Adhesive and cohesive failure analyses of bonded FRP composite tubular T-joints in combined axial compression and out-of-plane bending

Abstract

Tubular T-joints are the most widely used structural connectors. FRP composites, with their superior intrinsic properties, can be a more suitable choice than steel. Computational fracture mechanics has been employed to develop finite element models for predicting adhesive and cohesive failures in bonded composite T-joints under combined loading, with clamped chord and brace. Three-dimensional stress analysis, in conjunction with appropriate failure criteria (Tsai-Wu for adhesive failure and parabolic yield for cohesive failure), revealed that the saddle point (β = 0°) along the bottom toe line and middle fillet line of the bonded T-joint is particularly prone to adhesive and cohesive failures, respectively. Fracture growth analysis using VCCT demonstrated a predominant radial propagation of both adhesive and cohesive failures, primarily governed by opening mode. Simultaneous failure growth leads to mutual retardation, with cohesive failure significantly contributing to the suppression of adhesive failure propagation. Despite the interaction effects, fracture propagation within the bonded T-joint has intensified local stress concentrations, adversely impacting the structural integrity of the joint. Structural integrity of bonded T-joints to adhesive failure can be enhanced by using a circumferentially oriented ([0]₁₆) brace along with a cross-ply ([0/90]_4S) chord for smaller cracks (a_a/f_b ≤ 0.3), while circumferential ([0]₁₆) chord for larger cracks (a_a/f_b ≤ 0.3). Circumferentially ([0]₁₆) oriented fiber reinforced brace and chord are proposed for improving the overall joint fracture resistance against simultaneous influence of adhesive and cohesive failures.

Keywords

adhesive failure cohesive failure FRP composites SERR tubular T-joints VCCT

Introduction

Tubular T-joints (TTJs) are the most common and primary constituents of complex structural joints. TTJs find ample applications in onshore and offshore structures such as stadiums, long-span roofs, crane structures, bridges, ship loaders, towers, and oil platform jackets. These tubular joints used in structural applications under normal working conditions experience a combined state of loading. Laminated Fiber Reinforced Polymer (FRP) composites serve as a viable alternative to metallic structural (tubular) components in these applications, owing to their advantageous properties like corrosion resistance, environmental durability, and a low weight-to-strength ratio. Though joints are inevitable due to fabrication limitations, remain a primary factor in weakening structural integrity of tubular structures. However, with continuing industrial advancements, adhesive bonding has become the preferred method for assembling FRP composites, offering benefits such as simplified application, even load transfer, and effective vibration damping. Adhesive and cohesive failures are the primary modes of joint fractures developed in bonded FRP composite TTJ because of localized stresses in the joint regions. Accordingly, detailed three-dimensional stress, failure, and fracture (adhesive/cohesive) analysis have been carried out under combined loading. To closely simulate practical onsite loading scenarios, the current analysis incorporates clamped chord ends and a brace under combined loading involving axial compression and Out-of-Plane Bending (OPB). Identification of the regions susceptible to adhesive and cohesive failures, accompanied by a comprehensive analysis of joint fracture propagation to understand their interaction, influence on joint integrity, and to propose the most suitable ply orientation for enhancing the damage tolerance of TTJ structures.

Stress distribution in Tubular Single Lap Joint (TSLJ) composed of FRP laminates has been investigated by Chon¹ through closed-form analytical solutions and numerically. Parameters influencing joint stress concentration have been reported to be “wrap angle, adhesive thickness, and overlap length.” TSLJ comprised of steel and composite adherends has been examined for joint edge stress concentration effects by Hipol.² An analytical model was proposed by Yang et al.³ employing “first-order laminated anisotropic plate theory” to predict peel and shear stress distributions in bonded composite tubular joints, including socket and butt-and-strap configurations, subjected to tensile and bending loadings. Zou and Taheri⁴ conducted an in-depth investigation into the distribution of shear stresses within the adhesive layer of bonded sandwich composite pipe joints subjected to torsional loading. Joint stresses in composite tubular joints have been iteratively calculated by Oh,^5–7 considering nonlinear adhesive properties. Hosseinzadeh and Taheri⁸ analyzed the efficacy of geometrical and material non-linearity on the performance and efficiency of bonded TSLJ comprised of metallic and composite tubes under torsional loading. An analytical model grounded in “first-order laminated shell theory” have been proposed by Yang and Guan⁹ to assess the stress and strain distributions in adhesively bonded Tubular Socket Joint (TSJ) and butt-and-strap joints under combined tensile and torsional load. The influence of “laminate stacking sequence, overlap length, adhesive layer thickness, and adhesive stiffness” on peel and shear stress distribution in bonded tubular joints have been investigated analytically by Xu and Li.¹⁰ They concluded that all the six stress components have a remarkable impact on strength, emphasizing the necessity of the 3-D study of bonded composite tubular joints. Parashar and Mertiny¹¹ have numerically verified the efficacy of ply configuration on the bonding strength of adhesive bonded composite tubular joints through fracture mechanics and strength of material approaches. De Queiroz et al.¹² provided a brief discussion on recent advancements aimed at enhancing the mechanical performance of composite adhesive joints, along with the key factors influencing their behaviour.

However, the fracture analysis focusing on onset, growth, and improvisation of damage tolerant characteristics against joint fractures is a comparatively vital task. Stress analysis of laminated FRP composite bonded tubular joints under various loading conditions has been attempted by various researchers.^1–11 However, literature related to fracture analysis of bonded tubular joints composed of FRP composite laminates is limited. Hashim et al.¹³ experimentally and numerically analyzed the structural integrity of adhesive bonded composite pipe joints with and without artificial axisymmetric joint fractures (de-bonds). The influence of delamination pre-embedded in adherends of bonded composite TSLJ and TSJ under torsion have been investigated by Esmaeel and Taheri.^14,15 Onset and growth of fractures in joints (adhesive/cohesive) and FRP adherends (delamination) of composite TSLJ and TSJ under axial tensile loading have been rigorously analyzed numerically by Das and Pradhan.^16–18 Kumar et al.¹⁹ have numerically studied the effect of axial and circumferential growth of delamination damages on stress distribution corresponding to adhesive mid-layer of FRP composite made bonded TSLJ under torsional loading. A detailed review of damage tolerance, failure behaviour, and assessment techniques for fibre-reinforced composites was presented by Dalfi et al.²⁰ The study further highlighted strategies to improve damage tolerance, particularly through tailored fibre architectures and optimized resin systems. Goda et al.²¹ investigated the effect of fibre–matrix interfacial adhesion on the mechanical response of FRP composites. Their findings indicate that adequate adhesion significantly enhances stress transfer and load distribution, thereby influencing the fracture and failure mechanisms of the FRP composites. Three-dimensional stress and failure analysis along the critical joint regions of adhesive bonded tubular T and K-joint comprised of FRP composites subjected to axial compressive loading have been analysed numerically by Kumar and Das.^22,23 Rashnooie et al.²⁴ performed experimental analyses to determine the SCF-reduction effectiveness of FRP jacking in steel tubular T-joints exposed to in-plane bending loads. Most of the cited literature primarily address the fracture behavior of FRP composite bonded joints, including TSLJ, TSJ, and tubular butt-and-strap types, which typically feature planar bonding surfaces and fracture paths. Joint fracture analysis of bonded TTJs having non-planar joint region and fracture profiles under practical loading conditions (combined loading) needs the development of a thoroughly validated fracture mechanics based model capable of analyzing the structures under both intact and fractured boundary conditions.

Current research presents a computational fracture mechanics-based parametric FE model for analyzing the stress, failure, and fracture behavior of FRP laminate tubular T-joints under combined axial compression and OPB loading. The major objectives of this research are outlined below:

⁃ To investigate the effect of combined axial compressive and OPB loading on adhesive and cohesive failure initiation in the bonded T-joint (through 3-D stress and failure analyses).

⁃ To analyse radial/circumferential growth of adhesive and cohesive failures in terms of predominant fracture modes quantified through VCCT based SERR.

⁃ To analyse the structural integrity of the bonded tubular T-joints under combined loading through investigating the effect of (i) adhesive/cohesive failures on joint fracture, (ii) adhesive failure growth on cohesive failure, and vice-versa

⁃ To propose the optimum chord/brace stacking sequences for improving damage tolerance of the bonded tubular T-joints against both adhesive and cohesive failures under combined loading.

Specimen geometry and boundary conditions

Material properties and dimensions of the bonded TTJ (shown in Figure 1) have been considered same as of Kumar and Das.²² Chord and brace are considered to be composed of T300/934-Gr/E laminates having ply-configurations [0/90]₄. Geometrical dimensions and material properties of bonded TTJ have given in Tables 1 and 2, respectively. To analyze the joint region, two cylindrical coordinate systems have been adopted- (i) Brace Coordinate System (BCS): r, β, z and (ii) Chord Coordinate System (CCS): R, θ, Z. Both originating at the intersection point of the brace and chord longitudinal axes, as illustrated in Figure 1. The main objective of the present research is to assess the structural integrity and improve the joint fracture resistance of the adhesive bonded composite TTJs upon combined state of loading. Therefore, finite element simulations were conducted to estimate the critical load magnitudes of axial compression and OPB at which the failure index attains unity when applied indecently. From this analysis, axial compressive (5 MPa) and OPB (0.3 MPa) loads (slightly higher than critical loads) were selected, ensuring that each loading mode independently triggers the onset of failure in the joint region. This strategy provides a critical foundation for understanding initiation and propagation of joint fractures, essential for designing the chord/brace stacking sequences to enhance the fracture resistance and overall structural integrity. Loading and boundary conditions adopted in finite element modelling for the bonded TTJs are defined below:

• For nodes at $Z = (\pm \frac{l_{c}}{2})$ ; all DOFs are restrained (u_R = v_θ = w_Z = 0).

• For nodes at $z = l_{b,} r = (\frac{d_{b}}{2} - t_{b})$ ; radial and circumferential movements are restrained (u_r = v_β = 0).

• Nodes at z = l_b, are subjected to combined loading composed of axial compressive load: P₀ = 5.0 MPa and out-of-plane bending load: T_OPB = 0.3 MPa.

Where, u_r, v_β, w_z and u_R, v_θ, w_Z represents displacement along radial, circumferential, and axial directions corresponding to BCS and CCS, respectively.

Figure 1.

Geometrical details and boundary conditions of adhesive bonded tubular T-joint: (i) Sectional front view and (ii) Top view. (i) (ii).

Table 1.

Geometric dimensions of bonded tubular T-joint constituents.²².

Chord dimensions (in mm)	Brace dimensions (in mm)	Adhesive fillet dimensions (in mm)
d_c = 318.5	d_b = 139.8 mm	f_r = 15
t_c = 4.5	t_b = 4.4 mm	f_b = 10
l_c = 1593	l_b = 796.5 mm	f_h = 10

Note-d_b, t_b, l_b and d_c, t_c, l_c represents the diameter, thickness, length of chord and brace respectively. Whereas, f_r, f_b, f_h represents average fillet radius, fillet width and height at crown location, respectively.

Table 2.

Material properties and strengths of the T300/934Gr/E bonded tubular T-joint components²².

T-joint components	Strength properties	Material properties
Chord and brace made of T300/934 Graphite/Epoxy FRP composite (orthotropic)	k_T = 1586 MPa	E_k = 127.5 GPa
	k_C = 1517 MPa	E_i = 4.8 GPa
		E_j = 9.0 GPa
	j_T = j_C = 80 MPa	υ_ki = υ_jk = 0.28
	i_T = i_C = 49 MPa	υ_ij = 0.41
	S_ij = S_jk = S_ki = 96 MPa	G_ki = G_jk = 4.8 GPa
		G_ij = 2.55 GPa
Epoxy adhesive (isotropic)	Y_T = 65 MPa	E = 2.8 GPa
Epoxy adhesive (isotropic)	Y_C = 84.5 MPa	υ = 0.35

Note-i, j, k represents radial, circumferential and axial directions, respectively, in terms of BCS and CCS.

This investigation aims to systematically identify the locations within the bonded T-joint that exhibit high fracture sensitivity. Subsequently, a parametric fracture analysis has been performed to study crack propagation under combined mechanical loading and boundary constraints, providing insights for improving joint durability. Therefore, it is crucial to identify locations within the non-planar T-joint region likely to be censorious towards adhesive/cohesive failure. These regions are closely examined for stress concentrations that contribute to fracture initiation, using relevant failure criteria. Joint region of bonded TTJ consists of few interfaces, lines, and points that are expected to be sensitive from joint fracture under adopted loading and boundary conditions. Brace Adhesive Interface (BAI) and Chord Adhesive Interface (CAI) are expected to be sensitive towards adhesive failure. However, Adhesive Fillet Area (AFA) is expected to be the cohesive failure sensitive T-joint interface. Top Toe Line (TTL) & Bottom Toe Line (BTL) are generated at AFA-brace junction & AFA-chord junction, respectively, whereas the chord-brace junction is termed as Chord Brace Intersection Line (CBIL) have been expected to be the adhesive failure prone lines (Figure 2(ii)). Middle Fillet Line (MFL) the central line of AFA, is expected to line predisposed to cohesive failure. It is pertinent to note that each of these probable fracture prone lines has been considered to have eight perceptible points that are likely to be censorious from joint stress concentrations perspective. These points are designated as saddle points, crown points, and saddle-crown midpoints as demonstrated in Figure 2.

Figure 2.

(i) Zoomed top view of joint region depicting points and (ii) Zoomed front view of joint region depicting lines and interfaces.

Finite element modeling

Ansys Parametric Design Language (APDL) codes capable of analyzing both the intact and fractured states of the joint were developed to perform finite element analysis of bonded TTJs, in ANSYS thereby supporting a comprehensive fracture mechanics-based evaluation.

Structure modeling

Bonded TTJs were modeled using SOLID185 elements, which is three-dimensional structural brick element with eight nodes at corners. Solid elements were chosen over shell element (SHELL 181) to accurately model the geometry of adhesive fillets. Moreover, solid elements offer a significant advantage in fracture analysis, particularly in the decomposition of total SERR into its constituent modes (Mode I, Mode II, and Mode III), which is essential for a reliable fracture mechanics-based evaluation. To represent the distinct material behavior of each component, the adhesive fillet was modeled using homogeneous SOLID185 elements, reflecting the isotropic nature of epoxy resin. In contrast, the brace and chord sections, comprising laminated FRP composites were discretized using layered SOLID185 elements to capture the through-thickness variation in material properties due to ply orientations. This element strategy ensured the accuracy of stress distribution, interfacial response, and progressive fracture modeling under combined loading conditions. To simulate the bonded interface behavior between the adhesive fillet and the adjoining structural components, CONTA175 elements with the ‘Bonded Always’ contact formulation were implemented. This configuration ensured continuous displacement compatibility at the interfaces, thereby replicating ideal adhesive bonding conditions within the finite element model. In contrast, the interface of chord and brace (plug area) has been retained unbonded.

The key findings of this investigation are the stress and failure index profiles for intact joints, and the mode-separated SERR values for fractured joints. High stress gradients are expected in the bonded TTJ region under combined axial and OPB loading. Hence to capture these stiff stress gradients accurately, mesh convergence has been performed. Accordingly, the optimized mesh size adopted in BAI and CAI regions of the TTJ is defined as one part in the radial direction, 120 parts in the circumferential direction, and 20 parts in the axial direction, as shown in Figure 3. The converged finite element model has 76,200 elements and 124,388 nodes with each node having three translational degrees of freedom. The meshing pattern adopted near adhesive and cohesive failures is uniform in terms of finite element size. Type of finite elements used for discretizing the adhesive and cohesive failure front regions are however different, and can clearly be depicted in Figure 3(ii) and (iii), respectively. Finite Element (FE) mesh near adhesive failure fronts consists of homogeneous structural SOLID 185 (for adhesive) and layered structural SOLID 185 (for chord) elements (Figure 3(ii)). However, homogeneous structural SOLID 185 element-based FE mesh has been adopted near cohesive failure front region (Figure 3(iii)). To ensure the accuracy and reliability of the developed APDL-based finite element model, results corresponding to the optimized mesh size were validated through comparison with existing experimentally, analytically, and numerically obtained SCF distribution over TTL of steel tubular T-joints subjected to axial loading,^25,26 out-of-plane bending,^26,27 and combined (axial and OPB) loading,²⁷ and found to be in close agreement (Figure 4(i–iii)).

Figure 3.

Finite element meshing pattern adopted for stress and fracture analysis: (i) Meshing pattern in the TTJ (ii) Meshing pattern near adhesive failure and (iii) Meshing pattern at cohesive failure.

Figure 4.

SCF variation over top toe line of tubular T-joint subjected to (i) Axial loading, (ii) Out of plane bending, (iii) Combined (axial & OPB) loading, and (iv) Resistance curve for adhesively bonded composite DCB.

Fracture modeling

The application of suitable failure criteria, as detailed in “Three-dimensional stress and failure analysis” section, facilitates accurate identification of fracture initiation zones within the joint. To simulate adhesive failure around the saddle point (β = 0°, S_1C) in the CAI (Figure 5(i)), a node-wise matching strategy was adopted to ensure that every node on the chord surface corresponds exactly to a node on the adjacent adhesive layer. These matched nodes were connected via CONTA-175 contact elements with constant penalty and tangent stiffness values to represent bonding behavior. However, to realistically represent failure progression, nodes located within the adhesive failure surface were excluded from bonding. CONTA-175 elements are inherently linked with internal Multi-Point Constraint (MPC) elements, which facilitate the extraction of nodal reaction forces. These reaction forces are essential for evaluating various SERR modes using the VCCT. Hence, intact and failure conditions of bonded TTJ can be adequately modeled through the application of CONTA-175 elements. Cohesive failures about the saddle point at β = 0⁰ (S_1A) in AMS (as shown in Figure 5(ii)) have been simulated by ensuring that there are two nodes at each location of AMS; one belongs to top and another belongs to the bottom surface of AMS. As in the case of adhesive failure modeling, all nodes in the AMS region are bonded via CONTA-175 elements configured with constant penalty and tangent stiffness parameters, except those residing within the cohesive failure regions. Circumferential and radial growth of joint failures was simulated by incrementally shifting the CONTA-175 elements defining the adhesive/cohesive failure fronts to the next nodes along the circumferential and radial directions. Accuracy in estimating SERR modes at failure fronts under combined loading conditions has been ensured by validating the computational failure mechanics-based FE model with experimental observations. Accordingly, SERR modes at joint failure fronts, calculated via VCCT, were verified against the experimental resistance curve for an adhesively bonded double cantilever beam²⁸ (Figure 4(iv)), demonstrated close agreement. Figure 4 illustrates that the developed finite element model is capable of producing stress and SERR modes in close agreement with both analytical solutions and experimental data, confirming its accuracy and reliability.

Figure 5.

Top view of joint region showing fracture fronts: (i) Pre-embedded adhesive failure (2ϕ_a, a_a) and (ii) Pre-embedded cohesive failure (2ϕ_c, a_c) simulated about saddle point (β = 0°).

Onset and growth criteria for joint failure

The epoxy adhesive fillet, being mechanically weaker than the FRP composite adherends (brace/chord), is susceptible to failure under combined loading. Consequently, two principal types of joint failure, adhesive (interfacial) and cohesive (within the adhesive) are taken into account. Accurately locating the initiation of these failures is crucial for understanding and improving the structural integrity of bonded FRP TTJs. The Tsai-Wu criterion is employed for predicting failure in the adherend-adhesive interface, while the parabolic yield criterion is used for the adhesive. These criteria were commonly adopted in prior studies^{16,17,22,23,29–33} for identifying adhesive and cohesive failure initiation sites in bonded composite joints.

Criteria for the onset of adhesive failures

The expanded form of Tsai-Wu criterion (Equation (1)) has been implemented in the probable adhesive failure sensitive zones (CAI and BAI) to predict the exact location of adhesive failure initiation. The Tsai-Wu failure criterion, which is based on total strain energy, has been shown to predict failure stresses that are in close agreement with experimental results.^30,34–36 Therefore Tsai-Wu criterion for composites have been preferably used to predict the initiation of adhesive failure in bonded composite joints.^{16,17,22,23,29,30,32,33} The non-planar and mixed-mode nature of adhesive failure at the joint region indicates the predominance behavior of all the six stress components towards the fracture onset. Contrary to the planar and self-similar mode of adhesive failure growth observed in bonded TSLJ/TSJ,^16,17 joint fractures in bonded TTJ under the combined state of loading are expected to have contribution of all the six stress components.

\frac{σ_{i}^{2}}{i_{T} i_{C}} + \frac{σ_{j}^{2}}{j_{T} j_{C}} + \frac{σ_{k}^{2}}{k_{T} k_{C}} + \frac{τ_{i j}^{2}}{S_{i j}^{2}} + \frac{τ_{j k}^{2}}{S_{j k}^{2}} + \frac{τ_{k i}^{2}}{S_{k i}^{2}} + σ_{i} (\frac{1}{i_{T}} - \frac{1}{i_{C}}) + σ_{j} (\frac{1}{j_{T}} - \frac{1}{j_{C}}) + σ_{k} (\frac{1}{k_{T}} - \frac{1}{k_{C}}) +

f_{i j} σ_{i} σ_{j} + f_{j k} σ_{j} σ_{k} + f_{k i} σ_{k} σ_{i} = e^{2} {\begin{cases} e \geq 1, f a i l u r e \\ e < 1, n o f a i l u r e \end{cases}

(1)

Where ‘e’ represents the failure index. During adhesive failure prediction in critical T-joint regions, indices i, j, and k are represented in BCS for BAI, and in CCS for CAI. Here, i_C, j_C, and k_C refer to the allowable compressive strengths of the FRP in the radial, circumferential, and axial directions, while i_T, j_T, and k_T denote the respective tensile strengths. Shear strengths in different coupling directions are represented by S_ij, S_jk, and S_ki. Coupling coefficients f_ij, f_jk, and f_ki represent interactions among the i, j, and k directions, defined as (R, θ, Z) in CCS and (r, β, z) in BCS. The Mises-Hencky empirical relation (Equations (2)–(4)), which accounts for the tensile and compressive strengths of composite laminates, is utilized to determine the coupling coefficients.³⁴

f_{i j} = - \frac{1}{2} \sqrt{\frac{1}{(i_{C} * i_{T} {* j}_{C} {* j}_{T})}}

(2)

f_{j k} = - \frac{1}{2} \sqrt{\frac{1}{(j_{C} * j_{T} {* k}_{C} {* k}_{T})}}

(3)

f_{k i} = - \frac{1}{2} \sqrt{\frac{1}{(k_{C} * k_{T} {* i}_{C} {* i}_{T})}}

(4)

Criteria for the onset of cohesive failures

Macroscopic yield criterion for polymer proposed by Raghava et al.³⁷ popularly known as parabolic yield criterion, have been considered to predict the cohesive failure in the adhesive fillet (epoxy based polymer). By accounting for the disparity between tensile and compressive yield strengths of polymers, this criterion enables accurate prediction of failure stresses and has shown strong correlation with experimental observations.³⁷ The parabolic yield criterion has been widely applied to assess cohesive failure^16,17,22,23 initiation in adhesively bonded composite joints, such as lap and socket joints. Therefore, parabolic yield failure criterion (Equation (5)) has been employed to accurately capture the initiation of cohesive failure in the AFA region of the bonded composite T-joint. Where σ₁, σ₂, and σ₃ represent principal stresses, while Y_T and Y_C refer to the tensile and compressive yield strengths of the epoxy adhesive and e stands for failure index. Applied criterion converges to the standard von Mises yield criterion if both the tensile and compressive strengths are identical.

{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{3} - σ_{1})}^{2} + 2 (| Y_{C} | - Y_{T}) (σ_{1} {+ σ}_{2} {+ σ}_{3}) = 2 | Y_{C} | Y_{T} e

(5)

Criteria for growth of joint fractures

Fracture growth in adhesively bonded joints is often quantified using energy-based (SERR) or stress-based (SIF, J-integral, CTOD) parameters. The SERR approach is advantageous as it avoids crack-tip singularity issues and does not require fine meshing near the crack tip. Therefore, structural stability or integrity of bonded tubular joints typically assessed through SERR modes evaluated using VCCT.^{16–19,22,23} VCCT-based computation of SERR modes relies on nodal forces at the crack front and nodal displacements behind it. SERR modes in this study have been evaluated using the modified crack closure integral method, first proposed by Irwin and adopted by Rybicki and Kanninen.³⁸ It is based on the assumption that the energy (ΔE) required to close a virtually extended crack equals the energy released during its propagation. The non-planar shape of the CAI and AMS requires local coordinates to define the nodal orientations along the circumferential damage fronts of joint fractures. Hence separate coordinate systems are defined at each node of fracture fronts. A symbolic representation of local coordinates at circumferential damage front of adhesive failure is shown in Figure 6. For estimating SERR modes along circumferential damage front of cohesive failure, similar to adhesive failure, local coordinates are defined at each node. The local coordinates are defined such that nodes at circumferential damage fronts are taken as origin, node in front of circumferential damage front is taken as radial direction (r_i), and adjacent node is a taken as circumferential direction (β_i) and perpendicular to crack surface becomes axial direction (z_i). The SERR modes corresponding to the failure fronts (depicted in Figure 6) are determined using the following formulations:

G_{I} = \frac{1}{2 Δ A i} F_{z i} (w_{T i} - w_{B i})

(6)

G_{I I} = \frac{1}{2 Δ A i} F_{r i} (u_{T i} - u_{B i})

(7)

G_{I I I} = \frac{1}{2 Δ A i} F_{β i} (v_{T i} - v_{B i})

(8)

G_{T} = G_{I} + G_{I I} + G_{I I I}

(9)

Figure 6.

(a) Isotropic view of tubular T-joint, (b) CAI with simulated adhesive failure, (c) Local coordinates defined at radial and circumferential fracture fronts, and (d) Finite element mesh description at circumferential fracture front of the adhesive failure (2ϕ_a, a_a) for calculation of SERR modes using VCCT.

Where i represents node number (i.e. i = 1, 2, 3, .........) on circumferential damage fronts, ΔA = (Δϕ * Δa) represents the virtually expanded area of FE mesh maintained on fracture fronts (Figure 6). Nodal forces on the fracture fronts along the radial, circumferential and axial directions have been indicated through F_ri, F_βi, and F_zi, respectively. Nodal displacements in the radial (u), circumferential (v), and axial (w) directions behind the fracture front are denoted by u_Ti, v_Ti, w_Ti for the top surface (T) and u_Bi, v_Bi, w_Bi for the bottom surface (B). The nodal axial force (F_zi) plays a critical role in driving mode-I (opening) fracture at both radial and circumferential fronts in the CAI and AMS regions. For radial fracture propagation, the radial nodal force (F_ri) induces in-plane sliding (mode-II), while the circumferential nodal force (F_βi) contributes to out-of-plane tearing (mode-III). During circumferential fracture growth, the same forces act in reversed roles, leading to out-of-plane and in-plane shearing modes, respectively.

Results and discussion

Three-dimensional stress and failure analysis in adhesive failure prone regions

Three-dimensional stress and failure index distribution along the joint regions is essential prior to joint fracture (adhesive/cohesive) analysis. Failure analysis with appropriate failure criteria identifies the joint region susceptible to adhesive and/or cohesive failures whereas, stress analysis reveals the stress components primarily responsible for fracture. The non-planar shape of adhesive-adherend interfaces, orthotropic characteristics of the FRP brace/chord (adherends), and combined loading cause inconsistent stress profiles along the joint region. Contrary to the case of TSLJ/TSJ,^14–19 where the joint stress profiles are uniform along circumference.

Three-dimensional stress analysis of CAI and BAI regions of the bonded composite TTJ manifests that stresses over CAI are relatively more prominent, while negligibly small over BAI. Therefore, three-dimensional stress variation over CAI has been shown in Figure 7. Circumferential stress (σ_θ) has been noticed to be the most censorious normal stress (Figure 7(ii)), having maximal stress concentrations on saddle point (S_1C at β = 0⁰) whereas, minimal in saddle point (S_2C at β = 180⁰) and crown points. The peak magnitudes of circumferential stresses at BTL (σ_θ = −182.4 MPa) reduces with a steep gradient towards CBIL. Axial stress (σ_Z) is the next critical normal stress (Figure 7(iii)), which attains peak magnitudes at saddle-crown midpoints in BTL. However, the maximum stress magnitudes decrease with sharp gradients from BTL to CBIL, attains almost nominal stress concentrations towards CBIL. Radial stress (σ_R) variation over CAI exhibits a different pattern compared to circumferential (σ_θ) and axial (σ_Z) stresses, having peak tensile magnitudes about saddle points (at β = 0⁰) at CBIL. In contrast, the peak magnitude of compressive nature has occurred about saddle crown midpoints somewhere in between BTL and CBIL (Figure 7(i)). Circumferential-axial shear stress (τ_θZ) is the most significant shear component (Figure 7(v)), having peak identical tensile and compressive stress magnitudes at the saddle-crown midpoints at BTL. The peak stress magnitudes decrease gradually and achieve almost null magnitudes at CBIL. Radial-circumferential shear stress (τ_Rθ) is the subsequent censorious shear stress, having peak magnitude at saddle points (at β = 0⁰) at BTL (Figure 7(iv)). Moving towards CBIL from BTL, the positive shear stress magnitudes keep decreasing and become negative with the mutual shifting of censorial stress location to saddle-crown midpoints from saddle points. Radial-axial shear stress (τ_RZ) is minimal, peaking near the saddle–crown midpoints at BTL and dropping towards CBIL (Figure 7(vi)).

Figure 7.

Three-dimensional stress variation along the chord-adhesive interface of bonded TTJ: (i) Radial stress (ii) Circumferential stress (iii) Axial stress (iv) Radial-circumferential stress (v) Circumferential-axial and (vi) Radial-axial stress.

The peak magnitude of normal stress (σ_θ = −182.4 MPa) being relatively larger than the peak magnitude of shear stress (τ_θZ = ±63.5 MPa) are anticipated to significantly contribute to adhesive failure onset in BTL. Saddle point (S_1C at β = 0⁰) becomes the most censorious location in the BTL due to significant concentrations of circumferential stress (σ_θ). However, saddle-crown midpoints have been observed to be the next locations vulnerable towards adhesive failure onset as axial stress (σ_Z), and circumferential axial shear (τ_θZ) stresses have peak magnitudes in these locations. In contrast, the other saddle point (S_2C at β = 180⁰) on the BTL are the least susceptible to adhesive failure initiation as all the stress components at this location have almost zero magnitudes. All stress components along the CBIL (Figure 7) show significantly lower concentrations compared to the BTL.

Failure indices based on Tsai-Wu criterion (Equation (1)) along adhesive failure initiation prone regions (CAI and BAI) have been shown in Figure 8. It is evident that BAI remains safe from adhesive failure, as the maximum magnitude is less than unity (e_max < 1). The failure index distribution along the BTL and TTL shows pronounced symmetry. BTL of the bonded FRP TTJ is the most censorious to adhesive failure onset under combined state of loading on the basis of magnitude of failure indices (Figure 8). Failure index profile over the CAI reaffirms the BTL’s predisposition to adhesive failure initiation. Saddle point (S_1C at β = 0⁰) as the primary site for adhesive failure initiation in BTL, with failure index magnitude exceeding unity (e ≥1). The sharp decline towards the crown (C_1C at β = 90⁰), followed by a near-horizontal trend up to (C_2C at β = 270⁰), and a subsequent steep rise thereafter. This behavior reflects localized stress concentration effects and variation in adhesive stress transfer across the joint interface. Failure index profile over BTL replicates the circumferential stress profile, indicating predominance of the circumferential stress concentration towards adhesive failure initiation. TTL and CBIL show sub-critical failure indices (e < 1), confirming their safety against adhesive failure onset. However, the risk of adhesive failure onset diminishes considerably in the direction of the CBIL.

Figure 8.

Tsai-Wu failure criteria-based failure index profile corresponding to (i) CAI and (ii) BAI of bonded TTJ.

Three-dimensional stress and failure analysis in cohesive failure prone regions

3D normal and shear stress variations along the intermediate AFA (Figure 2(ii)) are the depicted in Figure 9. All the normal stress components (σ_r, σ_β, σ_z) are compressive in nature and exhibit nearly similar profiles with comparable magnitudes. Normal stresses attain their peak compressive magnitudes at saddle point (S_1A at β = 0⁰) corresponding to the MFL and decreases steeply towards TTL and BTL. Saddle points at β = 0⁰ (S_1A) and β = 180⁰ (S_2A) experience the maximum and minimum magnitudes of normal stresses, respectively at MFL. Among the shear stress components, the radial-axial shear stress (τ_rz) is the most significant, exhibiting a peak magnitude of 24.8 MPa at the saddle point (S_1A at β = 0⁰). The radial-axial shear stress (τ_rz) exhibits a distribution similar to that of the normal stresses; however, it is tensile throughout. Remaining shear stresses (τ_rβ, τ_βz) reach their peak values at the saddle-crown midpoints and exhibit minima at the saddle and crown points. All stress components exhibit peak concentrations in the MFL, making it the most vulnerable region for cohesive failure initiation within the intermediate AFA. Saddle point at β = 0⁰ (S_1A) is the most critical region in MFL towards cohesive failure initiation with significant axial (σ_z), circumferential (σ_β), radial (σ_r) and radial-axial (τ_rz) shear stress concentrations. Whereas saddle point at β = 180⁰ and crown points are the least vulnerable towards cohesive failure onset.

Figure 9.

Three-dimensional stress variation along intermediate adhesive fillet area of bonded TTJ: (i) Radial stress (ii) Circumferential stress (iii) Axial stress (iv) Radial-circumferential stress (v) Circumferential-axial stress and (vi) Radial-Axial stress.

3D variation of failure index based on parabolic yield criterion (Equation (5)) over critical cohesive regions (intermediate AFA) has been presented in Figure 10. Peak failure index (e_max = 0.68) occurs at the saddle point (S_1A at β = 0⁰) along the MFL, marking it as the critical location for cohesive failure onset. Potential for cohesive failure initiation in the AFA reduces markedly along steep gradients extending toward both the BTL and TTL. Similar to CAI, failure index distribution along MFL decreases steeply from saddle point (S_1A) at β = 0⁰ to crown point (C_1C) at β = 90⁰, remains almost flat (horizontal) up to crown point (C_2C) at β = 270⁰ and again starts increasing with a very steep gradient. Based on von Mises stress distribution in the adhesive fillet, it is observed that the cohesive failure at MFL tends to grow along AMS (Figure 2(ii)). Under the adopted combined state of loading and boundary constraints, the significantly higher adhesive failure index (e_max = 2.25) compared to the cohesive failure index (e_max = 0.68) implies that the adhesive bonded FRP composite TTJs are more likely to fail adhesively.

Figure 10.

Failure index variation corresponding to intermediate Adhesive Fillet Area of bonded composite TTJ.

Joint fracture analysis

Pre-embedded joint fractures have been simulated at their respective fracture susceptible regions (as per the failure analysis (Figures 8 and 10)) of the bonded TTJ through the incorporation of computational fracture mechanics based FE model. VCCT has been employed to determine mode-specific fracture growth parameters (SERRs) for both circumferential and radial fracture fronts.

Adhesive failure analysis of T-joint

Failure analysis (in pervious section) unveils that region: 0⁰ ≤ β ≤ 42⁰; 318⁰ ≤ β ≤ 360⁰ on the CAI is the most susceptible region to adhesive failure (e ≥ 1). Saddle point on BTL (S_1C, β = 0⁰) is the most critical adhesive failure point (e = 2.25). Hence pre-embedded adhesive failure (2ϕ_a = 30⁰, a_a = 2 mm) in CAI (Figure 5(i)) have been modeled such that the adhesive failure front completely lies within the adhesive failure susceptible region. Various modes of SERR have been shown in Figure 11. Opening mode (Mode I) is primarily responsible for failure growth along circumferential (AB) and radial (BC and AD) directions, supported by the observation that the Mode-I SERR (G_I) is markedly greater than the Mode-II and Mode-III components. Mode-II and Mode-III SERR components contribute minimally to the circumferential and radial growth of adhesive failure fronts. The close resemblance of total SERR profiles with the opening mode along both radial and circumferential fronts highlights the governing influence of Mode-I in adhesive failure propagation. Maximum total strain energy release rate (G_T-max) in the radial direction is 0.5 kJ/m², significantly higher than the 0.22 kJ/m² observed in the circumferential direction, indicating a higher tendency for adhesive failure to expand radially. Thus, the circumferential fracture front plays a more decisive role in adhesive failure propagation compared to the radial fronts.

Figure 11.

SERR modes along (i) Circumferential fracture front (AB) and (ii) Radial fracture fronts (BC & AD) of adhesive failure (2ϕ_a = 30°, a_a = 2 mm) pre-embedded in CAI.

Cohesive failure analysis of T-joint

Failure analysis across adhesive fillet area (Figure 10) revealed that MFL regions about saddle point (S_1A at β = 0⁰) are the most vulnerable towards cohesive failure onset. Accordingly, a pre-embedded cohesive failure (2ϕ_c = 30⁰, a_c = 1 mm) has been simulated in AMS (Figure 5(ii)) to estimate the SERR components along radial and circumferential fronts (Figure 12). Cohesive failure along both radial (FG & EH) and circumferential (EF) fronts primarily propagate in the opening mode (G_I). Sliding (G_II) and tearing (G_III) SERR modes along circumferential and radial fracture fronts have almost zero contributions to the overall cohesive failure growth. Hence, total SERR (G_T) along radial and circumferential cohesive fronts overlaps the opening mode (G_I). Pre-embedded cohesive failures in the AMS of bonded TTJs are more likely to propagate radially (G_T-max = 3.2 kJ/m²) than circumferentially (G_T-max = 2.3 kJ/m²). The risk associated with the onset of cohesive failure exceeds that of adhesive failure, primarily due to the significantly greater G_T-max (3.2 kJ/m²) observed along the circumferential fracture front in cohesive failure, compared to 0.5 kJ/m² in adhesive failure. This underscores the need for more robust design considerations in regions susceptible to cohesive damage.

Figure 12.

SERR modes along (i) Circumferential fracture front (EF) and (ii) Radial fracture fronts (FG & EH) of the pre-embedded cohesive failure (2ϕ_c = 30°, a_c = 1 mm).

Joint fracture growth analysis

To examine the influence of geometric parameters on the radial and circumferential expansion of adhesive and cohesive failures, the normalized radial lengths (a_a/f_b for adhesive and a_c/f_m for cohesive failure) have been systematically varied from 0.1 to 0.6 with a step size of 0.1. Correspondingly, the normalized circumferential widths (2ϕ_a/π for adhesive and 2ϕ_c/π for cohesive failures) are explored in the range of 0.17 to 0.83, with a step size of 0.17. In this context, a_a is the radial length of the adhesive failure, f_b is the width of the adhesive fillet, and 2ϕ_a is the circumferential width of the adhesive failure. Similarly, a_c represents the radial length of the cohesive failure, f_m is the width of the adhesive mid-surface, and 2ϕ_c denotes the circumferential width of the cohesive failure. The joint fracture analysis presented in previous section clearly demonstrates that the progression of both radial and circumferential damage fronts is predominantly by the opening mode. Thus, in subsequent investigations, only the total SERR has been considered, recognizing that both adhesive and cohesive failure fronts are governed by the opening mode.

Adhesive failure growth

Adhesive failures of a certain circumferential fracture front width (2ϕ_a = 30⁰) and radial fracture front length (a_a = 4 mm) have been pre-embedded about saddle points in CAI (Figure 5(i)) to simulate the radial and circumferential growth of adhesive failures. Thus, the total SERR (G_T) along the circumferential fracture front (AB) and radial fracture front-1 (AD) of the adhesive failure are presented in Figure 13. Fracture modes exhibit a gradual increase with the progression of radial adhesive failure, indicating heightened energy release with crack extension. Smaller sized adhesive failures (a_a/f_b ≤ 0.3) are noticed to have almost uniform variation over the circumferential fracture front. Whereas peak magnitudes for larger adhesive failure (a_a/f_b > 0.3) have been lied on a particular domain (2ϕ_a = −6⁰ to +6⁰), making these locations primarily susceptible to mode-I fracture growth. A decreasing proclivity of SERR have been observed as the adhesive failure propagates along the circumferential direction. The fracture growth susceptibility regions on the radial fracture front in have been noticed to shift towards the free surface of adhesive failure with the circumferential growth.

Figure 13.

Total SERR variation on the (i) Circumferential fracture front (2ϕ_a = 30°) for radial propagation of adhesive failure (a_a/f_b) and (ii) Radial fracture front (a_a = 4 mm) for circumferential propagation of adhesive failure (2ϕ_a/π).

Cohesive failure growth

To simulate radial and circumferential propagation of cohesive failures in the AMS region (Figure 5(ii)), certain circumferential fracture front width (2ϕ_c) of 30⁰ and radial fracture front length (a_c) of 2 mm were pre-embedded around the saddle point. Accordingly, variation of total SERR along the circumferential and radial fracture front of cohesive failure have been shown in Figure 14. As the cohesive failure grows radially, SERRs has been noted to be accelerating with a sharp gradient for smaller cohesive failures (a_c/f_m ≤ 0.3) and becomes gradual for larger cohesive failures (a_c/f_m > 0.3). With the progression of cohesive failure, radial fracture-prone region along the circumferential front shifts from 2ϕ_c = ±12⁰ (for a_c/f_m = 0.1) towards the centre (2ϕ_c = 0⁰), as the cohesive failure grows in size. Similar to adhesive failure, fracture mode has been noticed to reduce as cohesive failure expands circumferentially (2ϕ_c/π = 0.17 to 0.67). With circumferential growth of cohesive failure, fracture susceptible region on the radial fracture fronts of cohesive failure have been concentrated at r = 74.65 (with G_T-max), indicating vulnerability of radial position.

Figure 14.

Total SERR variation on the (i) Circumferential fracture front (2ϕ_c = 30°) for radial propagation of cohesive failure (a_c/f_m) and (ii) Radial fracture front (a_c = 2 mm) for circumferential propagation of cohesive failure (2ϕ_c/π).

Influence of adhesive failure radial propagation on the structural integrity of TTL and MFL

The effect of adhesive failure progression on the failures of MFL and TTL has been plotted in Figure 15. A gradual rise in failure indices in TTL and a steep decline in MFL have been observed with radial growth of the pre-embedded adhesive failure in CAI, reflecting behavior over the radial and circumferential fracture fronts. Failure index reduction along the MFL and TTL, corresponding to the circumferential fracture front (β = −15⁰ to +15⁰), reveals the stress-relieving effect induced by the presence of adhesive failure. In contrast, higher failure indices in the TTL/MFL zones suggest that adhesive failure within the CAI intensifies stresses in areas associated with radial fracture fronts. Thus, as adhesive failure propagates, it notably increases the risk of both adhesive and cohesive failures in the TTL/MFL zones associated with radial fracture fronts. Nonetheless, the influence is limited, given that the failure index in MFL due to radial adhesive failure growth (Figure 15(ii)) is comparatively lower than the maximum failure index of the intact TTJ. However, the magnitude of adhesive failure induced failure index (e_max = 0.165) along TTL (Figure 15(i)) is comparatively higher than that of joint without fracture; nevertheless, not censorious adequate to hamper the structure (e <<1). Although failure indices rise in both TTL and MFL due to radial adhesive failure growth, MFL experiences a more pronounced effect. This is substantiated by the e_max increment from 0.53 to 0.62 (15.68%) in MFL, whereas TTL shows only a minor increase from 0.154 to 0.166 (7.42%).

Figure 15.

Effect of radial adhesive failure growth (2ϕ_a = 30°) on failure index variation over (i) TTL and (ii) MFL of the bonded TTJ.

Influence of cohesive failure radial propagation on the structural integrity of BTL & TTL

The influence of cohesive failure radial propagation simulated about saddle point in AMS on adhesive failures along BTL and TTL are shown in Figure 16. The radial expansion of cohesive failure leads to a sharp decline in failure indices over the BTL and TTL corresponding to circumferential fracture fronts (β = −15⁰ to +15⁰), highlighting stress relaxation in these zones. Thus, it indicates that cohesive failure in AMS reduces the possibility of adhesive failure in BTL. However, a sharp increase in failure indices can be seen in TTL corresponding to radial fronts, indicating that stress concentrations in these TTL regions get enhanced as the cohesive failure expands radially. The radial propagation of cohesive failure appears to promote the risk of adhesive failure within the TTL regions aligned with the respective radial fronts. Cohesive failure radial propagation increases the failure index (e_max = 0.182) compared to the undamaged T-joint; however, this value is still significantly below the threshold required to trigger adhesive failure (e <<1). Notably, radial growth of cohesive failure results in elevated failure indices at both TTL and BTL near the radial fracture front. However, the TTL exhibits a more pronounced effect, with e_max increasing by approximately 15.8% (from 0.15 to 0.17), whereas the BTL shows only a marginal increase of 3.6% (from 1.85 to 1.92). Comparison of adhesive and cohesive failure radial growth reveals that cohesive failure has a more significant impact on TTL failure, as shown in Figure 16(iii).

Figure 16.

Effect of radial cohesive failure (2ϕ_c = 30°) growth on failure index variation in (i) TTL, (ii) BTL of the bonded TTJ, and (iii) Effect of radial adhesive/cohesive failure growth on TTL failure.

Mutual interaction between joint fractures

Effect of adhesive failure radial growth about saddle points in CAI on total SERR (G_T) distribution along the circumferential damage front of cohesive failure (2ϕ_c = 30⁰, a_c/f_m = 0.2) in AMS has been shown in Figure 17(i). A decrease in total SERR is observed as adhesive failure propagates radially (a_a/f_b = 0.0 to 0.6). The decreasing rate of total SERR along cohesive failure fronts is uniformly accelerating with radial propagation of adhesive failure. The concentration of peak total SERR at 2ϕ_c = ±12⁰ along the circumferential damage front suggests these regions serve as primary sites for cohesive failure initiation and progression. It may be noted that radial adhesive failure growth causes an overall reduction of total SERR through 37% by minimizing the G_T-max corresponding to the circumferential cohesive failure front from 3.1 kJ/m² to 1.98 kJ/m² (Figure 17(iii)). Effect of radial growth of cohesive failure simulated about saddle points in AMS on total SERR (G_T) distribution along the circumferential damage front of adhesive failure (2ϕ_a = 30⁰, a_a/f_b = 0.2) in CAI has been shown in Figure 17(ii). As the cohesive failure grows radially (a_c/f_m = 0.0 to 0.6), total SERR at circumferential damage front of adhesive failure has been gradually decreasing. It indicates that cohesive failure radial growth diminishes the rate of adhesive failure propagation. It is important to note that radial cohesive failure growth causes an overall reduction of total SERR through 57% by minimizing the G_T-max corresponding to the circumferential adhesive failure front from 0.4942 kJ/m² to 0.2126 kJ/m² (Figure 17(iii)).

Figure 17.

Total SERR variation on the circumferential damage fronts of (i) Cohesive failure (a_c = 1 mm, 2ϕ_c = 30°) with radial growth of adhesive failure (a_a/f_b), (ii) Adhesive failure (a_a = 2 mm, 2ϕ_a = 30°) with the radial growth of cohesive failure (a_c/f_m) and (iii) Cohesive/adhesive failure growth with radial adhesive/cohesive failure.

Joint failure resistance improvisation of T-joint

Enhancing the resistance to failure propagation in bonded TTJs has been accomplished by tailoring the stacking sequence of adherends to reduce the strain energy available for the advancement of failure fronts. As part of the investigation, eight different stacking sequences have been employed, representing a range of laminate configurations: cross-ply ([0/90]_4s & [30/60]_4s), angle-ply ([45/-45]_4s & [55/-55]_4s), unidirectional ([0]₁₆ & [90]₁₆) and quasi-isotropic ([90/±45/0]_2s & [90/±30/90]_2s).¹⁶ While evaluating joint failure sensitivity to chord and brace ply orientations, measures were implemented to ensure that inter-ply delamination did not occur, thereby isolating the effect of laminate architecture. It has been observed in “Joint fracture analysis” section that, under the given loading and boundary constraints, total SERR (G_T) profiles resemble with mode-I SERR for both adhesive and cohesive failures (Figures 11(i) and 12(i)) indicate the joint’s susceptibility to opening-mode-driven fracture. Consequently, maximum total SERR (G_T-max) is employed as the primary indicator to quantify the ability of the joint to arrest fracture growth.

Through chord stacking sequences

In order to explore the best possible ply-orientation for chord to enhance the damage resistance against joint fractures. The bonded composite TTJ is simulated for each chord ply-orientation keeping the stacking sequence of the brace ([0/90]₄). Radial lengths of the pre-embedded adhesive failures (a_a/f_b = 0.1 to 0.6) and cohesive failures (a_c/f_m = 0.1 to 0.6) were varied and the maximum total SERR (G_T-max) results have been depicted in Figure 18. Among the investigated ply orientations, the cross-ply ([0/90]_4S) configuration demonstrates superior fracture resistance for adhesive failures with a_a/f_b ≤ 0.45, by effectively minimizing the maximum SERR (Figure 18(i)). In contrast, when the adhesive failure increases beyond a_a/f_b > 0.45, the unidirectional ([0]₁₆) ply configuration oriented circumferentially becomes more favourable, providing improved control over the total SERR. Circumferentially oriented ([0]₁₆) and axially oriented ([90]₁₆) unidirectional ply-configurations seem to be the most censorious chord ply-orientation from arresting adhesive failures perspective in a_a/f_b ≤ 0.25 and a_a/f_b > 0.25, respectively. However, as the adhesive failure expands to a_a/f_b > 0.45, [0]₁₆becomes the most suitable ply configuration of the chord. In comparison, circumferentially oriented ([0]₁₆) unidirectional layups of chord are the prominent ply configuration for cohesive failures as it assures minimum possible maximum total SERR (Figure 18(ii)). In contrast, arresting cohesive failures, the axially oriented ([90]₁₆) unidirectional layup performs the poorest, indicating its unsuitability as a chord ply design. FRP composite chords with axially oriented fibres ([90]₁₆) exhibit high stiffness in the axial direction but comparatively low stiffness circumferentially. As a result, circumferential nodal deformation at the radial adhesion fracture front is expected to be more pronounced, which in turn leads to an increase in Mode-I SERR along the circumferential adhesive/cohesive fracture front.

Figure 18.

Effect of chord ply orientations on maximum total SERR (G_T-max) during radial growth of (i) Adhesive failure (2ϕ_a = 30°) and (ii) Cohesive failure (2ϕ_c = 30°) with brace ply orientation of [0/90]₄.

Through brace stacking sequences

To further enhance adhesive failure resistance, the bonded FRP composite TTJ has been re-analyzed by incorporating optimal chord stacking sequences of cross-ply ([0/90]_4S) and unidirectional ([0]₁₆) configurations. Brace stacking sequences have been varied alongside changes in adhesive failure length for each of these chord stacking sequences. The slight variation in G_T-max with brace stacking sequences, as shown in Figure 19(i–ii), underscores the primary role of chord ply configuration in resisting adhesive failure growth. The brace ply configuration has a negligible effect, primarily because the brace sustains only a minor fraction of the applied load. Among the evaluated brace layups, the circumferentially aligned unidirectional ([0]₁₆) configuration offers superior resistance to fracture growth throughout all considered ranges of adhesive failure. Based on the observed fracture behavior, the [0/90]_4S - [0]₁₆ chord–brace configuration is optimal for mitigating a_a/f_b ≤ 0.45 of adhesive failures, while the [0]₁₆ - [0]₁₆ configuration proves more effective in resisting adhesive failures of (a_a/f_b > 0.45).

Figure 19.

Effect of brace ply orientation on maximum total SERR (G_T-max) during radial growth of adhesive and cohesive failures (2ϕ_a = 30° & 2ϕ_c = 30°) with optimized chord configuration of (i) Cross-ply [0/90]_4S for adhesive failure (ii) Unidirectional [0]₁₆ for adhesive failure and (iii) Unidirectional [0]₁₆ for cohesive failure.

In line with the adhesive failure study, the investigation of cohesive failures involved varying the brace stacking sequences in combination with changes in radial failure lengths (a_c/f_m), while keeping the unidirectional [0]₁₆ chord configuration fixed as the optimal orientation (Figure 19(iii)). Consistent with adhesive failure behavior, cohesive failure resistance is influenced more significantly by chord ply orientation, while the impact of brace stacking sequences remains comparatively less substantial. Maximum cohesive failure resistance was achieved in the brace when circumferentially oriented fibers ([0]₁₆) were employed, across all radial growth ranges. In contrast, angle-ply [45/-45]_4S configuration is the most censorious brace ply orientation for the entire range of cohesive failure and should not be employed for better performance of the bonded TTJs. It is important to note that [0]₁₆ chord – [0]₁₆ brace is the best ply-configuration of the bonded TTJ to arrest the cohesive failures. Accordingly, it can be concluded in order to improve the damage tolerance of the bonded T-joint against the adhesive and cohesive failures [0]₁₆ chord-[0]₁₆ brace will be the optimum set of ply-configuration (Figure 18 -19).

Summary and conclusions

Adhesive bonded composite TTJs with fixed chord ends and brace under combined loading have been appraised for comprehensive stress, failure, and fracture analysis in the present research. To assess fracture behavior and improve structural resilience, APDL codes rooted in computational fracture mechanics have been implemented in ANSYS for simulating crack initiation, growth, and strategies to enhance damage tolerance in bonded joints. Joint fractures are most likely to initiate along the lines (BTL & MFL) and interfaces (CAI & AFA). For locating the critical zones for adhesive and cohesive failure initiation, three-dimensional stress and failure index profiles have been examined along key interfaces within the joint. Radial and circumferential propagation of the pre-embedded joint fractures in the fracture susceptible regions has been quantified through energy based fracture parameters, i.e., SERR (G_I, G_II, G_III & G_T), calculated using VCCT. Maximum total SERR (G_T-max) has been adopted as the vital criterion for improving joint fracture growth resistance of adhesive and cohesive failure to obtain optimal ply orientation of adherends. The principal outcomes of the current analysis are listed below:

• Saddle point at β = 0⁰ for BTL (S_1C) is the joint location vulnerable to adhesive failure primarily due to circumferential stress concentration, however same location corresponding to MFL is critical for cohesive failure because of all the normal and radial axial shear stress concentrations.

• Adhesive and cohesive failure at the bonded T-joint region are prone to grow radially than circumferentially, predominantly through opening mode.

• Radial growth of adhesive and cohesive failure causes shifting of the fracture susceptibility region (maximum SERR concentration) on circumferential damage fronts from edges towards the central region for opening mode of fracture.

• Simultaneous growth of adhesive and cohesive failures decelerate the growth of each other. However, cohesive failure has a predominant effect on adhesive failure in terms of fracture growth minimisation.

• Cross ply ([0/90]_4S)-circumferentially oriented ([0]₁₆) fibre direction is the best suited stacking sequence combination for chord-brace to enhance joint fracture resistance smaller sized adhesive failures (a_a/f_b ≤ 0.45) and circumferentially oriented ([0]₁₆) chord and brace for larger adhesive failures (a_a/f_b > 0.45) and cohesive failures of all ranges.

• Structural integrity of the bonded TTJ against simultaneous adhesive and cohesive failure can be improved through use of circumferentially oriented fibres ([0]₁₆) for both the brace and chord.

Footnotes

ORCID iDs

Umesh Kumar

Rashmi Ranjan Das

Funding

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

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The authors confirm that the data supporting the findings of this study are available within the article. Data that support the findings of this study may be provided by the corresponding author, upon reasonable request*.

Appendix

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