Editing Continuum mechanics (section)

==Forces in a continuum==
{{see also|Stress (mechanics)|Cauchy stress tensor}}
A solid is a deformable body that possesses shear strength, ''sc.'' a solid can support shear forces (forces parallel to the material surface on which they act). Fluids, on the other hand, do not sustain shear forces.

Following the classical dynamics of [[Isaac Newton|Newton]] and [[Leonhard Euler|Euler]], the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces <math>\mathbf F_C</math> and body forces <math>\mathbf F_B</math>.{{sfn|Smith|1993|p=97}} Thus, the total force <math>\mathcal F</math> applied to a body or to a portion of the body can be expressed as:

:<math>\mathcal F = \mathbf F_C + \mathbf F_B</math>

===Surface forces===

''[[Surface forces]]'' or ''contact forces'', expressed as force per unit area, can act either on the bounding surface of the body, as a result of mechanical contact with other bodies, or on imaginary internal surfaces that bound portions of the body, as a result of the mechanical interaction between the parts of the body to either side of the surface ([[Cauchy stress tensor|Euler-Cauchy's stress principle]]). When a body is acted upon by external contact forces, internal contact forces are then transmitted from point to point inside the body to balance their action, according to [[Newton's laws of motion|Newton's third law of motion]] of conservation of [[linear momentum]] and [[angular momentum]] (for continuous bodies these laws are called the [[Euler's laws|Euler's equations of motion]]). The internal contact forces are related to the body's [[deformation (mechanics)|deformation]] through [[constitutive equations]]. The internal contact forces may be mathematically described by how they relate to the motion of the body, independent of the body's material makeup.{{Citation needed|date=December 2022}}

The distribution of internal contact forces throughout the volume of the body is assumed to be continuous. Therefore, there exists a ''contact force density'' or ''Cauchy traction field''{{sfn|Smith|1993}} <math>\mathbf T(\mathbf n, \mathbf x, t)</math> that represents this distribution in a particular configuration of the body at a given time <math>t\,\!</math>. It is not a vector field because it depends not only on the position <math>\mathbf x</math> of a particular material point, but also on the local orientation of the surface element as defined by its normal vector <math>\mathbf n</math>.{{sfn|Lubliner|2008}}{{page needed|date=August 2020}}

Any differential area <math>dS\,\!</math> with normal vector <math>\mathbf n</math> of a given internal surface area <math>S\,\!</math>, bounding a portion of the body, experiences a contact force <math>d\mathbf F_C\,\!</math> arising from the contact between both portions of the body on each side of <math>S\,\!</math>, and it is given by

:<math>d\mathbf F_C= \mathbf T^{(\mathbf n)}\,dS</math>

where <math>\mathbf T^{(\mathbf n)}</math> is the ''surface traction'',{{sfn|Liu|2002}} also called ''stress vector'',{{sfn|Wu|2004}} ''traction'',{{sfn|Fung|1977}}{{page needed|date=August 2020}} or ''traction vector''.{{sfn|Mase|1970}} The stress vector is a frame-indifferent vector (see [[Cauchy stress tensor|Euler-Cauchy's stress principle]]).

The total contact force on the particular internal surface <math>S\,\!</math> is then expressed as the sum ([[surface integral]]) of the contact forces on all differential surfaces <math>dS\,\!</math>:

:<math>\mathbf F_C=\int_S \mathbf T^{(\mathbf n)}\,dS</math>

In continuum mechanics a body is considered stress-free if the only forces present are those inter-atomic forces ([[ionic bond|ionic]], [[metallic bond|metallic]], and [[van der Waals force]]s) required to hold the body together and to keep its shape in the absence of all external influences, including gravitational attraction.{{sfn|Mase|1970}}{{sfn|Atanackovic|Guran|2000}} Stresses generated during manufacture of the body to a specific configuration are also excluded when considering stresses in a body. Therefore, the stresses considered in continuum mechanics are only those produced by deformation of the body, ''sc.'' only relative changes in stress are considered, not the absolute values of stress.

===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>