Effects of Magnetic Field in Creep Behavior of Three-Phase Laminated Composite Cylindrical Shells

Effects of Magnetic Field in Creep Behavior of Three-Phase Laminated Composite Cylindrical Shells

K. Hosseinpour, A.R. Ghasemi^*

Composite and Nanocomposite Research Laboratory, Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-53153, Iran.

1. Introduction

In the recent years, use of the polymer composite cylinders in various industries with high temperature and pressure environments due to high mechanical strength, low weight and easy to shape increased [1,2]. Due to the time-dependent behavior of polymer matrix composites and nano-composites even at low temperatures, investigating the long-term creep behavior using the viscoelastic assumptions attracted many researchers. Zhang et al. [3] used the thermo-elasticity analysis theory to study of stress distribution in the wall of the composite cylinder. They showed that the results of the analytical method and finite element model (FEM) for multi-layer composite cylinder under the thermo-mechanical loads have acceptable agreement with each other. Time-dependent behavior of thick walled multi-layered composite cylinders made of carbon/epoxy assuming nonlinear viscoelastic Schapery’s model [4] was studied by Guedes [5]. The long-term performance of GRP after 50 years was predicted from failure pressure and time to failure through sustained internal pressure test by Yoon and Oh [6]. Due to the development of nano-industry and performed studies it was shown that the addition of nano-particles will improve the mechanical [7], thermal [8] and residual [9] properties of two-phase and three-phase nano-composites. Creep behavior of the polycarbonate reinforced by MWCNTs fiber nano-composite was studied by Zhou et al. [10]. They demonstrated that the results of experimental and Burger’s model [11] had a good agreement. Starkova et al. [12] predicted the long-term creep behavior of MWCNT/epoxy nano-composites and used the experimental work and modeling. Their results illustrated that the Schapery’s model [6] had a good agreement with their experimental results. Mohandes et al. [13] studied the influence of size dependency and volume percentage of CNTs on the mechanical behavior of the nano-composite cylinders. They used the Mori-Tanaka model [14] to obtain the thermal, mechanical and piezoelectric properties of nano-composite cylinders. In another work, Mohandes et al. [15] studied the behaviour of rotating cylinder made of composite reinforced by multi-walled carbon nanotubes (MWCNTs) subjected to mechanical loading. Ghasemi et al. [16] studied the influence of weight fraction of MWCNTs and lay-ups on the way of distribution creep strains in the wall of the MWCNTs /E-glass/vinylester three-phase nano-composites. Their results demonstrated that the addition of the MWCNT to the vinylester can reduce the absolute values of the radial and circumferential creep strains. Also, all the mechanical properties of nano-composites cylinder were obtained using micromechanical relations.

Despite studies on creep behavior of composite cylinders, there is no article that reviews the effect of adding nano-particles, temperature loads and magnetic field to long-term creep strain distribution of the wall of the three-phase nano-composite cylinders. The main purpose of the present article is to study the effect of thermal and magneto loading on creep strains and radial displacement in the three-phase nano-composite cylinder wall.

1. Mathematical Model

A SWCNTs/GF/Vinylester composite cylinder with the conditions having inner radius of  and outer radius of  was considered. Internal pressure in the inner wall of the cylinder is  and the cylinder was subjected to uniform distributed temperature field   and placed in a uniform magnetic field .

Prandtl-Reuss relation defined the creep strains increment in radial and circumferential direction and current stresses and also, creep constitutive model could be written as:

The constitutive creep model in this literature has been assumed the Schapery nonlinear viscoelastic model that results in [17]:

where  is the effective stress which is an octahedral stress:

Also, ,  and  are linear elastic coefficients and for GF/Vinylester are shown in Table 1 for different angles and , , ,  and are nonlinear coefficients of the Schapery constitutive model that following equations are used for the GF/vinylester [18].

By assumption of uniform magnetic field, the equilibrium equation for the thick walled cylinder would be the expressed as:

where  is Lorenz’s force and its equation is:

where μ is the magnetic permeability and    is the magnetic field intensity in the axial direction. Also, the considered total strain is the summation of elastic strain and creep strain, stress and strain relations would be written as:

where ,  and  denoted radial, circumferential and axial directions, respectively.

Table 1. Viscoelastic linear parameters of glass/vinylester [19]

Also,  is the modulus matrix in cylindrical coordinate as follows:

where  is the Cartesian coordinate and  is the transfer matrix as:

where ,   and  is lay-up direction. Also,  is the modulus matrix in cylindrical coordinate where superscript  denoted the number of layers. Superscript  in Eq. (7) specifies the creep. Assuming axial symmetry and linear strain relation, strain-displacement relation could be written as:

Also with assuming constant thermal gradient in the wall of the cylinder, thermal strain relation would be written as:

Substituting radial and circumferential stresses from Eq. (7) into Eq. (5), the following differential equation containing creep strains is obtained:

where constant coefficients  in Eq. (13) could be summarized as follow:

The solution for Eq. (13) can be obtained:

where and  are unknown integration constants and other parameters are:

In order to calculate the unknown constant coefficients for each layer, there is a need to use boundary conditions. For the -layered composite cylinder, there are  unknown constant coefficients which include  and  used in below boundary conditions:

Using the expressed relations and Mendelson’s approximation method for the long period of time, the history of strains in time can be calculated [16].

1. Micro-Mechanical Model

Three-phase nano-composite laminate was formed to the combination of isotropic matrix (vinylester resin), carbon nanotubes (SWCNTs) and fibers (E-Glass). It is assumed that SWCTs are homogeneously distributed in the matrix without the presence of air voids and have the same mechanical and thermal properties and are isotropic. The effective mechanical and thermal properties of the three-phase SWCNTPC multi-layered cylinder can be predicted according to Halpin–Tsai [20] and Schapery relations [21], respectively. Young’s moduli, shear moduli, Poisson’s ratio and the coefficient of thermal expansion are as follow:

where , , ,  and  are elastic and  shear modulus,   Poisson’s ratio, volume fraction and coefficient of thermal expansion (CTE) of the fiber, respectively. Also , , ,  and  are elastic and  shear moduli,  Poisson’s ratio, volume fraction and coefficient of thermal expansion of the SWCNTs/vinylester Two-phase nano-composite, respectively and are presented as below:

where  and  are the elastic moduli of the SWCNTs and the matrix, respectively and  and  are the orientation factor and aspect ratio, respectively. Also  and  are the CTE of the SWCNTs and matrix, respectively. For nano-composite with two-phase (MWCCNTs/vinylester) and no trapped air, volume fraction of the SWCNTs is obtained as [22]:

where  and  are the density of the SWCNTs and matrix, respectively, and  is the weight fraction of SWCNTs.

For the mentioned conditions, a composite cylinder made of SWCNTs/GF/vinylester with elastic properties for each one is shown in the Table 2.

1. Numerical Results and Discussion

The results discussed in the present section are based on the material properties, geometry, loading condition and introduced in previous section and Tables 1 and 2 as well. Effect of temperature and magnetic field on creep strains and radial displacement after 10 years are discussed in the three-phase nano-composite cylinder with  weight fraction of SWCNTs is considered and distribution of creep strains and radial displacement in the wall of three-phase nano-composite cylinder with lay-up [0/45/0/45] for a period of 10 years is plotted. Fig. 2 demonstrated the distribution of radial creep strains with and without the magnetic field. As it is shown in the presence of a magnetic field, the creep strain for every thermal loading is lower in magnitude. Also, Fig. 3 demonstrated that an increase in temperature, increases the radial creep strain and the increased creep strain in a magnetic field is lower without magnetic field.

Variation of circumferential creep strain in the wall of three-phase nano-composite cylinder demonstrated in Fig. 3. Values of circumferential decreased in each layer with increased dimensionless ratio of the radius. Also, values of circumferential creep strain in the same temperature and with magnetic field is lower than without magnetic field.

Table 2. Material properties of SWCNTs/GF/vinylester three-phase composites

a

b

Fig. 2. Radial creep strain of the nano-composite cylinder with lay-up [0/45/0/45] (a) without and (b) with the effect of magnetic field.

a

b

Fig. 3. Circumferential creep strain of the nano-composite cylinder with lay-up [0/45/0/45] (a) without and (b) with the effect of magnetic field.

Fig.s 4-a and 4-b illustrated the distribution radial displacement in the wall of three-phase nano-composite cylinder in two state of without and with the magnetic field, respectively. Fig. 4 demonstrated that with rising the temperatures, radial displacement increased. Also, radial displacement with magnetic field are lower than that of without magnetic field.

1. Conclusions

Creep response of a three-phase nano-composite cylinder made of SWCNTs /E-glass/vinylester with 4.5% weight fraction subjected to thermal, mechanical and magnetic loads has been investigated. Effects of operating temperature, magnetic field and lay-up on radial and circumferential creep strains and radial displacements are also studied. The magnetic field is reduced the radial and circumferential creep strain and radial displacement. An increase in temperature would increase the radial and circumferential creep strain and radial displacement. Furthermore, the temperature increase in the magnetic field is less effective on the values of creep strain and radial displacement.

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a

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Fig. 4. Radial displacement of the nano-composite cylinder with lay-up [0/45/0/45] (a) without and (b) with the effect of magnetic field.

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