Critical velocities of a two-layer composite tube under a moving internal pressure

Critical velocities of a two-layer composite tube under a uniform internal pressure moving at a constant velocity are analytically determined. The formulation is based on a Love–Kirchhoff thin shell theory that incorporates the rotary inertia and material anisotropy. The composite tube consists of two perfectly bonded axisymmetric circular cylindrical layers of dissimilar materials, which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form expressions for the critical velocities and radial displacement of the two-layer composite tube are first derived for the general case by including the effects of material anisotropy, rotary inertia and radial stress. The formulas for composite tubes without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry for each layer are then obtained as special cases. In addition, it is shown that the model for single-layer, homogeneous tubes can be recovered from the current model as a special case. To illustrate the new model, a composite tube with an isotropic inner layer and an orthotropic outer layer is analyzed as an example. All four critical velocities of the composite tube are calculated using the newly derived closed-form formulas. Six values of the lowest critical velocity of the two-layer composite tube are computed using three sets of the new formulas, which compare fairly well with existing results.


Introduction
It has been known that dynamic strains in a homogeneous tube under a moving internal pressure can exceed three times of the corresponding static values (e.g., [11,[30][31][32]). This occurs when the velocity of the moving pressure is very close to the lowest critical velocity of the tube. Therefore, it is necessary to find critical velocities of such tubes to ensure safe designs. Critical velocities are also needed in designing other structures subjected to moving loads (e.g., [7,8,23,28,33]).
Efforts have been made to determine critical velocities of single-layer, homogeneous tubes of materials with various types of symmetry, including isotropic, cubic, transversely isotropic and orthotropic (e.g., [10,11,15,27,30,35,36]). However, for composite tubes containing two or more cylindrical layers of dissimilar materials (e.g., [20,21,29]), very few studies have been conducted to find their critical velocities. Jones and Whittier [16] investigated the harmonic wave propagation in an infinitely long two-layered axisymmetric cylindrical shell using a Timoshenko-type shell theory similar to those of Lin and Morgan [19] and Herrmann and Mirsky [14], but they did not include any discussion on critical velocities of the composite cylinder. Chonan [5] studied the steady-state response of a two-layered cylindrical shell to a moving ring load and computed the lowest critical velocity under various bonding conditions at the interface by using the Fourier transform method. The critical velocity of an isotropic tube wrapped with an orthotropic layer was numerically determined by Simkins [29] based on the free vibration model for laminated orthotropic cylindrical shells developed in Dong [6] using a  Donnell-type shell theory. However, no closed-form formula has been provided in these and subsequent thin shell theory-based studies on critical velocities of composite tubes containing two or more bonded layers of dissimilar materials. This motivated the current work.
In the present study, closed-form formulas for critical velocities and radial displacement of a composite tube consisting of two cylindrical layers of dissimilar materials subjected to a uniform internal pressure moving at a constant velocity are derived by using a Love-Kirchhoff thin shell theory (e.g., [2,34,40]) that incorporates the rotary inertia and material anisotropy. Being based on the general 3D constitutive relations for orthotropic elastic materials, the current formulation provides a unified treatment of composite tubes containing two layers of dissimilar materials, each of which can have a different type of material symmetry, including orthotropic, transversely isotropic, cubic or isotropic. It is shown that the newly derived formulas for two-layer composite tubes recover those for single-layer, homogeneous tubes as special cases.
The rest of this paper proceeds as follows. In Sect. 2, the model for axisymmetric orthotropic cylindrical Love-Kirchhoff thin shells with the rotary inertia effect formulated in Gao and Littlefield [11] is briefly reviewed. In Sect. 3, closed-form formulas for critical velocities and radial displacement of a composite tube consisting of two layers of dissimilar materials under a moving internal pressure are derived by using the Love-Kirchhoff thin shell model reviewed in Sect. 2. In Sect. 4, the general analytical formulas obtained in Sect. 3 are applied to composite tubes made from two layers of dissimilar orthotropic, transversely isotropic, cubic or isotropic materials, leading to four sets of closed-form formulas for critical velocities of two-layer composite tubes. In Sect. 5, a numerical example is provided for a composite tube consisting of an isotropic inner layer and an orthotropic outer layer by directly applying the new formulas derived in Sect. 3. The critical velocity values predicted by the newly obtained analytical formulas are compared with each other and to those computationally determined in Simkins [29]. A summary is presented in Sect. 6.

Axisymmetric circular cylindrical shell model with the rotary inertia effect: review
Based on the Love-Kirchhoff kinematic relations, the displacement field in an axisymmetric circular cylindrical thin shell of uniform thickness h and mean radius R in the coordinate system {x, θ , z} shown in Fig. 1 can be written as (e.g., [18,40]) where u x , u θ and u z are, respectively, the x-, θ -and z-components of the displacement vector u of a point (x, θ , z) in the shell at time t, and u and w are, respectively, the x-and z-displacement components of the corresponding point (x, θ , 0) on the shell middle surface at time t. Note that u θ is identically zero and there is no dependence on θ for all kinematic and kinetic quantities in such an axisymmetric problem (e.g., [11,14]), which differ from those in general deformations of circular cylindrical shells (e.g., [37,40]). From Eq. (1), the components of the infinitesimal strain tensor in the axisymmetric circular cylindrical thin shell can be obtained as (e.g., [11]), with r ≈ R, ε xθ ε θ x 0, ε xz ε zx 0, ε θ z ε zθ 0.
The constitutive equations for an orthotropic linear elastic material in the cylindrical coordinate system {x, θ , z} have the general form (e.g., [4,9,22,39]): where σ ij (i, j ∈ {x, θ, z}) are the components of the Cauchy stress tensor, and C ij (i, j ∈ {1, 2, 3, 4, 5, 6}) are the components of the elastic stiffness matrix, which contains nine independent components for the orthotropic material, as shown in Eq. (3). Using Eq. (2) in Eq. (3) gives the stress components in terms of the kinematic variables u(x, t) and w(x, t) in the axisymmetric circular cylindrical thin shell as The equations of motion in terms of u and w for the axisymmetric circular cylindrical Love-Kirchhoff thin shell satisfying Eqs. (1), (2) and (4) are given by [11] h C 11 where f x and f z are, respectively, the x-and z-components of the body force resultant (force per unit area) through the shell thickness acting on the mid-surface. Note that in this axisymmetric case the rotary inertia effect is incorporated through the term " ρh 3 12 ∂ 4 w ∂ x 2 ∂t 2 " in Eq. (5b) (e.g., [17,38]).

Critical velocities of a two-layer composite tube under a moving pressure
The critical velocities of a two-layer composite tube subjected to an internal pressure moving at a constant velocity are derived herein using the axisymmetric circular cylindrical Love-Kirchhoff thin shell model reviewed in Sect. 2.

General case
Consider an axisymmetric two-layer composite tube, which consists of an inner cylindrical layer of mean radius R 1 and thickness h 1 and an outer cylindrical layer of mean radius R 2 and thickness h 2 . The two layers of dissimilar materials are perfectly bonded at the interface r R 1 + h 1 /2 R 2 − h 2 /2, and the composite tube is under a uniform internal pressure p 0 moving at a constant velocity V, as shown in Fig. 2.
The moving pressure can be represented by where p 0 is the magnitude of the pressure and H(·) is the Heaviside step function. Suppose that each layer is an orthotropic elastic material and satisfies the governing equations in Eqs. (5a) and (5b) for an axisymmetric circular cylindrical Love-Kirchhoff thin shell. Then, applying Eqs. (5a) and (5b) to the inner layer gives, with f where the superscript "(1)" denotes the inner layer (i.e., material 1) and use has been made of Eq. (6). Similarly, using Eqs. (5a) and (5b) for the outer cylindrical layer yields, with f where the superscript "(2)" stands for the outer layer (i.e., material 2). The continuity conditions at the perfectly bonded interface r R 1 + h 1 /2 R 2 − h 2 /2 of the two-layer axisymmetric composite tube read Substituting Eq. (1) into Eqs. (9a) and (9b) leads to where use has been made of the approximations h 1 2 ∂w (1) Equations (11a) and (11b) are the equations of motion for the composite tube, which can be solved to obtain u(x, t) and w(x, t). Note that for the current composite tube the rotary inertia effect is included through the term " When the tube is homogeneous with C h and h 2 0, Eqs. (11a) and (11b) reduce to which are identical to the governing equations for a single-layer axisymmetric orthotropic tube under a uniform internal pressure p 0 moving at a constant velocity V [11]. The steady-state solution of Eqs. (11a) and (11b) is derived next by starting with the following transformation (e.g., [15,30]: where ξ is a new variable. In terms of ξ , Eqs. (11a) and (11b) become where all derivatives are with respect to the new variable ξ . When C shows that w is a constant for any value of u. This defines one critical velocity of the composite tube given by which is also the dilatational wave velocity (denoted by V d ) of the two-layer shell. When C 11 − ρ (1) + ρ (2) V 2 0, Eq. (14a) can be integrated once with respect to ξ to obtain where c is an integration constant. Substituting Eq. (15b) into Eq. (14b) then results in where c ≡ − C (1) c is a constant. The solution of Eq. (16) has the form: where w p is a particular solution, and w h is the general solution of the homogeneous part of Eq. (16) given by The solution of Eq. (18) reads where W and α are constants. Substituting Eq.
When A 0, Eq. (20) shows that α 2 will be either positive or negative, leading to an exponentially changing or oscillating deflection w according to Eqs. (17) and (19), which is a critical state for tube deformations. Hence, the condition of A 0 defines a critical velocity for the composite tube, which is given by, upon using Eq. (21a), as one critical velocity. When C 0, α 0 will be a double repeated root of the characteristic equation in Eq. (20) and the other two roots of Eq. (20) will be either real or purely imaginary, giving an exponentially changing or oscillating deflection w according to Eqs. (17) and (19), which is a critical state. Therefore, the condition of C 0 defines another critical velocity, which can be readily obtained from Eq. (20c) as V cr 1 1 with the quantity in the curly brackets > 0. From Eqs. (15a) and (24), it is clearly seen that V cr1 < V cr3 . However, a mathematical comparison of Eq. (15a) with Eq. (23) shows that V cr2 V cr3 when h 1 h 2 or C (1) 11 /ρ (2) , V cr2 > V cr3 when h 1 > h 2 and C (1) 11 /ρ (1) > C (2) 11 /ρ (2) or h 1 < h 2 and C (1) 11 /ρ (1) < C (2) 11 /ρ (2) , and V cr2 < V cr3 when h 1 > h 2 and C (1) (2) . Finally, when the discriminant of Eq. (20) vanishes, α will be paired real or purely imaginary double roots, yielding an exponentially changing or oscillating deflection w, which is a critical state. Thus, this condition defines one more critical velocity. That is, from Eqs. (21a-c) and (22), it follows that which can be expanded to obtain the following cubic equation in V 2 : It can be readily shown that Eq. (26a), a cubic equation in the standard form, can be reduced to the following depressed cubic equation: where The discriminant of Eq. (27) is When d > 0, Eq. (27) has three distinct real roots given by Viète's trigonometric solution as (e.g., [24,25] From Eqs. (30) and (28), the three distinct real roots of Eq. (26a) can be obtained as Then, the critical velocity V cr0 in this case can be readily determined as the smallest real value among V 1 , V 2 and V 3 calculated from Eq. (31). When d < 0, Eq. (27) has one real root and two conjugated complex roots that are obtainable from the Cardano formula. The three roots of Eq. (26a) in this case can then be determined using the Cardano formula and Eq. (28) as (e.g., [26]) where V 2 1 in Eq. (32a) is the real root, which defines V cr0 in this case. When d 0, Eq. (27) has a triple root of 0 if P 0 or a single root of ω 1 3Q P and a double root of ω 2 ω 3 − 3Q 2P if P 0 (e.g., [25]). Then, it follows from these results and Eq. (28) that the roots of Eq. (26a) in this case are given by when P 0, and V cr0 is the smaller real value of V 1 and V 2 ( V 3 ) computed from Eq. (33b) when P 0.
Note that Eqs. (20) and (21a-c) can also be expressed as where Equation (35) can be rewritten as, with the help of Eqs. (15a), (23) and (24), It is seen from Eq. (36) that p and q each can be positive or negative, depending on the value of V. With the critical velocities V cr1 , V cr2 and V cr3 determined, the range of the pressure velocity V can be divided into four segments, for each of which the combination of p and q signs is different, resulting in different forms of the solution of Eq. (16) for the radial displacement w, as shown next.

Segment 1: 0 < V < V cr1
In this case, q > 0 and p > 0, which follows directly from Eq. (36) and the relations V cr1 < V cr3 and V < V cr2 . Then, the solution of Eq. (34) can be readily obtained as where Clearly, m > 0 in this case. From Eq. (37), the four roots of the characteristic equation in Eq. (34) or Eq. (20) can be determined as [9,12]: where i is the imaginary unit (with i 2 − 1) and the overhead bar represents the conjugate. From Eqs. (17), (19) and (39), the general solution of Eq. (16) in the current case (with c 0) can be obtained as where the superscripts "(I)" and "(II)" represent, respectively, region I (behind the pressure front; ξ < 0) and region II (ahead of the pressure front; ξ > 0), w 0 is the static displacement given by ϕ, β, τ , χ and κ are parameters defined by and W 4 are constants to be determined from the boundary and continuity conditions in the axial direction.
In this case, q < 0 and p > 0, which follows directly from Eq. (36) and the relation V < V cr3 .

Segment 4: V > V cr3
In this case, q < 0 and p < 0, which follows directly from Eq. (36)  With the critical velocities V cr0 -V cr3 and the radial displacement w( 10a)) can be obtained from Eq. (11a). Then, all of the displacement, strain and stress components in each layer of the composite tube can be readily computed from Eqs. (1), (2) and (4) for a given value of the pressure velocity V.

Composite tube without the rotary inertia effect
If the rotary inertia effect is neglected, Eq. (11b) reduces to Similar to what is done in solving Eqs. (11a) and (11b), the characteristic equation for the system of Eqs. (11a) and (50) can be obtained as which, as a quadratic equation in α 2 , has the roots: The critical velocity V cr0 in this case can be determined from the condition of vanishing discriminant of Eq. (51): which can be expanded to yield By following the procedure used in solving Eq. (26a), this cubic equation in V 2 can be solved to get the critical velocity V cr0 . It is found that the formulas derived in Eqs. (27)-(33a,b) can be used to obtain V cr0 in this case except that b and c in Eq. (26b) are now given by The critical velocities V cr1 and V cr3 remain to be given, respectively, by Eqs. (24) and (15a), but V cr2 is irrelevant in this case.
The radial displacement w continues to be provided by one of those expressions obtained in Sect. 3.1 except that p and q defined in Eq. (35) or Eq. (36) need to be replaced by

Composite tube with V/V cr2 < < 1 and V/V cr3 < <1
With the help of Eqs.
When V /V cr2 < < 1 and V /V cr3 < < 1, Eq. (56) simplifies to Note that Eqs. (58) and (51)  − ρ (1) The critical velocity V cr0 is given by the condition of vanishing discriminant of Eq. (58): which leads to This formula is valid for any two-layer composite tube under an internal pressure moving at a constant velocity V < < min(V cr2 , V cr3 ), which suppresses the rotary inertia effect. There is no other critical velocity in this case, as indicated in Eq. (58). As a result, 0 < V < V cr0 (< V cr1 ) is the only velocity range to be considered. The expression for w remains the same as one of those given in Eqs. (40a)-(40c) for this velocity range except that p and q defined in Eq. (35) need to be replaced by which follow directly from Eqs. (34) and (58).

Applications to composite tubes with layers exhibiting four types of material symmetry
The critical velocity and radial displacement formulas derived in Sect. 3 for composite shells are applied here to tubes consisting of two dissimilar orthotropic, transversely isotropic, cubic or isotropic layers. The general formulas obtained in Sect. 3 can also be directly used for composite tubes containing two layers with a combination of any two of these four types of materials.

Composite tube with two orthotropic thin layers
The general case of a composite tube made from two dissimilar orthotropic layers with different C ij and ρ has been discussed in detail in Sect. 3.1. For thin orthotropic cylindrical shells with σ zz ≈ 0 and under axisymmetric loading, the elastic stiffness constants are given by [3,9,11] where E xx and E θθ are, respectively, Young's moduli in the x-and θ -directions, ν xθ and ν θ x are Poisson's ratios, and μ xz is the shear modulus in the xz-plane. Applying Eq. (63) to each orthotropic layer and subsequently using Eq. (31), (32a), (33a) or (33b) will lead to the determination of the critical velocity V cr0 of a composite tube consisting of two dissimilar orthotropic thin layers.
Substituting Eq. (63) into Eqs. (24), (23) and (15a), respectively, yields the critical velocities V cr1 , V cr2 and V cr3 as V cr 1 1 for a composite tube made from two dissimilar orthotropic thin layers, which incorporate the rotary inertia effect (but exclude the radial stress effect) and have no restriction on the magnitude of the pressure velocity.
With V cr1 − V cr3 determined, the expressions of the radial displacement w in regions I and II for a given pressure velocity V belonging to one of the four segments can then be identified from those provided in Sect. 3.1, with C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) now given by Eq. (63).
When V < < V cr2 and V < < V cr3 , the substitution of Eq. (63) into Eq. (61) yields as the critical velocity for a composite tube consisting of two dissimilar orthotropic thin layers under an internal pressure moving at a constant velocity V < < min(V cr2 , V cr3 ), with both the rotary inertia and radical stress effects precluded. If the tube is homogeneous with E which is the same as that reported in Tzeng and Hopkins [36] and derived in Gao and Littlefield [11] for single-layer orthotropic thin tubes without including the rotary inertia effect. In this case with V < < min(V cr2 , V cr3 ), there is no other critical velocity for the composite tube consisting of two dissimilar orthotropic thin layers, and the expression for w remains the same as that given in Eq. (40a), (40b) or (40c) for the velocity range 0 < V < V cr0 (< V cr1 ) except that C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) are now given by Eq. (63).

Composite tube with two transversely isotropic thin layers
For thin transversely isotropic cylindrical shells with σ zz ≈ 0 and under axisymmetric loading, the elastic stiffness constants are given by [3,11,13,41] where E T and E L are, respectively, the Young's moduli in the transverse (isotropic) plane and longitudinal direction, ν TL and ν LT are Poisson's ratios, and μ LT is the shear modulus in the longitudinal direction. Using Eq. (67) for each transversely isotropic layer and then applying Eq. (31), (32a), (33a) or (33b) will give the critical velocity V cr0 , and substituting Eq. (67) into Eqs. (24), (23) and (15a) will yield the critical velocities V cr1 , V cr2 and V cr3 as V cr 1 1 for a composite tube consisting of two dissimilar transversely isotropic thin layers, which account for the rotary inertia effect (but exclude the radial stress effect) and have no restriction on the magnitude of the pressure velocity.
With V cr1 − V cr3 determined, the expressions of the radial displacement w for a given pressure velocity V can then be identified from those provided in Sect. 3.1, with C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) now given by Eq. (67).
When V < < V cr2 and V < < V cr3 , substituting Eq. (67) into Eq. (61) gives as the critical velocity V cr0 for a composite tube consisting of two dissimilar transversely isotropic thin layers under an internal pressure moving at a constant velocity V < < min(V cr2 , V cr3 ), which has precluded the rotary inertia and radial stress effects. If the tube is homogeneous with E which is the same as that first derived in Gao and Littlefield [11] for single-layer transversely isotropic thin tubes without including the rotary inertia effect. In this case with V < < min(V cr2 , V cr3 ), Eq. (58) shows that there is no other critical velocity for the composite tube consisting of two dissimilar transversely isotropic thin layers. In addition, the expression for w remains the same as that given in Eq. (40a), (40b) or (40c) for the velocity range 0 < V < V cr0 (< V cr1 ) except that C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) are now given by Eq. (67).

Composite tube with two cubic material layers
For cubic materials with three independent elastic constants E, ν and μ, the non-vanishing stiffness components are (e.g., [1,3] Applying Eq. (71) to each cubic material layer and then using Eq. (31), (32a), (33a) or (33b) will lead to the determination of the critical velocity V cr0 , and substituting Eq. (71) into Eqs. (24), (23) and (15a), respectively, will yield the critical velocities V cr1 , V cr2 and V cr3 as V cr 1 1 for a composite tube consisting of two dissimilar cubic material layers, which incorporate both the rotary inertia and radial stress effects and have no restriction on the magnitude of the pressure velocity.
With V cr1 − V cr3 determined, the expressions of w for a given pressure velocity V can then be identified from those provided in Sect. 3.1, with C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) now given by Eq. (71).
When V < < V cr2 and V < < V cr3 , using Eq. (71) in Eq. (61) leads to as the critical velocity V cr0 for a composite tube consisting of two dissimilar cubic material layers under an internal pressure moving at a constant velocity V < < min(V cr2 , V cr3 ), which has precluded the rotary inertia effect but accounts for the radial stress effect. If the tube is homogeneous with which is the same as that first derived in Gao and Littlefield [11] for single-layer cubic material tubes without including the rotary inertia effect. In this case with V < < min(V cr2 , V cr3 ), there is no other critical velocity for the composite tube consisting of two dissimilar cubic material layers, and the expression for w remains the same as that given in Eq. (40a), (40b) or (40c) for the velocity range 0 < V < V cr0 (< V cr1 ) except that C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) are now given by Eq. (71).
For thin cubic material cylindrical shells with σ zz ≈ 0 and under axisymmetric loading, the elastic stiffness constants are given by [3,11] Using Eq. (75) for each cubic material thin layer and then applying Eq. (31), (32a), (33a) or (33b) will result in the critical velocity V cr0 , and substituting Eq. (75) into Eqs. (24), (23) and (15a), respectively, will lead to the critical velocities V cr1 , V cr2 and V cr3 as V cr 1 1 V cr 2 1 for a composite tube consisting of two dissimilar cubic material thin layers, which include the rotary inertia effect (but exclude the radial stress effect) and have no restriction on the magnitude of the pressure velocity.
With V cr1 − V cr3 determined, the expressions of w for a given pressure velocity V can then be identified from those provided in Sect. 3.1, with C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) now given by Eq. (75).
When V < < V cr2 and V < < V cr3 , the use of Eq. (75) in Eq. (61) yields as the critical velocity V cr0 for a composite tube consisting of two dissimilar cubic material thin layers under an internal pressure moving at a constant velocity V < < min(V cr2 , V cr3 ), which has excluded both the rotary inertia and radial stress effects. If the tube is homogeneous with which is the same as that first derived in Gao and Littlefield [11] for single-layer cubic material thin tubes without including the rotary inertia effect. In this case with V < < min(V cr2 , V cr3 ), there is no other critical velocity for the composite tube consisting of two dissimilar cubic material thin layers. In addition, the expression for w remains the same as that given in Eq. (40a), (40b) or (40c) for the velocity range 0 < V < V cr0 (< V cr1 ) except that C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) are now given by Eq. (75).
With V cr1 − V cr3 determined, the expressions of w for a given pressure velocity V can then be identified from those provided in Sect. 3.1, with C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) now given by Eq. (79).
When V < < V cr2 and V < < V cr3 , substituting Eq. (79) into Eq. (61) gives as the critical velocity for a composite tube consisting of two dissimilar isotropic layers under an internal pressure moving at a constant velocity V < < min(V cr2 , V cr3 ), which has precluded the rotary inertia effect but accounts for the radial stress effect. If the tube is homogeneous with which is the same as that first derived in Gao and Littlefield [11] for single-layer isotropic tubes without including the rotary inertia effect. In this case with V < < min(V cr2 , V cr3 ), there is no other critical velocity for the composite tube consisting of two dissimilar isotropic layers, as dictated by Eq. (58). In addition, the expression for w remains the same as that given in Eq. (40a), (40b) or (40c) for the velocity range 0 < V < V cr0 (< V cr1 ) except that C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) are now given by Eq. (79).
For the composite tube with two dissimilar isotropic thin layers, σ zz ≈ 0 and the expressions of σ xx and σ θθ in each layer are the same as those for the composite tube with two dissimilar cubic material thin layers. As a result, the expressions for the stiffness components C 11 , C 12 , C 22 and C 55 are also the same as those listed in Eq. (75) except that μ E 2(1+v) in the current case. Applying Eq. (75) to each isotropic thin layer and then using Eq. (31), (32a), (33a) or (33b) will lead to the critical velocity V cr0 , and substituting Eq. (75) into Eqs. (24), (23) and (15a), respectively, will yield the critical velocities V cr1 , V cr2 and V cr3 that are, respectively, the same as those listed in Eqs. (76a), (76b) and (76c) for the composite tube consisting of two dissimilar cubic material thin layers, which include the rotary inertia effect (but exclude the radial stress effect) and have no restriction on the magnitude of the pressure velocity.
When V < < min(V cr2 , V cr3 ), substituting Eq. (75) into Eq. (61) will yield the critical velocity formula V cr0 for a composite tube consisting of two dissimilar isotropic thin layers under an internal pressure moving at a constant velocity V < < min(V cr2 , V cr3 ), which will be the same as that obtained in Eq. (77) for the composite tube with two dissimilar cubic material thin layers without the rotary inertia effect except that μ E 2(1+v) here. In this case, there is no other critical velocity, as in all other cases with V < < min(V cr2 , V cr3 ). That is, when both the rotary inertia and radial normal stress effects are suppressed and the material anisotropy is not considered, the newly derived formulas reduce to those for a composite tube containing two thin layers of dissimilar isotropic materials, which is a special (the simplest) case.
If the tube is homogeneous with E (1) E (2) E, ν (1) ν (2) ν, ρ (1) ρ (2) ρ, R 1 ≈ R ≈ R 2 , h 1 h and h 2 0, then the critical velocity formula for a single-layer isotropic thin tube under an internal pressure moving at a constant velocity V < < V cr3 ( V cr2 here) can be obtained from Eq. (77) as a special case, which will be the same as that derived in Eq. (78) for single-layer cubic material thin tubes.
In this case with V < < min(V cr2 , V cr3 ), there is no other critical velocity for the composite tube consisting of two dissimilar isotropic thin layers. In addition, the expression for w remains the same as that given in Eq. (40a), (40b) or (40c) for the velocity range 0 < V < V cr0 (< V cr1 ) except that C 11 , C 12 and C 22 for each layer involved in p and q defined in Eq. (35) are now given by Eq. (75).

Example
To quantitatively illustrate the formulas analytically derived in Sects. 3 and 4, a numerical example is presented herein.
Consider a composite tube consisting of an isotropic steel inner layer with R 1 31.524 mm, h 1 3.048 mm, ρ (1) 9.6527 GPa. These geometrical and material constants for the inner and outer layers are taken from Simkins [29], where the critical velocity V cr0 was numerically determined as the minimum phase velocity from a dispersion curve. The other critical velocities V cr1 -V cr3 were not discussed in Simkins [29], since the method used there was only capable of predicting V cr0 .
From Eqs. (15a), (23) and (24), the critical velocities V cr1 -V cr3 of this two-layer composite tube can be directly obtained as where the superscripts "M" and "P" denote, respectively, the values based on the measured and predicted material properties and geometrical parameters of the orthotropic outer layer provided in Simkins [29] and listed above. The critical velocity V cr0 can be computed using various formulas derived in Sect. 3. For the general case with both the rotary inertia and radial stress effects incorporated, it follows from Eqs. (26b), (28) and (29) where the subscript "SD" represents the special case with the velocity constraints.
The numerical values of V cr0 of the two-layer composite tube obtained above using various formulas derived in Sect. 3 are listed in Table 1, where they are also compared to the values of V cr0 for the same composite tube computationally determined in Simkins [29] by plotting the dispersion curves, which read A comparison of the values of V cr0 summarized in Table 1 with the values of V cr1 -V cr3 listed in Eq. (83) for the two-layer composite tube shows that V cr0 is much smaller than V cr1 , V cr2 or V cr3 , which is consistent with what was found for the single-layer isotropic steel tube without the orthotropic outer layer [10,11].
From Table 1, it is clear that the critical velocity V cr0 given by the formulas without including the rotary inertia effect is higher than that predicted by the general formulas incorporating the rotary inertia effect, which is true for both the cases with the measured and predicted properties for the orthotropic outer layer. In addition, in both cases the value of V cr0 obtained using the closed-form formula in Eq. (61), which is derived under the conditions of V < < V cr2 and V < < V cr3 that have also precluded the rotary inertia effect, is slightly larger than that predicted by the formulas without the rotary inertia effect. That is, the simplified formulas in the two cases excluding the rotary inertia effect over-predict the critical velocity V cr0 of the composite tube and may lead to unsafe designs. This finding is consistent with what was observed for single-layer homogeneous tubes [11]. Moreover, the more explicit and simpler formula in Eq. (61) predicts about the same value of V cr0 as that given by the formulas in Eqs. (31), (28), (26b), (54b) and (29). Hence, Eq. (61) can be used to quickly compute V cr0 when the rotary inertia effect can be neglected.
Finally, the values of the critical velocity V cr0 predicted by the three different sets of formulas derived in the current study are seen to agree fairly well with those numerically determined by Simkins [29] from dispersion curves of traveling flexural waves using the laminated orthotropic shell model of Dong [6], especially for the value based on the predicted properties of the orthotropic outer layer. This provides a validation of the newly developed analytical model.

Summary
Closed-form formulas are derived for critical velocities and radial displacement of a composite tube consisting of two laminated cylindrical layers of dissimilar materials under a uniform internal pressure moving at a constant velocity. A model for axisymmetric circular cylindrical Love-Kirchhoff thin shells incorporating the rotary inertia effect and general constitutive equations for orthotropic elastic materials is used in the formulation, leading to a unified treatment of composite tubes containing two layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic.
The formulas for two-layer composite tubes obtained for the general case include both the rotary inertia and radial stress effects and consider the material anisotropy for each layer, which are reduced to those for the special cases without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry. It is shown that the current new model for two-layer composite tubes recovers the model for single-layer, homogeneous tubes as a special case.
An example for a composite tube with an isotropic inner layer and an orthotropic outer layer is studied by directly applying the new model, which enables the determination of all four critical velocities of the composite tube. The numerical results reveal that six values of the lowest critical velocity given by three sets of the newly derived closed-form formulas are in fairly good agreement with those numerically determined by Simkins [29], thereby validating the current analytical model.