Stress vector

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 spatial stress was defined in terms of a contact stress vector and a vector normal to the area element on which it acts, both of which are assumed to be indifferent. From (A.51), t = Qt and i = Qn. Then under the transformation (A.50) t = s i, so that... 

A related problem in eomposites is the need to design optimal fiber orientations for a eomposite part given the set of stress vectors and levels to whieh the part will be subjected. These design eonsiderations would be useful in designing airframe eomponents sueh as parts for the tail, wing, or fuselage. A similar problem is assessment of the peiformanee penalties that might result from imperfections in manufacture. 

J. Crowe, L. Crowe, and J. Carpenter, Are freezing and dehydration similar stress vectors A comparison... 

J. H. Crowe, J. F. Carpenter, L. M. Crowe, and T. J. Anchordoguy, Are freezing and dehydration similar stress vectors A comparison of modes of interaction of stabilizing solutes with biomolecules, Cryobiology, 27, 219 (1990). 

Note 2 t is sometimes called true stress. The term traction (or stress vector) is preferred to... 

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,... 

The stress vectors on each of the orthogonal faces can be resolved into components that align with the coordinate axes,... 

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... 

The stress vector on the z plane has three components that can be determined from the projections of the principal stresses. These components, written to align with the principal axes, are... 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

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... 

Assuming that the stress vector t and the unit vector n are to be represented as column vectors, show that... 

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... 

Here r is the stress vector, with components that are typically taken to align with the coordinate directions. Recognize that both normal stress and shear stress contribute to work. That is, work is associated with both dilatation and deformation. It is important to note that there is not a -iid A construct in the work-rate integral, for example, as is the... 

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

Spelled out in operational detail, the components of the stress vector r may be written as... 

In either case, carrying out the matrix-vector multiplication reveals the meaning of the stress vector as... 

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 forces and stresses applied to a body may be resolved in three vectors, one normal to an arbitrarily selected element of area and two tangential. For the yz plane, the stress vectors are a, and on, a, respectively. Six analogous stresses exist for tile other orthogonal orientations, giving a total of nine quantities, of which three exist as commutative pairs (arl = crSr). The state of stress, therefore, is defined by three tensiic or normal components (pxx,oyy. o--f) and three shear or tangential components (crIy,cr.I ,CTy,), The shear components are most readily applicable to the determination of jj and G,... 

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. ... 

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