eFunda: Lamina Stress-Strain Relations for Principal Directions
engineering fundamentals Lamina Stress-Strain Relations
for Principal Directions

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Macromechanics of Lamina

From control surfaces of modern aircrafts, to hulls and keels of yachts, to racing car bodies, to tennis rackets, fishing rods, golf shafts and heads, laminated fiber reinforced composite is one of the the most widely used composites in industry.

Unless otherwise noted, the following assumptions are made in our discussion of the macro-mechanics of laminated composites.

  1. The matrix is homogeneous, isotropic, and linear elastic.
  2. The fiber is homogeneous, isotropic, linear elastic, continuous, regularly spaced, and perfectly aligned.
  3. The lamina (single layer) is macroscopically homogeneous, macroscopically orthotropic, linear elastic, initially stress-free, void-free, and perfectly bonded.
  4. The laminate is composed of two or more perfectly bonded laminae to act as an integrated structural element.
Stress-Strain Relations for Principal Directions

Before discussing the mechanics of laminated composites, we need to understand the mechanical behavior of a single layer -- lamina. Since each lamina is a thin layer, one can treat a lamina as a plane stress problem. This simplification immediately reduces the 6×6 stiffness matrix to a 3×3 one.

Since each lamina is constructed by unidirectional fibers bonded by a metal or polymer matrix, it can be considered as an orthotropic material. Thus, the stress-strain relations on the principal axes can be expressed by the compliance matrix [S] such that

[strain] = [S][stress]

or by the stiffness matrix [C] such that

[stress] = [C][strain]

Please note that the engineering shear strain is used in the stress-strain relations, and, the notation S for the compliance matrix and C for the stiffness matrix are not misprints. Please consult this page for more information.

For both stiffness and compliant matrices are symmetric, i.e.,

only four of E1, E2, G12, nu12, and nu21 are independent material properties. Again, the shear modulus G12 corresponds to the engineering shear strain gamma12 which is twice the tensor shear strain e12.

Please note that there can be many fibers across the thickness of a lamina and these fibers may not be arranged uniformly in most industrial practice. However, the combination of the matrix and the fibers forms an orthotropic and homogeneous material from a marcomechanics standpoint. Some literature therefore schematically illustrates a lamina with only one layer of uniformly distributed fibers as shown below.

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