Dong-mei Yin, Bao-ming Li, Hong-cheng Xiao

National Key Laboratory of Transient Physics, Nanjing University of Science & Technology, Nanjing, 210094, China

Keywords:Railgun Barrel Composite materials Filament winding Winding tension Residual prestress

ABSTRACT Based on the elastic theory of cylindrical shells and the theory of composite laminates, a prediction model for the residual prestress of the simplified round composite barrel for railgun is established.Only the fibre pretension is considered in this model. A three dimensional numerical simulation for the residual prestress in the railgun barrel is carried out, by combining the temperature differential method with the element birth and death technology.The results obtained by the two methods are compared.It reveals that the distribution trends of residual prestress are consistent. And the difference for residual prestress in the filament wound composite housing of barrel is relatively small. The same finite element method is used to analysis the residual prestress in the non-simplified composite barrels for railgun,which are under different control modes of winding tension.The results mean that the residual prestress in barrel will increase while the taper coefficient for winding is decreasing. Therefore, the sealing performance in bore is improved, but the strength of the filament wound composite housing drops. In addition,the axial and circumferential residual prestress in the filament wound composite housing with constant torque winding are close to the ones in iso-stress design for barrel.

1. Introduction

In order to meet the needs of lightweight and motility of railgun in the actual combat, filament wound composite barrel is a good choice in the design of railgun barrel, which has high specific stiffness and strength and design flexibility.However,cure reaction and winding tension in the manufacturing process,especially with the high tension winding technology, may cause initial residual stress in the winding products [1-4]. It will affect the mechanical properties of the winding products. Therefore many researchers have developed a series of methods, such as theoretical analysis methods [5-7], numerical simulation methods [8-11] and experimental technology[12-15],to investigate on the winding tension and the residual prestress caused by it for different winding products. And the pretension winding technology adopted in the manufacturing process of the filament wound composite barrel for railgun can increase the fiber volume fraction and arrange the fiber in orderly for the barrel.Also it can produce prestress in barrel[16],which can improve the stiffness and seal performance for the barrel bore.But the research for it is less reported.Therefore,the residual prestress caused only by winding tension in the filament wound composite barrel for railgun will be investigated in this paper.

In addition, if the interior structures of winding layers can be considered as the liner,rails and insulators in bore form the liner of filament wound composite barrel for railgun. It is different from most winding products’ liners with one kind of homogeneous material,which are often winded in circumferential direction with one kind fiber. And the composite barrels of railguns are winded with a variety of fiber types based on consideration of insulation and mechanical properties. These factors may make the residual prestress in the barrel more complex.Referring to the design theory of filament winding pressure vessel, a theoretical model for the residual prestress in the round composite barrel for railgun will be established in this work, which is based on the elastic theory of cylindrical shells and the theory of composite laminates. Through joining the temperature differential method and the element birth and death technology,a finite element numerical simulation will be done to analyze the residual prestress in the railgun barrel,which is leaded only by winding tension. Furthermore, the residual prestress in the composite barrels for railgun which are under different control modes of winding tension will be analyzed by the 3D numerical simulation.

2. Theoretical model of the residual prestress in composite barrel for railgun

The filament wound composite barrel of railgun is assumed to behave as a hybrid filament wound cylinder with an isotropic liner.The structural coordinate system(cylindrical coordinate system:θz-r) for the barrel is defined by rotating the material coordinate system (1-2-3) for the filament wound composite housing with winding angle α around 3-axis. And the 3-axis is paralleling to raxis.Here r,θ and z are denoted as radial,circumferential and axial coordinates.And 1-axis is along the fiber direction.The cylinder is composed of n layers,and the innermost layer is its liner.According to the composite laminate theory, the constitutive equation in the structural coordinate system for the kth filament wound layer is given as follows [17-19]:

where Qij(i,j = 1,2,3) are elastic constants, and k = 1,2, …,n-1.

If the radial, circumferential and axial displacements can be considered as [18,19]:

Based on the axisymmetric assumptions,it can be deduced from Eqs. (1) and (2):

Then the equilibrium equations for the kth layer are simplified as:

And the strains in the kth can be expressed as:

Thus,from Eqs.(1)-(5),the radial displacement of kth layer can be written as:

For the winding layers:

For the liner:

The interfaces for the adjacent layers in barrel should satisfy the continuity conditions:

The normal stresses in the inner surface of barrel and the outer surface of the kth layer are:

In addition, the conditions of equilibrium of axial forces and torque are shown in Eqs. (10) and (11) respectively:

After that, the stress and strain in every layer which are inside the newly winding layer can be obtained by Eqs.(1)and(5)-(7).At last, after all winding have been completed, the residual prestress and prestrain in each layer of barrel can be got by linear superposition.

3. Numerical simulation model of the residual prestress in composite barrel for railgun

Fig.1. The structure of railgun barrel.

A round railgun barrel is adopted in this work [20], and its geometric structure is shown in Fig. 1(a). It includes two copper rails, ceramic insulators and filament wound composite housing(glass fiber layers and carbon fiber layers). The radius of bore is 25 mm, the length of barrel L is 4 m, and the thickness of carbon fiber layers is 25 mm. Both thicknesses of glass fiber layers 1 and glass fiber layers 2 are 5 mm.And the winding angles in fiber layers are±45°.The copper rails and Ceramic insulators are considered as isotropic materials. Material properties for copper rails are:density = 8900.00 kg/m3, Poisson’s ratio = 0.31, yield stress= 320.00 MPa,Young’s modulus = 117.00 GPa.And the material parameters for ceramic insulator are: density = 3896.00 kg/m3, Poisson’s ratio = 0.218, Young’s modulus = 293.15 GPa. The filament wound composite housing are composed by carbon fiber layers and glass fiber layers, and their orthotropic material parameters are provided in Table 1.

The bonds among rails, insulators in bore and each winding layer are assumed to be perfect. A 3D finite element model of the filament wound composite barrel for railgun is built in the software ANSYS,as exhibited in Fig.1(b).The rails and insulators in bore are modeled with isotropic solid elements. While the filament wound composite housing is meshed with laminated elements,12 layers elements along its radial direction. And each laminated element has four layers.The liner of barrel is comprised of rails and ceramic insulators.All degrees of freedom on the two end faces of liner are restrained.

3.1. Loading of the winding tension

The effects of thermal loads in winding process, such as curing process, are supposed to be not considered. After the equivalent thermal expansion coefficient is defined for each layer, winding pretension in fiber can be modeled by loading certain temperature.Here, we assume that the stress of 1 MPa generated in the fiber direction with the change of temperature of 1°C. And the loading temperature(ΔT)can be gained according to the initial prestress in the fiber bundles.

Table 1 Material parameters of filament wound layers.

Thus the equivalent thermal expansion coefficients in all directions for each winding layer can be calculated:

where σiand Eiare initial fiber prestresses and elastic moduli in all directions for each winding layer, respectively.

3.2. Simulation for the winding process

Owing to the great number of layers,the finite element model is set up in one time to simulate the winding process by using the element birth and death technology. The birth and death of element will be achieved by modifying the element stiffness.If the element stiffness is multiplied with a small coefficient, the parameters of loading,mass and damping of this element are all set to zero.It means that this element is killed.Once the death element is activated, the above parameters for this element return to the original ones. When the jth is winding, the layers (≤j) are all activated,and the layers(＞j)are all killed.This method can be used to simulate the jth layer winding undisturbed by the subsequent winding layers. And when the (j+1)th layer is activated to be calculated,it can start with the outer diameter of the wrapped layer j rather than the one modeled at the beginning for the jth layer.

4. Calculation and discussion

In order to verify the results’ reliability, the liner in the above barrel model is assumed to be simplified as a copper liner,and the constant tension winding method is adopted in this model. The results gained from the theoretical model and numerical simulation model are compared with each other. The distributions of all direction components of residual prestress through the wall thickness for this simplified barrel model are shown in Fig. 2. It reveals that the distribution trends of residual prestress obtained by two methods are consistent,and the stresses in all directions of liner are mainly compressive stresses. Stresses in the interfaces among different materials appear various degrees of fluctuation,especially in the interfaces of liner and filament wound layers, inner glass fiber layers and carbon fiber layers.The differences of the stresses’ values in the liners for two kinds of models are relatively bigger,especially for the axial stress component.It is influenced by several factors,such as the constraints on the end faces of liner for the numerical simulation model, and theory of thin walled cylindrical shell for the theoretical model. But the distributions of residual prestress in the filament winding layers is close, so the numerical model is reliable to some extent, especially for the analysis of the residual prestress of the filament wound composite housing.

Fig.2. Distributions of Residual prestress through the wall thickness for the simplified barrel model. (Method 1-Finite element method, Method 2- theoretical model).

Then this numerical simulation method is applied to the simulation of residual prestress in the non-simplified filament wound composite barrel for railgun in Fig. 1(a). Three common winding tension control models,constant tension,constant torque and taper tension,are employed in the railgun’s barrels.In addition,with the taper tension winding, the initial fiber prestress along the fiber direction for the kth layer,which is also equal to the initial prestress in the fiber bundle listed in Eqs. (12) and (13), can be found as follows [6,21]:

According to Eq.(16),if β=0,the winding tension control mode is simplified to the constant tension mode. While β = 1, the one turns into the constant torque mode. Furthermore, other taper tension winding modes(β=0.25,0.75)are also used in this work.The distributions of initial fiber prestress in fiber direction through the wall thickness for the filament wound layers with different β are given in Fig. 3.

The distributions of residual stress caused by different winding tension control modes in the filament wound composite barrel for railgun are gained by the numerical simulation. Since the barrel liner consists of two different materials, two paths are defined on the section of barrel, which is located on the half length of barrel.Path oS1 is along the oy-axis and through the middle of the rail,while path oS2 is along the ox-axis and through the middle of the ceramic insulator, as shown in Fig.1(a). The variations of residual prestress components in all directions through the wall thickness of barrel for these two paths are displayed in Fig.4 and Fig.5.It can be also observed that the stresses in all directions of liner are mainly compressive stresses. And the stresses in the interfaces among different materials also present various degrees of fluctuation,especially in the interfaces of liner and filament wound composite housing, inner glass fiber layers and carbon fiber layers. With the taper coefficient increasing, the level of residual stress in barrel drops. It is because that the decreasing amplitude of winding tension enlarges gradually, which can be seen in Fig. 3. While β = 0(constant tension mode), the circumferential and axial residual prestresses become greater from the inner layer to the outer layer of the filament wound layers,obviously in the carbon fiber layers.It presents a feature of “internal looseness and external tightness”.And when β = 1 (constant torque mode), the circumferential and axial residual prestresses in the filament wound layers are relatively more uniform,especially in the carbon fiber layers.It is closer to the requirement of iso-stress design for filament winding.

Fig. 3. Distributions of initial fiber prestress in fiber direction through the wall thickness for the filament wound layers.

Fig. 4. Variations of residual prestress in all directions through the wall thickness of barrel for path oS1.

Fig. 5. Variations of residual prestress in all directions through the wall thickness of barrel for path oS2.

The residual prestress contours for all directions in railgun barrel, which is under the constant torque mode, are exhibited in Fig. 6. Clearly, residual stress in liner is variable along the circumference. It is due to the fact that the liner is made up of different materials which distributing along the circumference.It is also can be observed in Figs. 4 and 5. Therefore, there are differences for residual prestresses on the path oS1 and oS2.In which the axial and circumferential residual prestresses in rail liner are bigger than the ones in ceramic liner.But the differences for residual prestresses in filament wound layers on the two paths are relatively smaller.

Moreover, the electromagnetic load in the railgun barrel is incomplete axial symmetry. When the electromagnetic load is greater, the contact interfaces between the rails and ceramic insulators will appear separation because of larger normal stress and shear stress on the interfaces [20]. And it will cause the failure of bore seal. Here, a constant electromagnetic pressure (200Mpa) is assumed to be loaded on the inner sides of rails.Then the path AB is defined on the contact plane(in circumference 60°)along the radial direction, as depicted in Fig. 1(a). And it is also on the section of barrel, which is located on the half length of barrel.

The normal stress and shear stress on the path AB for the barrel only under the electromagnetic load, and the residual prestress in normal and shear direction on the path AB for the barrel after filament winding, are compared in Fig. 7. It presents that the normal stress on the contact interface for the liner only with the electromagnetic pressure is mainly tensile tress.And after filament winding with fiber prestress,the normal residual prestress on this contact interface is compressive stress.So the normal tensile stress leaded by the electromagnetic load can be weakened in a fiber prestress winding barrel during its launching. Similarly, the directions of the shear stresses on the contact interface under these two loading conditions are opposite to each other. Hence, it can enhance the bond strength for the interfaces and slow the separation of contact interfaces.Then the seal performance of bore will be improved. And with the decreasing of β, the normal residual prestress on the contact interface is larger, while the change of shear residual prestress is smaller.It means that reducing the value of β is better for improving the seal performance of bore. But the same kinds of stresses in the filament wound layers with two loading conditions respectively are in the same direction, so they are superimposed with each other.And the intensified stresses are bigger with the decline of β,which will lead to a higher stress level in the winding layers. This will reduce the strength of the composite housing for barrel.

5. Conclusions

In this work, a theoretical analysis model for the residual prestress only caused by the fibre pretension in the round composite barrel of railgun is developed. Based on the elastic theory and the theory of composite laminates,the barrel is simplified as a hybrid filament wound cylinder with an isotropic liner in this model. Then the residual prestress in this railgun barrel model is also simulated in a 3D finite element model through combining the temperature differential method with the element birth and death technology.

Fig. 6. Residual prestress contours for all directions in railgun barrel with β = 1(unit:Pa).

Fig. 7. Normal stress and Shear stress on the path AB.

Comparing the results of two methods, it indicates that the distribution trends of residual prestress are consistent. And the difference for residual prestress in the filament wound composite housing of barrel is relatively small. Furthermore, the residual prestresses in the non-simplified composite barrels for railgun,which are wound with different winding tension modes, are analyzed by using the same numerical simulation method. The results reveal some rules of the distributions of residual prestresses of the barrel with different control modes of winding tension, and their effects on some mechanical performances of barrel. It can provide a reference to the optimal design of filament winding for railgun barrel.

Declaration of competing interest

No conflict of interest exits in the submission of this manuscript,and manuscript is approved by all authors for publication.I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere.All the authors:Dong-mei Yin, Bao-ming Li and Hong-cheng Xiao.

We deeply appreciate your consideration of our manuscript,and we look forward to receiving comments from the reviewers.If you have any queries,please don’t hesitate to contact me at the address below.