Stress—Continue proportional limit

When a solid is subjected to a shearing force, the solid (simultaneously with the application of force) deforms, and internal stresses develop until a condition of static equilibrium is reached. Within the elastic limit of a substance, these internal stresses are proportional to the induced shearing strains (deformation). The ability of a material to reach static equilibrium, rather than deform continuously, is due to a property called shear strength. 

The ultimate/failure strength in torsion, the outer flbers of a section are the first to shear, and the rupture extends toward the axis as the twisting is continued. The torsion formula for round shafts has no theoretical basis after the shearing stresses on the outer fibers exceed the proportional limit, as the stresses along the section then are no longer proportional to their distances from the axis. It is convenient, however, to compare the torsional strength of various materials by using the formula to compute values of x at which rupture takes place. 

The existence of an elevated temperature, with or without long-term or continuous loading, would suggest the possibility that a material might exceed its elastic limits. As explained earlier concerning momentary loading, the properties to consider are the proportional limit and the maximum shear stress. 

An approximate sketch of the stress-strain diagram for mild steel is shown in Fig. 2.8(a). The numbers given for proportional limit, upper and lower yield points and maximum stress are taken from the literature, but are only approximations. Notice that the stress is nearly hnear with strain until it reaches the upper yield point stress which is also known as the elastic-plastic tensile instability point. At this point the load (or stress) decreases as the deformation continues to increase. That is, less load is necessary to sustain continued deformation. The region between the lower yield point and the maximum stress is a region of strain hardening, a concept that is discussed in the next section. Note that if true stress and strain are used, the maximum or ultimate stress is at the rupture point. 

As velocity continues to rise, the thicknesses of the laminar sublayer and buffer layers decrease, almost in inverse proportion to the velocity. The shear stress becomes almost proportional to the momentum flux (pk ) and is only a modest function of fluid viscosity. Heat and mass transfer (qv) to the wall, which formerly were limited by diffusion throughout the pipe, now are limited mostly by the thin layers at the wall. Both the heat- and mass-transfer rates are increased by the onset of turbulence and continue to rise almost in proportion to the velocity. 

Structural dements resist blast loads by developing an internal resistance based on material stress and section properties. To design or analyze the response of an element it is necessary to determine the relationship between resistance and deflection. In flexural response, stress rises in direct proportion to strain in the member. Because resistance is also a function of material stress, it also rises in proportion to strain. After the stress in the outer fibers reaches the yield limit, (lie relationship between stress and strain, and thus resistance, becomes nonlinear. As the outer fibers of the member continue to yield, stress in the interior of the section also begins to yield and a plastic hinge is formed at the locations of maximum moment in the member. If premature buckling is prevented, deformation continues as llic member absorbs load until rupture strains arc achieved. 

This means that in the elastic region, pressure and density are linearly related. Beyond the elastic region, the wave velocity increases with pressure or density and Pip is not linearly proportional. Wave velocity continues to increase with stress or pressure throughout the region of interest. Therefore, up to the elastic limit, the sound velocity in a material is constant. Beyond the elastic limit, the velocity increases with increasing pressure. Let us look at a major implication of this fact. Consider the pressure wave shown in Figure 14.3. 

The fundamental property of a classical fluid, to cite Lamb s treatise on hydrodynamics [1], is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the cominon surface. In other words, the only stress that a surface of an element of fluid at rest can sustain is a normal pressure. Pressure oriented other than normally can be resolved into a component perpendicular to the surface and a tangential component, the latter of which will induce motion. One of the fundamental distinctions between the response of an elastic solid and a classical fluid to tangential stress is that there is a limited displacement within the solid which is proportional to the stress whereas the motion of a fluid continues as long as the stress is maintained. 

As a melt is subjected to a fixed stress (or strain), the deformation vs. time curve will show an initial rapid deformation followed by a continuous flow (Fig. 1-6). The relative importance of elasticity (deformation) and viscosity (flow) depends on the time scale of the deformation. For a short time, elasticity dominates over a long time, the flow becomes purely viscous. This behavior influences processes when a part is annealed, it will change its shape or, with post-extrusion (Chapter 5), swelling occurs. Deformation contributes significantly to process flow defects. Melts with small deformation have proportional stress-strain behavior. As the stress on a melt is increased, the recoverable strain tends to reach a limiting value. It is in the high-stress range, near the elastic limit, that processes operate. 

In the proportional (elastic) region, Hooke s law applies, and Yoimg s modulus E = o/e. Beyond the elastic limit there is no stress-to-strain proportionality, and Hooke s law does not apply. Beyond the elastic limit, stress can be constant while strain continues to increase. A viscoelastic solid plastic has viscous fluid characteristics beyond the elastic limit. 

The cellular type of mechanical organization has certain limitations and is suitable mainly for static tissue situations. Animals lead active lives and their mechanical structures are adapted to various dynamic requirements, both in the stresses they have to meet and in the changes which tissues undergo in the course of a lifetime. As a result, in many animal tissues, the cells have lost their primary mechanical function and occupy a very much smaller proportion of the total volume of the tissue than in plants. In animal tissues the mechanical function is taken over by extracellular tissue elements, comprising protein fibres embedded in a polymeric aqueous gel. The restriction of mechanical function to extracellular tissue components permits larger spaces between cells and makes possible a greater freedom in tissue pattern. Thus more than one type of cell may occur in the same tissue and there is scope for continuous reorganization to meet the... 

Particulate materials, such as clay or particulate gels of the type discussed in Chapter 4, may be plasticy rather than viscoelastic. Two simple types of plastic behavior are illustrated in Fig. 18b a perfectly plastic material is elastic up to t)xt yield stress, Oy > but it deforms without limit if a higher stress is applied in a linearly hardening material there is a finite slope after the yield stress. In real plastic materials, the stress-strain relations are likely to be curved, rather than linear. If the stress is raised above Oy and then released, the elastic strain is recovered but the plastic strain is not. This differs from viscous behavior in its time-dependence if the stress on a linearly hardening plastic material is raised to Oh and held constant, the strain remains at a viscoelastic material would continue to deform at a rate proportional to Ou/ri-... 

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