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ELASTICITY Elasticity Elasticity of Composites Viscoelasticity Elasticity: Theory, Applications and Numerics Martin H. Sadd Elsevier Butterworth-Heinemann, Oxford (2005)

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Revise modes of deformation, stress, strain and other basics (click here) before starting this chapter etc.

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What kind of mechanical behaviour phenomena does one have to understand? Elasticity Plasticity Fracture Fatigue Mechanical Behaviour Creep Elongation at constant load at High temperatures Phenomenologically mechanical behaviour can be understood as in the flow diagram below. Multiple mechanisms may be associated with these phenomena (e.g. creep can occur by diffusion, grain boundary sliding etc.). These phenomena may lead to the failure of a material. Note: above is a broad classification for convenience. E.g. Creep is also leads to plastic deformation! Recoverable deformation Permanent deformation Propagation of cracks in a material Oscillatory loading

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Elastic Plastic Deformation Instantaneous Time dependent Instantaneous Time dependent Recoverable Permanent Anelasticity Viscoelasticity

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Elasticity Linear Non-linear E.g. Al deformed at small strains E.g. deformation of an elastomer like rubber Elastic deformation is reversible deformation- i.e. when load/forces/constraints are released the body returns to its original configuration (shape and size). Elastic deformation can be caused by tension/compression or shear forces. Usually in metals and ceramics elastic deformation is seen at low strains (less than ~10 –3 ). The elastic behaviour of metals and ceramics is usually linear.

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A,B,m,n constants m > n Attractive Repulsive Atomic model for elasticity Let us consider the stretching of bonds (leading to elastic deformation). Atoms in a solid feel an attractive force at larger atomic separations and feel a repulsive force (when electron clouds overlap too much) at shorter separations. (At very large separations there is no force felt). The energy and the force (which is a gradient of the energy field) display functional behaviour as below. The plots of these functions is shown in the next slide

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r Potential energy (U) Force (F) Attractive Repulsive r Attractive Repulsive r0r0 r0r0 r0r0 Equilibrium separation

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r Force r0r0 For displacements around r 0 Force-displacement curve is approximately linear THE LINEAR ELASTIC REGION Near r 0 the red line (tangent to the F-r curve at r = r 0 ) coincides with the blue line (F-r) curve Elastic modulus is the slope of the Force-interatomic spacing curve (F-r curve), at the

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Youngs modulus (Y / E) ** Youngs modulus is :: to the –ve slope of the F-r curve at r = r 0 ** Youngs modulus is not an elastic modulus but an elastic constant Stress strain Compression Tension

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Stress strain Compression Tension Stress-strain curve for an elastomer Due to efficient filling of space T due to uncoiling of polymer chains C T C T >

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Other elastic moduli = E. E Youngs modulus = G. G Shear modulus hydrodynami = K.volumetric strainK Bulk modulus

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Bonding and Elastic modulus Materials with strong bonds have a deep potential energy well with a high curvature high elastic modulus Along the period of a periodic table the covalent character of the bond and its strength increase systematic increase in elastic modulus Down a period the covalent character of the bonding in Y On heating the elastic modulus decrease: 0 K M.P, 10-20% in modulus Along the period LiBeBC diamond C graphite Atomic number (Z)34566 Youngs Modulus (GN / m 2 ) Down the row C diamond SiGeSnPb Atomic number (Z) Youngs Modulus (GN / m 2 )

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Anisotropy in the Elastic modulus In a crystal the interatomic distance varies with direction elastic anisotropy Elastic anisotropy is especially pronounced in materials with two kinds of bonds E.g. in graphite E [10 10] = 950 GPa, E [0001] = 8 GPa Two kinds of ordering along two directions E.g. Decagonal QC E [100000] E [000001]

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Property Material dependence Geometry dependence Elastic modulus Stiffness of a material is its ability to resist elastic deformation of deflection on loading depends on the geometry of the component. High modulus in conjunction with good ductility should be chosen (good ductility avoids catastrophic failure in case of accidental overloading) Covalently bonded materials- e.g. diamond have high E (1140 GPa) BUT brittle Ionic solids are also very brittle Elastic modulus in design Ionic solids NaClMgOAl 2 O 3 TiCSilica glass Youngs Modulus (GN / m 2 )

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METALS First transition series good combination of ductility & modulus (200 GPa) Second & third transition series even higher modulus, but higher density POLYMERS Polymers can have good plasticity but low modulus dependent on the nature of secondary bonds- Van der Walls / hydrogen presence of bulky side groups branching in the chains Unbranched polyethylene E = 0.2 GPa, Polystyrene with large phenyl side group E = 3 GPa, 3D network polymer phenol formaldehyde E = 3-5 GPa cross-linking

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Increasing the modulus of a material METALS By suitably alloying the Youngs modulus can be increased But E is a structure (microstructure) insensitive property the increase is fraction added TiB 2 (~ spherical, in equilibrium with matrix) added to Fe to increase E COMPOSITES A second phase (reinforcement) can be added to a low E material to E (particles, fibres, laminates) The second phase can be brittle and the ductility is provided by the matrix if reinforcement fractures the crack is stopped by the matrix

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COMPOSITES Laminate composite Aligned fiber composite Particulate composite Modulus parallel to the direction of the fiberes Volume fractions Under iso-strain conditions I.e. parallel configuration m-matrix, f-fibre, c-composite

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Composite modulus in isostress and isostrain conditions Under iso-strain conditions [ m = f = c ] I.e. ~ resistances in series configuration Under iso-stress conditions [ m = f = c ] I.e. ~ resistances in parallel configuration Usually not found in practice A B Volume fraction E c EmEm EfEf Isostress Isostrain For a given fiber fraction f, the modulii of various conceivable composites lie between an upper bound given by isostrain condition and a lower bound given by isostress condition f Voigt averaging Reuss averaging

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