ELASTICITY ELASTICITY COMPOSITES
Elasticity: Theory, Applications and Numerics Martin H. Sadd Elsevier Butterworth-Heinemann, Oxford (2005)
Recoverable
Instantaneous
Elastic Time dependent
Anelasticity
Deformation Instantaneous Plastic Permanent
Time dependent Viscoelasticity
Elasticity
Linear Elasticity Non-linear
Atomic model for elasticity
dU F =− dr
A B U =− n + m r r Attractive
Repulsive
A,B,m,n → constants m>n
nA mB F = − n +1 + m +1 r r A' B ' F =− p + q r r
A' B ' F =− p + q r r
Repulsive
Repulsive
r →
Force (F) →
Potential energy (U) →
A B U =− n + m r r
r0 r →
r0 Attractive r0
Equilibrium separation
Attractive
Near r0 the red line (tangent to the F-r curve at r = r0) coincides with the blue line (F-r) curve
Force →
r →
r0 For displacements around r0 → Force – displacement curve is approximately linear THE LINEAR ELASTIC REGION
Young’s modulus (Y / E)**
Stress →
Young’s modulus is :: to the –ve slope of the F-r curve at r = r0
Compression
dF Y ∝− dr 2
Tension
dF d U Y ∝− = 2 dr dr
strain →
** Young’s modulus is not an elastic modulus but an elastic constant
Stress-strain curve for an elastomer
Stress →
ε
Tension
C
strain → ε Due to efficient filling of space
Compression
ε
ε T due to uncoiling of polymer chains
T
>ε
C
T
Other elastic modulii σ = E.ε
E → Young’s modulus
τ = G.γ
G → Shear modulus
σ
hydrodynami
= K.volumetric strain
K → Bulk modulus
εt ν =− εl E G= 2(1 + ν ) E K= 3(1 − 2ν )
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 →
Li
Be
B
Cdiamond
Cgraphite
Atomic number (Z)
3
4
5
6
6
Young’s Modulus (GN / m2)
11.5
289
440
1140
8
Down the row →
Cdiamond
Si
Ge
Sn
Pb
Atomic number (Z)
6
14
32
50
82
Young’s Modulus (GN / m2)
1140
103
99
52
16
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 [1010] = 950 GPa, E [0001] = 8 GPa ► Two kinds of ordering along two directions E.g. Decagonal QC E [100000] ≠ E [000001]
Material dependence
Elastic modulus
Property Geometry dependence Elastic modulus in design 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 Ionic solids →
NaCl
Young’s Modulus (GN / m2)
37
MgO Al2O3 310
402
TiC
Silica glass
308
70
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
Increasing the modulus of a material METALS ► By suitably alloying the Young’s modulus can be increased ► But E is a structure (microstructure) insensitive property ⇒ the increase is α fraction added ► TiB2 (~ 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
COMPOSITES
Laminate composite
Aligned fiber composite
Particulate composite
Modulus parallel to the direction of the fiberes
Ec = E f V f + EmVm Volume fractions
Under iso-strain conditions I.e. parallel configuration m-matrix, f-fibre, c-composite
Composite modulus in isostress and isostrain conditions
E c = E f V f + E mVm
Under iso-strain conditions [ε
V f Vm 1 = + Ec E f Em
Under iso-stress conditions [σ
Ec →
Iso
A
= ε f = ε c]
I.e. ~ resistances in series configuration m
= σ f = σ c]
I.e. ~ resistances in parallel configuration Usually not found in practice n i a r st
Iso
Em
m
f Volume fraction →
Ef ss e str
B
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