Micromechanics- Classical Laminate Theory 3

Question 1

Kirchoff assumptions for laminates

Answer 1

Flat and thin laminate.
No through thickness stresses (i.e overall plane stress)
Edge effects can be neglected

Question 2

How to number plies

Answer 2

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

Question 3

Notation for referring to a composite, thickness, individual lamina

Answer 3

Subscript c refers to a composite of n laminae.
Thickness of each lamina is t sub k.
k refers to one lamina

Question 4

Angles between load direction and principal material direction notation for laminate

Answer 4

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.

Question 5

What can the average stress for the laminate in the 1’ direction be expressed via?

Answer 5

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

Question 6

Expressing the fact that the in-plane strains must be identical for all laminae

Answer 6

ε’1c=ε’1k

Question 7

Relationship between stiffness and compliance tensor

Answer 7

Reciprocal
[S]=[C]^-1 (inverse)

Question 8

Determinant of stiffness matrix

Answer 8

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

Question 9

Formulae for laminate moduli

Answer 9

For E’1c, E’2c and G’12c
All in databook

Question 10

Formulae for laminate major Poisson’s ratio and 1st interaction ratio

Answer 10

For ν’12c and η’121
Formulae in databook
Same thing about ignoring thicknesses and sums if thickness of each ply the same

Question 11

Formula for determinant of compliance matrix

Answer 11

Not in databook but of same form as that for stiffness matrix. Just change Cs to Ss

Question 12

Angles for different lay-up sequences

Answer 12

Single ply: 0
Cross-ply: 0/90
Angle-ply: 30/-30
Balanced: 0/45/90/-45 or 0/60/-60

Question 13

Laminate YM vs loading angle for different lay-up sequences

Answer 13

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)

Question 14

Quasi-isotropic laminate

Answer 14

The case for balanced lay-up. There is in-plane anisotropy (property doesn’t vary with loading angle). Out of plane still different

Question 15

Laminate shear modulus vs loading angle for different lay-up sequences

Answer 15

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)

Question 16

Laminate major Poisson’s ratio vs loading angle for different lay-up sequences

Answer 16

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

Question 17

Laminate interaction ratio vs loading angle for different lay-up sequences

Answer 17

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

Question 18

How is stiffness matrix for kth ply represented?

Answer 18

[C]sub k
Multiply by strains to get stresses

Question 19

Single expression for the in-plane stresses in the kth ply based on the applied stresses

Answer 19

[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)

Question 20

Order of terms in the Sc bar matrix

Answer 20

Always S sub ijc bar (is are two number)
11 12 16
12 22 26
16 26 65

Question 21

In-plane stresses in the 0° ply in the fibre direction vs loading angle for single, cross and balanced lay-up

Answer 21

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

Question 22

What direction is the through-thickness stress?

Answer 22

3 direction

Question 23

What are through-thickness stresses and what can they lead to?

Answer 23

The internal stresses acting through the thickness of the laminate. Can lead to distortions such as bending, twisting or shearing

Question 24

Which lay-up sequences are prone to bending and which remove the effect and how?

Answer 24

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

Question 25

What causes distortions in laminates?

Answer 25

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

Question 26

Which layup sequences are prone to shearing and which reduce this?

Answer 26

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