Kinetics
5.2.1 Force, Traction and Stress Vectors
38 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.
39 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).
40 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.
41 A stress, Fig 5.1 is a second order cartesian tensor, Oj where the 1st subscript (i) refers to the direction of outward facing normal, and the second one (j) to the direction of component force.
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Oil |
O12 |
O13 |
t1 | ||
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a = Oij = |
O21 |
O22 |
O 23 |
= |
t2 |
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O31 |
O32 |
O33 |
t3 |
42 In fact the nine rectangular components aj of a turn out to be the three sets of three vector components (an , a12 ,a13), (a21 , a22, a23), (a3i ,a32 ,a33) which correspond to the three tractions ti, 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. 5.2.
43 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.
- Figure 5.1: Stress Components on an Infinitesimal Element
X1 Stresses as components of a traction vector (Components of a tensor of order 2 are vectors)
X1 Stresses as components of a traction vector (Components of a tensor of order 2 are vectors)
Figure 5.2: Stresses as Tensor Components
5.2.2 Traction on an Arbitrary Plane; Cauchy's Stress Tensor
44 Let us now consider the problem of determining the traction acting on the surface of an oblique plane (characterized by its normal n) in terms of the known tractions normal to the three principal axis, ti, t2 and t3. This will be done through the so-called Cauchy's tetrahedron shown in Fig. 5.3, and will be obtained without any assumption of equilibrium and it will apply
- X
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(-K | |
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2 * | |
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3 / |
> |
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A |
2s |
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*\7 | |
Figure 5.3: Cauchy's Tetrahedron in fluid dynamics as well as in solid mechanics.
45 This equation is a vector equation, and the corresponding algebraic equations for the components of tn are
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tni |
= Oiim + cr2i n2 + cr31n3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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tjU2 |
= a\2n\ + (722«'2 + Cr32rc3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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tn3 |
= cr13m + 0-23^2 + & 3317-3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Indicial notation |
tni |
= CTjiUj | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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dyadic notation |
tn |
46 We have thus established that the nine components aij are components of the second order tensor, Cauchy's stress tensor. Example 5-1: Stress Vectors if the stress tensor at point P is given by a = 
We seek to determine the traction (or stress vector) t passing through P and parallel to the plane ABC where A(4,0,0), B(0, 2,0) and C(0,0,6). Solution: The vector normal to the plane can be found by taking the cross products of vectors AB and AC:
= 12ei + 24e2 + 8e3 The unit normal of N is given by Hence the stress vector (traction) will be 
5.2.3 Invariants 47 The principal stresses are physical quantities, whose values do not depend on the coordinate system in which the components of the stress were initially given. They are therefore invariants of the stress state. 48 When the determinant in the characteristic Equation is expanded, the cubic equation takes the form where the symbols Ia, IIa and IIIa denote the following scalar expressions in the stress components:
49 In terms of the principal stresses, those invariants can be simplified into
5.2.4 Spherical and Deviatoric Stress Tensors 50 If we let a denote the mean normal stress p a = -p= — (cru + cr22 + CT33J = ~gu = -tr a then the stress tensor can be written as the sum of two tensors: Hydrostatic stress in which each normal stress is equal to —p and the shear stresses are zero. The hydrostatic stress produces volume change without change in shape in an isotropic medium. _ Deviatoric Stress: which causes the change in shape.
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