Document Type : Research Paper
Authors
Department of Solid Mechanics, University of Kashan, Kashan, Iran
Abstract
Keywords
Main Subjects
Thermal Variations of ThermoMechanical Loading Effects on Creep Stress and Strain Distribution in Multilayered Composite Cylinder
Department of Solid Mechanics, University of Kashan, Kashan, Iran
KEYWORDS 

ABSTRACT 
Thermomechanical loading; Thermal variation; Multilayer composites; Integral model; Viscoelastic creep. 
In this research, thermal variation and internal pressure effects on stresses and strains of the creep behavior of composite laminates are investigated. Schapery's viscoelastic approach as a nonlinear model is used by Prandtl‐Reuss theory and Mendelson's approximation to explain the stress/strains creep treatment. History of the radial, circumferential, and axial creep stresses and strains of a laminated composite cylindrical shell made of the Eglass/Vinylester laminated composites are studied and examined. Different laminated sequences of layups, linear variation of thermal loading, various composite materials accomplished with internal pressure loading are investigated. For this purpose, two laminated layups as [ / / / ] and [ / / / ] are selected, and Schapery integral nonlinear viscoelastic model of creep stress and strain approach for the constitutive equation is used. It was believed that the circumferential creep strain and absolute values of axial and radial creep strains are increasing with time and Also, creep stresses/strains in the multilayer composite crossply [ / / / ] is more than the [ / / / ] laminated composite cylindrical shells. 
Laminated composite structures are widely used in structural engineering, vehicles, aerospace, building, and other industries. Researchers use linear and nonlinear viscoelastic approaches to describe the creep behavior of the polymer matrix composites (PMCs) under thermal and mechanical loading [1, 2]. However, PMCs generally exhibit nonlinear behavior and ratedependent treatments in creep rupture. Lou and Schapery [3] presented a single integral model, which has been used to predict the timedependent strains for PMCs [3]. Tuttle et al. [4] used the Schapery single integral model and classical laminate plate theory (CLPT) to study the creep behavior of graphiteepoxy composite materials. Their results demonstrated the longterm creep compliance for and offaxis unidirectional specimens is accurate to within 10 percent at minutes. The creep strain behavior response using the Burgers approach investigated by Berardi et al. [5], and numerical results are presented.
Gudes et al. [6] used a numerical method based on the Schapery integral constitutive equation to study the creep behavior of T300/5280 laminates. Their observations demonstrated a good agreement between computed and experimental results. Fiber orientation is one of the essential parameters that affected the PMCs' mechanical characterization and creep responses behavior, studied by Papanicolaou et al. [7]. They studied the effect of fiber orientation on the nonlinearity parameters of the Schapery integral model. Their results showed that the increase in the values of nonlinear parameters occurs with increasing stress, while this behavior is stabilized by decreasing the fiber direction, except for nonlinear parameters. In another study, Muliana et al. [8] obtained linear and nonlinearity parameters of the Schapery integral model for glass/vinyl ester composites. They did their experiments for 0 , 45 and 90 fiber angles. Sayyidmousavi et al. [9] studied the viscoelastic thermomechanical behavior of the hightemperature PMCs. Their results showed the importance of calculating the viscoelastic effect of the matrix material with increasing temperature.
Samarehmousavi and Taheribehrooz [10] proposed a new phenomenological residual resistance describing the residual strength data under arbitrary stress conditions and stable times for T300 / L20 unidirectional layers. Song et al. [11] constructed design stresses and design spaces for hightemperature carbon/polymer composites considering creep responses in polymers. They used the powerlaw model to model transient and steadystate creep to describe polyamide macroscopic creep responses.
Rathore et al. [12] studied the effect of temperature on the flexural and creep performance of glass/epoxy, carbon/epoxy, and glass/carbon/epoxy hybrid composites. Their results showed that with increasing temperature, both strength and stiffness of allfiber composites continued to decrease to the highest test temperature (110 ° C). They also represent the synergistic effect of glass and carbon fibers on hybrid composites at high temperatures.
Multilayer composite cylinders are spread in environments with high temperatures and high internal pressure. Due to the importance of stress and strain distribution in the walls of the composite cylinders, researchers have performed many studies. Ghasemi and Hosseinpour [1315] studied the creep stress and strain in laminated PMCs circular cylindrical shells. Thermal and mechanical loading on the filamentwinding composite pipe was performed by Bakaiyan et al. [16]. Their results showed that the hoop to the axial stress ratio depends on the change in the fiber directions. Hocine et al. [17] presented an experimental and analytical study for metal pressure vessels reinforced with carbonepoxy layers.
A numerical and analytical study of the functionally graded materials (FGMs) reinforced by multilayer composites was investigated by Ghasemi et al. [18] using the theory of infinitesimally small elasticity. Some researchers studied the thermal and mechanical nonlinear responses of T300/5208 composite circular cylindrical shells under the internal and external pressure accomplished with axial force [1921]. The timedependent radial, axial, hoop, and shear stress distributions in the thickness of the multilayered T300/5208 cylinder presented by Guedes [19]. His results showed the importance of the viscoelastic effect on the evolution of the timedependent internal stress field in thicksheet cylinders. Poirette et al. [22] presented a new approach based on the timedependent damage spectrum. They performed quasistatic, creep, and fatigue testing to characterize the composite structure's elastic, plastic, viscoelastic, and viscoplastic behavior. Furthermore, other researchers presented and examined the effect of seawater absorption on the longterm viscoelastic response of the glass/BMI laminated composites structures [2324].
Due to the importance of stress and strain distribution in laminated composite cylindrical shells, the effect of laminated sequences, type of composite materials, linear thermal variations, and nonlinear viscoelastic behavior are presented in this research. For this goal, using the nonlinear viscoelastic approaches, internal pressure as mechanical loading accomplished with linear thermal temperature distribution in the thickness of the cylindrical are considered. Moreover, using different models and relations, the distribution of the radial, axial and circumferential creep stress and strain in the thickness of the cylindrical shell for longterm creep behavior are examined and discussed.
A laminated multilayered composite circular cylinder made of Vinylester resin reinforced by Eglass fiber with an inner radius and outer radius is observed in Fig. 1 The length of the cylinder is high, and the plane strain condition can be used. The internal pressure of the cylinder is and the axial symmetric thermomechanical loading is established.
The steadystate conditions, long lengths, and axial symmetric assumptions, the differential equation of heat conduction in the cylindrical coordinate system are as follows [1516].
Fig. 1. Multilayered composite cylindrical shell

(1) 
By solving Eq. (1), the temperature equation is obtained:

(2) 
where" and " are constant coefficients. Assuming that the outer surfaces have free convection to air at ambient temperature And the inner surfaces have forced convection to the temperature of the hot fluid at . The boundary conditions are as follows [16]:
for for 
(3) 
Therefore

(4) 
where , and are thermal conduction (TC) of cylindrical shell, inner and outer average convection heat transfer coefficients. The following data are assumed in this research:




The material properties of the Eglass/vinyl ester matrix are given in Table 1 based on the experimental results reported by Sawant et al. [25].
Table 1. Mechanical and thermal properties of
Eglass/vinyl ester composites

19268 

13210 

4572 

0.31 

0.35 

10 

44 
TC through the thickness ( ) 
0.49 
A useful nonlinear viscoelastic model has been developed and demonstrated by Schapery [34, 1516]. This method is fascinating because it allows us to describe a fundamental nonlinear phenomenon based on an integral relation. To model the creep behavior of the PMCs based on the nonlinear viscoelastic approach, in this research, Schapery's single integral model has been used.
At lowstress levels, the creep strain and stress can be related by strain compliance parameter, which is defined as follows:

(5) 
where and are creep compliance and creep strain, respectively. The Boltzmann's superposition principle to express timedependent behavior for the lowstress levels can be presented as:

(6) 
The applied integral in this equation is known as the "hereditary integral" approach. The above equation shows that the strain at any given time depends on the entire stress history. Lou and Schapery [3] described the behavior of the nonlinear viscoelastic materials using expansion of the Eq. (6). The Schapery's single integral constitutive equation model under the uniaxial isothermal conditions can be expressed as [3]:

(7) 
where is transient creep compliance and, and indicated the effects of time (reduced times) and can be defined as [3]:

(8) 
where , and are stressdependent nonlinear material parameters, which depend on the stress and temperature. Substituting the applied pressure ( ) into the Eq. (7), the creep strain is obtained as [3]:

(9) 
Findley's power law is defined as follows:

(10) 
Therefore, the creep strain relation can be written as follow:

(11) 
where and are independent of time and temperature and are obtained according to the linear viscoelastic behavior in the reference temperature. The nonlinear parameters of the GF/vinyl ester materials based on the Schapery constitutive equations have been mentioned as [25]:

(12) 
where is the effective stress that is octahedral stress and calculated by:
.
Furthermore, the linear parameters for GF/vinyl ester composites in Eq. (11) are shown in Table 2 [8].
Table 2. The linear viscoelastic parameters of
the unidirectional Eglass/Vinylester
Offaxis angle (θ) 
0° 
45° 
90° 
* 
0.53 
0.79 
0.81 
* 
0.32 
0.16 
1.35 
n 
0.16 
0.20 
0.189 
The stress and strain relations for the K^{th} layer of the cylindrical shells can be presented as:

(13) 
where in the cylindrical system , the , and are the thermal strains, creep strains, and mechanical strains, respectively. Also, are the component of the stiffnesses matrix in the cylindrical coordinate for the number of layers, which is obtained by multiple transfer matrix in the modulus matrix in the Cartesian coordinate systems [25].
The straindisplacement by the emphasis on and radial displacement ( ) only depend on the radial direction, can be written as follows:

(14) 
Furthermore, the can be obtained by:

(15) 









According to the assumption considered, the equilibrium equations for the rotating cylinder are as follow:

(16) 

(17) 

(18) 
Substituting stressstrain relations from Eq. (14) into Eq. (13), and using Eq. (15), (2), and (16), the following differential equations are obtained:

(19) 

where

(20) 







The solution for Eq. (19) can be obtained as:

(21) 
where and are unknown constant integration coefficients, and other parameters are:

(22) 





To calculate the unknown constant coefficients for each layer, we must use boundary conditions. For the N layer of the composite cylinder under axial symmetrical loading, assuming that the surface between the core and skin layers are minimal, the boundary conditions are expressed as [17]:

(23) 









In addition, the additional below boundary conditions could be allowed to obtain the other constants:

(24) 





To obtain the timedependent creep analysis, the coefficients' creep stress and strain parameters must be considered. In this research, the PrandtlReuss relations are used. Also, the history of stress and strains vs. the time can be calculated using the expressed relations and Mendelson's approximation method for an extended period. In this approach, the thickness of the circular cylindrical shells is divided into the k increment. The authors explain the numerical procedure in the previous works [1315].
Using the assumptions considered in the previous sections for material properties and thermomechanical loading, the creep stress and strain distribution in the cylindrical wall for different time steps were obtained. The history of radial, circumferential and axial creep stress and strain for two laminated sequences as [ / / / ] and [ / / / ], are plotted in Figs. 2 to 5. Values of creep strains that grow with times for the crossply laminates are shown in Fig. 2. History of the distribution radial, circumferential and axial creep strains ( , ) in the dimensionless wall (r/b), the crossply laminates [ / / / ] showed in Figs. 2a, 2b, and 2c, respectively. The difference values of the creep strains between layers increased with increasing time.



a 

a 



b 

b 



C 

C 
Fig. 2. Radial (a), circumferential (b), and axial creep strains for the composite cylinder with [ / / / ] layups 

Fig. 3. Radial (a), circumferential (b), and axial creep stress for the composite cylinder with [ / / / ] layup 
Also, in each layer, by raising dimensionless radius, values of radial creep strains increased. Still, values of circumferential creep strains decreased that in layer with fiber angle 0 rate of change are lesser.
History of the dimensionless distribution radial, circumferential and axial stresses for the [ / / / ] crossply laminates are plotted in Figs. 3a, 3b, and 3c, respectively. Generally, the dimensionless effective stresses increase with time at a decreasing rate during the life of the cylindrical shell. Also, dimensionless values radial stress are continuous in the wall of the cylinder. On the other hand in dimensionless values of circumferential and axial stresses in the fiber angle are very close together.
Fig. 4a, 4b and 4c demonstrated the distribution of the axial, radial, and circumferential creep strains for laminated composite [ / / / ] sequences. Fig. 4 Shows that values of creep strain increased with time, although in the fiber angle 0 the values of creep strains are increased is significantly lower than fiber angle 45 .
Also, with raising in each layer, values of axial creep strains increased, but values of circumferential and radial creep strains decreased. Fig. 5 shows plotted variations of dimensionless radial, circumferential, and axial stresses. Changing procedure of the stresses are similar to [ / / / ] laminated sequences. Also, dimensionless values of stresses in [ / / / ] crossply sequences are more than dimensionless values of stresses in the [ / / / ] laminated composite cylindrical shells.



a 

a 



b 

b 



C 

C 
Fig. 4. Radial (a), circumferential (b), and axial creep strains for the composite cylinder with [ / / / ] layups 

Fig. 5. Radial (a), circumferential (b), and axial creep stresses for the composite cylinder with [ / / / ] layups 
Also, with raising in each layer, values of axial creep strains increased, but values of circumferential and radial creep strains decreased. Fig. 5 shows plotted variations of dimensionless radial, circumferential, and axial stresses. Changing procedure of the stresses are similar to [ / / / ] laminated sequences. Also, dimensionless values of stresses in [ / / / ] crossply sequences are more than dimensionless values of stresses in the [ / / / ] laminated composite cylindrical shells.



a 

b 
Fig. 6. Radial and circumferential creep strains variations versus the time for the composite cylinder with a) crossply [ / / / ], and (b) [ / / / ] layups 
Variation radial and circumferential creep strains in the cylinder's internal surface vs. time are demonstrated in Fig. 6a and 6b for two layups. In Fig. 6 it is observed that with increasing the time, values of creep strains are increased in two layups, but with increased time, the creep strain rate is slower. Figure 6 demonstrate at the beginning time, the rate of creep strain change in crossply [ / / / ] layups are more than that in [ / / / ] layups. Also, values of circumferential creep strain are higher than the values of the radial creep strains.
No research on the timedependent creep of the pressure vessels or laminate composite structures have been founded, but to validate the results of this research, this method has been compared with the work of You et al. [26], which expressed the steadystate creep of functionally graded materials (FGM) under internal pressure.
Fig. 7. Dimensionless radial stress variations in the wall of composite cylinder
Thus, using the analysis of You et al. [26] for the homogeneous material with the same geometry, our results for dimensionless compressive radial stress are shown in Fig. 7. In order to show the numerical results, the nondimensional radial coordinate system is defined as:

(25) 
Considering similar radial stresses calculated from both of these approaches, the validation of the present solution method can be examined in this respect.
The timedependent creep treatment for laminated composite cylinders under combined thermomechanical loading and internal pressure were studied. Through the use of nonlinear viscoelastic model analysis and increment time approximation, the creep stress/strain was investigated.
The distribution of histories of radial, circumferential, ad axial creep stress and strain in the cylinder wall was obtained. It has been shown that dimensionless stresses and strain are increasing with time, and the rate of increase between 10 and 15 years is less than that of between 5 and 10 years. Despite values, the creep strains in [ / / / ] layup are lower than [ / / / ] crossply laminates. In [ / / / ] layup and in fiber angle 45 changes of creep strains with time are very little. Also, in [ / / / ] layup creep strains with less slope to reach steadystate than [ / / / ] crossply laminated composite cylinder.
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