] Williams ML. Structural analysis of viscoelastic materials. AIAA J 1964; 2:785-808.
[13] Schapry RA. Application of Thermodynamics to Thermomechanical Fracture and Birefringent Phenomena in Viscoelastic Media. J Appl Phys 1964;35:1451-1465.
[14] Christensen RM, Naghdi PM. Linear Non-Isothermal Viscoelastic Solids. Acta Mech 1967;3.1:1-12.
[15] Schapery RA. Stress analysis of viscoelastic composite materials. J Compos Mater 1967;1:228-267.
[16] Poon H, Ahmad MF. A material point time integration procedure for anisotropic, thermo rheologically simple, viscoelastic solids. Comput Mech 1998; 21:236-242.
[17] Taylora ZA, Comasb O, Chengb M, Passengerb J, Hawkesa DJ, Atkinsona D, Ourselina S. On modelling of anisotropic viscoelasticity for soft tissue simulation: Numerical solution and GPU execution. Med Image Anal 2009;13:234-244.
[18] Pettermann HE, DeSimone A. An anisotropic linear thermo-viscoelastic constitutive law. Mech Time Depend Mater 2018;22:421-433.
[19] Tobolsky AV, Andrews RD. Systems manifesting superposed elastic and viscous behavior. J Chem Phys 1945;13:3-27.
[20] Schwarzl F, Staverman AJ. Time‐temperature dependence of linear viscoelastic behavior. J Appl Physi 1952;23:838-843.
[21] Morland LW, Lee EH. Stress analysis for linear viscoelastic materials with temperature variation. Trans Soc Rheol 1960;4:233-263.
[22] Williams ML, Landel RF, Ferry JD. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J Am Chem Soc 1955;77:3701-3707.
[23] Muki R, Sternberg E. On transient thermal stresses in viscoelastic materials with temperature-dependent properties. J Appl Mech 1961;28:193-207.
[24] Schapery RA. Approximation methods of transform inversion for viscoelastic stress analysis. Proceedings of the Fourth U.S. National Congress of Applied Mechanics 1962;2:1075.
[25] Lee EH, Rogers TG. Solution of viscoelastic stress analysis problems using measured creep or relaxation functions. J Appl Mech 1963;30:127-133.
[26] Humphreys JS, Martin CJ. Determination of Transient Thermal Stresses in a Slab with Temperature‐Dependent Viscoelastic Properties. Trans Soc Rheol 1963;7:155-170.
[27] Sternberg E. On the analysis of thermal stresses in visco-elastic solids," Fort Belvoir. Defense Technical Information Center; 1963.
[28] Valanis KC, Lianis G. A method of analysis of transient thermal stresses in thermorheologically simple viscoelastic solids. J Appl Mech 1964;31:47-53.
[29] Lee EH, Rogers TG. On the generation of residual stresses in thermoviscoelastic bodies. J Appl Mech 1965;32:874-880.
[30] Lockett FJ, Morland LW. Thermal stresses in a viscoelastic thin-walled tube with temperature dependent properties. Int J Eng Sci 1967;5:879-898.
[31] Selvadurai APS. Transient thermo-viscoelastic response of a crack in a layered structure. J Therm Stresses 1992;15:143-167.
[32] Zocher MA, Groves SE, Allen DH. A three-dimensional finite element formulation for thermoviscoelastic orthotropic media. Int J Numer Methods Eng 1997;40:2267-2288.
[33] White SR, Kim YK. Process-induced residual stress analysis of AS4/3501-6 composite material. Mech Compos Mater Struct 1998;5:153-186.
[34] Sunderland P, Yu W, Manson JA. Thermoviscoelastic Analysis of Process-Induced Internal Stresses in Thermoplastic Matrix Composites. Polym Compos 2001;22:579-592.
[35] Zhao LG, Warrior NA, Long AC. A thermo-viscoelastic analysis of process-induced residual stress in fibre-reinforced polymer–matrix composites. Mater Sci Eng A 2007;452-453:483-498.
[36] Anastasia HM, Rami HA. A multi-scale framework for layered composites with thermo-rheologically complex behaviors. Int J Solids Struct 2008;45:2937-2963.
[37] Muliana A, Khan KA. A time-integration algorithm for thermo-rheologically complex polymers. Comput Mater Sci 2008;41:576.
[38] Sawant S, Muliana A. A thermo-mechanical viscoelastic analysis of orthotropic materials. Compos Struct 2008;83:61-72.
[39] Ashrafi1 H, Keshmiri H, Bahadori MR, Shariyat M. An FEM Approach for Three – Dimensional Thermoviscoelastic Stress Analysis of Orthotropic Cylinders Made of Polymers. Adv Mat Res 2013;685:295-299.
[40] Sorvari J, Hämäläinen J. Time integration in linear viscoelasticity-a comparative study. Mech Time Depend Mater 2010;14:307–328.
[41] Crochon T, Schönherr T, Li C, Lévesque M. On finite-element implementation strategies of Schapery-type constitutive theories. Mech Time Depend Mater 2010;14:359–387.
[42] Sadeghinia M, Jansen KMB, Ernst LJ. Characterization of the viscoelastic properties of an epoxy molding compound during cure. Microelectron Reliab 2012;52:1711-1718.
[43] Abouhamzeh M, Sinke J, Jansen KMB, Benedictus R. A new procedure for thermo-viscoelastic modelling of composites with general orthotropy and geometry. Compos Struct 2015;133:871-877.
[44] Abouhamzeh M, Sinke J, Jansen KMB, Benedictus R. Thermo-viscoelastic analysis of GLARE. Compos Part B Eng 2016;99:1-8.
[45] Lee SS. Boundary element analysis of viscoelastic solids under transient thermal state. Eng Anal Bound Elem 1995;16:35-39.
[46] Lee SS. Boundary element analysis of a viscoelastic thin-walled cylinder subjected to thermal transient. Int J Press Vessels Piping. 1995; 63:195-198.
[47] Lee SS, Westmann RA. Application of high-order quadrature rules to time-domain boundary element analysis of viscoelasticity. Int J Numer Methods Eng 1995;38:607-629.
[48] Sladek V, Sladek J. Boundary integral equation method in two-dimensional thermoelasticity. Eng Anal 1984;1:135-148.
[49] Sladek V, Sladek J. Boundary integral equation method in thermoelasticity part III: uncoupled thermoelasticity. Appl Math Model 1984;8:413-418.
[50] Hwu C, Hsu CL, Hsu CW, Shiah YC. Fundamental solutions for two-dimensional anisotropic thermo-magneto-electro-elasticity. MMS 2019;24:3575-3596.
[51] Nguyen VT, Hwu C. Boundary element method for contact between multiple rigid punches and anisotropic viscoelastic foundation. Eng Anal Bound Elem 2020;118:295-305.
[52] Nguyen VT, Hwu C. Time-stepping method for frictional contact of anisotropic viscoelastic solids. Int J Mech Sci 2020;184:105836.
[53] Chen YC, Hwu C. Boundary Element Analysis for Viscoelastic Solids Containing Interfaces/Holes/Cracks/Inclusions. Eng Anal Bound Elem 2011;35:1010-1018.
[54] Hwu C. Anisotropic elastic plates. Springer; 2010.
[55] Brebbia CA, Telles JCF, Wrobel LC. Boundary element techniques: theory and applications in engineering. Springer; 1984.