Document Type: Research Paper
Authors
Composite and Nanocomposite Research Laboratory, Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran.
Abstract
Keywords

Mechanics of Advanced Composite Structures 6 (2019) 51–56


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Effects of Magnetic Field in Creep Behavior of ThreePhase 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, 8731753153, Iran.
Paper INFO 

ABSTRACT 
Paper history: Received 20190125 Received in revised form 20190215 Accepted 20190426 
Due to the importance effect of magnetic field on the history of longterm radial and circumferential creep strain and radial displacement for a threephase nanocomposite exposed to an internal pressure and placed uniform temperature,the present article subject has been proposed. Threephase nanocomposite made ofsinglewalled carbon nano tubes (SWCNTs)/ glass fiber (GF)/vinylester used to micromechanical models in order to calculate the mechanical and thermal properties. By assuming nonlinear viscoelastic based on Schapery integral model and using classical laminate theory, PrandtlReuss relations and Mendelson’s approximation method achieved results. Distribution of the radial creep strain, circumferential creep strain and radial displacement in two states including without and with magnetic field and three temperature conditions for laminated layups [0/45/0/45] described for 10 years. The results indicate that the magnetic field has reduced the radial and circumferential creep strain and radial displacement. Furthermore, the temperature increase in the magnetic field is less effective on the increased values of creep strain and radial displacement. Finally, It has been founded that magnetic field would reduce the creep strain of all case studies. 



Keywords: Thermomagnetomechanical loading Threephase composite cylinder Micromechanical model 


© 2019 Published by Semnan University Press. All rights reserved. 
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 timedependent behavior of polymer matrix composites and nanocomposites even at low temperatures, investigating the longterm creep behavior using the viscoelastic assumptions attracted many researchers. Zhang et al. [3] used the thermoelasticity 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 multilayer composite cylinder under the thermomechanical loads have acceptable agreement with each other. Timedependent behavior of thick walled multilayered composite cylinders made of carbon/epoxy assuming nonlinear viscoelastic Schapery’s model [4] was studied by Guedes [5]. The longterm 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 nanoindustry and performed studies it was shown that the addition of nanoparticles will improve the mechanical [7], thermal [8] and residual [9] properties of twophase and threephase nanocomposites. Creep behavior of the polycarbonate reinforced by MWCNTs fiber nanocomposite 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 longterm creep behavior of MWCNT/epoxy nanocomposites 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 nanocomposite cylinders. They used the MoriTanaka model [14] to obtain the thermal, mechanical and piezoelectric properties of nanocomposite cylinders. In another work, Mohandes et al. [15] studied the behaviour of rotating cylinder made of composite reinforced by multiwalled carbon nanotubes (MWCNTs) subjected to mechanical loading. Ghasemi et al. [16] studied the influence of weight fraction of MWCNTs and layups on the way of distribution creep strains in the wall of the MWCNTs /Eglass/vinylester threephase nanocomposites. 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 nanocomposites cylinder were obtained using micromechanical relations.
Despite studies on creep behavior of composite cylinders, there is no article that reviews the effect of adding nanoparticles, temperature loads and magnetic field to longterm creep strain distribution of the wall of the threephase nanocomposite 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 threephase nanocomposite cylinder wall.
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 .
PrandtlReuss relation defined the creep strains increment in radial and circumferential direction and current stresses and also, creep constitutive model could be written as:
(1) 
The constitutive creep model in this literature has been assumed the Schapery nonlinear viscoelastic model that results in [17]:

(2) 
where is the effective stress which is an octahedral stress:
(3) 
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].
(4) 
By assumption of uniform magnetic field, the equilibrium equation for the thick walled cylinder would be the expressed as:
(5) 
where is Lorenz’s force and its equation is:
(6) 
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:
(7) 
where , and denoted radial, circumferential and axial directions, respectively.
Table 1. Viscoelastic linear parameters of glass/vinylester [19]
90° 
45° 
0° 
Offaxis angle ( ) 
0.81 
0.79 
0.53 

1.35 
0.16 
0.32 

0.189 
0.20 
0.16 
n 
Also, is the modulus matrix in cylindrical coordinate as follows:
(8) 
where is the Cartesian coordinate and is the transfer matrix as:
(9) 

(10) 
where , and is layup 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, straindisplacement relation could be written as:
(11) 
Also with assuming constant thermal gradient in the wall of the cylinder, thermal strain relation would be written as:
(12) 

Substituting radial and circumferential stresses from Eq. (7) into Eq. (5), the following differential equation containing creep strains is obtained:
(13) 
where constant coefficients in Eq. (13) could be summarized as follow:
(14) 
The solution for Eq. (13) can be obtained:


where and are unknown integration constants and other parameters are:
(16) 

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:




(17) 


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].
Threephase nanocomposite laminate was formed to the combination of isotropic matrix (vinylester resin), carbon nanotubes (SWCNTs) and fibers (EGlass). 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 threephase SWCNTPC multilayered 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:
(18) 

(19) 

(20) 
(21) 

(22) 

(23) 
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 Twophase nanocomposite, respectively and are presented as below:
(24) 

(25) 

(26) 

(27) 

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 nanocomposite with twophase (MWCCNTs/vinylester) and no trapped air, volume fraction of the SWCNTs is obtained as [22]:
(28) 
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.
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 threephase nanocomposite cylinder with weight fraction of SWCNTs is considered and distribution of creep strains and radial displacement in the wall of threephase nanocomposite cylinder with layup [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 threephase nanocomposite 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 threephase composites
Nanofiller [23] 
Fiber 
Polymer matrix 
Property 
SWCNT 
Eglass 
Vinylester 

640 
71.78 
4.99 
Young's modulus ( ) 
0.33 
0.25 
0.3 
Poisson’s ratio ( ) 
3.45 
5 
62.46 
CTE ( ) 
0.66 
34 
Volume fraction (%) 
a
b
Fig. 2. Radial creep strain of the nanocomposite cylinder with layup [0/45/0/45] (a) without and (b) with the effect of magnetic field.
a
b
Fig. 3. Circumferential creep strain of the nanocomposite cylinder with layup [0/45/0/45] (a) without and (b) with the effect of magnetic field.
Fig.s 4a and 4b illustrated the distribution radial displacement in the wall of threephase nanocomposite 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.
Creep response of a threephase nanocomposite cylinder made of SWCNTs /Eglass/vinylester with 4.5% weight fraction subjected to thermal, mechanical and magnetic loads has been investigated. Effects of operating temperature, magnetic field and layup 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
b
Fig. 4. Radial displacement of the nanocomposite cylinder with layup [0/45/0/45] (a) without and (b) with the effect of magnetic field.
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