Laminates and Laminate Theory

Question 1

What are the diffference between a layer and a ply in a laminate?

Answer 1

• Ply: Always refers to a single sheet of composite material.
• Layer: Can refer to a single ply or a group of plies with the same orientation.

Question 2

What is a symmetric laminate?

Answer 2

A symmetric laminate has perfect symmetry about the mid-plane. Some examples:
* [ 0 / 90 / 0]
* [0 / 90 / -45 / 45 / 45 / -45 / 90 /0]

The B matrix is 0 for symmetric laminates.

Question 3

What is a balanced laminate?

Answer 3

A balanced laminate is such that for every layer with orientation θ, there exists another layer with the same material and same thickness and with an orientation -θ (other that 0 or 90)
Example:
* [-45 / 45 / -45 / 45]
* [0 / 90 / 0]

A_xs = A_ys = 0 for a balanced laminate

Question 4

What needs to be defined for a laminate layup?

Answer 4

• material properties
• thickness
• orientation angle

Question 5

What are the assumptions for the laminate theory?

Answer 5

1. The layers are perfectly bonded
2. Each layer is homogeneous
3. Individual layer properties can be isotropic, transverse isotropic or orthotropic
4. Each layer is in a state of plane stress
5. The laminate deforms according to the Kirchoff assumptions

Question 6

What are the Kirchoff assumptions?

Answer 6

• Normals to the midplane remain straight and normal to the deformed midplane after deformation
• Normals to the midplane do not change length

Question 7

What is the meaning of homogeneous layers?

Answer 7

A homogeneous layer has a set of properties that does not vary across the plane or through the thickness

Question 8

Can a laminate be homogeneous?

Answer 8

No, a laminate cannot be considered homogeneous because it is composed of multiple layers with different properties and orientations

Question 9

Explain the difference between plane stress and plane strain

Answer 9

• Plane Stress: Applicable to thin structures, assumes zero out-of-plane stresses.
• Plane Strain: Applicable to long structures, assumes zero out-of-plane strains.

Question 10

Is it possible to be in a state of plane stress and plane strain simultaneously? If so, how?

Answer 10

No, due to the contradictory nature of their definitions:
* Plane stress: σ_z = 0 ->material is free to deform in the z-direction, thus e_z cannot be 0.
* Plane strain: e_z = 0 -> material is constrained in the z-direcion, and it has to be stresses in z-direction to prevent deformation.

Question 11

What is the strain in position z (Kirchhoff assumption) in matrix form?

Answer 11

Question 12

What are the equations for the kurvatures k_x, k_y and k_xy in Kirchhoff assumption?

Answer 12

Question 13

What are the equations for the strains in the Kirchoff assumption?

Answer 13

Question 14

What is the sign of the k_x-value for this curve?

Answer 14

k_x > 0

Question 15

Which k-value
is responsible for the shape of this curve?

Answer 15

k_xy ≠ 0 gives the “twisting” shape.
(here k_xy>0)

Question 16

What is the definition of a curvature?

Answer 16

Curvature (κ) is a measure of how sharply a curve bends.
κ = 1/radius

Question 17

What are the relations between laminate loads/moments and midplane strans/curvatures?

Answer 17

Question 18

What is the unit of in-plane laminate forces, N?

Answer 18

Force per unit length [N/m]

Question 19

What is the unit of laminate moments?

Answer 19

Force multiplied by length (F*m)

Question 20

Describe the meaning and consequences of a quazi-isotropic laminate.

Answer 20

• Quasi-isotropic laminates behave nearly isotropically in their plane by having plies oriented in multiple directions.
• They offer uniform in-plane mechanical properties and uniform stress distribution, making them suitable for applications with varying or uncertain load directions.

Question 21

what are the effective in-plane elastic properties for symmetric and balanced laminates?

Answer 21

Question 22

How is the effective in-plane elastic moduli E_x for a symmetric and balanced laminate found?

Answer 22

Finding E_x:
1. Consider the loadcase N_x ≠0 , N_y = 0, and calculate N=Aε^0
2. The stress σ_x = Nx/h
3. E_x = σ_x / ε_x

Question 23

How to find the poissons ratio v_xy if the midplane-strains are known?

Answer 23

Question 24

What is the effective flexural stiffness along x-direction (symmetric, balanced, no bend-twist coupling)

Answer 24

Question 25

What primary idea behind sandwich structures?

Answer 25

To increase the flexural stiffness without adding much weight to the laminate. This is achieved by using a low-density core material sandwiched between two face sheets.

Question 26

What simplifications can be made for the mechanics of sandwich beams?

Answer 26

• The thickness of the face sheets is much less than the thickness of the core, i.e: t_f &laquo_space;t_c
• The modulus of the core is much less than the effective modulus of the face sheets, i.e: E_c &laquo_space;E_f , and E_c can then be neglected when computing the bending stiffness