Uniform Domain Equilibrium Equation with Finite Deformations

. In corner areas of structures, high stress values and gradients occur, and lead to stress concentrations. Infinite stress and deformations are determined by a solution of the linear elasticity theory problem in the area with a wedge-shape boundary notch. Infinite solutions of the elasticity problem occur under impact of forced deformations, when a surge of the deformation value reaches beyond the area boundary. Relative values of stress concentrations for corner area zones make no more sense. At finite displacements, high deformation and stress values occur in the corner zones of the area. For a linear statement of the elasticity theory problem, at minor deflections, not only first-order, but also second-order derivatives of the displacements function are significant. To account for finite deformations of such corner zones of the area, correct formulations of elasticity problems are required. Study objective: influence determination of the infinitesimal order of the deformation on the appearance of equilibrium equations of an area with induced (temperature) deformations. This allows for the analysis of the influence of linear, shear deformations, and of the swing on the solution of the elasticity problem with induced deformations.


Introduction
In corner zone areas of structures, high stress values and gradients occur, which lead to stress concentrations [1][2][3][4][5].Infinite stress and deformations are determined by a solution of the linear elasticity theory problem in the area with a wedge-shape boundary notch [6,7].Infinite solutions of the elasticity problem occur under impact of forced deformations, when a surge of the deformation value reaches beyond the area boundary.Relative values of stress concentrations for corner area zones make no more sense.At finite displacements, high deformation and stress values occur in the corner zones of the area [8][9][10][11][12].For a linear statement of the elasticity theory problem, at minor deflections, not only first-order, but also second-order derivatives of the displacements function are significant.To account for finite deformations of such corner zones of the area, correct formulations of elasticity problems are required.
Study objective: influence determination of the infinitesimal order of the deformation on the appearance of equilibrium equations of an area with induced (temperature) deformations.This allows for the analysis of the influence of linear, shear deformations, and of the swing on the solution of the elasticity problem with induced deformations.
Study objectives: 1) Derivation of expressions for finite deformations in a polar system of coordinates under recognitions of induced (temperature) deformations.
2) Derivation of an equilibrium equation for a flat area in the polar system of coordinates, under recognition of finite and induced deformations.
3) Reduction of the deformation infinitesimal order analysis to the appearance of equilibrium equations under recognition of finite and induced deformations.

Problem statement
The flat elasticity theory problem is analysed for an area, which can feature a wedge-shaped notch on the area boundary [7,9,11].In the flat area (Fig. 1), spatial forces are acting; the induced deformations are free temperature deformations ij T  .In  is constant and equal to  .The limit conditions of the area are uniform.
Problem statement: for  uniform area with induced deformations, an equilibrium equation shall be derived, under recognition of finite deformations.

Equilibrium equation
A spatial orthogonal curvilinear coordinate system [7,[9][10][11][12] i  is analysed, i=1,2,3, i k are unit vectors directed positively along the axes i  , that is, the basis vectors of the area before the deformation.Selected is an infinitesimal element limited by six coordinate  edges, thereby, i E  are the relative elongations along i  axes after the deformation, * i k are the basis vectors of the area after the deformation.
The equilibrium equation of all the forces acting onto the oblique-angled prism appears as follows [9,11,12]: SS is the surface area of the cuboid faces before and after the deformation, F are the summarized spatial forces after the deformation.Having specified the forces by the cuboid faces after the deformation in the initial vector basis i k , i=1,2,3, before the deformation, the equations (1) can be put down as follows: Thereby, i F is the projection of the summarized isometric force on A flat state of deformation is analysed [7,8], the body points experience displacement in the planes perpendicular to OZ axis: For orthogonal curvilinear system of coordinates: The equilibrium equations ( 1), ( 2), (3) are reduced: thereby, in equations ( 3), ( 4): thereby, the summarized stress values  are associated with the stress values ij  in the area point by the following ratios: . It should be noted that the appearance of the linear equations of equilibrium in the summarized stress values in the polar system of coordinates of appearance (3), ( 4) is similar to that of the 'classic' equilibrium equations for minor deformations: Now, we put down the equations of equilibrium (3), (4) in deformations.

Physical correlations
We assume that the connection form for the summarized stress values and deformations is the same, with that in the Hooke's law between the stress and the deformations for minor deformations.[9.11].Under impact of the induced (temperature) deformations, the Duhamel-Neumann dependence will acquire the following appearance: In the polar system of coordinates, the stress (18) shall appear as follows: Thereby, 22 1 )(1 )  are defined as (11), (12).
, e  are small in relation to the single unit, then alternatives are considered: E3S Web of Conferences 410, 03007 (2023) https://doi.org/10.1051/e3sconf/202341003007FORM-2023 А) the magnitude 3   is the infinitesimal of the same or higher infinitesimal order than ij e B) the magnitudes ij e are the infinitesimals of the same or higher infinitesimal order than.12 2

12
(2 ) ( ) (29) For deformations (23), (24) of case А), the equilibrium equations in the deformations (25), (26) match the classic equilibrium equations under impacts of induced deformations.For infinitesimal order relations (27) of case В), along with the 'classic' appearance of the components of equilibrium equations ( 28), (29), swings are additionally contained.For the obtained equilibrium equations in the deformations (21), ( 22), uniform boundary conditions may be put down as follows: i nk are the directional cosines of the angles between the boundary normals and the basis of i k area before the deformation.For elasticity boundary value problem with induced deformations, the boundary conditions can be expressed via deformations and displacements, putting down the expressions s ij using ( 5), ( 6), the deformations ( 11), (12), and the correlations (19), (20).

Results
Obtained were equations of the elasticity problem for a flat area with finite deformations under impact of induced temperature deformations.Obtained were the expressions for finite deformations (11), ( 12), (17), the physical correlations (19), (20), the equilibrium equations for flat area with finite deformations under impact of induced deformations (21), (22), the boundary conditions (30).Obtained were the equilibrium equations in the deformations (25), (26) for infinitesimal swing cases (23), (24) as well as the equilibrium equations (28), (29) for infinitesimal linear and shear deformations relative to the swings (27).

Conclusion
The formulations of the basic deformation elasticity problem equations contained in the paper allow for an analysis of the influence of the infinitesimal order on the appearance of equilibrium equations of an area with finite and induced (temperature) deformations.This allows for the analysis of the influence of linear, shear deformations, and of the swing on the solution of the geometrically non-linear elasticity problem with induced deformations.

 , whereas the second 1 
(finite discontinuity) of induced deformations can be set.The discontinuity of the deformations could occur, e.g., if for one of 2 subarea is not loaded.A homogeneous body is in flat state of deformation.Area  features E-modulus,  Poisson's ratio,1   line expansion factor of the areas, and 2 are the summarized stress on the faces of the oblique-angled prism, are Lame's parameters.After deformation, the rectangular prism of the cuboid is transformed to an oblique-angled one, with i H * (1 ) is the sum of the normal summarized stress values,  is the line expansion coefficient, ij  is Kronecker symbol.
111 are free temperature deformations, E is the elasticity module,  is