Inequality problems

Panagiotopoulos P.D. (1985) Inequality problems in mechanics and applications. Birkhauser, Boston, Basel, Stuttgart. 

It is not only the stream number that creates the need to split streams at the pinch. Sometimes the CP inequality criteria [Eqs. (16.1) and (16.2)] CEmnot be met at the pinch without a stream split. Consider the above-pinch part of a problem in Fig. 16.13a. The number of hot streams is less than the number of cold, and hence Eq. (16.3) is satisfied. However, the CP inequality also must be satisfied, i.e., Eq. (16.1). Neither of the two cold streams has a large enough CP. The hot stream can be made smaller by splitting it into two parallel branches (Fig. 16.136). 

Clearly, in designs different from those in Figs. 16.13 and 16.14 when streams are split to satisfy the CP inequality, this might create a problem with the number of streams at the pinch such that Eqs. (16.3) and (16.4) are no longer satisfied. This would then require further stream splits to satisfy the stream number criterion. Figure 16.15 presents algorithms for the overall approach. ... 

The network can now be designed using the pinch design method.The philosophy of the pinch design method is to start at the pinch and move away. At the pinch, the rules for the CP inequality and the number of streams must be obeyed. Above the utility pinch and below the process pinch in Fig. 16.17, there is no problem in applying this philosophy. However, between the two pinches, there is a problem, since designing away from both pinches could lead to a clash where both meet. 

Figure 16.215 shows an alternative match for stream 1 which also obeys the CP inequality. The tick-off" heuristic also fixes its duty to be 12 MW. The area for this match is 5087 m , and the target for the remaining problem above the pinch is 3788 m . Tlius the match in Fig. 16.216 causes the overall target to be exceeded by 16 m (0.2 percent). This seems to be a better match and therefore is accepted. 

The cold-utility target for the problem shown in Fig. 16.22 is 4 MW. If the design is started at the pinch with stream 3, then stream 3 must be split to satisfy the CP inequality (Fig. 16.22a). Matching one of the branches against stream 1 and ticking off stream 1 results in a duty of 8 MW. 

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

In the sequel, we consider concrete boundary conditions for the above models to formulate boundary value problems. Also, restrictions of the inequality type imposed upon the solutions are introduced. We begin with the nonpenetration conditions in contact problems (see Kravchuk, 1997 Khludnev, Sokolowski, 1997 Duvaut, Lions, 1972). 

Let us emphasize that not model can be presented as a minimization problem like (1.55) or (1.57). Thus, elastoplastic problems considered in Chapter 5 can be formulated as variational inequalities, but we do not consider any minimization problems in plasticity. In all cases, we have to study variational problems or variational inequalities. It is a principal topic of the following two sections. As for general variational principles in mechanics and physics we refer the reader to (Washizu, 1968 Chernous ko, Banichuk, 1973 Ekeland, Temam, 1976 Telega, 1987 Panagiotopoulos, 1985 Morel, Solimini, 1995). 

The inequality like (1.59) is called a variational inequality. It was obtained from a minimization problem of the functional J over the set K. In the sequel we will look more attentively at a connection between a minimization problem and a variational inequality. Now we want to underline one essential point. We see that the problem (1.58) is more general in comparison with the minimization problem on the whole space V. It is well known that the necessary condition in the last problem coincides with the Euler equation. The variational inequality (1.59) generalizes the Euler equation. Moreover, ior K = V the Euler equation follows from (1.59). To obtain it we take U = Uq +u and substitute in (1.59) with an arbitrary element u gV. It gives... 

Theorem 1.12. The solution of the problem (1.81) exists if and only if there exists a solution of the variational inequality... 

In this section we define trace spaces at boundaries and consider Green s formulae. The statements formulated are applied to boundary value problems for solids with cracks provided that inequality type boundary conditions hold at the crack faces. 

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. 

Considering the crack, we impose the nonpenetration condition of the inequality type at the crack faces. The nonpenetration condition for the plate-punch system also is the inequality type. It is well known that, in general, solutions of problems having restrictions of inequality type are not smooth. In this section, we establish existence and regularity results related to the problem considered. Namely, the following questions are under consideration ... 

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. 

Herewith the problem of minimizing H over the set is equivalent to the variational inequality... 

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 ... 

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). 

The properties of our measure jx depend on the regularity of the solution. The inequality (2.226) is actually a Signorini-type problem for finding W provided that the function w is already known. It can be written as follows ... 

We first note that the coercivity and weak lower semicontinuity of the functional n imply that the problem (2.248) has a (unique) solution The coercivity is provided by the following two inequalities,... 

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 Section 3.1.3 a complete system of equations and inequalities holding on F, X (0,T) is found (i.e. boundary conditions on F, x (0,T) are found). Simultaneously, a relationship between two formulations of the problem is established, that is an equivalence of the variational inequality and the equations (3.3), (3.4) with appropriate boundary conditions is proved. 

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 existence of two angular points on 7= = presents no problems since has a compact support. Hence, the inequality (3.35) with the equations (3.31) yield the identity... 

In this section we consider the boundary value problem for model equations of a thermoelastic plate with a vertical crack (see Khludnev, 1996d). The unknown functions in the mathematical model under consideration are such quantities as the temperature 9 and the horizontal and vertical displacements W = (w, w ), w of the mid-surface points of the plate. We use the so-called coupled model of thermoelasticity, which implies in particular that we need to solve simultaneously the equations that describe heat conduction and the deformation of the plate. The presence of the crack leads to the fact that the domain of a solution has a nonsmooth boundary. As before, the main feature of the problem as a whole is the existence of a constraint in the form of an inequality imposed on the crack faces. This constraint provides a mutual nonpenetration of the crack faces ... 

We prove the solvability of the problem. We also find boundary conditions holding on the crack faces and having the form of a system of equations and inequalities and establish some enhanced regularity properties for the solution near the points of the crack. Some other results on thermoelasic problems can be found in (Gilbert et al., 1990 Zuazua, 1995). 

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... 

Observe that variational inequality (3.106) is valid for every function X G 82- It means that a solution % to problem (3.106) with 9 G Si coincides with the unique solution to problem (3.100) with the same 9] i.e. problems (3.100) and (3.106) are equivalent. For small 5, we write down an extra variational inequality for which a solution exists, and demonstrate that the solution coincides with the solution of variational inequality (3.98). 

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