Cauchy stress principle

Broadly speaking, our description of continuum mechanics will be divided along a few traditional lines. First, we will consider the kinematic description of deformation without searching for the attributes of the forces that lead to a particular state of deformation. Here it will be shown that the displacement fields themselves do not cast a fine enough net to sufficiently distinguish between rigid body motions, which are often of little interest, and the more important relative motions that result in internal stresses. These observations call for the introduction of other kinematic measures of deformation such as the various strain tensors. Once we have settled these kinematic preliminaries, we turn to the analysis of the forces in continua that lead to such deformation, and culminate in the Cauchy stress principle. 

Fig. 2.7. Tetrahedral volume element used to illustrate the Cauchy stress principle.

The Cauchy stress principle arises through consideration of the equilibrium of body forces and surface tractions in the special case of the infinitesimal tetrahedral volume shown in fig. 2.7. Three faces of the tetrahedron are perpendicular to the Cartesian axes while the fourth face is characterized by a normal n. The idea is to insist on the equilibrium of this elementary volume, which results in the observation that the traction vector on an arbitrary plane with normal n (such as is shown in the figure) can be determined once the traction vectors on the Cartesian planes are known. In particular, it is found that = crn, where a is known as the stress tensor, and carries the information about the traction vectors associated with the Cartesian planes. The simple outcome of this argument is the claim that... 

Rrst principal stress Cauchy stress principle... 

The average force per unit area is Af,-/AS. This quantity attains a limiting nonzero value as AS approaches zero at point P (Cauchy s stress principle). This limiting quantity is called the stress vector, or traction vector T. But T depends on the orientation of the area element, that is, the direction of the surface defined by normal n. Thus it would appear that there are an infinite number of unrelated ways of expressing the state of stress at point P. 

The stress vector represents the force of the material outside of V(t) acting on the material inside V(t). The assumption, based on the short range of the inter-particle forces, is that this force which the material exerts on itself acts entirely on the surface S(t). Truesdell calls this stress principle the defining concept of continuum mechanics. The stress vector depends not only on time and space, but also on the surface orientation. However, according to Cauchy s fundamental theorem,... 

According to a postulate called the Cauchy s stress principle, for any closed surface S there is a distribution of the stress vector t with resultant and moment, equivalent to a force field acting on the continuum in volume V enclosed by a sur-... 

Cancellation of the Cauchy term may bring some discrepancies, the more evident one being that, whatever h is, it leads to a zero second normal stress difference. A more subtle one concerns the loss of the thermodynamic consistency of the model. Indeed, it is not possible to find any potential function in the form Udi, I2) with h2di, I2) = 0 unless hi only depends on Ii. As mentioned by Larson [27, 28], this can induce violation of the second principle in complex flows such as those encoxmtered in processing conditions. 

We see that application of the angular acceleration principle does reduce, somewhat, the imbalance between the number of unknowns and equations that derive from the basic principles of mass and momentum conservation. In particular, we have shown that the stress tensor must be symmetric. Complete specification of a symmetric tensor requires only six independent components rather than the full nine that would be required in general for a second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently independent unknowns and only four independent relationships between them. It is clear that the equations derived up to now - namely, the equation of continuity and Cauchy s equation of motion do not provide enough information to uniquely describe a flow system. Additional relations need to be derived or otherwise obtained. These are the so-called constitutive equations. We shall return to the problem of specifying constitutive equations shortly. First, however, we wish to consider the last available conservation principle, namely, conservation of energy. 

If we denote by T, Cauchy s stress tensor of our material and by b, the density of body forces, then by Truesdell s third principle the balances of linear momentum and of moment of momentum for the whole mixture in local form turn out to be... 

At this point, we have seen that 100 years was required for the concepts of kinematics, stress, and constitutive relations to be developed. It was Euler who provided a precise statement of the non-relativistic laws of mechanics, introduced the cut principle and the concept of stress, and developed important results concerning the kinematics of continua. This was done primarily in the period between 1750 and 1766. In 1822, Cauchy placed the concept of stress on a modem basis and in... 

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