Introduction to elastic properties

Elasticity basics: stress and strain

Material deforms when subjected to an external force. If the the shape deformation is reversible, the material is elastic, e.g. like a spring.
Here it is assumed that the material is isotropic.

Stress σ is defined as

σ = F/A,

where F is the force and A is the area.

Strain ε is obtained as

ε = ΔL/L.

 

 

Poisson ratio (isotropic material)

Poisson ratio gives information on the deformation of the sample during tensile testing. Poisson ratio is defined as -1 times the ratio of lateral to applied elastic strains.


Poisson ratio = -(ΔD/D) / (ΔL/L)

 

Poisson ratios for solids

Isotropic materials: one Poisson ratio

Example. Auxetic foam.


Anisotropic material: Poisson ratios form a tensor.

References

For samples containing crystallites which are oriented preferably parallel to the stretching direction,  Poisson ratio of the crystallites has been determined from the increase/decrease of the distance of lattice planes during tension using combined x-ray diffraction measurements and tensile testing.

References

 

Hooke's law

English physicist Robert Hooke (1635–1703)

Let F be the force  and A the cross-sectional area of the sample. The stress is written as

 σ = F/A

and the strain as

ε = ΔL/L.

Young’s modulus may be written as

E = σ/ ε

In other words, it is

E = (F/A) (L/ΔL).
 

If E is constant,

 F/A = E ΔL/L.

http://en.wikipedia.org/wiki/Stress%E2%80%93strain_curve


material

Young modulus GPa

steel

200

aluminium

70


http://en.wikipedia.org/wiki/Young%27s_modulus

Examples.


Stress-strain curves of never-dried birch wood


Stress-strain curves of norway spruce wood. A earlywood, juvenile, B latewood juvenile, C earlywood mature, D latewood mature

2. M. Peura et al. Wood Sci. Technol., 41:565-583, 2007
3. M. Peura et al. Biomacromolecules, 7:1521-1528, 2006.

Hooke’s law for an anisotropic system

For isotropic samples. the law is σ = E ε where E is the modulus of elasticity.

For anisotropic sample

σij = Σ cijkl εkl

Here  cijkl  are elastic coefficients. The stress tensor σij is symmetric (units of pressure).

The strain tensor εkl is also symmetric (dimensionless).

Orthotropic material

6 stress and 6 strain coefficients are needed. Here Cartesian base e1, e2, e3 is used.

If strain components are collected in a column vector ε = [ε11 ε22 ε33 ε12 ε13 ε23] and
stress components to another column vector σ = [σ11 σ22 σ33 σ12 σ13 σ23] one obtains
a linear equation

ε = Cσ.

The elastic components form then a matrix.

Table. Poisson ratios for some materials.

Material

Crystalline

macroscopic

reference

spruce (Picea abies) earlywood

0.3-0.8 (Peura)

0,53 ± 0,18

Sinn et al. 2001

spruce (Picea abies) latewood

-

0,70 ± 0,15

Sinn et al. 2001

Ramie cellulose

0,38 ± 0,04 (Ibeta)

-

Nakamura et al. 2004

Cellulose (MCC)

0.30

 

Roberts et al. 1994

Kevlar 29, 49

0,31

 

Nakamae et al. 1992

Kevlar 49

 

0,3

Kostar et al. 2000

Kevlar 49, braided

 

» 1

Kostar et al. 2000

Carbon (celion 4000)

 

0,28

Kostar et al. 2000

α-C3N4

-0.0527, -0.0252, -0.0308

 

Guo et al. 1995

References

Mechanical response at molecular level

The elastic properties of solids can be related to the intermolecular forces between the atoms forming the solids.

Example. Assume that atoms are in a 2-D square lattice with lattice constants a, a, π/2. Atoms are assumed to be at lattice points, thus the interatomic separation is a. The sample is stretched in one direction and interatomic spacing increases from a to r.

The interaction between the atoms is represented by springs.
 

Force for each spring  is

F = k(r-a),

where k is the effective spring constant for an interatomic bond.
The tensile stress is 

k(r-a)/a2,

where a2 is the area per spring. Tensile strain on the sample

ε = (r-a)/a

Young modulus E = (stress)/(strain) = k/a.

Young’s modulus in terms of interatomic potential

Interatomic potential

U(r) = U(a) + ½ (r-a)2 (d2U/dr2)(a)+ …

Thus

U(r) = constant + ½ k(r-a)2,

where k is the  spring constant k = (d2U/dr2)(a).

Generally one may write

U(r) = b f(r/a)

The constant b represents the bond energy and f is a function.

The minimum value of U is –b at r = a.

The function f(x) is dimensionless and f(1) = -1. In terms of it the
spring constant is

 k = b/a2 f’’(1)

and Young's modulus

E = f’’(1)/a3 b.

Young modulus depends on the bond energy.

Creep

Creep is the tendency of a solid material to slowly move or deform permanently under the influence of stresses.

Time dependent strain that accompanies an applied constant stress
long term exposure to levels of stress that are below the yield
Often in temperatures near its melting point.  Real solids can flow if a constant stress is applied to them.

Reasons:



Tests

Cyclic tests

Creep

Stress relaxation tests

 



Equipment at Department of Physics
http://www.tiniusolsen.com/products/bench-machines/bench-h5k-t.html

D. Loidl (a), H. Peterlik (a), O. Paris (b), M. Burghammer (c). Local Nanostructure of Single Carbon Fibres during Bending Deformation.
http://www.esrf.eu/UsersAndScience/Publications/Highlights/2004/SCM/SCM7

Ultrasonics

Ultrasound waves are mechanical waves that require a physical medium in order to support their propagation. The particles of the medium oscillate about their equilibrium position.

Cyclic sound pressure (frequency> 20 kHz)

Tool for nondestructive testing and imaging.

Longitudinal wave: the displacement will be in the direction of propagation of the wave
 


Shear waves: the displacement of the particles is at right angles to the propagation direction


The particle displacement from the equilibrium position is denoted by x and the frequency of the sound by f.
 

Velocity will be  u = ∂x/ ∂t and acceleration a = ∂u/ ∂t.

If the speed of sound is denoted by co  and angular frequency by w, then |u| = w|x| and |a| = w|u|.

The medium density of the material varies: If the equilibrium density is denoted by ρ0, the
pressure in the medium may be written as: p = p0 + ρ0 c0 u
 

Ultrasonic waves transport energy in the form of kinetic energy (particle motion) and potential energy (fluid compression). The acoustic intensity I of a sound wave is defined as the average rate of flow of energy through a unit area normal to the direction of propagation. For the case of a plane plane wave

I = ½ P2 (ρ0 c0 )

where P is the pressure amplitude of the wave
 

Attenuation

This attenuation is due to either absorption or scattering. Absorption is a mechanism that represents that portion of ultrasonic wave that is converted into heat, and scattering can be thought of as that portion of the wave, which changes direction.
Lambert-Beer law

I = I0 exp(-2az),

where I0 is the initial intensity, a the absorption coefficient and x the distance.
 

Loss of energy

Linearity or non-linearity?

Propagation of acoustic waves is often assumed linear. The pressure obeys the wave equation

(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2)p = 1/c2 ∂2p/∂t2

In linear acoustics it is assumed the particles of the medium vibrate about their equilibrium position with no net flow.

(Bulk movement of fluid away from the transducer in the direction of propagation known as acoustic streaming.)

Stiffness modulus

Ultrasound velocity v
The density of the sample ρ
Stiffness modulus E = v2 ρ
 

References