Chapter 2 Stiffness of Unidirectional Composites

1. Chapter 2Stiffness of Unidirectional Composites M. A. Farjoo

2. Preface 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 • 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: sx, sy and ss. • The sign convention shall be observed when we deal with composite materials.

4. Stress • 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 Strain is the special variation of the displacements. Du=relative displacement along x axis. Dv=relative displacement along y axis.

6. strain 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. Strain Shear makes distortion in the element.

8. Strain esis 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. Stress – Strain Relations • 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. Stress-Strain Relations Ex=Longitudinal Young’s modulus. nx= Longitudinal Poisson’s ratio.

11. Stress-Strain Relations

12. Stress-Strain Relations By applying the principle of super position, the longitudinal shear stress would be:

13. Stress-Strain Relations

14. Stress-Strain Relations 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. Stress-Strain Relations Or: The stress can be solved in terms of strain as:

16. Stress-Strain Relations So the components of Modulus would be defined as the following matrix:

17. Stress-Strain Relations • 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. Symmetry of Compliance and Modulus Considering elastic energy in the body: And substituting stress strain in terms of compliance: Recovering stress-strain relation by differentiating of the energy:

19. Symmetry of Compliance and Modulus • Comparing with compliance matrix definition the only condition that both equations match is: • Sxy = Syx • As same as this method, One can find: • Qxy=Qyx