Document Type: Research Paper
Authors
Faculty of Mechanical Engineering, University of Kashan, Kashan
Abstract
Keywords

Mechanics of Advanced Composite Structures 5 (2018) 141 – 147


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Creep Strain and Stress Analysis in Laminated Composite Pressure Vessels
A.R. Ghasemi^{*}, K. Hosseinpour
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 20170921 Received in revised form 20180107 Accepted 20180129 
This study investigates the timedependent longterm creep strain in a composite cylinder made of glass/vinylester with a unidirectional ply. The cylinder is subjected to an internal pressure and the boundary condition is free–free and acts as thermal insulation. The classical lamination theory (CLT) is used to derive the governing equations as a secondorder equation to determine the radial, circumferential, axial, and effective stresses in the cylinder wall. The distribution of the radial and circumferential creep strains is based on the Schapery’s single integral model for nonlinear viscoelastic materials. This study focuses on the effect of the orientation of the fibers on the creep strain distribution in the wall of a cylinder. The results show that the creep strain is lower when than at . As the angle of the fibers increases, the distribution of the creep strain becomes more uniform. 



Keywords: Longterm creep strain Schapery single integral Nonlinear viscoelastic Polymer matrix composites 


© 2018 Published by Semnan University Press. All rights reserved. 
Many studies have been done on the cylindrical structure because it is one of the most widely used structures in industry [1, 2]. Laminated composite materials (LCMs) have superior specific properties regarding their stiffness and strength, and tend to replace the traditional materials used for the construction of structural components in several engineering fields. Tsukrov and Drach [3] investigated the elastic deformation of laminated composite cylinders that were subjected to homogeneous boundary conditions. Zhang et al. [4] investigated the thermomechanical stresses in a multilayered composite pressure vessel. They derived an analytical solution for determining the stress–strain distribution of a laminated composite pressure vessel subjected to internal pressure and a thermal load. Ghasemi et al. [5] performed analytical and numerical investigations of functionally graded materials (FGMs) reinforced by LCMs using the infinitesimal theory of elasticity. Arefi et al. [6, 7] studied the twodimensional thermoelastic analysis of a functionally graded cylindrical shell subjected to uniform and nonuniform mechanical and thermal loads.
Designing for the longterm durability of composite structures requires an understanding of the viscoelastic behavior of the composite materials and how it affects strength. Compared with the elastic properties of the LCMs, the timedependent and creep properties are extremely weak. Violette and Schapery [8] studied the timedependent compressive strength of unidirectional viscoelastic composite materials and the longterm durability of unidirectional composite structures. Chio and Yuan [9] investigated the timedependent deformation of a pultruded glass fiber reinforced polymer (GFRP) composite. They concluded that Findley’s power law model [10] can be successfully used to estimate the timedependent deformation of pultruded GFRP composite columns. Papanicolaou et al. [11] described the effect of fiber orientation on the nonlinear viscoelastic response of a unidirectional fiber polymeric composite. Using Schapery’s nonlinear single integral constitutive equation, the researchers characterized the offaxis viscoelastic behavior of carbon fiber reinforced polymer (CFRP) composite materials. The generic function described each of the nonlinear parameters. Muliana et al. [12] tested and verified the viscoelastic responses of offaxis glass/vinylester multilayered composites under combined thermomechanical loadings. They performed isothermal creep tests under various stresses and temperatures on axial, transverse, and 45° offaxis specimens. They used the thermomechanical viscoelastic behaviors in order to characterize the timestresstemperature dependent properties of the material and to predict the longterm behaviors of the composite [12]. Sawant and Muliana [13] introduced a numerical algorithm for the nonlinear thermomechanical viscoelastic analysis of orthotropic composite materials. Using Schapery’s nonlinear single integral form, the effects of stress and temperature were considered along with the elastic and timedependent material properties, which allowed for the prediction of timedependent responses under general stress and temperature histories. Muddasani et al. [14] performed experimental work and a finite element analysis of nonlinear thermoviscoelastic behaviors of multilayered composites. A nonlinear viscoelastic constitutive model based on the convolution integral equation for orthotropic materials was adopted. The nonlinear parameters in the equation were calibrated in the uniaxial isothermal creep tests for offaxis specimens and modeled as a function of the effective stress and temperature [14].
The laminated cylinder is one of a very few structural cases for which an exact elasticity solution is available. Guedes [15] analyzed the effect of the nonviscoelastic behavior of the polymer matrix on the mechanical behavior of thick multilayered cylinders. The proposed solution matched the exact solution for the pressurization of a compressive linear viscoelastic cylinder constrained by an elastic case. In addition, the viscoelastic effect on the timedependent internal stress field was demonstrated. Faria and Guedes [16] discussed the possibility of reducing the test duration of a glassfiber reinforced plastic (GFRP) piping system, while maintaining a final conclusion of longterm properties. Yoon and Oh [17] predicted the longterm performance of GRP pipes after 50 years based on failure pressure and time to failure using sustained internal pressure tests. Lavergne et al. [18] estimated the viscoelastic behavior and creep failure of Eglass fiber reinforced polyvinyl chloride (PVC). They used homogenization methods to obtained viscoelastic properties of PVC that reinforced of chopped glass fibers and random orientations. Poirette et al. [19] presented a new model that was based on the idea of a timedependent damage spectrum; the researchers performed quasistatic, creep, and fatigue tests to characterize the elastic, viscoelastic, and plastic behaviors of the composite material. Yian et al. [20] investigated the effect of seawater absorption on the longterm viscoelastic response of glass/Bismaleimide composites. Monfared [21] used an analytical method to predict the creep behavior of fibrous composites under a tensile axial stress; the model showed good agreements between the results from the analytical and finite element methods.
The strain distribution in the wall of a composite cylinder, especially in the longterm, and the effect of layups are very important. In this paper, a composite cylinder made of glass/vinylester with unidirectional fibers is subjected to internal pressure and studied using nonlinear viscoelastic approaches. Different theories and relations are used to investigate the longterm circumferential and radial strains in the cylinder wall, as well as the effect of the angle of the fibers on the distribution of the strains.
A laminated cylinder made of glass/vinylester is shown in Fig. 1. The layups are considered unidirectional for different orientations. The internal and external surfaces of the cylinder are insulated so that there is no loss of heat. In addition, the displacement conditions in the internal and external boundaries are free–free and the cylinder is closed at both ends.
Figure 1. A composite cylinder with unidirectional fibers
After solving the homogeneous and particular solutions of the differential equation, the radial displacement relation for every layer can be written as follows:
(1) 
where and are the axial and circumferential stresses, respectively. Considering the displacementstrain relations with the axial symmetry assumptions and the linear strain relation, the following equations are obtained:
(2) 
where , and are the radial, circumferential, and axial strains, respectively, and is the displacement in the radial direction. The Cartesian coordinates for the stress–strain relations in a unidirectional composite are:
(3) 
where , and 3 denote the different parameters in the , and directions, respectively, and the superscript specifies the creep strain. In addition, is the modulus matrix with a nonlinear viscoelastic emphasis.
To convert the matrix from Cartesian coordinates to cylindrical coordinates, a transfer matrix is used, which is defined as follows [22]:
(4)

=

where , and is the fiber angle. Furthermore, the stress–strain relations in cylindrical coordinates can be obtained as:
(5) 
By substituting Equation (2) into Equation (5), the stressdisplacement relations for every layer can be obtained as:
(6) 
where the superscript specifies the number of layers. By substituting Equation (6) into Equation (1), the equilibrium equation for each layer can be considered as follows:
(7) 
This equation has homogeneous and particular solutions.
(8) 
Using linear elastic relations and axial symmetry assumptions, the equilibrium equation can be described as:
(9) 
To solve Equation (9), the creep strain values and the constant coefficients and are needed. To calculate the constant coefficients, the boundary conditions can be considered as follows:
(10) 
for and
To obtain the creep strain parameters, the relation between the strains and the current stresses and the uniaxial creep behavior is based on the Prandtl–Reuss equations:
(11) 
where is the amount of octahedral stress:
(12) 
In addition, is the creep strain obtained with the Schapery’s single integral model at a constant temperature ( ) using the power method [23].
(13) 
Using the expressed relations and Mendelson’s approximation method [24] for a long period of time, the history of strains in time can be calculated. In this method, the thickness of the cylinder is divided into K divisions, as shown in Fig. 2.
It is assumed that the total time is sum of time increments throughout the progress of the creep process. For example:
(14) 
In addition, the values for and as the primary circumferential and radial strains, are considered for all divisions. In the present study, initially and . The strain for each time intervals is calculated by adding the previous strains with the new strain for all the thickness points of the cylinder. Ghasemi et al. [25] explained the numerical procedure.
Using the assumptions considered in the previous sections about creep solution and the nonlinear viscoelastic solution, the strain distribution in the cylinder wall for different time intervals is obtained. Table 1 shows the properties of a composite cylinder made of glass/vinylester with orthotropic properties using the Schapery coefficients for glass/vinylester.
Figure 2. Division of the cylinder wall
Table 1. Elastic properties of the glass/vinylester composite cylinder [12]
Poisson’s ratio 

Modulus (MPa) 

_{23} 
_{12}= _{13} 

G_{23} 
G_{12}=G_{13} 
E_{22}=E_{33} 
E_{11} 
0.35 
0.31 

3700 
4572 
13210 
19268 
Furthermore, the composite cylinder is unidirectional at different orientations of ; the cylinder is hollow with an interior radius , outer radius and interior pressure .
Because the nonlinear parameters depend on the stress, to calculate these parameters in the Schapery constitutive equation for the reference temperature (25°C), the following equation can be used for the glass/vinylester composite [12]:
(15) 
The linear parameters in Equation (13) are shown in Table 2 for different angles [14].
3.1. Solution of the TimeDependent Creep Strain Distribution
In this section, timedependent creep behavior is studied for a unidirectional composite cylinder under internal pressure. To determine the composite behavior, the composite is considered to be nonlinear viscoelastic material in Schapery’s single integral method, which is a function of temperature, time, stress, and fiber layup. However, all analyses are considered for ambient temperature. The timedependent Mendelssohn numerical approximation method is used to estimate the longterm behavior. Figs. 3 and 4 show the radial and circumferential creep strain distribution in a composite cylinder wall under internal pressure for different fiber orientations. In Figs. 3 and 4, the diagrams are almost parallel to each other. Moreover, the values of the radial and circumferential strains strongly converge over time.
The variations in circumferential creep strain in the cylinder wall at a fiber orientation of from 5 to 30 years are shown in Fig. 3 (a). The values of the circumferential creep strain on the inner surfaces vary from 1.31% to 1.98%. As the time increase, slop of circumferential creep strain reduced. The values of the circumferential creep strain on the outer surface are less than on the inner surface. In Fig. 3 (b), the values of the radial creep strains vary on the inner surface from –0.97% to –1.33%. As depicted, the slope of the strain variations increases from 5 years to 30 years.
Table 2. Viscoelastic linear parameters of glass/vinylester composite
90̂° 
45° 
Offaxis angle ( ) 
0.81 
0.79 
* 
1.35 
0.16 
* 
0.189 
0.20 
n 
(a)
(b)
Figure 3. Diagram of circumferential creep strain (a) and radial creep strain (b) for the composite cylinder under internal pressure of the arrangement with the angel of
And the absolute values of the radial creep strain increased with time. In addition, the values of the radial creep strain on the outer surface vary from –0.597% to –0.81%.
The variations of the circumferential creep strain in the cylinder wall at a fiber orientation of are shown in Fig.4 (a). The values of the strain range from 0.23% to 0.27% on the inner surface. The maximum and minimum strains on the inner surface occur at 30 and 5 years, respectively. Furthermore, the values of the strain gradually decrease over the years; the values of the strain on the outer surface vary from 0.15% to 0.21%.
The values of the radial strain, which are shown in Fig 4 (b), vary from –0.19% to –0.25% on the inner surface. According to the figure, the absolute values of the strain decrease from the inner surface toward the outer surface and vary from –0.22% to –0.29%.
To clarify the extent of the changes in creep strains, the values of the radial and circumferential creep strains on the inner, middle, and outer wall of the cylinder with fiber angles of and are shown in Table 3 and Table 4, respectively.
(a)
(b)
Figure 4. Diagram of circumferential creep strain (a) and radial creep strain (b) for the composite cylinder under internal pressure of the arrangement with the angel of
Table 3. Values of radial and circumferential creep strain in the walls of a cylinder with a fiber orientation of
Creep strain 
Radius ratio 

After the number of years 




5 
10 
15 
20 
30 

0.833 
0.9798 
1.1050 
1.1860 
1.2410 
1.3360 

0.9083 
0.7753 
0.8743 
0.9378 
0.9856 
1.0750 

1.000 
0.5987 
0.6751 
0.7240 
0.7608 
0.8154 

(%) 
0.833 
1.3130 
1.4800 
1.5870 
1.8770 
1.9880 

0.9083 
1.0820 
1.2200 
1.3080 
1.374 
1.4730 

1.000 
0.8827 
0.9952 
1.0670 
1.1210 
1.2020 

Table 4. Values of radial and circumferential creep strain in the walls of a cylinder with a fiber orientation of
Creep strain 
Radius ratio 

After the number of years 




5 
10 
15 
20 
30 

0.8333 
0.1913 
0.2148 
0.2301 
0.2417 
0.2592 

0.9083 
0.1532 
0.1791 
0.1840 
0.1913 
0.2072 

1.000 
0.1196 
0.1341 
0.1435 
0.1507 
0.1615 

(%) 
0.833 
0.2317 
0.2600 
0.2785 
0.2926 
0.3138 

0.9083 
0.1924 
0.2159 
0.2311 
0.2427 
0.2602 

1.000 
0.1578 
0.1769 
0.1893 
0.1988 
0.2131 

3.2. Validation of study
The present study has not been founded on any research about the timedependent creep strain in laminated composite or pressure vessels. You et al. [26] studied the steadystate creep in FGM cylindrical vessels under internal pressure and in homogeneous radially polarized anisotropic piezoelectric material. Fig. 5 shows the dimensionless radial stresses over 10 years in the wall of a cylinder with a fiber angle of , which are based on the solution by You et al. [26] for the case of a zero material constant (i.e., homogeneous material). Considering the very similar radial stresses obtained in both approaches, the correctness of the present solution may be verified in this respect.
In this study, the distribution of longterm creep strain in a composite cylinder under internal pressure was investigated for different fiber orientations. Nonlinear viscoelastic approaches were used to study the distribution of the stresses and longterm creep strain. The distribution of longterm circumferential and radial strains in the cylinder wall and the effect of the fiber orientations on the distribution were considered using Schapery’s single integral model, Prandtl–Reuss relation, and Mendelson’s approximation method.
Figure 5. Dimensionless radial stress variations in the wall of a composite cylinder
The obtained results indicate that the parameters of fiber orientation, dimension radius, and time have considerable effects on the distribution of longterm creep strain in a composite cylinder
The results show that the values of creep strain for a fiber angle of are less than at a fiber angle of . The values of creep strain at a specific fiber angle over time indicate that as time increases, the absolute values of the creep strain increase. The values of the creep strains decrease from the inner surface toward the outer surface.
References
[1] Arefi M. Thermoelastic analysis of a rotating hollow cylinder made of arbitrary functionally graded materials. J Theorical and Applied Mechanic 2015; 45: 41–60.
[2] Loghman A, Nasr M and Arefi M. Nonsymmetric thermomechanical analysis of a functionally graded cylinder subjected to mechanical, thermal, and magnetic loads. J of Thermal Stresses 2017; 40(6): 765782.
[3] Tsukrov I and Drach B. Elastic deformation of composite cylinders with cylindrical orthotropic layers. Int J Solids and Structure 2010; 47: 2533.
[4] Zhang Q, Wang ZW, Tang CY, Hu DP, Liu PQ and Xia LZ. Analytical solution of the thermomechanical stresses in a multilayered composite pressure vessel considering the influence of the closed ends. Int J Pressure Vessels and Piping 2012; 98: 102110.
[5] Ghasemi AR, Kazemian A and Moradi M. Analytical and numerical investigation of FGM pressure vessel reinforced by laminated composite materials. J Solid Mechanic 2014; 6(1): 4353.
[6] Arefi M, Koohi Faegh R and Loghman A. The effect of axially variable thermal and mechanical loads on the 2D thermoelastic response of FG cylindrical shell. J Thermal Stresses 2016; 39(12): 15391559
[7] Arefi M, Abbasi AR and Vaziri Sereshk MR. Two dimensional thermoelastic analysis of FG cylindrical shell resting on the Pasternak foundation subjected to mechanical and thermal loads based on FSDT formulation. J Thermal Stresses 2016; 39: 554570
[8] Violette MG and Schapery RA. Timedependent compressive strength of unidirectional viscoelastic composite materials. Mechanic TimeDependent Material 2002; 6: 133145.
[9] Chio Y and Yuan RL. Timedependent deformation of pultruded fiber reinforced polymer composite columns. J Composite for Construct 2003; 7(4): 356362.
[10] Findley WN. Mechanism and mechanic of creep of plastic. J Polymer Engineerin 1960; 16: 5765.
[11] Papanicolaou GC, Zaoutsos Sp and Kontou EA. Fiber orientation dependence of continuous carbon/epoxy composites nonlinear viscoelastic behavior. Composite Science and Technology 2004; 64: 25352545.
[12] Muliana A, Nair A, Khan KA and Wagner S. Characterization of thermomechanical and long term behaviors of multilayered composite materials. Composite Science and Technology 2006; 66: 29072924.
[13] Sawant S and Muliana A. A thermomechanical viscoelastic analysis of orthotropic materials. Composite Structres 2008; 83: 6172.
[14] Muddasani M, Sawant S and Muliana A. Thermoviscoelastic responses of multilayered polymer composite experimental and numerical studies. Composite Structures 2010; 92: 26412652.
[15] Guedes RM. Nonlinear viscoelastic analysis of thickWalled cylindrical composite pipes. Int J Mechanic Science 2010; 52: 10641073.
[16] Faria H and Guedes RM. Longterm behaviour of GFRP pipes: Reducing the prediction test duration. Polymer Testing 2010; 29: 337–345.
[17] Yoon SH and Oh JO. Prediction of long term performance for GRP pipes under sustained internal pressure. Composite Structure 2015; 134: 185–189
[18] Lavergne F, Sab K, Sanahuja J, Bornert M and Toulemonde C. Estimation of creep strain and creep failure of a glass reinforced plastic by semianalytical methods and 3D numerical simulations. Mechanic of Materials 2015; 89: 130–150.
[19] Poirette Y, Dominique P and Frédéric T. A contribution to timedependent damage modeling of composite structures. Appiled Composite Materials 2014; 21(4): 677688.
[20] Yian Z, Zhiying W, Keey SL and Boay CG. Longterm viscoelastic response of Eglass/Bismaleimide composite in seawater environment. Appied Composite Materials 2015; 22(6): 693709.
[21] Monfared V. Circular functions based comprehensive analysis of plastic creep deformations in the fiber reinforced composites. Appied Composite Materials 2016:113.
[22] Xia M, Takayanagi H and Kemmochi K. Analysis of multilayered filamentwound composite pipes under internal pressure. Composite Structure 2001; 53: 483491.
[23] Lou YC, Schapery RA, Viscoelastic characterization of a nonlinear fiberreinforced plastic. J Composite Materials 1971; 5:208234.
[24] Mendelson A. Plasticity: theory and applications. New York: Macmillan. 1968.
[25] Ghasemi AR, Hosseinpour K, Mohandes M. Modelling creep behavior of carbonnanotubes/fiber/polymer composite cylinders. Part N: J Nanomaterials, Nanoengineering and Nanosystems 2017, DOI: 10.1177/2397791418768576
[26] You H, Ou H, Zheng ZY. Creep deformation and stresses in thickwalled cylindrical vessels of functionally graded materials subjected to internal pressure. Composite Structures 2007; 78: 285–91.