Analysis of the substrate nature on the strength of a single‑lap joint

Nowadays, the adhesive bonding process takes an important place in several industrial fields, especially in aeronautics. Given its advantages over other conventional mechanical processes, this process is being extended to be applied in composite materials and, in recent years, more particularly in the bonding of functionally graded materials. In bonded assemblies, the adhesive is the weak link due to its mechanical properties which are very weak compared to those of the two substrates. To this end, current research aims to reduce the concentration of stresses in the adhesive joint by choosing substrates whose mechanical properties are adequate with those of the adhesive in order to minimize load transfer and ensure a robust assembly. The present work analyzes, by the finite element method, the mechanical behavior of a single-lap joint with varying nature of the substrates. The analysis takes into account the variation of stresses in the adhesive of a bonded joint of different types (metal/metal, composite/composite, FGM/metal, FGM/FGM). On the other hand, an attempt has been made to introduce an adhesive with graded mechanical properties made up of two types to ensure efficient joining and reduce stress concentrations at the bond edges. The introduction of the FGM substrate mechanical properties is done using a USDFLD (User subroutine to redefine field variables at a material point) subroutine implemented in the ABAQUS computer code. Different damage approaches were used for the adhesive, namely the virtual crack closure technique (VCCT) and cohesive zone models (CZM) techniques. The effect of the mechanical properties of the substrates and adhesive were considered. The results show clearly that the value of the different stresses can be reduced if the mechanical properties of the substrates are optimized. On the other hand, the different techniques used to model the bonded joint converge towards the same results, emphasizing the agreement of the load–displacement curves with the experimental test, and the variation of the stresses according to the lap length with analysis by analytical models.


Introduction
Structural assembly in the field of transport, particularly in aeronautics, increasingly requires reliable solutions. There are many structural joining techniques, namely welding, riveting and bonding, each with advantages and disadvantages. The mastery of structural assemblies in an industrial application then turns out to be one of the most important factors for manufacturing a reliable product that can meet the various economic and technical requirements. The choice of the assembly method best suited to the application remains the key element to have a robust structure meeting the specifications of the product.
An adhesive bonded joint consists of using an adhesive to join two parts together. Research carried out over the last decades is traditionally focused on bonding metallic and composite materials. This joining technique has many 1 3 469 Page 2 of 27 advantages, such as the ability to join different materials, better distribution of stresses, weight gain, reduction in corrosion points, and absence of holes during machining, thus preventing the hole-induced stress concentration.
However, adhesive bonding remains a complex joining technique. Surface preparation must be meticulous, the cross-linking reactions of adhesives are sometimes long, the joining strength is limited by the adhesive properties, and difficulties arise on the lifespan prediction of these assemblies. Given these overall characteristics, adhesive bonding often replaces traditional joining processes such as bolting, riveting or even welding.
In adhesive bonded assemblies, the mechanical properties of the adhesive are weak compared to those of the substrates. Therefore, and to ensure the safety of the adhesive bonded structure, it is necessary to analyze the stress distributions in the adhesive layer. For an adhesive bonded structure, failure typically occurs at the free ends of the adhesive layer, which serve as stress concentration sites.
The stress distributions analysis in the adhesive layer was the concern of several studies. Volkersen [1], Goland et al. [2] and Hart-Smith [3] proposed the first approaches to analyze the adhesive's behavior in a joint, although taking into account some simplifying assumptions. Chen and Cheng [4] were able to establish a two-dimensional model giving rise to zero shear stresses at joint edges. Harris et al. [5] studied double-lap composite joints under bending and developed a model to describe the mechanical behavior of the joint. Rocha and Campilho [6] analyzed the effect of using different CZM conditions in modeling single-lap bonded joints under a tensile loading. El-Hannani et al. [7] carried out a three-dimensional numerical analysis by a nonlinear FEM technique, and proved the need for a three-dimensional stress analysis in single-lap joints. Madani et al. [8] used a threedimensional nonlinear FEM analysis to evaluate stress distributions in the adhesive layer used to join two aluminum 2024-T3 (AL2024-T3) plates.
Estimation of the stress fields and deformations in adhesive bonded joints presents difficulties due to the complex geometry and the various properties of the materials to be assembled. By reducing these stresses, which damage the joint, the strength of the joint is increased, resulting in a greater load carrying capacity of the adhesive bonded joint [9].
More recently, different authors [10,11] attempted to provide a new solution to reduce stresses in a composite/ composite adhesive bonded joint by modifying the nature of the fibers and using a hybrid composite. The obtained results were relevant from the joint strength point of view. Structural joining by adhesive bonding composite materials and respective durability are relevant research topics, and it is known that the adhesive bonded joint strongly depends on the nature of the fibers and composite layup.
Moreover, the introduction of hybrid composites in recent years brings a major advantage in stiffness flexibility compensation of composite plates. The behavior of composite adhesive bonded joints has been the concern of researchers for several years. While their analysis is generally studied in tension-shear, composite joints are also subjected to different load types.
More recently, several studies have been conducted by using FGM to reduce stress concentrations, which consequently leads to improved strength of adhesive bonded joints [12][13][14]. Numerical modeling of FGM joints has been studied [15]. Carbas et al. [16] introduced the idea of using mixed adhesive joints to increase joint strength and minimize peak adhesive stresses. Chen et al. [17] numerically analyzed the strength of adhesive bonded joints under the effect of graded substrates. Marques et al. [18] and Durodola et al. [19] used functionally graded adhesive joints. Apalak and Gunes [20] studied the effect of a functionally graded layer between a layer of pure ceramic (Al 2 O 3 ) and a layer of pure metal (Ni). Gannesh and Choo [21] investigated the effect of spatial gradation of the substrates' modulus of elasticity on peak stress and stress distribution in single-lap adhesive bonded joints. Most of the cited studies model the adhesive as a third material or as an interface.
The present work takes into account these two modeling approaches for the adhesive. The objective is to analyze by the FEM, using the ABAQUS computer code, the distribution of stresses in an adhesive layer used to join different types of materials (metal/metal, composite/composite and FGM/ FGM). On the other hand, the VCCT and CZM techniques were used to analyze the tensile response of the joints. The adhesive was modeled as a third material, interface and as a third material with interface. The numerical simulation results were compared with the different analytical models, namely Volkersen, and Goland and Reissner. As part of the study, it was deemed essential to undertake a mesh convergence analysis to ensure the reliability of the numerical model, which consisted of modifying the type and density of the elements of the mesh. In the second part of this work, and on the basis of FGM substrates joints, a functionally graded adhesive joint is proposed between an Araldite® adhesive and the Adekit®A-140. Two different gradation designs of the joint have been proposed, namely of circular and square shape.

Mathematical formulation
The analytical approaches for adhesive bonded joint analyses are numerous and their use is generally based on simplifying hypotheses made according with the problem posed, such as for example the hypotheses of constraints, which are constant or vary linearly with the adhesive thickness, or consideration or not of the substrates' rotation due to the load eccentricity. For obvious reasons of simplicity, most classical analytical studies choose a two-dimensional approach to analyze the stress state. To compare the different existing approaches for the analysis of stress distributions in an adhesive bonded joint, in this work the single-lap adhesive bonded joint presented in Fig. 1 was addressed.
In Fig. 1, t a is the adhesive layer thickness, H 1 and H 2 are the substrates thicknesses and L C = C∕2 is the overlap length. For asymmetric substrates, solutions for adhesive stresses can be found in the work of Luo and Tong [22]. In the present study, adhesive stresses only for symmetrical members were considered by introducing the following variables: The variables in Eq. (1) have the usual meanings used in Euler's beam theory, the indices 1 and 2 denote identical members in the region of overlap, and the force components are represented in the free-body diagrams of the infinitesimal elements, shown in Fig. 2.
The equilibrium equations are given as follows: where N i , Q i , and M i (i = 1, 2) are the longitudinal forces, transverse shear stresses and bending moments per unit width for the substrates, respectively. The differential Eqs. (2) and (3) can be combined, in order to determine the adhesive shear ( ) and peel stresses ( ) in the adhesive layer, which are given by Goland and Reissner [2], as:  where E a and G a are the Young's modulus and the shear modulus of the adhesive, respectively, u 2 is the longitudinal displacement, and t a is the adhesive layer thickness. By using the variables of Eqs. (1), (2), (3) and (4) lead to: Goland and Reissner [2] are the first to take into account the substrate bending. In this case, the analytical solution for the τ stress distribution in the adhesive is as follows: The bending moment factor for this case is defined as: With the variable needed for the k factor: And the variable needed in the τ stress formula: Volkersen [1] introduced a formulation for the stress analysis of a single lap joint that assumes that the adhesive only deforms in shear. The substrates are treated as an elastic and isotropic material deforming only under tension. The simplest approach is to assume that the substrates are rigid and that only the adhesive deforms in shear. For a sample of width w , thickness e and L C = C∕2 (Fig. 1), stress is given by: However, the stress distribution in the joint is not uniform due to the substrates' elongation, which increases in the overlapping area from the ends. The resulting effect is a high concentration of shear stress close to the overlap ends. Volkersen analysis [1] makes it possible to express the distribution of τ stresses in the adhesive as: where (x) is the average shear stress, E 1 and E 2 are the Young's moduli of the substrates, and H 1 and H 2 are their thicknesses, with: According to the mathematical equations proposed by Volkersen, in which the adhesive is considered as linear, and the substrates are isotropic with constant young's modulus, in this work the young's modulus was regarded variable as a function of the thickness of the substrate in order to analyze the case of an FGM assembly and determine the shear stress distribution in the adhesive joint.
If the FGM substrates are of different nature and graded, E(z) 1 ≠ E(z) 2 , Eq. (11) can be written by taking H 1 = H 2 . Under these conditions, the system is said to be balanced and τ(x) in the adhesive layer is given by: where the variable w(z) used in the proposed formula (see Eq. 13) for the shear stress for the single lap joint FGM/ FGM is given by Eq. (14).
Hart-Smith notes that the moment applied at the joint edge and calculated using the factor k defined by Goland and Reisner is strictly applicable only in the case of light loads and small L c , and then considers the substrates independently and not in a single block as Goland and Reisner do. In addition, Hart-Smith developed a perfect elastic-plastic approach. Thus, the shear stress by the analytical model of Hart-Smith is written as: in which the variables needed in the formula are given by: Hart-smith's [3] bending moment factor is: and the variables for k-factor: Numerical approaches make it possible to take into consideration certain hypotheses not taken into account by analytical models, such as the stress distribution which, along the thickness of the adhesive film, can be constant or vary linearly, or even the taking into account or not of the rotation of the joint at the overlap. However, given the high deformation graded, a fine mesh is necessary. Under these conditions, the numerical approach gives accurate solutions, although at the expense of the computation time. In the recent numerical analysis, Eqs. (6), (11) and (15) were introduced in MATLAB, to obtain the variation of the stresses in the adhesive joint to validate our numerical model.

Elastoplastic behavior of the adhesive and substrates
The introduction of the mechanical characteristics of the adhesive and the substrates proves to be of capital importance to have reliable results for the stress variations. The definition of the plastic limit for the adhesive is necessary for this study, and different types of existing criteria can be considered. The Ramberg and Osgood [23] model was used, defined as follows: Equation (21) can be used to describe the uni-axial tensile and compressive behavior of various aluminum metallic alloys, considering K and n as the strain hardening parameters. The terms E, K and n are all described as functions of the strain rate (d ∕dt) . Parameters such as the stiffness, stress, and plastic yield, are all affected by variations in strain rate. Ramberg-Osgood type equations [23] have also been used to describe the shear behavior of structural adhesives. Zabora et al. [24] used a velocity-dependent form of the Ramberg-Osgood equation to describe the shear behavior of structural adhesives in bonded structures.

Damage criterion in the 3D cohesive model
In the first part of this work, the FEM is used to simulate the separation in the adhesive joint without taking into account the composite delamination (case of a composite/composite assembly) nor the presence of bonding defects for the different techniques proposed. Cohesive laws were used to simulate the behavior of the adhesive joint, which are generally defined to model continuous failures. The choice of traction-separation model is available in ABAQUS and initially assumes linear elastic behavior followed by damage initiation and exponential evolution up to failure. Several modeling approaches have been proposed in this work, such as the Virtual Crack Closure Technique (VCCT), to predict the onset of delamination, which falls under the scope of linear elastic fracture mechanics (LEFM) and makes it possible to propagate the delamination, but requires knowing a priori the location and the front of the defect. The VCCT technique is available in ABAQUS. The most commonly used failure criterion in VCCT analyses is the Benzeggagh-Kenane law (BK) [25], which defines the evolution of damage. This criterion is based on the total energy release rate (G T ) to define the mixed mode fracture energy, as shown in the following equation: where G T is the mixed-mode fracture energy, G I , G II and G III represent the energy release rates in mode I, sliding II and III, respectively, G IC is the critical mode I energy release rate, and is the material parameter.
Other methods can be used to model debonding. Specifically, Cohesive Zone Models (CZM) allow to determine both the initiation and the propagation of the debonding. In the CZM technique, an initial defect is not required, because the crack can start and propagate anywhere in the layer of cohesive elements. Even if these models are widely developed in research laboratories, they remain little used in the industry. The elastic behavior of cohesive elements is coupled between all the components of the traction vector and the separation vector can depend on the field variables. The nominal strains can be defined as: The nominal strain components are equal to the respective components of the relative displacement n , s , and t divided by t a , the original thickness of the cohesive element. The elastic behavior is written in terms of an elastic constitutive matrix which relates the nominal stresses to the nominal strains across the interface. The nominal stresses are the force components divided by the original area at each integration point, while the nominal strains are the separations divided by the original thickness at each integration point. The nominal tensile stress vector consists of three components ( t n , t s , t t ), which represent the normal tension and the two shear tensions, respectively. The elastic behavior can then be written as: , which represent the normal stiffness and the two shear stiffness's, respectively. The chosen damage initiation criterion is based on the quadratic nominal stress criterion (QUADS), which was proposed in a material of cohesive. This criterion can be represented by: where t 0 n , t 0 s and t 0 t are the cohesive strengths in pure modes I, II and III. The damage propagation criterion can be defined according to the energy dissipated following the process of damage, also called energy of rupture. The dependence of the fracture energy on the mixed mode can be defined on the basis of a power law failure criterion. The power law criterion states that failure under mixed mode conditions is governed by a power law interaction of the energies required to cause failure in the individual modes (normal and twoshear), and it is given by: where is a parameter given by the experiment and G IC , G IIC , G IIIC are defined experimentally. G I , G II , G III represent the energy release rates in opening mode I, and sliding modes II and III, respectively. The damage variable (D) gives indication of the damage evolution. At complete separation, its value equals to one. The expression of D during exponential degradation is reduced to: Here, G o is the elastic energy at damage initiation, G T equivalent fracture toughness for complete damage (the fracture energy), T 0 eff as the effective traction at damage initiation, o m the effective displacement at damage initiation and f m the effective displacement at complete failure. Subscript m accounting for the contact separation is caused by mixed effect of deformation in normal mode and shear mode, which can be calculated as:

Bonded joint models choice
The numerical models of bonded assemblies are simulated considering different types of materials. Hence, the following cases are undergoing in the present paper:

The FE model of the bonded joint and material choice
The numerical model including the analysis of the stress distribution was carried out by 3D FEM analysis, using the ABAQUS computer code. The analysis carried out took into account the variation in the nature of the substrates. It should be noted that the type of mesh element, the contact properties and the boundary conditions used are the same for all the assembly parts. The substrates were bonded with two-component high strength epoxy adhesive type Adekit ® A-140 used in aerospace structural applications. Figure 3 shows the stress-strain curve outcome from the experiment tensile test of the Adekit ® A-140 adhesive carried at LASIE laboratory in France. Macroscopic properties, such as stiffness and strength, of the Adekit ® A-140 adhesive were measured experimentally and fed directly to the numerical model purposed (Table 1). To model adhesive debonding, other mechanical properties are required. Ezzine et al. [26] were able to characterize this adhesive in DCB and ENF tests. The final mechanical and fracture properties of the Adekit ® A-140 adhesive are presented in Table 1.

Description of the adhesive bonded joint model (AL2024-T3/AL2024-T3)
A single-lap joint was considered, bonded with the Adekit ® A-140, made of two substrates whose dimensions are: Length L = 125 mm, width W = 25 mm and thickness of H = 2 mm. The overlap length is L C = 25 mm. The dimensions of the two substrates and the adhesive layer are depicted in Fig. 4. For all models, one end of the joint is clamped, while the other end is subjected to a tensile force per unit width. Two values of applied forces have been chosen: the variation of stress are measured along the overlap length under a unite force equal to P = 100 N/ mm, and for the determination of the load-displacement tensile curves, a maximum unite force P = 200 N/mm was considered.
For the aluminum substrates, the mechanical properties presented in Table 2 were imported from the tensile curve shown in Fig. 5.

Description of the bonded joint model (composite/composite)
For the case of joint with composite material substrates, the carbon fiber substrates are considered, while the matrix is taken to be epoxy based. The laminate sequence were layered as [0] 16 . Each layer is composed of carbon fibers reinforcement and an epoxy resin. The thickness of each ply is 0.125 mm, making up a total thickness of 2 mm. The carbon fiber used in this work is HR carbon fiber, belonging to the group of high-strength fibers. The properties of the two constituents are given in Table 3.
To determine the mechanical properties of the composite substrate, the CADEC (Computer Aided Design Environment for Composite) calculation software was used, which is specially designed for composite materials and uses composite homogenization calculations. The global properties of the composite are presented in Table 4.
The dimensions and geometry of the adhesive bonded joint with two composite substrates are shown in Fig. 6. All dimensions, boundary conditions and adhesive (Adekit ® A-140) are the same as for the joint of metal substrates.

Description of the bonded joint model (metal/ FGM)
FGM are high performance and microscopically heterogeneous materials with specific properties in the preferred orientation [28]. This new family of materials is beginning to leave the academic field to enter the industrial field for high added-value applications. The choice for the FGM substrate joint aims to improve the distribution of stresses and increase the joint's strength, leading to possible applications in aeronautics and aerospace industries. The mechanical properties of FGM are shown in Table 5.
In the literature on FGMs, most research uses the simple mixing rule to obtain the effective material properties. Regarding the volume fraction distribution functions of the substrates, the equivalent material properties of FGM could be determined as functions of the power law variation along the thickness of the substrate [30]: where n is the non-negative volume fraction exponent and H the thickness of the substrate. Once the volume fraction V(z) is defined, the equation of the law of mixtures is written as: P(z) represents the effective material properties of FGM (SDV (Solution-dependent state variables); see Figs. 8, 9, 10, 11 and 12) and V m (z) is the volume fraction of the metal in the FGM substrates. The subscripts c and m represent the ceramic and metallic phases, respectively. The TTO model is a metal alloy homogenization method used to locally assess effective elastoplastic parameters of the FGM-AL/ SiC compound. In the TTO model, an additional parameter q is required, which represents the relationship between the stress and strain transfer, such as: In the TTO model, the mixture of materials is treated as elastoplastic with linear isotropic hardening, for which the effective Young's modulus E(z) , the initial yield stress Y0 (z) and tangent modulus H(z) of FGM are defined as follows: where Y0m the initial yield stress of metal H m is the tangent modulus of metal. To determine the effective properties of FGM with TTO model in the ABAQUS fine element code,  a USDFLD subroutine is implemented to define the material properties of the FGM according to the coordinates of the integration points in a FEM model. In this work, integration points arranged across the thickness direction of the FGM (ceramic/metal) substrate were used. This method also aims to improve the performance of the mesh element type in terms of continuity distribution material properties and continuity of stress at the interfaces for the purpose of calculating the resulting stress. A surface method is proposed, obtained by subdividing the interval − H 2 , H 2 of the plate in m: surface number (Fig. 7): where z i is the coordinate of the surface with respect to or global reference, i = 1, … , m is the position of the surface in the FGM substrate, h = H n is the distance between two successive surfaces, m = n + 1 is the number of surfaces in the substrate, and n is the number of layers. To apply the TTO model Eq. (20) in our technique UMM, it is supposed that the surface is located exactly on the point of integration.
where E USDFLD (z) is the effective Young's modulus of the FGM of layer k ( k = 1, 2, 3 … n ). Note that E z 1 and E z m are the Young's moduli of the upper and lower faces of the H interval, respectively. The gradation in the finite element model is done by layer according to the thickness of the substrate with the TTO model given by the following formulae (Fig. 7).
In this study, the continuous variation of the properties (e.g., Young modulus) according to the thickness of an FGM plate in shown in Figs. 8,9,10,11,12 with an exponent of the volume fraction for n = 1. For example in the Fig. 8 the face upper (black) 100% ceramic (SiC) while the lower face (white) 100% metal (AL2024-T3) with a graded transition between the two material. In order to define the effective Young's modulus of the FGM numerically, it is necessary to call the solution-dependent state variable command (SDV1).
In the first configuration of the FGM-1 joint (FGM AL2024-T3 /Al 2024-T3), it is assumed that the lower face of plate 2 rich in metal is in contact with the adhesive (Fig. 8). On the other hand, for the second FGM-2 configuration (AL2024-T3/FGM SiC ), the surfaces have been reversed, such that the ceramic face is in contact with the adhesive (Fig. 9).
In the FGM-3 (FGM AL2024-T3 /FGM AL2024-T3 ) configuration, depicted in Fig. 10, it was assumed that the two FGM substrates' contact with the adhesive is made on the metal faces of both substrate 1 and substrate 2.
In the FGM-4 (FGM SiC /FGM SiC ) configuration, shown in Fig. 11, the two FGM substrates are considered to contact the adhesive by the ceramic faces of both substrates.
The last configuration ensures contact through the adhesive by the ceramic side of substrate 1 and the metal side of substrate 2 (Fig. 12).
In the representation of the FEM models (Figs. 8,9,10,11,12), the zones rich in SiC ceramics are red and the zones rich in metal AL2024-T3 are blue. For the present study, the FGM substrate is decomposed into two ductile/fragile materials (AL2024-T3/SiC), whose material properties were defined in the previous section.

Different types of numerical joint models
Different types of numerical joint models were tested, aiming to select the best technique to validate the adhesive bonded joint tests: 1. Cohesive (adhesive with zero thickness) 2. Adhesive as third material (Tie) without VCCT and without cohesive CZM 3. Adhesive as third material (Tie) with VCCT and without cohesive CZM 4. Adhesive as third material without VCCT, without Tie and with two cohesive interfaces For the cohesive zone model CZM technique (1), the adhesive is considered with zero thickness (Fig. 13). For this approach, a set of nodes was created (solid layers of total thickness zero) from a connection between the surface of the 3D finite elements of the substrates (1 and 2) in an orphan model (Fig. 14). The numbering of the nodes and the direction of the normal to the surface are generated automatically.
For the CZM implementation, the cohesive mesh technique with zero thickness was used, considering the COHD3D FEM element, as illustrated in Fig. 14.
In the two techniques that follow (2 and 3) the adhesive is considered as a third 3D material with thickness of 0.2 mm, and a contact procedure (Tie) was applied between the plates  (Fig. 15a). This technique specifies four contact surfaces (Tie) and can be defined as: • The contact plane between the lower face of the substrate and the upper face of the adhesive.
• The contact plane between the underside of the adhesive and the upper side of the substrate.
In the models three, the adhesive is modeled with the VCCT approach. To predict the adhesive joints' behavior, damage criteria based on the traction-separation law of the adhesive were considered for the models 3. In the second model 2 (Fig. 15a) the maximum constrained damage initiation criterion MAXPS was used for the adhesive (see Sect. 4). The fracture criterion used for the adhesive in model three to determine damage initiation is the quadratic stress criterion QUADS (see Sect. 4). The damage evolution for both models is based on the failure energy criterion as a function of the mode mixture using the power law failure criterion for the adhesive. VCCT simulations generally provide high accuracy with low computation time. In the simulation model's last assembly (4), an adhesive was considered as the third material with two cohesive interfaces (Fig. 15b).
In the VCCT technique, to model damage in the adhesive layer, it is preferable to use the cohesive zone technique, which models the adhesive with an infinitely thin layer. In The definition of the mesh is one of the most important aspects to properly model the behavior of the joint and to obtain a good convergence. To obtain an optimal mesh it is necessary to choose the type of elements and their distribution. Several meshing strategies can be used. It is possible to mesh the whole model in the same way or to further refine the mesh in the areas of high stress concentrations. Figure 15 gives an overview of a 3D mesh of bonded assemblies composed of two substrates produced with the ABAQUS software. For both contact model and VCCT model, threedimensional-solid elements were used in all joint substrates with different finite element densities. The adhesive and the overlap ends must have a very fine mesh compared to the other areas, which can have a coarse mesh to avoid having an excessive number of degrees of freedom. Figure 15 shows the mesh of the different single-lap joint components with a mesh refinement of the lap area. A sensitivity study of the type of C3D8R elements was carried out, and it was concluded that the hexahedral element provides an improved solution since the discretization of the substrate is done in a uniform way apart from the adhesive.
The reliability of the results concerning the analysis of the different stresses in the adhesive joint requires a very fine mesh at the level of the overlap zone in the two substrates and the adhesive. The substrates is meshed with five layers of type elements C3D8R in the thickness direction, Cohesive is discretized with a single layer of COH3D8 type elements in its thickness (Fig. 14) and the adhesive with four layers of C3D8 type elements. The number of elements used in this analysis was 66,132 elements for the structure with Adhesive and 62,222 with cohesive (Fig. 14). The mesh includes a number of elements for each substrate 30,822, the cohesive elements 428 (Fig. 14) and the adhesive elements 4488 (Fig. 15).

Convergence and validation
A tensile test was carried out on the AL2024-T3/AL2024-T3 adhesive bonded joint with a speed of 0.3 mm/min, whose result is shown in Fig. 16. This test aims to serve as a benchmark to validate the numerical models. Figure 17 represents the different types of methods used to obtain the joint's response in tensile test, for assessment of the most suitable method in comparison with the experimental results.

Influence of the different types of techniques and mesh on the behavior of the joint in tensile test
It is clearly noted from Fig. 17 that the tensile response of the single-lap joint presents a straight line and that the respective failure takes place at a load of approximately 11,800 N. Additionally, the different numerical modeling techniques used to model the adhesive or the adhesive substrate contact converge towards the same value of the separation load. Following the different simulation techniques of the adhesive, it was possible to promote the separation of the two plates through the CZM technique (Fig. 18). The distribution of the von Mises stress in the adhesive as a function of the lap length is presented in Fig. 19 for the four modeling techniques (contact Tie without VCCT, contact Tie with VCCT, Cohesive single without VCCT, and two cohesive interface with VCCT). It is clearly noted that the shape of the curve is similar irrespectively of the technique, with peak stresses at the overlap edges. The VCCT and adhesive as third material techniques practically present the same values of von Mises stresses. The lowest values are noted for the cohesive technique 4. The von Mises stress plots are not symmetrical since one of the two edges of the adhesive is in direct contact with the plate, while the other edge is linked directly with the free edge of the substrate. The overall level of stress (joint and adhesive) is practically the same, except for the cohesive technique, case in which the two edges of the adhesive  -displacement curve by  different techniques on the  behavior of the single-lap joint  compared with the experimental  tests (AL2024-T3/AL2024-T3 single-lap joint, L C = 25 mm) Fig. 18 Presentation of the plate separation present a strong zone of stress concentration leading to a rapid joint failure after a high applied load.

Comparison between analytical techniques and numerical modeling
For the variation of shear stresses, the numerically obtained results were compared with analytical ones, namely the models of Hart Smith, Volkersen, and Goland and Reissner (Fig. 20). It is clearly observed that the results converge. A good agreement is noted between the analytical models and the numerical results. The maximum shear stress is similar between all models. At the inner overlap, the stress is almost zero for the analytical models, while it has a higher value for the numerical models. The analytical models are simple and do not take into consideration several parameters, namely the stress variation according to the thickness and the bending moment, which justifies this difference. For an adhesive bonded joint with FGM substrates, a validation attempt was carried out for the modified Volkersen model in Eq. (13) by comparison with the numerical analysis (Fig. 21). It is clearly noticed that the analytical model gives a shear stress variation close to the numerical analysis, except that the values differ. At the free edges, the predicted analytical stresses are lower than expected due to the simplification of the assumptions considered in the Volkersen analytical model, which does not take into consideration the variation of the stresses according to the width, while also neglecting the bending moment.

Response of different assemblies to tension load
The various numerical models were established in order to proceed to a comparison between the various techniques used in Abaqus to model the adhesive joint and on the one hand to try to compare the obtained results on the stresses distribution in the adhesive joint with the various existing analytical models taking into account the influence of the nature of the substrate. In this section, it was chosen to analyze the effect of the substrates on the shear stresses, considering the approach of the cohesive zone model with two cohesive interfaces (model 4) since the result of the force elongation curve with that of the experimental (case of AL2024-T3/AL2024-T3 assembly) is in good agreement. Figure 22 shows the tensile load-displacement curves for an adhesive bonded joint taking into account the variation of the substrates. The shape of the curves is practically the same, except that the stiffness and maximum load values differ depending on the nature of the joint. The AL/AL joint has an average strength between the conditions tested. The FGM/FGM joints show a higher stiffness and a smaller displacement at maximum load compared to the other types of joints. In FGM joints in general, if the adhesive contacts between AL/AL or AL/Ceramic, the load transfer will be done by the adhesive and by the aluminum substrate, which provides a higher joint strength. The composite/composite joints are the most compliant joints, although the achieved strength is similar to the AL/AL joints. In these joints, the adhesive absorbs most stresses and, therefore, the joint strength is naturally limited.

Stress distributions in the adhesive layer
• Stress distributions σ VM Figure 23 represents the variation of von Mises stresses in the adhesive layer for the different joint configurations. It is clearly noticed that the distribution of von Mises stresses is practically the same in the adhesive joint, whatever the configuration of the joint. The highest values arise at both edges of the adhesive. The center area of the adhesive, between approximately 10 and 25 mm of overlap length in Fig. 23 It is clearly noted that the stresses are concentrated at the two edges of the adhesive and that the core of the adhesive is almost inactive for the case of FGM substrate joint. The size of the zone of high stress concentration varies depending on the type of joint configuration. The variation of shear stresses in the adhesive joint according to the lap length is shown in Fig. 24.
The value of the shear stress varies according to the type of joint. The highest peak stresses are noted for the composite/composite type joint. Similarly to von Mises stress distributions, stresses are concentrated at the two edges of the adhesive joint. These peak stresses are different between both edges of the overlap.
The core of the adhesive is lightly stressed but, between joint configurations, the FGM-4 type joint promotes the highest values. The difference in the distribution and size of The variation of peel stresses in the adhesive layer along its length clearly shows that the highest values relate to the FGM-3 joint, case in which the substrate contact with the adhesive joint is made in the aluminum sides. On the other hand, the ceramic face will be on the outside and joint bending diminishes, which will transmit more stress to the adhesive and induce detachment from the substrate.
Similarly to the von Mises and shear stresses (Fig. 25), the highest peel stresses are found at the two overlap edges. A portion of the adhesive at the core exhibits zero peel stress values, while the adjacent portion of the edges exhibits compressive stresses. The highest values are noted for the case of the FGM-3 joint, which induces more separation stress in the adhesive joint. The smallest peak peel stresses are noted for the case of an FGM-4 type joint.

Graded adhesive
Several authors [18][19][20][21] proposed the use of FGM, essentially adhesives in the form of a functionally graded in the joint. Indeed, the graded cohesive method has been applied in the work of Moreira and Campilho [31], to evaluate the strength improvement of bonded repairs in aluminum structures with distinct external reinforcements bonded with the Araldite ® AV138 adhesive. In this context, the effect of using an adhesive with variable properties was studied by combining between the Adekit ® A-140 and Araldite ® AV138 adhesives. The choice for a functionally graded adhesive joint is becoming more important in modern technology, as this technique is used to meet industrial requirements, because it combines the best features of different adhesives.  The mechanical properties of the two adhesives used in the present study are presented in Table 6.
For the present analysis, the properties of the functionally graded adhesive (FGA) continuously vary according to the length per row of elements according to the user mesh method (UMM) [32] and per coordinate in the USDFLD subroutine-program (see Sect. 6.3 Eq. (18)). The adopted mixing law is given by the following equation: P(x, y) FGA represents the effective material properties of the FGA and V C1 (x, y) is the volume fraction of adhesive in the FGA layer, which is ensured by a power law along the (39) P(x, y) FGA = P C1 − P C2 .V C1 (x, y) + P C2 direction of variation. Two configurations are proposed to grade the effective material properties of the Adekit ® A-140/ Araldite ® AV138 FGA adhesive combination. The first case follows the radius R(r): where n is the exponent of the non-negative volume fraction, the indices C 1 and C 2 represent the phases of the Cohesive Adekit ® A-140 and Araldite ® AV138 FGA, respectively, r is the cohesive radius coordinate (Fig. 27a) and R C is the radius of the cohesive layer, determined as follows: With b and L C being the lap joint width and length of cohesive, respectively.
Based on the value of the normal stiffness obtained in Table 7, the distribution of the normal stiffness of the adhesive as a function of the radius of the adhesive joint (Fig. 26) for different values of the exponent of graded n presents a good agreement between the two methods (UMM and USDFLD).
The second approach for mechanical property distribution in the FGA consists of a square grading according to the table of the comparison, which shows a good convergence of results between the methods graded the material properties.
Thus, the UMM method is recommended in order to simplify the complexity of mathematical formulations. The distribution of the mechanical properties of the two adhesives according to the two concepts of grading, by radius and a square, is shown in Fig. 27. By the analysis of von Mises stress distributions (Fig. 28), it is visible that the shape of the curves is the same regardless of the adhesive properties graded method (along a radius or square). Major stress concentrations are noted at both edges, while the core of the adhesive in all cases is practically inactive. The highest values are noted for the case of the Araldite® adhesive. Regardless of the value of the gradation exponent n, the value of the shear stress only slightly varies.
The analysis of the load-displacement curves in the presence of an adhesive with a functionally graded between the properties of two adhesives (Adekit ® A-140/ Araldite ® AV138) is presented in Fig. 29. It is clearly noted that the strength of the assembly can be improved if, on the one hand, the choice between the two adhesives is optimized so that the one with the highest ductility is placed at the level of the two edges in order to attenuate the maximum high stress concentration and on the one hand the design form of the adhesive gradation is optimized.
The resistance of the single joint assembly in the presence of a graded adhesive where its mechanical properties are the combination of the two mechanical properties of the Fig. 28 Distribution of the von Mises stress in the adhesive joint according to the lap length for P = 100 N/mm for a gradation according to a radius and b square with different volume fraction exponent values (n) two adhesives Adekit ® A-140 and Araldite ® is presented in Fig. 29. It is clearly noted that the presence of simple adhesive of type Adekit ® A-140 joint has more resistance for the assembly compared to that of simple adhesive joint of type Araldite ® . However, if an FGA is used, the resistance of the assembly with an Araldite ® adhesive will be reinforced in the presence of the layers of Adekit ® A-140. The design of the gradation has an important role on the value of the maximum tensile force and therefore on the resistance of the connection. Graded radius concept design provides better strength for the assembly.

Conclusion
In this work, the effect of the substrates' nature on the strength of adhesive bonded joints was analyzed by finite element techniques. AL2024-T3, composite and FGM, plates were considered. TTO mixing laws were used to describe the variation of the material properties of the FGM (Young's modulus, density and Poisson's ratio) according to the substrates' thickness. Different techniques to model the adhesive joint have been demonstrated, namely the cohesive zone technique, the VCCT technique and taking into account an adhesive as a third material. Analysis of the results in the form of load-displacement curves, and variation of the constraints according to the overlap length made it possible to draw the following conclusions: • The results of the finite element numerical analysis using the ABAQUS computer code showed the effectiveness of this method in determining the various stresses in the adhesive joint. • The von Mises stresses are concentrated at the free edges of the adhesive layer, while the adhesive core remains practically inactive, whatever the nature of the two substrates. • The peel and shear stress distributions are symmetrical when the joint are made of similar material. • The mechanical behavior of the adhesive in a bonded joint depends essentially on the mechanical properties of the substrates. • A substrate with low mechanical properties transmits less stress to the adhesive, however, those with high mechanical properties transmit almost all of the applied load to the adhesive. • In the case of a joint made of FGM, the maximum value of the von Mises stress depends on the nature of the face of the adhesive which is in contact with the adhesive layer. • The maximum stress is found for the case where the ceramic layer is in contact with the adhesive, while the lowest value of the von Mises stress is for the case where the face of the aluminum substrate is in contact with the adhesive. • The different numerical techniques used to model the adhesive lead to the same results whether in terms of load-displacement curves with the experimental test and also the variation of stresses according to the length of covering with the different analytical models.