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A guide to tensor usage in Materials Science

The properties in circles are known as **field tensors**,
while the properties relating them are **matter tensors**. Some of
the more familiar tensor properties and their relationships are shown
below. Click "Next" to view a more complete and in-depth version of
the diagram.

This tensor relates the electric polarisation
and the electric field that induced it.

A scalar quantity which is a measure of the
average kinetic energy of particles "the hotness" of a system.

The tensor relating the strain of a body
to a change in temperature. It is purely a scaled version of the strain
tensor since the change in temperature is a scalar quantity.

The tensor representing the different forces acting
on a body.

This tensor represents the different deformations
of a body.

A tensor quantity that relates the temperature
change to the change in the electric dipole.

This relates the rate of enthalpy increase
of a material to its temperature.

A vector quantity arising due to an electric charge
or a time varying magnetic field. The field exerts a force on electrically
charged objects.

The heat content of the system (dependent on the
nature of the atoms and the bonding between them).

A vector measurement describing the separation
of electric charge within a material.

For some materials an electric polarisation
is generated when a stress is applied. The relationship between the
applied stress state and the resulting dipole is described by the direct
piezoelectric tensor.

The tensor relating applied stress in a material
to the strain induced.

A guide to tensor usage in Materials Science

The numbers in the square brackets represent the rank
of the tensor involved. Try clicking on some of the labels to link to
other parts of the DoITPoMS site that cover the specific properties
in more detail. This is still not an exhaustive list of tensor applications
as many field tensors have been left out.

The tensor relating the strain of a body
to a change in temperature. It is purely a scaled version of the strain
tensor since the change in temperature is a scalar quantity.

A tensor quantity that relates the temperature
change to the change in the electric dipole.

When an electric field is applied
to a piezoelectric material there is a change of shape of the material.
This is the converse piezoelectric effect. The relationship is a linear
one and so can be described with tensors. It is found that the coefficients
relating the field to the strain are the same as those relating the
stress to the electric dipole.

A vector measurement describing the separation
of electric charge within a material.

Due to the linear relationship between the stress
on a system and the strain of the system, the coefficients relating
these two 2nd rank tensors form a 4th rank tensor. The tensor relating
strain to stress is the compliance, with coefficients satisfying the
equation *ε*_{ij} = *s*_{ijkl} *σ*_{kl}.

The tensor relating stress to strain is the stiffness, with coefficients satisfying the equation*σ*_{ij} = *c*_{ijkl}
*ε*_{kl}

The tensor relating stress to strain is the stiffness, with coefficients satisfying the equation

For some materials an electric moment is
generated when a stress is applied. The relationship between stress
and electric moment is a linear one, and since the stress is a rank
2 tensor and the electric moment is a rank 1 tensor, the tensor describing
the relationship is of rank 3, a cubic array of coefficients.

The heat content of the system (dependent on the
nature of the atoms and the bonding between them).

A vector quantity arising due to an electric charge
or a time varying magnetic field. The field exerts a force on electrically
charged objects.

This is a scalar which relates the rate of
enthalpy increase of a material to its temperature

Relates the enthalpy (heat) produced
by a strain to that same strain.

Relates the enthalpy (heat) produced
by an electric dipole to the electric dipole.

The tensor relates the electric polarisation
and the electric field that induced it. It is a second rank tensor with
coefficients *ε*_{ij} that satisfy the equation P_{i}=
*ε*_{ij}E_{j}

Relates the enthalpy (heat) caused by a stress
to that same stress.

The tensor representing the different deformations
of a body. The tensor components originate from the deformations of
line elements, involving normal strain and shear strain. Normal strain
is the extension per unit length parallel to the basis axes. Shear strain
component *e*_{ij} is the rotation of a line element parallel
to *Ox*_{j} towards *Ox*_{i}. The strain tensor
is defined to be symmetric, as the anti-symmetric component of the general
tensor is merely a rotation

The tensor representing the different forces on
a body. The tensor components originate from the stress states on the
surface of a cube, involving normal and shear stresses. The components
of the tensor *σ*_{ij} are defined as the force in
the direction *Ox*_{i} direction transmitted across the
plane perpendicular to *Ox*_{j}. The stress tensor is defined
to be symmetric.

A measurement of the "hotness" of a system.

The tensor relating the strain of a body to a change
in temperature. It is purely a scaled version of the strain tensor since
the change in temperature is a scalar quantity.

This property occurs when a body is not
allowed to deform under a temperature change. The tensor relates the
stress tensor to the temperature change. It is purely a scaled version
of the stress tensor since the change in temperature is a scalar quantity.

This tensor relates the enthalpy (heat)
produced by an applied electric field to said electric field, and is
a rank 1 tensor. Research is currently being undertaken to use this
effect to cool electronic devices.