Shear stress solids

Figure 6. Schematic representation of the shear rate program, structural history and accompanying stress response of a shear-rate-increase flow. Shear stress (solid curve) is seen to exhibit a lower second overshoot than the corresponding one-step experiment (dashed curve). Model parameters are identical to those used in previous figures,...

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

Shear stresses are developed in a fluid when a layer of fluid moves faster or slower than a nearby layer of fluid or a solid surface. In laminar flow, the shear stress is equal to the product of fluid viscosity and velocity gradient or rate of shear. Under laminar-flow conditions, shear forces are larger than inertial forces in the fluid. 

T Solid-vapor interfacial energy dyn/cm dyn/cm z Pow der shear stress kg/cm psf... 

Powder Mechanics Measurements As opposed to fluids, powders may withstand applied shear stress similar to a bulk solid due to interparticle friction. As the applied shear stress is increased, the powder will reach a maximum sustainable shear stress T, at which point it yields or flows. This limit of shear stress T increases with increasing applied normal load O, with the functional relationship being referred to as a yield locus. A well-known example is the Mohr-Coulomb yield locus, or... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

So, for given strain rate s and v (a function of the applied shear stress in the shock front), the rate of mixing that occurs is enhanced by the factor djhy due to strain localization and thermal trapping. This effect is in addition to the greater local temperatures achieved in the shear band (Fig. 7.14). Thus we see in a qualitative way how micromechanical defects can enhance solid-state reactivity. 

Glasses, like metals, are formed by deformation. Liquid metals have a low viscosity (about the same as that of water), and transform discontinuously to a solid when they are cast and cooled. The viscosity of glasses falls slowly and continuously as they are heated. Viscosity is defined in the way shown in Fig. 19.7. If a shear stress is applied to the hot glass, it shears at a shear strain rate 7. Then the viscosity, ij, is defined by... 

In a fluid under stress, the ratio of the shear stress, r. to the rate of strain, y, is called the shear viscosity, rj, and is analogous to the modulus of a solid. In an ideal (Newtonian) fluid the viscosity is a material constant. However, for plastics the viscosity varies depending on the stress, strain rate, temperature etc. A typical relationship between shear stress and shear rate for a plastic is shown in Fig. 5.1. 

As a starting point it is useful to plot the relationship between shear stress and shear rate as shown in Fig. 5.1 since this is similar to the stress-strain characteristics for a solid. However, in practice it is often more convenient to rearrange the variables and plot viscosity against strain rate as shown in Fig. 5.2. Logarithmic scales are common so that several decades of stress and viscosity can be included. Fig. 5.2 also illustrates the effect of temperature on the viscosity of polymer melts. 

When a shearing stress is imposed on a solid, deformation occurs, until a point is reached when the internal stresses produced balance the shearing stresses. Provided the elastic limit for the material is not exceeded the solid will return to its original shape when the load is removed. 

A fluid, on the other hand, flows under the action of a shearing stress no matter how small this stress is. A fluid at rest has no shearing stresses, and all forces are at right angles to the surrounding surfaces. Materials such as glass and solid bitumen are fluids and, if stressed for a period of time, will tend to flow. 

When liquid flows along a solid surface (see Fig. 4.2) a shearing stress is set up (friction power/surface), which is expressed by... 

But, there is no need to rely on hugonium. The theory and practice of the deformation of solids under other, less intense, loadings are well developed and show that the fluidlike flow of shock deformation is the expected consequence of the motion of defects in response to applied shear stresses that exceed the shear strength of solids. In most shock loadings, the shear stresses are well in excess of that shear strength and there is certainly ample theory and experiment to qualitatively identify overall features of the defect genera-... 

The defect question delineates solid behavior from liquid behavior. In liquid deformation, there is no fundamental need for an unusual deformation mechanism to explain the observed shock deformation. There may be superficial, macroscopic similarities between the shock deformation of solids and fluids, but the fundamental deformation questions differ in the two cases. Fluids may, in fact, be subjected to intense transient viscous shear stresses that can cause mechanically induced defects, but first-order behaviors do not require defects to provide a fundamental basis for interpretation of mechanical response data. 

Equations (2.9) and (2.10) are representative of all isotropic, homogeneous solids, regardless of the stress-strain relations of a solid. What is strongly materials specific and uncertain is the appropriate value for shear stress, particularly if materials are in an inelastic condition or anisotropic, inhomogeneous properties are involved. The limiting shear stress controlled by strength is termed r. ... 

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

The idealization of a fixed shear stress at which a solid yields mechanically is often qualitatively correct, but yielding is perhaps better characterized as occurring over a range of stresses. For example, the x quartz does not exhibit a precursor until stresses exceed 6 GPa. Nevertheless, there is strong evidence that the yielding process begins to occur at stresses of 4 GPa [74G01]. 

Each stage of particle formation is controlled variously by the type of reactor, i.e. gas-liquid contacting apparatus. Gas-liquid mass transfer phenomena determine the level of solute supersaturation and its spatial distribution in the liquid phase the counterpart role in liquid-liquid reaction systems may be played by micromixing phenomena. The agglomeration and subsequent ageing processes are likely to be affected by the flow dynamics such as motion of the suspension of solids and the fluid shear stress distribution. Thus, the choice of reactor is of substantial importance for the tailoring of product quality as well as for production efficiency. 

If confined phases are exposed to a shear strain, their unique structure, analyzed in the previous section, permits them to sustain a remarkable stress. This is a consequence of mere confinement and is not necessarily coupled to the presence of any solid-like structures of the confined phase [133]. The effect of an exposure to shear stress(es) can be investigated experimentally with the SFA (see Sec. IIA 1). A key quantity determined (in principle) experimentally is the shear stress By using arguments similar to the ones for (see Sec. IV A 1), virial and force expressions for can... 

Bingham-plastic slurries require a shear stress diagram showing shear rate vs. shear stress for the slurry in order to determine the coefficient of rigidity, T], which is the slope of the plot at a particular concentration. This is laboratory data requiring a rheometer. These are usually fine solids at high concentrations. 

In what follows we shall always consider the pressure as having a uniform value for all directions through any point. Gases and liquids at rest satisfy this condition under some circumstances a solid may he treated thermodynamically as a fluid, e.g., when it is immersed in a liquid under pressure and is free from torsion or shearing stress. Conditions (2) and (3), however, very materially limit the range of applicability in such cases. 

Some materials have the characteristics of both solids and liquids. For instance, tooth paste behaves as a solid in the tube, but when the tube is squeezed the paste flows as a plug. The essentia] characteristic of such a material is that it will not flow until a certain critical shear stress, known as the yield stress is exceeded. Thus, it behaves as a solid at low shear stresses and as a fluid at high shear stress. It is a further example of a shear-thinning fluid, with an infinite apparent viscosity at stress values below the yield value, and a falling finite value as the stress is progressively increased beyond this point. 

Lin Q, Jiang F, Wang X-Q, Han Z, Tai Y-C, Lew J, Ho C-M (2000) MEMS Thermal Shear-Stress Sensors Experiments, Theory and Modehng, Technical Digest, Solid State Sensors and Actuators Workshop, Hilton Head, SC, 4—8 June 2000, pp 304-307 Lin TY, Yang CY (2007) An experimental investigation of forced convection heat transfer performance in micro-tubes by the method of hquid crystal thermography. Int. J. Heat Mass Transfer 50 4736-4742... 

The conhned liquid is found to exhibit both viscous and elastic response, which demonstrates that a transition from the liquid to solid state may occur in thin hlms. The solidihed liquid in the him deforms under shear, and hnally yields when the shear stress exceeds a critical value, which results in the static friction force required to initiate the motion. 

Fig. 47—Load-carrying and shearing behavior of confined ILs thin film. Liquid volume is decreased from volume 1 to volume 6 corresponding to a decreasing thickness of ILs films. The confined thin film of ILs exists at the contact area under a normal load of hundreds of MPa and undertakes shearing stress like a solid-solid contact.

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