A nonlinear cohesive/friction coupled model for shear induced delamination of adhesive composite joint

Abstract

This paper originally proposes a nonlinear cohesive/frictional contact coupled model for the mode-II shear delamination of adhesive composite joint based on a modified Xu and Needleman’s exponential cohesive model. First, the friction is assumed to increase nonlinearly at the delamination interface when the tangential cohesive softening appears. Second, a non-associative plasticity model based on the Mohr–Coulomb frictional contact law is proposed, which includes a frictional slip criterion and a slip potential function. Third, a return mapping algorithm based on the non-associative plasticity theory is proposed to solve the updated normal and tangential tractions and stiffnesses. It is shown the tangential cohesive traction and stiffness depend on the friction and dilatancy of the delamination interface. Finally, the proposed theoretical model is implemented using three-dimensional finite element analysis by ABAQUS-UEL (user element subroutine) and demonstrated by comparing the finite element results with the analytical results for the $[0^{\circ }]_{6}$, $[\pm 30^{\circ }]_{5}$, $[\pm 45^{\circ }]_{5}$ end-notched flexure adhesive composite joints with the mode-II shear delamination. The effects of the friction coefficient, cohesive strength, normal contact stiffness and mesh size on the load–displacement curves and delamination mechanisms of composites are studied. Numerical results show the shear delamination growth is governed by the transition from the decreased tangential cohesive traction to the increased tangential friction, and the frictional effect becomes distinct after unstable delamination for angle-ply laminates.

Appendices

Appendix 1: Cohesive/friction contact algorithm using FEA

By referring to the return mapping algorithm, the updated tangential traction $T_t$ and the consistent tangential stiffness $K_t$ are solved by the following Newton iterations:

• (a) After the tangential cohesive softening appears, the trial tangential traction is calculated as

  $$\begin{aligned}&T_{t }^{tr(i+1)} =T_{t }^{(i+1)} \left( {[[u]]_{t(e)}^{tr(i+1)} =[[u]]_t^{(i+1)} -[[u]]_{t(p)}^{(i)} } \right) \nonumber \\&\qquad +\,\mu \left| {T_n} \right| ^{(i+1)}\left( {[[u]]_{n(e)}^{tr(i+1)} =[[u]]_n^{(i+1)} -[[u]]_{n(p)}^{(i)} } \right) \nonumber \\ \end{aligned}$$

  (12)

  where $[[u]]_t>0$ and $[[u]]_n<0$ are the normal and tangential displacement jumps at the beginning of new ($i+$1)th increment, and $[[u]]_{n(p)}^{(i)}$ and $[[u]]_{t(p)}^{(i)}$ are the normal and tangential plastic displacement jumps at the end of ith increment.

• (b) If $T_{t }^{tr(i+1)} \le \mu _s \left| {T_n} \right| ^{(i+1)}$, the tangential stiffness $K_t \left( {[[u]]_t^{(i+1)} =[[u]]_{t(e)}^{tr(i+1)} } \right)$ and the tangential traction $T_t =T_{t}^{tr(i+1)} =T_t \left( [[u]]_t^{(i+1)}\right. \left. =[[u]]_{t(e)}^{tr(i+1)}\right)$ in Eq. (5) are calculated. There is no plastic loading and slip for the contacted interface.

• (c) If $T_{t }^{tr(i+1)} >\mu _s \left| {T_n} \right| ^{(i+1)}$, the plastic loading and frictional slip appear. After the initial value for $\Delta \lambda =0$ is given, the following Newton iterations are performed to solve the updated elastic displacement jump $[[u]]_{t(e)}^{(i+1)}$ and the plastic displacement jump $[[u]]_{t(p)}^{(i+1)}$

  $$\begin{aligned}&T_{t}^{{(i + 1)}} = \tau _{{\max }} \sqrt{2\exp \left( 1 \right) } \frac{{\left| {[[u]]_{{t(e)}}^{{(i + 1)}} } \right| }}{{\delta _{t} }}\nonumber \\&\quad \exp \left[ { - \frac{{\left( {[[u]]_{{t(e)}}^{{(i + 1)}} } \right) ^{2} }}{{\delta _{t}^{2} }}} \right] > 0,\nonumber \\&\left| {T_{n} } \right| ^{{(i + 1)}} = K_{n}\left| {\left( {[[u]]_{{n(e)}}^{{(i + 1)}} } \right) } \right| > 0, \nonumber \\&F = T_{{t}}^{{(i + 1)}} + \mu ^{{(i + 1)}} \left| {T_{n} } \right| _{{}}^{{(i + 1)}} - \mu _{s} \left| {T_{n} } \right| ^{{(i + 1)}} ,\nonumber \\&G = T_{{t}}^{{(i + 1)}} + \beta _{s} \left| {T_{n} } \right| ^{{(i + 1)}} , \nonumber \\&F+\frac{{\partial F}}{{\partial \Delta \lambda }}\hbox {d}\Delta \lambda + \cdots = 0\nonumber \\&\quad \Rightarrow \hbox {d}\Delta \lambda = - F/d_{t} > 0\nonumber \\&\quad \Rightarrow \Delta \lambda ^{{(i + 1)}} = \Delta \lambda ^{{(i)}} - F^{{(i + 1)}} /d_{t}^{{(i + 1)}} > 0, \nonumber \\&d_{t}^{{(i + 1)}} = \left. {\frac{{\partial F}}{{\partial \Delta \lambda }}} \right| ^{{(i + 1)}} \nonumber \\&\quad \qquad \;\, = K_{t}^{{(i + 1)}} + \beta _{s} \left( {\mu ^{{(i + 1)}} - \mu _{s} } \right) K_{{n}} < 0,\nonumber \\&\mu ^{{(i + 1)}} = \mu _{s} \frac{{\exp \left( {d^{{(i + 1)}} } \right) - 1}}{{\exp \left( 1 \right) - 1}}, \nonumber \\&K_{t}^{{(i + 1)}} = \frac{{\tau _{{\max }} }}{{\delta _{t}}}\sqrt{2\exp \left( 1 \right) }\nonumber \\&\qquad \qquad \quad \;\times \exp \left[ { - \frac{{\left( {[[u]]_{{t(e)}}^{{(i + 1)}} } \right) ^{2} }}{{\delta _{t}^{2} }}} \right] \nonumber \\&\qquad \qquad \quad \;\times \left( {1 - \frac{{2\left( {[[u]]_{{t(e)}}^{{(i + 1)}} } \right) ^{2} }}{{\delta _{t}^{2} }}} \right) < 0, \nonumber \\&{[}[u]]_{{t(p)}}^{{(i + 1)}} = [[u]]_{{t(p)}}^{i} + \Delta \lambda ^{{(i + 1)}} > 0,\nonumber \\&[[u]]_{{t(e)}}^{{(i + 1)}} = [[u]]_{t}^{{(i + 1)}} - {[[u]]}_{{t(p)}}^{{(i + 1)}} > 0, \nonumber \\&{[}[u]]_{{n(p)}}^{{(i + 1)}} = [[u]]_{{n(p)}}^{{(i)}} + \beta _{s} \Delta \lambda ^{{(i + 1)}} > 0,\nonumber \\&{ [[u]]}_{{n(e)}}^{{(i + 1)}} = [[u]]_{n}^{{(i + 1)}} - [[u]]_{{n(p)}}^{{(i + 1)}} <0,\nonumber \\&d^{{(i + 1)}} = \frac{{[[u]]_{{t(e)}}^{c} }}{{[[u]]_{{t(e)}}^{c} - [[u]]_{{t(e)}}^{0} }}\left( {1 - \frac{{[[u]]_{{t(e)}}^{0} }}{{[[u]]_{{t(e)}}^{{(i + 1)}} }}} \right) \nonumber \\ \end{aligned}$$

  (13)

  where $\Delta \left( \cdot \right)$ denotes the increment. It is shown the return mapping algorithm ensures that the frictional slip criterion meets the Kuhn–Tucker load conditions $\dot{\lambda }>0$ and $F=0$.

Remarks

Bathe and Chaudhary (1985), Simo et al. (1986), Perić and Owen (1992) and Weyler et al. (2012) proposed the penalty based frictional contact algorithms. In their work, the tangential displacement jump is divided into the elastic and plastic parts, and the positive normal and tangential penalty stiffnesses are used in the return mapping algorithm. However, these algorithms are not suitable for the cohesive/friction coupled problem because of tangential cohesive softening behavior. During numerical iterations in Eq. (13), the tangential elastic displacement jump $[[u]]_{t(e)}$ increases and the tangential traction $T_t$ decreases, but the normal elastic displacement jump $\left| {\left( {[[u]]_{n(e)}^{(i+1)} } \right) } \right|$ increases and the normal traction $\left| {T_n } \right|$ increases, which guarantees the convergence of the proposed algorithm at $\left| F \right| <tol(tol$ is the tolerance). In addition, it is noted the introduction of $\mu \left| {T_n } \right|$ helps to accelerate the convergence of the proposed algorithm in Eq. (13) because $\left| {d_t^{(i+1)} } \right|$ decreases and $\Delta \lambda$ increases and $\left| {T_n } \right| ^{(i+1)}$ increases. Finally, the normal traction $T_n=T_n \left( {[[u]]_{n(e)}} \right)$, the tangential traction $T_t=\mu _s \left| {T_n} \right|$ and the tangential stiffness $K_t=K_t\left( {[[u]]_{t(e)}} \right)$ are updated after convergence.

Appendix 2: 3D finite element formulation for cohesive models

The node displacement $\varvec{d}_{\mathrm{N}}$ in the global coordinate system $\left( {1,2,3} \right)$ is written as

$$\begin{aligned} \varvec{d}_{\mathrm{N}} =(d_x^1 ,d_y^1 ,d_z^1 ,d_x^2 ,d_y^2 ,d_z^2 \cdots d_x^8 ,d_y^8 ,d_z^8 ) \end{aligned}$$

(14)

The relative displacement between top and bottom surfaces is given by

$$\begin{aligned} \Delta \varvec{u}_{\mathrm{N}} ={\varvec{\phi }} \varvec{d}_{\mathrm{N}} = \left( {{\begin{array}{cc} {-\varvec{I}_{12\times 12} }&{} {\varvec{I}_{12\times 12} } \\ \end{array} }} \right) \varvec{d}_{\mathrm{N}} \end{aligned}$$

(15)

where $\varvec{I}$ is the identity matrix and ${\varvec{\phi }}$ is a $12\times 24$ matrix.

The displacement at any point within the cohesive element in the global coordinate system $\left( {1,2,3} \right)$ is calculated as

$$\begin{aligned} \Delta \varvec{u}\left( {\xi ,\eta } \right)= & {} \left( {{\begin{array}{l} {\Delta u_x \left( {\xi ,\eta } \right) } \\ {\Delta u_y \left( {\xi ,\eta } \right) } \\ {\Delta u_z \left( {\xi ,\eta } \right) } \\ \end{array} }} \right) \nonumber \\= & {} \varvec{H}\left( {\xi ,\eta } \right) \Delta \varvec{u}_{\mathrm{N}} =\varvec{H}\left( {\xi ,\eta } \right) \phi \varvec{d}_{\mathrm{N}} ={\varvec{Bd}}_{\mathrm{N}}\nonumber \\ \end{aligned}$$

(16)

where $\varvec{H}\left( {\xi ,\eta } \right)$ is the $3\times 12$ shape function and $\varvec{B}$ is the strain matrix.

The coordinate $\varvec{x}_R^N$ at any reference surface in the deformed configuration is interpolated as

$$\begin{aligned} \varvec{x}_R^N =\frac{1}{2}\left( {{\begin{array}{cc} {\varvec{I}_{12\times 12} }&{} {\varvec{I}_{12\times 12} } \\ \end{array} }} \right) (\varvec{x}_N +\varvec{d}_{N} ) \end{aligned}$$

(17)

where $\varvec{x}_N$ is the initial node coordinate in the cohesive element.

The displacement jump $[[\varvec{u}]]$ at any point in the cohesive element in the local coordinate system $\left( {\xi , \eta ,\zeta } \right)$ is written as

$$\begin{aligned} {[[\varvec{u}]]={\varvec{\Theta }} ^{T}{\varvec{\Delta }} {\varvec{u}}(\xi ,\eta ),}\quad {{\varvec{\Theta }} =({\varvec{n}},{\varvec{t}}_1 ,{\varvec{t}}_2 )} \end{aligned}$$

(18)

where ${\varvec{\Theta }}$ is the transformation matrix where three components are given by

$$\begin{aligned} {{\varvec{n}}=\frac{\left( {\frac{\partial {\varvec{x}}^{R}}{\partial \xi }\times \frac{\partial {\varvec{x}}^{R}}{\partial \eta }} \right) }{\left\| {\frac{\partial {\varvec{x}}^{R}}{\partial \xi }\times \frac{\partial {\varvec{x}}^{R}}{\partial \eta }} \right\| },} \quad {{\varvec{t}}_1 =\frac{\frac{\partial {\varvec{x}}^{R}}{\partial \xi }}{\left\| {\frac{\partial {\varvec{x}}^{R}}{\partial \xi }} \right\| },} \quad {{\varvec{t}}_2 ={\varvec{n}}\times {\varvec{t}}_1 } \end{aligned}$$

(19)

where ${\varvec{x}}^{R}={\varvec{H}}\left( {\xi ,\eta } \right) {\varvec{x}}_R^N$ is a point on the reference surface. ${\varvec{n}}$ is the normal direction and ${\varvec{t}}_1$ and ${\varvec{t}}_2$ are the tangential directions.

Finally, the node residual force vector ${\varvec{f}}_{24\times 1}$ and the stiffness tensor ${\varvec{K}}_{24\times 24}$ of 3D cohesive elements are defined by

$$\begin{aligned} {\varvec{f}}_{24\times 1}= & {} \int _{-1}^1 {\int _{-1}^1 {{\varvec{B}}^{T}{\varvec{\Theta }} ^{T}{\varvec{T}}} } \det J\hbox {d}\xi \hbox {d}\eta \end{aligned}$$

(20)

$$\begin{aligned} {\varvec{K}}_{24\times 24}= & {} \frac{\partial {\varvec{f}}}{\partial {\varvec{d}}_N }=\int _{-1}^1 {\int _{-1}^1 {{\varvec{B}}^{T}{\varvec{\Theta }} ^{T}{\varvec{D}}^s } } {\varvec{\Theta }} {\varvec{B}} \hbox {d}\xi \hbox {d}\eta \nonumber \\ {\varvec{D}}^s= & {} \left[ {{\begin{array}{cc} {K_n}&{} \\ &{} {K_t} \\ \end{array} }} \right] \end{aligned}$$

(21)

Appendix 3: Mode-II shear delamination fracture toughness of composites

Four methods for calculating the mode-II ERR are: 1 Compliance theory (Broek 1984), 2 Beam theory (Russell and Street 1982), 3 Modified beam theory (Carlsson et al. 1986) and 4 Associative beam theory and first-order shear deformation theory (Ozdil et al. 1998). The mode-II delamination fracture toughness $G_{\mathrm{II}}^c$ is determined using the initial delamination crack length $a=a_0$ and the corresponding experimental load value P(Theotokoglou and Vrettos 2006).

1.1 Compliance theory (Broek 1984)

Load-line compliance $C=\varDelta /P$ is defined, where $\varDelta$ is the displacement at the central loading point and P is the applied load. The ERR $G_{{\mathrm{II}}}$ takes the form

$$\begin{aligned} G_{\mathrm{II}} =\frac{P^{2}}{2b}\frac{\partial C}{\partial a} \end{aligned}$$

(22)

where a is the actual crack length and b is the width of the specimen.

1.2 Beam theory (Russell and Street 1982)

The load-line compliance $C_{\mathrm{BT}}$ and the mode-II ERR $G_{\mathrm{II}}^{{BT}}$ are given by the beam theory

$$\begin{aligned} C_{\mathrm{BT}} =\frac{2L^{3}+3a^{3}}{8E_1 bh^{3}} \end{aligned}$$

(23)

$$\begin{aligned} G_{\mathrm{II}}^{BT} =\frac{9a^{2}P^{2}}{16E_1 b^{2}h^{3}} \end{aligned}$$

(24)

where $E_1$ is the longitudinal elastic modulus.

1.3 Modified beam theory (Carlsson et al. 1986)

Equations (25) and (26) were modified to include the effect of transverse shear deformation

$$\begin{aligned} C_{\mathrm{SH}}= & {} C_{\mathrm{BT}} +\frac{1.2L+0.9a}{4bG_{13} h} \end{aligned}$$

(25)

$$\begin{aligned} G_{\mathrm{II}}^{\mathrm{SH}}= & {} G_{\mathrm{II}}^{\mathrm{BT}} \left[ {1+0.2\left( {\frac{E_1 }{G_{13} }} \right) \left( {\frac{h}{a}} \right) ^{2}} \right] \end{aligned}$$

(26)

where $G_{13}$ is the shear modulus.

1.4 Associated beam theory and first-order shear deformation theory (Ozdil et al. 1998)

By combining the laminate beam theory and the first order shear deformation theory, the compliance and the mode-II ERR are given by

$$\begin{aligned} C_{\mathrm{SBT}}= & {} \frac{L^{3}(d_{11} )_{\mathrm{BC}} }{6b}+\frac{L(a_{55} )_{\mathrm{BC}} }{2bk}+\frac{a^{3}\left[ {(d_{11} )_{\mathrm{AB}} -(d_{11} )_{\mathrm{BC}} } \right] }{12b}\nonumber \\&+\,\frac{a\left[ {(a_{55} )_{\mathrm{AB}} -(a_{55} )_{\mathrm{BC}} } \right] }{4bk} \end{aligned}$$

(27)

$$\begin{aligned} G_{\mathrm{II}}^{\mathrm{SBT}}= & {} \frac{P^{2}}{8b^{2}}\left\{ {a^{2}\left[ {(d_{11} )_{\mathrm{AB}} -(d_{11} )_{\mathrm{BC}} } \right] +\frac{(a_{55} )_{\mathrm{AB}} -(a_{55} )_{\mathrm{BC}} }{k}} \right\} \nonumber \\ \end{aligned}$$

(28)

where $k=5/6$ is the shear correction factor, AB is the delaminated section, BC is a part of the intact section of the ENF specimen in Fig. 6, and ($d_{11})_{\mathrm{AB}}$, ($d_{11})_{\mathrm{BC}}$, ($a_{55})_{\mathrm{AB}}$ and ($a_{55})_{\mathrm{BC}}$ are the effective bending and shear compliances, respectively.

To calculate $C_{\mathrm{SBT}}$ and $G_{\mathrm{II}}^{\mathrm{SBT}}$ for the ENF specimen, the effective bending and shear compliances are calculated according to the method proposed by Ozdil et al. (1998).

Appendix 4: Analytical formulas for the load–displacement curves for mode-II delamination of ENF composites

According to the analytical formulas derived by Mi et al. (1998), the load–displacement curve of the ENF composite specimen is divided into three parts:

$$\begin{aligned}&\hbox {Curve}-\overline{{\mathrm{O}}}\overline{{\mathrm{B}}}: \quad \varDelta =\frac{P(2L^{3}+3a_0^3)}{96E_1 I} \end{aligned}$$

(29)

$$\begin{aligned}&\hbox {Curve}-\overline{{\mathrm{A}}}\overline{{\hbox {B}}}\overline{{C}}(a<L): \nonumber \\&\quad \varDelta =\frac{P}{96E_1 I}\left[ {2L^{3}+\frac{\left( {64G_{\mathrm{II}}^c bE_1 I} \right) ^{3/2}}{\sqrt{3}P^{3}}} \right] \end{aligned}$$

(30)

$$\begin{aligned}&\hbox {Curve}-\overline{{\mathrm{D}}}\overline{{\mathrm{E}}}(a>L): \nonumber \\&\quad \varDelta =\frac{P}{24E_1 I}\left[ {2L^{3}-\frac{\left( {64G_{\mathrm{II}}^c bE_1 I} \right) ^{3/2}}{4\sqrt{3}P^{3}}} \right] \end{aligned}$$

(31)

where I is the second-order inertia moment and $a_0$ is the initial crack length.