Classical Continuum Dynamics

Abstract

If a deformable body is subjected to the action of forces distributed over its surface, these forces are transmitted to every point in the interior of the body. As a result, the relative positions of its physical elements change. Unlike body forces, which act on each volume element within the body (e.g., gravitational forces), the surface forces act at an internal point P across a surface in which P is embedded. Consider a vectorial surface element through P, ΔS = nΔS, where the normal n is given with respect to some fixed Cartesian coordinate system. We may define the resultant force ΔF, acting on ΔS, as the measure of the action of the continuum I (on the side of ΔS toward which n is directed) on continuum II(Fig. 1.1). The ratio ΔF/ΔS is the average stress vector on ΔS. If the limit of this quotient exists as ΔS → 0 (n fixed), we write

$$\mathop {\lim }\limits_{\Delta S \to 0} {{\Delta {\bf{F}}} \over {\Delta S}} = {{d{\bf{F}}} \over {dS}} = {\bf{T}}\left( {\bf{n}} \right)$$

and call T the stress vector (or traction) at P associated with the normal n. In this connection, the following may be noted:

• 1. By Newton’s law of action and reaction, side II exerts upon side I an equal and opposite force; i.e.,

  $${\bf{T}}\left( { - {\bf{n}}} \right) = - {\bf{T}}\left( {\bf{n}} \right).$$

• 2. We assume that the stress vector at P associated with the normal n is independent of the surface used for its definition (Cauchy’s stress principle). This is a fundamental hypothesis. It allows us to replace the unknown actual intermolecular forces by a single force that depends upon two geometric entities alone, viz., the coordinates of the point relative to the applied surface forces and the orientation of the normal.

• 3. Given a finite surface S within the body, the total force acting across it is determined by the integral

  $$\int_S {{\bf{T}}\left( {\bf{n}} \right)dS.} $$

  This integral will vanish if S is a closed surface, provided no body forces (including inertial forces) are acting on the volume bounded by S. If the body forces are present, we shall assume that the surface forces transmitted into the continuum are, at each moment, in equilibrium with the body forces.

• 4. Equation (1.1) does not specify the nature of the functional relationship T(n). However, this can be found by solving a canonical problem that relates to the surface force distribution alone, with no body forces. If the whole body is in equilibrium under the action of the surface forces, so is an infinitesimal volume element around P. For the sake of simplicity, we choose it to be a tetrahedron PABC three of whose faces dS (i = 1, 2, 3) coincide with the coordinate planes. Because the sides of the tetrahedron are small, Eqs. (1.2) and (1.3) yield (Fig. 1.2)

  $$\int_S {{\bf{T}}\left( {\bf{n}} \right)dS} = {\bf{T}}\left( {\bf{n}} \right)d{S_n} + {\bf{T}}\left( { - {{\bf{e}}_1}} \right)d{S_1} + {\bf{T}}\left( { - {{\bf{e}}_2}} \right)d{S_2} + {\bf{T}}\left( { - {{\bf{e}}_3}} \right)d{S_3} = {\bf{T}}\left( {\bf{n}} \right)d{S_n} - {\bf{T}}\left( {{{\bf{e}}_1}} \right)d{S_1} - {\bf{T}}\left( {{{\bf{e}}_2}} \right)d{S_2} - {\bf{T}}\left( {{{\bf{e}}_3}} \right)d{S_3}, $$

  where e _i (i = 1, 2, 3) is the unit vector along the x _i direction and dS _n is the area of the face ABC. Because dS _i = n _i dS _n = (n. e _i )dS _n and the matter inside PABC is in equilibrium, we may write Eq. (1.4) formally as

  $${\bf{T}}\left( {\bf{n}} \right) = {\bf{n}} \cdot \left[ {{{\bf{e}}_1}{\bf{T}}\left( {{{\bf{e}}_1}} \right) + {{\bf{e}}_2}{\bf{T}}\left( {{{\bf{e}}_2}} \right) + {{\bf{e}}_3}{\bf{T}}\left( {{{\bf{e}}_3}} \right)} \right].$$

  Here, for example, T(e ₁) denotes the stress vector across the x ₁-plane with normal e ₁. As the volume of the tetrahedron is allowed to shrink to zero, the plane ABC contains the point P and the vector T(n) becomes the stress vector at P associated with the normal n. Equation (1.5) shows that it is sufficient to know the value of the stress on three mutually perpendicular planes passing through the point P of the deformed body in order to evaluate the stress on an arbitrary plane passing through that point.

Man can only conquer nature by obeying her.

(Francis Bacon)