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A Layerwise Higher-Order Approach for the Free-Edge Effect in Angle-Ply Laminates

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Mechanics of Composite Materials Aims and scope

The free edges of layered structures require special attention in design processes. Due to the discontinuous material properties, singular stress concentrations occur where the layer interfaces meet the free edge. These bi-material points have to be closely examined as possible starting points for delamination processes. In the present study, we propose an efficient approximate closed-form method to analyze the interlaminar shear stresses in symmetric angle-ply laminates under uniform axial extension and homogeneous thermal loading. The method bases on a higher-order layerwise displacement approach. The resulting displacement and stress distributions of a first-order and a third-order analysis are compared and validated against numerical reference data gained by a self-developed finite element formulation. Especially, the results of the third-order analysis show a good agreement to the reference data along the interface, even if the singular behavior of the stresses in the very close vicinity of the free-edge cannot be reproduced exactly. A comparison of the displacement and stress distributions in thickness direction shows the benefit of the third-order over the first-order approach. In addition, the influence of the ply angle on the results is reproduced by both approaches.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Technical University Darmstadt, Franziska-Braun-Str. 7, 64287, Darmstadt, Germany

    C. Frey, J. Hahn, N. Schneider & W. Becker

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  1. C. Frey
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Correspondence to C. Frey.

Appendices

Appendix A. Boundary conditions of the proposed models at the free edge y = 0 and at the symmetry plane y = b.

For FSDT model:

y = 0

y = b

 

\({N}_{xy}^{\left(1\right)}+\frac{2}{t}{M}_{xy}^{\left(2\right)}=0\)

\({\widehat{u}}_{x}^{\left(1\right)}=0\)

 

\({N}_{xy}^{\left(2\right)}-\frac{2}{t}{M}_{xy}^{\left(2\right)}=0\)

\({\widehat{u}}_{x}^{\left(2\right)}=0\)

 

\({N}_{yy}^{\left(1\right)}+\frac{2}{t}{M}_{yy}^{\left(2\right)}=0\)

\({\widehat{u}}_{y}^{\left(1\right)}=0\)

 

\({N}_{yy}^{\left(2\right)}-\frac{2}{t}{M}_{yy}^{\left(2\right)}=0\)

\({\widehat{u}}_{y}^{\left(2\right)}=0\)

 

\({M}_{xy}^{\left(1\right)}-{M}_{xy}^{\left(2\right)}=0\)

\({\psi }_{x}^{\left(1\right)}=0\)

 

\({M}_{yy}^{\left(1\right)}-{M}_{yy}^{\left(2\right)}=0\)

\({\psi }_{y}^{\left(1\right)}=0\)

For TSDT model:

y = 0

y = b

 

\({N}_{xy}^{\left(1\right)}-{3c}_{2}\left({O}_{xy}^{\left(1\right)}-{O}_{xy}^{\left(2\right)}\right)+{6c}_{3}\left({P}_{xy}^{\left(1\right)}+{P}_{xy}^{\left(2\right)}\right)=0\)

\({\widehat{u}}_{x}^{\left(1\right)}=0\)

 

\({N}_{xy}^{\left(2\right)}+{3c}_{2}\left({O}_{xy}^{\left(1\right)}-{O}_{xy}^{\left(2\right)}\right)-{6c}_{3}\left({P}_{xy}^{\left(1\right)}+{P}_{xy}^{\left(2\right)}\right)=0\)

\({\widehat{u}}_{x}^{\left(2\right)}=0\)

 

\({N}_{yy}^{\left(1\right)}-{3c}_{2}\left({O}_{yy}^{\left(1\right)}-{O}_{yy}^{\left(2\right)}\right)+{6c}_{3}\left({P}_{yy}^{\left(1\right)}+{P}_{yy}^{\left(2\right)}\right)=0\)

\({\widehat{u}}_{y}^{\left(1\right)}=0\)

 

\({N}_{yy}^{\left(1\right)}+{3c}_{2}\left({O}_{yy}^{\left(1\right)}-{O}_{yy}^{\left(2\right)}\right)-{6c}_{3}\left({P}_{yy}^{\left(1\right)}+{P}_{yy}^{\left(2\right)}\right)=0\)

\({\widehat{u}}_{y}^{\left(2\right)}=0\)

 

\({M}_{xy}^{\left(1\right)}+{c}_{2}t\left({O}_{xy}^{\left(1\right)}-{O}_{xy}^{\left(2\right)}\right)-{c}_{3}t\left(7{P}_{xy}^{\left(1\right)}+2{P}_{xy}^{\left(2\right)}\right)=0\)

\({\psi }_{x}^{\left(1\right)}=0\)

 

\({M}_{xy}^{\left(2\right)}+{c}_{2}t\left({O}_{xy}^{\left(1\right)}-{O}_{xy}^{\left(2\right)}\right)-{c}_{3}t\left(2{P}_{xy}^{\left(1\right)}+7{P}_{xy}^{\left(2\right)}\right)=0\)

\({\psi }_{y}^{\left(2\right)}=0\)

 

\({M}_{yy}^{\left(1\right)}+{c}_{2}t\left({O}_{yy}^{\left(1\right)}-{O}_{yy}^{\left(2\right)}\right)-{c}_{3}t\left(7{P}_{yy}^{\left(1\right)}+2{P}_{yy}^{\left(2\right)}\right)=0\)

\({\psi }_{y}^{\left(1\right)}=0\)

 

\({M}_{yy}^{\left(2\right)}+{c}_{2}t\left({O}_{yy}^{\left(1\right)}-{O}_{yy}^{\left(2\right)}\right)-{c}_{3}t\left(2{P}_{yy}^{\left(1\right)}+7{P}_{yy}^{\left(2\right)}\right)=0\)

\({\psi }_{y}^{\left(2\right)}=0\)

Appendix B. Finite Element Formulation

The matrix \(\overline{\mathbf{B} }\) contains the partial derivatives of the shape functions in an averaged sense. It is given by

$$\overline{\mathbf{B} }=\mathbf{B}-{\mathbf{B}}_{m}+{\overline{\mathbf{B}} }_{m},$$
(A1)

where B, Bm and \({\overline{\mathbf{B}} }_{m}\) are given as

$$\begin{array}{r}\mathbf{B}=\left[\begin{array}{cccccccccccc}\frac{\partial {N}_{1}}{\partial x}& 0& 0& \frac{\partial {N}_{2}}{\partial x}& 0& 0& \frac{\partial {N}_{3}}{\partial x}& 0& 0& \frac{\partial {N}_{4}}{\partial x}& 0& 0\\ 0& \frac{\partial {N}_{1}}{\partial y}& 0& 0& \frac{\partial {N}_{2}}{\partial y}& 0& 0& \frac{\partial {N}_{3}}{\partial y}& 0& 0& \frac{\partial {N}_{4}}{\partial y}& 0\\ 0& 0& \frac{\partial {N}_{1}}{\partial z}& 0& 0& \frac{\partial {N}_{2}}{\partial z}& 0& 0& \frac{\partial {N}_{3}}{\partial z}& 0& 0& \frac{\partial {N}_{4}}{\partial z}\\ 0& \frac{\partial {N}_{1}}{\partial z}& \frac{\partial {N}_{1}}{\partial y}& 0& \frac{\partial {N}_{2}}{\partial z}& \frac{\partial {N}_{2}}{\partial y}& 0& \frac{\partial {N}_{3}}{\partial z}& \frac{\partial {N}_{3}}{\partial y}& 0& \frac{\partial {N}_{4}}{\partial z}& \frac{\partial {N}_{4}}{\partial y}\\ \frac{\partial {N}_{1}}{\partial z}& 0& 0& \frac{\partial {N}_{2}}{\partial z}& 0& 0& \frac{\partial {N}_{3}}{\partial z}& 0& 0& \frac{\partial {N}_{4}}{\partial z}& 0& 0\\ \frac{\partial {N}_{1}}{\partial y}& 0& 0& \frac{\partial {N}_{2}}{\partial y}& 0& 0& \frac{\partial {N}_{3}}{\partial y}& 0& 0& \frac{\partial {N}_{4}}{\partial y}& 0& 0\end{array}\right],\\ {\mathbf{B}}_{{\varvec{m}}}=\frac{1}{3}\left[\begin{array}{cccccccccccc}\frac{\partial {N}_{1}}{\partial x}& \frac{\partial {N}_{1}}{\partial y}& \frac{\partial {N}_{1}}{\partial z}& \frac{\partial {N}_{2}}{\partial x}& \frac{\partial {N}_{2}}{\partial y}& \frac{\partial {N}_{2}}{\partial z}& \frac{\partial {N}_{3}}{\partial x}& \frac{\partial {N}_{3}}{\partial y}& \frac{\partial {N}_{3}}{\partial z}& \frac{\partial {N}_{4}}{\partial x}& \frac{\partial {N}_{4}}{\partial y}& \frac{\partial {N}_{4}}{\partial z}\\ \frac{\partial {N}_{1}}{\partial x}& \frac{\partial {N}_{1}}{\partial y}& \frac{\partial {N}_{1}}{\partial z}& \frac{\partial {N}_{2}}{\partial x}& \frac{\partial {N}_{2}}{\partial y}& \frac{\partial {N}_{2}}{\partial z}& \frac{\partial {N}_{3}}{\partial x}& \frac{\partial {N}_{3}}{\partial y}& \frac{\partial {N}_{3}}{\partial z}& \frac{\partial {N}_{4}}{\partial x}& \frac{\partial {N}_{4}}{\partial y}& \frac{\partial {N}_{4}}{\partial z}\\ \frac{\partial {N}_{1}}{\partial x}& \frac{\partial {N}_{1}}{\partial y}& \frac{\partial {N}_{1}}{\partial z}& \frac{\partial {N}_{2}}{\partial x}& \frac{\partial {N}_{2}}{\partial y}& \frac{\partial {N}_{2}}{\partial z}& \frac{\partial {N}_{3}}{\partial x}& \frac{\partial {N}_{3}}{\partial y}& \frac{\partial {N}_{3}}{\partial z}& \frac{\partial {N}_{4}}{\partial x}& \frac{\partial {N}_{4}}{\partial y}& \frac{\partial {N}_{4}}{\partial z}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right],\\ {\overline{\mathbf{B}} }_{{\varvec{m}}}=\frac{1}{3}\left[\begin{array}{cccccccccccc}\frac{\partial {\overline{N} }_{1}}{\partial x}& \frac{\partial {\overline{N} }_{1}}{\partial y}& \frac{\partial {\overline{N} }_{1}}{\partial z}& \frac{\partial {\overline{N} }_{2}}{\partial x}& \frac{\partial {\overline{N} }_{2}}{\partial y}& \frac{\partial {\overline{N} }_{2}}{\partial z}& \frac{\partial {\overline{N} }_{3}}{\partial x}& \frac{\partial {\overline{N} }_{3}}{\partial y}& \frac{\partial {\overline{N} }_{3}}{\partial z}& \frac{\partial {\overline{N} }_{4}}{\partial x}& \frac{\partial {\overline{N} }_{4}}{\partial y}& \frac{\partial {\overline{N} }_{4}}{\partial z}\\ \frac{\partial {\overline{N} }_{1}}{\partial x}& \frac{\partial {\overline{N} }_{1}}{\partial y}& \frac{\partial {\overline{N} }_{1}}{\partial z}& \frac{\partial {\overline{N} }_{2}}{\partial x}& \frac{\partial {\overline{N} }_{2}}{\partial y}& \frac{\partial {\overline{N} }_{2}}{\partial z}& \frac{\partial {\overline{N} }_{3}}{\partial x}& \frac{\partial {\overline{N} }_{3}}{\partial y}& \frac{\partial {\overline{N} }_{3}}{\partial z}& \frac{\partial {\overline{N} }_{4}}{\partial x}& \frac{\partial {\overline{N} }_{4}}{\partial y}& \frac{\partial {\overline{N} }_{4}}{\partial z}\\ \frac{\partial {\overline{N} }_{1}}{\partial x}& \frac{\partial {\overline{N} }_{1}}{\partial y}& \frac{\partial {\overline{N} }_{1}}{\partial z}& \frac{\partial {\overline{N} }_{2}}{\partial x}& \frac{\partial {\overline{N} }_{2}}{\partial y}& \frac{\partial {\overline{N} }_{2}}{\partial z}& \frac{\partial {\overline{N} }_{3}}{\partial x}& \frac{\partial {\overline{N} }_{3}}{\partial y}& \frac{\partial {\overline{N} }_{3}}{\partial z}& \frac{\partial {\overline{N} }_{4}}{\partial x}& \frac{\partial {\overline{N} }_{4}}{\partial y}& \frac{\partial {\overline{N} }_{4}}{\partial z}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right],\end{array}$$
(A2)

The volume-averaged partial derivatives of the shape function are defined as

$$\frac{\partial {\overline{N} }_{m}}{\partial {x}_{i}}=\frac{1}{V}\int v\frac{\partial {N}_{m}}{\partial {x}_{i}}dV.$$
(A3)

Using the matrix \(\overline{\mathbf{B} }\) instead of \(\overline{\mathbf{B} }\) prevents the element stiffness matrix from getting singular for nearly incompressible materials [62, 63]. Furthermore, the matrix B0 is given as

$${\mathbf{B}}_{0}={\left[\frac{\partial {N}_{5}}{\partial x}00000\right]}^{{\varvec{T}}}.$$
(A4)

Appendix C. Effective Stiffness

The constitutive law of a symmetric laminate under uniform axial strain ε0 in x -direction according to the CLPT is given by

$$\left[\begin{array}{c}{N}_{x}\\ 0\\ 0\end{array}\right]=\left[\begin{array}{ccc}{A}_{11}& {A}_{12}& {A}_{16}\\ {A}_{12}& {A}_{22}& {A}_{26}\\ {A}_{16}& {A}_{26}& {A}_{66}\end{array}\right]\left[\begin{array}{c}{\varepsilon }_{0}\\ {\varepsilon }_{y}^{0}\\ {\gamma }_{xy}^{0}\end{array}\right].$$
(A5)

Consequently, this results in

$${N}_{x}=\frac{1}{{\left({\mathbf{A}}^{-1}\right)}_{11}}{\varepsilon }_{0},$$
(A6)

which implicates the definition of the effective longitudinal stiffness for symmetric laminates:

$${E}_{\mathrm{xx}}=\frac{1}{{\left({\mathbf{A}}^{-1}\right)}_{11}d}.$$
(A7)

Herein, d is the total laminate thickness and A-1 the inverse of the extensional stiffness matrix of the laminate. For the general case of arbitrary layups with bending extension coupling, (ABD)-1 must be used instead of A-1.

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Frey, C., Hahn, J., Schneider, N. et al. A Layerwise Higher-Order Approach for the Free-Edge Effect in Angle-Ply Laminates. Mech Compos Mater 59, 299–318 (2023). https://doi.org/10.1007/s11029-023-10097-8

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