Some typical examples of the interfacial stresses of a composite cylinder subjected to uniform heat flow (see Fig. 1) are illustrated in this section. In general, from Eq.
(37), the thermoviscoelastic constitutive equation can be expressed as
0 0
where
/ /
( ) i , ( ) ti
i i ti ti
g τ =η e−τ λ g τ =η e−τ λ . (92)
Considering a thermorheologically simple material associated with a Kelvin-Maxwell three-parameter model, Eq. (43) becomes
0 the creep compliance and thermal expansion at t=0 and ∞ , respectively. For isothermal uni-axial constant load σ0, the thermoviscoelastic model in Eq. (45)
Consider an isotropic material, the relations between elastic compliances and modulus are
0
where G and ν denote the elastic shear modulus and Poisson’s ratio, respectively.
Meanwhile, the thermal expansion coefficients are
[ ]
0 0 0 0 T
αm = α α α (97)
In the following discussion, we assume the constants of material a(D ) are a G G= a, ν =0.32, k k= , a α α= a, λ λ= , 0t η η= t = , ( ) 1b T =
where T denotes the referred temperature. The material constants without mention 0 here, except the symmetric terms, are assumed to be zero. Note that η η= t = of 0 material a(D ) or a c(D ) indicates that the material properties of the medium are c independent of time, or the medium is assumed to be elastic for simplicity. In the following cases, let the heat flow angle be 0° with respect to x -axis (1 γ = ) and 0 b=2a.
Fig. 2 shows the evolution of interfacial normal stresses in a three-phase cylinder
with Ga =Gc, ka = , kc αa =αc, /G Gc b = , /2 k kc b =G Gc/ b, and α αc/ b =G Gb/ c. It shows that the normal stress decreases as time increases because of thermoviscoelastic effects of material b. It also shows that the normal stress becomes negative at z be= i0 and beiπ/ 4, and positive at bei3 / 4π and beiπ. This indicates that the compressive (or tensile) normal stress prevails at the site with high (or low) temperature. Moreover, it indicates that the normal stress is symmetric to the x -axis 1 and skew-symmetric to the x -axis. Fig. 3 shows the evolution of interfacial shear 2 stresses in a three-phase cylinder with Ga =Gc, ka = , kc αa =αc, /G Gc b = , 2
/ /
c b c b
k k =G G , and α αc/ b =G Gb/ c. It indicates the shear stresses decrease with time and converge to constant values as t/λ>2. The shear stress becomes positive at z be= −iπ/ 2 and be−iπ/ 4 and negative at z =beiπ/ 4 and beiπ/ 2. On the contrary, the shear stress is symmetric to the x -axis and skew-symmetric to the 2 x -axis. 1
A three-phase cylinder can be reduced to a two-phase hollow cylinder by letting the inner medium vanish. Fig. 4 and Fig. 5 show that the evolution of interfacial normal and shear stresses in a two-phase hollow cylinder with G Gc/ b = , 2
/ /
c b c b
k k =G G , /α αc b =G Gb/ c, by assuming Ga → , 00 ka → . The results show that the interfacial stresses have the same tendency as those in a three-phase solid cylinder (comparing to Fig. 2 and Fig. 3) except that the values of the hollow cylinder case (Fig. 4 and Fig. 5) are larger. This happens because of an increase of the temperature on the interface L in the hollow cylinder.
In general, it is difficult to obtain the real time solution from inverse Laplace transform. The inverse operation often leads into instable results. For a Kelvin-Maxwell three-parameter model, we only have a pole at −1/λ (λ λt = ) in this work. Therefore, we can use Eqs. (41) and (42) to find the numerical solution
without any instability. Furthermore, there are few studies dealing with multilayered thermoviscoelatic problems. However, the thermoviscoelatic solution can be verified by comparing with the corresponding thermoelastic problem at the time t=0. The results shown in Figs. 2 ~ 5 agree well with the thermoelastic solution at t=0 [18].
7. Conclusions
Using analytic continuation associated with the successive alternating technique and Laplace transform, a thermoviscoelastic solution of a three-phase cylinder is found in this work. Some typical examples are discussed and the results are found to agree well with the thermoelastic solution at the time beginning. The results show the interfacial stresses decrease with time at the beginning and then converge to constants after a period of time because of the thermoviscoelastic effects. It also shows the interfacial stress in a hollow cylinder is larger than those in a solid cylinder.
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Fig. 1 A viscoelastic composite cylinder subjected to a uniform heat flow.
Fig. 2 Evolution of interfacial normal stresses in a three-phase cylinder with Ga =Gc,
a c
k = , k αa =αc, /G Gc b = , /2 k kc b =G Gc/ b, and α αc/ b =G Gb/ c.
Fig. 3 Evolution of interfacial shear stresses in a three-phase cylinder with Ga =Gc,
a c
k = , k αa =αc, /G Gc b = , /2 k kc b =G Gc/ b, and α αc/ b =G Gb/ c.
Fig. 4 Evolution of interfacial normal stresses in a two-phase hollow cylinder with
/ 2
c b
G G = , /k kc b =G Gc/ b, and α αc/ b =G Gb/ c.
Fig. 5 Evolution of interfacial shear stresses in a two-phase hollow cylinder with
/ 2
c b
G G = ,