Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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852 ANDREY V. PYATIGORETS, MIHAI O. MARASTEANU, LEV KHAZANOVICH AND HENRYK K. STOLARSKI be limited in the following cases: the solution of the corresponding elastic problem is a transcendental function of elastic properties; the viscoelastic medium exhibits strong aging effects; the time-temperature superposition principle is not applicable. These are topics for future research. Appendix Master relaxation modulus curve. The master relaxation modulus E(ξ) of asphalt binders is determined from experimental data for creep compliances, which are obtained with the use of the bending-beam rheometer (BBR) test. In this test [AASHTO 2008], a small beam made of an asphalt binder or asphalt mix is subjected to three-point bending at different low temperatures (for example, −18 ◦ C, −24 ◦ C, −30 ◦ C). The deflection and the applied load are measured in real time. According to Euler–Bernoulli beam theory, the maximum elastic deflection at the mid-span of the beam is δmax = P L3 , (A1) 48E I where P is the concentrated load, L the beam span, I the moment of inertia, and E the Young’s modulus of the material. Using (A1), the creep compliance of the viscoelastic beam D(t) can be determined to be 48 I D(t) = δ(t), P L3 where δ(t) is the mid-span deflection history of the beam caused by a constant load P applied at zero time. The creep compliance D(t) is converted to the relaxation modulus E(t) using the Hopkins–Hamming method [1957] of numerical integration of a convolution integral: � t 0 E(t)D(t − τ) dτ = t. After E(t) is obtained for each testing temperature, the shift function aT (T ) can be found by shifting each relaxation curve along time axis in such a way that they create a single smooth curve, called the master relaxation curve. The direction of shifting is chosen in correspondence with the reference and current temperatures. The master curve is fitted using least squares by the CAM model (15), and the shift function is fitted using expression (9). This finally yields the constants C1 and C2 and reduced time ξ(t). Outline of the solution of the composite elastic cylinder problem. (Refer to Figure 3.) The constitutive equations for total strains in each cylinder are written in polar coordinates (r, θ) as εrr = 1+ν � (1 − ν)σrr − νσθθ + αE�T E � , εθθ = 1+ν � (1 − ν)σθθ − νσrr + αE�T E � . (A2) It is assumed that only radial displacement u exists, and the strain components are expressed (see, e.g., [Ugural and Fenster 2003]) as εrr = u,r, εθθ = u r , εrθ = 0. where the subscript comma means derivative. Substituting these expressions into (A2) and using the equilibrium equation ∂σrr ∂r + σrr − σθθ = 0, r
MATRIX OPERATOR METHOD FOR THERMOVISCOELASTIC ANALYSIS OF COMPOSITE STRUCTURES 853 it is straightforward to obtain the solution for the unknown radial displacement. After cancellation of the terms containing temperature, the result is u = c1r + c2 r , where c1 and c2 are constants. The general solution for the stresses in each elastic cylinder is σrr = E 1 + ν � c1 c2 − 1 − 2ν r 2 � − E 1 − 2ν α�T, σθθ = σrr + rσr,r. Using the boundary conditions for the problem of composite cylinder (Figure 3), namely σ inc rr = 0 at r = r0, σ inc bind rr = σrr at r = r1, u inc r = ubind r at r = r1, σ bind rr = 0 at r = r2, the constants c1 and c2 are found for each cylindrical layer. Then the solution for the circumferential stresses in the binder is given by (27)–(28), in which b(r) = − r 2 1 · r 2 a(r) = − r 2 1 r 2 · r 2 0 −r 2 1 r 2 1 −r 2 2 r 2 +r 2 2 r 2 1 −2νbr 2 1 +r 2 2 r · 2 +r 2 2 r 2−2νir 2 1 +r 2 0 · (αi −1)(νi +1) , c = νb+1 r 2 0 −r 2 1 r 2 1 −r 2 2 · αb(νb+1)−(νi +1) , (A3) νi +1 · r 2 1 −2νbr 2 1 +r 2 2 r 2 −2νir 2 1 +r 2 0 · νb+1 . (A4) νi +1 The solution for the total circumferential strains in the inclusion is given by (29)–(30), in which d = 1 + (1−2νi) r 2 1 r 2 , 0 g = r 2 1 r 2 · 0 r 2 0 −r 2 1 r 2 1 −r 2 � 1−2νb+ 2 r 2 2 r 2 � 1+νb . 1+νi 1 (A5) References [AASHTO 2005] “Standard specification for performance graded asphalt binder”, standard M320-05, American Association of State Highway and Transportation Officials, Washington, DC, 2005. [AASHTO 2007] “Standard method of test for determining the fracture properties of asphalt binder in direct tension (DT)”, standard T314-07, American Association of State Highway and Transportation Officials, Washington, DC, 2007. [AASHTO 2008] “Standard method of test for determining the flexural creep stiffness of asphalt binder using the bending beam rheometer (BBR)”, standard T313-08, American Association of State Highway and Transportation Officials, Washington, DC, 2008. [Arutyunyan and Zevin 1997] N. K. Arutyunyan and A. A. Zevin, Design of structures considering creep, A. A. Balkema, Brookfield, VT, 1997. [Barber 1992] J. R. Barber, Elasticity, Kluwer, Dordrecht, 1992. [Bažant 1972] Z. P. Bažant, “Numerical determination of long-range stress history from strain history in concrete”, Mater. Struct. 5 (1972), 135–141. [Bykov et al. 1971] D. L. Bykov, A. A. Il’yushin, P. M. Ogibalov, and B. E. Pobedrya, “Some fundamental problems of the theory of thermoviscoelasticity”, Mech. Compos. Mater. 7:1 (1971), 36–49. [Chien and Tzeng 1995] L. S. Chien and J. T. Tzeng, “A thermal viscoelastic analysis for thick-walled composite cylinders”, J. Compos. Mat. 29:4 (1995), 525–548. [Ekel’chik et al. 1994] V. S. Ekel’chik, L. V. Konovalova, and V. M. Ryabov, “Use of the Laplace transform to calculate temperature stresses in viscoelastic bodies during uniform cooling”, Mech. Compos. Mater. 29:5 (1994), 516–519. [Ferry 1961] J. D. Ferry, Viscoelastic properties of polymers, 2nd ed., Wiley, New York, 1961. [Findley et al. 1976] W. N. Findley, J. S. Lai, and K. Onaran, Creep and relaxation of nonlinear viscoelastic materials, Series Appl. Math. Mech. 18, North-Holland, Amsterdam, 1976. Reprinted Dover, New York, 1989.
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708 BAHATTIN KILIC AND ERDOGAN MADE
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