Total stress components

If a filament or rod of a virgin (stress-free) material is stretched at a constant extension rate, the total stress component ajj, initially zero, will increase with time in a manner which depends on the nature of the polymer. It is then possible to define a tensile stress growth coefficient ... 

Let us now discuss what other parameters should be measured in order to determine normal stress differences (see Chapter 2). The total stress component normal to the direction of shear (that is, the surface of the cone or plate) is... 

Thus, the rest of the nonzero total stress components are equal to... 

In the simplest case, the vertical (total) stress component is given by the weight of the overburden... 

Note that since i and j each represent any of three possible directions, there are a total of nine possible components of the stress tensor (at any given point in a fluid). However, it can readily be shown that the stress tensor is symmetrical (i.e., the ij components are the same as the ji components), so there are at most six independent stress components. 

Thus the total stress, cry, at any point within a fluid is composed of both the isotropic pressure and anisotropic stress components, as follows ... 

These turbulent momentum flux components are also called Reynolds stresses. Thus, the total stress in a Newtonian fluid in turbulent flow is composed of both viscous and turbulent (Reynolds) stresses ... 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

Explicit forms for the stress tensors d1 are deduced from the microscopic expressions for the component stress tensors and from the scheme of the total stress devision between the components [164]. Within this model almost all essential features of the viscoelastic phase separation observable experimentally can be reproduced [165] (see Fig. 20) existence of a frozen period after the quench nucleation of the less viscous phase in a droplet pattern the volume shrinking of the more viscous phase transient formation of the bicontinuous network structure phase inversion in the final stage. 

It follows that consists of two stress components a crack tip debond stress, at, and a friction stress component, at is not only a function of the interfacial fracture toughness, G c, but is also dependent on the debond length, t, relative to the total... 

By dividing the constraint forces into components induced by the elastic and flow forces, the total stress may be expressed as the sum... 

Equations (5.139) to (5.142) are the basic equations for a gas-solid flow. More detailed information on both the fluid-particle interacting force Fa and the total stresses T and Tp must be specified before these equations can be solved. One approach to formulate the fluid-particle interacting force FA is to decompose the total stress into a component E representing the macroscopic variations in the fluid stress tensor on a scale that is large compared to the particle spacing, and a component e representing the effect of detailed variations of the point stress tensor as the fluid flows around the particle [Anderson and... 

Figure 7.4. Total, elastic, and viscous stress-strain curves for collagen fibers from rat tail tendon. The total stress-strain curve (open boxes) was obtained by collecting all the initial, instantaneous, force measurements at increasing time intervals and then dividing by the initial cross-sectional area. The elastic stress-strain curve (closed diamonds) was obtained by collecting all the force measurements at equilibrium and then dividing by the initial cross-sectional area. The viscous component curve (closed squares) was obtained as the difference between the total and the elastic stresses. Error bars represent one standard deviation of the mean.

Figure 7.7. Total, elastic, and viscous stress-strain curves for uncrosslinked self-assembled type I collagen fibers.Total (open squares), elastic (filled diamonds), and viscous (filled squares) stress-strain curves for self-assembled uncrosslinked collagen fibers obtained from incremental stress-strain measurements at a strain rate of 10%/min. The fibers were tested immediately after manufacture and were not aged at room temperature. Error bars represent one standard deviation of the mean value for total and viscous stress components. Standard deviations for the elastic stress components are similar to those shown for the total stress but are omitted to present a clearer plot. The straight line for the elastic stress-strain curve closely overlaps the line for the viscous stress-strain curve. Note that the viscous stress-strain curve is above the elastic curve suggesting that viscous sliding is the predominant energy absorbing mechanism for uncrosslinked collagen fibers.

Forces applied to a water-saturated porous medium will cause stresses which result in strain (deformation). The stress, strain and groundwater pressure in a water-saturated porous medium are coupled, as first recognized by Biot (1941). Under the assumed stress conditions, the vertical normal component of total stress (o ) that acts downwards on a horizontal plane at any depth is caused by the weight of the overlying water-saturated rock. This stress is born by the solid matrix of the porous medium (o ) and by the pressure of the groundwater in the pores (p ) (e.g. Hubbert andRubey, 1959)... 

The changes in vertical effective component of total stress are thus directly coupled to changes in pressure of the groundwater. 

Let <5 denote the length of the connector chain pulled out from the polymer by a force / as shown in Fig. 8. When 6=1, where l is the total connector chain length, the chain is completely pulled out and the force/vanishes. Assuming that the chains are pulled out normal to the interface so that the tangential component of the pullout force is zero, then a, the traction stress component normal to the planar interface, is related to the force/and 2, the number of chains crossing a unit area of the interface, by ct=/2. a is related to the rate of chain pullout <5(f, x) and the remaining chain length l - by ... 

The total stress tensor (i.e., force due to the stresses per surface unit, sometimes called traction ) has commonly been divided into two components the hydrostatic pressure, pe, and the viscous stress, o, tensors ([11]) ... 

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