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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? >>> 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…)

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

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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 Poisson’s 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:**

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

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**Both these laminates have the same in-plane stiffness**

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

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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!

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HAWT (2004) VAWT (1987)

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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 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.

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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,-60; 0,90,+45,-45; etc. ODE/BMT: FRP Design Guide

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**proportion of plies at 90o**

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 D11=Eh3/12(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 loads: where:

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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).**

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