Structural scales and types of analysis in composite materials

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Structural scales and types of analysis in
composite materials
Daniel Ishai Engineering Mechanics of
Composite Materials
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• Micromechanics- which fibre?- how much
  fibre?- arrangement of fibres?
  gtgtgt LAYER PROPERTIES
  
  (strength, stiffness)
• Laminate Theory- which layers?- how many
  layers?- how thick?
  gtgtgt LAMINATE PROPERTIES
• LAMINATE PROPERTIES
  gtgtgt BEHAVIOUR UNDER LOADS
  (strains, stresses, curvature, failure mode)

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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

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Typical laminate configurations for storage tanks
to BS4994
Eckold (1994)
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fibre direction
E2
E1
The unidirectional ply (or lamina) has maximum
stiffness anisotropy - E1E2
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90o
0o
We could remove the in-plane anisotropy by
constructing a cross-ply laminate, with UD
plies oriented at 0 and 90o. Now E1 E2.
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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
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This is analogous to a metal laminate consisting
of one sheet of steel (modulus 210 GPa) bonded
to one of aluminium (modulus 70 GPa)
P Powell Engineering with Fibre-Polymer
Laminates
Note the small anticlastic bending due to the
different Poissons ratio of steel and aluminium.
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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
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A single angle-ply UD lamina (ie fibre
orientation q ? 0o or 90o) will shear under
direct stress
q
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In a 2-ply laminate (q, -q), the shear
deformations cancel out, but result in
tension-twist coupling
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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
90/0/0/90 90,0s
0/90/90/0 0,90s
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Both these laminates have the same in-plane
stiffness. How do the flexural stiffnesses
compare?
0o
90o
90/0/0/90 90,0s
0/90/90/0 0,90s
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• The two laminates 0,90s and 90,0s have the
  same in-plane stiffness, but different flexural
  stiffnesses
• Ply orientations determine in-plane properties.
• Stacking sequence determines flexural properties.
• The 0,90s laminate becomes 90,0s if rotated.
  So this cross-ply laminate has flexural
  properties which depend on how the load is
  applied!

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VAWT (1987)
HAWT (2004)
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• To avoid all coupling effects, a laminate
  containing an angle ply must be balanced as well
  as symmetric - for every ply at angle q, the
  laminate must contain another at -q.
• Balance and symmetry are not the
  same0/30/-30/30/0 - symmetric but not balanced
  direct stress/shear strain coupling.30/30/-30/
  -30 - balanced but not symmetric direct
  stress/twist coupling.

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• 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,-600,90,45,-45 etc.

ODE/BMT FRP Design Guide
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UD (0o) laminate
proportion of plies at 90o
proportion of plies at 0o
proportion of plies at 45o
UD (90o) laminate
Carpet plot for tensile modulus of glass/epoxy
laminate
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0/90 (cross-ply)E 29 GPa
0/90/45 (quasi-isotropic)E 22 GPa
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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
  sx, etc.

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Classical Plate Analysis

• 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

where z is distance from centre of plate
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Classical Plate Analysis

• Stress stiffness x strain
• Giving

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• In simpler terms

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
D11Eh3/12(1-n2), etc.
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Classical Laminate Analysis

• 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.

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Classical Laminate Analysis

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

A, B and D are all 3x3 matrices.
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Classical Laminate Analysis

• Matrix inversion gives strains resulting from
  applied loadswhere

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Effective Elastic Properties of the Laminate
(thickness h)

Bending stiffness from the inverted D matrix
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Classical Laminate Analysis - assumptions

• 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
  (tyz tzx sz 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.

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Steps in Classical Laminate Analysis

• 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.

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