Path independence

This section is concerned with the two-dimensional elasticity equations. Our aim is to find the derivative of the energy functional with respect to the crack length. The nonpenetration condition is assumed to hold at the crack faces. We derive the Griffith formula and prove the path independence of the Rice-Cherepanov integral. This section follows the publication (Khludnev, Sokolowski, 1998c). 

We have to note that the result is obtained for nonlinear boundary conditions (4.128). The well-known path independence of the Rice-Cherepanov integral was previously proved in elasticity theory for linear boundary conditions a22 = 0,ai2 = 0 holding on Ef (see Parton, Morozov, 1985). 

Rice J.R. (1968) A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379-386. 

The dislocation multiplication law N = (l/L)N i is path-independent i.e., N depends only on y and not on the rate at which the deformation occurs. Show that the multiplication law given by... 

AV is the net change in gravitational potential energy. This term is path independent and depends only on the initial and final heights, h and hj, above some arbitrary reference height with respect to the surface of the earth. 

For hard sphere collisions, v(v) would be proportional to v9 and the mean free path independent of v A(v) is an equivalent mean free path for a- general force law. Cf. S. Chapman and T. G. Cowling, The Mathematical Theory of Non- Uniform Oases, pp. 91 and 348, Cambridge University Press, 1958. 

In the absence of K the enzyme exhibits a basal Mg -ATPase activity that can be reduced, but not completely removed, upon further purification of the enzyme by free-flow or zonal electrophoresis [66,89]. Wallmark et al. [104] demonstrated that the rate of spontaneous breakdown of phosphoenzyme corresponded very well to the Mg -ATPase activity at low ATP concentrations, implying that this activity was not due to a contaminating Mg -ATPase with a reaction path independent of the phosphoenzyme. This conclusion was confirmed by Reenstra et al. [129] in a study on the nonhyperbolic ATP dependence of ATPase activity and phosphoenzyme... 

Here d is the volume density at a point. For instance, at points where masses are absent div g = 0. Let us discuss the physical and mathematical content of these equations. The first one clearly shows that the attraction field does not have vortices and, correspondingly, the work done by this field is path independent. In other words, the circulation of the field is equal to zero. At the same time, the second equation demonstrates that the field g is caused by sources (masses) only. As illustration, consider the set of these equations in the Cartesian system of coordinates ... 

Since the field does not vanish at infinity, we assume that the potential is known at some point either outside or inside the layer. For instance, suppose that C7 = 0 at the plane z = 0. Taking into account that the integral in Equation (1.149) is path independent, we choose... 

Since enthalpy changes are path-independent quantities, one is at liberty to choose a convenient path for making the calculation. If we carry out the reaction isothermally at the inlet temperature and then heat the products at constant composition to the effluent temperature, we find that... 

J. R. Rice, A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks, J. Appl. Mech., 1968, 35,379 386. 

Even though the free energy difference is a path independent quantity, it is observed that certain sampling difficulties arise when a polar solute is transferred to a non polar solute accompanied by a large change in molecular volume. Under this circumstance, if one attempts mutation of both the partial charges and the non bonded parameters simultaneously, the solute-solvent energy increases enormously as a consequence of very close... 

At this point one question must be answered Is the potential calculated in the manner above path independent [21] Equivalently, is the field given by Equation 7.33 curl-free For one-dimensional cases and within the central field approximation for atoms, it is. For other systems, there is a small solenoidal component [21,22] and we will see later that it arises from the difference in the kinetic energy of the true system and the corresponding Kohn-Sham system (in this case the HF system and its Kohn-Sham counterpart). For the time being, we explore whether the physics of calculating the potential in the manner prescribed above is correct in the cases where the curl of the field vanishes. 

Despite the differences in soil properties, /7-nitrophenol was recovered in all the soils studied, together with water-soluble diethyl-thiphosphate. This result proves that hydrolysis is the main degradation path, independent of the nature of the soil. However, both the type of clay and the presence of organic matter affect the amount of degraded parathion. This behavior is illustrated in Fig. 16.16. 

If we calculate the above line integral for an arbitrary approximate 1 the result will, in general, be dependent on the path y used. However path independence is guaranteed [60] if v c satisfies the condition... 

The uniaxial failure envelope developed by Smith (95) is one of the most useful devices for the simple failure characterization of many viscoelastic materials. This envelope normally consists of a log-log plot of temperature-reduced failure stress vs. the strain at break. Figure 22 is a schematic of the Smith failure envelope. Such curves may be generated by plotting the rupture stress and strain values from tests conducted over a range of temperatures and strain rates. The rupture locus moves counterclockwise around the envelope as the temperature is lowered or the strain rate is increased. Constant strain, constant strain rate, and constant load tests on amorphous unfilled polymers (96) have shown the general path independence of the failure envelope. Studies by Smith (97) and Fishman (29) have shown a path dependence of the rupture envelope, however, for solid propellants. 

If one adopts McLennan s [78b] interpretation, then Eq. (21) is a realization of a standard theorem of Newtonian mechanics conservation of total energy = conservation of kinetic plus potential energy (see, e.g., Chap. 4 of Kleppner and Kolenkow, [80]). The reason is simple Coulomb electric force is central, then work is path independent, and total energy is function of position only. The time derivative of total energy is of course zero, as in Eq. (21). In this interpretation Qp and Qi are manifestations of kinetic energy. 

The internal energy and the enthalpy of a system depend only on the state of the system, as specified by parameters such as V, P, and T. When a system goes from an initial to a final state, AE and AH depend only on those two states, and are independent of the path taken between them. This path-independence implies two important rules ... 

The thermodynamic principles for evaluating the influences of T and P on equilibrium are well established and will not be dealt with in detail here. One basic concept is that the combined influences of temperature and pressure are path independent functions, which simply means that the order in which they are calculated is not important. In practice, however, the availability of data may determine the sequence of calculation. It is generally necessary to calculate the temperature effect first for most systems of practical concern to the sedimentary carbonate chemist. 

This work done is path-independent since V x R(r) = 0. For systems of certain symmetry such as closed shell atoms or open-shell atoms in the central-field approximation, the jellium and structureless-pseudopotential models of a metal surface considered here, etc., the work Wxc (r) and Wt (r) are separately path-independent since for these cases Vx xc(r) = VxZt (r) = 0. 

Properties that are path independent depend only on the conditions at the start and at the end stages of the change are referred... 

AX is then the same whatever route or path is selected to achieve the change and hence is path independent. 

Figure 1.1 Comparison of path dependent functions (q and w) and path independent change (AU) during the expansion of a gas from (V, T,) to (Vf, 7)) via two different states (V, T() (path 1) and (Vf, T) (path 2). 1/ and T represent the volume and temperature of the gas. q and w represent the heat absorbed by the gas and the work done by the gas on the surroundings in expanding against the external pressure, P. Wf and W2 are both negative (using the convention discussed in Frame 7) since work of expansion is expended by the gas and lost from the (gas) system.

The result of this calculation was In K12 = 0.039, which is the same as K12 = 1.04 [path c]. These three values for K12 agree, within the precision of the data, substantiating the path independence of Eq. 5.43. 

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