Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by finite element analysis. Our investigations show that the mechanics of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell. A general Maple code is provided as essential part of this paper.

When the torus was first studied, high-order and complicated governing equations of a torus under symmetric loads were reduced to a single lower-order, complex-form ordinary differential equation (ODE) by Hans Reissner (1912) [25] when he was a professor at ETH in Switzerland. His colleague at ETH, Meissner (1915) [26], derived a complex-form equations for the shell of revolution. Hence, the first complex-form equation of the shells of revolution including torus is called the Reissner-Meissner equation. Reissner supervised a doctoral candidate, Gustav Weihs 1911 [19], who was the first person to receive a Dr.-Ing. in the field of torus research, in 1911. Unfortunately, I have not been able to find Weihs' thesis, and became familiar with his work from a citation by Wissler in his dissertation (Wissler 1916 [20]). Wissler was supervised by Meissner, and was the second person to receive a Dr.-Ing in the topic of torus. The first exact series solution of the complex-form equation was obtained by (Wissler 1916 [20]), but his work had no followers, perhaps because his series solution was not linked to any special functions and some computational challenge without computer. Reissner 1949 [28]), and with his PhD syudent Clark, they published two papers (Clark 1950 [10, 11]) on the asymptotic solution and calculations. In 1953, Dahl studied toroidal-shell expansion joints (Dahl 1953 [12]). In 1959, Novozhilov published his celebrated monograph on the complex-form theory of shells, and gave an asymptotic solution to the symmetrical deformation of a torus (Novozhilov 1959 [13]). In 1959, Tao introduced a variable transformation and successfully transformed the Reissner-Meissner complexform equation of a torus to a Heun-type ODE, and was the first person to find an exact solution that can be expressed in terms of Heun functions (Tao 1959 [29]). However, Tao did not perform any numerical calculations using his exact solution. Nevertheless, both Wissler and Tao reached the academic peak in the research of a torus with small deformation. In 1965, Steele investigated toroidal pressure vessels, and was able to make a concise comparison of the volume and weight properties of enough shapes to provide a convenient basis for their design (Steele 1965 [30]). Qian and Liang 1979 [2], Xia and Zhang 1986 [3], Zhang, Ren and Sun 1990 [4], Zhang and Zhang 1991 [5], 1994 [6] conducted further study, sought an enhanced asymptotic solution valid for the full domain of θ ∈ [0, 2π]. Despite the complex-form formulation of all previous study, Sun was the first person to derive displacement-type equations of a torus and proposed a closed-form solution when the radius ratio tends to be null (Sun 2010 [8] 2012 [9]). The first monograph on torus was edited by Sun and published in 2012 (Sun 2012 [9]).
In 1959 Novozhilov systematically developed a comprehensive complex-form theory of thin shells and reformulated the symmetrical deformation of a torus (Novozhilov 1959 [13]), then he proposed an asymptotic solution for the thin torus. However, no exact solution in terms of special functions has been obtained for Novozhilov's complex-form ODE of symmetrical deformation of a torus. We will shoulder this burden and propose such a solution. Once we have the exact solution, the convergence over the full domain of θ ∈ [0, 2π] can then be guaranteed.
The remainder of this paper is organized as follows. Section 2 presents a formulation of the Novzhilov complex-form ordinary differential equation of a torus. Section 3 introduces a variable transformation and finds an exact solution of the complex ODE. Section 4 splits the complex-ODE into a real-form ODE. In Section 5, we carry out numerical simulations for three cases by using our own Maple code. Section 6 discusses the difference between the membrane and bending theory of a torus. Section 7 h verifies our results by both finite element analysis and Hans Reissner's formulation. Section 8 provides conclusions and recommendations.

II. NOVOZHILOV'S FORMULATION OF SYMMETRICAL DEFORMATION OF ELASTIC TORUS
For the torus shown in Fig. 1, the positions of points on the middle surface will be determined by the angles θ and φ. Further, let R 1 be the radius of curvature of the meridian and R 2 the radius of curvature of the normal section, tangential to the parallel circle. This second radius is equal to the segment of the perpendicular to the middle surface between this surface and the axis of the torus.
The Lamé parameters in this case are determined by the expressions The principal radii of curvature are given by where α = a/R. The complex-form governing equation of a torus for symmetrical deformation was obtained by Novozhilov (1959) [13] as follows: where V (θ) is a auxiliary complex function of real variable θ and where µ is the Poisson ratio, h is the thickness, q is the distributed load, and C is an integration constant that can be determined from static considerations. The analysis of the torus has thus been reduced to the problem of finding a solution for the ODE in (3).
Having the auxiliary function V (θ), all other quantities can be expressed in terms of V (θ). The resultant forces are where the Re and Im stand for the real and imaginary portion of the complex function V (θ). And resultant moments are resultant shear force is and angle of rotation of the tangent to the meridian is The component of the displacement of an arbitrary point on the meridian in the direction of the axis of the torus is and the component of the displacement in the direction perpendicular to this axis is The deformation displacement components of the middle surface in Fig. 3 can be obtained as The components of the middle surface forces are where Q z and Q x , respectively, are the components of the forces in the direction of the axis of the torus and perpendicular to this axis, which act through the contemplated point of the meridian, as shown in Fig. 4.
dx . Hence Eq. 3 is transformed as follows: where Eq. 19 is a Fuchian-type differential equation whose series solution was given by Wissler 1916 [20], which is an exact solution but has not linked to any known functions.
To establish a relationship between Eq. 19 and well-known equations, let us introduce another variable transformation, , and Eq. 19 can be written as where where It is easy to see that Eq. 21 or 22 is a Fuchian-type differential equation with four regular singular points: 0, 1, α−1 2ϵ , and ∞. Eq. 22 was studied by Heun (1889) ( Ronveaux 1995 [32], Marsden, Sirovich and Antman 1991 [34], and [33]), whose solutions can be represented by Heun's functions. Named after Karl Heun (1859-1929), these are unique local Frobenius solutions of a second-order linear ordinary differential equation of the Fuchsian type, which in the general case has four regular singular points.
The standard Heun ordinary differential equation is Its solution is given by where HeunG(c, p, β, σ, ρ, δ, z) is the Heun function, which can be expressed in series as where Heun functions generalize the hypergeometric function. Because of their wide range of applications, they can be considered the 21st-century successors to hypergeometric functions (Ronveaux 1995 [32]).
The exact solution of Eq. 21 is the sum of a homogenous solution V h (x) and particular solution V p (x), i.e., V = V h + V p , both of which can be expressed by Heun functions. The homogenous solution can be given as where where . Substituting Wissler's transformation x = sin θ into the above solutions, we have where A particular solution V p in some simpler case can be expressed analytically by Heun functions, which will not be discussed here due to its complicated form. If the loading condition is not constant, then some integrations involving the Heun function in V p may not be obtainable analytically, which unfortunately restricts the use of the exact solution.

IV. THE SPLIT OF THE NOVOZHILOV EQUATION
For further computation, the solution V (θ) must be split into real and imaginary parts, i.e., Re(V ) and Im(V ), respectively. However, this is not possible analytically due to the complicated expression of the Heun function. Without Re(V ) and Im(V ), no analytical components of force, moment, and displacement for the torus can be obtained, which limits the use of the complex-form analytical solution obtained in the previous section. Therefore, it is natural to seek a numerical solution based on the Novozhilov equation, i.e., Eq. 3.
Let us write V (θ) as Because we cannot analytically represent the integration of the displacement component of an arbitrary point on the meridian in the direction of the axis of a torus in Eq. 12, we use the corresponding differential equation, Now, the symmetrical deformation problem becomes one of finding the functions A(θ), B(θ), and △ z (θ) in Eq.40 and 41..
With these functions, we can compute all other quantities, such as T 1, T 2 , M 1 , M 2 and N 1 , as well as u, w and ϑ. They can be expressed in terms of A(θ) and B(θ) as follows The resultant moments are the resultant shear force is and the angle of rotation of the tangent to the meridian is The bending-related shell stresses are obtained by combining the direct stresses due to the stress resultants with the flexural stresses due to the bending moments, The coupled differential equations (40) can be decoupled into differential equations about A(θ) and B(θ), both fourth-order ODEs, but no analytical solutions about A(θ) and B(θ) can be obtained.

V. NUMERICAL STUDIES OF SYMMETRICAL DEFORMATION OF TORUS
We numerically solve equations 40 and 41 for some typical cases. Unless otherwise stated, numerical calculations in this paper are based on the data in Table II.
We vary the radius a = 0.18k, k = 1, 2, 3, 4, 5, while other quantities are unchanged. Table I lists the geometric and material properties of the torus. For simplification of picture presentation of our results, physical units will not be plotted in all drawings.
For this purpose, we list all physical units of in Table II:   A. Complete torus with a penetrate cut along the parallel θ = π 2 and loaded with vertical force Q 0 For a complete torus with a penetrate cut along the parallel θ = π 2 and loaded with vertical force Q 0 , the loading condition is shown in Fig. 6. The boundary condition is: Figure 6: Torus with a cut along its parallel at θ = π 2 or θ = − π 2 under load Q 0 For this problem, the constant C can be determined by the boundary condition of T 1 at θ = π 2 , namely Taking into account the uniform distribution load q = 0, the above equation gives C = 1 2πa Q 0 . With the C, the boundary loading condition at both θ = π 2 and θ = − 3π 2 are satisfied. The rest of the boundary condition can be expressed in terms of A(θ) and B(θ) as Now, the problem becomes to solve A(θ) and B(θ) under conditions (52).
A general code for this case is written by Maple [35]. With the help of the above Maple code, some numerical results have been obtained and are shown in Fig. 7, 8, 9, and Fig. 10. The above all figures indicate that all quantities such as bending moments, surface forces, shear force, and displacement are strongly effected by the radius ratio α = a/R, and vary dramatically with θ both near to and far from the edge.
B. Complete torus with penetrate cut along the parallel θ = π 2 and loaded with bending moment M 0 Fig. 11 shows a complete torus with a penetrate cut along the parallel θ = π 2 and loaded with distributed bending moment M 0 . The boundary condition is:  For this problem, the constant C can be determined by the boundary condition of T 1 at θ = π 2 , namely Taking into account the uniform distribution load q = 0, the above equation gives C = 0, and the boundary loading condition at both θ = π 2 and θ = − 3π 2 are satisfied. The rest of the boundary condition can be The problem now becomes to find A(θ) and B(θ) under conditions (57). The numerical results are shown in Fig. 12, 13, and 14, which indicate that all quantities such as the bending moments, surface forces, shear force, and displacement are strongly affected by the radius ratio α = a/R, and vary dramatically with the angle θ both near to and far from the edge. If we remove the inner region of a torus, we will have a half torus consisting of the outer region. The half torus is under a vertical load along its parallel at θ = π 2 and θ = π, as shown in Fig. 15. Taking into  account of symmetric nature, we only consider the top half. Thus we have boundary condition: For this problem, the constant C can be determined by the boundary condition of N 1 at θ = 0, namely We obtain C = − Q 0 2πa , in which case the boundary loading condition at both θ = 0 and θ = π are satisfied.  The boundary condition in 59 can be expressed in terms of A(θ) and B(θ), as follows: The problem now becomes to solve A(θ) and B(θ) under conditions (62) and (63). The numerical results are shown in Fig. 16, 17, and 18, and the stress is shown in Fig. 19.
These figures indicate that all quantities, such as bending moments, membranae forces, shear force, and displacement, are strongly affected by the radius ratio α = a/R, and vary dramatically with θ both near to and far from the edge.

VI. COMPARISON BETWEEN THE MEMBRANE THEORY AND BENDING THEORY OF TORUS
In the history of theory of shells, a membrane theory is developed to simplify the analysis of shell structures. In the study of the equilibrium of the shell element, this theory neglects all bending moments,  To obtain a clear picture of the difference between a membrane and the theory of a torus, let us perform comparisons for the above three case studies. For the case study in Section 5.1, the surface force is shown in Fig. 20. For the case study in Section 5.2, the comparison is shown in Fig. 21. For the case study in  These comparison studies reveal that the results from the membrane theory of the torus are questionable.
Therefore, the bending theory of shells should be used for the analysis of a torus.   Therefore, our results are more trustable since the finite element analysis supports our numerical prediction nicely.

B. Comparison with Hans Reisnner's formulation of torus
In the theory of shells of revolution under axisymmetric load, according to Timoshenko and Woinowsky-Krieger 1959 [14], the decisive step was the introduction of N 1 and χ as unknowns by H. Reissner 1912 [25]. The idea has been much extended by E. Meissner 1925 [26]. The formulation of Reissner-Meissner of shells of revaluation with constant thickness can be found in the masterpieces of Flügge 1973 [15] and presented as follows: where χ is the angle by which an element rdθ of the meridian rotates during deformation; r = R 2 sin θ, the load term P is constant to be determined by the value of T 1 (π/2) = − P 2πR 2 and The resultant membranae forces can be represented by shear force N 1 : Substituting the principal radii, ie., R 1 , R 2 into Eq.64 will give governing equation for torus as follows: R + a sin θ a sin θ R + a sin θ a sin θ Once we obtain the shear force N 1 and rotation χ, one can compute all other quantities, such as T 1, T 2 , M 1 , M 2 , as well as u, w.
The problem of elastic torus becomes to find the shear force N 1 and rotation χ from Eq.69 and 70. Obviously, Eq.69 and 70 are complicated and hard to be solved analytically. To verify our numerical results obtained from Novozhilov's formulation, we write a Maple code to compute the Eq.69 and 70 numerically. Detailed Reissner's formulation, Maple coding and numerical simulations will be presented in another paper. The data of torus is given in Table II. The numerical simulation here show that the results of two formulations proposed by Novozhilov and Hans Reissner are agree each other very well, which provides a good supportive evidence about the correctness of our simulation.

VIII. CONCLUSIONS AND FUTURE PERSPECTIVES
We have obtained an exact solution of the complex-form ODE for the symmetrical deformation of a torus. This solution is expressed by the Heun functions. However, as we stated, the Huen function cannot be split into real and imaginary parts explicitly. To solve this issue, we proposed a new strategy of transforming the complex-form ODE of torus into an real-form ODE in terms of A(θ), B(θ), and △ z (θ), which allowed us to easily deal with different kinds of boundary conditions.
To verify our formulation, we wrote a computational code in Maple and carried out some numerical simulations. The validation of our numerical results was confirmed and supported by both finite element analysis and Reissner's formulation. The numerical investigations show that the bending theory of shells should be used for torus deformation analysis due to its vary Gaussian curvature.
Regarding the future perspectives, it must be pointed out that the complex-form governing ODE of a torus is good for static deformation analysis but has some disadvantages, for instance, the the complexform governing ODE cannot be used to deal with dynamical and nonlinear problems. This definitely limits the use of complex-form modeling. To overcome this shortcoming, displacement-type ODE of shells must be used (Sun 2010 [8]), which remains an open problem.

Acknowledgement:
The author is honored to have benefits from my supervisor Prof. Dr.Ing Wei Zhang (Wei Chang ), who was the first Chinese scholar studied the deformation of torus, it is my privilege to dedicate this paper to the memories of Prof. Zhang for his great contribution to the analysis of elastic torus. The author appreciates my students: Mr. Guang-Kai Song, for providing finite element analysis data in Fig. 23, and preparation of Fig.2, Fig. 3 and 4; Mr. Xiang Li, for preparation of Fig.1 (a). I also wish to express my gratitude to anonymous reviewers for their high-level academic comments that helps me to enhance the quality of this paper.
Data availability: The data that support the findings of this study are available from the corresponding