Analytical Solution of Deformations for Two-Layer Timoshenko Beams Glued by a Viscoelastic Interlayer

An analytical solution of stresses and deformations for two-layer Timoshenko beams glued by a viscoelastic interlayer under uniform transverse load is presented.The standard linear solid model is employed to simulate the viscoelastic characteristics of the interlayer, in which the memory effect of strains is considered. Themechanical behavior of each layer is described by the first-order shear deformation theory (FSDT). By means of the principle of minimum potential energy, a group of equations for displacements and rotation angles are derived out.The final solution is obtained by conducting the Laplace transform and the inversion of Laplace transform to the equation group. Numerical comparison shows that the present solutions and the finite element results are in good agreement. It is shown that the present results are more accurate than those obtained from the Euler-Bernoulli beam theory, especially for thick beams. And the present solutions can accurately describe the variation of stresses and deformations of the beam with the time, compared with those ignoring the memory effect of strains. Finally, the effects of the geometric parameters and material properties of the interlayer on stresses and deformations of the beam are studied in detail.


Introduction
Owing to excellent combination advantages, layered composite structures are extensively used in civil engineering.The common applications are steel-concrete composite beams [1,2], wood-concrete composite panels [3,4], fiber reinforced concrete [5,6], laminated glasses [7,8], and so on.The existing analysis indicates that the interface connection property has considerable impact on the stresses and deformations in the layered structures [9,10].Behr et al. [11] studied the temperature effect on behavior of laminated glass.Galuppi and Royer-Carfagni [12] studied the effective width of equivalent slab for composite beam with shear connectors.In practical engineering, the adhesives are sometimes used to connect several structural components to form a composite structure.For the viscoelastic materials, the dependence of the stress state on the strain history is intrinsic.This study focuses on the mechanical behavior of two-layer beams glued by adhesives.The adhesives possess viscoelastic characteristics in nature, and thus the mechanical properties of the layered beams are time-varying and even subjected to static loads.
As is known to all, the Euler-Bernoulli (EB) beam theory assumes that the cross sections in the beam remain plane and perpendicular to the neutral axis during deformation.Based on this theory, several studies can be found for layered beams glued by viscoelastic interlayer.Zhang and Wang [13] analyzed the viscoelastic property of concrete beams reinforced by fiber reinforced polymers and gave the analytical solutions by the use of the numerical Laplace transformation method.Lei [14] studied the dynamic characteristics of damped viscoelastic nonlocal EB beams with different boundary conditions by the use of the transfer function method.Wang [15] presented the analytical solutions for bending behavior of viscoelastic beams under quasi-static loading.Bending and buckling behaviors of layered beams glued by viscoelastic interlayer were investigated by Galuppi and Royer-Carfagni [16,17].In their report, two kinds of analytical models for the interlayer were proposed: the fully viscoelastic model considering the memory effect of strains and the quasi-elastic model ignoring the memory effect of strains (the viscoelastic material is approximately modelled as an "aging" material).Li [18] investigated the time-varying behaviors of layered functionally graded beams with viscoelastic interlayer by using the analytical Laplace transformation.However, the transverse shear deformation is ignored in the EB beam theory, which will lead to imprecise results when the transverse shear deformations are considerable, such as deep beams.In order to consider the transverse shear deformations, the first-order shear deformation theory (FSDT) was proposed.This theory assumes that the transverse shear strain exists and is invariable along the beam thickness.We call such a beam as the Timoshenko beam.Based on the FSDT theory, Lenci and Clementi [19] studied the natural frequency of layered beams sandwiching an elastic interlayer.Murakami [20] presented a laminated Timoshenko theory which considers the interlayer slip in the beam.Wang [21] studied the vibration and damping effect in a three-layered composite annular plate with a viscoelastic interlayer based on the FSDT theory.Bardella [22] studied the reliability of FSDT model for sandwich beams.Mattei and Bardella [23] and Tessler et al. [24] developed the structural analysis for sandwich beams based on the zigzag model.Frostig et al. [25] studied the behavior of sandwich beams based on the high-order theory.The investigation of bending response for functionally graded viscoelastic sandwich beam with elastic core resting on Pasternak's elastic foundations was presented by Zenkour [26].Arikoglu [27] dealt with the vibration of a three-layered composite beam with a viscoelastic core by using the differential transform method.The analytical solutions of stresses and displacements of twolayer Timoshenko beams connected by viscoelastic interlayer, whose memory effect of strains was ignored, were studied by Wang et al. [28].Aenlle and Pelayo [29,30] studied the dynamic effective thickness of laminated-glass beams and plates.Serpilli and Lenci [31][32][33] developed the asymptotic expansion method to study the mechanical properties of three-layer beams and plates with soft adhesive.
In the present study, we extend the study from [16] on the slender layered beams with viscoelastic interlayer to the thick layered beams with viscoelastic interlayer.The motivation is to analyze the shear effect of length-to-thickness ratio on the deformations of the layered beams.An analytical solution based on the FSDT is developed for a two-layer Timoshenko beam glued by epoxy.The memory effect of the strains of the interlayer material is considered in the analysis.A very simple model, the standard linear solid model (SLSM), is used to model the viscoelasticity of the epoxy interlayer.The deformation and rotation angle of each beam layer are described in terms of the FSDT.The viscoelastic property in the interlayer is simulated by the standard linear solid model.By means of the principle of minimum potential energy and the Laplace transformation, the solutions of deformations and rotation angles are obtained analytically.

Analytical Model
Consider a simply supported layered beam of length  and width  as shown in Figure 1, consisting of two external elastic layers with thicknesses ℎ 1 and ℎ 2 adhesively glued by a thin epoxy with thickness h, subjected to transverse distributed load (, ).H is the distance between the geometric axes of upper and lower layers.A Cartesian coordinate system is established with origin at the bottom surface of the beam.The elastic moduli of two external layers are  1 and  2 , and Poisson's ratios are  1 and  2 , respectively.For isotropic materials, the shear modulus in the beam layers is expressed by   =  /2(1+  ), i=1, 2. The shear modulus of the epoxy interlayer is time-dependent and denoted by () [34].
. .Assumptions.The present study is carried out based on the following assumptions: (i) The strain and displacement in the beam are small.
(ii) The interlayer is very thin with respect to the external layers.The shear stress is constant along its thickness.
(iii) The transverse displacements and rotation angles of external elastic layers are the same, but the axial displacements about the geometric axis of each layer are different.
(iv) Since the interlayer, made of viscoelastic adhesives, is relatively soft than the beam layer, the longitudinal normal stress and strain in the interlayer are negligible.
. .Basic Equations.According to the first-order shear deformation theory (FSDT), the longitudinal displacement u(x, z, t) and the transverse displacement w(x, z, t) of the layered beam are given by  (, , ) where ( The longitudinal normal stress   and the transverse shear stress   in the beam are further obtained by where  1 and  2 are, respectively, the shear coefficients in FSDT for the upper and lower beam layers.For beams with rectangular cross-section, the shear coefficients can be given as [35] where  is Poisson's ratio of materials of the upper and lower elastic layer.We assume that the internal interlayer is very thin with respect to the external plies.This leads to constant shear stress along the thickness of the interlayer.For an elastic material, the longitudinal shear strain  0 (, ) and shear stress  0 (, ) in the interlayer can be expressed by the deformation and rotation angle in the upper and lower beam layers [36], as follows: where  is the distance between the geometric axes of the upper and lower layers and  = ℎ 1 /2 + ℎ + ℎ 2 /2.() is the shear modulus of the interlayer.
In Cartesian coordinates, the strain energy in the beam can be written as +   (, , )   (, , ) where  is the cross-sectional area of the beam.The energy of the system is given by where  is the Lagrange density function, which is where  1 and  2 are the extensional rigidities,  1 and  2 are the bending rigidities for upper and lower beam layers, and  1 is the shearing rigidity of the whole beam.These parameters can be expressed as Lagrange's equations for the beam are Equilibrium equations can be obtained by substituting ( 9) into (11), as follows: Considering the memory effect of strains in the interlayer, the constitutive equation for viscoelastic interlayer can be obtained by the Boltzmann superposition principle [37], as follows: It can be seen that the stresses at one point not only depend on the strains at that moment, but also depend on the whole strain history.According to the Riemann-Stieltjes convolution [38,39], (15) can be rewritten as  0 (, ) =  0 (, ) *  () (16) in which * means convolution.

Analytical Solution
. .Formulation of Solution.The boundary conditions of simply supported beams at  = 0,  are given as Here, we expand the displacements, rotation angles, and applied loads of the beam into Fourier series, as follows: where   (),   (),   (), and   () are the unknown Fourier coefficients and   = /.By substituting ( 21)-( 23) into ( 17)- (19), we obtain Several models, which can comprehensively describe the viscoelastic properties of materials, were developed for polymers [16,40].It is found that the standard linear solid (SLS) model is suitable for simulating the viscoelastic property of some adhesives such as epoxy [34].The SLS model, as shown in Figure 2, consists of a Maxwell Model and a Hookean spring in parallel.Notations  and  0 represent the viscosity coefficient and the relaxation shear modulus and G ∞ is the long-term shear modulus.By using this model, the timedependent shear modulus G(t) in the interlayer is given by where  = / 0 is the relaxation time.The shear stress  0 (, ) of the viscoelastic interlayer is further expressed by . .Laplace Transformation.By conducting the Laplace transforms to ( 24)-( 27), one has in which s is the Laplace transform variable, () =  ∞ / +  0 /( + 1/).The solution of (30) is () = −  5  ()   () where . .Determination of Unknowns.Making the inverse of Laplace transforms to (31)- (34) gives The analytical expressions of coefficients in the above equations are given in the Appendix.
Substituting these coefficients back to (21), ( 22), (3), and (29), the final expressions of the displacements, rotation angles, longitudinal normal stress of the beam, and the shear stresses in the interlayer can be obtained, respectively.

Numerical Examples
. .Convergence Study.Consider a simply supported beam subjected to the uniform load q=1 N/mm 2 .The two external  3. It can be seen from Figure 3 that the stresses and displacements rapidly converge to constants with the increase of .When M⩾30, a satisfied accuracy has been obtained.Thus, the terms of Fourier series is fixed at M=30 in the following studies unless stated.
. .Comparison Study.The present results (TFV) based on the Timoshenko beam theory are compared with those (EFV) based on the EB beam theory proposed by Galuppi and Royer-Carfagni [16] and those (FE) obtained from the finite element solutions by using the software ANSYS.The parameters of the beam are  1 = 2 =70 Gpa,  0 =470.529Mpa, G ∞ =0.471 Mpa, =1 s,  1 = 2 =0.3, ℎ 1 =ℎ 2 =50 mm, h=0.1 mm, and b=30mm, respectively.The load distribution is () = sin(/) N/mm 2 and the length  of the beam is variable.In the FE modelling, the viscoelastic property of the interlayer is simulated by VISCO-88 element with four nodes, which can consider the memory effect of viscoelasticity.The PLANE-183 element with four nodes is used for the external layers.
Considering the thickness of the interlayer only to be 0.1 mm, the length and width of rectangular elements for interlayer are fixed at 0.5 mm and 0.1 mm, respectively.Considering the external layers are much thicker than the interlayer, the length and width of rectangular elements for external layers are fixed at 0.5 mm and 2.5 mm, respectively.Taking the beam with the length-to-thickness ratio to be 10 as example, 2000 elements for the interlayer and 80000 elements for the external layers should be used in the FE computations.Table 1 gives the comparisons of mid-span deflection   between the TFV solutions with the EFV and FE ones for different length-tothickness ratios /ℎ 0 , where ℎ 0 is the total thickness of the whole beam, ℎ 0 =ℎ 1 +h+ℎ 2 .Three different time values, t=10 s, 100 s, 1000 s, are considered, respectively.It can be seen from Table 1 that there are good agreements between the TFV results and the FE results.The maximum error is less than 0.5641% for all cases.Furthermore, it is seen that the TFV results coincide with the EFV ones well for slender beams; however, the difference significantly increases in the decrease of /ℎ 0 , especially at the initial time of applying load.The maximum error of   reaches 16.806% for /ℎ 0 =2.5 when the load is at first applied to the beam.The maximum error of   reaches 7.075% for /ℎ 0 =2.5 when the interlayer is fully relaxed.Figure 4 gives the comparisons of   between the TFV results with the EFV and FE ones when t=10000 s with respect to length-to-thickness ratio /ℎ 0 .It can be seen from Figure 4 that, for the small length-to-thickness ratios l/ℎ 0 , the TFV results obviously have a better accuracy than the EFV ones.Tables 2 and 3 give the comparisons of the maximum longitudinal normal stress   and the maximum shear stress  0 between the TFV solutions with the EFV and FE ones for different length-to-thickness ratios /ℎ 0 , respectively.It can be seen from Tables 2 and 3 that the results from the TFV solutions are the same as those from the EFV solutions.This can be reasonably explained.For a simply supported layered beam considered here, the shear forces and moments at cross-section of the beam can be directly determined by only considering the force equilibrium.It is well known that both the Euler-Bernoulli beam theory and the Timoshenko beam theory follow the plane cross-section (a) t=300 s t=1000 t=1000 t=1000 t=1000  assumption.Therefore, the normal stress distributions on the cross-section for two different beam theories should be identical, if the moments on the cross-section are the same.Moreover, it can be easily deduced that the rotational angles and the longitudinal displacements of the simply supported layered beam are all independent of the shear coefficient, as seen in ( 31)- (33).Therefore, the shear stresses in the interlayer from the Timoshenko beam theory are also the same as those from the Euler-Bernoulli beam theory.Such a conclusion is valid only for statically determinate beams; however, it is invalid for statically indeterminate beams.Furthermore, we compare the present results (TFV) with those (TFV (NME)) proposed by Wang et al. [28] and those obtained from the finite element (FE) solution by using the software ANSYS.The TFV (NME) solutions are also based on the FSDT; however, the memory effect of strains in the interlayer is neglected; i.e., the viscoelastic material is approximately modelled as an "aging" material.We consider a simply supported two-layer Timoshenko beam glued by a viscoelastic interlayer and subjected to the uniform load q=1 N/mm 2 .The parameters of the beam are Mpa,  0 =500 Mpa, h=0.5 mm, =0.5 s, respectively.Figure 5 shows the comparisons of   ,  0 , u, and  from three different solutions with respect to time .It can be seen from Figure 5 that the present results (TFV) are in agreement with the FE ones.However, there are considerable differences between the present results (TFV) and those (TFV(NME)) without considering the memory effect of strains in the interlayer.Therefore, the results ignoring the memory effect of strains in the interlayer cannot accurately describe the varying of stresses and displacements of the beam with the time.The same responses have been obtained and recorded by Galuppi and Royer-Carfagni [16,36] for beam with external plies modelled as EB beams.
Figure 6 shows the distributions of stresses and displacements at the time t=10, 100, and 1000 s, respectively, for h=0.5 mm,  0 =500 Mpa, G ∞ =0.5 Mpa, and =0.5 s.It can be seen from Figure 6 that (1) the absolute values of the longitudinal normal stress   at the outside surfaces of the beam and the axial displacement  at the ends of the beam increase with time; (2) the transverse shear stress   in the external elastic layers is almost no change with the time; (3) the mid-span deflection  increases with time; (4) the maximum value of the shear stress  0 in the interlayer appears at the ends of the beam and decreases with time.Figure 7 shows the effects of parameters h,  0 , G ∞ , and  on the mid-span deflection   with respect to the time.It can be seen from Figure 7 that the mid-span deflection   increases with time and tends to be constants for all cases.
increases with the increase of glue thickness h; however, it decreases with the increase of long-term shear modulus G ∞ .The relaxation time  has no effect on the values of   at the limited times t=0 and t=∞.Further consider the layered beam with a viscoelastic interlayer as mentioned above.The thickness of the interlayer is h=0.1 mm.The mid-span deflection   (at x=500 mm) with respect to the time for different viscoelasticity parameters  0 and G ∞ are shown in Figure 8.The monolithic limit, which corresponds to the static solution of  → ∞, and the layered limit, which corresponds to the static solution of G=0, are also plotted for references.Physically, for the layered limit the beam is composed of free-sliding plies and for the monolithic limit the beam is composed of no relatively slipping plies [16,36].It is seen from Figure 8 that the deflections of the beam for all cases are between the monolithic limit and the layered limit.Namely, the deflection of the beam with a viscoelastic interlayer is always larger than its monolithic limit, but smaller than its layered limit.Moreover, it can be seen from Figure 8 that  0 has the significant influence on the short-term response of deflection of the layered beam and G ∞ has significant influence on the long-term response of deflection of the layered beam.

Conclusion Remarks
According to the FSDT, this paper studies the stresses and deformations of simply supported two layers of Timoshenko beams glued with a viscoelastic interlayer and subjected to arbitrary loads.The viscoelasticity of the interlayer is described by the standard linear solid model and the memory effect of strains in the interlayer is considered.Taking the sinusoidal load and uniform load for example, the effects of external layer thickness and the interlayer material properties on deflection of the laminated beam are checked in detail.The results show that there are good agreements between the present results and those obtained with the Euler-Bernoulli beam theory when the length-to-thickness ratio is large; however, the errors between the two results increase in the decrease of length-to-thickness ratio.Moreover, the error of deflection from the Euler-Bernoulli beam theory at the initial time of applying load is significantly larger than that when the interlayer is fully relaxed.Therefore, the present results have a better accuracy than those in terms of the Euler-Bernoulli beam theory.The effect of interlayer material property on the deflection of the laminated beam is similar to those obtained from the Euler-Bernoulli beam theory [16,36].

Appendix
The coefficients in (36)

Figure 3 :
Figure3: The convergences of   ,  0 , u, w with respect to the series term  for t=300 s and t=1000 s, respectively.

Figure 5 :
Figure 5: Comparisons of   ,  0 ,  and  from three different solutions with respect to the time .

Figure 6 :
Figure 6: Distributions of stresses and displacements of the laminated beam.

Figure 7 :
Figure 7: Effects of parameters ℎ,  0 ,  ∞ , and  on the mid-span deflection   with respect to the time .
Figure 1: Simply supported two-layer beam glued by a viscoelastic interlayer.
1 (, ) and  2 (, ) are displacements at geometric axes of upper and lower layers, respectively; (, ) is the rotation angle of transverse cross-section of the layered beam; (, ) is the deflection of the layered beam; t denotes time. 1 and  2 are the distances from the geometric axes of upper and lower layers to the bottom surface of the beam, and  1 = ℎ 1 /2 + ℎ + ℎ 2 ,  2 = ℎ 2 /2.The longitudinal normal strain   and the transverse shear strain   in the beam are given by

Table 1 :
Comparisons of maximum deflection   from the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios /ℎ 0 and time values t.