Stress, dimensions

Ctjki is a fourth order tensor that linearly relates a and e. It is sometimes called the elastic rigidity tensor and contains 81 elements that completely describe the elastic characteristics of the medium. Because of the symmetry of a and e, only 36 elements of Cyu are independent in general cases. Moreover only 2 independent rigidity constants are present in Cyti for linear homogeneous isotropic purely elastic medium Lame coefficient A and /r have a stress dimension, A is related to longitudinal strain and n to shear strain. For the purpose of clarity, a condensed notation is often used... 

Modulus of elasticity Material parameter describing the relation of tension to expansion, when a material is under mechanical stress. Dimension N/mm2 (MPa). 

This equation can be interpreted as requiring the normality of the plastic strain increment vector to yield the surface in the hyper-space of n stress dimensions. As before dl is the proportionality constant. 

Fig. 4. Measurement results of a pin of dimensions 46/15,5 as a function of tensile and compressive stress using the Wirotest .

In Fig, 4. measurement results of a pin of dimensions 46/15,5 have been shown as a function of tensile and compressive stress using the Wirotest , where the course of the graph of these dependence on the load in the linear scope of indications is similar to the compressive as well as the tensile stress. 

Actual crystal planes tend to be incomplete and imperfect in many ways. Nonequilibrium surface stresses may be relieved by surface imperfections such as overgrowths, incomplete planes, steps, and dislocations (see below) as illustrated in Fig. VII-5 [98, 99]. The distribution of such features depends on the past history of the material, including the presence of adsorbing impurities [100]. Finally, for sufficiently small crystals (1-10 nm in dimension), quantum-mechanical effects may alter various physical (e.g., optical) properties [101]. 

The elasticity of a fiber describes its abiUty to return to original dimensions upon release of a deforming stress, and is quantitatively described by the stress or tenacity at the yield point. The final fiber quaUty factor is its toughness, which describes its abiUty to absorb work. Toughness may be quantitatively designated by the work required to mpture the fiber, which may be evaluated from the area under the total stress-strain curve. The usual textile unit for this property is mass pet unit linear density. The toughness index, defined as one-half the product of the stress and strain at break also in units of mass pet unit linear density, is frequentiy used as an approximation of the work required to mpture a fiber. The stress-strain curves of some typical textile fibers ate shown in Figure 5. 

Creep. The creep characteristic of plastic foams must be considered when they are used in stmctural appHcations. Creep is the change in dimensions of a material when it is maintained under a constant stress. Data on the deformation of polystyrene foam under various static loads have been compiled (158). There are two types of creep in this material short-term and long-term. Short-term creep exists in foams at all stress levels however, a threshold stress level exists below which there is no detectable long-term creep. The minimum load required to cause long-term creep in molded polystyrene foam varies with density ranging from 50 kPa (7.3 psi) for foam density 16 kg/m (1 lb /ft ) to 455 kPa (66 psi) at foam density 160 kg/m (10... 

A more important effect of prestressiag is its effect on the mean stress at the bore of the cylinder when an internal pressure is appHed. It may be seen from Figure 6 that when an initially stress-free cylinder is subjected to an internal pressure, the shear stress at the bore of the cylinder increases from O to A. On the other hand, when a prestressed cylinder of the same dimensions is subjected to the same internal pressure, the shear stress at the bore changes from C to E. Although the range of shear stress is the same ia the two cases (distance OA = CE), the mean shear stress ia the prestressed cylinder, represented by point G, is smaller than that for the initially stress-free cylinder represented by point H. This reduction in the mean shear stress increases the fatigue strength of components subjected to repeated internal pressure. 

Elastic Properties. The abiUty of a fiber to deform under below-mpture loads and to return to its original configuration or dimension upon load removal is an important performance criterion. Permanent deformation may be as detrimental as actual breakage, rendering a product inadequate for further use. Thus, the repeated stress or strain characteristics are of significance in predicting or evaluating functional properties. 

The method makes use of the principle that a constant ratio of induced stress. s in the stored contents to the consolidating pressure p exists. Thus, for any hopper design for which th.eff curve is available, the shear-tester results can be potted, and the point where/= s is located. Since the distance at which this occurs above the hopper vertex is also known, these values become the hopper dimensions at that point. 

The parameters for the model were originally evaluated for oil shale, a material for which substantial fracture stress and fragment size data depending on strain rate were available (see Fig. 8.11). In the case of a less well-characterized brittle material, the parameters may be inferred from the shear-wave velocity and a dynamic fracture or spall stress at a known strain rate. In particular, is approximately one-third the shear-wave velocity, m has been shown to be about 6 for various brittle materials (Grady and Lipkin, 1980), and k can then be determined from a known dynamic fracture stress using an analytic solution of (8.65), (8.66) and (8.68) in one dimension for constant strain rate. 

Because strain is dimensionless, the moduli have the same dimensions as those of stress force per unit area (N m ). In those units, the moduli are enormous, so they are usually reported instead in units of GPa. 

A metal bar of width w is compressed between two hard anvils as shown in Fig. Al.l. The third dimension of the bar, L, is much greater than zu. Plastic deformation takes place as a result of shearing along planes, defined by the dashed lines in the figure, at a shear stress k. Find an upper bound for the load F when (a) there is no friction between anvils and bar, and (b) there is sufficient friction to effectively weld the anvils to the bar. Show that the solution to case (b) satisfies the general formula... 

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