M1 Anisotropic Elastic Behavior of the Laminate (Classical Laminate Theory)

Topic

In this model, the stiffness of the complete laminate is predicted by combining its constituent individual plies—each with its own orientation within the stack—according to classical lamination theory. Combining the reduced stiffness of each ply (established in a separate model describing the effective elastic properties of individual plies) with that ply's orientation and position within the stack results in the formation of three matrices characterizing the entire laminate: the extensional stiffness matrix, the coupling matrix, and the bending stiffness matrix.

Of these three, the coupling matrix links the laminate's extensional behavior to its bending behavior; if the ply stack is asymmetric with respect to the laminate's mid-plane, the coupling matrix becomes non-zero, meaning a force applied within the laminate's plane causes it to curve in addition to stretching or compressing it. Conversely, if the stack is symmetric, the coupling matrix becomes zero, completely decoupling the laminate's extensional and bending behaviors.

Among the three, the extensional stiffness matrix predicts the laminate's extensional stiffness—that is, the extent to which the laminate stretches or compresses within its own plane in response to a force applied within that same plane. Higher reduced stiffness in individual plies, or a larger proportion of the total laminate thickness accounted for by those plies, results in a higher extensional stiffness matrix and a stiffer laminate under that type of loading. The flexural stiffness matrix, on the other hand, predicts the laminate's flexural stiffness—that is, how much the laminate bends in response to an applied bending moment. Because plies located further from the laminate's mid-plane contribute more to flexural stiffness, this matrix is ​​particularly sensitive to those outer plies compared to those near the mid-plane; this is analogous to how a cross-section with a higher moment of inertia offers greater resistance to bending.

Consequently, this model—which addresses the laminate stiffness introduced at the outset—relies directly on another model within the same set: the one that determines the reduced stiffness of each individual ply. Because this model uses that reduced stiffness to calculate the laminate's flexural stiffness matrix, a separate model in the set—the one describing laminate buckling—can utilize that same matrix to predict the load at which buckling occurs.

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