Equilibrium inequalities

The sign of AG can be used to predict the direction in which a reaction moves to reach its equilibrium position. A reaction is always thermodynamically favored when enthalpy decreases and entropy increases. Substituting the inequalities AH < 0 and AS > 0 into equation 6.2 shows that AG is negative when a reaction is thermodynamically favored. When AG is positive, the reaction is unfavorable as written (although the reverse reaction is favorable). Systems at equilibrium have a AG of zero. 

In this chapter we analyse a wide class of equilibrium problems with cracks. It is well known that the classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, in particular, the crack faces are considered to be stress-free (Cherepanov, 1979, 1983 Kachanov, 1974 Morozov, 1984). This means that displacements found as solutions of these boundary value problems do not satisfy nonpenetration conditions. There are practical examples showing that interpenetration of crack faces may occur in these cases. An essential feature of our consideration is that restrictions of Signorini type are considered at the crack faces which do not allow the opposite crack faces to penetrate each other. The restrictions can be written as inequalities for the displacement vector. As a result a complete set of boundary conditions at crack faces is written as a system of equations and inequalities. The presence of inequality type boundary conditions implies the boundary problems to be nonlinear, which requires the investigation of corresponding boundary value problems. In the chapter, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, viscoelastic. 

Our aim is to analyze the solution properties of the variational inequality describing the equilibrium state of the elastic plate. The plate is assumed to have a vertical crack and, simultaneously, to contact with a rigid punch. 

The equilibrium problem for the plate can be formulated as variational, namely, it corresponds to the minimum of the functional H over the set of admissible displacements. To minimize the functional H over the set we can consider the variational inequality... 

The crack shape is defined by the function -ip. This function is assumed to be fixed. It is noteworthy that the problems of choice of the so-called extreme crack shapes were considered in (Khludnev, 1994 Khludnev, Sokolowski, 1997). We also address this problem in Sections 2.4 and 4.9. The solution regularity for biharmonic variational inequalities was analysed in (Frehse, 1973 Caffarelli et ah, 1979 Schild, 1984). The last paper also contains the results on the solution smoothness in the case of thin obstacles. As for general solution properties for the equilibrium problem of the plates having cracks, one may refer to (Morozov, 1984). Referring to this book, the boundary conditions imposed on crack faces have the equality type. In this case there is no interaction between the crack faces. 

L flc) be some given functions of the external forces. The equilibrium problem for a plate with a crack is formulated as the following variational inequality ... 

Then it follows that the solution of the variational inequality (2.165) is characterized by the equilibrium equation... 

Here inequalities (2.185), (2.186) are assumed to be satisfied almost everywhere in the Lebesgue sense on F, and in We assume that < 0 on F, so that the set Kg is nonempty. The equilibrium problem for a shallow shell with a solution satisfying the nonpenetration conditions (2.185), (2.186) can be formulated as follows ... 

We consider an equilibrium problem for a shell with a crack. The faces of the crack are assumed to satisfy a nonpenetration condition, which is an inequality imposed on the horizontal shell displacements. The properties of the solution are analysed - in particular, the smoothness of the stress field in the vicinity of the crack. The character of the contact between the crack faces is described in terms of a suitable nonnegative measure. The stability of the solution is investigated for small perturbations to the crack geometry. The results presented were obtained in (Khludnev, 1996b). 

By varying the test function W K, one can deduce that the variational inequality (2.265) is equivalent to the equilibrium equations in flc. 

We consider a boundary value problem for equations describing an equilibrium of a plate being under the creep law (1.31)-(1.32). The plate is assumed to have a vertical crack. As before, the main peculiarity of the problem is determined by the presence of an inequality imposed on a solution which represents a mutual nonpenetration condition of the crack faces... 

An existence theorem to the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. The solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C °° near crack points provided the crack opening is equal to zero. The results of this section are published in (Khludnev, 1996c). 

The structure of the section is as follows. In Section 3.1.2 we prove a solvability of the equilibrium problem. This problem is formulated as a variational inequality holding in Q. The equations (3.3), (3.4) are fulfilled in the sense of distributions. On the other hand, if the solution is smooth and satisfies (3.3), (3.4) and all the boundary conditions then the above variational inequality holds. 

In this subsection we prove an existence theorem of the equilibrium problem for the plate. The problem is formulated as a variational inequality which together with (3.2), (3.5) contains full information about other boundary conditions holding on x (0, T). An exact form of these conditions is found in the next subsection. 

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. ... 

We can now give an exact statement of the equilibrium problem for a plate. Suppose that / G L Q ). An element (0, x) G 17 is said to be a solution to the equilibrium problem for a thermoelastic plate with a crack if it satisfies the variational inequality... 

It is noteworthy that the original equilibrium problem for a plate with a crack can be stated twofold. On the one hand, it may be formulated as variational inequality (3.98). In this case all the above-derived boundary conditions are formal consequences of such a statement under the supposition of sufficient smoothness of a solution. On the other hand, the problem may be formulated as equations (3.92)-(3.94) given initial and boundary conditions (3.95)-(3.97) and (3.118)-(3.122). Furthermore, if we assume that a solution is sufficiently smooth then from (3.92)-(3.97) and (3.118)-(3.122) we can derive variational inequality (3.98). 

In fact, by the second Korn inequality this scalar product induces a norm which is equivalent to the norm given by (5.3). Hence, because (/, p) = 0 for all p G R fl), the identity (5.29) actually holds for every u G Therefore, the equilibrium equations... 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

Inequalities (6) and (7) show that the surface is, at every point which corresponds with a homogeneous phase in stable equilibrium, convex downwards in every direction. 

Equation (5.47) gives the criterion for reversibility or spontaneity within subsystem A of an isolated system. The inequality applies to the spontaneous process, while the equality holds for the reversible process. Only when equilibrium is present can a change in an isolated system be conceived to occur reversibly. Therefore, the criterion for reversibility is a criterion for equilibrium, and equation (5.47) applies to the spontaneous or the equilibrium process, depending upon whether the inequality or equality is used. 

Equation (5.52) is the first of our criteria. The subscripts indicate that equation (5.52) applies to the condition of constant entropy, volume, and total moles, with the equality applying to the equilibrium process and the inequality to the spontaneous process. 

The two main conditions (besides the stability of the a radical towards the solvent to observe such an electron catalysis are a sufficient high rate of addition of the nucleophile and the thermodynamic inequality E°ll >E°12 implying a fast displacement of the latter equilibrium to the direction of the formation of the anion radical of 71. 

It is clear that J-diffusion is a good approximation for rotational relaxation as a whole, if the centre of equilibrium distribution over J is within the limits of non-adiabatic theory. In the opposite case m-diffusion is preferable. Consequently, the J-diffusion model is applicable, if the following inequality holds ... 

Here X is the angle between the field and unit vector Tq, and 0 the angle between fields gs and —gs. Note, that the last inequality of the set (2.14) is necessary to provide equilibrium along the z-axis. From the last two equations we have for the magnitude of the surface force... 

The second and probably more common possibility is that the inequality (107) is reversed. In such case, the reasoning leading to (108) cannot be used, but we can now assume that H+, BH, and B are in local equilibrium with each other over much of the passivated region, since for all the cases listed in Table III the lifetime rBH of BH with respect to thermal breakup must have been rather less than the duration of the experiment (cf. the discussion of rBH in Section 3d below). Now from the equilibrium equations (2), (3), and (10), we have... 

The activity product Q ave corresponding to the averaged analysis (ignoring variation in activity coefficients) equals the equilibrium constant K only when fluids A and B are identical otherwise Qmc exceeds K and anhydrite is reported to be supersaturated. To demonstrate this inequality, we can assume arbitrary values for aCa++ and so4 that satisfy Equations 6.4—6.5 and substitute them into Equation 6.6. 

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