## Force Traction and Stress Vectors

1 There are two kinds of forces in continuum mechanics body forces: act on the elements of volume or mass inside the body, e.g. gravity, electromagnetic fields. dF = pbdVol.

surface forces: are contact forces acting on the free body at its bounding surface. Those will be defined in terms of force per unit area.

2 The surface force per unit area acting on an element dS is called traction or more accurately stress vector.

Most authors limit the term traction to an actual bounding surface of a body, and use the term stress vector for an imaginary interior surface (even though the state of stress is a tensor and not a vector).

3 The traction vectors on planes perpendicular to the coordinate axes are particularly useful. When the vectors acting at a point on three such mutually perpendicular planes is given, the stress vector at that point on any other arbitrarily inclined plane can be expressed in terms of the first set of tractions.

4 A stress, Fig 2.1 is a second order cartesian tensor, aj where the 1st subscript (i) refers to the direction of outward facing normal, and the second one (j) to the direction of component force.

 011 012 013 t1 a = aij = 021 022 023 = t2 0"31 032 033 I t3

5 In fact the nine rectangular components aij of a turn out to be the three sets of three vector components (a11,a12,a13), (021,022,023), (031,032,033) which correspond to the three tractions t1, t2 and t3 which are acting on the x1,x2 and x3 faces (It should be noted that those tractions are not necesarily normal to the faces, and they can be decomposed into a normal and shear traction if need be). In other words, stresses are nothing else than the components of tractions (stress vector), Fig. 2.2.

6 The state of stress at a point cannot be specified entirely by a single vector with three components; it requires the second-order tensor with all nine components.

 o. 13 ' i,'-'-»i o /fl/
 X 3 V3 \ \ V / : V2 > / \ i / X2 y V 1 N 1 ' >Xi (Components of a vector are scalars)

X1 Stresses as components of a traction vector (Components of a tensor of order 2 are vectors)

Figure 2.2: Stresses as Tensor Components o

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