Analysis of laminates after initial failure

Accounting for the modificationof stiffness in the failed layers

Common practice: set all lamina properties to zero when failure occurs(gives a conservative estimate of the load carrying capacity)

for the failed ply

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Example: 5 mm symmetric cross‐ply constructed from 15 identical laminae9 li 0° d 6 li 90° l d i i N9 plies at 0° and 6 plies at 90°, load in traction Nx

Step 1: Failure of the 90° plies

Extensional stiffness matrix before failure of the 90° plyExtensional stiffness matrix before failure of the 90° ply

Step 2: failure of the first ply (maximum strain theory)

Failure stresses

Calculation of the failure strains

Failure strains

The 90° ply will fail when x=0.00153

The load Nx leading to the failure of the 90° ply is solution of 

Step 3: Behaviour after failure of the first ply

After the failure of the 90° ply, 

Q = 0

Step 3: Behaviour after failure of the first ply

Q 90° = 0

The failure occurs for 

Flowchart for laminatestrength analysis

Short fiber composites

•Convenient for complex geometries•Convenient for complex geometries•Can be mixed with liquid resin and used in injection molding(in injection moulding, the final orientation of the fibers depends on the flow in the mould)•Random oriented fibers are nearly isotropic•Not suitable for critical structures (less strength and stiffness), 

but this may change with carbon nanotubes

Glass fiber reinforced nylon with random fiber orientation (Agarwall, Fig 4.11) 1

Stress transfer theory for aligned discontinuous fibers composites

Matrix shear deformation in a representative volume (Gibson)

Equilibrium along z:

0

The stress at the fiber end is neglected

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We assume a rigid‐plastic behaviour of the matrix.(the interface shear stress is constant y)For short fibers the fiber stress is linearFor short fibers, the fiber stress is linear

And the maximum occurs at the middle of the fiber

However, the fiber stress cannot exceed the stress thatwould occur in a continuous fiber , for whichThis corresponds to

The load transfer length is the Minimum length which allowsMinimum length which allowsto reach this value

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For a fixed length l, if one increases the composite stress cThe fiber stress cannot exceed the fiber ultimate stress The fiber stress cannot exceed the fiber ultimate stress fuThe critical fiber length is that which allows to reach fu:

lc is also called« Ineffective length »

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Numerical results: 1. Elastic analysis

r<0: friction forces ll l d f iallow load transfer in 

case of interface failure)

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Numerical results: 2. Elastic‐plastic  analysis

Interfacial shear stress near fiberend is not constant, because of 3‐D

yield stress criterion is used.

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Modulus of aligned discontinuous fiber composites

Halpin‐Tsai (curve fitting on numerical solutions)

(Same as for transverse modulus of continuous fibers with =2l/d ) 

(Same as for transverse modulus of continuous circular fibers : =2 )

Variation of EL of aligned short fibers as a functionof the aspect ratio l/d, for differnet values of Ef/Em

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Asymptotic value of EL for large l/dL

(Halpin‐Tsai converges towards the rule of mixture)

Modulus of short fiber composites with randomly oriented fibers

Empirical formulae:

With EL and ET corresponding to aligned short fibers withthe same aspect ratio l/d and the same volume fraction Vf

Since the material is isotropic, 8

1. Aligned short fibers (Halpin‐Tsai)

2. Random fiber orientation

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2. Random fiber orientation

Strength of aligned short‐fiber composites

If l>lc, the fibers will reach the failure stress fu

For very small fiber volume fractions: the value for continuous fibers

is further lowered by large stress concentrations

Strength of randomly oriented short fibers

Strength predictions obtained by replacing the randomly oriented composite by a quasi‐isotropicStrength predictions obtained by replacing the randomly oriented composite by a quasi isotropiccontinuous laminate with the same volume fraction correlate well with experiments.

Equivalent quasi‐isotropic laminate:Analysis up to the failure of all plies

Maximum strain theory10

Sandwich structures

Plates with similar flexural rigidity EI

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Simplified stress analysis in sandwich structures

Bending stresses are in the skin only, and uniform. Shear stresses are in the core only, and uniform.The bending deflection is dependent on the tensile and compressive moduli in the skin material.The shear deflection is dependent on the shear modulus in the core

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The shear deflection is dependent on the shear modulus in the core.

Sandwich panel failure modes (Hexcel composites)

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14http://www.hexcel.com/Resources/DataSheets/Brochure‐Data‐Sheets/Honeycomb_Sandwich_Design_Technology.pdf