Mechanics of aircraft materials 6 lamina theory cdio

We use the effective or average values of stress, strain and moduli when referring to lamina behavior... 2D Stress-Strain Relation of an Unidirectional Lamina... Lamina Stiffness Matrix

Second subscript refers to the direction in which the stress acts

 Corresponding to each stress

component, there is a strain component,

eij describing the deformation at a point.

 Normal strains describe the extension

per unit length.

 Shear strains describe distortional

deformation.

Stresses and strains are related to each other The most general

form of this relationship is:

9 Stresses x 9 Strains =

81 Components in relationship

    SC1

Inverse Relationship

[S] and [C] are symmetric matrices!

We use the effective (or average ) values of stress, strain and moduli when referring to lamina behavior

Orthotropic Material: 9 Constants

Three planes of symmetry

Transversely Isotropic Material

Three planes of symmetry 2 and 3 directions the same

MaterialNonzero TermsIndependent terms

3D Stress-Strain Relation of an Orthotropic Material in term of nine engineering

constants

Uniaxial Load in Fiber Direction

s

Transverse Load

2

Young's moduli in and directions

Poisson's ratios (extension-extension coupling)

Shear moduli in ,and directions

   SC1

Inverse Relationship

[S] and [C] are symmetric matrices!

2D Stress-Strain Relation of an Unidirectional Lamina

Plane Stress assumption

 e  S s

 s  Q e

Lamina Stiffness Matrix

Some Typical Properties

Are Carbon and Graphite the same?

“No, they are different Carbon Fibers have93-95% carbon content, while graphite hasmore than 99% carbon content Also,carbon fibers are produced at 2400oF(1316oC), while Graphite Fibers are typicallyproduced in excess of 3400oF (1900oC).”

2D Stress-Strain Relation of an Angle Unidirectional Lamina

Generally Angle Lamina

Generally Angle Lamina

x,y: Global axes 1,2: Local axes

cossin2sin cos

sin cossin coscossin

Transformation in Matrix Form

Condensed Matrix Form

Stress and Strain

Lamina Stiffness Matrix

General Stress-Strain Behavior

See: Example 2.5/ Page 92 “Mechanics of Composite

Materials”, Autar K.Kaw

Coefficients of Mutual Influence

Problem 1

Determine the stiffness matrix of a Laminate

Determine the stiffness matrix for a [+45/-45/-45/+45] symmetric angle-ply laminate

consisting of 0.25 mm thick unidirectional AS/3501 graphite epoxy lamina

Determine the stiffness matrix of a Laminate solution

-1 Find the value of the “reduced stiffness matrix” [Q] for each ply using its four elastic moduli E11, E22, G12

2 Find the value of the “transformed reduced stiffness

matrix” for each ply by using the [Q] and the angle of each ply

Q  

y x

Exploded view of a [+45/-45/-45/+45] symmetric laminate

Autar K Kaw, Mechanics of Composite Materials, CRC Press, 2006

2.2; 2.3; 2.6; 2.9; 2.10;

2.11; 2.15; 2.16; 2.17; 2.20; 2.21