Stress direct

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

Figure 5.24 The predicted direct stress concentration at different locations within the domain...

The shear stress, t, due to the assembly torque diminishes to zero with time, the preload, F, remaining constant, and so the stress on the solenoid section is only the direct stress,. v, as given in equation 4.75 (see Figure 4.41(b)) (Edwards and McKee, 1991). A second reliability can then be determined by considering the requirement that the pre-load stress remains above a minimum level to avoid loosening in service (0.5 S/)min from experiment) (Marbacher, 1999). The reliability, R, can then be determined from the probabilistic requirement, P, to avoid loosening ... 

The convention normally used is that direct stresses and strains have one suffix to indicate the direction of the stress or strain. Shear stresses and strains have two suffices. The first suffix indicates the direction of the normal to the plane on which the stress acts and the second suffix indicates the direction of the stress (or strain). Poisson s Ratio has two suffices. Thus, vi2 is the negative ratio of the strain in the 2-direction to the strain in the 1-direction for a stress applied in the 1-direction (V 2 = — il for an applied a ). v 2 is sometimes referred to as the major Poisson s Ratio and U2i is the minor Poisson s Ratio. In an isotropic material where V21 = i 2i. then the suffices are not needed and normally are not used. 

The fundamental analysis of a laminate can be explained, in principle, by use of a simple two-layered cross-ply laminate (a layer with fibers at 0° to the x-direction on top of an equal-thickness layer with fibers at 90° to the x-direction). We will analyze this laminate approximately by considering what conditions the two unbonded layers in Figure 4-3 must satisfy in order for the two layers to be bonded to form a laminate. Imagine that the layers are separate but are subjected to a load in the x-direction. The force is divided between the two layers such that the x-direction deformation of each layer is identical. That is, the laminae in a laminate must deform alike along the interface between the layers or else fracture must existl Accordingly, deformation compatibility of layers is a requirement for a laminate. Because of the equal x-direction deformation of each layer, the top (0°) layer has the most x-direction ress because it is stiffer than the bottom (90°) layer in the x-direction./ Trie x-direction stresses in the top and bottom layers can be shown to have the relation... 

The curves for 3M XP251S fiberglass-epoxy are shown in Figures C-1 through C-5 [C-1]. Curves are given for both tensile and compressive behavior of the direct stresses. Note that the behavior in the fiber direction is essentially linear in both tension and compression. Transverse to the fiber direction, the behavior is nearly linear in tension, but very nonlinear in compression. The shear stress-strain curve is highly nonlinear. The Poisson s ratios (not shown) are essentially constant with values v.,2 =. 25 and V21 =. 09. 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles, follows ... 

The second major assumption is that the material is elastic, meaning that the strains are directly proportional to the stresses applied and when the load is removed the deformation will disappear. In engineering terms the material is assumed to obey Hooke s Law. This assumption is probably a close approximation of the material s actual behavior in direct stress below its proportional limit, particularly in tension, if the fibers are stiff and elastic in the Hookean sense and carry essentially all the stress. This assumption is probably less valid in shear, where the plastic carries a substantial portion of the stress. The plastic may then undergo plastic flow, leading to creep or relaxation of the stresses, especially when the stresses are high. 

Isotropic construction Identifies RPs having uniform properties in all directions. The measured properties of an isotropic material are independent on the axis of testing. The material will react consistently even if stress is applied in different directions stress-strain ratio is uniform throughout the flat plane of the material. 

Anomalously high extended defect concentrations may be achieved in crystals by submitting them to directional stress. For example, Willaime and Gaudais (1977) induced dislocation densities of 10 cm in sanidine crystals (KAlSi30g triclinic), and Ardell et al. (1973) reached a dislocation density of 10 cm in quartz with the same method. 

Figure 5.118 Twisting, bending, and shear of an unsymmetrical laminate under direct stress. Reprinted, by permission, from P. C. Powell and A. J. I. Housz, Engineering with Polymers, p. 201. Copyright 1998 by Stanley Thomed Publishers.

Let us first consider the case of an isotropic material, then simplify it for the case of an orthotropic material (same properties in the two directions orthogonal to the fiber axis—in this case, directions 2 and 3), snch as a nnidirectionally reinforced composite lamina. Eqnation (5.128) can be written in terms of the strain and stress components, which are conpled dne to the anisotropy of the material. In order to describe the behavior in a manageable way, it is cnstomary to introdnce a reduced set of nomenclature. Direct stresses and strains have two snbscripts—for example, an, 22, ti2, and Y2i, depending on whether the stresses and strains are tensile (a and s) or shear (t and y) in natnre. The modnli should therefore also have two subscripts En, E22, and G 2, and so on. By convention, engineers nse a contracted form of notation, where possible, so that repeated snbscripts are reduced to just one an becomes a, En becomes En but Gn stays the same. The convention is fnrther extended for stresses and strains, such that distinctions between tensile and shear stresses and strains are... 

An externally applied stress will affect the internal strain and the domain structures will respond this process is termed the ferroelastic effect. Compression will favour polar orientations perpendicular to the stress while tension will favour a parallel orientation. Thus the polarity conferred by a field through 90° domain changes can be reversed by a compressive stress in the field direction. Stress will not affect 180° domains except in so far as their behaviour may be coupled with other domain changes. 

What if we could make perfectly extended chain fibers, so that the covalent bonds were being directly stressed These should be very stiff and strong in the draw direction, but much weaker and with a much lower modulus perpendicular to this direction. 

Stress-Strain Relations for a Fluid. The medium is hydrostatic when the direct stresses in three orthogonal directions are equal and the shear stresses are zero a.. = a.. = (3A + 2p) e... 

Steady state acoustic response of the unit cell occurred for the composites considered after the passage of some five acoustic oscillations. Patterns of direct stress, and shear stress as shown in Figures 2 and 3 were obtained. As expected, the corners of the cavities concentrated the stresses. Viscoelastic energy loss calculations, not discussed here, also show that the corners of the cavities are concentrations of energy losses. 

The direct stress due to the weight of the vessel, its contents, and any attachments. The stress will be tensile (positive) for points below the plane of the vessel supports, and compressive (negative) for points above the supports, see Figure 13.18. The dead-weight stress will normally only be significant, compared to the magnitude of the other stresses, in tall vessels. 

When a tensile stress is applied to the solid containing the particle direct stresses CTjj and appear at point A. They are... 

The direct stress due to the weight of the vessel, its contents, and any attachments. The stress will be tensile (positive) for points below the plane of... 

For a confined powder which is composed of the particles the response to the application of stress follows the steps discussed above. That is, before the transmission of stress to an individual particle in the body of the powder is the same as if it were directly stressed, the particles have to rearrange to allow multiple points of inter-particulate contact and approach the Tully dense" state (relative to the individual particles). This results in a lag in the stress-strain prorile however the yield point and energy of deformation should remain close to that of the individual particle property once corrected for the number of particles in the powder. A schematic stress-strain prorile for a particle and for a powder made up of the particles is illustrated in Figure 2A. Figure 2B shows the same plots normalized. 

Stress is defined as force per unit area. The elastic response of a solid consists of deformation which occurs in all the directions. Since force has three components in three (chosen) x, y and z directions and since the deformation also occurs in all three directions, stress, a, is a tensor of the second rank having 9 components described by a (3 x 3) matrix cry, y-i. 3. Alternately it may be easier to visualize why a is a tensor of second rank if one recalls that area itself can be described by a vector perpendicular to the surface which has three components. 

Figure 5. Residual stresses perpendicular to the weld direction also crack growth direction stresses measured on the M(T) specimen used for obtaining the crack growth rates.

The optical properties of semicrystalline polymers are often anisotropic. On the other hand, amorphous polymers are normally isotropic unless directional stresses are frozen in a glassy specimen during fabrication by a process such as injection molding. Anisotropy can often be induced in an amorphous polymer by imposing an electric field (Kerr effect), a magnetic field (Cotton-Mouton effect), or a mechanical deformation. Such external perturbations can also increase the anisotropy of a polymer that is anisotropic even in the absence of the perturbation. 

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