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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.
> Direct piezoelectric effect Elasticity Susceptibility Pyroelectric effect Heat capacity Thermal expansion Electric field Polarisation Strain Enthalpy Stress Temperature
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.
> Direct piezoelectric effect[3] Elasticity[4] Susceptibility[2] Pyroelectric effect[1] Heat capacity[0] Thermal expansion[2] ElectricField[1] Polarisation Strain Enthalpy Stress Temperature [2] [2] [1] [0] [0] Electromechanical effects (piezoelectricity) [3] Electrothermal effects (pyroelectricity ) [1] Thermoelastic effects [2] Heat of deformation [2] Thermal pressure [2] Piezocaloric effect [2] Converse piezoelectric effect [3] Electrocaloric effect [1] Heat of polarisation [1]
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 εij = sijkl σkl.
The tensor relating stress to strain is the stiffness, with coefficients satisfying the equation σij = cijkl εkl
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 εij that satisfy the equation Pi= εijEj
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 eij is the rotation of a line element parallel to Oxj towards Oxi. 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 σij are defined as the force in the direction Oxi direction transmitted across the plane perpendicular to Oxj. 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.