174 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES It is conventional in composite mechanics to use the above subscript notation for the point of which becomes evident only when three-dimensional anisotropic problems are encountered.The subscript 6 is for the sixth component of stress or strain that includes three direct terms and three shear terms. 6.2.2 Stress-Strain Law for Single Ply in Laminate Axes: Off-Axis Laminates As already noted,when a ply is incorporated in a laminate,its fibers will make some prescribed angle 0 with a reference axis fixed in the laminate.Let this be the x-axis,and note that the angle 0 is measured from the x-axis to the 1-axis and is positive in the counterclockwise direction;the y-axis is perpendicular to the x- axis and in the plane of the ply (See Fig.6.2.).All subsequent calculations are made using the x-y,or"laminate"axes,therefore it is necessary to transform the stress-strain law from the material axes to the laminate axes.If the stresses in the laminate axes are denoted by ox and T,then these are related to the stresses referred to the material axes by the usual transformation equations, c2 231 6.5) where c denotes cos 6 and s denotes sin 6.Also,the strains in the material axes are related to those in the laminate axes,namely,ex,and yo,by what is essentially the strain transformation: 2-2 -0 (6.6) Txy.7xy Ox.Ex Fibres ---- Fig.6.2 Laminate axes for a single ply174 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES It is conventional in composite mechanics to use the above subscript notation for Q, the point of which becomes evident only when three-dimensional anisotropic problems are encountered. The subscript 6 is for the sixth component of stress or strain that includes three direct terms and three shear terms. 6.2.2 Stress-Strain Law for Single Ply in Laminate Axes: Off-Axis Laminates As already noted, when a ply is incorporated in a laminate, its fibers will make some prescribed angle 0 with a reference axis fixed in the laminate. Let this be the x-axis, and note that the angle 0 is measured from the x-axis to the/-axis and is positive in the counterclockwise direction; the y-axis is perpendicular to the x￾axis and in the plane of the ply (See Fig. 6.2.). All subsequent calculations are made using the x-y, or "laminate" axes, therefore it is necessary to transform the stress-strain law from the material axes to the laminate axes. If the stresses in the laminate axes are denoted by trx, try, and "l~y, then these are related to the stresses referred to the material axes by the usual transformation equations, Dry I = if S 2 C 2 csl[ l 2CS 0"2 1 (6.5) "lxy CS --CS C 2 -- S 2 "/'12 where c denotes cos 0 and s denotes sin 0. Also, the strains in the material axes are related to those in the laminate axes, namely, 8x, ey, and Yxy, by what is essentially the strain transformation: e2 1 = s 2 c s 2 -cs csl[ x ey 1 (6.6) 3/12 --2CS 2CS C2 -- S 2 %y I Oy, ey ......_..=, / . / / / / / /Yl / // / , / / ~/ ... / ./ / A /~'~ / e // /// ///" ~/// / / / / / /~ / / / / / / / / / / / i I f # / rxy, 7xy t ~ aXo ~X Fibres Fig. 6.2 Laminate axes for a single ply