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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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fibre direction E1E1 E2E2 The unidirectional ply (or lamina) has maximum stiffness anisotropy - E 1 »E 2

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0o0o 90 o We could remove the in-plane anisotropy by constructing a cross-ply laminate, with UD plies oriented at 0 and 90 o. Now E 1 = E 2.

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But under the action of an in-plane load, the strain in the relatively stiff 0 o layer is less than that in the 90 o 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): Note the small anticlastic bending due to the different Poissons ratio of steel and aluminium. P Powell: Engineering with Fibre-Polymer Laminates

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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 0 o or 90 o ) will shear under direct stress:

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In a 2-ply laminate (, - ), 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: 0o0o 90 o 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. How do the flexural stiffnesses compare? 0o0o 90 o 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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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, 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.

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The [0,90] cross-ply laminate (WR) has equal properties at 0 o and 90 o, 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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Carpet plot for tensile modulus of glass/epoxy laminate proportion of plies at 90 o proportion of plies at 45 o proportion of plies at 0 o UD (0 o ) laminate UD (90 o ) 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, N x = h x, etc.

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Plate bending is described by curvatures k x, k y, k xy. The curvature is equal to 1 / radius of curvature. Total plate strain results from in-plane loads and curvature according to: Classical Plate Analysis where z is distance from centre of plate

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Stress = stiffness x strain: Giving: Classical Plate Analysis

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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] h 3 /12, so that D 11 =Eh 3 /12(1- 2 ), etc.

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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

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The result is a so-called constitutive equation, which describes the relationship between the applied loads and laminate deformations. Classical Laminate Analysis [A], [B] and [D] are all 3x3 matrices.

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Matrix inversion gives strains resulting from applied loads: where: Classical Laminate Analysis

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Effective Elastic Properties of the Laminate (thickness h) Bending stiffness from the inverted D matrix:

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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

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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

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

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