 Elasticity  Composites

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 [1010] = 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