The classical lamination theory (CLT) has some assumptions that must be considered before its application. First, the straight line orthogonal to the mid-plane (Figure 1) remains orthogonal to the mid-plane even after deformation. The laminate only perform small strains and deformations, which means that, the laminate can account rigid motion. The last hypothesis is that the plies are rigidly connected. Actually, this one is not totally correct, because the interface has a small layer of matrix between the plies. However, in a simple CLT, this consideration is neglected. Although these assumptions simplify the calculations, there are some consequences. First, the continuity of the displacement across interfaces are negligible shear deformations. Otherwise the straight assumption would not be possible. There are another laminations theories which accounts that, the shear deformations may not be negligible, specially for the composite beams.

Overview

The angle α (Figure 1) is the rotation of the mid-plane with respect to the non-deformed configuration. There is a displacement in the z direction of the mid-plane and the rotation, in this case, is the partial derivative of the displacement with respect to the x direction (Figure 1). Since the problem states about a plate, there is a y-direction that has its respective angle of rotation with respect to the mid-plane. In this case, the partial derivative would be respective to y. The displacement of a material point is given by the displacement of the mid-plane, approximately the distance of the material point from the mid-plane, the coordinate z times the rotation angle. This is a 2D analysis, thus the same is accounted for the y direction.

u(x,y) = u₀(x,y) – z(∂w₀(x,y)/∂x) = u₀ + z∙α

v(x,y) = v₀(x,y) – z(∂w₀(x,y)/∂y) = u₀ + z∙β

w(x,y) = w₀(x,y) + z

These equations underline that, these displacements depend on the distance z from the mid-plane and the angle of rotation. The strain comes from the displacement, by definition it is the derivative of the displacement by the reference axis. The shear strain is the sum of the cross derivatives, which means the derivatives of the displacement by the orthogonal axis respective to it.

ε_x = ∂u/∂x = ∂u₀/∂x – z(∂²w/∂x²) = ε_x0 + Z∙K_x

ε_y = ∂v/∂y = ∂v₀/∂y – z(∂²w/∂y²) = ε_y0 + Z∙K_y

γ_xy = [(∂v/∂x) + (∂u/∂y)] = [(∂v₀/∂x) + (∂u₀/∂y)] – 2z∙∂²w/(∂x∙∂y) = γ_xy0 + Z∙K_xy

If the terms ∂v/∂x and ∂u/∂y are substituted inside γ_xy, thus it comes out a term which is the derivative of the displacement at the mid-plane ∂v₀/∂x and ∂u₀/∂y. Another term is the position of the material times the negative of the second derivative of w, which is basically the curvature. Actually, this equation resembles the beam theory, where Z is the second derivative of the beam displacement, which is related to the light bending moment. In addition, there is also the curvature of the beam, which is given by the term K_xy. The last term of those equations is approximately the same, the difference is that the term of the mixed derivative of the shear strain is a twist curvature. This one occurs when the shear comes together with torsion. Hence, it is possible to represent those equations in a tensorial form.

{ε_x ε_y γ_xy}^T = {ε_x0 ε_y0 γ_xy0}^T + Z∙{K_x K_y K_xy}^T → {ε_xy} = {ε_xy}₀ + Z∙{K_xy}

Inside the laminate, the effects are illustrated by Figure 2

The strains are continuous, they start from zero at the mid-plane and increase linearly until the edges. This means that, strains depends linearly from the coordinate z. However, in this case there are different plies across z direction and possibly those have different elastic moduli. Hence, if the strain is multiplied by the elastic moduli, it is obtained the available strength. This is not fully correct, but regarding qualitative terms, is a reasonable result. Actually, this approach resembles to multiply ε by E, ply per ply, which results in a discontinuous stress. In other words, each ply, different from homogeneous pile of material, for a laminate has its own effect. At the same time, a laminate has a constant strain ε and an asymmetric elastic modulus E. For instance, in the case of a layup which is non-symmetrical with respect to the mid-plane. This means that, the stress distribution will be asymmetric with respect to the mid-plane, thus a moment with respect to it is generated. Depending on the lamination sequence it may be expected also some trending behaviour. This is the reason why some lamination sequences are very much over cured.

References

1. P.K. Mallick, Fiber-Reinforced Composites: materials, manufacturing and design – 3° Ed., CRC Press, 2008