Contact Force and Friction of Generally Layered Laminates with Residual Hygrothermal Stresses under Mode II In-Plane-Shear Delamination

Abstract

Mode II (in-plane-shear) delamination tests are more complex than mode I (opening) due to the presence of a contact force between the two arms. This force is essential for the calculation of the energy release rate (ERR) and is closely linked to friction effects. A novel formulation is presented in this article to estimate the contact force analytically. Specifically, the contact force is derived within the context of the rigid, semi-rigid, and flexible joint models. The analytical solutions consider the case of a generally layered composite laminate with residual hygrothermal stresses and are used to evaluate the ERR. The new formulation is compared with numerical models created using the Virtual Crack Closure Technique (VCCT) and the Cohesive Zone Method (CZM) for a fiber–metal laminate. The results show that the new formulation provides nearly identical ERR predictions to those of the VCCT and CZM models. Additionally, it is demonstrated that the effect of friction on the ERR is less than 1%.

1. Introduction

Composite laminates are widely used as a structural material as they are lightweight and bear high loads. Slender composite laminates commonly fail under delamination which has been widely studied, and many analytical theories have been developed to understand this complex phenomenon [1,2]. The theories focus primarily on the energy release rate (ERR), the energy that is required for a unit length of crack to propagate.

The analytical theories are based on different assumptions that are made to simplify the 3D problem where delamination takes place on a plane (2D delamination). Here, the crack propagates between two sub-laminates (upper and lower part of the composite laminate) and can be simplified by transforming the 2D delamination problem into a 1D problem [3]. In particular, the two sub-laminates are considered as composite beams, and the crack propagates on a single line between the two sub-beams. Beam kinematic equations are derived using the conventional plate theory (CLT), where shear effects are not considered [4,5], or the first-order shear deformation theory (FSDT), where shear effects are considered. Both CLT and FSDT are formulated under the plane strain or plane stress assumptions. The 1D delamination problem is the most common case found in the literature and many theories have been presented considering different interface boundary conditions [6], i.e., different assumptions for the continuity of displacements and rotations of the bonded surface between the two sub-beams.

The most straightforward case is when the interface between the two sub-beams is considered to be totally rigid [7]. Here, both the relative displacements and the relative rotations at the crack tip are zero. This case is mentioned as the conventional composite beam theory or rigid joint model [8]. Even though the assumption of zero relative displacements and rotations simplifies significantly the formulation, it can highly influence the predictions of the ERR [9,10] and should be avoided.

The shear deformable theory (or semi-rigid joint model) is a generalization of the conventional composite beam theory and is able to capture the crack-tip rotations [11]. This model considers that only the relative displacements at the crack tip are zero, and the relative rotations are released. This considerably improves the predictions of the ERR and has been extensively used [3,12,13,14].

However, when local phenomena are considered, the imposition of zero relative displacements can lead to inaccuracies regarding the distribution of stresses close to the crack tip [15]. This problem is addressed by the interface deformable theory (or flexible joint model) [15], where both relative displacements and rotations are allowed between the two sub-beams. The flexible joint model is a general version of the “beams on elastic foundation” model developed for symmetric beams [9,16,17].

The three aforementioned theories are initially formulated under generic loading conditions and then expressed for specific delamination tests. The boundary conditions that are considered for delamination tests express the forces that are applied on each sub-beam. For mode I opening tests, such as the Double Cantilever Beam (DCB) [18] test, only external loads are considered since there is no contact between the two plates. However, for mode II in-plane shear tests, such as the End-Notched Flexure (ENF) [19] and the End-Loaded Split (ELS) [20] test, a contact force appears between the two sub-beams and should be considered in the formulation.

The effects of this force on the ERR were investigated in the work of Nairn [21]. The approach that Nairn followed for the estimation of the contact force was based on the rigid joint model and was specifically applied for a three-layered laminate where the upper part and the lower part were considered to be individually symmetric. Residual thermal stresses were considered in the formulation but extension–bending coupling phenomena and shear stresses were neglected.

Tsokanas and Loutas [14] used the same approach and presented explicit formulas for laminates with general stacking sequence utilizing the FSDT and considered also the extension–bending coupling of the upper and lower sub-beams. This formulation for the contact force, although not specified in the texts, was based on the rigid joint model as well, therefore considering a rigid interface on the crack tip with no relative rotations and translations.

An important outcome of the contact force is the presence of friction between the two sub-beams [21]. Friction can cause extra resistance to the crack development, which can increase the critical load and the critical ERR. These variations can also affect the predictions of ERR and multiple studies have been presented where these issues are considered.

For instance, Kageyama et al. [22] discussed the effects of various test conditions, such as specimen size, loading conditions, and friction between delamination surfaces on the mode II interlaminar fracture toughness at the initiation and during delamination growth, through experimental and theoretical methods.

A few years later, Davidson and Sun [23] used numerical simulations to study the effects of friction, test geometry, and fixture compliance on the perceived mode II interlaminar fracture toughness of laminated composites. A newly developed “direct energy balance approach” was employed to obtain the true energy release rate for a range of specimens, test geometries, and coefficient of friction. Additionally, a compliance calibration technique was utilized and was combined with experimental results from fixture compliance tests to obtain the perceived toughness. By varying the different parameters, the individual and combined effects of friction, test geometry, and fixture compliance on the ratio of perceived to true toughness were obtained.

A study with a similar aim was performed by Sun and Davidson [24]. This research utilized nonlinear finite element analyses to study the impact of friction and geometric nonlinearities on the ERR in three- and four-point ENF tests. The study used the direct energy balance approach to determine the ERR and found that the effects of nonlinearities on the ERR were greater in the four-point test than in the three-point test.

Bing and Sun [25] examined the impact of transverse normal stress on mode II fracture toughness of unidirectional fiber-reinforced composites. The study combined experimental testing and finite element analyses to study the S2/8552 glass/epoxy composite using off-axis specimens with a through-the-width crack. The authors concluded that transverse normal compressive stress had a significant effect on the mode II fracture toughness of the composite.

Parrinello et al. [26] presented an experimental and numerical study of the effect of friction on mode II delamination in composite laminates. The study highlighted that frictional stresses between the crack edges could absorb and dissipate significant energy during delamination. Their research included experiments on a set of unidirectional ENF specimens. A numerical analysis that employed a cohesive-frictional constitutive model with only frictional strength was performed to aid the understanding of the experiments and evaluate the frictional constitutive parameters.

Finally, Mencattelli et al. [27] evaluated the impact of friction on four-point ENF tests for carbon fiber–epoxy composite materials. The authors examined the hysteresis loop in the experimentally obtained load versus displacement curve due to an unloading–loading cycle. The effects of friction were considered from external (pins) and internal (the interface between the two arms) sources, and numerical simulations were performed aiming to better understand the experimental results.

Considering the above, it is evident that analytical and numerical methods can aid the understanding of mode II delamination tests. In this study, all three analytical models (the rigid joint model, the semi-rigid joint model, and the flexible joint model) are used to derive analytical solutions regarding the contact force. The formulation is based on the FSDT and takes into account extension–bending coupling phenomena, shear deformations, and residual hygrothermal stresses. The solutions for each joint model using the boundary conditions of the ENF and ELS tests are reconsidered and additional terms that account for the presence of the friction force are added. The derived formulas of the contact force are then used to compute the ERR. The results of the analytical models are compared with regard to the contact force and the ERR with finite element (FE) models developed using the Cohesive Zone Method (CZM) and the Virtual Crack Closure Technique (VCCT).

In this study, the analytical formulation of the contact force, including the effects of friction, is derived in order to address three key points. Firstly, the analytical solution of the contact force is determined for three distinct models and the effects of different interface boundary conditions are evaluated. This is crucial because the existing literature often uses the contact force from the rigid joint model to predict the ERR in any of the three joint models. However, the rigid joint model presents errors with respect to the prediction of the ERR; hence, it is necessary to derive the contact force using the semi-rigid and flexible joint models. Secondly, the effect of residual stresses on the contact force is examined, as the residual stresses are a significant driving force in inducing delamination. Finally, the importance of friction at the interface between two arms on the ERR is investigated, since it can impact the accuracy of experimental data.

2. Theoretical Background

This section presents the theoretical background used for the study of a composite laminate under delamination and generic loading conditions as it is necessary to introduce important aspects of physics behind the rigid, semi-rigid, and flexible joint models as well as establish the terminology used in the ensuing analysis of mode II tests. Only the important aspects of the three models are presented and the reader is referred to [28] for more information.

2.1. Problem Description

Figure 1 showcases a composite laminate with residual hygrothermal stresses. The laminate contains a crack along the width (B) which splits the laminate into two asymmetric and unbalanced sub-laminates (s) of thickness $h^{(s)}$ which are nonidentical. In the ensuing analysis, the upper and lower sub-laminates are denoted by the superscripts $s=1,2$. For the study of the stresses along the length of the laminate $(L)$, two portions are used, namely, the cracked part and the uncracked part. The uncracked part has a length of $(L-\alpha )$ with $\alpha$ denoting the crack length, and $(L-\alpha )$ is much larger than the thickness of the laminate $(h^{(1)}+h^{(2)})$. Note that the composite laminate and the two sub-laminates are generally layered and can be comprised of any type of linear elastic material.

Plane strain and plane stress assumptions have been used before in formulating the delamination problem. Here, the plane strain case is considered, and the 2D delamination problem reduces to 1D. The crack now propagates on a line situated at the interface between two beams (the upper and the lower sub-beams) as illustrated in Figure 2a.

2.2. Static Equilibrium

The crack-tip element [15] is a system of two beams. Here, the cracked and uncracked parts of the system are studied together, and generic loading conditions are imposed (Figure 2a) [6]. For the problem presented in the previous section, two coordinate systems are considered: $x^{\prime }-z^{\prime }$ ($x^{\prime }\in [0,{\alpha }^{-}]$) and $x-z$ ($x\in [0^{+},L-\alpha ]$). The first coordinate system corresponds to the cracked part and the second to the uncracked. The tip of the crack is situated at $x^{\prime }=\alpha$ (or $x=0$).

Taking into account the aforementioned considerations, one can express the overall equilibrium equations of the crack-tip element as follows (Figure 2b):

$$\begin{array}{ccc}\hfill {\mathcal{N}}^{(1^{\prime })}+{\mathcal{N}}^{(2^{\prime })}& =& {\mathcal{N}}^{(1)}+{\mathcal{N}}^{(2)}={\mathcal{N}}^{(0)}\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\mathcal{Q}}^{(1^{\prime })}+{\mathcal{Q}}^{(2^{\prime })}& =& {\mathcal{Q}}^{(1)}+{\mathcal{Q}}^{(2)}={\mathcal{Q}}^{(0)}\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\mathcal{M}}^{(1^{\prime })}+{\mathcal{M}}^{(2^{\prime })}+{\mathcal{N}}^{(1^{\prime })}{\displaystyle \frac{h^{(1)}+h^{(2)}}{2}}+{\mathcal{Q}}^{(0)}x& =& {\mathcal{M}}^{(1)}+{\mathcal{M}}^{(2)}+{\mathcal{N}}^{(1)}{\displaystyle \frac{h^{(1)}+h^{(2)}}{2}}={\mathcal{M}}^{(0)}\hfill \end{array}$$

(3)

Here, $\mathcal{N}$, $\mathcal{Q}$, and $\mathcal{M}$ represent the axial forces, shear forces, and bending moments. The superscripts $(s^{\prime })$ and $(s)$ are used to distinguish between the uncracked part and the cracked part. Finally, the superscript $(0)$ refers to the neutral axis of the total beam at a large distance from the crack tip.

2.3. Constitutive Equations

The FSDT [29] is considered to establish the constitutive equations of the system. The constitutive equations for the plane strain case can be expressed in terms of the extensional a, bending–extension coupling b, shear l, and bending d compliance and the axial strain ${\mathsf{e}}_1$ and curvature ${\mathsf{e}}_2$ induced by the residual hygrothermal effects as:

$$\left[\begin{array}{c}\epsilon \\ \kappa \end{array}\right]=\left[\begin{array}{cc}a& b\\ b& d\end{array}\right]\left[\begin{array}{c}\mathcal{N}\\ \mathcal{M}\end{array}\right]+\left[\begin{array}{c}{\mathsf{e}}_1\\ {\mathsf{e}}_2\end{array}\right],\gamma =l\mathcal{Q}$$

(4)

The superscript of $\mathcal{N}$, $\mathcal{Q}$, and $\mathcal{M}$ is here omitted as they can refer to the cracked and the uncracked parts. The above formulas have been derived considering the kinematic assumptions of the FSDT. These kinematic assumptions allow one to express the axial strain ($\epsilon$), the shear strain ($\gamma$), and the curvature ($\kappa$) as:

$$\epsilon ={\displaystyle \frac{du}{dx}},\gamma ={\displaystyle \frac{dw}{dx}}+\varphi ,\kappa ={\displaystyle \frac{d\varphi }{dx}}$$

(5)

where u, w, and $\varphi$ represent the axial displacement, the transverse displacement, and the cross-sectional rotation of the middle plane of both sub-beams. Note that only thermal effects are taken into account in this study and the effects of moisture conditions on the beams are omitted. However, one can simply superpose their effects in a similar manner.

2.4. Analysis of the System of Two Sub-Beams

The equilibrium equations of the system of the two sub-beams (applicable to all joint models) can be derived considering the free-body diagram illustrated in Figure 3a:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{\mathcal{N}}^{(1)}}{dx}}& =& B\tau ,{\displaystyle \frac{d{\mathcal{N}}^{(2)}}{dx}}=-B\tau \hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{\mathcal{Q}}^{(1)}}{dx}}& =& B\sigma ,{\displaystyle \frac{d{\mathcal{Q}}^{(2)}}{dx}}=-B\sigma \hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{\mathcal{M}}^{(1)}}{dx}}& =& {\mathcal{Q}}^{(1)}-B{\displaystyle \frac{h^{(1)}}{2}}\tau ,{\displaystyle \frac{d{\mathcal{M}}^{(2)}}{dx}}={\mathcal{Q}}^{(2)}-B{\displaystyle \frac{h^{(2)}}{2}}\tau \hfill \end{array}$$

(8)

where $\sigma$ denotes the normal and $\tau$ the shear stress located on the interface between the two sub-beams. Both $\sigma$ and $\tau$ are functions of x.

2.5. Continuity Conditions

The continuity conditions regarding the displacement and the rotation of the bonded section of the system of the two sub-beams are different for each of the three models (denoted by m). In this study, three different models of joints are considered: the rigid joint model ($m=1$) in which both translations and rotation are not allowed, the semi-rigid joint model ($m=2$) in which only the translations are not allowed, and the flexible joint model ($m=3$) where both are allowed. Figure 3b presents the resulting moments and forces and the interface’s boundary conditions. To distinguish the variables (displacements, rotations, etc.) among the three models, the superscript $(ms)$ is used when a variable of the uncracked part of a sub-beam s is considered using each model m. The interface models refer only to the interface, and therefore, the variables that refer to the cracked part of the system do not depend on the assumptions of the model and are distinguished using the superscript $(s^{\prime })$. Finally, the value of $s=0$ is used for variables that correspond to the total beam. The continuity conditions of the three models are based on [8,28,30].

2.5.1. Rigid Joint Model

For the rigid joint model, there is no relative translation and rotation between the two sub-beams. Therefore, the displacements and rotation of the interface are equal which can be mathematically expressed as:

$$\begin{array}{c}\hfill w^{(11)}=w^{(12)},u^{(11)}-{\displaystyle \frac{h^{(1)}}{2}}{\varphi }^{(11)}=u^{(12)}+{\displaystyle \frac{h^{(2)}}{2}}{\varphi }^{(12)}\end{array}$$

(9)

Here, $w^{(ms)}$, $u^{(ms)}$, and ${\varphi }^{(ms)}$ with $m=1$ denote the transverse and the axial displacement and the rotation.

Furthermore, the relative displacements ($\Delta u^{(1)},\Delta w^{(1)}$) and the relative rotation ($\Delta {\varphi }^{(1)}$) can be expressed as:

$$\begin{array}{c}\hfill \Delta u^{(1)}=u^{(12)}-u^{(11)}=0,\Delta w^{(1)}=w^{(12)}-w^{(11)}=0,\Delta {\varphi }^{(1)}={\varphi }^{(12)}-{\varphi }^{(11)}=0\end{array}$$

(10)

2.5.2. Semi-Rigid Joint Model

Regarding the semi-rigid joint model ($m=2$), the continuity is still valid since displacements are not permitted. This can be expressed as:

$$\begin{array}{c}\hfill w^{(21)}=w^{(22)},u^{(21)}-{\displaystyle \frac{h^{(1)}}{2}}{\varphi }^{(21)}=u^{(22)}+{\displaystyle \frac{h^{(2)}}{2}}{\varphi }^{(22)}\end{array}$$

(11)

However, allowing rotations implies a change in the condition of the relative rotation. The relative displacements ($\Delta u^{(2)},\Delta w^{(2)}$) and rotation ($\Delta {\varphi }^{(2)}$) are given by:

$$\begin{array}{c}\hfill \Delta u^{(2)}=u^{(22)}-u^{(21)}=0,\Delta w^{(2)}=w^{(22)}-w^{(21)}=0,\Delta {\varphi }^{(2)}={\varphi }^{(22)}-{\varphi }^{(21)}\ne 0\end{array}$$

(12)

2.5.3. Flexible Joint Model

The flexible joint model ($m=3$) was developed to explain the deformation appearing at the interface of each sub-beam. This deformation is, in general, nonlinear as explained in [8]. A proportional relationship between the interface stress and the deformation was considered in [15,31]. This assumption requires including additional terms in the continuity conditions. In particular, these terms express the contribution from the interface stress to the total displacement and are mentioned as “interface compliance coefficients” ${\mathsf{C}}_i^{(s)}$ ($i=1$: normal, $i=2$: shear) in [32]. Considering the above, the continuity conditions read:

$$\begin{array}{ccc}\hfill w^{(31)}-{\mathsf{C}}_1^{(1)}\sigma & =& w^{(32)}+{\mathsf{C}}_1^{(2)}\sigma ,u^{(31)}-{\displaystyle \frac{h^{(1)}}{2}}{\varphi }^{(31)}-{\mathsf{C}}_2^{(1)}\tau =u^{(32)}+{\displaystyle \frac{h^{(2)}}{2}}{\varphi }^{(32)}+{\mathsf{C}}_2^{(2)}\tau \hfill \end{array}$$

(13)

with ${\mathsf{C}}_1^{(s)}=h^{(s)}/10E_{33}^{(s)}$ and ${\mathsf{C}}_2^{(s)}=h^{(s)}/15G_{13}^{(s)}$.

The interface compliance coefficients are estimated by a semi-analytical and semi-numerical calibrating process [33]. $E_{33}^{(s)}$ and $G_{13}^{(s)}$ are the Young’s and shear moduli of the sub-beam s along its thickness. Note that for zero compliance coefficients, Equation (13) reduces to the one presented for the semi-rigid joint model.

The relative displacements ($\Delta u^{(3)},\Delta w^{(3)}$) and the relative rotation ($\Delta {\varphi }^{(3)}$) for the flexible joint model ($m=3$) can be expressed as:

$$\begin{array}{c}\hfill \Delta u^{(3)}=u^{(32)}-u^{(31)}\ne 0,\Delta w^{(3)}=w^{(32)}-w^{(31)}\ne 0,\Delta {\varphi }^{(3)}={\varphi }^{(32)}-{\varphi }^{(31)}\ne 0\end{array}$$

(14)

3. Mode II Delamination Tests

The above analysis has been established for generic loading conditions in [8] and in this article, it is applied to two mode II delamination tests, the ENF and the ELS (Figure 4). The ENF and the ELS are considered here since they are commonly used for the study of fracture toughness in mode II.

Using the forces presented in Figure 4, the static equilibrium of the crack portion (Equations (1)–(3)) is utilized to obtain the axial forces, the shear forces, and the bending moments as:

• ENF ($P_1=0$, $P_2=P/2$)

  $$\begin{array}{c}\hfill {\mathcal{N}}^{(1^{\prime })}=-\mu P_c,{\mathcal{N}}^{(2^{\prime })}=\mu P_c,{\mathcal{Q}}^{(1^{\prime })}=-P_c,{\mathcal{Q}}^{(2^{\prime })}=-P/2+P_c\end{array}$$

  (15)
• ELS ($P_1=0$, $P_2=P$)

  $$\begin{array}{c}\hfill {\mathcal{N}}^{(1^{\prime })}=-\mu P_c,{\mathcal{N}}^{(2^{\prime })}=\mu P_c,{\mathcal{Q}}^{(1^{\prime })}=-P_c,{\mathcal{Q}}^{(2^{\prime })}=-P+P_c\end{array}$$

  (16)

and

$$\begin{array}{c}\hfill {\mathcal{M}}^{(s^{\prime })}={\mathcal{Q}}^{(s^{\prime })}{x}^{\prime }-{\displaystyle \frac{h^{(s)}}{2}}{\mathcal{N}}^{(s^{\prime })}\end{array}$$

(17)

In the above equations, $\mu$ is the coefficient of friction, $P_c$ is the contact force developed between the two sub-laminates, and $P,P_1,$ and $P_2$ are the external acting forces as illustrated in Figure 4.

3.1. Contact Force between the Two Sub-Beams

The contact force $P_c$ transfers the load from the lower to the upper sub-beam, and in general, it is not equal to the reaction of the left support (for the ENF) or of the external load (for the ELS). However, the two forces have been previously considered equal in many studies [8,30,34,35], which is a correct assumption for symmetric laminates with individually symmetric sub-laminates.

For instance, Nairn [21] derived explicit formulas that related the displacement and contact force of the two arms for end-loaded specimens to the imposed loads and the thermal effects. Figure 4c illustrates the problem presented in the work of Nairn [21]. The approach for the estimation of the contact force was based on integrating the curvature equation $\kappa$ deriving from the CLT and obtaining the vertical displacement w of each of the two sub-laminates. Then, the contact force was derived by imposing a zero relative deflection at the end tip of the arms (${\Delta w|}_{x=0}$). Nairn specifically applied it for a three-layered laminate where the upper part and the lower part were considered to be individually symmetric.

Tsokanas and Loutas [14] used the same approach and presented explicit formulas for laminates with general stacking sequence utilizing FSDT and also considered the extension–bending coupling of the upper and lower sub-laminates.

The aforementioned formulations for the contact force are based on the rigid joint model. In the present work, the process of Nairn [21] is followed and the contact force equations are derived using the boundary conditions of all three analytical models, the rigid, the semi-rigid, and the flexible joint model, therefore extending the formulas of [14,21].

3.2. Static Equilibrium

The static equilibrium for each sub-beam s, illustrated in Figure 4c, is given by:

$$\begin{array}{ccc}\hfill {\mathcal{N}}^{(s^{\prime })}& =& n^{(s)}\mu P_c\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\mathcal{Q}}^{(s^{\prime })}& =& -(P_i-n^{(s)}{P}_c)\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill {\mathcal{M}}^{(s^{\prime })}& =& -(P_i-n^{(s)}{P}_c)x^{\prime }-n^{(s)}\mu P_c{\displaystyle \frac{h^{(s)}}{2}}\hfill \end{array}$$

(20)

where $n^{(s)}$ expresses the direction of the contact force for each sub-beam s:

$$\begin{array}{c}\hfill n^{(1)}=-1,n^{(2)}=1\end{array}$$

(21)

3.3. General Solution of Displacements and Rotation of the Cracked Part

Combining Equation (4) with Equations (18) and (20) and integrating with respect to $x^{\prime }$, the axial displacement $u^{(s^{\prime })}$ of the cracked part of each sub-beam can be derived as:

$$\begin{array}{cc}\hfill u^{(s^{\prime })}=& a^{(s)}(n^{(s)}\mu P_c)(x^{\prime }-\alpha )\hfill \\ \hfill & +b^{(s)}\left[-(P_i-n^{(s)}{P}_c){\displaystyle \frac{{(x^{\prime })}^2-{\alpha }^2}{2}}-n^{(s)}\mu P_c{\displaystyle \frac{h^{(s)}}{2}}(x^{\prime }-\alpha )\right]+{\mathsf{e}}_1^{(s)}(x^{\prime }-\alpha )+u^{(s)}{|}_{x^{\prime }=\alpha }\hfill \end{array}$$

(22)

and similarly for the rotation ${\varphi }^{(s^{\prime })}$:

$$\begin{array}{cc}\hfill {\varphi }^{(s^{\prime })}=& b^{(s)}(n^{(s)}\mu P_c)(x^{\prime }-\alpha )\hfill \\ \hfill & +d^{(s)}\left[-(P_i-n^{(s)}{P}_c){\displaystyle \frac{{(x^{\prime })}^2-{\alpha }^2}{2}}-n^{(s)}\mu P_c{\displaystyle \frac{h^{(s)}}{2}}(x^{\prime }-\alpha )\right]+{\mathsf{e}}_2^{(s)}(x^{\prime }-\alpha )+{\varphi }^{(s^{\prime })}{|}_{x^{\prime }=\alpha }\hfill \end{array}$$

(23)

The vertical displacement $w^{(s^{\prime })}$ is derived by substituting Equations (19) and (23) in Equation (4) and integrating with respect to $x^{\prime }$:

$$\begin{array}{cc}\hfill w^{(s^{\prime })}=& \left[l^{(s)}(n^{(s)}{P}_c-P_i)\right](x^{\prime }-\alpha )+n^{(s)}\mu P_c\left(b^{(s)}-{\displaystyle \frac{h^{(s)}}{2}}{d}^{(s)}\right)\left(-{\displaystyle \frac{{(x^{\prime })}^2}{2}}+\alpha x^{\prime }-{\displaystyle \frac{{\alpha }^2}{2}}\right)\hfill \\ \hfill & +\left[d^{(s)}(n^{(s)}{P}_c-P_i)\right]\left(-{\displaystyle \frac{{(x^{\prime })}^3}{6}}+{\displaystyle \frac{{\alpha }^2{x}^{\prime }}{2}}-{\displaystyle \frac{{\alpha }^3}{3}}\right)+{\mathsf{e}}_2^{(s)}\left(-{\displaystyle \frac{{(x^{\prime })}^2}{2}}+\alpha x^{\prime }-{\displaystyle \frac{{\alpha }^2}{2}}\right)\hfill \\ & +{\varphi }^{(s^{\prime })}{{|}_{x^{\prime }=\alpha }(\alpha -x^{\prime })+w^{(s^{\prime })}|}_{x^{\prime }=\alpha }\hfill \end{array}$$

(24)

where $u^{(s^{\prime })}{|}_{x^{\prime }=\alpha },{\varphi }^{(s^{\prime })}{{|}_{x^{\prime }=\alpha },w^{(s^{\prime })}|}_{x^{\prime }=\alpha }$ denote the axial displacement, the rotation, and the vertical displacement of the sub-beam s at the crack tip and are evaluated based on the boundary conditions that are imposed at the interface between the two sub-beams, i.e., by the different joint models.

It is worth mentioning that the vertical displacement $w^{(s^{\prime })}$ has the general form:

$$w(x)=w^{bending}(x)+w^{shear}(x)+w^{axial}(x)+w^{thermal}{(x)+w|}_{x=\alpha }$$

(25)

The bending term $w^{bending}(x)$ (terms of Equation (24) with bending compliance d) is equal to the one derived by the classical beam theory (Euler–Bernoulli theory for beams). The shear term $w^{shear}(x)$ (terms of Equation (24) with shear compliance l) corresponds to the addition of the shear effects and is equal to the one derived by Timoshenko’s beam theory. The axial term $w^{axial}(x)$ expresses the effect of the axial force (here, the contact force $P_c$) on the sub-beam’s deflection. For positive values, the axial force lowers the stiffness of the sub-beam and the deflection increases, whereas for negative values, the stiffness of the sub-beam increases and the deflection decreases. Notice also that the coupled effect of the axial forces and moments, represented by the appearance of the extension–bending coupling compliance $b^{(s)}$ in Equation (24), is captured by the present formula. This part was also neglected in the original work of Nairn [21]. The thermal term $w^{thermal}(x)$ (terms of Equation (24) with a middle plane curvature induced by the residual hygrothermal effects ${\mathsf{e}}_2$) expresses the influence of residual thermal stresses on the sub-beam’s deflection. Finally, the initial conditions at the crack tip ${w|}_{x=\alpha }$ are added and evaluated based on the different boundary conditions that are assumed at the interface of the two sub-beams, i.e., based on the rigid joint model, the semi-rigid joint model, or the flexible joint model.

3.4. Relative Displacement between the Two Sub-Beams

Utilizing Equation (24), the relative vertical displacement between the two sub-beams can be expressed as:

$$\begin{array}{cc}\hfill \Delta w=& w^{(2^{\prime })}-w^{(1^{\prime })}=\hfill \\ \hfill & \left[l^{(2)}(P_2-P_c)-l^{(1)}(P_c+P_1)\right](\alpha -x^{\prime })\hfill \\ \hfill & -\mu P_c\left(b^{(2)}-{\displaystyle \frac{h^{(2)}}{2}}{d}^{(2)}-b^{(1)}+{\displaystyle \frac{h^{(1)}}{2}}{d}^{(1)}\right)\left(-{\displaystyle \frac{{(x^{\prime })}^2}{2}}+\alpha x^{\prime }-{\displaystyle \frac{{\alpha }^2}{2}}\right)\hfill \\ \hfill & +\left[d^{(2)}(P_2-P_c)-d^{(1)}(P_c+P_1)\right]\left({\displaystyle \frac{{(x^{\prime })}^3}{6}}-{\displaystyle \frac{{\alpha }^2{x}^{\prime }}{2}}+{\displaystyle \frac{{\alpha }^3}{3}}\right)\hfill \\ \hfill & {+\Delta \varphi |}_{x^{\prime }=\alpha }(\alpha -x^{\prime }){+\Delta w|}_{x^{\prime }=\alpha }+{\displaystyle \frac{1}{2}}({\mathsf{e}}_2^{(1)}-{\mathsf{e}}_2^{(2)}){(\alpha -x^{\prime })}^2\hfill \end{array}$$

(26)

or in a matrix form:

$$\begin{array}{cc}\hfill \Delta w=& \left[\begin{array}{ccc}{\Omega }_1& {\Omega }_2& {\Omega }_3\end{array}\right]\left[\begin{array}{c}{P}_c\\ P_1\\ P_2\end{array}\right]+\Delta \varphi {{|}_{x^{\prime }=\alpha }(\alpha -x^{\prime })+\Delta w|}_{x^{\prime }=\alpha }+{\displaystyle \frac{1}{2}}({\mathsf{e}}_2^{(1)}-{\mathsf{e}}_2^{(2)}){(\alpha -x^{\prime })}^2\hfill \end{array}$$

(27)

where

$$\begin{array}{cc}\hfill {\Omega }_1& =(x^{\prime }-\alpha )(l^{(1)}+l^{(2)})-\mu \left(-b^{(1)}+{\displaystyle \frac{h^{(1)}}{2}}{d}^{(1)}+b^{(2)}-{\displaystyle \frac{h^{(2)}}{2}}{d}^{(2)}\right)\left(-{\displaystyle \frac{{(x^{\prime })}^2}{2}}+\alpha x^{\prime }-{\displaystyle \frac{{\alpha }^2}{2}}\right)\hfill \\ \hfill & +\left(d^{(1)}+d^{(2)}\right)\left(-{\displaystyle \frac{{(x^{\prime })}^3}{6}}+{\displaystyle \frac{{\alpha }^2{x}^{\prime }}{2}}-{\displaystyle \frac{{\alpha }^3}{3}}\right)\hfill \\ \hfill {\Omega }_2& =(x^{\prime }-\alpha )l^{(1)}+d^{(1)}\left(-{\displaystyle \frac{{(x^{\prime })}^3}{6}}+{\displaystyle \frac{{\alpha }^2{x}^{\prime }}{2}}-{\displaystyle \frac{{\alpha }^3}{3}}\right)\hfill \\ \hfill {\Omega }_3& =-(x^{\prime }-\alpha )l^{(2)}-d^{(2)}\left(-{\displaystyle \frac{{(x^{\prime })}^3}{6}}+{\displaystyle \frac{{\alpha }^2{x}^{\prime }}{2}}-{\displaystyle \frac{{\alpha }^3}{3}}\right)\hfill \end{array}$$

(28)

with ${\Delta w|}_{x^{\prime }=0}=0$ since the two ends must be in contact in order for a contact force to be generated.

A special solution of Equation (26) was presented in the work of Tsokanas and Loutas [14], where the relative deflections ${\Delta w|}_{x^{\prime }=\alpha }$ at the crack tip, the relative rotations ${\Delta \varphi |}_{x^{\prime }=\alpha }$ at the crack tip, and the friction force between the two sub-beams $\mu P_c$ were neglected. Equation (26) expressing the relative behavior between the two sub-beams at the cracked part of a composite laminate is provided for the first time in the literature.

The relative deflections ${\Delta w|}_{x^{\prime }=\alpha }$ and the relative rotations ${\Delta \varphi |}_{x^{\prime }=\alpha }$ at the crack tip are evaluated considering the displacement and rotation continuity conditions at the crack tip, where $x^{\prime }=a$ for the left-hand side of the crack tip (i.e., the cracked part), and $x=0$ at the right-hand side of the crack tip (i.e., the uncracked part):

$${\left[\begin{array}{c}\Delta u^{(m)}\\ \Delta {\varphi }^{(m)}\\ \Delta w^{(m)}\end{array}\right]}_{x^{\prime }={\alpha }^{-}}={\left[\begin{array}{c}\Delta u^{(m)}\\ \Delta {\varphi }^{(m)}\\ \Delta w^{(m)}\end{array}\right]}_{x^{\prime }=\alpha }={\left[\begin{array}{c}\Delta u^{(m)}\\ \Delta {\varphi }^{(m)}\\ \Delta w^{(m)}\end{array}\right]}_{x=0^{+}}={\left[\begin{array}{c}\Delta u^{(m)}\\ \Delta {\varphi }^{(m)}\\ \Delta w^{(m)}\end{array}\right]}_{x=0}$$

(29)

where the superscript m denotes one of the joint models. The solutions of $\Delta u^{(m)}$, $\Delta {\varphi }^{(m)}$, and $\Delta w^{(m)}$ can be evaluated from the solutions of the interface compliance of each model according to Equations (A4), (A11) and (A24). The relative axial displacements $\Delta u^{(m)}{|}_{x=0}$, the relative deflections $\Delta w^{(m)}{|}_{x=0}$, and the relative rotations $\Delta {\varphi }^{(m)}{|}_{x=0}$ for each model are given by:

$$\begin{array}{ccc}\hfill {\left[\begin{array}{c}\Delta u^{(m)}\\ \Delta {\varphi }^{(m)}\\ \Delta w^{(m)}\end{array}\right]}_{x=0}& =& \Delta \left[{\mathcal{S}}^{(m)}\right]\left[\begin{array}{c}N\\ M\\ Q\end{array}\right]\hfill \end{array}$$

(30)

where $\Delta \left[{\mathcal{S}}^{(m)}\right]=\left[{\mathcal{S}}^{(m2)}\right]-\left[{\mathcal{S}}^{(m1)}\right]$ expresses the relative compliance between the two sub-beams for each model m. Note that for the rigid model, where no relative translations or rotation are possible at the joint, the relative compliance is a zero matrix (i.e., $\Delta \left[{\mathcal{S}}^{(m)}\right]=\left[0\right]$).

$N,M,$ and Q are directly related to the calculation of the ERR (as shown in [28]) and need to be explicitly evaluated. Therefore, considering the overall equilibrium (Equations (1)–(3)), the continuity of resultant forces and moments at the crack tip (Equation (A3)) and the static equilibrium (Equations (18)–(20)), $N,M,$ and Q are evaluated after a lengthy process as:

$$\begin{array}{c}\hfill \left[\begin{array}{c}N\\ M\\ Q\end{array}\right]=\left[\begin{array}{c}\Phi \end{array}\right]\left[\begin{array}{c}{P}_c\\ P_1\\ P_2\end{array}\right]+\left[\begin{array}{c}\Theta \end{array}\right]\end{array}$$

(31)

Considering the boundary conditions of ENF and ELS tests Equations (15)–(17), the contact force can be expressed as

$$\begin{array}{cc}\hfill P_{cont}^{ENF}=& {\displaystyle \frac{-\left(({\Omega }_2+\Delta {\mathcal{S}}_{2j}^m\alpha {\Phi }_{j2}+\Delta {\mathcal{S}}_{3j}^m{\Phi }_{j3})\frac{P_2}{2}+(\Delta {\mathcal{S}}_{2j}^m\alpha +\Delta {\mathcal{S}}_{3j}^m){\Theta }_j+\frac{1}{2}({\mathsf{e}}_2^{(1)}-{\mathsf{e}}_2^{(2)}){\alpha }^2\right)}{{\Omega }_1+\Delta {\mathcal{S}}_{2j}^m\alpha {\Phi }_{j2}+\Delta {\mathcal{S}}_{3j}^m{\Phi }_{j3}}}\hfill \end{array}$$

(32)

for the ENF and

$$\begin{array}{cc}\hfill P_{cont}^{ELS}=& {\displaystyle \frac{-\left(({\Omega }_2+\Delta {\mathcal{S}}_{2j}^m\alpha {\Phi }_{j2}+\Delta {\mathcal{S}}_{3j}^m{\Phi }_{j3})P_2+(\Delta {\mathcal{S}}_{2j}^m\alpha +\Delta {\mathcal{S}}_{3j}^m){\Theta }_j+\frac{1}{2}({\mathsf{e}}_2^{(1)}-{\mathsf{e}}_2^{(2)}){\alpha }^2\right)}{{\Omega }_1+\Delta {\mathcal{S}}_{2j}^m\alpha {\Phi }_{j2}+\Delta {\mathcal{S}}_{3j}^m{\Phi }_{j3}}}\hfill \end{array}$$

(33)

for the ELS test.

The elements ${\Phi }_{ij}$, ${\Theta }_j$ of $[\Phi ]$, $[\Theta ]$ are given by:

$$\begin{array}{cc}\hfill & {\Phi }_{11}=-\mu \left({\displaystyle \frac{{\mathsf{a}}_1}{{\mathsf{a}}_2}}\right),{\Phi }_{12}=-\alpha {\displaystyle \frac{{\mathsf{a}}_1}{{\mathsf{a}}_2}},{\Phi }_{21}=-\alpha -\mu {\displaystyle \frac{h^{(1)}}{2}},{\Phi }_{22}=-\alpha {\displaystyle \frac{1}{\xi }}\left(b^{(2)}+{\displaystyle \frac{h^{(2)}}{2}}{d}^{(2)}+\eta {\displaystyle \frac{{\mathsf{a}}_1}{{\mathsf{a}}_2}}\right),\hfill \\ \hfill & {\Phi }_{31}=-1,{\Phi }_{32}=-\left[\left({\displaystyle \frac{\eta }{\xi }}+{\displaystyle \frac{h^{(1)}}{2}}\right){\displaystyle \frac{{\mathsf{a}}_1}{{\mathsf{a}}_2}}+{\displaystyle \frac{1}{\xi }}\left(b^{(2)}+{\displaystyle \frac{h^{(2)}}{2}}{d}^{(2)}\right)\right],\hfill \\ \hfill & {\Theta }_1=-{\displaystyle \frac{\mathsf{E}}{{\mathsf{a}}_2}},{\Theta }_2={\displaystyle \frac{\eta \mathsf{E}}{{\mathsf{a}}_2}}+{\mathsf{E}}_1,{\Theta }_3=0\hfill \end{array}$$

(34)

Note that coefficients ${\mathsf{a}}_1$, ${\mathsf{a}}_2$, $\eta$, $\xi$, and $\mathsf{E}$ are used in the governing equation and are explicitly given in Appendix A.

Equation (32) is a general solution of the one that was derived in the work of Tsokanas and Loutas [14] for the ENF test. In fact, their expression coincides with the present one for the rigid joint model without friction effects. A similar expression of the contact force without friction effects was also presented in [36]. The corresponding solution of Equation (33) for the ELS test using the rigid joint model and without friction effects was derived in [37]. The present solution provides, additionally, the solutions for the semi-rigid and the flexible joint models.

4. Case Study

4.1. Description

This study focused primarily on providing analytical solutions for the contact force in mode II delamination tests and utilizing this contact force to evaluate the ERR. The ERR can be estimated by utilizing the J-integral solutions outlined in [28]. In order to evaluate the J-integral, one needs to simply substitute the solutions for $N, M,$ and Q from Equation (31) in the final equation of the J-integral from [28] while considering the appropriate contact force from Equations (32) and (33) for the ENF and ELS tests, respectively.

Here, a case study was performed for the ENF test. The ELS test was omitted since the analytical solution for the contact force is almost identical. A generally layered composite laminate composed of layers of E-glass fiber prepregs in epoxy resin (E-GFRP) of thickness 0.191 mm bonded with aluminum 2014-T6 of thickness 0.4 mm was considered. The material properties were provided by the study of Bhat and Narayanan [38] and are presented in

Table 1. Properties (elastic and thermal) of aluminum 2014-T6, E-GFRP, and epoxy resin used in the simulation.

	Aluminum	E-GFRP	Epoxy Resin
$E_{11}$ (GPa)	72.00	38.73	3.50
$E_{22}$ (GPa)	72.00	6.94	3.50
$G_{12}$ (GPa)	27.06	2.50	1.25
${\nu }_{12}$	0.33	0.27	0.33
$a_{11}$ (10−6/°C)	23.00	7.26	57.50
$a_{22}$ (10−6/°C)	23.00	37.70	57.50

. A crack length of 25 mm and a total length of 100 mm were considered for the laminate. Load values were considered until 25 kN/m, and a temperature difference of ±135 degrees was considered, similar to [14]. The formulas of the contact force and the ERR were additionally compared with results of numerical FE models developed using the ABAQUS software version 22 [39].

4.2. Numerical Implementation

The CZM [40] and the VCCT [41,42] were both implemented for the numerical analysis. A brief introduction of the two methods is presented here to aid the understanding of the numerical simulation.

The CZM, introduced in [43,44], utilizes cohesive elements at the interface. These elements are situated along the direction of the crack and are governed by a constitutive law that combines a fracture mechanics and a stress-based criterion. In general, the load in one cohesive element increases with increasing load until a maximum stress level is reached. Then, the stiffness of the element degrades following a specific softening law. A linear softening law is the simplest one and is provided by ABAQUS. These elements were used here and the necessary parameters for their implementation were evaluated using the methodology of Turon et al. [45]. Finally, after loading the sample, the ERR was evaluated by integrating the stress of the first element at the tip of the crack [40].

The VCCT, introduced in [46], directly uses the nodes of the elements to evaluate the ERR, and no cohesive elements are needed at the interface. Implementing the VCCT in ABAQUS necessitates the definition of the crack propagation region. The nodes of this region were restrained using fixed node-to-node conditions for the displacement and rotation. The fixed condition on the nodes was released when the ERR exceeded a specific value, i.e., meeting a failure criterion. The B-K criterion [39,47] was used for this article. The reader is referred to [41,42] for more information concerning the implementation, history, and potential of the VCCT.

Unfortunately, the implementation of the VCCT comes with certain limitations such as inaccuracies due to the oscillatory nature of the stresses. To prevent oscillations, a thin artificial layer of epoxy was added between the two sub-beams as suggested by [3,14]. The thickness of this layer was taken sufficiently low (0.01 mm) so that the stiffness of the composite was not affected. For reasons of consistency, this thin epoxy layer was maintained for the CZM model as well, even though it was not necessary. Finally, the critical ERR for the CZM and the VCCT was taken as 10,000 so that the crack did not propagate since this study considered a stationary crack.

For the modeling of the composite laminate, the eight-node reduced integration plane strain quadrilateral elements (CPE8R in ABAQUS) were used to prevent shear locking from affecting the predictions of the model. Figure 5 illustrates the model in a meshed (a) and an unmeshed (b) state. Note that for both the CZM and the VCCT, the mesh was uniform and obtained by convergence studies, and approximately 15,500 and 12,000 elements were used for the upper and the lower sub-beams, respectively. Figure 5 also contains a magnified view of the crack tip. All simulations were performed in ABAQUS using the open-source Delamination Plug-In [48], which automatizes the simulation process.

5. Results

The analysis was carried out in two steps: one where only thermal loads were applied, and another where a mechanical load was imposed on the geometry that had already been deformed by the thermal loads. The deformed state of the FE models for the cases with only thermal loads of +135 (b) and −135 (d) degrees and the cases with thermal and mechanical loads (c) and (e) are presented in Figure 6. The results of both CZM and VCCT models were similar, and only the VCCT models are depicted in this figure.

The results of Figure 6 clearly show that imposing a +135-degree thermal load led to an inward rotation of the two arms of the cracked part of the laminate. On the other hand, a −135-degree thermal difference led to an outward rotation of the two arms. The different rotations resulted in different contacts at the end tip, as shown in a magnified form in Figure 6. This outcome has implications on the contact force, which is discussed further below. Note that the case without thermal loads presented similar contact as the case with +135 degrees.

5.1. Contact Force

The distribution of the contact force along the interface of the two sub-beams is presented in Figure 7 for the CZM and VCCT models for values of the external load equal to 5 ((a), (b), (c)), 15 ((d), (e), (f)), and 25 ((g), (h), (i)) kN/m as indicative values within the considered range. Furthermore, note that (a), (d), and (g) represent the results for the case without a thermal load, (b), (e), and (h) represent the results for the case with +135 degrees, and (c), (f), and (i) represent the results for the case with −135 degrees. Each dot on the lines of CZM and VCCT represents one nodal value of the contact force with the x coordinate counted from the left end tip of the composite laminate, as depicted in Figure 4c, and refers to the upper sub-beam. The coefficient of friction for the results presented here was considered to be 0.01 to limit the influence of the friction effects and focus only on the contact force.

Figure 7 illustrates that the distribution of the contact force was not identical for the three cases. In particular, the results of the +135-degree case and the case without thermal loads demonstrated increased effects at the edge with larger magnitudes than the −135-degree case. The −135-degree case further presented a spread of the values over a larger area. This area moved towards the left edge of the laminate as the external load increased.

This result is physically explained by the deformation caused by the residual thermal stresses on the structure. The application of +135 degrees of temperature difference resulted in an inward rotation of the lower sub-beam, leading to a point contact of the edge of the lower sub-beam and the upper sub-beam. On the other hand, a −135-degree temperature difference resulted in an outward rotation of the lower sub-beam, leading to the contact of a more extended area of the lower and upper sub-beam, located further away from the edge tip due to the rotation. However, an increase in external load led to an inward rotation of the lower sub-beam, moving the area towards the edge. In both cases, the upper sub-beam remained unaffected by the residual thermal stresses, being symmetric with respect to its middle plane. The case without thermal stresses did not present any additional rotation, leading to a point contact of the edge of the lower sub-beam and the upper sub-beam.

Figure 8 presents the contact force versus the external load for the cases without thermal loads (a), and with +135-degree (b) and −135-degree (c) temperature differences. The values of the VCCT and CZM models represent the sum of all contact nodal forces acting on the upper sub-beam. It is evident that the flexible joint model better followed the results of the CZM and the VCCT, which could be attributed to the release of the translation degree of freedom. The rigid and semi-rigid joint models presented a slightly higher estimation of the contact force. Moreover, the comparison of the analytical theories with the numerical results revealed that the contact force for the case without thermal loads (a) and a +135-degree (b) degree temperature difference was almost perfect for all the range of the considered load. However, for a −135-degree (c) temperature difference, there was a higher difference for the lower loads, still remaining relatively low. This implies that the location of contact plays an important role in the evaluation of the contact force. Nevertheless, as explained below, this difference did not result in large deviations in the prediction of the ERR. Additionally, it is evident that the contact force was not equal to the reaction force of the left end, which was half of that of the external load P. Finally, it was demonstrated that the temperature affected the contact force and led to an upward (case of +135 degrees) or a downward (case of −135 degrees) shift of the curve. This shift is related to the term $+{\displaystyle \frac{1}{2}}({\mathsf{e}}_2^{(1)}-{\mathsf{e}}_2^{(2)}){\alpha }^2$ included in Equations (32)–(33). Physically, this implies that the thermal effects are added as a constant parameter that is not influenced by the external load. The magnitude of the effect on the contact force depends on both the relative difference of the curvatures and the crack length.

5.2. Energy Release Rate

Figure 9 presents the ERR versus the external load for the cases without thermal loads ((a) and (d)), and with +135-degree ((b) and (e)), and −135-degree ((c) and (f)) temperature differences. Figure 9a–c present the solutions of the ERR considering the contact force evaluated by each respective joint model; for example, the formulation of the flexible joint model for the ERR was estimated by using the formulation of the flexible joint model for the contact force. On the other hand, Figure 9a–c present the solutions of the ERR considering the contact force evaluated by the rigid joint model, as it is commonly done in the literature. The value of the coefficient of friction was considered to be 0.01 to limit the influence of the friction effects on the ERR and we only considered the effects of the contact force that is presented in Section 5.1.

In Figure 9a–c, the comparison of the joint models, including the rigid, semi-rigid, and flexible models, with the VCCT and CZM models shows that the rigid and semi-rigid models were in good agreement with the VCCT, as they assumed no relative translation at the interface between the upper and lower sub-beams. Similarly, the flexible joint model was found to be comparable to the CZM, as it used cohesive elements between the two sub-beams that behaved like a series of springs. All three models were found to be consistent with the analytical theories and the numerical simulations. The addition of −135 degrees of temperature difference resulted in a decrease in the ERR whereas the addition of a −135-degree temperature difference resulted in an increase in the ERR. The decrease and increase in the ERR due to the thermal effects were of equal magnitude.

Figure 9d–f demonstrates that the consideration of the contact force evaluated by the rigid model in the evaluation of the ERR for all joint models did not influence the predictions of the semi-rigid model since the contact forces estimated by both the rigid and the semi-rigid model were almost equal. However, considering the contact force in the formulation of the flexible joint model considerably changed the estimation of the ERR. In fact, the ERR estimated by the flexible joint model was then almost identical to the one of the rigid and the semi-rigid joint model. Therefore, the application of the appropriate contact force is indeed important to obtain accurate results when using analytical models.

The impact of the coefficient of friction on the ERR is illustrated in Figure 10. The value of the ERR when a coefficient of friction of 0.01 was used is represented by $G_0$, whereas G represents the ERR when a different value of the coefficient of friction was applied. The study examined coefficient of friction values of up to 0.2, which is a typical value for an epoxy composite in contact with aluminum [49]. The results were only obtained for the case of +135 degrees, which considered both residual thermal stresses and a point force, consistent with the assumptions of the analytical models. The results showed that an increase in friction led to a linear decrease in the ERR. However, the decrease was minimal and for a coefficient of friction of 0.2, the expected reduction in the ERR was 0.8 %. It is worth noting that the external load was taken as 10 kN/m as the percentage-wise difference in the ERR was constant for all mechanical loads.

6. Conclusions

This article focused on the contact force appearing between the two arms of mode II delamination tests, specifically the ENF and ELS tests. A novel theoretical formulation was presented using rigid, semi-rigid, and flexible joint models by revisiting the formulation presented in [8,30]. Novel solutions for crack-tip displacements and rotations were derived and used for the relative compliance of the interface of the two sub-beams. Additionally, novel solutions for end-tip displacements and rotation were presented which took into account the movement of the end tip due to bending, axial, shear, and thermal effects, as well as the initial boundary conditions at the crack tip. These solutions were combined to derive explicit expressions of the contact force for the ENF and ELS tests while considering friction effects at the interface between the two sub-beams.

A case study was presented, and the analytical formulas were used to evaluate the ERR of a generally layered composite laminate with residual thermal stresses tested using a mode II ENF test. The results of the analytical formulation were compared with numerical models developed using the VCCT and CZM in terms of contact force, the evaluation of the ERR, and the effects of friction on the ERR.

The comparison illustrated that the analytical formulation was able to accurately capture the magnitude of the contact force for cases where the imposed external mechanical loads and thermal effects led to a point contact between the lower and upper sub-beams. However, one of the main limitations of the analytical formulation was its inability to capture the distribution of the contact force over a wide area. This resulted in a slight error in the prediction of the magnitude of the contact force, and future research should focus on including this in the analytical formulation. Nevertheless, the evaluation of the ERR was not affected by this difference. It was also shown that the contact force was not identical to the reaction of the lower left-end pin.

Using this contact force, the J-integral formulation of [30] was implemented to estimate the ERR. The comparison with numerical models clearly showed that the analytical formulas were capable of evaluating the ERR with and without friction effects. Finally, it was demonstrated that the effects of friction at the interface between the two sub-beams accounted for a small percentage of error in the prediction of the ERR.