Editing Tensor (section)

=== Continuum mechanics ===

Important examples are provided by [[continuum mechanics]].  The stresses inside a solid body or [[fluid]]<ref>{{cite book |last1=Schobeiri |first1=Meinhard T. |date=2021 |title=Fluid Mechanics for Engineers |publisher=Springer |pages=11–29 |chapter=Vector and Tensor Analysis, Applications to Fluid Mechanics}}</ref> are described by a tensor field.  The [[Stress (mechanics)|stress tensor]] and [[strain tensor]] are both second-order tensor fields, and are related in a general linear elastic material by a fourth-order [[elasticity tensor]] field.  In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be conveniently represented as a 3 × 3 array.  The three faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force.  The force's vector components are also three in number.  Thus, 3 × 3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment.  Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe.  Thus, a second-order tensor is needed.

If a particular [[Volume form|surface element]] inside the material is singled out, the material on one side of the surface will apply a force on the other side.  In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner.  This is described by a tensor of [[type of a tensor|type {{nowrap|(2, 0)}}]], in [[linear elasticity]], or more precisely by a tensor field of type {{nowrap|(2, 0)}}, since the stresses may vary from point to point.