In Classical Laminate Theory, what do the A, B, and D matrices represent and what physical properties do they relate?

In Classical Laminate Theory, what do the A, B, and D matrices represent and what physical properties do they relate?

• A. A is the extensional stiffness (in-plane), B is the extensional–bending coupling (through-thickness), and D is the bending stiffness. They are 3×3 matrices relating in-plane forces/moments to strains/curvatures.
• B. A is shear stiffness, B is torsional coupling, D is thermal stiffness; they are 4×4 matrices.
• C. A is mass matrix, B is damping matrix, D is stiffness matrix; they are 2×2.
• D. A and B are zero for all laminates; D is infinite.

Answer: (A)

In Classical Laminate Theory, the three matrices organize how a laminated plate resists in-plane and bending actions and how those actions are coupled. The first matrix, called the extensional stiffness, captures how in-plane forces relate to mid-plane in-plane strains: the in-plane force resultants are proportional to the mid-plane strains through that matrix. The second matrix, the extensional–bending coupling, links in-plane behavior to bending behavior and appears when the laminate is not symmetric about its mid-plane; it creates a coupling between mid-plane strains and curvatures, and between in-plane forces and curvatures. The third matrix, the bending stiffness, describes how bending moments relate to curvatures of the mid-plane. All three are 3×3 matrices because there are three independent in-plane components to consider (xx, yy, and xy). In compact form, the in-plane force resultants and bending moments together relate to the mid-plane strains and curvatures as: N and M form a 6-component vector that equals the block matrix [A B; B^T D] acting on the 6-component vector [epsilon^0; kappa], where epsilon^0 are mid-plane strains and kappa are curvatures. When the laminate is symmetric, coupling vanishes and B is zero; for asymmetric laminates, B is nonzero, producing extensional–bending coupling.

In Classical Laminate Theory, the three matrices organize how a laminated plate resists in-plane and bending actions and how those actions are coupled. The first matrix, called the extensional stiffness, captures how in-plane forces relate to mid-plane in-plane strains: the in-plane force resultants are proportional to the mid-plane strains through that matrix. The second matrix, the extensional–bending coupling, links in-plane behavior to bending behavior and appears when the laminate is not symmetric about its mid-plane; it creates a coupling between mid-plane strains and curvatures, and between in-plane forces and curvatures. The third matrix, the bending stiffness, describes how bending moments relate to curvatures of the mid-plane.

All three are 3×3 matrices because there are three independent in-plane components to consider (xx, yy, and xy). In compact form, the in-plane force resultants and bending moments together relate to the mid-plane strains and curvatures as: N and M form a 6-component vector that equals the block matrix [A B; B^T D] acting on the 6-component vector [epsilon^0; kappa], where epsilon^0 are mid-plane strains and kappa are curvatures. When the laminate is symmetric, coupling vanishes and B is zero; for asymmetric laminates, B is nonzero, producing extensional–bending coupling.