172 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES 6.2 Laminate Theory Classical laminate theory defines the response of a laminate with the following assumptions: For two-dimensional plane stress analysis,the strain is constant through the thickness. .For bending,the strain varies linearly through the thickness. The laminate is thin compared with its in-plane dimensions. .Each layer is quasi-homogeneous and orthotropic. .Displacements are small compared with the thickness. The behavior remains linear. With these assumptions satisfied,the laminate theory allows the response of a laminate to be calculated,engineering constants to be determined to substitute into standard formulas for stresses and deflections,and material properties of the laminate to be defined for substitution into finite element analysis as described in Chapter 16. 6.2.1 Stress-Strain Law for a Single Ply in the Material Axes: Unidirectional Laminates Consider a rectangular element of a single ply with the sides of the element parallel and perpendicular to the fiber direction(Fig.6.1).Clearly,the direction of the fibers defines a preferred direction in the material;it is thus natural to introduce a cartesian set of material axes 0-1,2,3 with the /-axis in the fiber direction,the 2-axis perpendicular to the fibers of the ply plane,and the 3-axis perpendicular to the plane of the ply.Here,interest is in the behavior of the ply when subjected to stresses acting in its plane,in other words,under plane stress conditions.These stresses (also referred to the material axes)will be denoted by o1,2,T12 and the associated strains by s1,s2,and y12.(Note that in composite mechanics,it is standard practice to work with"engineering"rather than"tensor" shear strains.)Although a single ply is highly anisotropic,it is intuitively evident that the coordinate planes 012,023,and 031 are those of material symmetry,there being a mirror image symmetry about these planes. 02.∈2 T12.712 01,∈1 Fibres Fig.6.1 Material axes for a single ply.172 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES 6.2 Laminate Theory Classical laminate theory defines the response of a laminate with the following assumptions: • For two-dimensional plane stress analysis, the strain is constant through the thickness. • For bending, the strain varies linearly through the thickness. • The laminate is thin compared with its in-plane dimensions. • Each layer is quasi-homogeneous and orthotropic. • Displacements are small compared with the thickness. • The behavior remains linear. With these assumptions satisfied, the laminate theory allows the response of a laminate to be calculated, engineering constants to be determined to substitute into standard formulas for stresses and deflections, and material properties of the laminate to be defined for substitution into finite element analysis as described in Chapter 16. 6.2.1 Stress-Strain Law for a Single Ply in the Material Axes: Unidirectional Laminates Consider a rectangular element of a single ply with the sides of the element parallel and perpendicular to the fiber direction (Fig. 6.1). Clearly, the direction of the fibers defines a preferred direction in the material; it is thus natural to introduce a cartesian set of material axes 0-1, 2, 3 with the /-axis in the fiber direction, the 2-axis perpendicular to the fibers of the ply plane, and the 3-axis perpendicular to the plane of the ply. Here, interest is in the behavior of the ply when subjected to stresses acting in its plane, in other words, under plane stress conditions. These stresses (also referred to the material axes) will be denoted by trl, tr2, r12 and the associated strains by el,/32, and 712. (Note that in composite mechanics, it is standard practice to work with "engineering" rather than "tensor" shear strains.) Although a single ply is highly anisotropic, it is intuitively evident that the coordinate planes 012, 023, and 031 are those of material symmetry, there being a mirror image symmetry about these planes. I 02. ~2 0 L I r12, 712 ] ~-~ °I, ~I Fibres 3 Fig. 6.1 Material axes for a single ply