Displacement Formulations for Deformation and Vibration of Elastic Circular Torus

The formulation used by most of the studies on an elastic torus are either Reissner mixed formulation or Novozhilov’s complex-form one, however, for vibration and some displacement boundary related problem of a torus, those formulations face a great challenge. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will carry on my previous work [ B.H. Sun, Closed-form solution of axisymmetric slender elastic toroidal shells. J. of Engineering Mechanics, 136 (2010) 1281-1288.], and with the help of my own maple code, I am able to simulate some typical problems and free vibration of the torus. The numerical results are veriﬁed by both ﬁnite element analysis and H. Reissner’s formulation. My investigations show that both deformation and stress response 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, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of a torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency. One of the most intestine discovery is that the crowns of a torus are the turning point of the Gaussian curvature at the crown where the mechanics’ response of inner and outer torus is almost separated.

Among regular shells, such as circular cylindrical shells, conical shells, spherical shells, and tori, the deformation of the torus is one of the most difficulty topics due to its complicated topology, where the torus geometry has both Positive, negative Gaussian curvature and turning points.
When the torus was first studied, high-order and complicated governing equations of a torus under symmetric loads were reduced to lower-order, 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, which is a ODE system about shear force Q and rotation χ (Flügge 1973 [15]). In 1959, Novozhilov published his celebrated monograph on the complex-form theory of shells, and formulated symmetrical deformation of a torus in his complex-form (Novozhilov 1959 [13]).
Regarding formulation of elastic torus, governing equations that all publications adopted are either Reissner-Meissner's mixed formulation or Novozhilov's complexform one. Although those formulations have some advantages, however they can not be used for vibration analysis and/or have difficulty when you have to deal with displacement boundary conditions. Therefore, it is desired to have displacement formulations for elastic torus. Sun 2010 [8] who derived displacement-type equations for torus and obtained a closed-form solution for special case as radius ratio a R → 0. This special solution can not capture full mechanics nature of torus and therefore are needed to be improved further.
This study wishes to carry on my previous works (Sun 2010 [8]), and investigates mechanical response for arbitrary ratio a R . The content is organised as follows: Section 2 reformulate displacement-type formulation of torus. Section 3 applies the displacement formulation to three problems. The first and 3rd problem can be calculated by both Reissner's and Novozhilov's formulation, while the 2nd can not be solved by any exiting formulation of torus except displacement-type one. For numerical simulation, we write a general code by Maple. Section 4 obtains eigenfrequency and eigenmode. Section 5 verifies our results by both finite element analysis and Hans Reissner's formulation. Section 6 provides conclusions.

II. DISPLACEMENT-TYPE FORMULATION OF SYMMETRICAL DEFORMATION OF ELASTIC TORUS
The displacement-type formulation was derived by Sun 2010 [8,9], but for completeness of representation, we will reformulate again in this section using different notations. For convenience of readers, all quantities notation and definition please refers to well-known book of J.N. Reddy 2007 [36].
For the torus shown in Fig. 3, 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.
With the above strains and curvature change, the resultant membranae force and bending moments can be Substituting the shear force Q ϕ in Eq.4 into Eq.3 and produce d dϕ The Eq. 6 can be further simplified by substituting the constitutive relations into Eq.bal-2, which will generate a final equations that is a six order ordinary differential equation system about displacement u(ϕ) and w(ϕ).
Eq.7 is the final governing equations in terms of displacements u, w of elastic thin torus and was derived by Sun 2010 [8,9].

III. NUMERICAL STUDIES OF SYMMETRICAL DEFORMATION OF TORUS
Eq. 7 will be solved numerically. Unless otherwise stated, numerical calculations in this paper are based on the data in Table I. We vary the radius a = 3k, k = 1, 2, 3, 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: The geometry of these tori are shown in Fig. 4.
A. Complete torus with a penetrate cut along the parallel θ = π 2 and loaded with vertical force Q0 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. 5. The boundary condition is: 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. In particular, Fig. 9 reveals that the opening relative displacement ∆ z is mainly contributed by the deformation of outer torus (ϕ ∈ [0, π]) while the most portion of inner torus (ϕ ∈ [0, −π]) remains undeformed. However, Zhang did not give either analytical or numerical solutions of the problem. After revisiting Zhang's problem of torus, it is found that the reduced formulations such as Hans Reissner's mixed formulation and Novozhilov's complex-form ones are not able solve the problem, because both formulations can not provide a proper displacement and rotation boundary conditions. To the best of my knowledgeless, the study of this difficulty problem has not been seen in the literature and worth to be investigated.

Preprints
If we cut the torus into two symmetrical parts as shown in Fig. 11. Let's apply the displacement-form formulation to the bottom portion of the torus, ie., the portion within ϕ ∈ [π/2, 3π/2], and taking into account of symmetry, the shear force Q 1 , displacement u and rotation 1 a (u − dw dϕ )in two boundaries are: The data of the torus is given in Table III: We wrote a maple code for Zhang's problem of torus, and produced some results as shown in Fig. 12,13 and 14

FIG. 12: Shear force and membrane force
The numerical studies show that the deformation is mainly contributed from the inner torus while outer torus almost undeformed. It seams that inner torus is separated from the outer one by the crowns (the turning point of the Gaussian curvature at ϕ = 0, π).
To make sure the correctness of this discovery, let's investigate another version of Zhang's problem of torus as shown in Fig. 15: The above numerical show that the turning point of the Gaussian curvature is the point of separation of me-  Let's apply the displacement-form formulation to the inner torus, ie., the portion within ϕ ∈ [−π/2, 0], and taking into account of symmetry, the shear force Q 1 , displacement u and rotation 1 a (u − dw dϕ ) in two boundaries of the inner torus are : The data of the torus is given in Table III. With the Maple code, we obtained some figures as shown in Fig.  19, 20, 21, 22 and 23.
An outer torus loaded with vertical force as shown in Fig. 24: Let's apply the displacement-form formulation to the outer torus, ie., the portion within ϕ ∈ [0, π/2], and taking into account of symmetry, the shear force Q 1 , displacement u and rotation 1 a (u − dw dϕ ) in two boundaries of the inner torus are : The data of the torus is given in Table III If we compare the results of both inner and out torus, it is easy to discover that the inner torus is much stronger than the outer one under same loading condition. For instance, the vertical deformation or elongation: Outer The deformation ∆ inner < ∆ outer reveals that the inner torus is stronger or stiffer than the outer one under same loadings, the geometric reason behind this phenomena is that the Gaussian curvature of the inner torus is negative while the outer's Gaussian curvature is positive. Therefore, from structural mechanics point of view, the shape of inner torus should be the first option when you design a torus.

IV. FREE VIBRATION, EIGENFREQUENCY AND EIGENMODE OF ELASTIC TORUS
Besides finite element method, both Reissner's mixed formulation and complex-form one can not be used to deal with vibration of torus, because the acceleration of motion must be taken into account in the formulation. In this case, displacement-type formulation is the only one can be accepted. Therefore, in this section, we will study symmetric free vibration of the torus. In this case, the initial namely −ρh ∂ 2 u ∂t 2 and −ρh ∂ 2 w ∂t 2 must be introduced to the Eq. 7 and lead to a free vibration equation system of elastic torus as follows: where the mass density ρ and time t. For free vibration analysis, the displacement u, w be set as [u, w] = [U (ϕ), W (ϕ)] exp(iωt, where ω is angular frequency, U (ϕ), W (ϕ) are the eigenmode of u, w, respectively. We have simulated the torus as shown in Fig. 5 without load Q 0 , which means that we are dealing with free vibration. With our general Maple code, eigenfrequencies are obtained as follows:

A. Taking into account of acceleration in both direction of u and w
For free vibration of torus, if we take into account of acceleration in both direction of u, w, namely −ρh ∂ 2 u ∂t 2 and −ρh ∂ 2 w ∂t 2 . The eigenfrequences are obtained as follows: The eigenfrequency value indicates that the torus with smaller ratio a/R will generate higher frequency.

B. Taking into account of acceleration in direction of w
For free vibration of torus, if we only take into account of acceleration in direction of w, namely −ρh ∂ 2 w ∂t 2 . The eigenfrequency are obtained as follows:

C. Eigenmode
The eigenmodes corresponding to the eigenfrequencies are The results indicate that the eigenfrequency in case of 4.1 and 4.2 has a big difference, while the eigenmode has little difference for literal motion w and small off alignment for mid-surface motion u. The big frequency difference implies that the acceleration in u direction can not be omitted for vibration of torus.
It is worth to mention that such analysis has not been reported in the literature, other shell's vibration can be seen in a famous book by A W Leissa 1993. [38] V. VERIFICATION OF OUR RESULTS BY FINITE ELEMENT ANALYSIS

A. vs. Finite element analysis
To verify our result, we carried out a finite element analysis for the case study in Section 5.1. The comparisons are shown in Fig. 31. The finite element results were simulated by ABAQUS with shell element S4R. We applied the boundary condition △ z (−3π/2) = 0 to cancel the rigid body motion, which might be the reason for minor differences in the range θ ∈ [−3π/2, −5π /4]. From physics point of view, our result is symmetric respect to the axes of θ = −π/2 while the FEA has slightly symmetry breaking. 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]. with constant thickness can be found in the masterpieces of Flügge 1973 [15]. All notations in this subsection can be found in Flügge's book [15]. The balance equations are: 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 2πR2 and g(θ) = 1 2π sin 2 θ [ The resultant membranae forces can be represented by shear force N 1 : Substituting the principal radii, ie., R 1 , R 2 into Eq.23 will give governing equation for torus as follows: 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.25 and 26. Obviously, Eq.25 and 26 are complicated and hard to be solved analytically.  coding and numerical simulations will be presented in another paper.
The numerical simulation here show that the results Hans Reissner agrees with our displacement formulation, which provides a good supportive evidence about the correctness of our simulation.

VI. CONCLUSIONS
We have reformulated elastic torus in terms of displacement and successfully solved three typical problems and free vibration of torus. 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. Our investigations show that both deformation and stress response 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, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Concerning the free vibration of torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors if initial in u direction were omitted. One of the most amazing discovery is that the crowns of a torus are the point of turning point of the Gaussian curvature at ϕ = 0, π, where the mechanics response of inner and outer torus is almost separated. 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. 31; Mr. Xiong Li, for preparation of Fig.2 (a), Jie Wei for drawing of Fig.2 (b) and Fig.3 (a,b). 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 author upon reasonable request.

Conflict of interest
The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.