Analysis of an orthotropic lamina

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Stress tensor

Convention

t = n

dA1=dA.n1Equilibrium:

dA2=dA.n2

dA3=dA.n3

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Principal stresses

The principal directions are solutions ofThe principal directions are solutions of

St i i tStress invariants:

In a coordinate system oriented along the principal directionsIn a coordinate system oriented along the principal directions,the stress tensor is diagonal:

Stress invariants in principal coordinates3

Generalized Hooke’s law

4th order tensor of elastic constants

Because of the symmetryof the strain tensor

Because of the symmetryof the stress tensor ConstitutiveConstitutive

Equation:Because of the 3 symmetry relationships, the number

of independent elastic constants is reducedfrom 34=81 to 21 in the most general anisotropic material

The order of partial differentiationMay be changed

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Change of coordinates in the elastic constants

Let two coordinate systems x and x’ related by the rotation matrix A=aij

Change of coordinates of the stress tensor:g

Applies to any second order tensorApplies to any second order tensor(same rule for the strain tensor)

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Tensor of elastic constant:

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O th t i it 3 f tOrthotropic composite: 3 axes of symmetry

The elastic constants do not change under coordinate transformations that preserve symmetry

(x1,x2)plane  Direction cosines:( 1, 2)pof symmetry

One findsOne finds:

Must be = 0

Similarly, 8 constants must be equal to 0:

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Direction cosines:(x2,x3)plane of symmetry

The following contantsMust also be equal to 0: 

There is no additional condition coming from the third plane of symmetry (x x )There is no additional condition coming from the third plane of symmetry (x1,x3).Overall, there are 21‐12= 9 independent elastic constants for an orthotropic material.

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Orthotropic materials: 9 independent elastic constants

Hooke’s law may be written inmatrix formHooke’s law may be written inmatrix form

Vector of engineering Engineering g gstress components

g gstrain components

Stiffness matrix

[the axes 1,2,3 coincide with the natural (orthotropy) axes of the material]9

In two dimensions:In two dimensions:

Stiffness matrix

Compliance matrix

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Stress‐strain relations and engineering constants for orthotropic lamina

The compliance matrix may be constructedThe compliance matrix may be constructedcolumn by column by considering 3 load cases:

1.

One gets the first columnof the compliance matrix

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2.

2nd column of the Compliance matrix

3.

3rd column of the C li iCompliance matrix

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Orthotropic lamina in natural axesp

1. Compliance matrix

For an isotropic lamina, EL=ET , G=E/2(1+)

2. Stiffness matrix

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Change of reference frame

We seek the transformation matrix [T]

Change of coordinatesFor a 2nd order tensor:

[T] is not a rotation  matrix !!

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Stress‐strain relationship

In L‐T axes:

In arbitrary axes:

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Stiffness matrix in arbitrary axes

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Compliance matrix in arbitrary axes:

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Example: find the strains in the lamina =60°

Elastic constants:Stresses:

Step 1 compute the stresses in orthotropy axesStep 1: compute the stresses in orthotropy axes

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Step 2: compute the strains in natural axes:

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[T(‐]

Step 3: compute the strains in the axes (x,y)

[T( ]

LT

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Engineering constants

The compliance matrix in bit b ittarbitrary axes may be written:

All the elastic constants may be expressed inAll the elastic constants may be expressed in terms of the 4 constants: EL, ET, GLT, LT.

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Graphite‐epoxy system Boron‐epoxy system

Ex < ET

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Balanced lamina: ET=EL,LT=TL

Not isotropic !!

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