Isotropic stress field

The loading of the structure is mechanically very simple since it consists in an initial isotropic stress field related to the dead weight. Concerning the mechanical boundary conditions, a zero normal stress is prescribed on the free boundaries of the pattern (wall of the drifts and well, and ground surface), and the symmetry planes are characterized by a zero normal displacement. Before excavation, the rock mass is supposed to be in a compressive stress state, and the principal minor stress 03, indicator of maximum compression, is equal to CTh (and 0i=O2=o =ayy). This stress increases (in absolute value) with depth, from -1.1 MPa at the top of the wells to -1.6 MPa at its base, and to -3.1 MPa in lower limit of the model. The excavation of drifts and wells causes a disturbance of this initial stress field (see fig.3). It is noticed that, apart... 

When the probe makes contact with the film, it generates a radial stress field around the point of contact. If the film is isotropic, it deforms in a uniform ring around the probe, as shown in Fig. 8.11 a). If the film is oriented, it deforms in a non-uniform manner. When the film is mildly oriented, the deformation area becomes ellipsoidal, as we see in Fig. 8.11 b), with its long axis... 

Next, let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be... 

Figure 3.8 Edge dislocation in an isotropic elastic body. Solid lines indicate isopotential cylinders for the portion of the diffusion potential of any interstitial atom present in the hydrostatic stress field of the dislocation. Dashed cylinders and tangential arrows indicate the direction of the corresponding force exerted on the interstitial atom.

The particulate phase in the annular zone of a spouted bed can be described as an isotropic, incompressible, rigid plastic, non-cohesive Coulomb powder. Assuming that this material is in a quasi-static critical condition, the stress field can be described by equations developed for a static material element. 

The superscripts e and 5 refer to edge and screw and serve as an instruction to use the isotropic linear elastic stress fields for the edge and screw dislocation, respectively, but with the Burgers vector adjusted to account for relevant trigonometric weighting factors. 

The analyses emphasize the idea that when a volume-element changes volume by diffusive mass transfer at constant density, the change is not isotropic the strain components Cyy, and e are linked by three separate equations to the components and gradients of the driving stress field. 

Another factor of anisotropic design analysis is greater dependence of stress distributions on materials properties. For isotropic materials, whether elastic, viscoelastic, etc., static values often result in stress fields which are independent of material stiffness properties. In part, this is due to the fact that Poisson s ratio is the only material parameter appearing in the compatibility equations for stress. This parameter does not vary widely between materials. However, the compatibility equations in stress for anisotropic materials depend on ratios of Young s moduli for different material axes, and this can introduce a strong dependence of stress on material stiffness. This approach can be used in component design, but the product and material design analysis become more closely related. 

A stress field applied transversely to the grating modifies the structure of the previously isotropic optical fibre, generating two perpendicular directions with different indices of refraction. For a single Bragg grating spacing... 

The fracture toughness, a term defined by Irwin (1956, 1960) to characterize brittleness, provides a measure of the conditions required for catastrophic crack propagation in a material (see Section 1.6). One fracture toughness parameter is the surface fracture energy y, defined as one-half G, the critical strain energy release rate above which catastrophic failure occurs. In turn G is related to another convenient toughness parameter, the critical stress intensity factor a measure of the stress field at the crack tip. For fracture of an isotropic material in a plane strain modet (Baer, 1964, p. 946) ... 

The Mori-Tanaka model is derived based on the principles of Eshelby s inclusion model for predicting an elastic stress field in and around elUpsoidal filler in an infinite matrix. The complete analytical solufions for longitudinal SI and transverse elastic moduh of an isotropic matrix filled with aligned spherical inclusion are [45,... 

The SMP system is considered to be macroscopicaUy isotropic and homogeneous. A uniform stress field assumption is held. This assumption suggests that when comparing with experimental results, uniaxial compression test results are preferred because specimens under uniaxial loading create a uniform stress distribution within the gauge length of the specimens. 

The Mori-Tanaka model is uses for prediction an elastic stress field for in and around an ellipsoidal reinforcement in an infinite matrix. This method is based on Eshebly s model. Longitudinal and transverse elastic modulus, Ejj and for isotropic matrix and directed spherical reinforcement are ... 

Using a commercially available finite element analysis program, ABAQUS, models representing SiC particulates in a ZrBa matrix were created. The SiC phase was modeled as a round particle in a two dimension (2D) ZrB matrix. Material properties were assumed to be isotropic for both the ZrB and the SiC after initial modeling efforts indicated only small changes in stress fields as a result of the anisotropic properties of the a-SiC (hexagonal polytype). The material properties used in the models, as well as other key model input variables, are included as Table II. 

The explanation of the effect of secondary component on the spreading of shear yielding (i.e., delocalization of shear banding) is based on a concept of local stress fields and stress concentrations in the matrix due to the presence of inclusions. This leads to a reduction of the external load needed to plastically deform the material. The original Goodier s solution (7) for an isolated particle in an isotropic matrix resulted in a maximum stress concentration of about 1.9 at the equator of the inclusion (8). It should be borne in mind that this solution... 

Cases with arbitrary orientations in an isotropic material (i.e., when stress depends on the orientation of the dislocation and the Burgers vector and the crystal axes are significant), stress fields and stress distributions around edge dislocations are shown and discussed in detail in Read s book. 

For all flaws, the local stress is a maximum adjacent to the flaw and decreases rapidly away from the flaw. Irwin obtained an expression for the stresses in the vicinity of a crack tip that exemplifies how these local stresses vary (Figure 9.1). In these equations, K-[ is the stress intensity factor and a function of the applied stress and the flaw severity. One of the features of these equations that is important in understanding how cracks propagate is that the stress is a maximum in the plane in front of the crack tip. As a consequence, for an isotropic material, the bonds direcdy in front of the crack tip are the ones that will fail that is, the crack grows in the plane perpendicular to the apphed stress. Moreover, in a uniform, planar, applied stress field (where O, = = constant), once the crack has started to grow in a given plane, it will... 

Here p l represents the isotropic pressure field, G(t) is the stress relaxation shear modulus, Gg is the equilibrium shear modulus which has a finite value for crosslinked rubbers, but equals zero for melts, and CT under the inte-gral stands for " (t ). Often dG(t-t )/dt is identified with a memory function m(t-t ). In that case the equation is transformed into ... 

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