Stress cartesian coordinates

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

While the foregoing discussion of stress and strain is based on a Cartesian coordinate system, any orthogonal coordinate system may be used. 

Any or all of these forces may result in local stresses within the fluid. Stress can be thought of as a (local) concentration of force, or the force per unit area that bounds an infinitesimal volume of the fluid. Now both force and area are vectors, the direction of the area being defined by the normal vector that points outward relative to the volume bounded by the surface. Thus, each stress component has a magnitude and two directions associated with it, which are the characteristics of a second-order tensor or dyad. If the direction in which the local force acts is designated by subscript j (e.g., j = x, y, or z in Cartesian coordinates) and the orientation (normal) of the local area element upon which it acts is designated by subscript i, then the corresponding stress component (ay) is given by... 

The components of the stress tensor in Cartesian coordinates are as follows ... 

The stress tensor describes the forces transmitted to an element of material through its contacts with adjacent elements (78). Traction is the force per unit area acting outwardly on the material adjacent to a material plane, and transmitted through its contact with material across the plane. If the components of traction are known for any set of three planes passing through a point, the traction across any plane through the point can be calculated. The stress at a material point is determined by an assembly erf nine components of traction, three for each plane. If the orientations of the three planes are chosen to be normal to the coordinate directions of a rectangular Cartesian coordinate system, the Cartesian components of the stress are obtained ... 

In the most general case, stresses on any of the six control-volume faces can potentially contribute to a force in any direction. In a cartesian coordinate system, only stresses in a certain direction can contribute to a force in that direction. In cylindrical coordinates and other noncartesian systems, the situation is more complex. As an example of this point, consider Fig. 2.15, which is a planar representation of the z face of the cylindrical differential element. Notice two important points that are revealed in this figure. One is that the the area of the 0 face varies from rdO on one side to (r + dr)dO on the other. Therefore, in computing net forces, the area s dependence on the r coordinate must be included. Specifically,... 

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... 

Notice that in these equations the terms on the right-hand side are written with a certain resemblance to the rows of the stress tensor, Eq. 2.180. The pressure gradients have been written as a separate terms. In the r and 9 equations, the final term collects some of the left-overs in going from Eqs. 3.57 and 3.58, yet maintaining the other terms in a form analogous to the stress tensor. The z equation has no left-over terms, which is also the case for the Navier-Stokes equations in cartesian coordinates. 

The state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

Figure H3.1.2 The oscillatory response of a food material that possesses both elastic and viscous properties can be represented by a complex variable G. This variable has two components that can be expressed as either the Cartesian coordinates G and G or the polar coordinates IG I and 8. IG I is the magnitude of the imaginary G and is measured as the ratio of the amplitudes of stress and strain.

The flow equation can be written in terms of stress components (in Cartesian coordinates) in the following form ... 

For an element in equilibrium with no body forces, the equations of equilibrium were obtained by Lame and Clapeyron (1831). Consider the stresses in a cubic element in equilibrium as shown in Fig. 2.3. Denote 7y as a component of the stress tensor T acting on a plane whose normal is in the direction of e and the resulting force is in the direction of ej. In the Cartesian coordinates in Fig. 2.3, the total force on the pair of element surfaces whose normal vectors are in the direction of ex can be given by... 

Consider a plane-strain problem in the x-z plane, as shown in Fig. 8.2. The stress tensor, expressed in Cartesian coordinates, takes the form... 

We present first a brief discussion of the stress tensor and the concepts of its mean and deviatoric parts. A rectangular Cartesian coordinate system (x, i 1,2,3) is used throughout. The stress tensor referred to this coordinate system is denoted... 

For a three-dimensional body, discussions of elastic responses in the framework of Hooke s law become more complicated. One defines a 3 x 3 stress tensor P [12], which is the force (with emits of newtons) expressed in a Cartesian coordinate system ... 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

The remaining six quantities are called shear stresses. They have two subscripts associated with the coordinates, and are referred to as the components of the molecular momentum flow tensor, or the components of the molecular stress tensor, as they are associated with molecular motion. Usually, the viscous stress tensor, t, and the molecular stress tensor, it, are simply referred to as stress tensors. For a Newtonian fluid, we may express the stresses in terms of velocity gradients and viscosities in Cartesian coordinates as follows ... 

The induced polarization in a piezoelectric, Pj, is a first-rank tensor (vector), and mechanical stress, is a second-rank tensor (nine components), which is represented in a Cartesian coordinate system with axes x, y, and z, as ... 

Kc and Gc are the parameters used in linear elastic fracture mechanics (LEFM). Both factors are implicitly defined to this point for plane stress conditions. To understand the term plane stress, imagine that the applied stress is resolved into three components along Cartesian coordinates plane stress occurs when one component is = 0. Such conditions are most likely to occur when the specimen is thin. 

The total stress tensor is a symmetric tensor (i.e., Tij = Tji), and in Cartesian coordinates it can thus be written ... 

In Cartesian coordinates the Reynolds shear stress pv Vy represents a flux of rr-momentum in the direction of y. Prandtl assumed that this momentum was transported by discrete lumps of fluid, which moved in the y direction over a distance I without interaction conserving the momentum and then mixed with the existing fluid at the new location. The mixing length, /, is supposed to be a variable analogous to the mean free path of kinetic theory in this process. 

For a generalized shear flow in the vicinity of a flat horizontal solid wall, the boundary layer flow can be described in Cartesian coordinates. The stress, —Oxy,eff, associated with direction y normal to the wall is apparently dominant, thus the stream-wise Reynolds averaged momentum equation yields ... 

The interior of a continuous sample contains many small volumes and small areas, on any of which attention can be focused. A small internal area has the property that, across it, the material on one side exerts a normal force and a tangential force on the material on the other side. Let the normal force be F and the area A then the ratio F/A approaches a limit as the size of A approaches zero. Thus we define the magnitude of the normal stress at a point across an infinitesimal area of a particular orientation. If we set up Cartesian coordinates so that the orientation of the area can be specified by the direction of its normal then, at a point, for every direction vector there is a normal-stress magnitude. The stress may be compressive or tensile, and in this text we treat compressions as positive. 

A stress applied to a crystal results in a strain. A phenomenological description of the electron energy levels under elastic strain was developed by Bardeen and Shockley [12]. It is referred to as the deformation potential approximation (DPA), in which the one-electron Hamiltonian is developed in a Taylor s series of the strain components The perturbation is written in cartesian coordinates, for a linear order in strain, as ... 

The maximum principal stress criterion for failure simply states that failure (by yielding or by fracture) would occur when the maximum principal stress reaches a critical value (ie., the material s yield strength, ays, or fracture strength, a/, or tensile strength, aurs)- For a three-dimensional state of stress, given in terms of the Cartesian coordinates x, y, and z in Fig. 2.1 and represented by the left-hand matrix in Eqn. (2.1), a set of principal stresses (see Fig. 2.1) can be readily obtained by transformation ... 

By using the Westergaard approach and the Airy stress function, the stresses near the tip of a crack may be considered (Fig. 3.2). A set of in-plane Cartesian coordinates X and y, or polar coordinates r and 0, is chosen, with the origin at the crack tip. The boundary conditions are as follows (i) stresses at the crack tip are very large and (ii) the crack surfaces are stress free. 

Let us select a point P within the body of the respect to the Cartesian coordinate axes as shown struct the plane ABC with direction cosines I, m and n relative to PA, PB, and PC. On applying the stress analysis to be found in any standard text on elasticity or rheology [2], we obtain the orthogonally oriented system of stress components illustrated in Fig.3-1, where a symbol of the form denotes a tensile stress normal to the plane of reference, and a symbol t- denotes a shear stress in that plane. 

Ivanov Dimitrov (1988) wrote the components of the surface viscous stress tensor in surface Cartesian coordinates... 

Rock Strength and Stress Factors. The rock of a petroleum reservoir is subjected to stresses because of the weight of the rock above the reservoir (overburden). The stress field can be resolved into vertical and horizontal stresses in cartesian coordinates. In the general case, the stress experienced by an element of reservoir rock may be resolved into both normal and shear stresses. However, the coordinate axis system can be oriented such that the shear stresses are zero and the stress state of the rock is described by three orthogonal principal stresses, [Pg.413]

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