Study on the Deformation and Stress Analysis of the Sponge/Rubber Adapter

The analysis of adaptors’ deformation and stress is an important prerequisite for the design of the adapters. This paper proposed the mathematical description and analytical method for the sponge cylinder and rubber cylinder under pressure. The analytical formulas on deformation and stress of interference fit between adapters and launchers are deduced. Also, the correctness of the analytical formula is verified by numerical simulation. Furthermore, the effect of precompression and the sponge’s thickness ratio on adapter’s deformation and stress is studied.The following results can be concluded: the results of numerical simulation are basically consistentwith that of analytical formula. After installing the adapter into the launcher, the rubber layerwillmove inward as awhole, then the sponge layer and the rubber layer are subjected to radial and tangential compressive stress. When the precompression is smaller than 0.0075 or larger than 0.0015, the deformation and stress of adapter are approximately in proportion to precompression. If the precompression keeps constant, the deformation and stress of adapter will have extremumwhen the thickness ratio of sponge equals to a certain value.


Introduction
In the cold launching system, there are elastic cushions for adapters between the missile and the metal cylinder.The adapter is usually glued by the sponge cylinder and the rubber cylinder [1][2][3], as shown in Figure 1.During the ejection of the missile, the friction between the adapter and the launcher could reduce the ejection height of the missile and is determined by the radial stress caused by the adapter's compression deformation [4].Therefore, the study of adapters' deformation and stress has an important engineering application value for the design of adapters.
As the adapter's deformation and stress analytical research is based on the study on sponge cylinder and the rubber cylinder, many studies have been conducted in recent years.The modeling of mechanical characteristics of sponge material is studied by Koohbor B et al. [5][6][7] and is verified with experiments.Liang et al. [8] simulate the damping performance of the sponge components using the neural network theory.Based on the study on identification process of mechanical properties of rubber materials, the constitutive equations for small strain and large strain are proposed by Hao H et al. [9,10].Lee H S et al. [11,12] studied the deformation characteristics of rubber cylinders subjected to torsion, axial force, radial force, and skew load.The radial stiffness formula of rubber cylinder is derived by Horton J M [13], and the theoretical analytical formula and experimental data are compared and verified.Yu Z et al. [14,15] give an analytical solution for the interference fit between superelastic incompressible rubber and linear elastic material under a certain plane strain condition.Nevertheless, there are few reports on the stress and deformation modes of the sponge/rubber adapters.
In this paper, the interference fit between the adapter and the launcher is simplified as an axisymmetric plane strain problem.Based on the Blatz-Ko strain energy function of the sponge material and the three time reduced polynomial constitutive model of rubber material, the differential equation description and the plane axisymmetric analytical solution of the sponge cylinder and the rubber cylinder are proposed.Then, the analytical formulas on deformation and stress of interference fit between adapters and launchers are deduced and the correctness of the analytical formula is verified by numerical simulation.Furthermore, the effect of precompression and the sponge's thickness ratio on adapter's deformation and stress is studied.The research methods and results in this paper can provide reference for the design of adapters

Mathematical Description of Sponge
Cylinder under Compression Poisson's ratio of sponge is approximately 0 when pressed, assuming that the load and boundary conditions are axisymmetric, it can be simplified as an axisymmetric plane strain problem.In the initial stage, the inner and out radius of sponge cylinder are, respectively,   and   , and then turn into   and   after deformation under the action of inner pressure of   and out pressure of   .The cylindrical coordinate system is adopted with the material coordinate of (, Θ, ) and the spatial coordinate of (, , ).Because of the axisymmetric plane strain problem, the deformation mode can be set as where  is the elongation of the cylinder in  direction.
According to (1), the main elongations of the deformation gradient are Sponge and rubber can produce large nonlinear deformation.The strain energy function is usually used to describe the deformation properties of the material [14].Sponge is a hyperelastic compressible material whose strain energy function  is a function of 3 main elongations, namely,  = (  ,   ,   ).The Cauchy stress in the 3 main elongation directions is [17] The axisymmetric plane strain problem should satisfy the static equilibrium equation with ignorance of gravity [18].
Equation ( 3) is substituted into (4): At present, the strain energy functions for compressible materials mainly consist of Ogden-Hill and Blatz-Ko strain energy function.The Blatz-Ko strain energy function has a simple form and can describe the deformation characteristics of compressible sponge material, and its strain energy function is [19] where  is the shear modulus of sponge in a natural state.Equation ( 6) is substituted into (5): where    =   /.

Differential Equation Analysis.
Set  =   /  =   /, and then take derivative of R to obtain The expression of intermediate variable  and (8) are substituted into (7): The integral solution is applied on the ordinary differential equations (9), and the deformation function of the sponge cylinder can be obtained.
where  0 and  0 are integral constant and are related to boundary conditions.The Cauchy stress in three directions could be obtained by combining (3) and (6).11) could be

Differential Equation Description of Rubber Cylinder.
If the axial length is far greater than the thickness, in the axisymmetric problem, the axial direction of rubber cylinder can not be extended or shortened, so it can be simplified as axisymmetric plane strain problem, the calculation method of deformation mode, and main elongation are consistent with sponge cylinder.As rubber is superelastic material, its constitutive relation is defined by the partial derivative of the corresponding strain energy function [20,21].
where  is the Cauchy stress deviation,  is volume ratio before and after deformation, dev(⋅) is tensor deviation operator,  is the strain energy density converted by unit volume before deformation,  is the deformation gradient tensor, then  =  ⋅   ,  =  −2/3 ⋅  [22]; For incompressible sponge material,  = 1.In terms of axisymmetric plane strain problem, (15) can be written as It could be obtained from (15) as where  is the hydrostatic pressure.Equation ( 4) is the partial derivative of present coordinate of ; combining with (2), it could be written as the equation of initial coordinate .
The differential equation of axisymmetric plane strain could be obtained by submitting ( 16) into (17).

Differential Equation Analysis.
In this paper, the threereduced polynomial constitutive model is used as the strain energy function of the rubber material.Because the volume of the rubber is incompressible, the strain energy function of the rubber is as follows [23,24]: In terms of axisymmetric plane strain,   = 1 and ( 19) is submitted into (18) to obtain As the rubber is incompressible and combining with (2), The deformation function of rubber cylinder could be obtained by integral.
where  0 is integral constant and is determined by the boundary condition.Equations ( 21) and ( 22) are substituted into (20): By integrating (23), the radial stress of rubber cylinder is as follows: where  0 is integral constant and is determined by the boundary condition.Combining ( 16) with (24), the tangential stress of the rubber cylinder is as follows: the sponge.Then the corresponding deformation mode and stress can be written as

Analysis of Interference Fit between Adapters and Launchers
4.1.Geometric Description.Schematic diagram of interference fit between adapters and metal launchers is shown in Figure 2. The materials from inside to the outside are sponges, incompressible rubber, and isotropic metals and are, respectively, assigned as material 1, material 2, and material 3. The subscript numbers 1, 2, and 3 in variable correspond to material 1, material 2, and material 3. The variable with subscript I has relationships with inner surface of adapter, which with subscript M has relationships with adhesive surface; the variable related to outer surface of launchers is marked with subscript O.The outer radius of launcher before deformation is  3 =   = 1 and the inner radius is  3 .Similarly, the adapter's outer radius is  2 before deformation, inner diameter is  1 =   , and the adhesive surface radius is   .The interference fit is adopted between rubber's outer diameter and launcher's inner diameter, and the precompression is .After the installation of adapter into the launcher, the radius after deformation is, respectively, as follows, for the launchers, the outer radius is  3 =   and the inner is  3 .For the adapter, the outer radius is  2 , the adhesive radius is   and the inner radius is  1 =   .Nevertheless,  3 and   are all unknown and the outer surface pressure of launcher is 0.

Axisymmetric Plane Strain Solution for
Launcher.Considering that the mechanical properties of the launcher are isotropic and without volume force, the boundary value problem is attributed to [25-28] where  is the stress function and is the function of  and  under polar coordinates.For the axisymmetric issue,  is the function of  and has nothing with .
If the stress function  is known, then its stress could be expressed as [25][26][27][28] Considering the operator of  2 / 2 + (1/)(/) = (1/)(/)(/), then (29) could be written as The integrating on (32) to obtain Then the corresponding stress component is as follows: where  0 ,  0 ,  0 , and  0 are the integral constant, which are determined by the boundary condition and single value condition of displacement.
Considering the axial symmetry, the small deformation geometric function is as follows: And the constitutive function of plane stress is as follows: The strain component could be calculated by constitutive (35b) and then be submitted into boundary value (30).Due to the single value condition of the displacement, the N0 = 0 and the integral constants can be obtained.

Equation Solving of Interference Fit
Problem.Before the adapter installing into the launcher, the sizes of adapter are, respectively,  2 ,   ,  1 =   .The size of launcher before deformation is  3 =   = 1.As  3 is known,  3 =  2 is obtained after the adapter installing into launcher.From (39), the relation between the inner surface pressure and the radial displacement of the launcher could be obtained.
The inner radius of launcher after deformation is Equation ( 41) could be written as where  =  3 and  = −(( According (22), the radius expression of adapter and adhesive surface after deformation is After the deformation of adapter, the radial pressure of the outer diameter of the rubber layer and the inner diameter of the launcher must be equal.According to (24), the radial pressure expression of the inner layer of the launcher can be written out.
Equation ( 42) is submitted into (45), then The following two equations can be obtained by bringing the initial internal and external radius of the material 1 in the adapter into (12).
Considering the radius of the adapter's adhesive surface after deformation, ( 44) is brought into ( 13): Due to the assumption that the body is rigid, the inner diameter of the adapter will not deform, so the inner diameter of the adapter is known and can be obtained: As the spongy and rubber materials are in a continuous radial stress in the adhesive surface, a similar equation ( 46) can be written out.
The nonlinear equation group (53) is calculated by using Newton-Raphson method [29,30], and the , , , ,  M , and  I could be obtained, and then the deformation and stress of the interference fit for the adapter and the launcher could be obtained.

Example Verification and Analysis. The precompression
of the adapter is 0.005161 in a certain missile launching system, the dimensionless sizes are  3 = 0.967742,  2 = 0.971613,   = 0.918710, and   =   = 0.903226.The material parameters of adapter and launcher are  1 = 185.0, 2 = 6.5660,  3 = 0.03488,   = 2.954595 × 10 6 , and V = 0.340.The finite element software ANSYS is adopted and the radial displacement, radial stress, and tangential stress curves of adapter and launcher are obtained, as shown in Figure 5.
From Figures 3-5, the displacement, radial stress, and tangential stress of analytical solution are consistent with that of the finite element.As a result, the analytic formula of interference fit of deformation and stress is deduced correctly in this paper, which is applicable to the sponge cylinder, the rubber cylinder, the adapter, and the launcher.
The maximum absolute value of adapter's displacement is 0.0040772 in Figure 3, which appears on the adhesive surface and is larger than the precompression of adapter.As indicated, the rubber layer will move onward as a whole when the adapter is subjected to pressure.The maximum absolute value of adapter's outer surface displacement is 0.0038593 with an inward movement.After the installation of adapter into the launcher, every point in the launcher moves outward; the maximum displacement of launcher appears on the interference fit surface with the value of 1.663775 × 10 −5 , which is far smaller than that of adapter.The outer surface displacement of launcher is 1.66361 × 10 −5 .
It could be concluded in Figure 4 that the absolute value of sponge's radial stress in the adapter turns small from the inside to the outside, while that of rubber layer turns large.From inside to outside, the absolute value of radial stress in launcher reduced to 0, and the three materials are all subjected to compression.The absolute value of the radial stress of the inner adapter is 1.5141, that on the adapter surface is 1.4999, and that on the interference fit between the adapter and the launcher is 1.8462.
In view of Figure 5, the absolute value of tangential stress of inner sponge layer in the adapter is 0.36058, which increase along the radial direction, and that of outer sponge layer reaches 0.37337.For the absolute value of tangential stress in the rubber layer, it is 0.36058 in the inner surface and decreases radially and then reduces to 7.7368 on the outer surface.The tangential stress for the whole adapter is compressive stress while that of launcher is tensile stress.The tensile stress is 56.481 in the inner surface of launcher and decrease radically, which is 54.664 in the outer layer.

Analysis on the Deformation and Stress
Effects of Adapters ) and outer surface ( M1 ), the radial displacement of outer sponge ( rM ), the radial stress of outer sponge ( rM ), the tangential stress of inner rubber ( M2 ) and outer surface ( O2 ), the tangential stress of the outer rubber surface ( rO2 ), the radial displacement of inner launcher ( rI3 ) and the outer radial displacement ( rO3 ), the tangential stress of the inner launcher ( I3 ), and tangential stress of outer surface ( O3 ).From Figures 6-13, the effects of  to  rI1 ,  rM ,  M2 ,  O2 ,  rO2 ,  rI3 ,  rO3 ,  I3 , and  O3 are divided onto three stages: (1) When 0 ≤  ≤ 0.0075 at the first stage,  rI1 ,  rM ,  M2 ,  O2 ,  rO2 ,  rI3 ,  rO3 ,  I3 , and  O3 is approximately proportion to the precompression .The absolute value of proportional coefficient is small and the precompression  varies largely, while the corresponding stress and displacement change little.
(3) When  ≥ 0.0105 at the third stage, the precompression  changes little while the corresponding stress and displacement change largely.defined as Δ  =   (  +   ), where   is radial thickness of sponge layer and   is the radial thickness of the rubber.Δ  =   /  is defined as the ratio of precompression of adapter to sponge layer's thickness,   is the precompression of adapter, and   =   +  .When Δ O = 0.2, 0.3, and 0.4, Figures 14-25 are, respectively, the effect curves of thickness ratio Δ E to the radial stress of the inner sponge surface ( rI1 ), the tangential stress of inner sponge ( I1 ), the radial displacement of outer sponge surface ( rM ), the radial stress of outer sponge surface ( rM ), the tangential stress of outer sponge surface ( M1 ), the tangential stress of inner rubber surface ( M2 ), the radial stress of outer rubber surface ( rO2 ), the tangential stress of the outer rubber surface ( O2 ), the radial displacement of the inner launcher surface ( rI3 ), the tangential stress of the inner launcher surface ( I3 ), the radial displacement of outer launcher surface ( rO3 ), and tangential stress of outer surface ( O3 ).

Effect of Thickness Ratio of the
As shown in Figures 14,15

Figure 1 :
Figure 1: Schematic diagram of the launcher and the adapter.

Figure 6 :
Figure 6: Effects to the  rI1 of adapter.
Sponge Layer on the Deformation and Stress.The thickness ratio of sponge layer is