Editing Continuum mechanics (section)

===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.