Chapter 2Stiffness of Unidirectional

CompositesM. A. Farjoo

The stiffness can be defined by appropriate stress – strain relations.

The components of any engineering constant can be expressed in terms of other ones.

Preface

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.

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” )

Stress

Strain is the special variation of the displacements.

Du=relative displacement along x axis. Dv=relative displacement along y axis.

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.

strain

Shear makes distortion in the element.

Strain

es 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.

Strain

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:

Stress – Strain Relations

Stress-Strain Relations

Ex=Longitudinal Young’s modulus.nx= Longitudinal Poisson’s ratio.

Stress-Strain Relations

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

Stress-Strain Relations

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.

Stress-Strain Relations

Or:

The stress can be solved in terms of strain as:

Stress-Strain Relations

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

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.

Stress-Strain Relations

Considering elastic energy in the body:

And substituting stress strain in terms of compliance:

Recovering stress-strain relation by differentiating of the energy:

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

Symmetry of Compliance and Modulus