Analytical solution for first ply failure analysis of laminated plates using Reddy–Legendre higher-order theory

Abstract

The existing finite element formulation generally leads to overestimation on the first ply failure (FPF) load. Reasons of this phenomenon have been less cleared in detail. Therefore, it requires to propose an analytical solution to provide an accurate prediction of the FPF with the high efficiency-cost ratio. To this end, the following work has been carried out: (1)(1) $$P_0(x)=1,P_n(x)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}|{\left(x^2-1\right)}^n|\hspace{1em}(n=1,2,\dots ).$$(1) Legendre polynomials as local terms are first introduced into Reddy-type higher-order theory, which provide a novel alternative form of global-local displacement field. (2)(2) $$\begin{array}{c}{u}^k(x,y,z)=\underset{u_G(x,y,z)}{\underbrace{u_0(x,y)+{\Phi }_1(z){\gamma }_x(x,y)+{\Phi }_2(z)w_{0,x}(x,y)}}+\underset{u_L^k(x,y,z)}{\underbrace{\sum _{i=1}^3{P}_i(z)u_i^k(x,y)}},\\ v^k(x,y,z)=\underset{v_G(x,y,z)}{\underbrace{v_0(x,y)+{\Psi }_1(z){\gamma }_y(x,y)+{\Psi }_2(z)w_{0,y}(x,y)}}+\underset{v_L^k(x,y,z)}{\underbrace{\sum _{i=1}^3{P}_i(z)v_i^k(x,y)}},\\ w^k(x,y)=\underset{w_G(x,y)}{\underbrace{w_0(x,y)}},\end{array}$$(2) Compared with the classical first-order and higher-order shear deformation theories, the proposed model can predict transverse shear stresses directly by constitutive equations which is close to the results obtained by the three-dimensional finite element method (3D-FEM). Therefore, the accuracy and efficiency of the FPF analysis can be improved, attributing to the accurate prediction of the quasi-three-dimensional stress field. (3)(3) $${\Phi }_1(z)={\Psi }_1(z)=z-4z^3/3h^2,\hspace{1em}{\Phi }_2(z)={\Psi }_2(z)=-z,$$(3) The static response and the FPF of laminated composite plates are analyzed, which have been compared with the results of selected models in the published literature and 3D-FEM. Influences of different failure theories, transverse loads, stacking sequence on prediction of the FPF loads are also explored. (4)(4) $$\begin{array}{c}{P}_1(z)={\xi }_k,\hspace{1em}{P}_2(z)=\left(3{\xi }_k^2-1\right)/2,\hspace{1em}{P}_3(z)=\left(5{\xi }_k^3-3{\xi }_k\right)/2,\\ {\xi }_k=a_{k}z-b_k,\hspace{1em}{a}_k=2/\left(z_{k+1}-z_k\right),\hspace{1em}{b}_k=\left(z_{k+1}+z_k\right)/\left(z_{k+1}-z_k\right),\end{array}$$(4) By visualizing full-field displacement, stress and damage variables rather than those obtained from key points, it is easier to find the failure tendency and the failure modes. As a result, it is necessary to propose an analytical solution for predicting the FPF of composite laminated plates, which can provide a reference for other investigators.

Keywords:

• Composite laminate
• first ply failure
• Reddy–Legendre
• analytical solution

Disclosure statement

No potential conflict of interest was reported by the author (s).

Additional information

Funding

The work described in this paper was supported by the National Natural Sciences Foundation of China (No. 12172295), SKLLIM1902, and the Natural Science Foundation in Shaanxi Province (No. 2019JQ-909).