MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2016/1237493 1237493 Research Article Full Debonding Process of Adhesively Bonded Composite and Metallic Pipe Joints under Torsion http://orcid.org/0000-0002-5802-1256 Han Jun 1 http://orcid.org/0000-0002-1942-6744 Yuan Hong 1 Paipetis Alkis S. MOE Key Lab of Disaster and Control in Engineering Institute of Applied Mechanics Jinan University Guangzhou China jnu.edu.cn 2016 29122016 2016 02 09 2016 27 11 2016 13 12 2016 29122016 2016 Copyright © 2016 Jun Han and Hong Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Bonded joints are very common in many pipeline systems, in which the bond behavior of the joint interface is of crucial importance. This paper presents two analytical solutions for the debonding process of a pipe joint under torsion, assuming that the bond interface follows either an exponential softening bond-slip law or a simplified bilinear model. The solutions are general, applicable to composites and metallic and indeed other pipes. Based on the analytical solutions, the influences of the bond length and stiffness on the torque-displacement curve and ultimate load are investigated. The solutions can be used to explain the stress transfer mechanism, the interface crack propagation, and the ductility of the joint.

Science and Technology Scheme of Guangzhou City 201510010013 Science and Technology Scheme of Guangdong Province 2012A030200003 National Natural Science Foundation of China 11032005
1. Introduction

Pipes are very important structures in many industries such as energy, aerospace, and construction . In consideration of whole weight, strength, and maintenance workload, it is commonly accepted that there should be less joints in a piping system at first design. Due to the limitation of transportation, installation, and rehabilitation, a joint seems essential for a large structure system containing different components. The limitations of the overall system performance usually come from the capacity of pipe joints. Therefore, the pipe joints play the most important role in the overall integrity of most piping systems .

For metallic piping system, the joints can be divided into three types: flange coupling, welding, and adhesive bonding. The first two traditional connections have the same shortage, such as high stress concentration. However, the adhesively bonded pipe joint can effectively lower the stress concentration . While for composite piping system, connecting methods can be roughly summarized as adhesive bonding, butt-strap, heat-activated coupling, and flange coupling. The first three joints are considered as permanent joints and the same joint mechanism is found. Most composite flanges are connected to composite pipe with one of the three permanent methods. Therefore, a general adhesively bonded pipe joint analysis can be used to analyze all the three types of permanent composite pipe joints .

Adhesively bonded pipe joints provide uniform load transfer over the bonded area. Nevertheless, stress distribution in the bondline adhesive is nonuniform. Among all the possible loading configurations, such as tension, torsion, and bending, torsion is one of the fundamental types. Due to the difficulties in the analysis of interfacial behavior, few theoretical studies of mode III (torsion) interfacial fracture problems are available in the previous references.

Based on the mechanics of composite materials and the maximum strain failure criterion, an analytical and experimental study was conducted to investigate the elastic and failure behavior of composite laminated pipe under torsion . Based on the general composite shell theory, Zou and Taheri  studied the stress concentrations at and near the end of the joints as functions of various parameters, such as the overlap length and thickness of the adhesive layer. Pugno and Surace  confirmed that the maximum stresses were attained at the ends of the adhesive and that the peak of maximum stress was reached at the end of the stiffer tube and does not tend to zero as the adhesive length approaches infinity. Cheng and Li  developed an adhesively bonded smart composite pipe joint system by integrating electromechanical coupling piezoelectric layers with the connection coupler.

Finite element results are good supplements to analytical research. A finite element analysis was used to calculate the residual thermal stresses generated by cooling down from the adhesive cure temperature and a nonlinear analysis incorporating the nonlinear adhesive behavior was performed . Taheri et al.  developed a simple method for assessing the behavior of adhesively bonded tubular joints under torsion, based on a parametric study conducted by 2D and 3D finite element analysis. A finite difference method was utilized to solve the system of equilibrium equations and it was modeled as a separate 3D elastic body without the uniform stress assumption .

In all previous solutions, linear elastic properties are assumed for the entire pipe joints. The linear elastic behavior may be appropriate for the pipes themselves. While for the adhesive layer which is usually the weakest link in the bonded joints and often suffers from microcracking and local damage, nonlinear modeling may be necessary. When the crack tip experiences inelastic damage, the concepts based purely on the theory of elasticity are not valid. Therefore, different bond-slip models with softening region were proposed in the previous references, such as rigid-softening, bilinear, trapezoidal, and exponential models . The main differences between these models lie in the shape of the traction-displacement response and the parameters used to describe that shape. Based on the existing models, some studies have also been presented to solve interfacial debonding problem for the adhesively bonded joints and closed-form solutions are obtained .

However, there are very few studies focusing on the analytical solution of full debonding process of pipe joints in the literatures. To the best of the authors’ knowledge, linear elastic properties are assumed for the entire pipe joints and very few researchers have taken interfacial softening and debonding into consideration. In the present study, there are two analytical solutions for the failure process of adhesively bonded interface, assuming that the bond interface follows either an exponential softening bond-slip law or a simplified bilinear model. The expressions for the interface slip and shear stress are derived for the different failure stages. The present research improves and clarifies the understanding of the interfacial debonding problem of bonded pipe joints under torsion. By modifying different shear modulus, the present results may be further extended to composite pipe joints, composite-metal pipe joints, or metallic pipe joints.

2. Interface Model of Pipe Joint 2.1. The Bonded Pipe Joint

Figure 1 shows a bonded pipe joint where two pipes (inner pipes) of the same diameter are joined together symmetrically with a pipe coupler (outer pipe) adhesively bonded to both pipes. Here the inner and outer pipes are defined as Pipes 1 and 2, respectively. Due to symmetry, only the right half of the pipe joint is considered. Let the length of the pipe coupler be 2L; the distance between the left end of Pipe 1 and the right end of the Pipe 2 is L. Assuming that the pipe coupler is fully bonded with the inner pipes, the bond length between Pipe 1 and Pipe 2 is also L for the right half of the pipe joint.

Cross section

Side view

Right half of the pipe joint

2.2. Assumptions

The following assumptions are made for the simplicity:

The adherents are homogeneous and linear elastic.

The adhesive is only under pure shear.

The pipe joint is under pure torsion which is resisted by the main pipe and coupler pipe; that is, the adhesive layer is assumed to only transmit shear stresses between Pipes 1 and 2, not contributing to any direct resistance to the torque.

Local bending effects in the pipe joint under the torsional load are neglected.

According to the classical torsion theory, the internal torsion T1 and T2 in Pipes 1 and 2 can be expressed, respectively, as follows:(1)T1=φ1G1J1(2)T2=φ2G2J2,where subscripts 1 and 2 represent Pipes 1 and 2, respectively; G is the shear modulus of the pipe material; φ is the first derivation of the rotational angle with respect to x (Figure 1(c)); and J is polar moment of inertia of the thin-walled pipes which can be expressed as follows:(3)J1=2πR13t1J2=2πR23t2,in which t is the thickness of the pipes and R is the radius the midline of the pipe cross sections (Figure 1(a)).

As the torque resultant in the soft and thin adhesive layer is ignored following assumption number 3 above, the equilibrium between external and internal torsion load in the pipe joint requires(4)T1+T2=0dT1dx+dT2dx=0.

2.3. Governing Equations

Let the relative rotation between Pipes 1 and 2 at x be φ as illustrated in Figure 2. Consider the torsional equilibrium of an infinitely small section dx in Pipe 1 (Figure 2):(5)2πRτRdx=dT1,where τ is the interfacial shear stress in the circumferential direction and R is the distance between the center of the pipe and midline of the adhesive layer which can be calculated by(6)R=12R1+t12+R2-t22.The relative slip at the bond interface in the circumferential direction as δ can be found from the relative interfacial rotation φ:(7)δ=Rφ=Rφ1-Rφ2.After substituting (1)–(4) into (5) and introducing the parameter λ, local bond strength τf, and interfacial fracture energy Gf, we have(8)d2δdx2-2Gfτf2λ2fδ=0(9)φ1=τf22Gfλ22πR2G1J1dδdx,where(10)λ2=2πR3G1J1+G2J2G1J1G2J2τf22Gf.Substituting (9) into (1), the relationship of T1 and derivative of δ can be obtained:(11)T1=2πR2τf22Gfλ2dδdx.Equation (8) is the governing differential equation of the adhesively bonded joint. It can be solved once the local bond-slip model is given.

Equilibrium of local interfacial shear stresses.

2.4. Bond-Slip Model

As introduced before, researchers have proposed various bond-slip models. Experimental results indicate that the bilinear model which features a linear ascending branch followed by a linear descending branch provides a close approximation . However, the use of linear softening law may lead to overestimating the mechanical properties of the joint. Therefore, a bond-slip law characterised by a linear phase, with slope α, followed by an exponential softening branch is used . And the results based on the two models are compared in order to provide better understanding of failure process.

For the exponential bond-slip law (Figure 3), the interfacial shear stress increases linearly with slip within the elastic range. The shear stress peaks at τf at which the value of the slip is denoted by δ1. This stage is defined as elastic stage which is simplified as E. The interface enters the softening stage thereafter and the interfacial shear stress decays exponentially with the interfacial slip. This stage is defined as softening stage which is simplified as S. The mathematical expressions of the interfacial bond-slip law in Figure 3 are(12)τ=fδ=τfδ1δ0δδ1τfe-2α2δ/δ1-1δ>δ1.The bilinear model (Figure 3) features a linear ascending branch followed by a linear descending branch to zero at a slip δf. This stage is defined as debonding stage which is simplified as D. It is usually treated as a simplified model of the exponential softening model. The results are usually very close if the fracture energy Gf is the same for the two models . This bond-slip model is mathematically described by the following:(13)τ=fδ=τfδ1δ0δδ1τfδf-δ1δf-δδ1<δδf0δ>δf.The positive coefficient α2 in (12) characterises the exponential decay can be related to the slip parameters of the bilinear model by letting the interfacial fracture energy Gf be equal for the two models:(14)α2=δ1δf-δ1.

Bond-slip models.

3. Debonding Process: The Exponential Model 3.1. Elastic Stage

At small loads, there is no interfacial softening or debonding along the interface, so the entire length of the interface is in an elastic stress state. Substituting the first expression in (12) into (8) gives the following differential equation:(15)δx-λ12δx=00δδ1,where(16)λ12=2πR3G1J1+G2J2G1J1G2J2τfδ1.The inner pipe is free from stress at x=0 and subjected to a torque T at x=L, so the boundary conditions can be expressed as (17)φ10=0(18)φ1L=TG1J1.Based on these boundary conditions, the solution of (15) for the relative shear displacement and shear stress of the adhesive layer can be written in the form(19)δx=Tδ1λ12πR2τfcoshλ1xsinhλ1L(20)τx=Tλ12πR2coshλ1xsinhλ1L.The interfacial slip at the free end of the outer pipe (i.e., the value of δ at x=L) is defined as the displacement of the bonded joint for convenience and denoted by Δ here. From (19), the torque-displacement is expressed as(21)T=2πR2τfδ1λ1tanhλ1LΔ.

3.2. Elastic-Softening Stage

As the load increases, the interfacial slip reaches δ1 at x=L and the bond enters the softening stage there. The whole interface is in an elastic-softening stage and the length of the softening region a increases with the increase of torque T. Substituting (12) into (8) gives (15) and the following equation for this stage:(22)δx-λ22e-2α2δx/δ1=0δ>δ1,where(23)λ22=2πR3G1J1+G2J2G1J1G2J2τfe2α2.Applying the boundary conditions of (17) and (18) and the following continuity conditions:(24)δL-a=δ1δxiscontinuesatx=L-a,the solution for the elastic region of the interface (0δδ1, i.e., 0xL-a) is obtained as(25)δx=δ1coshλ1xcoshλ1L-aτx=τfcoshλ1xcoshλ1L-aand the solution for the softening region of the interface (δ>δ1, i.e., L-axL) is (26)δx=1nln2mnc1+2nlncoshn2c1x-c2(27)τx=τfe-2α2δx/δ1-1,where(28)m=λ22n=2α2δ1.Using (24), the constants c1 and c2 can be obtained as(29)c1=δ1λ1tanhλ1L-a2+2mne-nδ1(30)c2=L-a-2nc1arccoshnc12menδ1.The expression of slip at x=L can be obtained from (26) as(31)Δ=δL=1nln2mnc1+2nlncoshn2c1L-c2.Substituting (18) and (26) into (9) yields(32)T=2πR2τfδfλ2-2mne-nΔ+c1.For exponential model, by substituting (31) into (32), the expression of T above can be rewritten as(33)T=2πR2τfδfλ2c1tanhn2c1L-c2.When L is sufficiently large (approaches to infinity), (33) and (29) converge to(34)Tu=2πR2τfδfλ2c1,where(35)c1=δ1λ12+2mne-nδ1in which Tu is the maximum possible value (the asymptotic value) T can reach.

In bonded joints, there is usually an effective bond length le, which is defined as the length beyond which an increase of the bond length the loading capacity of the bonded joint does not increase further . For the exponential bond-slip law employed here, the loading (torque) capacity always increases with the bond length L, but the rate of increase would be very small if L is sufficiently large. Following Yuan et al. 2004, the effective bond length is defined here as the bond length which provides a resistance of 97% of Tu. Following this definition, le can be obtained from (30) and (33)–(35) as(36)le=ae+2λ1,where (37)ae=4nc1-2nc1arccoshnc12menδ1.

3.3. Softening Stage

As the load increases, the peak of the shear stress moves towards the middle of the joint (x=0). When the interfacial slip at x=0 reaches δ1, the whole interface enters into softening. This stage is governed by (22) with the boundary conditions of (17) and (18). The solution for the interfacial slip and the shear stress of the bond interface can be obtained as(38)δx=1nln2mnc3+2nlncoshn2c3x-c4(39)τx=τfe-2α2δx/δ1-1,where the constants c3 and c4 are(40)c3tanh2nc32L-c4=Tδfλ22πR2τf2.c4=0Substituting (18) and (38) into (9) yields(41)T=2πR2τfδfλ2-2mne-nΔ+c3.The expression of the slip at x=L can be obtained from (38): (42)Δ=1nln2mnc3+2nlncoshn2c3L-c4.

4. Debonding Process: Bilinear Model 4.1. Elastic Stage

The solution for the elastic stage is the same as that in Section 3.1.

4.2. Elastic-Softening Stage

As the load increases, softening commences at x=L once the shear stress reaches τf there. T increases as the softening length a increases. Substituting the first or the second expression in (13) into (8) gives (15) and the following:(43)δx-λ32δf-δx=0δ1<δδf,where(44)λ32=2πR3G1J1+G2J2G1J1G2J2τfδf-δ1.With the same boundary and continuity conditions as in Section 3.2, the solution for the elastic region of the interface (0δδ1, i.e., 0xL-a) is the same as that in Section 3.2, and the solution for the softening region of the interface (δ1<δδf, i.e., L-axL) is given by (45)δx=δf-δ1λ3λ1tanhλ1L-asinλ3x-L+a-cosλ3x-L+a+δfδf-δ1(46)τx=-τfλ3λ1tanhλ1L-asinλ3x-L+a-cosλ3x-L+a.Substituting (18) and (45) into (9) yields(47)T=2πR2τfλ3λ3λ1tanhλ1L-acosλ3a+sinλ3a.The expression of the slip at x=L can be obtained from (45):(48)Δ=δf-δ1λ3λ1tanhλ1L-asinλ3a-cosλ3a+δfδf-δ1.During this stage, the torque-displacement curve can be determined from (47) and (48). When the interfacial slip increases to δf at x=L but the slip at x=0 is less than δ1, the interface enters into elastic-softening-debonding stage. When the slip at x=0 reaches δ1 and the slip at x=L is less than δf, the interface enters into softening stage. Therefore, there exists a critical bond length to distinguish the failure process:(49)Lcr=π2λ3.When the bond length is larger than Lcr, the interface enters into elastic-softening-debonding stage. When the bond length is shorter than Lcr, the interface enters into softening stage. For the bilinear model, T reaches its maximum when the derivative of (47) with respect to a equals zero. Therefore, a at the ultimate load can be found from the following relationship:(50)tanhλ1L-a=λ3λ1tanλ3a.Substituting (50) into (47) yields(51)T=2πR2τfλ3δfδf-δ1sinλ3a.It can be shown from (50) and (51) that T approaches the following value when L is large:(52)Tu=2πR2τfλ.Following the same definition in (36) and considering that tanh(2)0.97, the effective bond length when 0.97Tu is reached can be obtained from (50)–(52) to give(53)le=ae+12λ1lnλ1+λ3tanλ3aeλ1-λ3tanλ3ae,where(54)ae=1λ3arcsin0.97δf-δ1δf.

4.3. Joints with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M134"><mml:mi>L</mml:mi><mml:mo>></mml:mo><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> 4.3.1. Elastic-Softening-Debonding Stage

If L>Lcr, as the torque increases, the interfacial slip at x=L reaches δf which represents the initiation of debonding (or macrocracking or fracture). Because Δ=δf at the initiation of debonding, the corresponding value of a, denoted by ad, can be obtained from (48) as (55)λ3λ1tanhλ1L-adsinλ3ad-cosλ3ad=0.Debonding propagates hereafter and the peak shear stress moves towards x=0. Assuming that the debonded length of the interface starting at x=L is d, (25) and (45)-(46) are still valid if replacing L by L-d. Therefore, the torque-displacement relationship can be written as(56)T=2πR2τfλ3λ3λ1tanhλ1L-d-acosλ3a+sinλ3a(57)Δ=δf+δfTλ22πR2τfd.As the interfacial shear stress at x=L-d is zero, the following relationship can be obtained:(58)λ3λ1tanhλ1L-d-asinλ3a-cosλ3a=0.Substituting (58) into (56) yields the following simplified form:(59)T=2πR2τfλ31sinλ3a.At the end of this stage, the softening-debonding stage starts when L-d=au. Substituting this into (58) yields(60)au=π2λ3.Moreover, by substituting (60) into (59), (59) can be written as(61)T=2πR2τfλ3.

4.3.2. Softening-Debonding Stage

This stage is governed by (43) with the boundary conditions of (17) and (62)φ1a=TG1J1(63)δa=δf.Based on the boundary conditions, the following solution can thus be found by solving the governing equation (43):(64)a=π2λ3=au(65)δx=δf-δfTλ22πR2τfλ3cosλ3x0xau(66)τx=λ3T2πR2cosλ3x0xau.From (64) it is seen that the length of softening zone remains constant during this stage. The torque-displacement relationship can be obtained by solving (8) for the case of δ>δf or directly displacement superposition along the bonded joint:(67)Δ=δf+δfTλ22πR2τfL-au.

4.4. For Joints with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M158"><mml:mi>L</mml:mi><mml:mo><</mml:mo><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> 4.4.1. Softening Stage

As the load increases, the shear stress peak moves towards x=0. When the interfacial slip at x=0 reaches δ1, the whole interface enters into the softening zone. This stage is governed by (43) with boundary conditions of (17) and (18). Based on the boundary conditions, the following solution can be obtained by solving the governing equation (43):(68)δx=δf-Tλ2δfcosλ3x2πR2τfλ3sinλ3L(69)τx=Tλ3cosλ3x2πR2sinλ3L.The expression of the slip at x=L can be obtained from (68): (70)Δ=δf-Tλ2δf2πR2τfλ3cotλ3L.As shown in the bond-slip law, when the interface slip reaches δf, debonding occurs. Substituting this condition into (70), the external load reduces to zero. Hence, it can be concluded that the softening stage here always represents the ultimate failure stage.

5. Numerical Simulations

The results of an adhesively bonded metal-composite pipe joint are presented here. The typical inner diameter and thickness are assumed to be 300 and 10 mm for the metallic main pipe, respectively. And the inner diameter and thickness of the composite coupler are assumed to be 320.4 and 5 mm, respectively. For metallic main pipe, the mechanical properties are chosen as E11=E22=200 GPa, G12=75 GPa, and μ12=0.3. For composite coupler, the mechanical properties are chosen as E11=25.2 GPa, E22=7.5 GPa, G12=2.4 GPa, and μ12=0.32. The parameters for interfacial bond-slip laws are identified as τf=7.2 MPa, δ1=0.034 mm, and δf=0.16 mm.

According to the material properties and geometry parameters given above, the critical bond length for bilinear model can be calculated as Lcr=23 mm. Therefore, bond lengths of 20 and 80 mm are chosen.

The load-displacement curve for exponential model is shown in Figure 4. OA, AB, and BC are elastic, elastic-softening, and softening stages, respectively.

L = 20  mm

L = 80  mm

5.1.2. Load-Displacement Curves for Bilinear Model

When the bond length is shorter than Lcr, take L=20 mm and the load-displacement curve is shown in Figure 5(a). OA, AB, and BC are elastic, elastic-softening, and softening stages, respectively. When the bong length is longer than Lcr, take L=80 mm and the load-displacement curve is shown in Figure 5(b). OA, AB, BC, and CD are elastic, elastic-softening, elastic-softening-debonding, and softening-debonding stages, respectively.

L = 20  mm

L = 80  mm

5.2. Shear Stress Distribution

The shear stress distribution for exponential model is shown in Figure 6. When the load is small, the interfacial shear stress at loaded end is less than peak stress and the interface is in an elastic stage. When the interfacial shear stress reaches peak stress at loaded end, the interface enters into elastic-softening stage. As load increases, the peak stress moves from loaded end to unloaded end and the length of softening zone increases. When the interfacial shear stress at unloaded end reaches peak stress, the interface enters into softening stage. For simplified legend, the critical status is defined as the end. For example, the critical status between elastic stage and elastic-softening stage is defined as E-end in Figure 6(a). And the following abbreviations are the same.

Shear stress distribution: exponential model.

L = 20  mm

L = 80  mm

The shear stress distribution of L=20 mm and L=80 mm for bilinear model are shown in Figures 7(a) and 7(b), respectively. When the load is small, the interfacial shear stress at loaded end is less than peak stress and the interface is an in elastic stage. When the interfacial shear stress reaches peak stress at loaded end, the interface enters into elastic-softening stage. As load increases, the peak stress moves from loaded end to unloaded end and the length of softening zone increases. When the interfacial shear stress at unloaded end reaches peak stress while Δ<δf, the interface enters into softening stage shown as in Figure 7(a). When Δ=δf and the interfacial shear stress at unloaded end is less than peak stress, then the interface enters into elastic-softening-debonding stage as shown in Figure 7(b). The length of debonding zone increases as the peak stress moves to unloaded end. When the interfacial shear stress at unloaded end reaches peak stress, the interface enters into softening-debonding stage.

Shear stress distribution: bilinear model.

L = 20  mm

L = 80  mm

5.3. Parametric Study

Figure 8(a) shows the influence for different bond lengths of bilinear model on the load-displacement curves. From the figure, the significant influence for bond length on the curves could be observed. In the range of the effective bond length, as the bond length increases, not only the interface failure processes change but also the ultimate load and interfacial slip. Specifically, the increase of the bond length can increase damage ductility. However, when the bond length reaches a certain length (effective bond length), the ultimate load will hardly change. Figure 8(b) shows the influence for different bond lengths of exponential model on the load-displacement curves. From the figure we know that as the bond length increases the ultimate load increases. But the failure processes are all the same. Namely, the increase of the bond length can also increase damage ductility.

Effect of bond length on load-displacement.

Bilinear model

Exponential model

Figure 9 shows the comparison of the load-displacement curves between the two models for different bond lengths. The load-displacement curves are different when a softening area exists. In addition, since there is no debonding situation in the exponential model, the displacement can increase unlimitedly, with this being different from bilinear model in which the slip would approach δf. From the figures, the load increases faster in the bilinear model as the bond lengths increases. However, when the bond length is long, the ultimate loads between the two models seem no more different.

Comparison of the load-displacement curves predicted from using the two bond-slip models.

L = 20  mm

L = 40  mm

L = 60  mm

L = 80  mm

Figure 10 shows the load-displacement curves for different ratios of torsion stiffness (β=(G2J2)/(G1J1)). From the figures we can see that as the ratio increases, the ultimate load increases but the slip decreases; namely, the ductility reduces. Moreover, the larger the ratio is, the smaller the change magnitude will be.

Effect of torsion stiffness on the load-displacement curve.

Bilinear model

Exponential model

Figure 11 shows the relationship of effective bond length and ratio of torsion stiffness. From the figure we can see that a stiffer coupler leads to a longer effective bond length. But as the ratio is getting larger, the effective bond length does not obviously increase. As the ratio increases, the effective bond lengths of two models have the similar trend, but the effective bond length of exponential model is longer than that of bilinear model.

Effect of the torsion stiffness ratio on the effective bond length.

Through the numerical computation, the ultimate load of exponential and bilinear models for different bond lengths could be obtained. Figure 12 shows the ultimate load for different bond lengths. From the figure we can see that, for bilinear model, when the bond length is short, the ultimate load increases significantly with the bond length. When the bond length is long, the ultimate load stays essentially unchanged. For the exponential model, when the bond length is short, the trend is similar to the bilinear model, but the ultimate load is relatively smaller. Both models have the same ultimate load when the bond length is relatively long.

Effect of the bond length on the ultimate load.

6. Conclusions

On the basis of fully understanding the mechanical behavior of the joint interface, this paper gives a further understanding of the key factors of interfacial debonding. By modifying different shear moduli, the present results may be further extended to composite pipe joints, composite-metal pipe joints, or metallic pipe joints. Based on the derivations in the current study, some important conclusions are summarized as follows:

Through the nonlinear fracture mechanics, the analytical expressions of the interfacial shear stress and the load-displacement relationship at loaded end of pipe joints under torsion loads could be gotten. Thus the shear stress propagation and the debonding progress of the whole interface for different bond lengths could be predicted.

The influences of different bond length on the load-displacement curve and the ultimate load are studied through the analytical solutions. The stress transfer mechanism, the interface crack propagation, and the ductility behavior of the joints could be explained.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors gratefully acknowledge the financial support provided by the Science and Technology Scheme of Guangzhou City (201510010013), Science and Technology Scheme of Guangdong Province (2012A030200003), and the Natural Science Foundation of China (National Key Project no. 11032005).