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Structural scales and types of analysis in composite materials

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## 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**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**E2 E1 The unidirectional ply (or lamina) has maximum stiffness anisotropy - E1»E2**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.**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): P Powell: Engineering with Fibre-Polymer Laminates Note the small anticlastic bending due to the different Poisson’s ratio of steel and aluminium.**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**q 0o or 90o) will shear under direct stress: q**In a 2-ply laminate (q, -q), 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 90/0/0/90 [90,0]s 0/90/90/0 [0,90]s**Both these laminates have the same in-plane stiffness. How**do the flexuralstiffnesses compare? 0o 90o 90/0/0/90 [90,0]s 0/90/90/0 [0,90]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 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.**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**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**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 sx, etc.**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**Classical Plate Analysis**Stress = stiffness x strain: Giving:**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.**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.**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.**Classical Laminate Analysis**• Matrix inversion gives strains resulting from applied loads:where:**Effective Elastic Properties of the Laminate (thickness h)**Bending stiffness from the inverted D matrix:**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.**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.**Use of LAP software to calculate effect of cooling from cure**temperature (non-symmetric laminate).