Cubes

TENSORS: Stress and Strain

• To fully define the state of strain or stress in a material requires a magnitude and 2 directions. This is because for example in the case of stress, the direction of the applied force and the direction of the area upon which is applied need to be specified, i.e. Whether the stress is tensile or shear, and likewise for strain.

• For this reason stress and strain are 2nd rank tensors. The relationship between them is in general a complex one, described by the stiffness (or inversely the compliance) tensor of 4th rank

• Strain is often measured using strain gauges, small objects bonded to a sample surface with a resistance that changes significantly with even very small strains. This change can then easily be measured and related back to the sample strain.

TENSORS: Electric Field and Polarisation

• Electric polarisation is a measure of the density of dipole moments in a dielectric material. In an electric field, positive and negative charges in the material will separate, and it will become polarised.

• Electric field and polarisation arc both vectors (1st rank tensors). Due to the anisotropy in a general crystalline solid, the polarisation may not be parallel to the applied field. As such, each component of the polarisation can depend on each component of the electric field vector, and so the tensor describing the relationship between them has 9 coefficients and is of 2nd rank, and is know as the electric susceptibility.

• Polarisation and susceptibility are typically measured using optical techniques such as ellipsometry.

TENSORS: The Piezoelectric Effect

• In some special non-centrosymmetric materials, when a stress is applied a polarisation is induced, and conversely when an electric field is applied a strain is induced. This is known as the piezoelectric effect, and it is another material property that can be described by tensors.

• The piezoelectric tensor relates a 2nd rank and 1st rank tensor, and so in general relates 3 independent directions, and is a 3rd rank tensor. Similarly with the other matter tensors discussed, the piezoelectric tensor can be simplified for systems with some symmetry, for example in the case of quartz (trigonal), there are only 2 independent coefficients in the tensor.

• Piezoelectricity has countless applications, for example in ultrasonics, scanning probe microscopes, and in everyday devices such as cigarette lighters.

MECHANICAL

Stress
(MPa)
i

Young's Modulus (GPa)
i Poisson's Ratio
Shear Modulus (GPa))

Strain
(millistrain)
i

1x

• Pick a material or choose custom to enter your own values for the elastic moduli, and see how the strain varies with the inputted stress tensor.

• For the stiffest materials such as diamond, it is recommended to use some strain magnification so that the effect of an applied stress can be seen.

• If you wish to know more about any of the individual tensors, click the little information icons next to them.

Material

Strain
Magnification

INFO

x

ERROR

x

ELECTRICAL

Electric
Field
i (Vm⁻¹)

Electric
Susceptibility
i (Fm⁻¹)

Electric
Polarisation
i (E₀Cm⁻²)

• Again, pick a material or enter custom values for the electric susceptibility, and see how the resultant polarisation is affected by the field applied.

Material:

INFO

x

ERROR

x

MECHANICAL

Stress
(MPa)

Compliance
i (GPa⁻¹)

Strain
(millistrain)

ELECTRICAL

Electric
Field
(Vm⁻¹)

Electric
Polarisation
(E₀ Cm⁻²)

Electric Susceptibiliy
i (Fm⁻¹)

PIEZOELECTRICAL

d₁₁
d₁₄
i

(×10⁻¹² CN⁻¹)

• Using quartz as an example, we shall investigate the piezoelectric effect.

• Try inputting values into the stress and electric field tensor to see the effect on the polarisation.

INFO

x

ERROR

x