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

Authors: Jianlin Zhong; Jie Ren; Dawei Ma
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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 consistent with that of analytical formula. After installing the adapter into the launcher, the rubber layer will move inward as a whole, 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 extremum when the thickness ratio of sponge equals to a certain value.


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## Body

## 1. 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 [1–3], 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.Figure 1
Schematic diagram of the launcher and the adapter.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. [5–7] 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


## 2. Mathematical Description of Sponge Cylinder under Compression

### 2.1. Differential Equation Description of Sponge Cylinder

According to the study of Ju et al. [16], 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, RI and RO, and then turn into rI and rO after deformation under the action of inner pressure of σrI and out pressure of σrO. The cylindrical coordinate system is adopted with the material coordinate of (R,Θ,Z) and the spatial coordinate of (r,θ,z). Because of the axisymmetric plane strain problem, the deformation mode can be set as(1)r=fR,θ=Θ,z=λZwhere λ is the elongation of the cylinder in z direction.According to (1), the main elongations of the deformation gradient are(2)λr=drdR,λθ=rR,λz=1Sponge 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 W is a function of 3 main elongations, namely, W=W(λr,λθ,λz). The Cauchy stress in the 3 main elongation directions is [17](3)σr=1λθλz∂W∂λr,σθ=1λrλz∂W∂λθ,σz=1λrλθ∂W∂λzThe axisymmetric plane strain problem should satisfy the static equilibrium equation with ignorance of gravity [18].(4)dσrdr+1rσr-σθ=0Equation (3) is substituted into (4):(5)R∂λr∂R∂2W∂λr2+∂2W∂λr∂λθλr-λθ+∂W∂λr-∂W∂λθ=0At 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](6)Wλr,λθ,λz=μ2λr-2+λθ-2+λz-2+2λrλθλz-5where μ is the shear modulus of sponge in a natural state.Equation (6) is substituted into (5):(7)3Rλr′+λr4λθ3-λr=0where λr′=dλr/dR.


### 2.2. Differential Equation Analysis

Sett=λr/λθ=Rr′/r, and then take derivative of R to obtain(8)Rλr′=λθRt′-λr+λr2λθThe expression of intermediate variablet and (8) are substituted into (7):(9)dRR=3dt-t4-3t2+4tdrr=-3dtt3+3t-4The integral solution is applied on the ordinary differential equations (9), and the deformation function of the sponge cylinder can be obtained.(10)R8=A0t6e6/45tan-12t+1/151-t-4t2+t+4r4=A0t6e6/45tan-12t+1/15t2+t+41-t2whereA0 and B0 are integral constant and are related to boundary conditions.The Cauchy stress in three directions could be obtained by combining (3) and (6).(11)σr=μ1-1λr3λθ,σθ=μ1-1λθ3λr,σz=μ1-1λrλθ


### 2.3. Dimensionless of Deformation and Stress

η = R / R 0 represents the dimensionless radius before deformation, ξ=r/RO represents dimensionless radius after deformation and ξi=σi/μ (i=r,θ,z) is the dimensionless stress, A=AO/RO8, B=BO/RO4, h(t)=exp⁡6/15tg-1(2t+1)/15, and then (10) and (11) could be(12)η8=At6ht1-t4t2+t+4ζ4=Bt2+t+4ht1-t2(13)ζr=1-1λr3λθ,ζθ=1-1λθ3λr,ζZ=1-1λrλθ


## 2.1. Differential Equation Description of Sponge Cylinder

According to the study of Ju et al. [16], 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, RI and RO, and then turn into rI and rO after deformation under the action of inner pressure of σrI and out pressure of σrO. The cylindrical coordinate system is adopted with the material coordinate of (R,Θ,Z) and the spatial coordinate of (r,θ,z). Because of the axisymmetric plane strain problem, the deformation mode can be set as(1)r=fR,θ=Θ,z=λZwhere λ is the elongation of the cylinder in z direction.According to (1), the main elongations of the deformation gradient are(2)λr=drdR,λθ=rR,λz=1Sponge 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 W is a function of 3 main elongations, namely, W=W(λr,λθ,λz). The Cauchy stress in the 3 main elongation directions is [17](3)σr=1λθλz∂W∂λr,σθ=1λrλz∂W∂λθ,σz=1λrλθ∂W∂λzThe axisymmetric plane strain problem should satisfy the static equilibrium equation with ignorance of gravity [18].(4)dσrdr+1rσr-σθ=0Equation (3) is substituted into (4):(5)R∂λr∂R∂2W∂λr2+∂2W∂λr∂λθλr-λθ+∂W∂λr-∂W∂λθ=0At 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](6)Wλr,λθ,λz=μ2λr-2+λθ-2+λz-2+2λrλθλz-5where μ is the shear modulus of sponge in a natural state.Equation (6) is substituted into (5):(7)3Rλr′+λr4λθ3-λr=0where λr′=dλr/dR.


## 2.2. Differential Equation Analysis

Sett=λr/λθ=Rr′/r, and then take derivative of R to obtain(8)Rλr′=λθRt′-λr+λr2λθThe expression of intermediate variablet and (8) are substituted into (7):(9)dRR=3dt-t4-3t2+4tdrr=-3dtt3+3t-4The integral solution is applied on the ordinary differential equations (9), and the deformation function of the sponge cylinder can be obtained.(10)R8=A0t6e6/45tan-12t+1/151-t-4t2+t+4r4=A0t6e6/45tan-12t+1/15t2+t+41-t2whereA0 and B0 are integral constant and are related to boundary conditions.The Cauchy stress in three directions could be obtained by combining (3) and (6).(11)σr=μ1-1λr3λθ,σθ=μ1-1λθ3λr,σz=μ1-1λrλθ


## 2.3. Dimensionless of Deformation and Stress

η = R / R 0 represents the dimensionless radius before deformation, ξ=r/RO represents dimensionless radius after deformation and ξi=σi/μ (i=r,θ,z) is the dimensionless stress, A=AO/RO8, B=BO/RO4, h(t)=exp⁡6/15tg-1(2t+1)/15, and then (10) and (11) could be(12)η8=At6ht1-t4t2+t+4ζ4=Bt2+t+4ht1-t2(13)ζr=1-1λr3λθ,ζθ=1-1λθ3λr,ζZ=1-1λrλθ


## 3. Mathematical Description of Rubber Cylinder under Compression

### 3.1. 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].(14)s=2Jdev∂W∂I-1+I-1∂W∂I-2B--∂W∂I-2B-·B-wheres is the Cauchy stress deviation, J is volume ratio before and after deformation, dev(·) is tensor deviation operator, W is the strain energy density converted by unit volume before deformation, F is the deformation gradient tensor, then B=F·FT, B¯=J-2/3·B [22]; I1=tr(B), I2=I12-tr(B·BT); I1¯=J-2/3I1, I2¯=J-4/3I2; J=det(F).For incompressible sponge material,J=1. In terms of axisymmetric plane strain problem, (15) can be written as(15)s=2Jdev∂W∂I-1+I-1∂W∂I-2B--∂W∂I-2B-·B-i=r,θ,zIt could be obtained from (15) as(16)σr-σθ=σr-p-σθ-p=sp-sθ=2∂W∂I-1+I-1∂W∂I-2λrλθ-λθλr-∂W∂I-2λr2λθ2-λθ2λr2where p is the hydrostatic pressure.Equation (4) is the partial derivative of present coordinate of r; combining with (2), it could be written as the equation of initial coordinate R.(17)dσrdR+λrRλθσr-σθ=0The differential equation of axisymmetric plane strain could be obtained by submitting (16) into (17).(18)RdσrdR+2λrλθ∂W∂I-1+I-1∂W∂I-2λrλθ-λθλr-∂W∂I-2λr2λθ2-λθ2λr2=0


### 3.2. Differential Equation Analysis

In this paper, the three-reduced 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]:(19)W=C10I1-3+C20I1-32+C30I1-33In terms of axisymmetric plane strain,λz=1 and (19) is submitted into (18) to obtain(20)RdσrdR+2C10+C20I1-3+C30I1-32·λrλθ2-1=0As the rubber is incompressible and combining with (2),(21)drdR=RrThe deformation function of rubber cylinder could be obtained by integral.(22)r2=R2+C0where C0 is integral constant and is determined by the boundary condition.Equations (21) and (22) are substituted into (20):(23)dσrdR=-6C30R7R2+C04-4C20-6C30R5R2+C03-2C10-4C20+15C30R3R2+C02+6C30R2+C02R5+4C20-6C30R2+C0R3+2C10-4C20+15C30RBy integrating (23), the radial stress of rubber cylinder is as follows:(24)σr=-C30C03R2+C03+2C20-3C30C022R2+C02-C10C0R2+C0-3C30C022R4-2C20-3C30C0R2-C10-2C20+6C30ln⁡1+C0R2+D0where D0 is integral constant and is determined by the boundary condition.Combining (16) with (24), the tangential stress of the rubber cylinder is as follows:(25)σθ=σr+6C30R2+C05R6+R6R2+C03+4C20-6C30R2+C02R4-R4R2+C02+2C10-4C20+15C30R2+C0R2-R2R2+C0


### 3.3. Dimensionless of Deformation and Stress

Settingη=R/RO, which represents the dimensionless radius before deformation, ξ=r/R represents the dimensionless radius after deformation, ξi=σi/μ represents the dimensionless stress, C1=C10/μ, C2=C20/μ, C3=C30/μ, and D=D0/μ, where μ is the parameter of the Baltz-Ko constitutive model of the sponge. Then the corresponding deformation mode and stress can be written as(26)ξ2=η2+C(27)ξr=-C3C3η2+C3+2C2-3C3C22η2+C2-C1Cη2+C-3C3C22η4-2C2-3C3Cη2-C1-2C2+6C3ln⁡1+Cη2+D(28)ξθ=ξr+6C3η2+C3η6-η6η2+C3+4C2-6C3η2+C2η4-η4η2+C2-2C1-4C2+15C3η2+Cη2-η2η2+C


## 3.1. 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].(14)s=2Jdev∂W∂I-1+I-1∂W∂I-2B--∂W∂I-2B-·B-wheres is the Cauchy stress deviation, J is volume ratio before and after deformation, dev(·) is tensor deviation operator, W is the strain energy density converted by unit volume before deformation, F is the deformation gradient tensor, then B=F·FT, B¯=J-2/3·B [22]; I1=tr(B), I2=I12-tr(B·BT); I1¯=J-2/3I1, I2¯=J-4/3I2; J=det(F).For incompressible sponge material,J=1. In terms of axisymmetric plane strain problem, (15) can be written as(15)s=2Jdev∂W∂I-1+I-1∂W∂I-2B--∂W∂I-2B-·B-i=r,θ,zIt could be obtained from (15) as(16)σr-σθ=σr-p-σθ-p=sp-sθ=2∂W∂I-1+I-1∂W∂I-2λrλθ-λθλr-∂W∂I-2λr2λθ2-λθ2λr2where p is the hydrostatic pressure.Equation (4) is the partial derivative of present coordinate of r; combining with (2), it could be written as the equation of initial coordinate R.(17)dσrdR+λrRλθσr-σθ=0The differential equation of axisymmetric plane strain could be obtained by submitting (16) into (17).(18)RdσrdR+2λrλθ∂W∂I-1+I-1∂W∂I-2λrλθ-λθλr-∂W∂I-2λr2λθ2-λθ2λr2=0


## 3.2. Differential Equation Analysis

In this paper, the three-reduced 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]:(19)W=C10I1-3+C20I1-32+C30I1-33In terms of axisymmetric plane strain,λz=1 and (19) is submitted into (18) to obtain(20)RdσrdR+2C10+C20I1-3+C30I1-32·λrλθ2-1=0As the rubber is incompressible and combining with (2),(21)drdR=RrThe deformation function of rubber cylinder could be obtained by integral.(22)r2=R2+C0where C0 is integral constant and is determined by the boundary condition.Equations (21) and (22) are substituted into (20):(23)dσrdR=-6C30R7R2+C04-4C20-6C30R5R2+C03-2C10-4C20+15C30R3R2+C02+6C30R2+C02R5+4C20-6C30R2+C0R3+2C10-4C20+15C30RBy integrating (23), the radial stress of rubber cylinder is as follows:(24)σr=-C30C03R2+C03+2C20-3C30C022R2+C02-C10C0R2+C0-3C30C022R4-2C20-3C30C0R2-C10-2C20+6C30ln⁡1+C0R2+D0where D0 is integral constant and is determined by the boundary condition.Combining (16) with (24), the tangential stress of the rubber cylinder is as follows:(25)σθ=σr+6C30R2+C05R6+R6R2+C03+4C20-6C30R2+C02R4-R4R2+C02+2C10-4C20+15C30R2+C0R2-R2R2+C0


## 3.3. Dimensionless of Deformation and Stress

Settingη=R/RO, which represents the dimensionless radius before deformation, ξ=r/R represents the dimensionless radius after deformation, ξi=σi/μ represents the dimensionless stress, C1=C10/μ, C2=C20/μ, C3=C30/μ, and D=D0/μ, where μ is the parameter of the Baltz-Ko constitutive model of the sponge. Then the corresponding deformation mode and stress can be written as(26)ξ2=η2+C(27)ξr=-C3C3η2+C3+2C2-3C3C22η2+C2-C1Cη2+C-3C3C22η4-2C2-3C3Cη2-C1-2C2+6C3ln⁡1+Cη2+D(28)ξθ=ξr+6C3η2+C3η6-η6η2+C3+4C2-6C3η2+C2η4-η4η2+C2-2C1-4C2+15C3η2+Cη2-η2η2+C


## 4. 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 Figure2. 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.Figure 2
Schematic diagram of interference fit between adapters and metal launchers.The outer radius of launcher before deformation isη3O=ηO=1 and the inner radius is η3I. Similarly, the adapter’s outer radius is η2O before deformation, inner diameter is η1I=ηI, and the adhesive surface radius is ηM. The interference fit is adopted between rubber’s outer diameter and launcher’s inner diameter, and the precompression is δ. After the installation of adapter into the launcher, the radius after deformation is, respectively, as follows, for the launchers, the outer radius is ξ3O=ξO and the inner is ξ3I. For the adapter, the outer radius is ξ2O, the adhesive radius is ξM and the inner radius is ξ1I=ξI. Nevertheless, ξr3I and ξO are all unknown and the outer surface pressure of launcher is 0.


### 4.2. 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](29)∇4F=d2dR2+1RddR2F=0(30)σrR=RO=-σrO,τrϑR=RO=0,σrR=RI=-σrI,τrϑR=RI=0whereF is the stress function and is the function of r and θ under polar coordinates. For the axisymmetric issue, F is the function of r and has nothing with θ.If the stress functionF is known, then its stress could be expressed as [25–28](31)σr=1R∂F∂R+1R2∂2F∂R2σθ=∂2F∂R2τrθ=τθr=1R2∂F∂θ-1R∂2F∂R∂θ=-∂∂R1R∂F∂θConsidering the operator ofd2/dR2+1/Rd/dR=1/Rd/dRRd/dR, then (29) could be written as(32)∇4F=1RddRRddR1RddRRdFdR=0The integrating on (32) to obtain(33)FR=J0R2ln⁡R+K0ln⁡R+M0R2+N0Then the corresponding stress component is as follows:(34)σr=J01+2ln⁡R+K0R2+2M0,σθ=J03+2ln⁡R-K0R2+2M0,τrθ=τθr=0whereJ0, K0, M0, and N0 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:(35a)εr=∂ur∂R,εθ=urR+1R∂uθ∂θ,γrθ=γθr=1R∂ur∂θ+∂uθ∂R-uθRAnd the constitutive function of plane stress is as follows:(35b)εr=1-ν2Eσr-ν1-νσθ,εθ=1-ν2Eσθ-ν1-νσr,γrθ=1GτrθThe 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.(36)L0=RI2RO2σr1-σrORO2-RI22M0=RO2σrO-RI2σr1RO2-RI2σr=RI2RO2σr1-σrORO2-RI2·1R2RO2σrO-RI2σr1RO2-RI2σθ=-RI2RO2σr1-σrORO2-RI2·1R2RO2σrO-RI2σr1RO2-RI2σz=νσr+σθ(37)ur=1+νERI2RO2σr1-σrORO2-RI2·1R+1-2νRO2σrO-RI2σr1RO2-RI2·R,uθ=0Assuming thatξi=σi/μ, ψ=ξ-η, and E′=E/μ, the dimensionless is carried out on (36) and (37) to obtain(38)ζr=ηI2ηO2ζr1-ζrOηO2-ηI2·1η2+ηO2ζrO-ηI2ηr1ηO2-ηI2,ζθ=-ηI2ηO2ζr1-ζrOηO2-ηI2·1η2+ηO2ζrO-ηI2ηr1ηO2-ηI2,ζz=νζr+ζθ(39)ψr=1+νE′ηI2ηO2ζrO-ζrIηO2-ηI2·1η+1+2νηO2ζrO-ηI2ηr1ηO2-ηI2·η,uθ=0


### 4.3. Equation Solving of Interference Fit Problem

Before the adapter installing into the launcher, the sizes of adapter are, respectively,η2O, ηM, η1I=ηI. The size of launcher before deformation is η3O=ηO=1. As η3I is known, ξ3I=ξ2O 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.(40)ψr3I=-1+νE′η3O2+1-2νη3I2η3O2-η3I2η3Iζ3IThe inner radius of launcher after deformation is(41)ξ3I=η3I+ψr3I=η3I-1+νE′η3O2+1-2νη3I2η3O2-η3I2η3Iζ3IEquation (41) could be written as(42)ξ3I=a+bζr3Iwherea=η3I and b=-(1+v)/E′η3O2+(1-2v)η3I2/η3O2-η3I2η3I.According (22), the radius expression of adapter and adhesive surface after deformation is(43)ξ3I2=ξ2O2=η2O2+C(44)ξM2=ηM2+CAfter 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.(45)ξr3I=-C3C3η2O2+C3+2C2-3C3C22η2O2+C2-C1Cη2O2+C-3C3C22η2O4-2C2-3C3Cη2O2-C1-2C2+6C3ln⁡1+Cη2O2+DEquation (42) is submitted into (45), then(46)η2O2+C-ab=-C3C3η2O2+C3+2C2-3C3C22η2O2+C2-C1Cη2O2+C-3C3C22η2O4-2C2-3C3Cη2O2-C1-2C2+6C3ln⁡1+Cη2O2+DThe following two equations can be obtained by bringing the initial internal and external radius of the material 1 in the adapter into (12).(47)ηM8=AtM6htM1-tM4tM2+tM+4(48)ηI8=AtI6htI1-tI4tI2+tI+4Considering the radius of the adapter’s adhesive surface after deformation, (44) is brought into (13):(49)ηM2+C2=BtM2+tM+4htM1-tM2Due 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:(50)ξI4=BtI2+tI+4htI1-tI2As the spongy and rubber materials are in a continuous radial stress in the adhesive surface, a similar equation (46) can be written out.(51)1-1λrM3λθM=-C3C3ηM2+C3+2C2-3C3C22ηM2+C2-C1CηM2+C-3C3C22ηM4-2C2-3C3CηM2-C1-2C2+6C3ln⁡1+CηM2+DAccording tot=λr/λθ=Rr′/r, λrM=tMλθM could be obtained at the cohesive surface, where λθM=ξM/ηM. λrM=tM1+C/ηM2 could be obtained based on (44); thus (51) could be written as (51).(52)1-1tM31+C/ηM2=-C3C3ηM2+C3+2C2-3C3C22ηM2+C2-C1CηM2+C-3C3C22ηM4-2C2-3C3CηM2-C1-2C2+6C3ln⁡1+CηM2+DEquations (46), (47), (48), (49), (50), and (51) are transformed properly, and they are rewritten into equations forms.(53)FX=f1Xf2Xf3Xf4Xf5Xf6XT=0where X=ABCDtMtIT.(54)f1X=η2O2+C-ab+C3C3η2O2+C3-2C2-3C3C22η2O2+C2+C1Cη2O2+C+3C3C22η2O4+2C2-3C3Cη2O2+C1-2C2+6C3ln⁡1+Cη2O2-Df2X=1-ηM4tM3ηM2+C+C3C3ηM2+C3-2C2-3C3C22ηM2+C2+C1CηM2+C+3C3C22ηM4+2C2-3C3CηM2+C1-2C2+6C3ln⁡1+CηM2-Df3X=AtM6htM-ηM81-tM4tM2+tM+4f4X=AtI6htI-ηI81-tI4tI2+tI+4f5X=BtM2+tM+4htM-ηM2+C1-tM2f6X=BtI2+tI+4htI-ξI41-tI2The nonlinear equation group (53) is calculated by using Newton-Raphson method [29, 30], and the A, B, C, D, tM, and tI could be obtained, and then the deformation and stress of the interference fit for the adapter and the launcher could be obtained.


### 4.4. Example Verification and Analysis

The precompressionδ of the adapter is 0.005161 in a certain missile launching system, the dimensionless sizes are η3I=0.967742, η2O=0.971613, ηM=0.918710, and ηI=ξI=0.903226. The material parameters of adapter and launcher are C1=185.0, C2=6.5660, C3=0.03488, E′=2.954595×106, 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 Figures3–5, 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.Figure 3
Radial displacement curve.Figure 4
Radial stress curve.Figure 5
Tangential stress curve.The maximum absolute value of adapter’s displacement is 0.0040772 in Figure3, 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 Figure4 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 Figure5, 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.


## 4.1. Geometric Description

Schematic diagram of interference fit between adapters and metal launchers is shown in Figure2. 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.Figure 2
Schematic diagram of interference fit between adapters and metal launchers.The outer radius of launcher before deformation isη3O=ηO=1 and the inner radius is η3I. Similarly, the adapter’s outer radius is η2O before deformation, inner diameter is η1I=ηI, and the adhesive surface radius is ηM. The interference fit is adopted between rubber’s outer diameter and launcher’s inner diameter, and the precompression is δ. After the installation of adapter into the launcher, the radius after deformation is, respectively, as follows, for the launchers, the outer radius is ξ3O=ξO and the inner is ξ3I. For the adapter, the outer radius is ξ2O, the adhesive radius is ξM and the inner radius is ξ1I=ξI. Nevertheless, ξr3I and ξO are all unknown and the outer surface pressure of launcher is 0.


## 4.2. 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](29)∇4F=d2dR2+1RddR2F=0(30)σrR=RO=-σrO,τrϑR=RO=0,σrR=RI=-σrI,τrϑR=RI=0whereF is the stress function and is the function of r and θ under polar coordinates. For the axisymmetric issue, F is the function of r and has nothing with θ.If the stress functionF is known, then its stress could be expressed as [25–28](31)σr=1R∂F∂R+1R2∂2F∂R2σθ=∂2F∂R2τrθ=τθr=1R2∂F∂θ-1R∂2F∂R∂θ=-∂∂R1R∂F∂θConsidering the operator ofd2/dR2+1/Rd/dR=1/Rd/dRRd/dR, then (29) could be written as(32)∇4F=1RddRRddR1RddRRdFdR=0The integrating on (32) to obtain(33)FR=J0R2ln⁡R+K0ln⁡R+M0R2+N0Then the corresponding stress component is as follows:(34)σr=J01+2ln⁡R+K0R2+2M0,σθ=J03+2ln⁡R-K0R2+2M0,τrθ=τθr=0whereJ0, K0, M0, and N0 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:(35a)εr=∂ur∂R,εθ=urR+1R∂uθ∂θ,γrθ=γθr=1R∂ur∂θ+∂uθ∂R-uθRAnd the constitutive function of plane stress is as follows:(35b)εr=1-ν2Eσr-ν1-νσθ,εθ=1-ν2Eσθ-ν1-νσr,γrθ=1GτrθThe 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.(36)L0=RI2RO2σr1-σrORO2-RI22M0=RO2σrO-RI2σr1RO2-RI2σr=RI2RO2σr1-σrORO2-RI2·1R2RO2σrO-RI2σr1RO2-RI2σθ=-RI2RO2σr1-σrORO2-RI2·1R2RO2σrO-RI2σr1RO2-RI2σz=νσr+σθ(37)ur=1+νERI2RO2σr1-σrORO2-RI2·1R+1-2νRO2σrO-RI2σr1RO2-RI2·R,uθ=0Assuming thatξi=σi/μ, ψ=ξ-η, and E′=E/μ, the dimensionless is carried out on (36) and (37) to obtain(38)ζr=ηI2ηO2ζr1-ζrOηO2-ηI2·1η2+ηO2ζrO-ηI2ηr1ηO2-ηI2,ζθ=-ηI2ηO2ζr1-ζrOηO2-ηI2·1η2+ηO2ζrO-ηI2ηr1ηO2-ηI2,ζz=νζr+ζθ(39)ψr=1+νE′ηI2ηO2ζrO-ζrIηO2-ηI2·1η+1+2νηO2ζrO-ηI2ηr1ηO2-ηI2·η,uθ=0


## 4.3. Equation Solving of Interference Fit Problem

Before the adapter installing into the launcher, the sizes of adapter are, respectively,η2O, ηM, η1I=ηI. The size of launcher before deformation is η3O=ηO=1. As η3I is known, ξ3I=ξ2O 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.(40)ψr3I=-1+νE′η3O2+1-2νη3I2η3O2-η3I2η3Iζ3IThe inner radius of launcher after deformation is(41)ξ3I=η3I+ψr3I=η3I-1+νE′η3O2+1-2νη3I2η3O2-η3I2η3Iζ3IEquation (41) could be written as(42)ξ3I=a+bζr3Iwherea=η3I and b=-(1+v)/E′η3O2+(1-2v)η3I2/η3O2-η3I2η3I.According (22), the radius expression of adapter and adhesive surface after deformation is(43)ξ3I2=ξ2O2=η2O2+C(44)ξM2=ηM2+CAfter 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.(45)ξr3I=-C3C3η2O2+C3+2C2-3C3C22η2O2+C2-C1Cη2O2+C-3C3C22η2O4-2C2-3C3Cη2O2-C1-2C2+6C3ln⁡1+Cη2O2+DEquation (42) is submitted into (45), then(46)η2O2+C-ab=-C3C3η2O2+C3+2C2-3C3C22η2O2+C2-C1Cη2O2+C-3C3C22η2O4-2C2-3C3Cη2O2-C1-2C2+6C3ln⁡1+Cη2O2+DThe following two equations can be obtained by bringing the initial internal and external radius of the material 1 in the adapter into (12).(47)ηM8=AtM6htM1-tM4tM2+tM+4(48)ηI8=AtI6htI1-tI4tI2+tI+4Considering the radius of the adapter’s adhesive surface after deformation, (44) is brought into (13):(49)ηM2+C2=BtM2+tM+4htM1-tM2Due 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:(50)ξI4=BtI2+tI+4htI1-tI2As the spongy and rubber materials are in a continuous radial stress in the adhesive surface, a similar equation (46) can be written out.(51)1-1λrM3λθM=-C3C3ηM2+C3+2C2-3C3C22ηM2+C2-C1CηM2+C-3C3C22ηM4-2C2-3C3CηM2-C1-2C2+6C3ln⁡1+CηM2+DAccording tot=λr/λθ=Rr′/r, λrM=tMλθM could be obtained at the cohesive surface, where λθM=ξM/ηM. λrM=tM1+C/ηM2 could be obtained based on (44); thus (51) could be written as (51).(52)1-1tM31+C/ηM2=-C3C3ηM2+C3+2C2-3C3C22ηM2+C2-C1CηM2+C-3C3C22ηM4-2C2-3C3CηM2-C1-2C2+6C3ln⁡1+CηM2+DEquations (46), (47), (48), (49), (50), and (51) are transformed properly, and they are rewritten into equations forms.(53)FX=f1Xf2Xf3Xf4Xf5Xf6XT=0where X=ABCDtMtIT.(54)f1X=η2O2+C-ab+C3C3η2O2+C3-2C2-3C3C22η2O2+C2+C1Cη2O2+C+3C3C22η2O4+2C2-3C3Cη2O2+C1-2C2+6C3ln⁡1+Cη2O2-Df2X=1-ηM4tM3ηM2+C+C3C3ηM2+C3-2C2-3C3C22ηM2+C2+C1CηM2+C+3C3C22ηM4+2C2-3C3CηM2+C1-2C2+6C3ln⁡1+CηM2-Df3X=AtM6htM-ηM81-tM4tM2+tM+4f4X=AtI6htI-ηI81-tI4tI2+tI+4f5X=BtM2+tM+4htM-ηM2+C1-tM2f6X=BtI2+tI+4htI-ξI41-tI2The nonlinear equation group (53) is calculated by using Newton-Raphson method [29, 30], and the A, B, C, D, tM, and tI could be obtained, and then the deformation and stress of the interference fit for the adapter and the launcher could be obtained.


## 4.4. Example Verification and Analysis

The precompressionδ of the adapter is 0.005161 in a certain missile launching system, the dimensionless sizes are η3I=0.967742, η2O=0.971613, ηM=0.918710, and ηI=ξI=0.903226. The material parameters of adapter and launcher are C1=185.0, C2=6.5660, C3=0.03488, E′=2.954595×106, 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 Figures3–5, 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.Figure 3
Radial displacement curve.Figure 4
Radial stress curve.Figure 5
Tangential stress curve.The maximum absolute value of adapter’s displacement is 0.0040772 in Figure3, 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 Figure4 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 Figure5, 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.


## 5. Analysis on the Deformation and Stress Effects of Adapters

### 5.1. Effect of Precompression on the Deformation and Stress

Figures6–13 are effect curves of precompression δ, respectively, to radial stress of inner sponge (ζrI1), to the tangential stress of inner sponge (ζθI1) and outer surface (ζθM1), the radial displacement of outer sponge (ψrM), the radial stress of outer sponge (ζrM), the tangential stress of inner rubber (ζθM2) and outer surface (ζθO2), the tangential stress of the outer rubber surface (ζrO2), the radial displacement of inner launcher (ψrI3) and the outer radial displacement (ψrO3), the tangential stress of the inner launcher (ζθI3), and tangential stress of outer surface (ζθO3).Figure 6
Effects to theζrI1 of adapter.Figure 7
Effects to theζθI1 and ζθM1 of adapter.Figure 8
Effects to theψrM of adapter.Figure 9
Effects to theζrM of adapter.Figure 10
Effects to theζθM2 and ζθO2 of adapter.Figure 11
Effects ofδ to the ζrO2 of adapter.Figure 12
Effects to theψrI3 and ψrO3 of adapter.Figure 13
Effects ofδ to ζθI3 and ζθO3 of adapter.From Figures6–13, the effects of δ to ζrI1, ζrM, ζθM2, ζθO2, ζrO2, ψrI3, ψrO3, ζθI3, and ζθO3 are divided onto three stages:(1) When0≤δ≤0.0075 at the first stage, ζrI1, ζrM, ζθM2, ζθO2, ζrO2, ψrI3, ψrO3, ζθI3, and ζθO3 is approximately proportion to the precompression δ. The absolute value of proportional coefficient is small and the precompression δ varies largely, while the corresponding stress and displacement change little.(2) When0.0075≤δ≤0.0105 at the second stage, with the change of precompression δ, the change rates of ζrI1, ζrM, ζθM2, ζθO2, ζrO2, ψrI3, ψrO3, ζθI3, and ζθO3 are getting faster and faster.(3) Whenδ≥0.0105 at the third stage, the precompression δ changes little while the corresponding stress and displacement change largely.


### 5.2. Effect of Thickness Ratio of the Sponge Layer on the Deformation and Stress

The thickness ratio of sponge layer is defined asΔE=TE(TR+TE), where TE is radial thickness of sponge layer and TR is the radial thickness of the rubber. ΔO=TO/TS is defined as the ratio of precompression of adapter to sponge layer’s thickness, TO is the precompression of adapter, and TS=TR+TE. When ΔO=0.2, 0.3, and 0.4, Figures 14–25 are, respectively, the effect curves of thickness ratio ΔE to the radial stress of the inner sponge surface (ζrI1), the tangential stress of inner sponge (ζθI1), the radial displacement of outer sponge surface (ψrM), the radial stress of outer sponge surface (ζrM), the tangential stress of outer sponge surface (ζθM1), the tangential stress of inner rubber surface (ζθM2), the radial stress of outer rubber surface (ζrO2), the tangential stress of the outer rubber surface (ζθO2), the radial displacement of the inner launcher surface (ψrI3), the tangential stress of the inner launcher surface (ζθI3), the radial displacement of outer launcher surface (ψrO3), and tangential stress of outer surface (ζθO3).Figure 14
Effect curve ofΔE to ζrI1.Figure 15
Effect curve ofΔE to ζθI1.Figure 16
Effect curve ofΔE to ψrM.Figure 17
Effect curve ofΔE to ζrM.Figure 18
Effect curve ofΔE to ζθM1.Figure 19
Effect curve ofΔE to ζθM2.Figure 20
Effect curve ofΔE to ζrO2.Figure 21
Effect curve ofΔE to ζθO2.Figure 22
Effect curve ofΔE to ψrI3.Figure 23
Effect curve ofΔE to ζθI3.Figure 24
Effect curve ofΔE to ψrO3.Figure 25
Effect curve ofΔE to ζθO3.As shown in Figures14, 15, 18, 20, and 20–25, the absolute values of stresses ζrI1, ζθI1, ζrM, ζrO2, ψrI3, ζθI3, ψrO3, and ζθO3 change with ΔE, which first increase and then decrease with the maximum extremum point. When ΔO=0.2, 0.3, 0.4, the horizontal ordinates of ζrI1 and ζθI1 with the maximum value are, respectively, 0.14499, 0.16984, and 0.20795. The horizontal ordinates of the corresponding maximum point of ζrM, ζrO2, ψrI3, ζθI3, ψrO3, and ζθO3 are, respectively, 0.10936, 0.13422, and 0.17482. Furthermore, the horizontal ordinate of the maximum point is smaller than that of ζrI1 and ζθI1 and increases with the ΔO. In view of Figures 16, 19, and 21, ψrM, ζθM2, and ζθO2 are approximately in proportion to ΔE, when ΔE>0.1, ζθM1 is approximately linear with ΔE.


## 5.1. Effect of Precompression on the Deformation and Stress

Figures6–13 are effect curves of precompression δ, respectively, to radial stress of inner sponge (ζrI1), to the tangential stress of inner sponge (ζθI1) and outer surface (ζθM1), the radial displacement of outer sponge (ψrM), the radial stress of outer sponge (ζrM), the tangential stress of inner rubber (ζθM2) and outer surface (ζθO2), the tangential stress of the outer rubber surface (ζrO2), the radial displacement of inner launcher (ψrI3) and the outer radial displacement (ψrO3), the tangential stress of the inner launcher (ζθI3), and tangential stress of outer surface (ζθO3).Figure 6
Effects to theζrI1 of adapter.Figure 7
Effects to theζθI1 and ζθM1 of adapter.Figure 8
Effects to theψrM of adapter.Figure 9
Effects to theζrM of adapter.Figure 10
Effects to theζθM2 and ζθO2 of adapter.Figure 11
Effects ofδ to the ζrO2 of adapter.Figure 12
Effects to theψrI3 and ψrO3 of adapter.Figure 13
Effects ofδ to ζθI3 and ζθO3 of adapter.From Figures6–13, the effects of δ to ζrI1, ζrM, ζθM2, ζθO2, ζrO2, ψrI3, ψrO3, ζθI3, and ζθO3 are divided onto three stages:(1) When0≤δ≤0.0075 at the first stage, ζrI1, ζrM, ζθM2, ζθO2, ζrO2, ψrI3, ψrO3, ζθI3, and ζθO3 is approximately proportion to the precompression δ. The absolute value of proportional coefficient is small and the precompression δ varies largely, while the corresponding stress and displacement change little.(2) When0.0075≤δ≤0.0105 at the second stage, with the change of precompression δ, the change rates of ζrI1, ζrM, ζθM2, ζθO2, ζrO2, ψrI3, ψrO3, ζθI3, and ζθO3 are getting faster and faster.(3) Whenδ≥0.0105 at the third stage, the precompression δ changes little while the corresponding stress and displacement change largely.


## 5.2. Effect of Thickness Ratio of the Sponge Layer on the Deformation and Stress

The thickness ratio of sponge layer is defined asΔE=TE(TR+TE), where TE is radial thickness of sponge layer and TR is the radial thickness of the rubber. ΔO=TO/TS is defined as the ratio of precompression of adapter to sponge layer’s thickness, TO is the precompression of adapter, and TS=TR+TE. When ΔO=0.2, 0.3, and 0.4, Figures 14–25 are, respectively, the effect curves of thickness ratio ΔE to the radial stress of the inner sponge surface (ζrI1), the tangential stress of inner sponge (ζθI1), the radial displacement of outer sponge surface (ψrM), the radial stress of outer sponge surface (ζrM), the tangential stress of outer sponge surface (ζθM1), the tangential stress of inner rubber surface (ζθM2), the radial stress of outer rubber surface (ζrO2), the tangential stress of the outer rubber surface (ζθO2), the radial displacement of the inner launcher surface (ψrI3), the tangential stress of the inner launcher surface (ζθI3), the radial displacement of outer launcher surface (ψrO3), and tangential stress of outer surface (ζθO3).Figure 14
Effect curve ofΔE to ζrI1.Figure 15
Effect curve ofΔE to ζθI1.Figure 16
Effect curve ofΔE to ψrM.Figure 17
Effect curve ofΔE to ζrM.Figure 18
Effect curve ofΔE to ζθM1.Figure 19
Effect curve ofΔE to ζθM2.Figure 20
Effect curve ofΔE to ζrO2.Figure 21
Effect curve ofΔE to ζθO2.Figure 22
Effect curve ofΔE to ψrI3.Figure 23
Effect curve ofΔE to ζθI3.Figure 24
Effect curve ofΔE to ψrO3.Figure 25
Effect curve ofΔE to ζθO3.As shown in Figures14, 15, 18, 20, and 20–25, the absolute values of stresses ζrI1, ζθI1, ζrM, ζrO2, ψrI3, ζθI3, ψrO3, and ζθO3 change with ΔE, which first increase and then decrease with the maximum extremum point. When ΔO=0.2, 0.3, 0.4, the horizontal ordinates of ζrI1 and ζθI1 with the maximum value are, respectively, 0.14499, 0.16984, and 0.20795. The horizontal ordinates of the corresponding maximum point of ζrM, ζrO2, ψrI3, ζθI3, ψrO3, and ζθO3 are, respectively, 0.10936, 0.13422, and 0.17482. Furthermore, the horizontal ordinate of the maximum point is smaller than that of ζrI1 and ζθI1 and increases with the ΔO. In view of Figures 16, 19, and 21, ψrM, ζθM2, and ζθO2 are approximately in proportion to ΔE, when ΔE>0.1, ζθM1 is approximately linear with ΔE.


## 6. Conclusions

In this paper, the deformation and stress analytical formula of interference fit among the sponge cylinder, rubber cylinder, the adapter, and the launcher is deduced based on the assumption of axisymmetric plane. The deformation of adapter and the stress analytical solution are obtained and are verified by numerical simulation. The effects of precompression and the thickness ratio of sponge layer on the adapter’s deformation and stress are studied. The following conclusions can be obtained.(1) When the adapter carried out interference fit with launcher, the numerical simulation of radial displacement, radial stress, and tangential stress is consistent with that analytical solution, indicating that the derivation of the analytical formula for deformation and stress is correct.(2) After the installation of adapter into the launcher, its outer surface moves inward and the displacement with the maximum absolute value appears on the adhesive surface. The rubber layer moves inward as a whole and each point in the launcher moves outward; the maximum displacement of the launcher is on the interference fit surface. For the sponge layer, the absolute value of the radial stress decreases from inside to outside, while for the rubber layer, the absolute value of the radial stress increases from inside to outside, and that of the launcher reduces from inside to outside until 0. The three materials are all subjected to pressure. In terms of the tangential stress, that of the sponge layer increases along the radius while that of rubber layer decreases. The tangential stress of adapter is compressive stress while that of the launcher is tensile stress and decreases radially.(3) When the precompressionδ is smaller than 0.0075 or larger than 0.0105, ζrI1, ζrM, ζθM2, ζθO2, ζrO2, ψrI3, ψrO3, ζθI3, and ζθO3 are approximately in proportion to precompression δ. When δ is smaller than 0.0075, the absolute value of corresponding stress and displacement increases slightly, while δ is larger than 0.0105 that increases sharply.(4) The absolute values of stressesζrI1, ζθI1, ζrM, ζrO2, ψrI3, ζθI3, ψrO3, and ζθO3 vary with ΔE, which increase firstly and then decrease with a extremum. The horizontal ordinate increases with ΔO. ψrM, ζθM2, and ζθO2 are approximately in proportion with ΔE, and when ΔE>0.1, ζθM1 is approximately linear with ΔE.



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*Source: 9730540-2018-07-10.xml*