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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 (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 third law of motion of conservation of linear momentum and angular momentum (for continuous bodies these laws are called the Euler's equations of motion). The internal contact forces are related to the body's 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.

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 T (n, x, t) {\displaystyle \mathbf {T} (\mathbf {n},\mathbf {x},t)} that represents this distribution in a particular configuration of the body at a given time t {\displaystyle t\,\!}. It is not a vector field because it depends not only on the position x {\displaystyle \mathbf {x} } of a particular material point, but also on the local orientation of the surface element as defined by its normal vector n {\displaystyle \mathbf {n} }.

Any differential area d S {\displaystyle dS\,\!} with normal vector n {\displaystyle \mathbf {n} } of a given internal surface area S {\displaystyle S\,\!}, bounding a portion of the body, experiences a contact force d F C {\displaystyle d\mathbf {F} _{C}\,\!} arising from the contact between both portions of the body on each side of S {\displaystyle S\,\!}, and it is given by

where T (n) {\displaystyle \mathbf {T} ^{(\mathbf {n})}} is the surface traction, also called stress vector, traction, or traction vector. The stress vector is a frame-indifferent vector (see Euler-Cauchy's stress principle).

The total contact force on the particular internal surface S {\displaystyle S\,\!} is then expressed as the sum (surface integral) of the contact forces on all differential surfaces d S {\displaystyle dS\,\!}:

In continuum mechanics a body is considered stress-free if the only forces present are those inter-atomic forces (ionic, metallic, and van der Waals forces) required to hold the body together and to keep its shape in the absence of all external influences, including gravitational attraction. 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.