Elasto-plastic analysis and finite element simulation of thick-walled functionally graded cylinder subjected to combined pressure and thermal loading

1 Introduction

Functionally graded materials (FGMs) are the new generation of alloys in which the material properties are varied continuously through the thickness or radial directions with gradual changes in the volume fraction of their constituents. In the FGMs, The continuing change of material properties can be tailored to meet the requirements of different uses and working environments. In recent years, FGMs are being used increasingly in aerospace and many engineering applications [1, 2].

The FGMs were originally designed as thermal barrier materials for aerospace structures [3, 4]. Typical applications of FGMs include coatings for protection of metallic or ceramic substrates against heat penetration, wear, corrosion and oxidation [5]. Kieback et al. [6] provided different processing techniques for FGMs. They explained different powder processes and techniques considering recent developments in the field of graded polymer processing. As the use of FGMs increases, new methodologies have to be developed to characterize them as well as to analyze and design the structural components made of these materials.

Having investigated the literature of functionally graded (FG) cylinders, it is known that there are a number of studies dealing with the stress analysis of cylindrical FG vessels subjected to pressure and temperature gradient loadings. Fukui and Yamanaka [7] examined the effects of gradation of components on the strength and deformation of thick-walled FG tubes under internal pressure in the case of the plane stress state. Using the infinitesimal theory of elasticity, Tutuncu and Ozturk [8] provided closed-form solutions for stresses and displacements in FG cylindrical vessels subjected to internal pressure. They assumed that material stiffness obeys a power law through the wall thickness while the Poisson’s ratio remains constant. Liew et al. [9] presented an analysis of thermal stress behavior of FG circular cylinders. Assuming the state of the plane strain, they used the exponential volume fraction model and developed an analytical solution to deal with FG cylinders that are subjected to an arbitrary steady-state or transient temperature field. Jabbari et al. [10] developed a general elastic analysis of one-dimensional steady-state stresses in a thick-walled FG cylinder. They assumed the temperature distribution to be a function of the radius and used the direct method to solve heat conduction and Navier equations.

Tarn [11] determined exact solutions for the temperature distribution, thermoelastic deformations and stress fields in inhomogeneous anisotropic hollow and solid cylinders subjected to an axial force and a torque at the ends and the surface loads, which may vary circumferentially but not axially. Oral and Anlas [12] investigated the stress distribution in a nonhomogeneous anisotropic FG cylindrical body. They obtained closed-form solutions for stress potentials and stress distribution for an axisymmetric and orthotropic FG cylinder. Based on the theory of laminated composites and using a multilayered approach, Shao [13] presented the solutions of temperature, displacements and thermal/mechanical stresses in an FG circular hollow cylinder which has finite length and is subjected to axisymmetric thermal and mechanical loads. Shao and Ma [14] carried out the thermomechanical analysis of a FG hollow circular cylinder subjected to mechanical loads and linearly increasing boundary temperature.

Abrinia et al. [15] determined an analytical solution for computing the radial and circumferential stresses in an FG thick cylindrical vessel under the influence of internal pressure, temperature and combined loadings. Changing dimensionless parameters of analysis, they concluded that the properties of an FG thick cylindrical vessel could be modified and the lowest stress levels could be reached. Nie et al. [16] presented a technique to tailor materials for linear elastic FG cylinders and spheres in order to attain through-the-thickness either constant circumferential or in-plane shear stresses. They found the required radial variation of the volume fractions of constituents to make a linear combination of the radial and the circumferential stresses throughout the thickness. Based on the third-order shear deformation plate theory of Reddy, Mehrabadi and Aragh [17] studied the thermoelastic response of a two-dimensional functionally graded open cylindrical shell with temperature-dependent material properties.

As the yielding in the metallic part can significantly affect the stress distribution within the metal-ceramic FGMs, the stress analysis based only on elastic material behavior will not be exact [18]. To consider both elastic and plastic effects, Shabana and Noda [19] obtained the elasto-plastic thermal stresses in a FG rectangular plate subjected to different kinds of temperature conditions using the finite element method. Aboudi et al. [20] presented a higher-order micro-mechanical theory for thermoelastic/thermoplastic problems in order to obtain the response of materials FG in two directions. They concluded that the efficiency of two-dimensional FGMs under such separate thermal loading conditions is considerable. Cho and Choi [21] introduced a yield-criterion optimization of the volume fraction distribution in metal-ceramic FGMs. They combined the peak effective stress and the total strain energy to define an objective function and did the volume fraction optimization.

Akis and Eraslan [22] provided the elastic stress distributions in rotating FG hollow shafts with an emphasis on the onset of material yielding. They developed a generalized plane strain analytical model for this purpose and used the Tresca’s yield criterion to monitor the commencement of plastic deformation. Based on Tresca’s yield criterion, Ozturk and Gulgec [23] derived elastic-plastic deformation of a solid cylinder with fixed ends made of FGM and imposed to uniform internal heat generation. Supposing the elastic perfectly plastic materials, they obtained the stress distribution as well as the development of plastic region radius. Parvizi et al. [24] obtained an analytical elastic-plastic solution for thick-walled cylinders made of FGMs subjected to internal pressure and thermal loading. They showed that under the temperature gradient loading, there is a point in the FG cylinder where the circumferential stress changes from compressive to tensile. Sadeghian and Toussi [25] analyzed an axisymmetric thermal elasto-plastic stress in cylindrical vessels made of FGM. Based on Tresca’s yield criterion, they introduced the separate distribution of elastic and plastic stresses and deformation for an inhomogeneous cylinder.

Investigating the literature reveals that no research concerning the elasto-plastic analysis of FG cylindrical vessels under the influence of combined pressure and temperature loadings has been reported. In this paper, exact closed-form analytical elasto-plastic solution for a thick-walled FG cylinder subjected to combined pressure and temperature gradient loading is presented. For this purpose, a mathematical model based on the experimental yielding results of the Al A359/SiCp FG material [24] is developed for a radially graded cylinder that is bonded at both ends. Using the Tresca yield criterion and elastic perfectly plastic behavior of the material, different combinations of pressure and temperature necessary to start yielding are determined. Also, the relations given the plastic zone radii as well as the radial, circumferential, longitudinal and effective stresses in both elastic and plastic zones are obtained for a combined loading. Furthermore, the FG cylindrical vessel under pressure and temperature gradient loadings is simulated in the ABAQUS/Explicit software (University of Tehran, Tehran, Iran). Making a comparison of the present theoretical results with those from finite element simulation has proved the accuracy of the present analysis.

2 FGM modeling

Power law [11, 26] and volume fraction [27, 28] are two models which were presented in the literature to describe the properties variation in FGMs. Similar to our previous study [24], the experimental data of Rodríguez-Castro et al. [29] for Al A359/SiCp FG material with different SiC volume fractions is considered in this study. Hence, the analogous mathematical model based on the experimental results of Rodríguez-Castro et al. [29] is used. Therefore, the elastic modulus and yield strength variation through the thickness of the FG cylinder are modelled with the power law as follows:

where r is radius of the cylinder and can be obtained from Eq. (2) of [24]. In addition, for a specific radius, the constants n₁, n₄, E₀ and σy0 can be obtained from Table 1 of [24].

Moreover, other properties such as coefficients of linear thermal expansion and thermal conductivity are described by power law functions of the radial coordinate as:

Due to lack of data for α and K at the inner and outer surfaces of the cylinder in the results of Rodríguez-Castro et al. [29], the rule of mixture is used to determine α and K as follows:

where P(r) denotes a generic material property.

3 Theoretical formulation

3.1 Governing equation

As shown in Figure 1, consider an axisymmetric thick-walled FG cylinder with inner radius a and outer radius b. The cylindrical coordinate system (r, θ, z) is used and the axial symmetry in the geometry and loading is assumed. The equilibrium equation, the linear relations between strain and displacement components and the compatibility equation are the same as Eqs. (5), (6) and (7) in [24], respectively. It is assumed that the cylinder is bonded on both ends. Consequently, the linear constitutive thermoelastic equations in the cylindrical coordinate system are as follows:

Figure 1:

Configuration of the thick-walled FG cylinder.

(4)εr=1E(r)[(1-υ2)σr-υ(1+υ)σθ]+(1+υ)α(r)ΔT(r)εθ=1E(r)[-υ(1+υ)σr+(1−υ2)σθ]+(1+υ)α(r)ΔT(r) (4)

4 Finite element simulation

In Figure 2, an finite element (FE) model of an FG cylinder bonded at both ends is shown. Having some simplification in modeling of the FG cylinder, material in the radial direction is divided into 10 layers. The material properties in each layer are supposed to be constant, but vary from one layer to the next one in the radial direction according to Eqs. (1) and (2). The quantities of 29520 3D CAX4T type elements are used to accomplish the meshing of the vessel. The coupled temperature-displacement boundary conditions are applied in the finite element method (FEM) model.

Figure 2:

The finite element model of the FG cylinder.

5 Results and discussion

As a case study, a thick-walled FG cylindrical vessel with the specific properties which are mentioned in Table 1 is considered. Similar to our previous study [24], these values are extracted based on the experimental data of Rodríguez-Castro et al. [29]. The results are presented for an FG cylinder A359 aluminum alloy matrix reinforced with 30% silicon carbide particles at the inner surface and 20% silicon carbide particles at the outer surface. The Poisson’s ratio is supposed to be constant and equal to 0.3.

First, the radial, circumferential and effective stresses in the elastic FG cylinder subjected to pressure, temperature gradient and combined loadings are presented. Next, different combinations of the pressure and temperature which lead to onset of yielding are shown. Later, different values of the combined pressure and temperature loadings leading to the specific elastic-plastic boundary radius are demonstrated. After that, the distribution of radial, circumferential and effective stresses in the elasto-plastic FG cylinder subjected to different combined pressure and temperature gradient loadings are shown. Finally, the result of present analysis for FG cylinders under combined pressure and temperature gradient loadings are compared with those from FEM simulation.

In Figure 3, the distribution of radial, circumferential and effective stresses in the elastic part of an FG cylinder subjected to pressure, temperature gradient and combined pressure and temperature gradient loadings are presented. It can be seen that up to almost the middle radius of the cylinder, the pressure has a further effect on the radial stress, while in the second half, the opposite condition exists. In addition, the radial stresses based on the pressure and temperature loadings are compressive in any point in the FG cylinder and reach zero in the outer surface. Furthermore, the variations trend of circumferential stresses through the radial direction is mostly similar to the temperature gradient stress component. Note that the maximum effective stress for combined loading condition happens at the outer radius of the FG cylinder. Similarly, it can also be seen that the effective stress has just about the same trend as the stress component according to the temperature gradient loading.

Figure 3:

The (A) radial, (B) circumferential and (C) effective stresses in an FG cylinder subjected to pressure, temperature and combined pressure and temperature gradient loadings (p_a=40 MPa, p_b=0, T_a=400°K, T_b=300°K).

Different combinations of the pressure and temperature loadings that lead to onset of yielding at the outer radius of the FG cylinder are presented in Figure 4. Considering Eq. (21), the relation between pressure and temperature gradient to start yielding in the FG cylinder is essentially nonlinear. However, for the assumed domain of pressure and temperature gradient in Figure 4, the relation has approximately a linear trend, while by increasing the critical temperature loading, the critical pressure is reduced. In the case of combined loading, the subsequent stresses are obtained as a result of stresses from both temperature gradient and pressure loadings. Therefore, a higher temperature gradient requires a lower pressure loading to reach a yielding condition and vice versa.

Figure 4:

Different combinations of the critical pressure and temperature loadings lead to onset of yielding at outer radius of the FG cylinder.

Having investigated the effects of temperature gradient loading on the stress field in the FG cylinder under combined loading, distributions of the radial, circumferential and effective stresses in the FG cylinder subjected to different quantities of temperature gradient are presented in Figure 5. The pressures at the inner and the outer surfaces of the FG cylinder have been considered to be constant and equal to p_a=40 Mp_a and p_b=0. It is interesting that for different values of temperature, the points with radii 0.145 and 0.135 have the common circumferential and effective stresses equal to 40 MPa and 60 MPa, respectively. It can also be seen in the effective stresses that there are some points with zero value in which the direction of the graphs is changed.

Figure 5:

Distribution of the (A) radial, (B) circumferential and (C) effective stresses in an FG cylinder subjected to different temperature gradients and constant p_a=40 MPa, p_b=0.

Similarly, distributions of the radial, circumferential and effective stresses in the FG cylinder subjected to different quantities of internal pressure are presented in Figure 6. The temperature gradient for this case has been considered to be constant and equal to ΔT=100°K. For this case also, there are some points in the radial direction that the effective stresses become zero.

Figure 6:

Distribution of the (A) radial, (B) circumferential and (C) effective stresses in an FG cylinder subjected to different quantities of internal pressure and constant T_a=400°K, T_b=300°K.

Based on different combinations of pressure and temperature gradient loadings and using a three-dimensional mesh format and contour plot, the boundary radius of elasto-plastic zones for an FG cylinder with 0.1 m inner and 0.2 m outer radii are shown in Figure 7. In this study, the temperature is changed from 300°K to 700°K and the pressure from 0 MPa to 140 MPa. It is clear that the yielding first occurred at the outer layer and spread towards the inner layer by increasing the temperature and/or pressure loading. Moreover, at the start of yielding, both pressure and temperature gradient impact on the initiation of plastic layer and its spread. By contrast, having developed the plastic zone towards the inner surface of the FG cylinder, the impact of temperature loading on the position of elasto-plastic boundary (r_p) decreases significantly. In other words, for a constant temperature, the position of elasto-plastic boundary (r_p) is gradually changed by pressure. However, for a constant pressure, there is very slight change in the position of elastic-plastic boundary (r_p) with respect to temperature loading.

Figure 7:

Boundary radius of elasto-plastic zones for different values of the combined pressure and temperature loadings (A) three-dimensional mesh and (B) contour plot.

In Figure 8, the elasto-plastic radial, circumferential and effective stresses in the FG cylinder subjected to combined pressure and temperature gradient loadings are shown. The dashed-lines are used to represent the elastic region and the solid lines represent the plastic region.

Figure 8:

Elasto-plastic (A) radial, (B) circumferential and (C) effective stresses in an FG cylinder subjected to combined pressure and temperature gradient loadings p_a=155 MPa, p_b=0 and ΔT=60, 70, 80°K.

In Figure 9, comparisons of the present analytical results with the FE results of ABAQUS simulation are shown. It can be seen that the analytical results have a good agreement with those from FE simulation.

Figure 9:

The radial stresses in an FG cylinder subjected to (A) internal pressure p_a=118 MPa, (B) temperature gradient ΔT=290°K and (C) combined loading of p_a=30 MPa, ΔT=100°K.

6 Conclusion

An exact closed-form analytical elasto-plastic solution for a thick-walled FG cylinder subjected to combined pressure and temperature gradient loadings is presented for the first time. For any combined loading condition, the plastic zone radius is determined and the radial, circumferential, longitudinal and effective stresses in both elastic and plastic zones are calculated. In addition, the FG cylinder is simulated using ABAQUS/Explicit software. It is concluded that the variations of circumferential and effective elastic stresses through the radial direction have mostly a similar trend with the temperature gradient component of the stress. It is observed that the relation between the critical pressure and temperature that lead to the onset of yielding is approximately linear. For this case study, it is also concluded that onset of yielding happened from the outside surface and by developing the plastic zone towards the inner surface of the FG cylinder, the effect of the temperature gradient on the position of the elasto-plastic boundary radius decreases considerably. The results from the present theoretical analysis of the FG cylinder are compared with those of the finite element simulation and good agreements are observed between them.