Triangular distribution of in-plane stresses

II.1.6.2 Triangular distribution of in-plane stresses

In most expressions below, the components of tensors are expressed in laminate axes. Therefore, the “lam” underscore is often added to the different quantities used in the equations.

First, one calculates the components of Cauchy stress tensor. However, a few simplifying assumptions shall be done. The strain tensor components are calculated from the laminate average strain tensor and curvature as follows:

{\{ϵ\}}_{lam} = {\{ϵ^{0}\}}_{lam} + z {\{κ\}}_{lam},

{\{ϵ^{0}\}}_{lam} = {[a]}_{lam} \cdot {\{N\}}_{lam} + {[b]}_{lam} \cdot {\{M\}}_{lam},

{\{κ\}}_{lam} = [b] T_{lam} \cdot {\{N\}}_{lam} + {[d]}_{lam} \cdot {\{M\}}_{lam} .

(The thermo-elastic contributions have been neglected.) In most out-of-plane shear theories presented in the literature, one assumes a decoupling between in-plane load response and out-of-plane shear response. This allows us to neglect a few terms in the equations:

{\{ϵ^{0}\}}_{lam} \approx {[b]}_{lam} \cdot {\{M\}}_{lam},

{\{κ\}}_{lam} \approx {[d]}_{lam} \cdot {\{M\}}_{lam} .

One then writes a simple expression of the in-plane laminate deformation tensor:

{\{ϵ\}}_{lam} = {\{ϵ^{0}\}}_{lam} + z {\{κ\}}_{lam} .

Then, the components of Cauchy stress tensor are given by:

{\{σ\}}_{lam} (x, y, z) = {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) \cdot {\{M\}}_{lam} (x, y) .

(II.1.38)

In this last expression, the $3 \times 3$ matrix ${[C]}_{lam} (z)$ corresponds to the plies in-plane moduli expressed in laminate axes. It depends on $z$ because the components generally change from one ply to another. However, one shall assume that the components of the moduli matrix are constant in each ply.

Note that, in the local and global equilibrium relations (II.1.34) to (II.1.37), only partial derivatives of bending moments and Cauchy stress tensor components appear. One assumes the decoupling between the out-of-plane shear behavior and the absolute bending in laminate. However, as shown by expressions (II.1.36) and (II.1.37), the out-of-plane shear is related to the gradient of bending moment. One derives equation (II.1.38) wrt to $x$ and $y$ :

{\{\begin{matrix} σ_{x x, x} (x, y, z) \\ σ_{y y, x} (x, y, z) \\ τ_{x y, x} (x, y, z) \end{matrix}\}}_{lam} = {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) \cdot {\{\begin{matrix} M_{x x, x} (x, y) \\ M_{y y, x} (x, y) \\ M_{x y, x} (x, y) \end{matrix}\}}_{lam},

{\{\begin{matrix} σ_{x x, y} (x, y, z) \\ σ_{y y, y} (x, y, z) \\ τ_{x y, y} (x, y, z) \end{matrix}\}}_{lam} = {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) \cdot {\{\begin{matrix} M_{x x, y} (x, y) \\ M_{y y, y} (x, y) \\ M_{x y, y} (x, y) \end{matrix}\}}_{lam} .

At this point, one no longer needs to assume a dependence of the gradient of bending moments wrt $x$ and $y$ . The same is true for the gradient of Cauchy stress tensor. One also introduces a new notation:

{[F]}_{lam} (z) = {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) .

Then, the components of Cauchy stress tensor gradient are obtained from the components of bending moments gradient with the following expression:

{\{\begin{matrix} σ_{x x, x} (z) \\ σ_{y y, x} (z) \\ τ_{x y, x} (z) \\ σ_{x x, y} (z) \\ σ_{y y, y} (z) \\ τ_{x y, y} (z) \end{matrix}\}}_{lam} = [\begin{matrix} {[F]}_{lam} (z) & \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} & {[F]}_{lam} (z) \end{matrix}] \cdot {\{\begin{matrix} M_{x x, x} \\ M_{y y, x} \\ M_{x y, x} \\ M_{x x, y} \\ M_{y y, y} \\ M_{x y, y} \end{matrix}\}}_{lam} .

(II.1.39)

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