ὅδε οἶκος, ὦ ἑταῖρε, μνημεῖον ἐστιν ζῴων τῶν σοφῶν ἀνδρῶν, καὶ τῶν ἔργων αὐτῶν

The two main focuses of this research are the viscoelastic materials of the fractional type and the problem of the optimization of the shape of elastic rod, subject to certain constraints. We shall formulate of the constitutive equations for the viscoelastic materials, containing either arbitrary number of the fractional derivatives, or the distributed-order fractional derivative, so that the Second Law of Thermodynamics formulated in the terms of the Calusius-Duhamel inequality is satisfied. Namely, we shall determine the restrictions on the constitutive parameters and functions. Constitutive equations obtained as described previously will be coupled with the equations of deformable body and energy balance equations. Initial-boundary value problems will be analyzed, i.e. we shall solve and prove the existence and regularity of the solutions in the appropriate function and distribution spaces. Obtained results will be applied on description of the composites, materials widely used in the dentistry. By the use of the same models, we shall treat the problem of the marginal adaptation. We shall optimize the shape of the elastic rods by the use of the variation principles and the Pontryagin Principle of Maximum. The restrictions on the cross-section area (maximal or minimal value is prescribed) will receive due attention. Post-critical behavior of the rods will be analyzed, especially the stability of the equilibrium configurations that bifurcates from the initial configuration.

We shall describe the details of the investigations within the project that will branch into two directions. The first direction is the analysis of the behaviour of the viscoelastic materials, i.e. the materials that exhibit both elastic and viscous properties. Lately, among the many theories describing the viscoelastic body, the theory that uses the fractional derivatives, i.e. the derivative of non-integer order, of stress and strain is of the considerable interest. Participants of the project are engaged in this field for a quite a while and they have developed a specific and original approach to these problems. Namely, we employ fundamental principles of physics, such as the Second Law of Thermodynamics, in order to formulate the constitutive equations of the viscoelastic body of the fractional order. More precisely, we shall obtain restrictions on the parameters, orders of the fractional derivatives, and functions occurring in the constitutive equations that will follow, in the case of the isothermal deformations, from the Calusius-Duhamel inequality (the positivity of the dissipation work). We shall determine the restrictions on the parameters in the constitutive model of the viscoelastic body that includes the arbitrary number of the fractional derivatives of stress and strain. According to our knowledge, restrictions on the parameters following from the Second Law of Thermodynamics are still not determined. Besides, constitutive equations containing distributed-order fractional derivatives (i.e. the sum of all the fractional derivatives of order between zero and two multiplied by some constitutive function) of stress and strain will receive our attention. Note that this includes elastic, viscoelastic, and viscoinertial effects. Project participants already have some results in this field (see the reference list). The constitutive equations formulated as described previously will be coupled with the equations of deformable body and energy balance equations and initial-boundary problems will be analyzed. As far as the one dimensional bodies of finite length (rods) are concerned, three important problems, namely stress relaxation problem, creep, and forced oscillations induced by the harmonic stress (including the resonance), will be solved. In each of the mentioned cases existence, asymptotic behaviour and quasi-static approximation of the solution will be analyzed. Some attention will be paid to the effect of wave reflection on the ends of a rod and its influence on the stress and strain field. In the case of the infinite viscoelastic medium we shall investigate the wave propagation, especially the disturbance propagation speed, since these systems in the limiting cases reduce to the wave (speed of disturbance propagation is finite) and diffusion equation (speed is infinite). We shall also deal with the problem of formulation of the fractional differential equations arising from the variation principle. It is well-known that the conditions under which the Hamilton action is stationary lead to the Euler-Lagrange equations that contain both left and right fractional derivatives. We shall prove the existence of the solution of such equations and develop the tools to obtain the explicit form of the solution. We shall formulate the complementary principle in the case of the Lagrangean containing fractional derivatives. This will enable us to use the direct methods of the variation calculus in order to obtain the approximate solutions of the Euler-Lagrange equations, as well as to a-priori estimate the error of the approximate solutions. Obtained results of these parts of our research will be applied on the mathematical description of the composites, viscoelastic materials used in the dentistry in order to fill up the cavities. In our previous investigations we have shown that the composites can be well described by the model of the fractional viscoelastic body. Since the composites display the property of volume contraction, the effect of the stress relaxation, if not controlled, may cause the occurrence of the cracks on the borders composite-dentine, known as the marginal adaptation. We shall investigate this effect and propose polymerization techniques that can reduce this effect. We shall investigate the possibility to describe some materials (like nitinol) by the use of the pseudo-elastic constitutive equations. The second direction is the analysis of the optimal shape of an elastic rod, loaded with an axial force, strongest against the buckling, by the means of the Pontryagin Maximum Principle. Due attention will be paid to the boundary conditions that ensure the occurrence of the bi-modal optimization. Mathematically, this means that the eigenvalue of the spectral problem described by the four differential equations of the first order has geometrical multiplicity two, or more. Theoretical background for such analysis is obtained previously by the members of this project. We shall introduce restrictions on the cross-section area. Namely, it is greater, or less that the fixed value or it is between two fixed values. Constraints on the minimal value are known in the literature, while the constraints on the both maximal and minimal values of the area are sill not formulated. We shall formulate the Conservation Law and use it in order to estimate the error of the numeric integration. We shall analyze the post-critical behaviour of the optimally shaped rod by the use of the procedure based on the Pontryagin Principle. We shall use the energy method in order to determine the solution between the number of solutions that bifurcate form the equilibrium position.

There is both theoretical and practical significance of the proposed investigations. Theoretical significance:

1. Investigation of the constitutive equations of fractional order type is of great theoretical importance. Constitutive equations containing ordinary derivatives, in the viscoelasticity, are formulated in the accordance with the basic physical principles, such as the Principle of Determinism, Second Law of Thermodynamics, and Principle of equipresence. It is of the theoretical importance to formulate the fractional order type constitutive equations of viscoelasticity as well in the accordance with the mentioned principles. The importance of the theoretical results is that only those constitutive equations that are admissible will be used in order to fit the experimental data for the materials used in the dentistry. This means that, for example, in fitting the data by the use of the fractional Zener model we shall take care that stress relaxation time is less than the strain relaxation time. Using the constitutive equations formulated in this manner we shall solve several characteristic test problems analyzing the practical importance of the theoretical results. Due attention will be paid to the relation between restrictions induced by the fundamental laws of physics and their mathematical manifestations. By this we mean that e.g. the Second Law of Thermodynamics reflects the fact that all zeros of the characteristic equation lie in the left complex half-plane. This is the manifestation of connection between laws of physics and existence of solutions. It is well known that we can apply direct methods of variation calculus in order to obtain approximate solutions of the Euler-Lagrange equations if the mechanical system can be described by the Lagrangean. It is of the great importance to formulate the procedures that are used to estimate the error of the approximate solution a-priori. Complementary variation principle ensures the procedure for a-priori error estimate, and this principle will be formulated for the equations of the fractional order. This is of the theoretical interest, since the equations that contain both left and right Riemann-Liouville fractional derivatives (in many cases) cannot be treated by the usual means of the integral transformations.

2. Analysis of the optimal shape of the elastic rods has both theoretical and practical aspects. Namely, the weight reduce of the construction becomes a key point in the contemporary engineering. Although light, the construction still has to be stable. In order to determine the optimal shape of the rod with the least mass and yet stable, we shall use the modern mathematical techniques based upon the Pontryagin Maximum Principle. Since there are other various constraints (e.g. maximal stress should be less than a given value) it is necessary to incorporate them in the optimization procedure. Constraints change the mathematical structure of the problem, for example, space of zeros of the linear operator changes its structure and the geometrical multiplication of eigenvalues. Practical significance:

3. Results obtained from the theoretical considerations will be applied in mathematical modelling of the materials used in the dentistry. It is well known that the composites can be modelled as viscoelastic materials. Theory of the viscoelasticity of the fractional order type will be used in order to estimate the marginal adaptation of the composites. Also, we shall propose the procedure of the polymerization, applicable in the practice, which will improve the marginal adaptation. The material fatigue, due to the rotation of the devices made of pseudoelastic materials will be estimated theoretically and checked experimentally.

Key results expected in our investigations can be divided as follows:

1. We shall formulate and analyze constitutive equations of the linear viscoelasticity that contain fractional order derivatives of stress and strain, in particular, when the orders are not the same. By the use of the Second Law of Thermodynamics, we shall investigate the restrictions on coefficients and orders of the fractional derivatives in those equations. Constitutive equations of viscoelasticity, containing distributed-order fractional derivatives, will be analyzed in the cases of the weight functions and distributions. Constitutive equations will be coupled with the equations of the deformable body and the energy balance equations. We shall obtain systems of partial and fractional partial differential equations. Initial-boundary value problems will be analyzed for such systems. In the case of the finite spatial domain (i.e. rods), we shall investigate stress relaxation, creep and forced oscillation problems. Existence, uniqueness and regularity of the solutions will be investigated for each of problems. The asymptotic behaviour of the solutions will analyzed by the use of the Tauberian-type theorems. We shall investigate the wave propagation in the viscoelastic media of the infinite length, as well as the problems of the disturbance propagation speed and its attenuation. We shall formulate the generalized principle of the Hamilton least action in the case when the Langrangean function contains fractional derivatives of constant and variable order. We shall determine the conditions that guarantee the existence of the complementary variation principles of fractional order. Then they will be used to determine the a-priori error estimate for the approximate solution of the Euler-Lagrange equations. Obtained results will be used in order to model materials in the dentistry which display memory and viscoelastic effects. We shall develop mathematical models of the dental composites, based upon viscoelasticity of fractional type. In the case of the pseudo-elastic materials used in the dentistry (nitinol), we shall develop the procedure that will estimate the occurrence of the material fatigue.

2. We shall apply the Pontryagin Maximum Principle to optimize the shape of the elastic strongest rod against the buckling. The procedure will imply the possibility of analyzing both uni- and multi-modal buckling. We shall introduce restrictions on the cross-section area. Namely, it is greater, or less that the fixed value, or it is between two fixed values. Uni- and multi-modal buckling will be analyzed in the each case. If both types of buckling are possible, the separating limit will be determined. We shall analyze post-critical behaviour of the optimally shaped rod and numerically determine number and qualitative characteristics of the solutions that bifurcates form the trivial equilibrium position. We shall use the energy method to investigate the stability of the solutions.

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