Stress vector, magnitude

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. 

The minus sign in this equation is a matter of convention t(n) is considered positive when it acts inward on a surface whereas n is the outwardly directed normal, andp is taken as always positive. The fact that the magnitude of the pressure (or surface force) is independent of n is self-evident from its molecular origin but also can be proven on purely continuum mechanical grounds, because otherwise the principle of stress equilibrium, (2 25), cannot be satisfied for an arbitrary material volume element in the fluid. The form for the stress tensor T in a stationary fluid follows immediately from (2 59) and the general relationship (2-29) between the stress vector and the stress tensor ... 

On the basis of the complex representation for the dynamic mechanical modulus, the relationships between stress and strain can be considered on a complex plane in terms of vectors (74), as shown in Figure 32c. The applied strain of magnitude o is out of phase with the resulting stress of magnitude ctq by an amoimt represented by angle 5. The stress components of a are resolved as a, in-phase, and a", 90° out-of-phase with e = o-... 

We designate the magnitude of these stress components with a capital T and use two subscripts to identify each one. The first subscript refers to the plane on which the components are acting the second indicates the direction of the component on that plane. If we take another cut, say with a normal vector m, through the same point in the body, then the stress vector acting on m will be with... 

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... 

The spatial Cauchy stress tensor s is defined at time by f = sn, where t(x, t, n) is a contact force vector acting on an element of area da = n da with unit normal i and magnitude da in the current configuration. The element of area... 

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... 

In an isotropic medium, vectors such as stress S and strain X are related by vector equations such as, S = kX, where S and X have the same direction. If the medium is not isotropic the use of vectors to describe the response may be too restrictive and the scalar k may need to be replaced by a more general operator, capable of changing not only the magnitude of the vector X, but also its direction. Such a construct is called a tensor. 

A vector field, such as force, F(f,t) or flux, J(r, t), requires specification of a magnitude and a direction in reference to a fixed frame. A rank-two tensor field such as stress, er(r, t), relates a vector field to another vector often attached to the material in question for example, cr = F(r, t)/A, where F(r, t) is the force exerted by the stress, cr, on a virtual area embedded in the material and represented by the vector A = An, where n is the unit normal to the area and A is the magnitude of the area. 

Magnitudes of binary and ternary force interactions required in Eq. (62) are calculated by simple elastic or visco-elastic constitutive equations and then projected into force vectors. Elastic model is the direct implementation of the Hook s law with the stress between micro-elements A and B dependent linearly on the strain eAB ... 

Tlie strain vector OB can be resolved into vector OE along the direction of OA and OF perpendicular to OA. Then the projection OH of OE on the vertical axis is the magnitude of the strain which is in phase with the stress at any lime. Similarly, projection 01 of vector OF is the magnitude of the strain which is 90 (one-quarter cycle) out of phase with the stress. The stress can be similarly resolved into two components with one along the direction of OB and one leading the strain vector by n/2 rad. 

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