Composites: Part A 34 (2003) 135–150 www.elsevier.com/locate/compositesa Thermal and geometrically non-linear stress analyses of an adhesively bonded composite tee joint M. Kemal Apalak*, Recep Gunes, M. Onder Turaman, A. Alper Cerit Department of Mechanical Engineering, Erciyes University, Muhendislik Fakultesi, Kayseri 38039, Turkey Received 3 January 2002; revised 24 October 2002; accepted 5 November 2002 Abstract Adhesive bonding technique is used successfully for joining the carbon fibre reinforced plastics to metals or composite structures. A good design of adhesive joint with either simple or more complex geometry requires its stress and deformation states to be known for different boundary conditions. In case the adhesive joint is subjected to thermal loads, the thermal and mechanical mismatches of the adhesive and adherends cause thermal stresses. The plate-end conditions may also result in the adhesive joint to undergo large displacements and rotations whereas the adhesive and adherends deform elastically (small strain). In this study, the thermal and geometrically non-linear stress analyses of an adhesively bonded composite tee joint with single support plus an angled reinforcement made of unidirectional CFRPs were carried out using the non-linear finite element method. In the stress analysis, the effects of the large displacements were considered using the small displacement – large displacement theory. The stress states in the plates and the adhesive layer of the tee joint configurations bonded to a rigid base and a composite plate were investigated. An initial uniform temperature distribution was attributed to the adhesive joint for a stress free state, and then variable thermal boundary conditions, i.e. air flows with different velocity and temperature were specified along the outer surfaces of the tee joints. The thermal analysis showed that a non-uniform temperature distribution occurred in the tee joints, and high heat fluxes took place along the free surfaces of the adhesive fillets at the adhesive free ends. Later, the geometrical non-linear thermal-stress analysis of the tee joint was carried out for the final temperature distribution and two edge conditions applied to the edges of the vertical and horizontal plates (HP). High stress concentrations occurred around the rounded adherend corners inside the adhesive fillets at the adhesive free ends, and along the adhesive – composite adherend interfaces due to their thermal – mechanical mismatches. The most critical joint regions were adhesive fillets subjected to high thermal gradients, the middle region of HP, the region of the vertical plate corresponding to the free end of the vertical adhesive layer – left support interface. In addition, the support length had a small effect of reducing the peak stresses at the critical adherend and adhesive locations. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: B. Adhesion; E. Thermal analysis; Tee joint 1. Introduction Adhesive bonding technique has been used successfully to join structural components with different mechanical and thermal properties, such as composites and metals. A good design and a suitable adhesive type can provide a significant strength for the adhesive joint. However, adhesion mechanism, design parameters, i.e. overlap length, adhesive thickness, adherend edge geometry, adhesive fillet geometry influence the strength of adhesive joints. In order to design a suitable adhesive joint for specific purposes, the adhesion * Corresponding author. Tel.: þ90-352-437-4901; fax: þ 90-352-4375784. E-mail address: apalakmk@erciyes.edu.tr (M. K. Apalak). mechanism and the deformation and stress states of the adhesive and adherends based on joint geometry, loading conditions and especially material properties should be known in detail. Therefore, extensive analytical and experimental studies have been carried out on the stress and deformations of the adhesive joints, especially on the single, double-lap joints and their modifications due to their simple geometries and the ease of testing. The elastic stress analyses of the different types of adhesive joints in which the adhesive and adherends were treated as elastic materials showed that the stress concentrations occurred around the adhesive free ends, thus at the free ends of the adhesive –adherend interfaces, a large region in the middle of the adhesive overlap region experienced low stress distributions. Therefore, the studies 1359-835X/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 8 3 5 X ( 0 2 ) 0 0 2 3 3 - 6 136 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 have concentrated on the stress concentrations at the adhesive free ends, called edge effect, and on reducing the peak stresses [1 –3]. The effects of the geometrical factors, such as adhesive thickness, overlap length, adherend thickness and adhesive/adherend modulus ratio on the peak stresses at the adhesive free ends were investigated in detail. However, most studies assumed the adhesive free ends to have square shape. Adams and Wake showed that in case the adhesive layer have adhesive accumulations at its free ends, called adhesive spew fillets, the peak stresses could be reduced depending on the shape and size of the adhesive fillets [2]. In addition, it is possible to reduce peak stresses modifying the free edges of the adherends, i.e. tapering partly or sharpening a knife-edge. The adhesive joints may undergo thermal loads as well as structural loads in practice. In this case, thermal strain distributions may cause thermal stress distributions in the adhesive joints as a result of thermal and mechanical mismatches of the adhesive and adherends. Some of the recent studies have accounted for the effect of the thermal loads on the strength of the adhesive lap joints with some limitations. Thus, a uniform temperature distribution is attributed to the adhesive joints or a constant temperature along their outer boundaries is specified, the effect of a nonuniform temperature distribution in the adhesive joint as a consequence of variable thermal boundary conditions, i.e. fluid-adherend or adhesive interaction was not considered. Ioka et al. [4] used the boundary element method to predict thermal residual stresses on the adhesive –adherend interfaces and on the free surfaces of bonded dissimilar materials, and showed that thermal stress singularity disappeared for certain range of wedge angles of a pair of materials. Reedy and Guess [5] studied thermal residual stresses arising due to the cooling of the adhesive joint, and their effect on the joint strength. They showed that the peak adhesive stresses in the yield zone at the interface corner could decay significantly when sufficient time is given. Kim et al. [6] also investigated thermal stresses in adhesively bonded tubular single lap joint considering non-linear adhesive properties and presented a failure model. Kim and Lee [7] presented a method for determining an optimum design of an adhesive tubular lap joint based on failure modes due to thermal stresses induced by fabrication. Nakano et al. [8] investigated thermal stress distributions around the circular holes and rigid fillers in the adhesive layer of an adhesive butt joint subjected to a non-uniform temperature distribution, and found that the size and location of the circular holes and rigid fillers affected evidently the thermal stresses occurring on the adhesive – adherend interface and at the hole and filler peripheries. Nagakawa et al. [9] showed that thermal stresses around hole defects located near the centre of the adhesive layer were larger than those around the hole defects located near the free surface of the adhesive in an adhesive butt joint with adhesive layer including adhesive defects subjected to uniform temperature changes. Katsuo et al. [10] investigated the effects of thermal expansion coefficient and Young modulus ratios of the adhesive and adherends on the transient thermal stress distribution in an adhesively bonded butt joint whose upper and lower surfaces are assumed to have different constant temperatures. In case the adhesive joints experience large loads, two types of non-linearities affect the stress and deformation states of the adhesive joints. First, the loading and boundary conditions may cause large displacements and rotations in the adhesive joints with complex geometry, i.e. tee and corner joints, whereas the strains are still small. Therefore, stress concentration regions are affected considerably from the large displacements in the adhesive joint. In case the stress and strain variations are dependent on deformed geometry, this non-linearity is termed as the geometrical non-linearity. The small strain –small displacement (SSSD) approach of the elasticity theory cannot predict the elastic stress and deformation states in the adhesive joints. Therefore, the small strain– large displacement (SSLD) approach, which is an extension of the SSSD theory, should be applied to the adhesive joint problem. Secondly, the material properties of the adhesive and adherends may introduce a non-linearity, called the material non-linearity. In this case, the stress variations should obey constitutive laws of adhesive and adherend materials. Sawyer and Cooper [11] took into account the effect of the geometrical non-linearity on the stress distribution in an adhesive single lap joint whose adherends were preformed in order to reduce the angle between the line of the applied in-plane force and the bond line. The applied load has an eccentricity since the adherends are deformed as the load is applied. That the bending moment is dependent on the applied load makes the problem geometrical non-linear. Their analyses showed that pre-forming the adherends reduced the peel and shear stresses at both free ends of the overlap region and provided a more uniform shear stress distribution in the adhesive layer. Adams [12], Adams and Harris [13,14] investigated the failure modes and loads of an adhesive single lap joint and the effects of the local geometry of the adherend edges and adhesive accumulations at the free ends on the peak stresses using the non-linear finite element method. They considered the non-linear behaviour of both adhesive and adherends and the large displacement and rotation effects (geometrical non-linearity). They showed that the non-linear constitutive behaviour of the adhesive and adherends played important role on the joint strength, i.e. failure modes and loads, and that significant increases in the joint strength might be achieved by filleting the adhesive accumulations at the free edges of the of the overlap region. Adams et al. [15] also investigated the shear and transverse tensile stresses in carbon fibre reinforced plastic/steel double lap joints using an elasto-plastic constitutive law for the adhesive material (rubber-modified epoxy), and achieved considerable increases in the joint strength modifying the local geometry M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 of the critical zones at the edges of the overlap region of the composite-steel double lap joints. Czarnocki and Pierkaski [16] carried out three dimensional stress analysis of an adhesive single lap joint treating the adhesive as a non-linear material, and showed that the joint width also played an important role on the peak adhesive stresses. Reddy and Roy [17] investigated the effect of large displacement gradients on the adhesive stresses at the free ends of the adhesive layer in single lap joint for different loading conditions. They applied the geometrically non-linear finite element method based on the large displacement-small strain approach to an adhesive single lap joint for different plate end conditions, and showed that the large displacement gradients had an evident effect on the peak stresses at the adhesive free ends. Edlund and Klarbring [18,19] developed a method for analysing the three dimensional stress and deformation states of an adhesive single lap joint considering both the geometrical non-linearity and the non-linear material behaviour of the adhesive and plates. Due to their complex and unbalanced geometry the stress analyses of the adhesive joints, i.e. corner or tee joints are more complicated than those of the single and double-lap joints. Shenoi and Violette [20] also carried out the theoretical and experimental stress analyses of adhesively bonded composite tee joints used in small boats and investigated the influence of joint geometry on the ability to transfer out-of-plane loads. Apalak et al. [21,22] carried out the elastic stress analyses of two types of tee joints based on the SSSD theory for different loading conditions. Apalak and Engin [23] analysed the effect of the large displacement gradients on the elastic adhesive and adherend stresses of an adhesively bonded double containment cantilever joint using the non-linear finite element technique based on the SSLD theory and showed that the SSSD theory might be misleading for predicting the elastic stress and strain states of the adhesive joints. Apalak [24] investigated the elastic stress and deformations in the various adhesively bonded corner joints considering large displacement effects, and showed that the geometrical non-linearity affected the elastic stress levels occurring in the adherends and adhesive layer. The previous thermal stress analyses of the adhesive joints assumed that the heat transfer to occur by conduction through the adhesive joint members. A uniform temperature distribution was prescribed along the boundaries of the adhesive joints, or a constant temperature distribution was set through the adhesive joint. However, the effects of variable thermal boundary conditions and the heat transfer by convection were neglected. The adhesive joint may interact with fluids at a specific velocity, and temperature along its boundaries. In this case, the heat transfer takes place by convection between the outer surfaces of the adhesive joint and the fluid, and by conduction through the adhesive joint. The flow direction, velocity and temperature of 137 the fluid, the free-surface temperatures of adherends and adhesive play important role on the heat transfer by convection between the fluid and the outer surfaces of the adhesive joints since the convective heat transfer is dependent on the film coefficient being function of these factors. In this case, a non-uniform temperature distribution through the adhesive joint would be expected to occur contrary to the constant temperature distribution considered in the previous studies. Non-uniform thermal strain and stress distributions would occur as a result of non-uniform temperature distribution and the thermal and mechanical mismatches of the adhesive and adherend materials. Apalak and Gunes [25] investigated elastic thermal stress distributions in the adhesive layer and adherends of an adhesively bonded single lap joint considering the effects of variable thermal boundary conditions and the large displacement gradients for the different plate end conditions. They showed that non-uniform temperature and thermal strain distributions occurred through the adhesive joint and that the stress concentrations occurred inside the adhesive fillets around the adhesive free ends and along the adhesive – adherend interfaces. As a result the variable thermal conditions affect considerably the stress and deformation states of the adhesive joints their consideration in the analysis and design of the adhesive joints would give more realistic results. In this study, the steady state thermal analysis and geometrically non-linear stress analysis of an adhesively bonded tee joint were carried out using the non-linear finite element method in which the variable thermal boundary conditions were prescribed along the outer surfaces of the adhesive joint, and for different plate end conditions. 2. Thermal model and finite element formulation Application of the first law of thermodynamics to a differential control volume yields
 ›T þ vT 7T ¼ ffl q þ 7ðD7TÞ ð1Þ rc ›t with the Fourier’s law q ¼ 2D7T where v T is velocity vector for mass transport of heat, q is heat flux vector, ffl q is heat generation rate per unit volume, and D is conductivity matrix. The present problem assumes that the adhesive joint is subjected to the specified convection surfaces (Newton’s law of cooling) as q·n ¼ 2hm ðT1 2 TS Þ ð2Þ T n D7T ¼ hm ðT1 2 TÞ where n is unit outward normal vector, hm is film coefficient, T1 is bulk temperature of the adjacent fluid and TS is temperature at the surface of the model. Pre-multiplying Eq. (1) by a virtual change in temperature dT, integrating 138 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 over the volume of the element, and combining with Eq. (2) yield 
 ð
›T T T þ v LT þ L dTðDLTÞ dV rc dT ›t V ð ð ¼ dThm ðT1 2 TÞdS þ dTffl q dV ð3Þ S V where dX is an arbitrary infinitesimal line vector in the initial position. The small strain tensor and the rotation tensor are ! ›uj 1 ›ui ð10aÞ þ 1ij ¼ 2 ›xj ›x i and where LT ¼  › ›x › ›y 1 Vij ¼ 2  › ›z is a vector operator. The temperature is dependent on both space and time. Therefore, the temperature can be written in terms of element shape functions N and nodal temperature vector Te as T ¼ NT Te ð4Þ and the time derivatives of Eq. (4) may be written as ›T ›T ¼ NT e ¼ NT T_ e ›t ›t ð5Þ ›uj ›u i 2 ›x j ›xi ! ð10bÞ respectively. The large displacement-gradient components make the strain to be characterized from the initial state more difficult than in the small-strain case. Lagrangian formulation allows the finite-strain to be defined based on the material co-ordinates in the undeformed configuration. The deformation equations for the movement of a particle from its initial position X to the current position x are given as x ¼ xðX; tÞ or xi ¼ xi ðX1 ; X2 ; X3 Þ ð11Þ An arbitrary infinitesimal material vector dX at X can be associated with a vector dx at x as follows dT and LT also become dT ¼ {dTe }T N and LT ¼ BTe ð6Þ dx ¼ F·dX ¼ dX·FT ð12Þ T where B ¼ LN : The variational form of Eq. (3) can be written as ð ð ðrc{dTe }T NNT T_ e ÞdV þ ðrc{dTe }T NvT BTe ÞdV V V þ ð T V ¼ ð T ð{dTe } B DBTe ÞdV T S T {dTe } Nhm ðT1 2 N Te ÞdS þ ð ¼ ð V {dTe } Nffl q dV ð7Þ V V S T1 hm N dS 2 ð S hm NNT Te dS þ z F ¼ x7 ð13Þ In terms of the strain tensor the change in the squared length of the material vector dX is given as follows ðdsÞ2 2 ðdSÞ2 ¼ 2 dX·E·dX or ðdsÞ2 2 ðdSÞ2 T If r is assumed to remain constant over the volume of the element, and {dTe}T is dropped, Eq. (7) is reduced to the final form as [26] ð ð ð r cNNT T_ e dV þ r cNvT BTe dV þ BT DBTe dV V using the deformation gradient vector ð Nffl q dV ð8Þ V ¼ 2 dXI EIJ dXJ ð14Þ 2 The new squared length (ds) of the element is written in terms of the Green deformation tensor C referred to the undeformed configuration as follows ðdsÞ2 ¼ dX·C·dX or ðdsÞ2 ¼ dXI CIJ dXJ Comparison of the Eqs. (14) and (15) gives the relationship between the strain tensor E and the Green deformation tensor C 2E ¼ C 2 1 or 2EIJ ¼ CIJ 2 dIJ 3. Small strain – large displacement theory ð16Þ 2 The displacements and their gradients are assumed to be infinitesimal in the SSSD theory, and the current configuration of a body is compared with its initial state. The components of the unit relative displacement vector are defined as dui ›ui dXj ¼ ›Xj dS dS ð15Þ ð9Þ The new squared length (ds) of the element is written using the deformation gradient tensor as follows ðdsÞ2 ¼ dx·dx ¼ ðdX·FT ÞðF·dXÞ ¼ dX·ðFT ·FÞ·dX ð17Þ Comparison of Eqs. (15) and (17) shows that the Green deformation tensor C ¼ FT ·F or CIJ ¼ ›x k ›x k › XI › XJ ð18Þ M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 From Eq. (16) the strain tensor becomes
 1 ›x k ›x k T 1 2 dIJ E ¼ 2 ðF ·F 2 1Þ or EIJ ¼ 2 › XI › XJ ð19Þ Using the same reference axes for both xi and Xi, and using lower-case subscripts for both, we have x i ¼ X i þ ui ð20Þ with displacement components ui ¼ ui ðX1 ; X2 ; X3 ; tÞ: In terms of the displacements, the general expression for Eij in Eq. (19) takes the form " # ›uj 1 ›ui ›uk ›uk þ þ ð21Þ Eij ¼ 2 › Xj › Xi › Xi › Xj When Eqs. (10a) and (21) are compared it is evident that the large displacement theory is an extension of the small displacement theory since it includes the squares of the displacement gradients. In case the displacement gradients are not small compared to unity, the large displacement theory should be used in order to consider their non-linear effects on the stress and deformation states of the structures [27,28]. 4. Non-linear equilibrium equations 139 where N is a shape function array whose coefficients are functions of the initial position x within the element included in the expression of displacement within an element u ¼ Np ð27Þ where the nodal displacement vector p ¼ ½u1 ;u2 ;…;un T ð28Þ and the strain matrix B ¼ B0 þ BL ð29Þ where B0 is the linear strain matrix being a function of the shape functions only, while BL is the non-linear strain matrix being a function of the shape functions and displacements [23]. The components of Green’s strain vector E can be written as h i E ¼ B0 þ 12 BL p ð30Þ in terms of the linear and non-linear Green’s strain vectors E0 ¼ B0 p and EL ¼ 12 BL p ð31Þ The non-linear equilibrium Eq. (26) can be written as ð cðpÞ ¼ BT s dV 2 R ¼ w 2 R ¼ 0 ð32Þ V In order to establish the non-linear equilibrium equation for the deformed body subjected to the external loads, the virtual internal and external works done on the body are equated. The non-linear equilibrium equation from the virtual-work equation given in terms of the Lagrangian coordinate system is given as [26,29 –31] ð ð ð dET s dV ¼ rduT q dV þ duT q0 dA ð22Þ V V A where the external work is due to the virtual displacements du acting on the surface tractions, extending over the initial undeformed surface A and, given by q0 ¼ ½ q01 q03  q02 T ð23Þ and the body forces per unit mass, acting within the undeformed volume V given by q ¼ ½ q1 q2 q3  T ð24Þ r being the density of the undeformed body. The total Lagrangian virtual work equation (22) can be approximated by the finite element idealization as follows ð ð ð ð25Þ dpT BT s dV ¼ dpT rNT q dV þ dpT NT q0 dA V V A Since the virtual nodal displacements dp are arbitrary, Eq. (25) can be written as ð ð ð ð26Þ BT s dV ¼ rNT q dV þ NT q0 dA V V A where R is the right hand side of Eq. (26) for convenience and c(p) is termed the residual. The Newton – Raphson method is used for the solution of the assembled non-linear equations. The solution is achieved when c(p) is reduced to zero or a given convergence criterion is satisfied [23,26, 29 – 31]. 5. Joint configuration Since the adhesive tee joint consists of a HP, a support, a vertical plate (VP), and an angled reinforcement member being an extension of VP and adhesive layer bonding composite (CFRP) adherends, it is called an adhesively bonded composite tee joint with single support plus angled reinforcement as shown in Fig. 1. The VP and HP lengths L ¼ 360 mm, plate thickness t ¼ 4 mm, support length a ¼ 30 mm and adhesive thickness (d ¼ 0.5 mm was taken as main joint dimensions (see Fig. 1). In practice, since the adhesive layer is pressed between the plates by applying pressure to the plates some amount of adhesive are accumulated around the free ends of the adhesive layer, called adhesive fillets. The shape and size of the adhesive fillets play important role on the magnitude and location of the peak stresses in the adhesive layer [2]. Therefore, the adhesive fillets at the adhesive free ends were considered as shown in Fig. 1, and their shape was idealized to a triangle with a height and width ft twice the adhesive thickness d. 140 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 Fig. 1. Dimensions of a tee joint with single support plus angled reinforcement bonded to a flexible base. The previous studies on the stress analyses of the adhesive lap joints showed that the stress concentrations occurred around the adherend corners at the free ends of the adhesive layer [2,3]. The geometry of the adherend corners affects considerably the local stress distributions. In general, the adherend corners are assumed to be sharp for the simplicity of the analyses. However, the geometrical and material discontinuities result in stress singularities at the adherend corners. Adams and Harris [14] investigated the effect of the local geometry on the joint strength of a single lap joint using a model consisting adhesive layer around a rigid rounded corner in order to simplify the analysis, and found that the significant increases in the joint strength may be achieved in single lap joints by filleting the adhesive layer at the ends of the overlap region. Therefore the adherend corners were rounded with a radius R3 of 0.1d. The plates, support and angled reinforcement member of the adhesive tee joint were made of carbon fibre reinforced plastic, IM6/3501-6. An epoxy based adhesive was used to bond plates, support and angled reinforcement member. Each unidirectional ply consists of (IM6) carbon fibres and (3501-6) epoxy with thickness of 0.1335 mm (Table 1). The CFRP laminates (plates and supports) were made of thirty unidirectional IM6/3501-6 plies. The engineering constants of the unidirectional fibre-reinforced lamina IM6/3501-6 were computed by using the micro-mechanics approach based on the engineering constants of the IM6 fiber and the 3501-6 epoxy in the material coordinates ðx1 ; x2 ; x3 Þ as [32] E 1 ¼ Ef Vf þ Em Vm ; E2 ¼ y 12 ¼ y f Vf þ y m Vm ; Ef E m ; E f Vm þ Em Vf G12 ¼ Gf Gm Gf Vm þ Gm Vf ð33Þ where E1 is the longitudinal modulus, E2 is the transverse modulus, y 12 is the major Poisson’s ratio, G12 is the shear modulus, and the shear modulus of the fiber and the matrix Gf ¼ Ef ; 2ð1 þ y f Þ Gm ¼ Em 2ð1 þ y m Þ ð34Þ respectively. The following reciprocal relations also exist y 21 y y 31 y y 32 y ¼ 12 ; ¼ 13 ; ¼ 23 or E2 E1 E3 E1 E3 E2 ð35Þ y ij y ji ¼ Ei Ej In case of a transversely isotropic material with the 2– 3 plane as the plane of isotropy E 2 ¼ E3 ; G12 ¼ G13 y 12 ¼ y 13 ð36Þ Table 1 Base properties of graphite fibre (IM6), epoxy (3501-6) and lamina (IM6/3501-6) Property Unit Graphite fibre (IM36) Epoxy (3501-6) Lamina (IM6/3501-6) E11 E22 and E33 G12 and G13 G23 y 12 and y 13 y 23 a11 a22 and a33 Heat capacity (cp) Heat conduction (k11) GPa GPa GPa GPa 259.3020 13.9380 50.9910 8.2800 0.2600 0.3300 0.8543 3.2374 0.2000 83.5090 4.347 4.347 1.598 1.598 0.360 0.360 40.470 40.470 0.250 0.179 166.100 9.618 6.316 3.584 0.297 0.342 20.460 19.530 m 1/8C m 1/8C cal/g 8C watt/m 8C 82.726 141 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 The generalized Hooke’s law for an anisotropic material under isothermal conditions is given by or sij ¼ Cijkl 1kl In order to determine the strain – stress relations of a laminate, it is necessary to convert strain components referred to the material (lamina) coordinate system ðx1 ;x2 ;x3 Þ to those referred to the problem (laminate) coordinate system ðx;y;zÞ as 3 2 2 32 3 2 1xx 11 m n2 0 0 0 2mn 7 6 76 7 6 7 61 7 6 2 12 7 6 yy 7 6 n m2 0 0 0 mn 76 7 7 6 76 6 7 7 6 76 6 6 6 1zz 7 6 0 13 7 0 1 0 0 0 7 7 7 6 76 6 7 ð41aÞ 7 ¼6 76 6 7 6 6 21 7 6 0 7 1 0 0 m n 0 7 76 6 yz 7 6 4 7 7 6 76 6 7 7 6 76 6 7 76 6 21xz 7 6 0 1 0 0 2n m 0 5 5 4 54 5 4 ð37Þ or in contracted notation [32] si ¼ Cij 1j ð38Þ If the coordinate planes are chosen parallel to the three orthogonal planes of material symmetry, the Eq. (38) can be written in matrix form in the material coordinates ðx1 ; x2 ; x3 Þ as 32 3 2 2 3 11 C11 C12 C13 0 0 0 s1 76 7 6 6 7 76 7 6 6 7 6 C21 C22 C23 0 6 s2 7 0 0 7 6 12 7 76 7 6 6 7 76 7 6 6 7 76 7 6 6 7 0 0 7 6 13 7 6 C31 C32 C33 0 6 s3 7 76 7 6 6 7 ð39Þ 76 7 6 7 ¼6 76 7 6 6 7 0 0 C44 0 0 7 6 14 7 6 0 6 s4 7 76 7 6 6 7 76 7 6 6 7 6 7 6 0 6s 7 0 0 0 C55 0 7 7 6 15 7 6 6 57 54 5 4 4 5 s6 m 0 0 0 0 0 C66 16 m The substitution of the engineering constants in to the inverse relation of Eq. (39) gives 3 2 1 y 21 y 31 2 0 0 0 2 7 6 E1 E2 E3 7 6 7 6 72 3 2 3 6 7 s 6 y 1 y 12 32 11 7 1 62 0 0 0 2 76 7 6 7 6 E E E 1 3 2 76 7 6 6 7 6 7 6 s2 7 6 12 7 6 76 7 6 7 6 y 76 7 6 7 6 2 13 2 y 23 1 0 0 0 7 7 6 7 6 76 s 7 6 13 7 6 E1 E E 76 3 2 6 7 6 6 37 7 7 ð40aÞ 6 7 ¼6 76 7 6 7 6 76 1 6 14 7 6 6 s4 7 7 0 0 0 0 76 7 6 7 6 0 7 6 7 6 G23 76 7 61 7 6 76 6 57 6 6 s5 7 7 5 4 5 6 4 7 1 7 6 0 0 0 0 0 7 6 16 m 6 G13 7 s6 m 7 6 7 6 4 1 5 0 0 0 0 0 G12 {1}m ¼½Sm {s}m 21xy p 2mn 22mn 0 0 0 m2 2n2 ð40bÞ 16 m where m¼cos u; n¼sin u; and similarly {1}p ¼½RT {1}m ð41bÞ {s}p ¼½RT {s}m ð42Þ In order to relate compliance coefficients in the two coordinate systems, lets substitute Eqs. (40b) and (42) into Eq. (41b) as {1}p ¼½RT {1}m ¼½RT ð½Sm {s}m Þ¼½RT ½Sm ð½R{s}p Þ {1}p ¼½Sp {s}p ð43Þ  and ½Sm ;½S where ½Sp ;½S T  ½S¼½R ½S½R The thermal expansion coefficients aij are also transformed ða12 ¼ a13 ¼ a23 ¼0Þ as axx ¼ a11 m2 þ a22 n2 ; ayy ¼ a11 n2 þ a22 m2 ; ð44Þ 2axy ¼2ða11 2 a22 Þmn; 2axz ¼0; 2ayz ¼0; azz ¼0 In the thermal stress analysis of the adhesive tee joint, an eight-noded isoparametric quadratic quadrilateral plane element with four integration points was used to model Fig. 2. Mesh details of a tee joint with single support plus angled reinforcement bonded to a flexible plate. 142 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 the horizontal and VPs, support, angled reinforcement member and adhesive layers. The compliance matrix of the adherends made of the unidirectional IM6/3501-6 laminate was computed as mentioned before. Since the stress concentrations occur around the adhesive free ends, mesh areas around the adhesive free ends were refined in order to obtain a reasonable accuracy in the computations as shown in Fig. 2. 6. Thermal analysis The adhesive joints consist of adherends and adhesive layer with different mechanical and thermal properties. The thermal loads prescribed along the boundaries of the adhesive joints cause a non-uniform temperature distribution in the adhesive joint as a result of the heat transfer by conduction throughout the adhesive joint. However, the temperature distributions, consequently thermal strain distributions are non-uniform due to mismatches in the thermal conductivities of the adhesive and adherends. Since the thermal strains are not compatible along the bi-material interfaces, the thermal stresses arise along these interfaces and in their neighbourhoods as a result of the thermal and mechanical mismatches of the adhesive and adherends. The final temperature distribution of the adhesive joint should be known in order to compute the thermal strains, so this requires the thermal analysis of the adhesive joint subjected to thermal conditions. In the previous thermal stress analyses of the bonded lap joints considering the conductive heat transfer, a uniform temperature difference all over the adhesive joint or a constant temperature along its boundaries was prescribed for the simplicity of the analysis. In fact, the adhesive joints interact with an ambient having temperature and flow conditions along their outer surfaces. However, these variable thermal boundary conditions require a thermal analysis considering the convective heat transfer between the fluid and the outer surfaces of the adhesive joints and the conductive heat transfer throughout the adhesive joint members. In this study, two configurations of the adhesive tee joint are considered; one bonded to a rigid base and the other is bonded to a flexible HP. The variable thermal boundary conditions are prescribed along the outer surfaces of the adhesive bonded tee joints as shown Fig. 3. It is assumed that the left surfaces of the VP and the left support, and the upper surface at the left side of the HP are subjected to an air flow at a temperature T1 ¼ 80 8C and a velocity U1 ¼ 1 m/s whereas the right surface of the VP and the angled reinforcement member, and the upper surface at the right side of the HP experience an air flow at a temperature T1 ¼ 20 8C and a velocity U1 ¼ 1 m/s. In case of the rigid base, the lower surface of the horizontal adhesive layer was assumed to be an adiabatic surface whereas in case of the flexible base (Fig. 3(b)), an air stream with a temperature T1 ¼ 40 8C and a velocity U1 ¼ 1 m/s is prescribed along the bottom surface of the HP. In both cases, all members of the adhesive tee joint were assumed to have an initial temperature distribution at Ti ¼ 20 8C. During the fluidadherend or adhesive interaction, the flow direction of the air with respect to the adherend or adhesive surfaces play important role on the convective heat transfer taking place between the fluid (air) and the outer surfaces of the adhesive tee joint since the film coefficients along the adherend surfaces are dependent on the flow direction, the fluid temperature and velocity, and the surface temperature. The air flow was considered as normal to the adherend and adhesive surfaces pointed by surface numbers 1 –2, 2– 3, 4 – 5, 5 –6, 7– 8, 8 – 9 and 10 – 11, 11 –12, 13– 14, 14 – 15 and as horizontal along the other surfaces (Fig. 3(a) and (b)). The averaged film coefficients can be computed using the different empirical formula depending on the flow direction of the air flows [33]. In the case of the vertical air Fig. 3. Thermal boundary conditions of a tee joint with single support plus angled reinforcement bonded to (a) a rigid base, and (b) a flexible plate. 143 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 stream the film coefficient is given as
 U1 Deqv 0:731 l0 hm ¼ 0:205 Deqv v Table 2 Thermal properties of air ð45Þ where U1 is air velocity (m/s), Deqv is equivalent diameter (m), v is kinematic viscosity of the air (m2/s) and l0 is thermal conductivity of the air (w/m 8C). In the case of the horizontal air flow, the film coefficient can be computed as follows hm ¼ Nul L ð46Þ (T1 þ TS)/2 (8C) l0 (w/m 8C) l (kcal/m 8C) v (m2/s) £ 106 m (kg/ms) £ 105 cp (kcal/kg 8C) 20 0.0257 0.0221 15.11 1.82 0.240 30 0.02640 0.02270 16.04 1.865 0.2405 50 0.0278 0.0239 17.935 1.955 0.241 temperature T1 as follows: T1 þ TS 2 where l is thermal conductivity of the air (kcal/m h 8C), and Nusselt number Tf ¼ Nu ¼ 0:836Re1=2 Pr1=3 The steady-state thermal analyses of the adhesive tee joints were carried out for the thermal boundary conditions (Fig. 3). The film coefficients along the adherend surfaces were computed considering the thermal properties of the air based on the averaged temperature Tf (Table 2). The temperature distributions in the adherends and in the adhesive fillets of the joint region are plotted in Figs. 4 and 5 for the joints with rigid base and flexible base, respectively. In case of the rigid base, since a higher fluid temperature is specified at the left of the adhesive joint the heat transfer takes place from the left side of the adhesive tee joint to its right side; therefore, the temperature levels decrease uniformly to its right side (62.5 – 52.3 8C, Fig. 4(a)). Similar temperature ð47Þ where Reynolds and Prandtl numbers, respectively Re ¼ U1 L ; v Pr ¼ cp m l ð48Þ where cp is specific heat (kcal/kg 8C) and m is dynamic viscosity (kg/m s). The substitution of Eqs. (47) and (48) into Eq. (46) yields


 U L 1=2 cp m 1=3 l hm ¼ 0:836 1 ð49Þ v l L The thermal properties of the air were determined based on the averaged value of the surface temperature TS and the air ð50Þ Fig. 4. The temperature distributions in critical regions: (a) the joint region, (b) the vertical adhesive fillet, (c) the left horizontal adhesive fillet, and (d) the right horizontal adhesive fillet (rigid base, all temperatures in 8C). 144 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 Fig. 5. The temperature distributions in critical regions: (a) the joint region, (b) the vertical adhesive fillet, (c) the left horizontal adhesive fillet, and (d) the right horizontal adhesive fillet (flexible base, all temperatures in 8C). distributions are observed inside the vertical and horizontal adhesive fillets. The higher temperature levels are observed in the adhesive zones close to the left side of the adhesive joint, and they decrease uniformly (Fig. 4(b) – (d)). In case of the flexible base, the temperature levels decrease continuously from the leftmost edge of the support to the rightmost edge of the angled reinforcement member (57 – 47 8C, Fig. 5(a)). The temperature levels in the vertical adhesive fillet (VAF) decrease from the free surface of the adhesive fillet and the left support – adhesive interface to the VP – adhesive interface (Fig. 5(b)). In the left horizontal adhesive fillet (LHAF), the temperature distribution reaches high levels in the adhesive zones close to free surface of the adhesive fillet, and lower levels in the adhesive regions near the HP – adhesive layer (Fig. 5(c)). In the right adhesive fillet, the temperature distribution varies uniformly from the high levels in the angled reinforcement and HP – adhesive interfaces to lower levels along the free surface of the adhesive fillet (Fig. 5(d)). The thermal analyses of the adhesively bonded tee joints with rigid base and flexible base subjected to variable thermal boundary conditions show that the temperature distributions in the members of the adhesive tee joint are not uniform, and the thermal properties of each joint member (adherends and adhesive layer) affect considerably the heat transfer and its final temperature distribution. 7. Geometrically non-linear stress analysis This section concentrates on the geometrical non-linear stress analysis of the adhesively bonded tee joint. The stress analyses were carried out using the non-linear finite element method based on the SSLD theory. The temperature distributions determined based on the thermal analyses of the tee joints with rigid base and flexible base were applied to two adhesive tee joint configurations as thermal loads. In Fig. 6. Structural bodyconditions of a tee joint with single support plus angled reinforcement bonded to (a) a rigid base (BC-I), and (b) a flexible plate (BC-II). M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 145 Fig. 7. The deformed geometries of the adhesively bonded tee joints bonded to (a) a rigid base (BC-I), and (b) a flexible plate (BC-II). addition, the base of the adhesive tee joint and the upper edge of the VP are fully fixed in the boundary condition-I (BC-I, Fig. 6(a)) whereas the left and right bottom corners of the flexible plate are fully fixed in the boundary condition-II (BC-II, Fig. 6(b)). In the analyses, convergence criteria of 0.01 and 0.0001 were used for forces and displacements, respectively. The deformed and undeformed geometries of the adhesive tee joints bonded to a rigid base and flexible plate are compared in Fig. 7(a) and (b), respectively, based on their geometrical non-linear stress analyses. In the rigid base, (BC-I, Fig. 7(a)) the VP is forced to buckle since the upper edge of the VP is fixed in all directions. The large displacements are observed in its middle region and the free end of the adhesive layer – VP interface is subjected to large bending moment effects. Therefore, the large stresses and deformations can be expected around the adhesive layer– VP interface, and propagate throughout the vertical adhesive layer to the joint base. In case of the tee joint bonded to a flexible base (BC-II, Fig. 7(b)), both the VP and HPs buckle significantly. The VP exhibits lower displacements and rotations than those in case of the rigid base (Fig. 7(a)). However, since the left and right regions of the HP buckle considerably, the high bending moments causing peeling stresses are expected around the free ends of the horizontal adhesive layer. In addition, the free end of the vertical adhesive layer is also subjected to high displacements and rotations, consequently, stress and strain concentrations are expected around this region and propagate throughout the vertical adhesive layer to the joint base similar to the tee joint bonded to a rigid base. The normal stress sxx, szz and shear stress sxz distributions in the magnified joint region are plotted in Fig. 8(a) –(c), respectively, in order to show the stress states of the tee joint bonded to a rigid base (BC-I) in detail. The normal stresses sxx concentrated around the free ends of horizontal adhesive layer, and propagate through the horizontal adhesive layer to Fig. 8. The normal stress (a) sxx, (b) szz, and (c) shear sxz stress distributions in the joint region for the boundary condition BC-I (rigid base, all stresses in MPa). 146 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 Fig. 9. The von Mises stress distributions in critical adhesive regions: (a) the left horizontal adhesive fillet, (b) the right horizontal adhesive fillet, (c) the vertical adhesive fillet, (d) the bottom adhesive layer (rigid base, BC-I, all stresses in MPa). the middle joint region (Fig. 8(a)). The horizontal regions of the left support, the angled reinforcement member and especially the elbow regions of the support and angled reinforcement member also experience high normal stress sxx distributions. The normal stress szz distributions reach considerably high levels through the vertical region of the tee joint in which the left support and the angled reinforcement member being an extension of the VP are bonded by the vertical adhesive layer (Fig. 8(b)). They concentrate around the free end of the vertical adhesive layer, and in the elbow regions of the support and the reinforcement member. In addition, the shear stress sxz concentrations occur around the free ends of the vertical and horizontal adhesive layers and in the elbow region of the tee joint (Fig. 8(c)). However, when the normal and shear stress components are compared, the normal stresses appear as more critical stresses. Since the tee joint bonded to a rigid base experiences considerable stress levels in the joint region, the first adherend failure can be expected from the outer layers of the laminate plates, especially from the outer surfaces of elbows, and in the horizontal sections of the support and the angled reinforcement member. Conversely the first adhesive failure can initiate from the adhesive fillets at the adhesive free ends, and then propagate along the adherend –adhesive interfaces or through the adhesive layer. In this case, the critical adhesive region would be the adhesive fillet at the free end of the vertical adhesive layer. The von Mises stress distributions are also plotted in the left and right horizontal adhesive fillets (RHAF), in the VAF and in the bottom adhesive layer (BAL) of the tee joint bonded to a rigid base (BC-I) as shown in Fig. 9(a) – (d), respectively. The stress concentrations in the LHAF occur around the free end of the adhesive –rigid base interface, and propagate towards the rounded support corner (Fig. 9(a)). The similar stress distributions are observed in the RHAF, such that, the stress concentrations occur around the free end of the adhesive –rigid base interface and spread through the rounded support corner (Fig. 9(b)), but at lower levels 50% in comparison with the LHAF (Fig. 9(a)). In the VAF, the stress distribution varies from high levels at the free end of the adhesive – VP interface to lower levels at the rounded corner of the support (Fig. 9(c)). The adhesive regions close to the support and angled reinforcement – adhesive interfaces experience high stress levels in the BAL (Fig. 9(d)). The LHAF and VAF are more critical adhesive regions. However, since the stresses in the BAL and in the RHAF are close to those in the VAF, the first adhesive failures can be expected in the LHAF and in the VAF. In addition, the joint region of the tee joint bonded to a flexible base (BC-II) is magnified in order to show the stress M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 Fig. 10. The normal stress (a) sxx, (b) szz, and (c) shear sxz stress distributions in the joint region for the boundary condition BC-II (flexible base, all stresses in MPa). distributions in detail, and the normal stress sxx, szz and shear stress sxz distributions in the joint region are plotted in Fig. 10(a) – (c), respectively. The normal stresses sxx concentrate through the horizontal joint region along which the HP, the left support and the angled reinforcement member are bonded by the horizontal adhesive layer (Fig. 10(a)). The left and right free ends of the horizontal adhesive layer and especially the middle region near the horizontal adhesive layer of the HP experience considerably high stress concentrations. Due to high normal stress sxx the middle region of the HP, and its neighbourhood are more critical regions, and the adhesive free ends. The normal stresses sxx are 2– 3 times higher than the normal stress szz (Fig. 10(b)) and the shear stress sxz (Fig. 10(c)), respectively. On the contrary, the substantial normal stress szz distributions occur the joint region through which the support and the VP are bonded the vertical adhesive layer, 147 and in the elbow regions of the support and the angled reinforcement member (Fig. 10(b)). In addition, the VAF undergoes the normal stress szz concentrations propagating through the vertical adhesive layer towards the base of the tee joint. Therefore, the VAF and the elbow regions of the support and the angled reinforcement member are critical regions in terms of the normal stress szz (Fig. 10(b)). The shear stress sxz concentrations are observed around the horizontal and VAFs, and the elbow regions of the support and the angled reinforcement member (Fig. 10(c)). However, the shear stress sxz levels are 50 – 77% lower than the normal stresses. When the normal and shear stress distributions in the joint region are considered the horizontal and vertical adhesive layers, the middle region of the HP and the elbow regions are critical regions. However, the first adhesive failure can be expected inside the adhesive fillets at the adhesive free ends, and especially along the adhesivesupport, the adhesive – HP and the adhesive-angled reinforcement member interfaces in the bottom of the adhesive tee joint. The most critical adherend regions appear as the elbows of the support and the VP, and particularly the middle regions of the HP are bonded to other joint members. It would be better to support the lower surface of the HP by a thin metal sheet (by bonding) in order to prevent the failure of the HP along its lower and upper surfaces. When the normal and shear stresses (Fig. 10) are compared with those in the tee joint bonded a rigid base (Fig. 8) the normal stress sxx levels are eight times higher than those in the rigid base configuration (Fig. 8) whereas the normal stress szz and the shear stress szx distributions remain at close levels. The von Mises stress distributions are plotted inside the LHAF and RHAFs, the VAF and in the BAL as shown in Fig. 11(a) –(d), respectively, in order to show the critical locations at which the first adhesive cracks may initiate. In case of the left and right adhesive fillets, the stress concentrations occur at the free ends of the adhesive – HP interface, and the von Mises stresses propagate uniformly through the adhesive caps to lower levels around the rounded support or angled reinforcement corners (Fig. 11(a) and (b)). Similarly, the free end of the adhesive – VP interface in the VAF experiences high von Mises stress concentrations (Fig. 11(c)), and stress levels decrease uniformly towards the rounded corner of the support. In the BAL, whereas a large adhesive region experiences comparatively low von Mises stress distributions, the adhesive regions close to the adhesive-support, adhesiveangled reinforcement, and adhesive – HP interfaces undergo considerably high stresses (Fig. 11(d)). Although all these adhesive regions exhibit close stress levels, the first adhesive cracks may be expected to initiate inside the LHAF and the VAF. When the adhesive stresses at the critical locations in the rigid base and flexible base configurations are compared, the von Mises stress levels are higher by 32% than those in the LHAF of the rigid base configuration (Fig. 9(a)) whereas the von Mises stresses in 148 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 Fig. 11. The von Mises stress distributions in critical adhesive regions: (a) the left horizontal adhesive fillet, (b) the right horizontal adhesive fillet, (c) the vertical adhesive fillet, (d) the bottom adhesive layer (flexible base, BC-II, all stresses in MPa). the other critical adhesive regions remain at similar levels (Figs. 9 and 11). The stress concentrations occur around the adhesive free ends due to geometrical and material discontinuities is known as the edge effects in the adhesive joints. Since the first probable crack initiations can be expected from these adhesive regions, the stress concentrations should be reduced by modifying adherend edges or increasing bonding area [1,2]. For this reason, the effect of the support and angled reinforcement lengths on the peak stresses at the critical locations inside the adhesive fillets, and in the horizontal and VPs of the tee joint configurations bonded to a rigid base and a flexible base were investigated for the support and angled reinforcement lengths a ¼ 20, 30, 40, 50 and 60 mm based on both the thermal analysis and the geometrically non-linear elastic stress analysis. In case of the tee joint bonded to a rigid base, the variations of the normal stresses sxx and szz, and the shear stress sxz at the critical locations inside LHAF, the RHAF, the VAF, in BAL and at the VP are tabulated in Table 3 for different support and angled reinforcement lengths. Increasing the support length resulted in decrease of 2.4, 17.6 and 7.4% in the peak normal sxx and szz, and shear sxz stresses in the LHAF, respectively, while they are reduced by 5.9, 16.5 and 7.1% in the RHAF, respectively. However, the support length has a negligible effect on the normal and shear stress components in the VAF, BAL and VP, respectively, except the shear stress in the BAL (a decrease of 6.8%). In addition, after the support length a ¼ 30 mm corresponding to a ratio of the VP length to the support length L/a ¼ 12 its effect on the peak normal and shear stresses become negligible. In case of the flexible base, the variations of the normal sxx, szz and shear sxz stress components at the critical locations inside the LHAF, the RHAF, the VAF, in the BAL, at the VP and the HP are tabulated in Table 4 for different support and angled reinforcement lengths. In the LHAF, the normal stress szz is reduced by 4.2% whereas increasing the support length causes increase of 10 and 5.8% in the normal sxx and shear sxz stresses, respectively. In the RHAF, increasing the support length causes decrease of 2.6, 4.4 and 5.3% in the normal and shear stresses, respectively. Its effect on the normal and shear stresses in the VAF, BAL and at the VP is negligible. However, it results in increase of 14.5, 15.8 and 15% at the normal and shear stresses in the HP. The LHAF, the VAF (Fig. 11(a) and (c)), and the HP (Fig. 10(a) and (c)) are the most critical regions in the tee joint with flexible base. The peak normal and shear stresses in these critical regions cannot be reduced reasonably by increasing the support and reinforcement lengths. Therefore, bonding a thin metal sheet along the lower surface of 149 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 Table 3 Effect of the support and angled reinforcement length a on the peak stress components (in MPa) at the critical adhesive and plate locations of the adhesively bonded tee joint with a rigid base (BC-I, rigid base; Fig. 6(a)) a (mm) sxx PVa szz PVa sxz PVa Table 4 Effect of the support and angled reinforcement length a on the peak stress components (in MPa) at the critical adhesive and plate locations of the adhesively bonded tee joint with a flexible base (BC-II, flexible base; Fig. 6(b)) szz PVa sxz Left horizontal adhesive fillet 20 214.00 0.00 30 214.40 2.86 40 214.90 6.43 50 215.90 13.57 60 215.40 10.00 212.00 211.50 211.50 210.50 211.70 0.00 24.17 24.17 212.50 22.50 28.85 28.81 29.04 29.50 29.36 0.00 20.45 2.15 7.34 5.76 0.00 24.25 26.52 24.82 27.08 Right horizontal adhesive fillet 20 11.60 0.00 30 11.40 21.72 40 11.40 21.72 50 12.20 5.17 60 11.30 22.59 9.50 9.10 9.08 8.09 9.15 0.00 24.21 24.42 214.84 23.68 6.93 6.64 6.62 6.87 6.56 0.00 24.18 24.47 20.87 25.34 29.14 29.20 29.21 29.45 29.15 0.00 0.66 0.77 3.39 0.11 Vertical adhesive fillet 20 11.20 0.00 30 11.20 0.00 40 11.20 0.00 50 10.10 29.82 60 11.10 20.89 13.20 13.30 13.30 14.00 13.20 0.00 0.76 0.76 6.06 0.00 8.19 8.22 8.21 8.45 8.17 0.00 0.37 0.24 3.17 20.24 0.00 0.66 0.66 0.66 0.00 24.41 24.24 24.17 24.13 24.11 0.00 23.85 25.44 26.35 26.80 Bottom adhesive layer 20 11.30 30 11.20 40 11.20 50 11.20 60 11.20 0.00 20.88 20.88 20.88 20.88 11.90 11.70 11.70 11.70 11.60 0.00 21.68 21.68 21.68 22.52 4.09 4.04 4.02 4.01 4.00 0.00 21.22 21.71 21.96 22.20 0.00 0.59 1.48 3.85 3.25 23.79 23.81 23.82 23.76 23.82 0.00 0.53 0.79 20.79 0.79 Horizontal plate 20 22.70 30 25.80 40 26.00 50 20.00 60 23.20 0.00 13.66 14.54 211.89 2.20 3.93 5.30 5.11 4.59 4.55 0.00 34.86 30.03 16.79 15.78 3.52 4.24 4.04 3.12 3.56 0.00 20.45 14.77 211.36 1.14 0.00 0.00 0.00 22.08 20.52 23.60 23.60 23.60 24.00 23.50 0.00 0.00 0.00 1.69 20.42 3.22 3.23 3.23 3.16 3.21 0.00 0.31 0.31 21.86 20.31 a (mm) Left horizontal adhesive fillet 20 221.20 0.00 30 220.90 21.42 40 220.70 22.36 50 222.30 5.19 60 220.80 21.89 217.60 215.70 214.90 213.30 214.50 0.00 210.80 215.34 223.86 217.61 213.60 213.00 212.80 213.10 212.60 0.00 24.41 25.88 23.68 27.35 Right horizontal adhesive fillet 20 211.80 0.00 30 211.20 25.08 40 211.10 25.93 50 211.10 25.93 60 211.20 25.08 212.70 211.40 210.80 29.46 210.60 0.00 210.24 214.96 225.51 216.54 27.06 26.76 26.60 26.72 26.56 Vertical adhesive fillet 20 212.30 0.00 30 212.40 0.81 40 212.30 0.00 50 210.10 29.76 60 211.10 21.63 214.50 214.70 214.70 215.50 214.60 0.00 1.38 1.38 6.90 0.69 Bottom adhesive layer 20 215.30 30 215.40 40 215.50 50 215.50 60 215.50 0.00 0.65 1.31 1.31 1.31 215.10 215.20 215.20 215.10 215.10 0.00 0.45 0.45 21.57 20.22 233.80 234.00 234.30 235.10 234.90 Vertical plate 20 24.47 30 24.49 40 24.49 50 24.40 60 24.48 a Percent variation ðPV; %Þ ¼ lStress value of any al 2 lStress value of a ¼ 20 mml £ 100: Stress value of a ¼ 20 mm the HP, in which high normal stress concentrations occur, would increase the stiffness of the HP. Vertical plate 20 30 40 50 60 a 8. Conclusions In this study, the thermal analysis and the geometrically non-linear stress analysis of an adhesively bonded tee joint were carried out using the non-linear finite element method. The tee joint configurations bonded to a rigid base and a flexible base were considered. The variable thermal boundary conditions were specified along the outer boundaries of the adhesive tee joints, i.e. the air flows with different velocity, temperature and at the different flow directions to the plate surfaces. Since the adhesive tee joints consist of adhesive layer and adherends having different mechanical and thermal properties, the non-uniform temperature distributions occurred in the adhesive tee joints, PVa sxx 3.85 3.85 3.85 3.77 3.83 PVa Percent variation ðPV; %Þ ¼ lStress value of any al 2 lStress value of a ¼ 20 mml £ 100: Stress value of a ¼ 20 mm consequently, non-uniform thermal strain distributions. In addition, the edges of the HP and VPs were restrained partly or completely, and then the thermal stress distributions of two adhesive tee joints were determined based on their non-uniform thermal strain distributions using the SSLD theory. In case of the rigid base, the VP of the adhesive tee joint was buckled and the upper free end of the vertical adhesive layer experienced large displacements. The tee joint with flexible base had similar deformations for the VP and the free ends of the vertical adhesive layer; moreover, its HP was deformed (buckled) considerably. Thus, the left and right free ends of the horizontal adhesive layer and 150 M. K. Apalak et al. / Composites: Part A 34 (2003) 135–150 the middle region of the HP had evident deformations. The high normal stresses occurred in the horizontal and vertical sections of the joint regions of both tee joints. Furthermore, the horizontal and vertical adhesive fillets and the BAL were subjected to high stress concentrations in both cases of the rigid and flexible bases. In general, the stress levels reached a maximum at the free ends of the HP or VP – adhesive interfaces and distributed uniformly through adhesive fillets towards the rounded corners of the support and the angled reinforcement member. However, the most critical adhesive regions were the LHAF, VAF and the BAL. The effect of the support length on the peak stresses in the adhesive layer and in the HP and VPs was also investigated for a certain range of the support length. It was observed that in case of the adhesive tee joint with a rigid base, the support length has an effect of decreasing the peak stresses in the RHAFs whereas its effect on the peak stresses in the VAF, in the BAL and at the VP. The support length had an effect of decreasing the peak stresses at the critical locations inside the LHAF and RHAF, and in the HP whereas its effect on the peak stresses in the VAF, in the BAL and in the VP is insignificant. The thermal stress analyses for certain range of the support and angled reinforcement length showed that a ratio of the plate length to the support length L=a ¼ 12 is reasonable for the design of the adhesive tee joints subjected to similar thermal and structural boundary conditions. Finally, the thermal loads result in non-uniform temperature distributions in the adhesive joints due to the different thermal properties of the adhesive and composite plates, consequently non-uniform thermal strain distributions arise. When some additional structural constraints are attributed to the adhesive joints, they may experience considerably high stress and strain distributions. In case large displacements and displacement gradients are observed, the SSLD theory would predict reasonably accurate stress and deformation states of the adhesive joints providing that the adhesive and composite adherends have small strains. References [1] Semerdjiev S. Metal to metal adhesive bonding. London: Business Books Ltd; 1970. [2] Adams RD, Wake WC. Structural adhesive joints in engineering. London: Elsevier; 1984. [3] Kinloch AJ. Adhesion and adhesives. London: Chapman & Hall; 1987. [4] Ioka S, Kubo S, Ohji K, Kishimoto J. Thermal residual stresses in bonded dissimilar materials and their singularities. JSME Int J Ser A: Mech Mater Engng 1996;39(2):197–203. [5] Reedy ED, Guess TR. Butt joint strength effect of residual stress and stress relaxation. J Adhes Sci Technol 1996;10(1):33–45. [6] Kim YG, Lee SJ, Lee DG, Jeong KS. Strength analysis of adhesively bonded tubular single lap steel – steel joints under axial loads considering residual thermal stresses. 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