Surface stress vector

We may now attempt to simplify (2-23) to a differential form, as we did for the mass conservation equation, (2-6). The basic idea is to express all terms in (2-23) as integrals over Vm (t), leading to the requirement that the sum of the integrands is zero because Vm (l) is initially arbitrary. However, it is immediately apparent that this scheme will fail unless we can say more about the surface-stress vector t. Otherwise, there is no way to express the surface integral of t in terms of an equivalent volume integral over Vm(i). 

We can actually go one step further than this general observation and use the stress equilibrium principle applied to the tetrahedron to obtain a simple expression for the surface-stress vector t on an arbitrarily oriented surface at x in terms of the components of t on the three mutually perpendicular surfaces at the point x. We denote the stress vector on a surface with normal n as t(n). The area of the surface with unit normal n is denoted as A A . Then, applying the surface-stress equilibrium principle to the tetrahedron, we have... 

We see from (2-27) and (2-28) that the second-order tensor T is just the linear vector operator that operates on the unit normal to a surface at a point P to produce the surface-stress vector acting at that point. Indeed, rewriting (2-27), we obtain... 

The brute force way to calculate the force on a body is to integrate the surface stress vector over the body surface - in this case,... 

The solution of this integral equation gives us the unknown surface-stress vector t( j = T( ,) n. Then the general solution of the creeping-flow equations is... 

Now, generally, it is the motion of the particle that is specified, and the surface stress vector T( ,) n is unknown. However, if we evaluate (8-197) at points x,v on the boundary of the particle, it is converted into an integral equation from which the surface-stress distribution can be determined. Specifically, taking account of the condition (8-195), we obtain from (8-197)... 

A useful simphfication of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m through a single planar entrance and a single planar exit (Fig. 6-4), to whi(m the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tu is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + pgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. 

The stress tensor plays a prominent role in the Navier-Stokes and the energy equations, which are at the core of all fluid-flow analyses. The purpose of the stress tensor is to define uniquely the stress state at any (every) point in a flow field. It takes nine quantities (i.e., the entries in the tensor) to represent the stress state. It is also be important to extract from the stress tensor the three quantities needed to represent the stress vector on a given surface with a particular orientation in the flow. By relating the stress tensor to the strain-rate tensor, it is possible to describe the stress state in terms of the velocity field and the fluid viscosity. 

Fig. 2.11 The stress tensor describes the stress state at a point in space. It involves nine components, which are interpreted as components of the stress vectors on three orthogonal surfaces at the point where the three surfaces intersect.

The force acting on any differential segment of a surface can be represented as a vector. The orientation of the surface itself can be defined by an outward-normal unit vector, called n. This force vector, indeed any vector, has direction and magnitude, which can be resolved into components in various ways. Normally the components are taken to align with coordinate directions. The force vector itself, of course, is independent of the particular representation. In fluid flow the force on a surface is caused by the compressive (or expansive) and shearing actions of the fluid as it flows. Thermodynamic pressure also acts to exert force on a surface. By definition, stress is a force per unit area. On any surface where a force acts, a stress vector can also be defined. Like the force the stress vector can be represented by components in various ways. 

Seeking to find the relationship between stress vectors and tensors, consider Fig. 2.12, which shows an infinitesimally small, arbitrarily oriented surface A whose orientation is defined by the outward-pointing normal unit vector n. As illustrated, the unit vector can be resolved into components nz,nr, and hq,... 

Take the stress vector acting on surface A to be r, the stress on Az to be tz, and so on. Each of the four stress vectors has three components and the objective is to determine if there is any special relationship among them. Assuming that there may be a volumetric body force f (force per unit volume), the net force on the tetrahedron is determined from the contributions of the forces on each surface and the body force,... 

Combining the two equations above, the stress vector at any point on any surface A with orientation n can be written in terms of the nine stress components on three orthogonal surfaces that intersect at the point ... 

While the stress vector may be determined on any arbitrary surface, we are most often concerned with the stresses that act on the six surfaces of a differential control volume. On each surface there are normal and shearing stresses, as indicated in Fig. 2.13. The stress tensor... 

The force F and the stress r are both vectors, which are typically represented in components that align with a coordinate system. Since the stress vector at any surface whose orientation is represented by the outward normal n may be determined from the stress tensor, it follows that... 

Working in cartesian coordinates, determine the stress vector r on a differential surface whose orientation is represented by a unit vector n = nxtx + nyey + nzez. The stress state is represented by a tensor... 

On any arbitrary surface dA, the resultant stress can be represented as a vector r. The velocity at the surface is represented as a vector V. At any point in the flow field, the stress state is represented by a second-order tensor T. On a surface, which may represent some portion of the control surface that bounds a control volume, the stress is represented as a vector. The relationship between the stress tensor at a point T and the stress vector r on a particular surface that passes through the point is given as... 

The stress vector on any surface that is described by its outward-normal unit vector can be found by the operation... 

Assume that a stress tensor T is known at a point. Assume further that there are three orthogonal surfaces passing through the point for which the stress vectors r are parallel to the outward-normal unit vectors n that describe the orientation of the surfaces. In other words, on each of these surfaces the normal stress vector is a scalar multiple of the outward-normal unit vector,... 

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. 

It turns out, however, that the state of stress at P can be completely specified by giving the stress vector components in any three mutually perpendicular planes passing through the point. That is, only nine components, three for each vector, are needed to define the stress at point P. Each component can be described by two indices ij, the first denoting the orientation of the surface and the second, the direction of the force. Figure 2.3 gives these components for three Cartesian planes. The nine stress vector components form a second-order Cartesian tensor, the stress tensor8 n. ... 

For an arbitrary volume x of the multicomponent continuum, the total rate of change of linear momentum in the 7th coordinate direction must equal the sum of the following (1) the surface integral of the stress vector Y,k where afj equals the component in the direction Xj of the stress vector acting on that face of an elemental parallelepiped of species K which has an... 

The introduction of the concepts of stress and deformation at a point has been a fundamental concept in the development of the mechanics of continuum media. From a physical point of view, only the displacement is a real quantity, while stress imphes an idealized situation that is not directly measurable the value of a stress can only be inferred from its effects. The effects of the force at a point P depend on the orientation of the element surface SA comprising the point, which in turn is characterized by a vector rij (j =1,2,3) normal to the surface at P, as shown in Figure 4.1. The stress vector at the point P can be written as... 

From Newton s law of action and reaction the stress vector resulting from the force exerted by the material inside an arbitrary volume V upon the material surrounding it across the element surface 8A is... 

With the aid of equation (13.10) and Hooke s law, it can be demonstrated that the operator L transforms the displacement vector U at the given point into the stress vector arising on an infinitely small surface element with the normal vector n. That is why this operator is called a stress operator. 

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