Editing Continuum mechanics (section)

===Body forces===

''[[Body forces]]'' are forces originating from sources outside of the body{{sfn|Irgens|2008}} that act on the volume (or mass) of the body. Saying that body forces are due to outside sources implies that the interaction between different parts of the body (internal forces) are manifested through the contact forces alone.{{sfn|Liu|2002}} These forces arise from the presence of the body in force fields, ''e.g.'' [[gravitational field]] ([[gravitational force]]s) or electromagnetic field ([[electromagnetic force]]s), or from [[fictitious force|inertial forces]] when bodies are in motion. As the mass of a continuous body is assumed to be continuously distributed, any force originating from the mass is also continuously distributed. Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body,{{sfn|Chadwick|1999}} ''i.e.'' acting on every point in it. Body forces are represented by a body force density <math>\mathbf b(\mathbf x, t)</math> (per unit of mass), which is a frame-indifferent vector field.

In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density <math>\mathbf \rho (\mathbf x, t)\,\!</math> of the material, and it is specified in terms of force per unit mass (<math>b_i\,\!</math>) or per unit volume (<math>p_i\,\!</math>). These two specifications are related through the material density by the equation <math>\rho b_i = p_i\,\!</math>. Similarly, the intensity of electromagnetic forces depends upon the strength ([[electric charge]]) of the electromagnetic field.

The total body force applied to a continuous body is expressed as

:<math>\mathbf F_B=\int_V\mathbf b\,dm=\int_V \rho\mathbf b\,dV</math>

Body forces and contact forces acting on the body lead to corresponding moments of force ([[torque]]s) relative to a given point. Thus, the total applied torque <math>\mathcal M</math> about the origin is given by

:<math>\mathcal M= \mathbf M_C + \mathbf M_B</math>

In certain situations, not commonly considered in the analysis of the mechanical behavior of materials, it becomes necessary to include two other types of forces: these are ''couple stresses''{{refn|group=note|Maxwell pointed out that nonvanishing body moments exist in a magnet in a magnetic field and in a dielectric material in an electric field with different planes of polarization.{{sfn|Fung|1977|p=76}}}}{{refn|group=note|Couple stresses and body couples were first explored by Voigt and Cosserat, and later reintroduced by Mindlin in 1960 on his work for Bell Labs on pure quartz crystals.{{Citation needed|date=December 2022}}}} (surface couples,{{sfn|Irgens|2008}} contact torques){{sfn|Chadwick|1999}} and ''body moments''. Couple stresses are moments per unit area applied on a surface. Body moments, or body couples, are moments per unit volume or per unit mass applied to the volume of the body. Both are important in the analysis of stress for a polarized dielectric solid under the action of an electric field, materials where the molecular structure is taken into consideration (''e.g.'' bones), solids under the action of an external magnetic field, and the dislocation theory of metals.{{sfn|Wu|2004}}{{sfn|Fung|1977}}{{page needed|date=August 2020}}{{sfn|Irgens|2008}}

Materials that exhibit body couples and couple stresses in addition to moments produced exclusively by forces are called ''polar materials''.{{sfn|Fung|1977}}{{page needed|date=August 2020}}{{sfn|Chadwick|1999}} ''Non-polar materials'' are then those materials with only moments of forces. In the classical branches of continuum mechanics the development of the theory of stresses is based on non-polar materials.

Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) in the body can be given by

:<math>\mathcal F = \int_V \mathbf a\,dm = \int_S \mathbf T\,dS + \int_V \rho\mathbf b\,dV</math>
:<math>\mathcal M = \int_S \mathbf r \times \mathbf T\,dS + \int_V \mathbf r \times \rho\mathbf b\,dV</math>