Total strain

If we assume that there are certain ideal val ues for bond angles bond distances and so on itfol lows that deviations from these ideal values will destabilize a particular structure and increase its po tential energy This increase in potential energy is re ferred to as the strain energy of the structure Other terms for this increase include steric energy and steric strain Arithmetically the total strain energy ( ) of an alkane or cycloalkane can be considered as... 

At Its most basic level separating the total strain of a structure into its components is a qualita tive exercise For example a computer drawn model of the eclipsed conformation of butane using ideal bond angles and bond distances (Figure 3 8) reveals that two pairs of hydrogens are separated by a distance of only 175 pm a value considerably smaller than the sum of their van der Waals radii (2 X 120 pm = 240 pm) Thus this conformation is destabilized not only by the torsional strain associ ated with its eclipsed bonds but also by van der Waals strain... 

In this chapter we explored the three dimensional shapes of alkanes and cycloalkanes The most important point to be taken from the chapter is that a molecule adopts the shape that minimizes its total strain The sources of strain m alkanes and cycloalkanes are... 

Another type of experiment involves a fluid filament being drawn upward against gravity from a reservoir of the fluid (101,213,214), a phenomenon often called the tubeless siphon. The maximum height of the siphon is a measure of the spinnabiUty and extensional viscosity of the fluid. Mote quantitative measures of stress, strain, and strain rate can be determined from the pressure difference and filament diameter. A more recent filament stretching device ia which the specimen is held between two disks that move apart allows measurements ia low viscosity Hquids (215). AH of these methods are limited to spinnable fluids under small total strains and strain rates. High strain rates tend to break the column or filament. 

Shock loading in most metals and alloys produces greater hardening than quasi-static deformation to the same total strain, particularly if the metal undergoes a polymorphic phase transition, such as is observed in pure iron [1]-[10]. Figure 6.1 compares the stress-strain response of an annealed... 

The total strain rate in the material is assumed to be the sum of elastic and plastic contributions... 

Steady-propagating plastic waves [20]-[22] also give some useful information on the micromechanics of high-rate plastic deformation. Of particular interest is the universality of the dependence of total strain rate on peak longitudinal stress [21]. This can also be expressed in terms of a relationship between maximum shear stress and average plastic shear strain rate in the plastic wave... 

For example, a 10 GPa (total strain = 0.06) shock wave in copper has a maximum total strain rate 10 s [21] the risetime would thus be (eje) 0.6 ns. For uniaxial-strain compression, y averaged over the entire shock front. The resolution of the shock wave in a large-scale, multidimensional finite-difference code would be computationally expensive, but necessary to get the correct strength f behind the shock. An estimate of the error made in not resolving the shock wave can be obtained by calculating dt/dy)o with y 10 s (the actual plastic strain rate) and y 10 s (the plastic strain rate within the computed shock wave due to a time step of 0.06 qs). From (7.41) with y = 10 s (actual shock wave) and y = 10 s (computation) ... 

The relaxation time (arbitrarily defined as the time taken for the stress to relax to half its original value) can be calculated from the power-law creep data as follows. Consider a bolt which is tightened onto a rigid component so that the initial stress in its shank is CTj. In this geometry (Fig. 17.3(c)) the length of the shank must remain constant - that is, the total strain in the shank e must remain constant. But creep strain can rqjiace elastic strain e - , causing the stress to relax. At any time t... 

In this case the total strain is equal to the strain in each of the elements, i.e. 

Now consider the situation in which the stress, ai, was applied at time, ti, and an additional stress, Boltzmanns Superposition Principle states that the total strain at time, t, is the algebraic sum of the two independent responses. 

Tests have shown that when total strain is plotted against the logarithm of the total creep time (ie NT or total experimental time minus the recovery time) there is a linear relationship. This straight line includes the strain at the end of the first creep period and thus one calculation, for say the 10th cycle allows the line to be drawn. The total creep strain under intermittent loading can then be estimated for any combinations of loading/unloading times. 

Therefore after the 11th cycle the total creep time is 11x100=1.1x10 hours. If the total strain at this time is plotted on Fig. 2.51 then a straight line can be drawn through this point and the point edT), and this line may be extrapolated to any desired number of cycles. For the case in question the line must be extrapolated to (1001 x 1(X)) hours at which point the total strain may be obtained as 1.09%. Thus the accumulated residual strain after KXX) cycles would be 1.09 — 0.747 = 0.343% as calculated on the computer. 

That is, the total strain will be the sum of the tensile strain due to and the negative strain due to the Poisson s ratio effect caused by Oy. 

In both Equations (4.100) and (4.101), the six Oj are the coefficients of thermal deformation (expansion or contraction and distortion, I.e., shear), and AT is the temperature difference. In Equation (4.101), the terms CjjCXjAT are the thermal stresses if the total strain is zero. 

Note that the so-called thermal forces, N, are true thermal forces only when the total strains and curvatures are perfectly restrained, that is, zero. 

Problem 3.18 Draw a Newman projection along the C2-C3 bond of the following conformation of 2,3-dimethylbutane, and calculate a total strain energy ... 

Cyclobutane has less angle strain than cyclopropane but has more torsional strain because of its larger number of ring hydrogens. As a result, the total strain for the two compounds is nearly the same—110 kj/mol (26.4 kcal/mol) for cyclobutane versus 115 kj/mol (27.5 kcal/mol) for cyclopropane. Experiments show that cvclobutane is not quite flat but is slightly bent so that one carbon atom lies about 25° above the plane of the other three (Figure 4.5). The effect of... 

Problem 4,10 How many H eclipsing interactions would be present if cyclopentane were planar Assuming an energy cost of 4.0 kj/mol for each eclipsing interaction, how much torsional strain would planar cyclopentane have Since the measured total Strain of cyclopentane is 26 kj/mol, how much of the torsional strain is relieved by puckering ... 

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