Micromechanics- Classical Laminate Theory 3 Flashcards
Kirchoff assumptions for laminates
Flat and thin laminate.
No through thickness stresses (i.e overall plane stress)
Edge effects can be neglected
How to number plies
Often top one is ply 1 and the fibre direction in this ply chosen as reference direction. Next one down is ply 2. Any ply is ply k. Total of n plies
Notation for referring to a composite, thickness, individual lamina
Subscript c refers to a composite of n laminae.
Thickness of each lamina is t sub k.
k refers to one lamina
Angles between load direction and principal material direction notation for laminate
Load direction in 1’. Principal direction is that for the top ply and is 1. Principal direction for any other ply is 1k (not sub). Angle between 1’ and 1 is θ. Angle between 1 and 1k is θ sub k.
What can the average stress for the laminate in the 1’ direction be expressed via?
This is σ’1c and can be expressed via the stresses in the individual laminae in the 1’ direction.
Equal to sum from k=1 to n of σ’1k x tk over
sum from k=1 to n of tk
Can also be expressed in terms of stiffnesses and strains
Expressing the fact that the in-plane strains must be identical for all laminae
ε’1c=ε’1k
Relationship between stiffness and compliance tensor
Reciprocal
[S]=[C]^-1 (inverse)
Determinant of stiffness matrix
In the databook. Like the normal way of getting determinant from 3x3 matrix. Ignore the sums and thicknesses if thickness is constant for each ply
Formulae for laminate moduli
For E’1c, E’2c and G’12c
All in databook
Formulae for laminate major Poisson’s ratio and 1st interaction ratio
For ν’12c and η’121
Formulae in databook
Same thing about ignoring thicknesses and sums if thickness of each ply the same
Formula for determinant of compliance matrix
Not in databook but of same form as that for stiffness matrix. Just change Cs to Ss
Angles for different lay-up sequences
Single ply: 0
Cross-ply: 0/90
Angle-ply: 30/-30
Balanced: 0/45/90/-45 or 0/60/-60
Laminate YM vs loading angle for different lay-up sequences
Balanced: stays flat in the middle
Single: starts higher but steep decline to lower and then settles about 60°
Cross: starts bit higher, then bit more lower (min at 45) then back to bit higher (symmetric).
Angle: starts but higher, increases slightly then crosses balanced at 45 and gets lower in near straight line (even lower than single)
Quasi-isotropic laminate
The case for balanced lay-up. There is in-plane anisotropy (property doesn’t vary with loading angle). Out of plane still different
Laminate shear modulus vs loading angle for different lay-up sequences
Balanced: stays flat in the middle
Cross: normal distribution curve shape starting lower then peak higher than balanced line.
Single: normal distribution start and end at same as cross but never reaches balanced line.
Angle: upside down normal distribution starting above and min below balanced (broader than for cross)
Laminate major Poisson’s ratio vs loading angle for different lay-up sequences
Balanced: stays flat in middle.
Cross: very broad normal distribution starting lower and peaking higher (crossing about 15 and 75).
Single starts just below balanced, then higher by more, crosses at 60 then lower by more (end just lower than cross).
Angle: starts on balanced, curve down lower to min at 45, crosses at 65, up to way higher by 90
Laminate interaction ratio vs loading angle for different lay-up sequences
Balanced: stays at 0 (all other start and end at 0 on y-axis too).
Cross: goes lower then rises less steep, cross at 45, higher then back down steeper to 0.
Single: goes lower than cross, rises less steep, crosses at 60, higher by tiny bit then 0.
Angle: goes higher tiny bit, crosses at 30, steeper curve down, min above for single, curve back up less steep to 0
How is stiffness matrix for kth ply represented?
[C]sub k
Multiply by strains to get stresses
Single expression for the in-plane stresses in the kth ply based on the applied stresses
[σ1k σ2k τ12k]=[C]k[U]k[Sc bar][σ1’ σ2’ τ12’]
Where U in databook is matrix multiplied by applied strains to get strain matrix without ‘
Sc bar is matrix of S bar tensors for c like S11c bar (top left)
Order of terms in the Sc bar matrix
Always S sub ijc bar (is are two number)
11 12 16
12 22 26
16 26 65
In-plane stresses in the 0° ply in the fibre direction vs loading angle for single, cross and balanced lay-up
All start high and curve down with inflexion in middle.
Balanced starts highest and ends lowest.
Single starts lowest and ends highest (of the 3).
Cross starts in middle of 2 and ends just below single.
Effect due to more plies in laminate not being in 0° orientation (for not single) so the 0° ply is a smaller proportion of the whole and therefore bears a higher stress at 0° loading
What direction is the through-thickness stress?
3 direction
What are through-thickness stresses and what can they lead to?
The internal stresses acting through the thickness of the laminate. Can lead to distortions such as bending, twisting or shearing
Which lay-up sequences are prone to bending and which remove the effect and how?
Asymmetric layup sequences prone to bending. Like 0/90 or 0/60/120.
Removed by using symmetric layup sequences. A bending force in one direction is matched by an equal and opposite bending force in the other direction resulting in no bending. E.g 0/90/90/0 or 0/60/120/120/60/0
What causes distortions in laminates?
Poisson’s contractions cause a bent shape: different contractions in different plies, transverse ply also has strong through-thickness contraction.
Thermal expansion causes saddle shape: different for longitudinal and transverse directions in each ply
Which layup sequences are prone to shearing and which reduce this?
Unbalanced layup sequences prone to shearing.
Balanced reduce shearing, these are lay-up sequences with an equal number of +-θ not including 0 and 90s
