Theoretical solution for the axial vibration of functionally graded double-lap adhesive joints

Abstract

Adhesively bonded joints have widely been used in a number of engineering applications, owing to their improved mechanical performance as compared with other mechanical joining techniques, such as rivets or bolts. In this study, a theoretical solution of double-lap joints is established, with functionally graded adhesive and isotopic adherends, under harmonic loads. Assuming a parabolic distribution of the adhesive shear modulus along the overlap length, the analytical harmonic solution is expressed in terms of the solution of the nonlinear Heun differential equation. Furthermore, a two-dimensional finite-element model was developed to validate the analytical solution. We show that using softer adhesive at the edges of the bond line leads to less stress concentration. We also conclude that the adhesive’s shear modulus gradient along the bond line should be greater as the ratio of the adhesive’s shear stiffness to the adherends’ shear stiffness increases. An equation has also been established to determine the optimal shear modulus gradient in terms of the stiffness ratio.

Keywords

Balanced double-lap joints functionally graded adhesive confluent Heun function Heun differential equation closed-form solution stress concentration

1. Introduction

Adhesively bonded joints have been used extensively in automobile, aircraft, marine, and civil engineering to bond structural components as they have several advantages, compared with other bonding techniques. Mainly, bonded joints offer the possibility of assembling dissimilar material. Moreover, they lead to the least stress concentration compared with other bonding methods. The use of adhesive joints yields lightweight structures that can reduce energy consumption in transportation vehicles.

Bonded joints are the most vulnerable regions in the structure, no matter what type of bonding is used. Adhesive bonding induces the least stress concentration. However, the joint’s stress concentration is sufficiently severe for it to be an important point of onset of failure. Several methods have been applied to reduce the stress concentration and to smooth the stress distribution in a lap joint. One of these methods involves geometrical modification by rounding adherend corners or by modifying the joint end with spew fillet. Raphael [1] was the first to propose the mixed adhesive idea, with the aim of reducing the adhesive joint stress concentration. This could be achieved by using a brittle adhesive in the middle of the overlap and a ductile adhesive at the ends. Marques and da Silva [2] reported that a dual-adhesive double-strap joint is as strong as a double-strap joint employing one stiff adhesive. Moreover, the former is much more flexible than the latter. Working with single-lap joints, da Silva and Lopes [3] have found that mixed (brittle–ductile) adhesive joints have superior strength over adhesive joints with only one adhesive, either brittle or ductile.

Nowadays, there is a strong trend toward using functionally graded materials [4, 5]. They are mainly used to achieve a high strength-to-weight ratio. The main idea of using these advanced materials is to vary the mechanical properties of the adhesive gradually along the overlap length, in order to enhance the joint’s performance and realize cost-effective and lightweight structures [6]. More precisely, a stiff and brittle adhesive is used in the middle of the bond line; then, the adhesive’s flexibility and ductility is gradually increased while moving toward the bond line extremities. The mechanical properties of the adhesive were graded using an induction heating technique [7] and graded mixing of carbon nanoparticles [8]. Several works have been devised to approach this problem either theoretically, experimentally, or numerically. The reader is referred to Durodola [9] for a more comprehensive state-of-the-art review. In this work, we are mainly interested in developing a closed-form solution to the response of a functionally graded double-lap adhesive joint to harmonic loads. Thus, only theoretical works will be briefly discussed here.

Theoretical models that were already developed for adhesive joints with a uniform adhesive [10] led to partial or ordinary differential equations with constant parameters. Since functionally graded adhesives are used, the adhesive’s shear modulus and Young modulus start to depend on the position along the bond line. The former partial or ordinary differential equations with constant parameters change to partial or ordinary differential equations with variable parameters. This makes solving these differential equations more complex.

Carbas et al. [11] were interested in single-lap joints with functionally graded adhesive. They used the shear-lag model [12]. To solve the differential equations, they expressed the normal stress in the substrates as power series of the axial coordinate. They considered function types for the variation of the adhesive’s shear modulus along the bond line. The results of this study show that varying the mechanical properties along the overlap will give a smooth stress distribution and low stress concentrations at the ends of the overlap. Also dealing with single-lap joints, Stein et al. [13] have provided a theoretical solution for a functionally graded adhesive layer based on the theoretical approach of Goland and Reissner [14]. Consequently, the study takes bending moments into consideration. Hence, the peel shear stresses are also considered in their theoretical model. To solve the differential equations with variable parameters, Stein et al. [13] have also expanded the solutions in terms of power or Taylor series. The theoretical approach can work with various types of adhesive joint, such as single-lap joints, T joints, corner joints, and L joints [15]. Several profiles have been considered for the variation of the adhesive shear modulus along the bond line. The cosine distribution was reported to yield the best uniform stress distribution [13]. Stein et al. [16] have presented a comparative study between four theories of single-lap joints. However, the solution was expressed as a power series each time. Stein et al. [17] also considered thermal stresses but using a homogenization procedure.

In this work, we are interested in developing a theoretical solution for a functionally graded adhesive double-lap joint. Closed-form theoretical solutions are of great importance, giving valuable insight into the most influential parameters and yielding a better understanding of the physical background. Volkersen [12] was the first to deal with closed-form solutions for adhesive joints. This was done by assuming that the adherend deformation occurs along the axial axis and that the adhesive is only deformed under a shear state. However, in this analytical model, the bending effect and the transverse force caused by the eccentric force in single-lap joints were not considered. Goland and Reissner [14] were the first to consider these effects. They introduced a transverse force factor ( $k^{'})$ and a bending moment factor ( $k)$ . Oplinger [18] has studied the influence of the adherends’ deflection on the shear stress distribution of the adhesive layer. Tsai et al. [19] improved the shear-lag model of Volkersen by considering that the shear deformation in the adherends is linear through the thickness. This approach has since been applied in several applications [20, 21]. Adams and Mallick [22] assumed that the axial normal stress is linear through the thickness. Though the model is general and includes bending, shearing, hygrothermal stresses, and nonlinear adhesive behavior, the solution is calculated numerically and a closed-form solution is hard to obtain. Recently, Jiang et al. [23] derived a solution for unbalanced composite double-lap joints but accounting for shear and peeling stresses. Weißgraeber et al. [24] derived an analytical model for single-lap, double-lap, T, and L adhesive joints, considering plane-strain framework. They accounted for shear deformation using a first-order shear deformation assumption. For a more thorough review, the reader is referred to da Silva et al. [10].

In this work, we are interested in deriving the response of functionally graded balanced double-lap adhesive joints under harmonic loads. To the best of the authors’ knowledge this has never been undertaken before. Consequently, the theoretical developments will be undertaken based on the model of Tsai et al. [19] and assuming isotropic adherends. The extension of this work to orthotropic materials is not tedious. However, the examples that are considered here are limited to the case of isotropic materials. Indeed, the simple shear-lag model [12] and the improved shear-lag model [19] are the only methods that have been extended to include dynamic loads, as reviewed by Othman [25]. Sato [26] has derived an analytical model based on Volkersen’s study dealing with the impact loads of a semi-infinite adhesive joint. Gafar et al. [27] have solved Volkersen’s model for unbalanced single-lap joints under harmonic loads. Challita and Othman [28] have extended the improved shear-lag model of Tsai et al. [19] to provide the harmonic response of double-lap joints. Their study argued that the shear deformation of the adherends can be neglected if the axial stiffness of the adherends is much greater than the adhesive shear stiffness. This work was later improved by Almitani and Othman [29] to include damping of the adhesive and adherends. More recently, Challita [30] derived a theoretical–numerical solution to study the effects of voids on the natural frequencies of the adhesive joint. Wang et al. [31] also extended the model to account for peel stresses. Hazimeh et al. [32] also used the improved shear-lag model to obtain the stress response under impact loads at the adhesive layer edge. In comparison with the finite-element solution, their model has shown closer results than Sato’s model.

Therefore, the aim of this study is to derive an analytical solution for the harmonic response of balanced double-lap joints with functionally graded adhesives. This work can be used in future works for the optimization and design of this type of adhesive joint.

2. Theoretical model

2.1. Problem statement

The investigation of the dynamic response of the balanced double-lap joint and the adhesive’s shear stress distribution along the overlap joint will be established. As the double-lap joint is balanced, symmetry is considered and only half of the joint is studied, as shown in Figure 1. Furthermore, the analytical solution is based here on the work of Challita and Othman [28]. However, a major difference between this study and the study presented in [28] is the fact that the adhesive bond line will be considered as a functionally graded layer. Namely, the adhesive’s shear modulus is considered to be varying along the bond line, i.e., along the $x$ or axial direction. To be more precise, the adhesive’s shear modulus is represented by a symmetric function $G_{a} (x)$ that depends on variable $x$ . As recommended in several references [2, 4, 15] the adhesive is considered to be stiffest in the middle of the joint and softest at the edges (Figure 1).

Figure 1.

Adhesive shear modulus varying gradually along the $x$ -direction (half of the double-lap joint is represented).

In this work, only the overlap region is modeled, for the sake of simplicity (Figure 1). The left edge of the upper adherend is assumed fixed, whereas the right edge of the lower adherend is subjected to a unit harmonic load. Moreover, the adherends and the adhesive kinematics are assumed to take effect only in the $x$ -direction. The shear stress of the adherends is assumed to have a linear variation through the thickness [19, 28]. Conversely, a constant shear stress is assumed through the adhesive’s thickness. Thus, the adhesive shear stress is a proportional function of the displacements at the adherend–adhesive interfaces. The thickness of the outer adherend is half that of the inner adherend. Also, the same material is used for all adherends. Thus, the current work is limited to balanced double-lap joints.

2.2. Establishment of the differential equations

In this section, the procedure for obtaining the differential equation of the adhesive shear stress $(τ_{a})$ is detailed. Following Tsai et al. [19], the shear stress is assumed to be linear through the thickness of the adherends. Considering that the upper and lower joint surfaces are stress free, the adherend shear stresses are expressed as

τ_{i} = \frac{y}{t_{s}} τ_{a}

(1)

and

τ_{o} = τ_{a} - \frac{y - (t_{a} + t_{s})}{t_{s}} τ_{a},

(2)

where $τ_{i}$ , $τ_{o}$ and $τ_{a}$ are the shear stresses in the lower adherend, upper adherend, and adhesive layer, respectively, $t_{a}$ is the adhesive thicknesses, and $t_{s}$ corresponds to the outer adherends’ thickness and also half of the inner adherend thickness. Assuming small strain deformation and elastic behavior of the adherends, the axial displacements are established by first dividing equations (1) and (2) by the shear modulus of the adherends and then integrating with respect to the transverse coordinate $y$ . After simplification, the axial displacements read

u_{i} = u_{ia} + \frac{y^{2} - t_{s}^{2}}{2 t_{s} G_{s}} τ_{a}

(3)

and

u_{o} = u_{oa} - \frac{{(y - 2 t_{s} - t_{a})}^{2} - t_{s}^{2}}{2 t_{s} G_{s}} τ_{a},

(4)

where $G_{s}$ is the shear modulus of the adherends, $u_{i}$ and $u_{o}$ are the axial displacements of the inner and outer adherends, respectively, $u_{ia}$ is the displacement at the inner adherend–adhesive interface, and $u_{oa}$ is the displacement of outer adherend–adhesive interface. To obtain the axial forces, equations (3) and (4) are first differentiated with respect to $x$ , to obtain the axial strains. Next, they are multiplied by the Young modulus of the adherends to get the axial stresses. Finally, the expressions of axial stresses are integrated along the adherends’ thickness. After simplification, the axial forces can be written as

N_{i} = E (t_{s} \frac{d u_{ia}}{d x} - \frac{t_{s}^{2}}{3 G_{s}} \frac{d τ_{a}}{d x})

(5)

and

N_{o} = E (t_{s} \frac{d u_{oa}}{d x} + \frac{t_{s}^{2}}{3 G_{s}} \frac{d τ_{a}}{d x}),

(6)

where $E$ is the Young modulus of the adherends.

Now, by applying the Newton’s second law on a $d x$ -long infinitesimal adherend element and considering that the solution is harmonic and has the same frequency as the applied load, the gradients of the axial forces are expressed as [28]

\frac{d N_{i}}{d x} = - ρ_{s} t_{s} ω^{2} u_{ia} - (1 - \frac{ρ_{s} t_{s}^{2} ω^{2}}{3 G_{s}}) τ_{a}

(7)

and

\frac{d N_{o}}{d x} = - ρ_{s} t_{s} ω^{2} u_{oa} + (1 - \frac{ρ_{s} t_{s}^{2} ω^{2}}{3 G_{s}}) τ_{a},

(8)

where $ω$ and $ρ_{s}$ are the load’s angular frequency and the substrates’ material density, respectively.

The adhesive’s behavior is assumed elastic. Moreover, the adhesive’s shear stress is assumed constant through the thickness. Hence, it can be expressed in terms of the displacements at the adhesive–adherend interfaces. Namely,

τ_{a} = \frac{G_{a} (x)}{t_{a}} (u_{oa} - u_{ia}),

(9)

where $t_{a}$ is the adhesive layer thickness and $G_{a} (x)$ is the adhesive’s shear modulus, which varies along the bond line. It is worth noting that equations (1) to (9) are similar to the ones established by Challita and Othman [28]. However, the subsequent analytical developments are substantially different, as the adhesive’s shear modulus $G_{a} (x)$ is depending on the axial coordinate $x$ .

To obtain the differential equation governing the shear stress $τ_{a}$ , equations (5) and (6) are first substituted in equations (7) and (8), respectively. Subsequently, the obtained equations are subtracted one from the other. Finally, considering equations (9) and simplifying yields

\frac{d^{2} τ_{a}}{d x^{2}} - \frac{d τ_{a}}{d x} (\frac{2 \frac{G^{'} (x)}{G_{a} {(x)}^{2}}}{\frac{1}{G_{a} (x)} + \frac{2}{3} \frac{t_{s}}{G_{s} t_{a}}}) + τ_{a} (\frac{\frac{ρ_{s} ω^{2}}{E G_{a} (x)} - \frac{2}{E t_{s} t_{a}} + \frac{2}{3} \frac{ρ_{s} t_{s} ω^{2}}{G_{s} E t_{a}} - \frac{G (x)}{G_{a} {(x)}^{2}} + 2 \frac{G^{'} {(x)}^{2}}{G_{a} {(x)}^{3}}}{\frac{1}{G_{a} (x)} + \frac{2}{3} \frac{t_{s}}{G_{s} t_{a}}}) = 0 .

(10)

This is a homogeneous second-order differential equation in terms of $τ_{a}$ . However, its parameters are highly dependent on $x$ . Solving this equation is not straightforward and is mainly dependent on the expression of the shear modulus distribution $G_{a} (x)$ .

Stein et al. [13] have argued that the cosine distribution of the shear modulus leads to the best smoothing of the stress concentration within the adhesive layer. Consequently, the adhesive’s shear modulus distribution is considered, in this work, as the second-order approximation of the cosine distribution. More precisely,

G_{a} (x) = G_{o} (1 - θ \frac{x^{2}}{l^{2}}),

(11)

where $G_{o}$ is the shear modulus at the middle of the bond line, $θ$ is a dimensionless constant, and $l$ is the overlap length. Equation (11) has two main advantages. First, it simplifies the hard task of solving equation (10). Second, it gives a good approximation of the optimal cosine distribution, as $\sqrt{θ} (x / l)$ should be less than one to have positive shear modulus.

The dimensionless constant $θ$ allows the shear modulus $G_{a} (x)$ to have a number of distributions and $l$ is the overlap length. So, by changing $θ$ , the gradient of the adhesive layer’s stiffness can be increased or decreased and then the shear stress distribution and profile can be optimized, as well as the frequency response. The value of $θ$ according to equation (11) must be zero and four; this is explained as follows:

If $θ$ is equal to zero, the shear modulus is constant, the adhesive layer is uniform, and equation (10) is equivalent to the solution obtained by Challita and Othman [28].

If $θ$ is less than zero, the shear modulus has larger values at the edges. Specifically, the adhesive is stiff at the edges and soft in the middle. This opposes the recommendations of Raphael [1], da Silva and Lopes [3], and Stein et al. [13]. Having $θ$ negative leads to greater shear stress concentration at the edges.

If $θ$ is greater than or equal to four, the shear modulus is less than or equal to zero in some parts of the adhesive layer, which is nonphysical.

Now to go forward with equation (10), we substitute it in equation (11). After simplification, we obtain

\frac{d^{2} τ_{a}}{d x^{2}} - \frac{d τ_{a}}{d x} (\frac{\frac{4 θ x}{l^{2} {(1 - \frac{θ x^{2}}{l^{2}})}^{2}}}{\frac{1}{(1 - \frac{θ x^{2}}{l^{2}})} + κ}) + τ_{a} (\frac{\frac{l^{2} ξ^{2}}{(- θ x^{2} + l^{2})} - μ + κ ξ^{2} - \frac{2 θ l^{2}}{{(- θ x^{2} + l^{2})}^{2}} + 2 \frac{8 θ^{2} x^{2} l^{2}}{{(- θ x^{2} + l^{2})}^{3}}}{\frac{1}{(1 - \frac{θ x^{2}}{l^{2}})} + κ}) = 0,

(12)

where

ξ^{2} = (\frac{ρ_{s} ω^{2}}{E}), κ = (\frac{2}{3} \frac{G_{o} t_{s}}{G_{s} t_{a}}), μ = (\frac{2 G_{o}}{E t_{s} t_{a}}),

and $ξ$ is the wavenumber of one-dimension waves propagating in the adherends, $G_{o} t_{s} / G_{s} t_{a}$ is the ratio of adhesive shear stiffness to adherends’ shear stiffness in the middle of the joint and $G_{o} / E t_{s} t_{a}$ is the ratio of adhesive shear stiffness to adherends’ axial stiffness, also in the middle of the joint.

Equation (12) gives the governing differential equation of the shear stress $τ_{a}$ . Considering equation (9), it is also the governing equation of $(u_{oa} - u_{ia})$ , recall that $u_{ia}$ is the displacement of the adhesive–inner adherend interface and $u_{oa}$ is the displacement of the adhesive–outer adherend interface. To be able to calculate $u_{ia}$ and $u_{oa}$ separately, one more equation is needed. To this aim, equations (5) and (6) are substituted in equations (7) and (8), respectively. Subsequently, the obtained equations are summed. The shear stress is canceled from the solution and we obtain

\frac{d^{2} (u_{oa} + u_{ia})}{d x^{2}} + ξ^{2} (u_{oa} + u_{ia}) = 0 .

(13)

Solving equation (12) yields the shear stress $τ_{a}$ . Solving equations (12) and (13) yields $u_{ia}$ and $u_{oa}$ . Substituting the solutions in equations (3) to (6) gives the axial displacements and axial forces.

2.3. General solutions

Equation (13) is much simpler than equation (12). It is a homogeneous, ordinary, linear second-order differential equation having constant coefficients. The general solutions can be expressed as

u_{oa} + u_{ia} = A e^{- i ξ x} + B e^{i ξ x},

(14)

where $A$ and $B$ are frequency-dependent functions that will be determined later using boundary conditions. Equation (12) is also a homogeneous, ordinary, second-order differential equation. However, the coefficients are variable, i.e., they depend on $x$ . Equation (12) has the form of the following general confluent Heun differential equation [33 –35]:

(\frac{d^{2}}{d x^{2}} y (x)) + \frac{((a + b + 1) x - c)}{x^{2} - x} (\frac{d}{d x} y (x)) + \frac{ab}{x^{2} - x} y (x) = 0 .

(15)

The confluent Heun differential equation appears when two or more regular singularities coalesce to form an irregular singularity [36]. The regular singularities in equation (12) are

{\frac{l}{\sqrt{θ}}, \frac{\sqrt{θ κ l (κ + 1)}}{θ κ}, - \frac{l}{\sqrt{θ}}, - \frac{\sqrt{θ κ l (ϑ + 1)}}{θ κ}}

(16)

and the irregular singularity is

{\infty} .

(17)

The general solution of equation (12) is established by using Maple. It can be written as

τ_{a} = C (θ x^{2} + l^{2}) f (x) + D x (θ x^{2} + l^{2}) g (x),

(18)

where

f (x) = HC (0, - \frac{1}{2}, 1, - \frac{l^{2} (κ + 1) (- ξ^{2} κ + μ)}{4 κ θ^{2}}, \frac{(- l^{2} ξ^{2} + 2 θ) + l^{2} (- ξ^{2} + μ)}{4 θ κ}, \frac{θ κ x^{2}}{(κ + 1) l^{2}}),

(19)

g (x) = HC (0, \frac{1}{2}, 1, - \frac{l^{2} (κ + 1) (- ξ^{2} κ + μ)}{4 κ θ^{2}}, \frac{(- l^{2} ξ^{2} + 2 θ) + l^{2} (- ξ^{2} + μ)}{4 θ κ}, \frac{θ κ x^{2}}{(κ + 1) l^{2}}),

(20)

$HC$ is the confluent Heun function [34], and $C$ and $D$ are two frequency-dependent functions that are evaluated later using boundary conditions.

Using equations (9), (14), and (18), the displacements, at the adhesive–adherend interfaces $u_{ia}$ and $u_{oa}$ , are expressed as

u_{ia} = \frac{1}{2} A e^{- i ξ x} + \frac{1}{2} B e^{i ξ x} - \frac{1}{2} \frac{t_{a} (C (- θ x^{2} + l^{2}) f (x) + D x (- θ x^{2} + l^{2}) g (x))}{G_{o} (1 - \frac{θ x^{2}}{l^{2}})}

(21)

and

u_{oa} = \frac{1}{2} A e^{- i ξ x} + \frac{1}{2} B e^{i ξ x} + \frac{1}{2} \frac{t_{a} (C (- θ x^{2} + l^{2}) f (x) + D x (- θ x^{2} + l^{2}) g (x))}{G_{o} (1 - \frac{θ x^{2}}{l^{2}})} .

(22)

To obtain the average displacements of the adherends, equations (21) and (22) are substituted into equations (3) and (4). Then, the obtained equations are integrated along the y-axis. After simplification, the average displacement in the inner and outer adherends read

{\bar{u}}_{i} = \frac{1}{2} A e^{- i ξ x} + \frac{1}{2} B e^{i ξ x} - \frac{(- 2 t_{s} (θ x^{2} - l^{2}) G_{o} + 3 t_{a} l^{2} G_{s}) (C f (x) + D x g (x))}{G_{o} G_{s}}

(23)

and

{\bar{u}}_{o} = \frac{1}{2} A e^{- i ξ x} + \frac{1}{2} B e^{i ξ x} + \frac{(- 2 t_{s} (θ x^{2} - l^{2}) G_{o} + 3 t_{a} l^{2} G_{s}) (C f (x) + D x g (x))}{G_{o} G_{s}} .

(24)

2.4. Boundary conditions

The four frequency-dependent functions $A$ , $B$ , $C$ and $D$ are expressed by using the four boundary conditions of the lower and upper adherends. These four boundary conditions are as follows.

The left edge of the lower adherend is free. Then, $N_{i} (x = - l / 2) = 0$ .

Similarly, the right edge of the upper adherend is free. Then, $N_{o} (x = l / 2) = 0$ .

The right edge of the lower adherend is subjected to the external harmonic load. Thus, $N_{i} (x = l / 2) = P$ , where $P$ is the magnitude of the harmonic load.

The left edge of the upper adherend is cantilevered. So, ${\bar{u}}_{o} (x = - l / 2) = 0$ . In this work, this equation will be satisfied in two steps. First, an unknown reaction force $R$ is applied at the left edge of the upper adherend instead of the cantilevered end. Later, the reaction force is calculated to satisfy the cantilevered end. Therefore, the fourth boundary condition is expressed as $N_{2} (x = - l / 2) = 0$ .

Solving these four boundary conditions yields

A = \frac{i P e^{- \frac{1}{2} i ξ l} - i R e^{\frac{1}{2} i ξ l}}{t_{s} E ξ (e^{- \frac{1}{2} i ξ l} + e^{\frac{1}{2} i ξ l}) (e^{- \frac{1}{2} i ξ l} - e^{\frac{1}{2} i ξ l})},

(25)

B = \frac{i P e^{\frac{1}{2} i ξ l} - i R e^{- \frac{1}{2} i ξ l}}{t_{s} E ξ (e^{- \frac{1}{2} i ξ l} + e^{\frac{1}{2} i ξ l}) (e^{- \frac{1}{2} i ξ l} - e^{\frac{1}{2} i ξ l})},

(26)

C = \frac{3 G_{s} G_{o} (P + R) (κ + 1)}{t_{s} El κ (G_{o} (((- 4 + κ) K_{1} + 4 K_{2}) κ + 4 K_{2}) t_{s} - 6 G_{s} K_{1} t_{a} κ)},

(27)

and

D = \frac{6 G_{s} G_{o} (P - R) (κ + 1)}{(6 (κ + 1) ((θ - \frac{4}{3}) t_{s} G_{o} - 2 G_{s} t_{a}) K_{3} + κ θ (t_{s} (- 4 + θ) G_{o} - 6 G_{s} t_{a}) K_{4}) E t_{s} l^{2}},

(28)

where

K_{1} = HC' (0, - \frac{1}{2}, 1, - \frac{l^{2} (κ + 1) (- ξ^{2} κ + μ)}{4 κ θ^{2}}, \frac{(- l^{2} ξ^{2} + 2 θ) + l^{2} (- ξ^{2} + μ)}{4 θ κ}, \frac{1}{4} \frac{θ κ}{(κ + 1)}),

(29)

K_{2} = HC (0, - \frac{1}{2}, 1, - \frac{l^{2} (κ + 1) (- ξ^{2} κ + μ)}{4 κ θ^{2}}, \frac{(- l^{2} ξ^{2} + 2 θ) + l^{2} (- ξ^{2} + μ)}{4 θ ϑ}, \frac{1}{4} \frac{θ ϑ}{(ϑ + 1)}),

(30)

K_{3} = HC (0, \frac{1}{2}, 1, - \frac{l^{2} (κ + 1) (- ξ^{2} κ + μ)}{4 κ θ^{2}}, \frac{(- l^{2} ξ^{2} + 2 θ) + l^{2} (- ξ^{2} + μ)}{4 θ ϑ}, \frac{1}{4} \frac{θ κ}{(κ + 1)}),

(31)

K_{4} = HC' (0, \frac{1}{2}, 1, - \frac{l^{2} (κ + 1) (- ξ^{2} κ + μ)}{4 κ θ^{2}}, \frac{(- l^{2} ξ^{2} + 2 θ) + l^{2} (- ξ^{2} + μ)}{4 θ κ}, \frac{1}{4} \frac{θ κ}{(κ + 1)}),

(32)

and $HC'$ is the first derivative of the confluent Heun function [35].

Now, to match the cantilevered left end of the upper adherend we need to satisfy ${\bar{u}}_{o} (x = - l / 2) = 0$ . However, as mentioned earlier, the shear stress is assumed to have a linear variation through the thickness. Consequently, the adherends’ displacements vary parabolically through the thickness. Thus, the average displacement ${\bar{u}}_{2}$ is considered instead of $u_{2}$ [32]. Subsequently, the fourth boundary condition can also be expressed as ${\bar{u}}_{o} (x = - l / 2) = 0$ . Then, the reaction $R$ can be derived through the average displacement boundary condition. The average displacement ${\bar{u}}_{o}$ is determined in equation (24). Knowing the reaction force $R$ , the four frequency-dependent functions can be evaluated. Subsequently, the adhesive shear stress, the adherends’ axial displacements, and the forces can also be computed.

3. Numerical model

The finite-element method has been increasingly used for examining the behavior of joints. Indeed, it provides accurate analysis and results. A numerical analysis of the balanced double-lap joint with functionally graded adhesive was carried out using ANSYS. Owing to symmetry, only half of the geometry is considered. The numerical modeling was used to study the validation of the proposed analytical model by comparing the numerical and theoretical responses to harmonic loads. The simulations were undertaken based on a two-dimensional finite-element analysis. The adhesive and the material of the adherends was considered to be isotropic. Defining the adhesive as functionally graded material was the main difficulty in the numerical approach, since ANSYS software cannot define the adhesive’s shear modulus as a function in terms of the axial coordinate, as defined in equation (11), i.e.,

(G_{a} (x) = G_{o} (1 - θ \frac{x^{2}}{l^{2}})) .

To overcome this problem, the adhesive was divided into tiny segments to get a suitable distribution compared with the desired theoretical profile (Figure 1). More precisely, the adhesive was divided into $25$ segments. Each segment has a constant shear modulus, which depends on the segment’s position and the assumed shear modulus graded function $G_{a} (x)$ . In this work, three simulations were implemented with three different shear modulus distributions, by considering three different values of the constant $θ$ . Specifically, the constant $θ$ was considered equal to $10^{- 4}$ , 2, and 3.9. The first value of $θ$ , i.e., $θ = 10^{- 4}$ was considered to model the case of a uniform adhesive. If $θ = 10^{- 4}$ , $θ << 1$ , and consequently $G_{a} (x) \approx G_{0}$ . This case was mainly studied to validate the current solution based on the confluent Heun functions with respect to the conventional solution of the linear differential equation used in Challita and Othman [28]. The two other values were considered to model a moderate and extreme shear modulus variation, as $0 < θ < 4$ . For all three simulations, the shear modulus in the middle of the bond line was considered as $G_{o} = 2$ GPa. The values of the shear modulus of the tiny segments are given in Table 1.

Table 1.

Shear modulus values along the overlap that satisfy the assumed shear modulus graded function: G_o = 2 GPa.

Position (m)	Shear modulus, G_a (GPa)
	θ = 0	θ = 2	θ = 3.9
−0.024	2	1.0784	0.20288
−0.022		1.2256	0.48992
−0.02		1.36	0.752
−0.018		1.4816	0.98912
−0.016		1.5904	1.20128
−0.014		1.6864	1.38848
−0.012		1.7696	1.55072
−0.01		1.84	1.688
−0.008		1.8976	1.80032
−0.006		1.9424	1.88768
−0.004		1.9744	1.95008
−0.002		1.9936	1.98752
0		2	2
0.002		1.9936	1.98752
0.004		1.9744	1.95008
0.006		1.9424	1.88768
0.008		1.8976	1.80032
0.01		1.84	1.688
0.012		1.7696	1.55072
0.014		1.6864	1.38848
0.016		1.5904	1.20128
0.018		1.4816	0.98912
0.02		1.36	0.752
0.022		1.2256	0.48992
0.024		1.0784	0.20288

The element SURF153 was used for both adherends and adhesive. The two-dimensional plane-strain assumption was considered in this analysis. A mesh size of 0.25 mm was used (Figure 2). The left edge of the upper adherend was fixed. Also, a unit force was applied to the right edge of the lower adherend.

Figure 2.

The two-dimensional finite-element mesh.

Regarding the geometry, the adhesive layer had a thickness of $0.5$ mm and a length of $50$ mm. The Poisson ratio and density of the adhesive were $0.4$ and 1800 kg/m³, respectively. The joints were considered to be of the same material. The adherends were either steel or aluminum. The mechanical properties of the steel were 200 GPa, 7850 kg/m³, 0.3, and 76.9 GPa for the Young modulus, density, Poisson ratio, and shear modulus, respectively. For aluminum, the mechanical properties were 68.8 GPa, 2800 kg/m³, 0.34, and 25.7 GPa for the Young modulus, density, Poisson ratio, and shear modulus, respectively. The outer adherend was 2 mm thick and had the same length as the adhesive layer. Owing to symmetry, the vertical displacement at the lower boundary of the half-inner adherend was blocked.

4. Results and discussion

4.1. Validation

4.1.1. Case 1: Uniform adhesive $(θ << 1)$

In this section, the adhesive is considered to be uniform or almost uniform. The main advantage of this case is that it is possible to compare the solution of Challita and Othman [28] with the current solution expressed in terms of confluent Heun functions. The constant shear modulus of the adhesive is a specific case of the functionally graded materials model. It can be achieved by considering $θ = 0$ . However, this makes the regular singularities of equation (16) converge to the irregular singularity, $\infty$ . To surmount this problem, the constant $θ$ is considered to be very low but greater than zero, i.e., $0 < θ << 1$ . More precisely, it is assumed that $θ = 10^{- 4}$ or $G_{a} (x) \approx G_{o}$ . The solution obtained using confluent Heun functions is compared with, first, the solution obtained by Challita and Othman [28] assuming a uniform adhesive, i.e., $G_{a} (x) = G_{o}$ . Second, the case of functionally graded materials analytical solution, using insignificant $(0 < θ << 1)$ , is also compared with the finite-element solution where all tiny adhesive segments have the same shear modulus $G_{o}$ . In this work, the frequency ranges between 10.1 and 200 kHz.

Figures 3 and 4 depict the results obtained using aluminum adherends and an (almost) uniform adhesive layer. Figure 3 shows the frequency response in terms of the displacement at the right edge of the lower adherent, i.e., where the harmonic load is applied. Figure 4 presents the frequency response in terms of the maximum adhesive shear stress. Figure 3(a) compares the current work frequency response with the frequency response obtained by Challita and Othman [28], whereas Figure 3(b) compares the current work’s frequency response with the one obtained using the finite-element method.

Figure 3.

Displacement frequency response: (a) comparison between current work and Challita and Othman [28]; (b) comparison between current work and finite-element method.

Figure 4.

Maximum shear stress frequency response: comparison between current work and finite-element method (aluminum adherends).

The current solution matches well with the analytical solution of Challita and Othman [28] and the finite-element solution. All natural frequencies are captured with excellent accuracy (Figure 3). For example, the first natural frequency is recorded at 22.7, 22.7, and 22.9 kHz by the current work, the solution of Challita and Othman [28], and the finite-element method, respectively. The mismatch is insignificant (Table 2). The theoretical maximum shear stress also approximates the studied frequency range with good accuracy (Figure 4). Similar results are observed with a steel joint with a uniform adhesive layer (Figure 5). The error is always less than 1% (Table 2).

Table 2.

First and fifth natural frequency values and the corresponding error between the analytical and numerical models.

	$θ << 1$	$θ = 2$		$θ = 3.9$
	Aluminum adherends	Aluminum adherends	Steel adherends	Aluminum adherends	Steel adherends
1st natural frequency
Analytical (kHz)	22.7	22.2	21.2	21.2	20.2
Numerical (kHz)	22.9	22.3	21.4	21.1	20.2
Error (%)	0.87	0.44	0.93	0.47	$<< 1$
5th natural frequency
Analytical (kHz)	174.8	156.3	126.6	142.6	123.3
Numerical (kHz)	173.1	153.4	126.2	142.5	122.9
Error (%)	0.98%	1.89%	0.32%	0.07%	0.33%

Figure 5.

Maximum shear stress frequency response: comparison between current work and finite-element method (steel adherends).

4.1.2. Case 2: Functionally graded adhesive using moderate value of $θ$

In this section, we consider that $θ = 2$ . This is a moderate value of $θ$ , as this constant is constrained in the range $[0, 4]$ . The analytical model catches the overall behaviors of the displacement and maximum shear stress frequency responses very well (Figures 6 and 7). The natural frequencies are all captured with a good accuracy (Figures 6(a) and 7(a) and Table 2).

Figure 6.

Comparison between analytical and numerical frequency responses of aluminum joint: (a) displacement frequency response; (b) maximum shear stress response.

Figure 7.

Comparison between analytical and numerical frequency responses of steel joint: (a) displacement frequency response; (b) maximum shear stress response.

As an effect of the functional grading of the adhesive $(θ = 2)$ , the peaks of the maximum shear stress are now lower than the peaks obtained with almost uniform adhesive $(θ << 1)$ . For the aluminum adhesive joint, the height of the first peak of the maximum shear stress frequency response is $2.5 \times 10^{4}$ Pa/N if $(θ = 2)$ (Figure 6(b)). If $(θ << 1)$ , the height of the first peak of the maximum shear stress frequency response is $8 \times 10^{4}$ Pa/N (Figure 4). The functional grading of the adhesive has then reduced the first peak by more than three times. Similar results can be observed for the other peaks.

To check how well the theoretical model predicts the shear stress distribution in the adhesive layer, Figure 8 depicts a comparison between the finite-element model and the theoretical model based on Heun functions. The two models match, except at the edges of the adhesive layers. This is quite expected, as the theoretical model, which is based on the shear-lag assumption, does not account for the zero-shear stress boundary condition.

Figure 8.

Shear stress distribution along the overlap region: comparison between analytical and numerical models.

4.1.3. Case 3: Functionally graded adhesive using high value of $θ$

In this section, we consider that $θ = 3.9$ . This is a high value of $θ$ , as this constant is constrained in the range $] 0, 4 [$ . The functional grading of the adhesive is extreme. Here also, the analytical model catches the overall behavior of the displacement and frequency responses for both aluminum and steel joints very well (Figures 9 and 10). The theoretical model captures with a good accuracy the natural frequencies and the frequencies where the maximum shear stress achieves the highest values.

Figure 9.

Comparison between analytical and numerical frequency responses of aluminum joint: (a) displacement frequency response; (b) maximum shear stress response.

Figure 10.

Comparison between analytical and numerical frequency responses of steel joint: (a) displacement frequency response; (b) maximum shear stress response.

In Section 4.1, a comparison between the theoretical model and finite-element model was made for several values of the parameter $θ$ . In all cases, there is good matching between the two models in terms of frequency response and stress distribution. It is worth pointing out that the computational time of the theoretical model is a few seconds (2–5 s), whereas, the computational time of the finite-element model is some minutes (5–10 min). The theoretical model is then much faster than the numerical model, which is an important advantage, especially in the case of a parametric study.

4.2. Shear stress distribution

Next, we were interested in studying the effect of the functional grading of the adhesive on the shear stress distribution along the overlap. Changing $θ$ leads to different distributions of the adhesive shear modulus along the overlap. Therefore, the influence of changing $θ$ on the adhesive shear stress distribution was mainly considered. Moreover, the effects of changing the material of the adherends and the adhesive shear modulus in the middle of the joint, $G_{o}$ , were also analyzed. Aluminum and steel were used as adherends. Also, two values of $G_{o}$ were considered. For each $G_{o}$ used, and depending on the material (aluminum or steel), different shear stress distributions were established through different $θ$ values. Finally, two overlap lengths, 0.05 m and 0.025 m, were analyzed to showcase the influence of changing the overlap length on the distributions. The harmonic frequency was fixed at 10 kHz.

4.2.1. Aluminum adhesive joints

Figure 11 reports the effect of the parameter $θ$ on the shear stress distribution along the adhesive layer. Figure 11(a) and (b) were obtained with $G_{o} = 0.71$ GPa, whereas Figure 11(c) and (d) were obtained with $G_{o} = 2$ GPa. Also, the overlap length is considered to be 50 mm in Figure 11(a) and (c) and 25 mm in Figure 11(b) and (d). In each case, four values of the parameter $θ$ were used: $10^{- 4}$ , 2, 2.9, and 3.9.

Figure 11.

Shear stress distribution along the overlap for several values of $θ$ : (a) $G_{o}$ = 0.71 GPa and l = 0.05 m; (b) $G_{o}$ = 0.71 GPa,l = 0.025 m; (c) $G_{o}$ = 2 GPa; l = 0.05 m; (d) $G_{o}$ = 2 GPa, l = 0.025 m.

In the case of $G_{o} = 0.71$ GPa and $l = 50$ mm, the maximum shear stress is reduced as the parameter $θ$ increases and approaches four (Figure 11(a)). Also, the points where the shear stress reaches its maximum value are moved from the edges toward the joint’s middle as the parameter $θ$ increases. Using a uniform joint $(θ = 10^{- 4})$ , the maximum shear stress is about 70 Pa/N. It is reduced to about 30 Pa/N if $θ = 3.9$ . It is 2.3 times lower than the maximum shear stress of the uniform joint.

In the case of $G_{o} = 0.71$ GPa and $l = 25$ mm, the points where the shear stress reaches its maximum value are also moved from the edges toward the joint’s middle as the parameter $θ$ increases (Figure 11(b)). However, the best shear distribution is obtained for $θ = 2.9$ . Indeed, the least maximum shear stress is achieved when $θ = 2.9$ . Using a uniform joint $(θ = 10^{- 4})$ , the maximum shear stress is about 73.5 Pa/N. It is reduced to about 32 Pa/N if $θ = 2.9$ . It is also 2.3 times lower than the maximum shear stress of the uniform joint.

Figure 11(c), which was obtained by considering $G_{o} = 2$ GPa and $l = 50$ mm, shows a similar behavior to Figure 11(a). Namely, the maximum shear stress is reduced as the parameter $θ$ increases and approaches four. Furthermore, as the parameter $θ$ increases, the points where the shear stress reaches its maximum value are moved from the edges toward the joint’s middle. Using the optimal shear modulus distribution, it is possible to reduce the maximum shear stress by a factor of 2.65.

In the last case, where $G_{o} = 2$ GPa and $l = 25$ mm, the points where the shear stress reaches its maximum value are also moved from the edges toward the joint’s middle as the parameter $θ$ increases (Figure 11(d)). This behavior has been observed independently of any value of $G_{o}$ and $l$ . However, the least maximum shear stress is achieved when $θ = 3.9$ . The maximum shear stress achieved by $θ = 2.9$ is comparable to the optimal value. Using the optimal shear modulus distribution, the maximum shear stress is almost half of that obtained by a uniform adhesive.

Figure 11 shows that for long overlap lengths ( $l = 50 mm$ ), the greater $θ$ is, the lower is the maximum shear stress (Figure 11(a) and (c)). Conversely, there should be an optimal value for $θ$ in the case of shorter overlap lengths ( $l = 25 mm$ ) (Figure 11(b) and (d)).

4.2.2. Steel adhesive joints

Figure 12 reports the effect of the parameter $θ$ on the shear stress distribution along the adhesive layer in steel adhesive joints. Figure 12(a) and (b) were obtained using $G_{o} = 0.71$ GPa, whereas Figure 12(c) and (d) were obtained with $G_{o} = 2$ GPa. In addition, the overlap length was taken as 50 mm for Figure 12(a) and (c) and 25 mm for Figure 12(b) and (d). As for Figure 11, four values of the parameter $θ$ were used: $10^{- 4}$ , 2, 2.9, and 3.9.

Figure 12.

Shear stress distribution along the overlap for several values of $θ$ : (a) $G_{o}$ =0.71 GPa and l = 0.05 m; (b) $G_{o}$ = 0.71 GPa,l = 0.025 m; (c) $G_{o}$ = 2 GPa; l = 0.05 m; (d) $G_{o}$ = 2 GPa, l = 0.025 m.

As for the aluminum adhesive joints, the points where the shear stress reaches its maximum value are moved from the edges toward the joint’s middle as the parameter $θ$ increases. In the case of $G_{o} = 0.71$ GPa and $l = 25$ mm, the two points where the shear stress reaches its maximum value merge into one point: the middle of the joint (Figure 12(b)). Except for Figure 12(c), $G_{o} = 0.71$ GPa and $l = 25$ mm, increasing the value of $θ$ does not always improve the shear stress distribution. Indeed, there should be an optimal value of $θ$ that gives the least maximum shear stress. As shown in Figure 12(a) and (d), the least maximum shear stress is obtained for $θ = 2.9$ , whereas, the least maximum shear stress is obtained for $θ = 2$ in Figure 12(b). Using the optimal value of $θ,$ it is possible to reduce the maximum shear stress by a factor ranging between 1.3 (Figure 12(b)) and 2.3 (Figure 12(c)).

4.2.3. Discussion

The shear stress distribution in a uniform adhesive joint is mainly governed by the ratio of adhesive shear stiffness to the adherends’ shear or compression stiffness. It is possible to improve the shear stress distribution using a functionally graded adhesive. Table 3 reports the reductions (expressed as percentages) obtained as a result of using functionally graded adhesive in aluminum adhesive joints. Improvements can be as high as 62% when $G_{o} = 2 GPa, l = 0.05 m$ $and θ = 3.9$ . Consequently, the load that can be supported by the functionally graded adhesive is 2.6 times as high as the one supported by a uniform joint (Figure 13). For steel adhesive joints (Table 4), the highest reduction is obtained when $G_{o} = 2 GPa, l = 0.05 m$ $and θ = 3.9,$ and is equal to 57%.

Table 3.

Shear stress concentration reduction (%) between constant adhesive shear modulus model and graded adhesive shear modulus model for aluminum at different values of θ.

	$θ = 2$	$θ = 2.9$	$θ = 3.9$
$G_{o} = 0.71 GPa, l = 0.05 m$	36	53	57
$G_{o} = 0.71 GPa, l = 0.025 m$	36.4	41	36
$G_{o} = 2 GPa, l = 0.05 m$	31.2	53	61.9
$G_{o} = 2 GPa and l = 0.025 m$	34.4	49	51.4

Figure 13.

Adhesive shear stress concentration has same value in both models at different loads. FGM: functionally graded material.

Table 4.

Shear stress concentration reduction (%) between constant adhesive shear modulus model and graded adhesive shear modulus model for steel at different values of θ.

	$θ = 2$	$θ = 2.9$	$θ = 3.9$
$G_{o} = 0.71 GPa, l = 0.05 m$	38.5	46.5	44.3
$G_{o} = 0.71 GPa, l = 0.025 m$	22.4	13	0.5
$G_{o} = 2 GPa, l = 0.05 m$	36.9	53.5	57.2
$G_{o} = 2 GPa and l = 0.025 m$	36.3	41.4	35.4

It is critical to note that the overlap’s length plays a major role in changing the reduction percentage of the shear stress concentration. Also, while maintaining the length of the overlap, changing the value of $G_{o}$ will result in changes in the reduction percentage of the stress concentration. Additionally, it is worth mentioning that the adhesive shear stress distribution can be smoothed by changing the values of $G_{o}$ , $θ$ , and the overlap’s length, as shown in Figures 11 and 12.

Functionally graded adhesives give a more uniform shear stress profile in the adhesive layer than uniform adhesives do. However, the extent of improvement is also dependent on the same ratio. For adherends much stiffer than the adhesive, the stress distribution is the least nonuniform. Using functionally graded adhesive moderately improves the shear stress distribution and reduces the maximum shear stress. However, there is also an optimal value of $θ$ . Too great an increase in $θ$ highly decreases the stiffness of the adhesive at the edges and transfers all the load on the central part. In the extreme case, a high maximum shear stress is observed in the middle of the joint (Figure 12(b)).

Reducing the stiffness of the adherends or increasing the stiffness of the adhesive increases the shear stress nonuniformity and increases the maximum shear stress. Thus, great effort should be made to smooth the shear stress distribution. In this case, the least maximum shear stress is recorded for the largest value of $θ$ , i.e., $θ = 3.9$ (Figures 11(a), (c), and (d) and 12(c)).

To emphasize the effect of the ratio of adhesive stiffness to adherend stiffness, more calculations were made for $θ$ varying from 0.1 to 3.9 in steps of 0.05. Each time the optimal value of the gradation rate constant $θ$ , $θ_{opt}$ , which gives the lowest maximum shear stress, was evaluated. Table 5 shows that $θ_{opt}$ increases as the adherend becomes softer, the overlap becomes longer, and the adhesive becomes stiffer. This is interpreted in terms of the ratio of shear adhesive stiffness to axial adherend stiffness. Indeed, as this ratio increases, the stress concentration becomes more severe and a higher gradation rate of the adhesive stiffness is required. Figure 14 also depicts the variation of $θ_{opt}$ in terms of the ratio of adhesive shear stiffness to adherend axial stiffness. The adherend axial stiffness is defined as $s_{s} = E t_{s} / l$ , whereas the adhesive shear stiffness is defined as $s_{o} = G_{o} l / t_{o}$ , and $θ_{opt}$ increases as the ratio of adhesive stiffness to adherend stiffness increases. The increase rate is more important at small values of $s_{o} / s_{s}$ . At large values of $s_{o} / s_{s}$ , $θ_{opt}$ approaches the limit value of four. The following equation for $θ_{opt}$ is then proposed:

θ_{opt} = 4 (1 - e^{k {(\frac{G_{o} l^{2}}{E t_{s} t_{o}})}^{n}}),

(33)

where $k$ and $n$ are constants. The best fit is obtained for $k = 0.3085$ , $n = 0.7235$ , and an average error of 2.07%.

Table 5.

Optimal value of the gradation rate $θ$ .

Adherend	Adhesive $G_{o}$ (GPa)	Overlap $l$ (mm)	$θ_{opt}$
Aluminum	0.71	25	2.85
Aluminum	0.71	50	3.75
Aluminum	2	25	3.65
Aluminum	2	50	3.9
Steel	0.71	25	1.65
Steel	0.71	50	3.15
Steel	2	25	2.8
Steel	2	50	3.75

Figure 14.

Variation of $θ_{opt}$ as a function of ratio of adhesive shear stiffness to adherend axial stiffness.

Footnotes

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ramzi Othman

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Theoretical solution for the axial vibration of functionally graded double-lap adhesive joints

Abstract

Keywords

1. Introduction

2. Theoretical model

2.1. Problem statement

2.2. Establishment of the differential equations

2.3. General solutions

2.4. Boundary conditions

3. Numerical model

4. Results and discussion

4.1. Validation

4.1.1. Case 1: Uniform adhesive ( θ << 1 )

4.1.2. Case 2: Functionally graded adhesive using moderate value of θ

4.1.3. Case 3: Functionally graded adhesive using high value of θ

4.2. Shear stress distribution

4.2.1. Aluminum adhesive joints

4.2.2. Steel adhesive joints

4.2.3. Discussion

Footnotes

Funding

ORCID iD

References

4.1.1. Case 1: Uniform adhesive $(θ << 1)$

4.1.2. Case 2: Functionally graded adhesive using moderate value of $θ$

4.1.3. Case 3: Functionally graded adhesive using high value of $θ$