1. M. A. Farjoo

2.  The stiffness can be defined by appropriate stress – strain relations.  The components of any engineering constant can be expressed in terms of other ones.

3.  Stress is a measure of internal forces within the body.  Stress can be derived from: ◦ Applied forces using stress analysis. ◦ Measured displacements and stress analysis. ◦ Measured strains using stress strain relations.  The state of tress in a ply is predominantly plane stress.  The nonzero components are:  x,  y and  s.  The sign convention shall be observed when we deal with composite materials.

4.  The difference between tensile and compressive strength may be several hundred percent! ◦ (when the x-axis is towards the fibers longitudinal coordination, we call this is “on axis orientation” )

5.  Strain is the special variation of the displacements.   u=relative displacement along x axis.   v=relative displacement along y axis.

6.  The normal strain components are associated with changes in the length of an infinitesimal element.  The rectangular element after deformation remains rectangular although its length and width may change.  There is no distortion produced by the normal strain component.  Distortion is measured by the change of angles.

7.  Shear makes distortion in the element.

8.   s is the engineering shear strain which is twice the tensorial strain.  Eng. Shear strain is used because it measures the total change in angle or the total angle of twist in the case of a rod under torsion.

9.  Our study is limited to the Linearly Elastic Materials, so: ◦ The superposition rule is active here. ◦ And the elasticity is reversible. We can load and unload the structure without any hysteresis.  This assumptions are close to experimental.  The strain-stress relation can be derived by superposition method.  The on-axis stress-strain relations can be derived by superpositioning the results of the following simple tests:

10. E x =Longitudinal Young’s modulus. x = Longitudinal Poisson’s ratio.

12.  By applying the principle of super position, the longitudinal shear stress would be:

14.  All the material constants of the stress – strain relation shown in previous slide are called Engineering Constants.  A change of notation from Eng. Const. to Components of Compliance have been done.

15.  Or:  The stress can be solved in terms of strain as:

16.  So the components of Modulus would be defined as the following matrix:

17.  3 sets of material constants were shown. Any of which can completely describe the stiffness of on-axis unidirectional composites.  Each of them has the following characteristics: ◦ Modulus is used to calculate the stress from strain. This is the basic set needed for the stiffness of multidirectional laminates. ◦ Compliance is used to calculate the strain from stress. This is the set needed for calculation of Engineering Constants. ◦ Engineering Constants are the carry over from the conventional materials.

18.  Considering elastic energy in the body:  And substituting stress strain in terms of compliance:  Recovering stress-strain relation by differentiating of the energy:

19.  Comparing with compliance matrix definition the only condition that both equations match is: ◦ S xy = S yx  As same as this method, One can find: ◦ Q xy = Q yx