Tensor: Difference between revisions

Revision as of 05:00, 12 July 2024

Revision as of 05:00, 12 July 2024

Line 363:

=== Continuum mechanics ===

Important examples are provided by [[continuum mechanics]]. A tensor field describes 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> 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.

=== Other examples from physics ===