ELASTICITY

Elasticity

Elasticity of Composites

Viscoelasticity

Elasticity of Crystals (Elastic Anisotropy)

Elasticity: Theory, Applications and NumericsMartin H. Sadd

Elsevier Butterworth-Heinemann, Oxford (2005)

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

Elastic

Plastic

Deformation

Instantaneous

Time dependent

Instantaneous

Time dependent

Recoverable

Permanent

Anelasticity

Viscoelasticity

When you pull a rubber band and release it, the band regains it original length.

It is much more difficult (requires more load) to extend a metal wire as compared to a

rubber string.

It is difficult to extend a straight metal wire; however, if it is coiled in the form of a spring,

one gets considerable extensions easily.

A rubber string becomes brittle when dipped in liquid nitrogen and breaks, when one tries

to extend the same.

A diver gets lift-off using the elastic energy stored in the diving board. If the diving board

is too compliant, the diver cannot get sufficient lift-off.

A rim of metal is heated to expand the loop and then fitted around a wooden wheel (of say

a bullock cart). On cooling of the rim it fits tightly around the wheel.

Let us start with some observations

Click here to know about all the mechanisms by which materials fail

Elasticity

ElasticityLinear

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 ~103).

The elastic behaviour of metals and ceramics is usually linear.

mn r

B

r

AU

A,B,m,n constants

m > n

Attractive Repulsive

Atomic model for elasticity

dr

dUF

11

mn r

mB

r

nAF

qp r

B

r

AF

''

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

r

Po

tenti

al e

ner

gy

(U

)

Fo

rce

(F)

Attractive

Repulsive

r

Attractive

Repulsive

r0

r0

r0 Equilibrium separation

mn r

B

r

AU

qp r

B

r

AF

''

r

Fo

rce

r0

For displacements around r0 Force-displacement curve is approximately linear

THE LINEAR ELASTIC REGION

Near r0 the red line (tangent to the F-r curve at r = r0)

coincides with the blue line (F-r) curve

Elastic modulus is the slope of the Force-Interatomic spacing curve (F-r curve), at the equilibrium interatomic separation (r0).

In reality the Elastic modulus is 4th rank tensor (Eijkl) and the curve below captures one aspect of it.

dr

dFY

2

2

dr

Ud

dr

dFY

Youngs modulus (Y / E)**

Youngs modulus is proportional to the ve slope of the F-r curve at r = r0

** Youngs modulus is a measure of the stiffness of a material

Str

ess

strain Compression

Tension

Str

ess

strain

Compression

Tension

Stress-strain curve for an elastomer

Due to efficient

filling of space

T due to uncoiling

of polymer chains

CT

CT >

Other elastic moduli

= E. E Youngs modulus

= G. G Shear modulus

hydrodynamic = K.volumetric strain K Bulk modulus

l

t

)1(2

EG

)21(3

EK

We have noted that elastic modulus is a 4th rank tensor (with 81 components in general in

3D). In practice not all these components are independent. E.g. for a cubic crystal there are

only 3 independent constants (in two index condensed notation these are E11, E12, E44). For

an isotropic material the number of independent constants is only 2.

ij ijkl klE In tensor notation

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 andits 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

Youngs 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

Youngs 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]

PropertyMaterial dependent

Geometry dependent

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 (goodductility 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 NaCl MgO Al2O3 TiC Silica glass

Youngs Modulus (GN / m2) 37 310 402 308 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 Youngs modulus can be increased

But E is a structure (microstructure) insensitive property

the increase is 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

mmffc VEVEE

Modulus parallel to the direction of the fiberes

Volume fractions

Under iso-strain conditions

I.e. parallel configuration

m-matrix, f-fibre, c-composite

Composite modulus in isostress and isostrain conditions

mmffc VEVEE Under iso-strain conditions [m = f = c]

I.e. ~ resistances in series configuration

m

m

f

f

c E

V

E

V

E

1 Under iso-stress conditions [m = f = c] I.e. ~ resistances in parallel configuration

Usually not found in practice

A BVolume fraction

Ec

Em

Ef

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

For small stress, strain is proportional (linearly varies with) strain Hookes law.

The proportionality constants are 4th order (rank) tensors.

Elasticity of Single Crystals and Anisotropy

Usage-1 Usage-2 Dimensions

S (or s) Elastic Compliance constant Elastic Modulus (N/m2)1 [stress1]

C (or c) Elastic Stiffness constant Elastic Constant (Youngs modulus) (N/m2) [stress]

C ij ijkl klC

S ij ijkl klS Hookes law

Tensor form Number of components of Cijkl (or Sijkl)

in most general case (in practice the number

of components reduces considerably):

2D: 2222 = 16, (i = 1,2)

3D: 3333 = 81 , (i = 1,2,3)

If we apply just one component of stress (say 11), this in general will produce not only 11but also other components of strain (which other components will depend on symmetry of

the crystal- which we shall consider shortly). A glimpse of this we have already seen (in the

topic on understanding stress and strain) if we pull a cylinder along y-direction, it will contract along the

x-direction 11 exists (apart from elongation in the y-direction 22).

But, this may not be the only other strain produced.

What does the existence of these Cijkl components physically mean?

11 1111 11 1112 12

1121 21 1122 22

C C

C C

This implies that the body may shear also (12 = 21 exists). This occurs due

to anisotropy in the crystal.

So (in general) a bar: might shear if pulled (in addition to elongating)

may twist if bent (in addition to bending)

may bend if twisted (in addition to twisting).

11 1111 11 1112 12 1113 13

1121 21 1122 22 1133 33

1131 31 1132 32 1133 33

C C C

C C C

C C C

2D

3D

We saw that a single stress component 11 may be related to many strains.

Similarly, many stress components may give rise to a single strain (11).

11 1111 11 1112 12

1121 21 1122 22

S S

S S

2D 3D

11 1111 11 1112 12 1113 13

1121 21 1122 22 1123 23

1131 31 1132 32 1133 33

S S S

S S S

S S S

There are 4 such equations

(for each component ij)There are 9 such equations

The total number of expected components are: 2D: 2222 = 16, (i = 1,2);

3D: 3333 = 81 , (i = 1,2,3). But, not all of these are independent.

Cijkl = Cijlk (also, Sijkl = Sijlk) Cijkl = Cjikl (also, Sijkl = Sjikl)

only 36 of the 81 components are independent (for even the most general crystal)*.

From energy arguments it can further be seen that: Cijkl = Cklij (also, Sijkl = Sklij)*

of the 36 components only 21 survive (even for the crystal of the lowest symmetry i.e. triclinic crystal).

Given that crystals can have higher symmetry than the triclinic crystal, the number of

independent components of Cijkl (or Sijkl) may be significantly reduced.

All cubic crystals (irrespective of their point group symmetry) have only 3 independent

components of Cijkl (or Sijkl) these are C1111, C1122, C2323.

This is further reduced in the case of isotropic materials to 2 independent constants (C1111,

C1122). Usually, these independent constants are written as E & .

Hence, for second rank tensor properties (like magnetic susceptibility, ij) cubic

crystals are isotropic; but, not for forth rank tensor properties like Elastic

Stiffness.

No of independent components of Cijkl and Sijkl

* The reasoning is postponed for later.

44

11 12

2CA

C C

Anisotropy factory (A)

A > 1 direction stiffest

A < 1 direction stiffest

1011 N/m2

Structure C11 C12 C44 A

BCC Li 0.135 0.114 0.088 8.4

Na 0.074 0.062 0.042 7.2

K 0.037 0.031 0.019 607

DC C 10.20 2.50 4.92 1.3

Si 1.66 0.64 0.80 1.6

Ge 1.30 0.49 0.67 1.7

NaCl type NaCl 0.485 0.125 0.127 0.7

KCl 0.405 0.066 0.063 0.37

RbCl 0.363 0.062 0.047 0.31