The effect of viscoelasticity on the stress distribution of adhesively single-lap joint with an internal break in the composite adherends

Abstract

The aim of this research is to study the effect of a break in the laminated composite adherends on stress distribution in the adhesively single-lap joint with viscoelastic adhesive and matrix. The proposed model involves two adherends with E-glass fibers and poly-methyl-methacrylate matrix that have been adhered to each other by phenolic-epoxy resin. The equilibrium equations that are based on shear-lag theory have been derived in the Laplace domain, and the governing differential equations of the model have been derived analytically in the Laplace domain. A numerical inverse Laplace transform, which is called Gaver–Stehfest method, has been used to extract desired results in the time domain. The results obtained at the initial time completely matched with the results of elastic solution. Also, a comparison between results obtained from the analytical and finite element models show a relatively good match. The results show that viscoelastic behavior decreases the peak of stress near the break. Finally, the effect of size and location of the break, as well as volume fraction of fibers, on the stress distribution in the adhesive layer is fully investigated.

Appendix

$\mathbf{L}(s)$ is defined as

$$\begin{aligned} &\mathbf{L} (s)_{N\times(M+Q),N\times(M+Q)} \\ &\quad = \left [ \textstyle\begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c} \mathbf{L}_{N\times N}^{3} & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} & \mathbf{0}_{N\times N} & \cdots & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{6} & \mathbf{0}_{N\times N} & \cdots & \cdots & \cdots & \mathbf{0}_{N\times N} \\ \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{4} & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} & \ddots & \vdots & \mathbf{0}_{N\times N} & \mathbf{0}_{N\times N} & \vdots & \vdots & \vdots & \vdots \\ \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{5} & \ddots & \ddots & \ddots & \mathbf{0}_{N\times N} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \mathbf{0}_{N\times N} & \ddots & \ddots & \ddots & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \vdots & \ddots & \ddots & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{4} & \mathbf{L}_{N\times N}^{5} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \mathbf{0}_{N\times N} & \cdots & \mathbf{0}_{N\times N} & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{3} & \mathbf{0}_{N\times N} & \cdots & \cdots & \cdots & \cdots & \mathbf{0}_{N\times N} \\ \mathbf{L}_{N\times N}^{6} & \mathbf{0}_{N\times N} & \cdots & \cdots & \cdots & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{3} & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} & \mathbf{0}_{N\times N} & \cdots & \mathbf{0}_{N\times N} \\ \mathbf{0}_{N\times N} & \mathbf{0}_{N\times N} & \vdots & \vdots & \vdots & \vdots & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{4} & \mathbf{L}_{N\times N}^{5} & \ddots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{5} & \ddots & \ddots & \ddots & \mathbf{0}_{N\times N} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \mathbf{0}_{N\times N} & \ddots & \ddots & \ddots & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{4} & \mathbf{L}_{N\times N}^{5} \\ \mathbf{0}_{N\times N} & \cdots & \cdots & \cdots & \cdots & \mathbf{0}_{N\times N} & \mathbf{0}_{N\times N} & \cdots & \mathbf{0}_{N\times N} & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{3} \end{array}\displaystyle \right ] \end{aligned}$$

(41)

where $\mathbf{0}_{N\times N}$ is a zero matrix, and $\mathbf{L}_{N\times N}^{3}$, $\mathbf{L}_{N\times N}^{4}$, $\mathbf{L}_{N\times N}^{5}$, and $\mathbf{L}_{N\times N}^{6}$ are defined as follows:

$$\begin{aligned} \mathbf{L}_{N\times N}^{3} &= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \chi_{5} & \chi_{3} & 0 & \cdots & 0 \\ \psi & \chi_{6} & \psi & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & \psi & \chi_{6} & \psi \\ 0 & \cdots & 0 & \chi_{3} & \chi_{5} \end{array}\displaystyle \right ],\quad \mathbf{L}_{N\times N}^{4} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} -3\psi & \psi & 0 & \cdots & 0 \\ \psi & -4\psi & \psi & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & \psi & -4\psi & \psi \\ 0 & \cdots & 0 & \psi & -3\psi \end{array}\displaystyle \right ], \end{aligned}$$

(42)

$$\begin{aligned} \mathbf{L}_{N\times N}^{5} &= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \psi & 0 & 0 & \cdots & 0 \\ 0 & \psi & 0 & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & 0 & \psi & 0 \\ 0 & \cdots & 0 & 0 & \psi \end{array}\displaystyle \right ],\quad \mathbf{L}_{N\times N}^{6} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \chi_{4} & \varphi_{2} & 0 & \cdots & 0 \\ 0 & \varphi_{3} & 0 & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & 0 & \varphi_{3} & 0 \\ 0 & \cdots & 0 & \varphi_{2} & \chi_{4} \end{array}\displaystyle \right ] \end{aligned}$$

(43)

where $\chi_{1}$ to $\chi_{6}$ are defined as

$$\begin{aligned} \begin{aligned} \chi_{1} &= -\psi- \varphi_{2} - \varphi_{3},\qquad \chi_{2} =-2\psi- \varphi_{3}, \qquad \chi_{3} =\psi- \varphi_{2}, \\ \chi_{4} &= \varphi_{1} + \varphi_{2},\qquad \chi_{5} =-2\psi- \varphi_{1} - \varphi_{2},\qquad \chi_{6} =-3\psi- \varphi_{3}. \end{aligned} \end{aligned}$$

(44)

And also $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$ and $\psi$ are given by

$$\begin{aligned} \begin{aligned} \varphi_{1} &=s \widehat{G}_{a} ( s ) \frac{t_{f}}{E_{f} A_{f} h},\qquad \varphi_{2} = \frac{1}{4} s \widehat{G}_{a} ( s ) \frac{t_{m}}{E_{f} A_{f} h},\qquad \varphi_{3} =s \widehat{G}_{a} ( s ) \frac{t_{f} +0.5 t_{m}}{E_{f} A_{f} h}, \\ \psi&=s \widehat{G}_{m} ( s ) \frac{t_{f}}{E_{f} A_{f} t_{m}}. \end{aligned} \end{aligned}$$

(45)

Also

$$\begin{aligned} \begin{aligned} \mathbf{L}^{C} (s)_{N\times M,N\times M} &= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \mathbf{L}_{N\times N}^{7} & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} & \cdots & \mathbf{0}_{N\times N} \\ \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{4} & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} & \vdots \\ \mathbf{0}_{N\times N} & \ddots & \ddots & \ddots & \mathbf{0}_{N\times N} \\ \vdots & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{4} & \mathbf{L}_{N\times N}^{5} \\ \mathbf{0}_{N\times N} & \cdots & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{7} \end{array}\displaystyle \right ], \\ \mathbf{L}^{D} (s)_{N\times Q,N\times Q} &= \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \mathbf{L}_{N\times N}^{7} & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} & \cdots & \mathbf{0}_{N\times N} \\ \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{4} & \mathbf{L}_{N\times N}^{5} & \mathbf{0}_{N\times N} & \vdots \\ \mathbf{0}_{N\times N} & \ddots & \ddots & \ddots & \mathbf{0}_{N\times N} \\ \vdots & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{4} & \mathbf{L}_{N\times N}^{5} \\ \mathbf{0}_{N\times N} & \cdots & \mathbf{0}_{N\times N} & \mathbf{L}_{N\times N}^{5} & \mathbf{L}_{N\times N}^{7} \end{array}\displaystyle \right ]. \end{aligned} \end{aligned}$$

(46)

$\mathbf{L}_{N\times N}^{7}$ is defined as

$$ \mathbf{L}_{N\times N}^{7} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} -2\psi & \psi & 0 & \cdots & 0 \\ \psi & -3\psi & \psi & 0 & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & 0 & \psi & -3\psi & \psi \\ 0 & \cdots & 0 & \psi & -2\psi \end{array}\displaystyle \right ]. $$

(47)