Structural scales and types of analysis in composite materials

Daniel & Ishai: Engineering Mechanics of Composite Materials

• Micromechanics- which fibre?- how much fibre?- arrangement of fibres? >>> LAYER PROPERTIES (strength, stiffness)

• Laminate Theory- which layers?- how many layers?- how thick? >>> LAMINATE PROPERTIES

» LAMINATE PROPERTIES >>> BEHAVIOUR UNDER LOADS (strains, stresses, curvature, failure mode…)

Polymer composites are usually laminated from several individual layers of material. Layers can be ‘different’ in the sense of:

• different type of reinforcement

• different geometrical arrangement

• different orientation of reinforcement

• different amount of reinforcement

• different matrix

Typical laminate configurations for storage tanks to BS4994

Eckold (1994)

fibre direction

E1

E2

The unidirectional ply (or lamina) has maximum stiffness anisotropy - E1»E2

0o

90o

We could remove the in-plane anisotropy by constructing a ‘cross-ply’ laminate, with UD plies oriented at 0 and 90o. Now E1 = E2.

But under the action of an in-plane load, the strain in the relatively stiff 0o layer is less than that in the 90o layer.

Direct stress thus results in bending:

This is analogous to a metal laminate consisting of one sheet of steel (modulus ~ 210 GPa) bonded to one of aluminium (modulus ~ 70 GPa):

Note the small anticlastic bending due to the different Poisson’s ratio of steel and aluminium.

P Powell: Engineering with Fibre-Polymer Laminates

In this laminate, direct stress and bending are said to be coupled.

Thermal and moisture effects also result in coupling in certain laminates - consider the familiar bi-metallic strip:

A single ‘angle-ply’ UD lamina (ie fibre orientation 0o or 90o) will shear under direct stress:

In a 2-ply laminate (, -), the shear deformations cancel out, but result in tension-twist coupling:

To avoid coupling effects, the cross-ply laminate must be symmetric - each ply must be mirrored (in terms of thickness and orientation) about the centre.

Possible symmetric arrangements would be:

0o

90o

0/90/90/0 [0,90]s

90/0/0/90 [90,0]s

Both these laminates have the same in-plane stiffness.

How do the flexural stiffnesses compare?

0o

90o

0/90/90/0 [0,90]s

90/0/0/90 [90,0]s

• The two laminates [0,90]s and [90,0]s have the same in-plane stiffness, but different flexural stiffnesses

• Ply orientations determine in-plane properties.

• Stacking sequence determines flexural properties.

• The [0,90]s laminate becomes [90,0]s if rotated. So this cross-ply laminate has flexural properties which depend on how the load is applied!

VAWT (1987)HAWT (2004)

• To avoid all coupling effects, a laminate containing an angle ply must be balanced as well as symmetric - for every ply at angle , the laminate must contain another at -.

• Balance and symmetry are not the same:

0/30/-30/30/0 - symmetric but not balanced = direct stress/shear strain coupling.

30/30/-30/-30 - balanced but not symmetric = direct stress/twist coupling.

• The [0,90] cross-ply laminate (WR) has equal properties at 0o and 90o, but is not isotropic in plane.

• A ‘quasi-isotropic’ laminate must contain at least 3 different equally-spaced orientations: 0,60,-60;0,90,+45,-45; etc.

ODE/BMT: FRP Design Guide

Carpet plot for tensile modulus of glass/epoxy laminate

proportion of plies at 90o

proportion of plies at 45o

proportion of plies at 0o

UD (0o) laminate

UD (90o) laminate

0/90 (cross-ply)E = 29 GPa

0/90/±45 (quasi-isotropic)E = 22 GPa

Classical Plate Analysis• Plane stress (through-thickness and interlaminar

shear ignored).• ‘Thin’ laminates; ‘small’ out-of plane deflections• Plate loading described by equivalent force and

moment resultants.

• If stress is constant through thickness h, Nx = h x, etc.

2h/

2h/x σ(z).dzN 2h/

2h/x dz.σ(z).zM

• Plate bending is described by curvatures kx, ky, kxy.

• The ‘curvature’ is equal to 1 / radius of curvature.

• Total plate strain results from in-plane loads and curvature according to:

Classical Plate Analysis

[ ( )] [ ] [ ] z z ko

where z is distance from centre of plate

Stress = stiffness x strain:

Giving:

Classical Plate Analysis

( ) [ ][ ] [ ][ ]z Q z Q ko

2/

2/

2/

2/

2/

2/

2/

2/

]][[]][[]][[]][[][h

h

h

h

oh

h

h

h

o zdzkQdzQzdzkQdzQN

2/

2/

2/

2/

2]][[]][[][h

h

h

h

o dzzkQzdzQM

In simpler terms:

[ ] [ ][ ] [ ][ ]

[ ] [ ][ ] [ ][ ]

N A B k

M B D k

o

o

[A] is a matrix defining the in-plate stiffness. For an isotropic sheet, it is equal to the reduced stiffness multiplied by thickness (units force/distance).

[B] is a coupling matrix, which relates curvature to in-plane forces. For an isotropic sheet, it is identically zero.

[D] is the bending stiffness matrix. For a single isotropic sheet, [D] = [Q] h3/12, so that D11=Eh3/12(1-2), etc.

• Combines the principles of thin plate theory with those of stress transformation.

• Mathematically, integration is performed over a single layer and summed over all the layers in the laminate.

Classical Laminate Analysis

n

j

h

h

h

h

oj

j

j

j

zdzkQdzQN1

1 1

]][[]][[][

• The result is a so-called constitutive equation, which describes the relationship between the applied loads and laminate deformations.

Classical Laminate Analysis

][

][.

]][[

]][[

][

][

kDB

BA

M

N o

[A], [B] and [D] are all 3x3 matrices.

• Matrix inversion gives strains resulting from applied loads:

where:

Classical Laminate Analysis

][

][.

]][[

]][[

][

][

M

N

dc

ba

k

o

]][[

]][[

]][[

]][[1

dc

ba

DB

BA

Effective Elastic Properties of the Laminate (thickness h)

22

12

11

21

332211

;;1

;1

;1

a

a

a

a

haG

haE

haE yxxyxyyx

223

113

12;

12

dhE

dhE flex

yflexx

Bending stiffness from the inverted D matrix:

2211

1)(;

1)(

dEI

dEI yx

1 Layers in the laminate are perfectly bonded to each other – strain is continuous at the interface between plies.

2 The laminate is thin, and is in a state of plane stress. This means that there can be no interlaminar shear or through-thickness stresses (yz = zx = z = 0).

3 Each ply of the laminate is assumed to be homogeneous, with orthotropic properties.

4 Displacements are small compared to the thickness of the laminate.

5 The constituent materials have linear elastic properties.

6 The strain associated with bending is proportional to the distance from the neutral axis.

Classical Laminate Analysis - assumptions

1. Define the laminate – number of layers, thickness, elastic and strength properties and orientation of each layer.

2. Define the applied loads – any combination of force and moment resultants.

3. Calculate terms in the constitutive equation matrices [A], [B] and [D].

4. Invert the property matrices – [a] = [A]-1, etc.

5. Calculate effective engineering properties.

6. Calculate mid-plane strains and curvatures.

7. Calculate strains in each layer.

8. Calculate stresses in each layer from strains, moments and elastic properties.

9. Evaluate stresses and/or strains against failure criteria.

Steps in Classical Laminate Analysis

Use of LAP software to calculate effect of cooling from cure temperature (non-symmetric laminate).