Computational Structural Mechanics

1. What problem is being solved here?

A laminate is a stack of plies, each with its own material orientation \(\theta_k\) and thickness \(t_k\). Even when every ply comes from the same material system, the laminate does not behave like an isotropic plate. Its stiffness depends on the lamina elastic constants, the stacking sequence, the thickness distribution, and whether the laminate is symmetric or unsymmetric.

The practical questions are:

How stiff is the laminate in membrane loading?
How stiff is it in bending?
Does the laminate exhibit membrane–bending coupling?
Under a given load, what strains and curvatures develop?
Which ply sees the highest stress, and is that stress still below its strength limit?

2. The central constitutive statement

\[ \begin{bmatrix} \mathbf{N}\\ \mathbf{M} \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \mathbf{B}\\ \mathbf{B} & \mathbf{D} \end{bmatrix} \begin{bmatrix} \boldsymbol{\varepsilon}_0\\ \boldsymbol{\kappa} \end{bmatrix}. \]

This equation is the laminate-level constitutive law of Classical Lamination Theory (CLT). It states that the applied in-plane force resultants and bending/twisting moment resultants are related to the laminate mid-plane strains and curvatures through the stiffness blocks \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{D}\).

\(\mathbf{N}=[N_x\;N_y\;N_{xy}]^T\): in-plane force resultants, units N/m. These represent stretching, compression, or in-plane shear acting on the laminate.
\(\mathbf{M}=[M_x\;M_y\;M_{xy}]^T\): bending and twisting moment resultants, units N. In CLT these are moments per unit width, so their unit is \((\text{N}\cdot\text{m})/\text{m}=\text{N}\).
\(\boldsymbol{\varepsilon}_0=[\varepsilon_x^0\;\varepsilon_y^0\;\gamma_{xy}^0]^T\): mid-plane strain vector, dimensionless. It describes stretching or shear of the laminate mid-surface.
\(\boldsymbol{\kappa}=[\kappa_x\;\kappa_y\;\kappa_{xy}]^T\): curvature vector, units 1/m. It describes bending or twisting of the laminate.
\(\mathbf{A}\): extensional stiffness matrix, units N/m. It controls resistance to in-plane stretching and shear.
\(\mathbf{B}\): membrane–bending coupling matrix, units N. It measures how strongly stretching couples to bending.
\(\mathbf{D}\): bending stiffness matrix, units N·m. It controls resistance to bending and twisting.

Physically, the equation says that a laminate can respond in two ways: it can deform in its plane and it can bend out of plane. The matrix \(\mathbf{A}\) governs the first response, \(\mathbf{D}\) governs the second, and \(\mathbf{B}\) tells us whether these two responses are coupled.

In the usual CLT workflow, the matrices \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{D}\) are not prescribed independently. They are computed from the laminate itself: the lamina elastic properties, ply orientations, stacking sequence, and thickness coordinates.

By contrast, the vectors \(\mathbf{N}\) and \(\mathbf{M}\) are usually treated as the externally applied load resultants. Once the laminate stiffness and the applied loading are known, the main unknowns to be solved are the mid-plane strains \(\boldsymbol{\varepsilon}_0\) and the curvatures \(\boldsymbol{\kappa}\).

Typical interpretation. In a forward CLT problem, we first define the laminate, which gives \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{D}\). We then prescribe the applied force and moment resultants \(\mathbf{N}\) and \(\mathbf{M}\). The constitutive system is then solved for \(\boldsymbol{\varepsilon}_0\) and \(\boldsymbol{\kappa}\).

Example 1: symmetric laminate. If the stacking sequence is symmetric about the mid-plane, then typically \(\mathbf{B}=0\). In that case, applying only \(N_x\) produces mainly in-plane strain \(\varepsilon_x^0\), without inducing bending curvature.

Example 2: unsymmetric laminate. If the laminate is unsymmetric, then \(\mathbf{B}\neq 0\) in general. A purely in-plane tensile load can then produce not only extension, but also bending. In other words, pulling the laminate may cause it to curve.

This is the main equation of the module. The following page explains how the stiffness blocks are built from ply properties, ply orientations, and thickness coordinates.

3. How these notes are organized

CLT Theory and MATLAB: the core page, where ply, lamina, laminate, material properties, laminate matrices, strength parameters, and the code workflow are introduced carefully.
COMSOL Validation and Fracture Bridge: the extension page, where CLT is connected to finite element analysis, local stress concentration, and later fracture-mechanics thinking.

4. Module pages

CLT Theory and MATLAB

Basic definitions of ply, lamina, and laminate; material constants; reduced stiffness matrix; transformed reduced stiffness; laminate A, B, and D matrices; worked carbon/epoxy example; strength parameters; and MATLAB implementation.

Open notes

COMSOL Validation and Fracture Bridge

How to use CLT as a baseline, what should be compared between CLT and FEM, what kinds of local effects require COMSOL, how strength differs from fracture, and how the discussion extends toward damage initiation and propagation.

Open notes