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

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846 ANDREY V. PYATIGORETS, MIHAI O. MARASTEANU, LEV KHAZANOVICH AND HENRYK K. STOLARSKI The solutions for circumferential strains in the binder and in the inclusion can be obtained in a form similar to (28). In particular, the total circumferential strain εinc θθ (r) in the inclusion is ε inc θθ (r) = � � � � �� αi F1 r, Eb + αb F2 r, Eb · �T (29) � � � � where we omit the subscript “tot”, and the matrix functions F1 r, Eb and F2 r, Eb are given by � � F1 r, Eb = (1 + νi)I − 2 r 2 1 r 2 (1 − ν2 i )� d I + gEi � � r F2 r, Eb = 2 2 1 r 2 (1 − νi)(1 + νb) � d I + gEi with constants d and g given as (A5) in the Appendix. � �−1�−1, Eb � �−1�−1, Eb Comparison with the analytical solution. The calculations presented in the analysis of the bar problem (Section 6) were simple and did not involve evaluating the inverse of ˜E. The present problem of a composite cylinder (Figure 3) is a more realistic test of accuracy. The plane strain elastic solution for this problem is given by (28)–(30). The results are compared with an analytical solution obtained by the application of the Laplace transform. To be able to use the Laplace transform, the expression for the shift factor aT given by (9) is replaced with linear function (30) âT (T ) = Ĉ1 − Ĉ2T, (31) (The hat over the quantities means that they have a different interpretation from before — say in (9).) Using (31) and (10), the reduced time becomes ˆξ = − 1 � � � ln � � 1 + Ĉ2(T0 � − T ) � � � (32) � and we have ˆt(ξ) = Ĉ1 − Ĉ2T0 Ĉ0 Ĉ2 Ĉ0 Ĉ2 Ĉ1 − Ĉ2T0 � 1 − exp(−Ĉ0Ĉ2ξ) � � � Ĉ1 �1 , � ˆT (ξ) = − T0 − exp(−Ĉ0 Ĉ2 Ĉ2ξ) � . (33) Despite the fact that the expressions (31)–(33) do not have a physical interpretation (e.g., Ĉ1 − Ĉ2T0 �= 0, which follows from (32)), the choice of a linear form for âT allows the determination of the Laplace transform of � ˆT (ξ): � ˆT ∗ � � Ĉ1 Ĉ0 (s) = − T0 Ĉ2 Ĉ2 s � s + Ĉ0Ĉ2 �. (34) Similarly, the CAM model used for the description of the master relaxation modulus is replaced by a function for which the Laplace transform can be easily found. A series of several exponents (Prony series) is used instead of (15): Ê(ξ) = � Êi exp(ˆγ iξ), Ê ∗ (s) = � Êi . (35) s − ˆγ i i For illustration purposes it is enough to use only a few terms in the series; we preserve them up to i = 3. i
MATRIX OPERATOR METHOD FOR THERMOVISCOELASTIC ANALYSIS OF COMPOSITE STRUCTURES 847 Even though, for comparison purposes, the parameters Ĉ1, Ĉ2, Êi, and ˆγ i could be chosen arbitrarily, we chose them via a least squares fitting of the functions aT (T ) and E(ξ) experimentally obtained for a modified asphalt binder (PG58-40). (It is not essential to have perfect fitting of experimental data with the new functions âT (T ) and Ê(ξ); we just need to use the same functions in solving the problem using the two approaches being compared.) The values of the parameters involved in âT (T ) are Those appearing in Ê(ξ) are Ĉ1 = 41.156640, Ĉ2 = 2.276346 ◦ C −1 . (in MPa) Ê0 = 38.598647, Ê1 = 349.348243, Ê2 = 195.939985, Ê3 = 126.336051, (in sec −1 ) ˆγ 0 = 0, ˆγ 1 = −0.517286 · 10 9 , ˆγ 2 = −0.265315 · 10 8 , ˆγ 3 = −0.137463 · 10 7 . We next present the parameters of the composite cylinder, chosen according to an ABCD specimen described in detail in the next subsection. The specimen consisted of an elastic ring with thickness r1 − r0, surrounded by a viscoelastic binder with thickness r2 − r1 (compare Figure 3 on page 845). These geometric parameters are while the remaining parameters are r0 = 23.75 · 10 −3 m, r1 = 25.4 · 10 −3 m, r2 = 31.75 · 10 −3 m, (36) T0 = 18 ◦ C, C0 = −1 ◦ C/hour, α1 = 1.4 · 10 −6 ◦ C −1 , α2 = 2.0 · 10 −4 ◦ C −1 , E1 = 141 GPa, ν1 = 0.3, ν2 = 0.33. Figure 4 compares the analytical plane strain solution for σ bind θθ (r1) obtained with the use of the Laplace �� Figure 4. Comparison of circumferential stress σ bind θθ (r1) in the composite cylinder problem obtained analytically (Laplace transform method) and numerically by the matrix representation of the relaxation operator ˜E of (22)–(24). (37)
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Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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