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A part of a mechanics of composites textbook, focusing on isotropic linear elasticity. It covers topics such as small deformations, extensional and shear strains, strain tensor, material element in equilibrium, stress tensor, indicial notation, and elastic strain energy. It also discusses hooke's law for isotropic and anisotropic materials, as well as the transformation of stress and strain. Equations and formulas related to these topics.
Typology: Study notes
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REINFORCEMENT + MATRIX + INTERFACE = COMPOSITE
- Properties and performance depend on: - properties of the reinforcement and the matrix - size, shape and distribution of the reinforcement **- reinforcement/matrix interface
Deformation
symmetric
i = 1, 2, 3 j = 1, 2, 3
Material Element in Equilibrium
dx 2
dx 1
dx 3
Traction vectors on faces perpendicular to the coordinate axes
t (1)^ = 11 e 1 + 12 e 2 + 13 e 3 t (2)^ = 21 e 1 + 22 e 2 + 23 e 3 t (3)^ = 31 e 1 + 32 e 2 + 33 e 3
ij = (^) ji
Constitutive Response:
Uniaxial Loading (i=j) 11 = E 11 11 = 1 E 11
Young Modulus (extensional stiffness)
Extensional compliance
11
11
11
E
11
11
12 = 13 = 23 = 0
Equivalent response for uniaxial loading in the 2 and 3 directions
Depends on material properties
ij = f ( ij ) Hookes Law - Isotropic Material
Hooke’s Law Isotropic Material
12
12
G
Pure Shear Loading (ij) 12
12
Shear modulus (stiffness) Shear compliance
Equivalent response for shear loading in the 1-3 and 2-3 planes
2(1+ v )
ij = Cijkl kl ij = Sijkl (^) kl
Elastic Stiffness Constants
Elastic Compliance Constants Cijkl = Sijkl^ ^1 81 independent elastic constants? stress and strain are symmetric: ^ ij =^ ^ ji^ ij =^ ^ ji Cijkl = C (^) jikl = Cijlk 36 independent elastic constants
**- Properties depend on direction
n X 1
n = ni e (^) i = n 1 e 1 + n 2 e 2 + n 3 e 3 n = n i e i = n 1 e 1 + n 2 e 2 + n 3 e (^) 3 ni e i ni e (^) i
For a vector n:
Define Direction Cosines
ij = e i e (^) j = e (^) i e (^) j cos ij
e i = ij e (^) j { e } =^ [ ]{ } e n i = ij n (^) j { n } = [ ]{ } n
e (^) i = (^) ji e (^) j { } = e [ ] t^ { } e ni = ji n (^) j { } = n^ [ ] t^ { n }
[ ] t^ = [ ]^1
11
12
13 22
33 23
t (3)
t (1) t (2)
ij = e (^) i e (^) j = e (^) i e (^) j cos ij
X (^3)
X (^1)
X (^2)
X (^) 3 X 2
X 1
[ ( , , )] = [ 3 ( )] [ 1 ( )] [ 3 ( )]
11 12 13 21 22 23 31 32 33
cos cos cos sin sin sin sin cos cos sin sin sin sin + cos cos sin cos cos sin cos + sin sin cos sin sin sin cos cos cos
[ ] = [ ] t^ [ ][ ] [ ] = [ ][ ][ ] t
[ ] = [ ] t^ [ ] [ ] [^ ] =^ [ ][^ ] [ ] t
(^) ij = im (^) jn (^) mn kl = kr ls rs
[ ] { } = [ T ]^1 { } {^ } =^ [ T ]{^ ^ }
1 { }
[T ] is the stress transformation matrix
[T] is the engineering strain transformation matrix*
Simplified Stress Transformation Matrix for an In-plane Rotation,
[ T 3 ( )] =
m^2 n^2 0 0 0 2 mn n^2 m^2 0 0 0 2 mn 0 0 1 0 0 0 0 0 0 m n 0 0 0 0 n m 0 mn mn 0 0 0 m^2 n^2
[ T 1 ( )] =
1 0 0 0 0 0 0 m^2 n^2 2 mn 0 0 0 n^2 m^2 2 mn 0 0 0 mn mn m^2 n^2 0 0 0 0 0 m n 0 0 0 0 n m
About the 3 axis: About the 1 axis:
m = cos ; n = sin
where
(*)
Can show that: (^) [ T *] ^1 = [ T ] t^ [ C ] = [ T ][ C ][ T ] t
(*)
{ } = [ C ] { }
Cijkl = im (^) jn kr ls Cmnrs
No Symmetry
21 Independent constants
Plane of Material Symmetry : Properties are unchanged by a rotation of 180° about a given axis.
Properties are unchanged by a rotation of 180° about the “3” axis
[ C ] = [ T ][ C ][ T ] t^ = [ C ]
[ C ] = [ C ] for = 180 °
