Solids strain

Here, we present in suminary form the equations that describe solid strain, solid stress, and solid internal energy of the porous medium. First, the internal energy statement ... 

Post-yield large-strain plastic response of glassy solids strain softening and strain hardening... 

Ribbed smoked sheets (RSS) are made from intentionally coagulated latex. In this process, strained latex is diluted with water to 15% solids, strained again to remove particulate matter, and then coagulated by addition of an acid. The coagulum is compressed into 2-3 mm thick sheets and dried for 4-7 days in large smoke houses. 

Raw material Type of pretreatment Amount of solids Strain Cultivation on hydrolyzate Product Mode of operation... 

Constitutive equation n. In material science, an equation that relates stress in a material to strain or strain rate. Simple examples are (1) Hooke s law, which states that, in elastic solids, strain is directly proportional to stress, and (2) Newton s law of flow, which states that, in laminar shear flow, the shear rate is equal to the shear stress divided by the viscosity. Few plastic solids and liquids obey either of these laws. 

The equations of electrocapillarity become complicated in the case of the solid metal-electrolyte interface. The problem is that the work spent in a differential stretching of the interface is not equal to that in forming an infinitesimal amount of new surface, if the surface is under elastic strain. Couchman and co-workers [142, 143] and Mobliner and Beck [144] have, among others, discussed the thermodynamics of the situation, including some of the problems of terminology. 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

The procedures used for estimating the service life of solid rocket and gun propulsion systems include physical and chemical tests after storage at elevated temperatures under simulated field conditions, modeling and simulation of propellant strains and bond tine characteristics, measurements of stabilizer content, periodic surveillance tests of systems received after storage in the field, and extrapolation of the service life from the detailed data obtained (21—33). 

A useful approximation of B for a conical hopper is B = 22f/a, where a is the bulk density of the stored product. The apparatus for determining the properties of solids has been developed and is offered for sale by the consulting firm of Jenike and Johansen, Winchester, Massachusetts, which also performs these tests on a contract basis. The flow-factor FF tester, a constant-rate-of-strain, direct-shear-type machine, gives the locus of points for the FF cui ve as well as ( ), the... 

Decompositions may be exothermic or endothermic. Solids that decompose without melting upon heating are mostly such that can give rise to gaseous products. When a gas is made, the rate can be affected by the diffusional resistance of the product zone. Particle size is a factor. Aging of a solid can result in crystallization of the surface that has been found to affect the rate of reaction. Annealing reduces strains and slows any decomposition rates. The decompositions of some fine powders follow a first-order law. In other cases, the decomposed fraction x is in accordance with the Avrami-Erofeyev equation (cited by Galwey, Chemistry of Solids, Chapman Hall, 1967)... 

Wave interaetions will often eause a solid to be shoeked more than onee. For materials initially at standard eonditions, the state aehieved by the first shoek wave must lie on the prineipal Hugoniot. Any subsequent shoek wave is eentered on that preshoeked state, and in general will lie on a different reeentered, or seeond Hugoniot. However, for small strains in most materials it ean be shown that the differenee between the prineipal and seeond Hugoniots is negligible for most purposes (see Problems, Seetion 2.20). 

We imagine a finite-duration shock pulse arriving at some point in the material. The strain as a function of time is shown as the upper diagram in Fig. 7.11 for elastic-perfectly-plastic response (solid line) and quasi-elastic response generally observed (dash-dot line). The maximum volume strain = 1 - PoIp is designated... 

So, for given strain rate s and v (a function of the applied shear stress in the shock front), the rate of mixing that occurs is enhanced by the factor djhy due to strain localization and thermal trapping. This effect is in addition to the greater local temperatures achieved in the shear band (Fig. 7.14). Thus we see in a qualitative way how micromechanical defects can enhance solid-state reactivity. 

Plastic strain localization and mixing due to void collapse in porous materials works in the same way, with perhaps an even greater degree of actual mixing due to jetting, and other extreme conditions that can occur at internal free surfaces in shock-loaded solids. 

R.J. Clifton, On the Analysis of Elastic/Visco-Plastic Waves of Finite Uniaxial Strain, in Shock Waves and the Mechanical Properties of Solids (edited by J.J. Burke and V. Weiss), Syracuse University Press, 1971, pp. 73-119. 

This linear relationship between stress and strain is a very handy one when calculating the response of a solid to stress, but it must be remembered that most solids are elastic only to very small strains up to about 0.001. Beyond that some break and some become plastic - and this we will discuss in later chapters. A few solids like rubber are elastic up to very much larger strains of order 4 or 5, but they cease to be linearly elastic (that is the stress is no longer proportional to the strain) after a strain of about 0.01. 

Well, that is the case at the low temperature, when the rubber has a proper modulus of a few GPa. As the rubber warms up to room temperature, the Van der Waals bonds melt. (In fact, the stiffness of the bond is proportional to its melting point that is why diamond, which has the highest melting point of any material, also has the highest modulus.) The rubber remains solid because of the cross-links which form a sort of skeleton but when you load it, the chains now slide over each other in places where there are no cross-linking bonds. This, of course, gives extra strain, and the modulus goes down (remember, E = [Pg.61]

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic behaviour. This is the behaviour characterised by Hooke s Law (Chapter 3). All solids are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1%. The slope of the stress-strain line, which is the same in compression as in tension, is of... 

Fig. 8.1. Stress-strain behaviour for a linear elastic solid. The axes are calibrated for a material such as steel.

Figure 8.2 shows a non-linear elastic solid. Rubbers have a stress-strain curve like this, extending to very large strains (of order 5). The material is still elastic if unloaded, it follows the same path down as it did up, and all the energy stored, per unit volume, during loading is recovered on unloading - that is why catapults can be as lethal as they are. 

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