Material shear stress

Shear stress n. Force per imit area acting in the plane of the area to which the force is applied. In an elastic body, shear stress is equal to shear modulus times shear strain. In an inelastic fluid, shear stress is equal to viscosity times the shear rate. In viscoelastic materials, shear stress will be a function of both shear strain and shear rate. 

For the filling phase it is known that many surface defects may be caused by sudden changes in the flow front velocity. Hence a common constraint is that the flow front velocity be constant. Of course in attempting to achieve this, one must be careful to ensure that the material shear stress limits are not exceeded and that the temperature of the material is within permissible limits. 

In an inelastic fluid, shear stress is equal to viscosity times the shear rate. In viscoelastic materials, shear stress will be a function of both shear strain and shear rate. 

Criteria of Elastic Failure. Of the criteria of elastic failure which have been formulated, the two most important for ductile materials are the maximum shear stress criterion and the shear strain energy criterion. According to the former criterion, from equation 7... 

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

Fig. 4. (a) Shear stress diagram for elastic ideal plastic material (b) partiady plastic thick-waded cylinder. 

If it is assumed that yield and subsequent plastic flow of the material occurs in accordance with the maximum shear stress criterion, then /2 may be substituted for in the above and subsequent equations. For the shear strain energy criterion it may be assumed, as a first approximation, that the corresponding value is G j fz. Errors in this assumption have been discussed (11). 

It may be shown (33) that when the inner surface of a cylinder made of components of the same material is subjected to an internal pressure, the bote of each component experiences the same shear stress provided all components have the same diameter ratio. For these optimum conditions,... 

If the sum of the mechanical allowances, c, is neglected, then it may be shown from equation 15 that the pressure given by equation 33 is half the coUapse pressure of a cylinder made of an elastic ideal plastic material which yields in accordance with the shear stress energy criterion at a constant value of shear yield stress = y -... 

A material s flow function is usually measured on the same tester as the wall friction angle, although the cell arrangement is somewhat different (Fig. 6). ConsoHdation values are easily controUed, and the cohesive strength of the bulk soHd is determined by measuring interparticle shear stresses while some predeterrnined normal pressure is being appHed. 

A Hquid is a material that continues to deform as long as it is subjected to a tensile and/or shear stress. The latter is a force appHed tangentially to the material. In a Hquid, shear stress produces a sliding of one infinitesimal layer over another, resulting in a stack-of-cards type of flow (Fig. 1). 

The square root of viscosity is plotted against the reciprocal of the square root of shear rate (Fig. 3). The square of the slope is Tq, the yield stress the square of the intercept is, the viscosity at infinite shear rate. No material actually experiences an infinite shear rate, but is a good representation of the condition where all rheological stmcture has been broken down. The Casson yield stress Tq is somewhat different from the yield stress discussed earlier in that there may or may not be an intercept on the shear stress—shear rate curve for the material. If there is an intercept, then the Casson yield stress is quite close to that value. If there is no intercept, but the material is shear thinning, a Casson plot gives a value for Tq that is indicative of the degree of shear thinning. 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Effect of Temperature. In addition to being often dependent on parameters such as shear stress, shear rate, and time, viscosity is highly sensitive to changes in temperature. Most materials decrease in viscosity as temperature increases. The dependence is logarithmic and can be substantial, up to 10% change/°C. This has important implications for processing and handling of materials and for viscosity measurement. 

When an electric field is appHed to an ER fluid, it responds by forming fibrous or chain stmctures parallel to the appHed field. These stmctures greatly increase the viscosity of the fluid, by a factor of 10 in some cases. At low shear stress the material behaves like a soHd. The material has a yield stress, above which it will flow, but with a high viscosity. The force necessary to shear the fluid is proportional to the square of the electric field (116). 

Additional complications can occur if the mode of deformation of the material in the process differs from that of the measurement method. Most fluid rheology measurements are made under shear. If the material is extended, broken into droplets, or drawn into filaments, the extensional viscosity may be a more appropriate quantity for correlation with performance. This is the case in the parting nip of a roUer in which filamenting paint can cause roUer spatter if the extensional viscosity exceeds certain limits (109). In a number of cases shear stress is the key factor rather than shear rate, and controlled stress measurements are necessary. 

Initiation. Free-radical initiators are produced by several processes. The high temperatures and shearing stresses required for compounding, extmsion, and molding of polymeric materials can produce alkyl radicals by homolytic chain cleavage. Oxidatively sensitive substrates can react directly with oxygen, particularly at elevated temperatures, to yield radicals. 

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