5.3 effective moduli of cfrp lamina

Composite Laminate 1

Effective Moduli of a Continuous Fiber-Reinforced Lamina

Review of Linear Constitutive RelationsGeneral anisotropic material behavior is given by a relationship wherein all 6 tensor stress components are related to all 6 tensor strain components:

or


[C] is symmetric, but the matrix is full. The material constants in [C] are called the stiffness or elastic constants (or moduli).

Inverting the last relation gives

[S] is usually called the compliance matrix. Note that .

The determination of the material constants for a general anisotropic material is extremely difficult since the material has mechanical properties (Young's modulus, Poisson's ratio, etc.) that vary with the direction in which they are measured, and all stresses are coupled with all strains.


Orthotropic Material (3-D)A material that has mechanical properties that can be associated with an orthogonal principal material coordinate system is called orthotropic. A typical example is a unidirectional composite lamina shown below:

Orthotropic lamina with principal material (1,2,3) and non-principal (x,y,z) coordinates. Note that 1 is generally taken as the fiber direction, and 2 is transverse to the fiber but in the plane of a fiber layer.

This unidirectional composite lamina has three mutually orthogonal planes of material property symmetry and is called an orthotropic material. In the above figure, the 123 coordinates axes are referred to as the principal material coordinates since they are associated with the reinforcement directions. One can show that


for specially orthotropic materials wherein the 123 axes are principal material directions, the compliance matrix has the form:

Note that shear stresses (and strains) are now uncoupled from normal stresses (and strains).

Using the appropriate sequence of material uniaxial and shear tests (see Gibson, or any composite mechanics text), one can show that the compliance terms can be written in terms of engineering material constants so that we have the following relation between


engineering strains and stress. [Recall that engineering shear strain is twice the tensor shear strain, i.e., .]

are Young's moduli in the 1, 2, 3 directions, = Poisson's ratio for transverse normal strain in the j

direction when a normal stress is applied in the i direction, and

are shear moduli in the i-j plane.

Since the compliance matrix [S] must be symmetric, we see that


Hence, the strain-stress relation could also be written as:

Note that for the specially orthotropic material, there are 9 engineering constants: .Orthotropic Lamina in Plane Stress


For a single laminae, the lamina is often assumed to be in a simple two-dimensional state of plane stress (in the 1-2 plane) such that

. The lamina compliance relation simplifies to

or

Hence, there are only 5 non-zero compliances (only are 4 are independent since [S] is symmetric) and 4 independent material constants ( ).


The last equation can be inverted to obtain the lamina stiffness relation, but is written in terms of tensor strains as:

or

where the are components of the lamina stiffness matrix and are given by:


Transformation of Material Properties (1-2 to x-y) (or the Generally Orthotropic Lamina)

In order to analyze laminates having multiple laminae with fibers in different directions, it is necessary to determine material properties in an arbitrary x-y coordinate system in terms of material properties in the 1-2 principal material directions. This is a simple transformation similar to stress transformation in combined stresses.

Consider a lamina that has the principal 1 material axes at angle to the x axis (+ counterclockwise) as shown below. We can transform forces from x-y to 1-2 coordinates using the simple relationship:


1,2 is the local (material) coordinate system

x,y is the global coordinate system

The stress transforms as a second order tensor, i.e.,


This can be expanded to give the familiar set of i.e.,

or, in matrix notation as

The strain transforms the same as stress if we use tensor strain (engineering shear strain is not a tensor quantity). Recall that


engineering shear strain is twice the tensor shear strain, i.e., .

This last relation can be written in matrix notation as

Note that the square matrices in and are identical. Define the transformation matrix [T] as


Note that [T] is not the square matrix in and , but is similar. Inverting this matrix, we obtain

Comparing equation and , we see that can be written in terms of :


Likewise for the strain,

Note that equation can be inverted to obtain:


Now we are ready to transform the compliance from material (1,2) directions to global (x,y) directions. First substitute equation into equation to obtain:

Now substitute equation into equation to obtain

So we have now the stress-strain relation in x-y directions but written in terms of the stiffness in 1-2 material directions:


The triple matrix product must then be the transformed lamina stiffness matrix in x-y global directions. Hence, we define the transformed lamina stiffness matrix in x-y global directions by:

and becomes

Carrying out the matrix multiplication gives:


Note that the stiffness matrix now looks like an anisotropic material since the 3x3 has nine non-zero terms. However, the material is still orthotropic because the stiffness matrix can be expressed in terms of 4 independent lamina stiffness terms (

).

The compliance matrix in can be similarly written. From ,


Transform to global direction (similarly to that done for Q) to obtain:

where

5.3 effective moduli of cfrp lamina

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