An algorithm for full parametric solution of problems on the statics of orthotropic plates by the method of boundary states with perturbations

The article substantiates the possibility of building full parametric analytical solutions of mathematical physics problems in arbitrary regions by means of computer systems. The suggested effective means for such solutions is the method of boundary states with perturbations, which aptly incorporates all parameters of an orthotropic medium in a general solution. We performed check calculations of elastic fields of an anisotropic rectangular region (test and calculation problems) for a generalized plane stress state.


Introduction
Historically, there are three phases in the methodology for solving problems that aim to determine the strength of linear-elastic bodies.
Classical (pre-computer) method. It is characterized by an analytical approach to the formulation and solution of problems. The results are presented in a strict (closed or quadrature) or approximate (series-based) form and contain all the parameters of the problem (FPS: analytical full parametric solution). The obvious features and advantages of this approach are: wide use and concomitant development of the mathematical apparatus (special functions expanded into series, functions of a complex variable, integral transforms, ...); simplicity of computational procedures used in engineering and design work (including for optimization of solutions as regards geometric and physical parameters). The main disadvantages are: a relative narrowness of the classes available for the analysis of bodies in relation to their "geometry" (unlimited 3D or 2D space, half-space, layers, cylindrical bodies, cones (wedges), etc.); limited options for types of boundary conditions (BC; first main, second main, main combined, contact problems); and rareness of problems focusing on the interaction of homogeneous elastic bodies of different materials.
Computational (computer) method. The widespread use of computer technology has practically removed restrictions with regard to classes of geometric configurations of bodies considered. It became possible to find precise solutions for problems with numerous boundary conditions and problems involving piecewise-homogeneous and inhomogeneous bodies. However, the advantages inherent in the classical approach turned into disadvantages: a change of any parameter (outside the Ptheorem [1]) requires time-consuming and resource-intensive re-computation; this spurred the need to compare the results of the application of various computational techniques; the energy intensity of

Reduction of the model of weakly orthotropic environment by the perturbation method
The state of the environment is governed by the generalized Hooke's law (the coordinate axes coinciding with the axes of orthotropy), which expresses the stresses are the modulus of volume expansion and the shift moduli, respectively. In the case of a generalized plane stress state it is convenient to use the following isotropic medium parameters: E ,  are the moduli of elasticity and Poisson's ratio for an orthotropic medium, respectively.
Such an assignment allows us to describe real elastic constants using small parameters with the parameter  used twice: . Now Hooke's law (2.1) is rewritten in a convenient form: Zero values of the parameters correspond to isotropy.
Detailed summary of features on the asymptotic perturbation method is given in the monograph of Nayfeh A. [6] and Minaeva N. [7].  Below in asymptotic series the upper indices, which correspond to the powers of the small parameters, identify particular elements of the expansion: After a change of notation (tensorial and indicial notation combined) [8]  we present the elements of the expansion in the form conventional for the generalized Hooke's law for isotropic bodies: The Cauchy formula is put in the same notation: Equations (2.4) -(2.6) formally describe the plane deformation state of isotropic elastic bodies (or generalized plane stress, in which case, though, the modulus of volumetric strain will have a different value [9]). Assuming the elastic field formed based on the fictitious body forces through the biharmonic equation for the Airy stress function [9], for which the Kolosov-Muskhelishvili general representation formula is obtained. This allows to obtain a basis of space of internal states by the MBS method [3]. Since the original basis is independent from the parameters , an orthonormal basis ("body in the sense of MBS") is obtained only once with high precision and then used in each iteration.

The basics of the method of boundary states with perturbations
The MBS is based on the isomorphism of spaces  and  ,  being the space of internal states and  being the space of boundary states. The main provisions of the perturbation method as applied to the medium state concept are as follows. Operator A governs the constitutive relations of a medium. By making operator A act on the internal state  , we obtain the following right-hand members of constitutive relations f : where  is a small parameter. Nonlinear operator L generates a set of boundary conditions  from attributes of the boundary state  : We will analyze a class of operators A representable by series in terms of powers  involving linear operators k A : Let the right-hand member (2.6) be presented by the following series: then the operator equation (3.1) takes the following form: The internal state is treated as a series in terms of the small parameter  : where j  is a state of jorder in the method of perturbations. Ratios (3.3) adjusted for equation (3.4) deliver the following: which is later converted as follows: (3.5) After bringing (3.5) to a common index s we obtain the following: For approximation s we have: These calculations are also valid for operator L , which is applied to boundary state  . The resultant formula of the boundary state is: The result of the decomposition of the original nonlinear boundary value problem (3.1), (3.2) is reduced to a sequence of linear problems (3.6), (3.7).

The method of boundary states with perturbations for a multiple-parameter physical medium
Any internal state of a linear isotropic elastostatic medium is a set of displacements, strains, and stresses that correlate with each other through constitutive relations: ; moreover, this is a Hilbert isomorphism [3]: Any well-defined problem is reduced to an infinite system of linear algebraic equations (ISE) [3]  where all the coefficients l c are related to any component of the element basis. The matrix Q ("the skeleton of the problem") is determined structurally only by the type of boundary conditions and numerically through an orthonormal basis (with identity matrices used in basic problems). The vector of the right-hand members includes the information about the specific boundary conditions.
When using the MBSP, it is convenient to organize the enumeration of exponents of small parameters by assigning the consolidated index . An ISE is formulated at each step (4.1) in accordance with the boundary condition of the iteration. In practice, it suffices to take the real boundary conditions into account only when 0  p , solve only the basic problem with E Q  in subsequent iterations, and take account of the corrections brought about by the new fictitious volume forces (which are not actually potential forces, but are polynomial) in the right-hand members. A general method for searching for internal states for a class of such forces has been recently described [10].
There are certain steps to be taken before the iterations: the bases of the spaces  and  are formed using the general solution of Lamé's equations and the basis of functions harmonic in 6 1234567890 ''"" After performing a sufficient number of approximations it is necessary to make a final substitution and turn to dimensional values. The result is an FPS.

Full parametric solution of problems with rectangular plates
The FPS methodology described above is tested on a fairly simple first basic problem featuring an aluminium plate occupying the area . Following a non-dimensionalization  Test problem. The lateral faces of the plate are subjected to forces ( figure 1,a): Using the above real values of  ,  ,  , we obtain: . The relative asymptotic errors for are 0.59, 0.79, and 0.23%, respectively.
Calculation problem. As an example of how the proposed method for taking account of physical parameters for media is to be refined, we are going to solve the first basic problem for an orthotropic plate ( fig.1,b) in its heterogeneous state, it being understood that its orthotropic parameters are known. The non-dimensionalized load parameters are matched by the following equilibrium: We propose a second-order solution of the problem. To get around the awkwardness of the formulas involved, we will provide the components of the displacement vector in a truncated form.
The accuracy of the approximate parametric solution of the problem mainly depends on the accuracy of the solution in the zero approximation in which given boundary conditions are set on the boundary of the body. We present several explicit and implicit characteristics of the convergence of the problem in the zero approximation (table 2). Table 2. Nonzero Fourier coefficients.       As we can see from the drawings, the greatest tensile stress xx  along the axis of symmetry is focused along the left edge in the center of the plate, whereas compressive stress is localized at its edges. Stress yy  has approximately the same character. The nonzero shear stress xy  is concentrated near the left side and each side approximately the same distance (20% of the width of the plate). The character of the maximum shear stress Tmax differs little from the character of the axial tensile stresses, except that its highest concentration is not adjacent to the left side.
The foregoing analysis leads to the conclusion that MBSP proves to be an effective means of explicit full parametric solution of mechanics problems containing a finite number of physical parameters of the medium.