Inequality condition

The first difficulty derives from the fact that given any values of the macroscopic expected values (restricted only by broad moment inequality conditions), a probability density always exists (mathematically) giving rise to these expected values. This means that as far as the mathematical framework of dynamics and probability goes, the macroscopic variables could have values violating the laws of phenomenological physics (e.g., the equation of state, Newton s law of heat conduction, Stokes law of viscosity, etc.). In other words, there is a macroscopic dependence of macroscopic variables which reflects nothing in the microscopic model. Clearly, there must exist a principle whereby nature restricts the class of probability density functions, SF, so as to ensure the observed phenomenological dependences. 

PROBLEM L2.8 Assuming the worst-case situation, a metallic sphere for which a = 4jra3, and using the center-to-center distance z between spheres as a measure of number density, N is one sphere per cubic volume z3, show that the inequality condition Na<3 becomes 4 ra3 < 3z3. 

Both of the new gop(y4, B) and f piA, B) functions are proper metrics. We shall present a proof only for the chemically more important metric f p(A,B) the proof is entirely analogous for ggp(.A,B). Evidently, conditions (10), (11), and (12) of metric apply for the versions A. and 5,. realizing the optimum mutual arrangements. For a proof of the triangle inequality, condition (13), one can first consider a fixed version B. . of fuzzy set B and generate the optimum versions A. and C,.. of fuzzy sets A and C, which realize the optimum mutual arrangements with B,. ... 

To verify the third condition, consider first any two square matrices A and B which are real and nonsingular. Now pick a vector X0 such that X0 m = 1 and such that the vector norm (A + B)X0, is a maximum. Then the triangle inequality (condition 3 for vector norms) may be used to show that... 

Let us define the objective function the inequality condition and the bounds for bothD andL. Figure 10.45 shows the MATLAB definition for the objective function given by Eq. tl0.96L... 

Note that the CP inequalities given by Eqs. (16.1) and (16.2) apply only at the pinch when both ends of the match are at pinch conditions. 

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 a punch shape be described by the equation z = ip(x), and xi,X2,z be the Descartes coordinate system, x = xi,X2). We assume that the mid-surface of a plate occupies the domain fl of the plane = 0 in its non-deformable state. Then the nonpenetration condition for the plate vertical displacements w is expressed by the inequalities... 

The boundary inequality (1.46) is called a Signorini condition (Fichera, 1972). 

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 1.1. The inequality (1.62) gives necessary and sujficient conditions of the minimum over the set K for a convex and differentiable functional J. 

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. 

The arguments of Section 1.4.4 applied to the inequality (1.165) provide the conditions... 

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. 

In particular, the strict inequality in (2.105) provides m w) = 0. We have to note at this point that the boundary conditions (2.107), (2.110) hold on and... 

Besides, (2.107) holds good irrespective of the inequality w > in W, i.e. this condition takes place in the general case w >. At the same time, to derive (2.110), we make use of the equation (2.103) in W F,, which takes place provided that w > in >V. Moreover, the inequality w > in >V provides one more relation... 

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 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. 

In this section we proceed to study the plate model with the crack described in Sections 2.4, 2.5. The corresponding variational inequality is analysed provided that the nonpenetration condition holds. By the principles of Section 1.3, we propose approximate equations in the two-dimensional case and analytical solutions in the one-dimensional case (see Kovtunenko, 1996b, 1997b). 

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). 

K. Indeed, for this inclusion to be valid one has to verify the inequalities (2.246), (2.247). The relation (2.246) obviously takes place since V = 0 in some neighbourhood of x. So, it suffices to examine (2.247). To do this, we take into account that the graph has a rectilinear section near x. The last condition implies [w ] > 0 on T, n B2r(x )- Hence, it suffices to prove... 

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... 

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. 

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