Upload
truongkhanh
View
242
Download
4
Embed Size (px)
Citation preview
Analysis of an orthotropic lamina
1
Stress tensor
Convention
t = n
dA1=dA.n1Equilibrium:
dA2=dA.n2
dA3=dA.n3
2
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
4
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)
5
Tensor of elastic constant:
6
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:
7
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.
8
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
10
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
11
2.
2nd column of the Compliance matrix
3.
3rd column of the C li iCompliance matrix
12
Orthotropic lamina in natural axesp
1. Compliance matrix
For an isotropic lamina, EL=ET , G=E/2(1+)
2. Stiffness matrix
13
Change of reference frame
We seek the transformation matrix [T]
Change of coordinatesFor a 2nd order tensor:
[T] is not a rotation matrix !!
14
Stress‐strain relationship
In L‐T axes:
In arbitrary axes:
15
Stiffness matrix in arbitrary axes
16
Compliance matrix in arbitrary axes:
17
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
18
Step 2: compute the strains in natural axes:
19
[T(‐]
Step 3: compute the strains in the axes (x,y)
[T( ]
LT
20
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.
21
Graphite‐epoxy system Boron‐epoxy system
Ex < ET
22
Balanced lamina: ET=EL,LT=TL
Not isotropic !!
23