Yielding onset in FGM thick-walled spherical shells under thermo-mechanical loading

Abstract

Based on von Mises yield criterion, a thermoelasto-plastic analysis of a thick-walled spherical shell composed of functionally graded material (FGM) subjected to internal pressure and thermal gradient is conducted in order to accurately predict the onset radius of yielding within the vessel wall. The modulus of elasticity, thermal conductivity and thermal expansion coefficients, and yield stress of FGM are assumed to follow power-law functions in radius according to Erdogan’s model. The equilibrium differential equation of the shell under thermo-mechanical loading in steady-state conduction heat transfer are derived analytically and then solved to determine through-thickness variations of the radial and circumferential stresses and radial displacement in elastic and perfectly plastic zones in the vessel wall. The presented approach leads to the definition of new formulation to predict the onset radius of yielding. Moreover, a neural network (NN) solution to reliable quantify the influence of all uncertain input parameters with respect to uncertain output parameter is utilized. The numerical results showed that for various FGM parameters and specific thermal gradient, the plastic zone can commence from inside radius, outside radius, simultaneously in both radii, or may be launched in the intermediate radius of the spherical shell wall.

Keywords

Functionally graded material thick-walled spherical shell neural network solution thermal gradient thermoelasto-plastic analysis yielding onset

Introduction

Thin and thick-walled spherical shells are widely used in different engineering structures. The real engineering applications of the hollow spherical vessels can be addressed in the oil, gas and petrochemical, marine and aerospace industries such as high pressurized vessels operating in low and/or high temperature working conditions for storing liquid and gaseous types of materials like LPG, LNG, ammonia, etc., explorer submarines working in highly deep seas/oceans, unmanned or manned atmospheric re-entry heat shielded capsules, and pressure vessels working in the cryogenic low temperature/freezing conditions. Commonly, these hollow spherical vessels are under applied internal or external pressures or both and exposed to highly cold or hot temperatures in their inside or outside surfaces. These hollow spheres are mostly made up of isotropic materials such as carbon or stainless steels or may be composed of modern materials like super alloys or novel nonhomogeneous functionally gradient materials. The FGM hollow spheres/tubes can tolerate applied thermo-mechanical loads in appropriate fashion in highly hot or cold operating conditions. The FGM hollow pressurized vessels are able to withstand the severe temperature gradient in their shell wall. Moreover, the FGMs have high fracture strength, fracture toughness, corrosion resistance against fluids in contact with the exterior surfaces of the shell and erosion resistance as well. The physical and mechanical properties of FGM based shell type structures can gradually change continuously along any directions between the side wall surfaces of the shell. This characteristic of FGM properties can be modeled as functions along specific directions of the shell type structure. Because of spherical symmetry of the hollow sphere the variations of physical and mechanical properties of nonhomogeneous functionally gradient materials can be considered in radial direction of the vessel wall. The hollow spheres subjected to the thermo-mechanical loads may experience thermal stresses beyond the yield stress in which through-thickness layers of the vessel are kept in plastic zone. So, the thermoelasto-plastic analysis of these hollow spheres is really a crucial issue from both the structural analysis and engineering design points of view. From these points of view, this study has been initiated. Below, some of the research works conducted in the field of elasto-plastic analysis of hollow spherical vessels are reviewed.

In Ref. 1, elastic analysis of thick-walled spherical vessel made up of two different kinds of FGM under influence of internal pressure has been utilized. One hollow sphere consisted of two homogeneous layers near the inner and outer wall of the vessel and one FG layer in the middle; the other consisted of the FGM only. Analytical solution for an FGM thick-walled sphere exposed to one-dimensional steady-state thermal conduction in radial direction was performed in Ref. 2 and radial relations for distribution of radial displacement, radial, and hoop stresses were obtained. In Ref. 3, thermo-mechanical buckling analysis of thin shallow spherical shells made up of temperature-dependent for a metal-ceramic FGM for various types of loading was performed. The governing equations were derived using the first-order shell theory of Love and Kirchhoff and the Donnell–Mushtari–Vlasov kinematics relations. Thermo-mechanical analysis of an FG hollow sphere with nonlinearly variation for the material properties in radial direction subjected to internal pressure and temperature gradient along the shell wall was carried out in Ref. 4. In Ref. 5, thermo-elastic analysis of an FG hollow sphere was carried out and numerical solutions for radial displacement, stress components and thermal field were obtained using the polynomial differential quadrature (PDQ) method. The elasto-plastic analysis of a thick-walled FGM spherical pressure vessel under internal pressure and temperature difference was investigated in Ref. 6. Different combinations of internal pressure and temperature gradient loading were applied to study their effects on the yield onset radius and stress components. In Ref. 7, elastic analysis of a thick-walled spherical shell with an inner FGM coating subjected to internal and external hydrostatic pressures was performed using three-dimensional elasticity theory. A semi-analytical iterative method was used for elastic analysis of FGM thick-walled spherical shells under the effect of internal pressure in Ref. 8. For different values of inhomogeneity constant, distributions of radial displacement, radial and circumferential stresses, and von Mises equivalent stress were obtained. In Ref. 9, analytical elastic solutions for radial and circumferential stresses of cylindrical and spherical pressure vessels composed of FGMs were presented. In Ref. 10, the unsteady-state thermo-mechanical analysis of thick-walled spheres of FGMs with linearly time dependent temperature characteristics were studied using Laplace transform method. In Ref. 11, thermoelasto-plastic analysis of FGM thick-walled spherical storage tanks under the effect of internal pressure and temperature gradient was investigated. The von Mises yield criterion and elastic-perfectly-plastic (EPP) assumptions were used to define the material behavior in plastic zones. The patterns of plastic zones spreading for various combinations of internal pressure and positive or negative temperature gradient were investigated. In Ref. 12, based on von Mises yield criterion associated with the flow rules for perfectly plastic material behavior, thermoelasto-plastic analytical solution of FGM thick-walled spherical shells under one-dimensional heat conduction was presented while the inner and outer surfaces were exposed to a uniform heat flux and to an airstream, respectively. In Ref. 13, based on Kelvin–Voigt model, the effect of viscoelastic interfaces on thermo-mechanical behavior of a layered FGM spherical shell subjected to thermo-mechanical loading at the inner and outer surfaces was analytically investigated. In Ref. 14, based on small deformations and von Mises yield criterion, analytical solution for the yielding onset of two-layer composite spherical shells under either internal or external pressure was presented. In Ref. 15, based on classical laminate theory (CLT), thermo-mechanical nonlinear buckling and post-buckling analyses of clamped FGM shallow spherical vessels under uniform external pressure were studied with assumption of temperature independent for mechanical and thermo-mechanical properties. Governing equations were derived using the FSDT considering geometrical nonlinearity. In Ref. 16, the elasto-plastic analysis for a spherical shell under external pressure was presented. The radial and circumferential stresses were computed for the spherical shell for compressible and additionally incompressible materials based on Seth’s transition speculation of elastic-plastic transitions. Based on the spherically symmetric plane strain assumptions, thermo-mechanical analysis in a simple power-law graded spherical shell subjected to steady-state thermal and internal/external pressure loads was utilized in Ref. 17. In Ref. 18, a thick-walled spherical vessel of homogeneous material subjected to both internal and external pressure was analyzed and solution was obtained using Seth’s transition theory. In Ref. 19, using Tresca’s yield criterion elasto-plastic solution for thick-walled spherical shells with an inner FGM layer under internal and external pressures was developed analytically. The modulus of elasticity and the uniaxial yield limit of the FG coating layer were considered to vary nonlinearly through the thickness. In Ref. 20, based on the Mori–Tanaka material model for elastic material properties of the composite at the inner and external radius of the FGM spheres and with assumption of radially variation of yield stress according to the rule of mixture, the elasto-plastic analysis of thick-walled spherical shells coated internally with FGM under the effect of the temperature gradient and pressure was utilized. In Ref. 21, elastic and perfectly plastic thermal stress analyses for spherical FGM shells were performed and the effect of pressure and temperature upon the growth of plastic zone was studied. In Ref. 22, elasto-plastic analysis of FGM spherical shells with no temperature-dependent material properties under internal pressure without thermal gradient was conducted. It is worthy to be mentioned that several computational methods are capable of dealing with a much wider application range of structural analysis.^23–25 For example, in Ref. 23, a mesh-free method with assumption of finite strains and arbitrary evolving cracks in thin shell type of structures was employed for several elastic and elasto-plastic examples with no need to discretization of the director field. In Ref. 24, a deep neural networks (DNNs) scheme via machine learning method as an option for approximation in computational mechanics was utilized to solve PDEs of several problems with applications in engineering based on energy approach. In Ref. 25, to study the thin plate bending deflection, a deep collocation method (DCM) based on a feed-forward DNN algorithms involved in deep learning for governing partial differential equations (PDEs) was employed. Further studies close to the present topic can be found in Refs. 26–29. In Ref. 26, dynamic modeling and vibration analysis of bolted flange joint disk-drum structures (BFJDSs) employing the Kirchhoff thin plate, Sanders’ shell, and Euler–Bernoulli beam theories were utilized. Dynamic analysis of axially loaded thin-walled shaft-disk rotors exposed to a non-uniform temperature gradient with temperature-dependent material properties was investigated in Ref. 27. In Ref. 28, frequency response of spinning cylindrical shells with discontinuous boundary conditions and under both point and distributed harmonic loads was studied using a semi-analytical method. In Ref. 29, using FSDT for shells, the traveling wave free vibration analysis of a spinning functionally graded spherical-cylindrical-conical (SCC) shell with arbitrary boundary conditions subjected to thermal environment was performed to calculate the critical speed of the spinning shells. In Ref. 30, free vibration analysis of porous FGM cylindrical shells based on modified power-law formulation containing porosities under the effect of different thermal load types was performed.

This paper deals with a classical discretization on the derived governing differential equations for the thermoelasto-plastic radial displacement and radial and hoop stress components of a thick-walled spherical shell made up of nonhomogeneous FGM with different material parameters based on the Erdogan’s model under thermo-mechanical loads in the steady-state conduction heat transfer condition. To predict commence of the yielding radius and spread of the elastic and plastic regions, the von Mises yield criterion is used. Closed form solutions in elastic and fully plastic zones of thermoelasto-plastic analyses of a hollow spherical vessel under internal pressure and thermal gradient between inner and outer surfaces have been presented. Obtained results of numerical simulations in this study are validated and compared with the ones reported in the literature in special cases. Moreover, to avoid a classical discretization and to explain how we deal with uncertainties in the input parameters a variance based uncertainty analysis quantifying the influence of all uncertain input parameters with respect to all uncertain output parameters has been carried out. To do this, a neural network (NN) scheme which naturally takes care of the uncertainties has been employed in which is extremely efficient, routinely account for uncertainties and is a natural framework for inverse analysis. These issues can be considered as the novelty and main contribution of this study compared to the above reviewed works in which to the best of the author knowledge, there are no prior publications investigating thermoelasto-plastic analysis of FGM thick-walled spherical shells simultaneously with influence of temperature-dependent physical and mechanical properties of material and also accompanied with a NN scheme to reliable prediction of the value of yield onset variable under considered input parameters of applied loadings, FGM parameters, and geometry of hollow sphere. Thus, this paper endeavors to fill this void by conducting an in-depth investigation of these types of thick-walled spherical shells.

Problem definition

A thick-walled spherical shell made up of nonhomogeneous FGM with inside radius a, outside radius b, and thickness h in which h/a>1/20 is shown in Figure 1. The spherical coordinate system rθφ with origin O is located at the center of hollow sphere where r, θ, and φ are the radial, hoop (or circumferential), and meridional coordinates, respectively. The spherical vessel is subjected to internal pressure P_a and external pressure P_b. It is also assumed that the spherical shell is exposed to one-dimensional steady-state conduction heat transfer due to thermal gradient along the shell wall. Moreover, the shell undergoes thermal expansion in radial direction with temperature distribution of T(r) where both the inside and outside surfaces of hollow sphere are maintained at constant temperature T_a and T_b, respectively, in which T_a>T_b. In Figure 1, the radial, circumferential, and meridional stress components applied on a differential element of the shell are indicated with σ_r, σ_θ, and σ_φ, respectively. In addition, it is assumed that the physical and mechanical properties for the spherical shell of FGM including elastic modulus E, thermal expansion coefficient α, thermal conductivity coefficient k, and yield stress Y are nonlinearly varied in radial direction defined by power-law functions in terms of radial distance r based on the Erdogan’s model as follows^4,20,31:

E (r, T) = \bar{E} {(r / b)}^{m_{1}} = E_{0} r^{m_{1}},

(1)

α (r, T) = \bar{α} {(r / b)}^{m_{2}} = α_{0} r^{m_{2}},

(2)

k (r) = \bar{k} {(r / b)}^{m_{3}} = k_{0} r^{m_{3}},

(3)

Y (r) = \bar{Y} {(r / b)}^{m_{4}} = Y_{0} r^{m_{4}},

(4)

where

E_{0} = \bar{E} / b^{m_{1}}

α_{0} = \bar{α} / b^{m_{2}}

k_{0} = \bar{k} / b^{m_{3}}

, and

Y_{0} = \bar{Y} / b^{m_{4}}

are the referential parameters for the elastic modulus, thermal expansion coefficient, thermal conductivity, and yield stress of the outer surface of the hollow sphere, respectively. Moreover m₁, m₂, m₃, m₄, and ν are considered as temperature independent parameters. Furthermore, the value of Poisson’s ratio ν has been taken as constant because although the material is not homogenous but considering that all materials almost have a constant Poisson’s ratio value in the elastic range, which is a reasonable assumption.^2,4,5,7,31

Elastic analysis of FGM thick-walled spherical shell

Derivation of governing equations

Because of the spherical symmetry for the one-dimensional problem under study, the stress and strain components in the θ and φ directions are identical and the shearing stress and strain components are zero, that is^4,32–34

\begin{array}{l} σ_{θ} = σ_{φ}, σ_{r θ} = σ_{r φ} = σ_{θ φ} = 0, \\ ε_{θ} = ε_{φ}, ε_{r θ} = ε_{r φ} = ε_{θ φ} = 0, \end{array}

(5)

According to generalized Hooke’s law the alternate form of constitutive equations of linear elastic materials in spherical coordinate system to describe the thermo-elastic radial and hoop stress-strain relations for a hollow sphere with through thickness temperature distribution T(r) as added thermal load are⁴

σ_{r} = [λ (r) + 2 μ (r)] ε_{r} + 2 λ (r) ε_{θ} - β (r) α (r, T) T (r),

(6)

σ_{θ} (= σ_{φ}) = 2 [λ (r) + μ (r)] ε_{θ} + λ (r) ε_{r} - β (r) α (r, T) T (r),

(7)

in which

λ (r) = \frac{ν E (r)}{(1 + ν) (1 - 2 ν)}, μ (r) = \frac{E (r)}{2 (1 + ν)}, β (r) = 3 λ (r) + 2 μ (r) = \frac{E (r)}{1 - 2 ν},

(8)

where λ and μ are Lame’s constants.⁴ For symmetric problems in spherical coordinates, the radial and circumferential components of strain-displacement relations are³²

ε_{r} (r) = \frac{d u (r)}{d r},

(9)

ε_{θ} (r) = ε_{φ} (r) = \frac{u (r)}{r},

(10)

in which u(r) is radial displacement of any point in the shell wall. Neglecting the body force and due to symmetry of the geometry and loading the governing equilibrium differential equation is written as follows³²:

\frac{d σ_{r}}{d r} + \frac{2 (σ_{r} - σ_{θ})}{r} = 0,

(11)

It is to be pointed out that the equilibrium equation (11) is always valid in both elastic and plastic zones. By substituting equations (1), (2), and (8)–(10) into equations (6) and (7), the following relations for the stress components are obtained

σ_{r} = [\frac{ν E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)} + \frac{E_{0} r^{m_{1}}}{(1 + ν)}] \frac{d u}{d r} + \frac{2 ν E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)} \frac{u}{r} - [\frac{3 ν E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)} + \frac{E_{0} r^{m_{1}}}{(1 + ν)}] α_{0} r^{m_{2}} T (r),

(12)

σ_{θ} (= σ_{φ}) = [\frac{2 ν E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)} + \frac{E_{0} r^{m_{1}}}{(1 + ν)}] \frac{u}{r} + [\frac{ν E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)}] \frac{d u}{d r} - [\frac{3 ν E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)} + \frac{E_{0} r^{m_{1}}}{(1 + ν)}] α_{0} r^{m_{2}} T (r),

(13)

By differentiating the relation (12) with respect to r and then substituting the obtained result along with equations (12) and (13) into equation (11) and doing some mathematical simplifications, the governing second order ordinary differential equation (ODE) in terms of radial displacement is obtained as follows:

r^{2} \frac{d^{2} u}{d r^{2}} + A r \frac{d u}{d r} + B u = \frac{1 + ν}{1 - ν} α_{0} (T) r^{m_{2}} [(m_{1} + m_{2}) r T (r) + r^{2} \frac{d T (r)}{d r}],

(14)

in which constant coefficients A and B are

A = m_{1} + 2, B = 2 [ν (m_{1} - 1) - (1 - 2 ν)] / (1 - ν),

(15)

Thermal analysis of FGM thick-walled spherical shell

The governing equation for the steady-state conduction heat transfer in radial direction for a spherical shell is expressed as^21,31,32

\frac{1}{r^{2}} \frac{\partial}{\partial r} [k (r) r^{2} \frac{\partial T (r)}{\partial r}] = 0,

(16)

in which equation (16) can be rewritten as

\frac{d^{2} T (r)}{d r^{2}} + [\frac{d k (r) / d r}{k (r)} + \frac{2}{r}] \frac{d T (r)}{d r} = 0,

(17)

Thermal boundary conditions for a hollow sphere exposed to temperature gradient between the inner surface at r = a and the outer surface at r = b are

D_{11} {\frac{d T (r)}{d r} |}_{r = a} + D_{12} T (a) = t_{1},

(18)

D_{21} {\frac{d T (r)}{d r} |}_{r = b} + D_{22} T (b) = t_{2},

(19)

in which D₁₁ and D₂₁ are conduction heat transfer coefficients, D₁₂ and D₂₂ are convection heat transfer coefficients and t₁ and t₂ are specific constants in the inside and outside radii, respectively. Substituting equation (3) into equation (17) yields

\frac{d^{2} T (r)}{d r^{2}} + (\frac{m_{3} + 2}{r}) \frac{d T (r)}{d r} = 0,

(20)

The general solution of the second order ODE in equation (20) is as follows:

T (r) = D_{1} r^{- (m_{3} + 1)} + D_{2},

(21)

By substituting equation (21) into equations (18) and (19), the constant coefficients D₁ and D₂ are obtained as follows:

D_{1} = \frac{t_{1} D_{22} - t_{2} D_{12}}{D_{12} [(m_{3} + 1) D_{21} b^{- (m_{3} + 2)} - D_{22} b^{- (m_{3} + 1)}] - D_{22} [(m_{3} + 1) D_{11} a^{- (m_{3} + 2)} - D_{12} a^{- (m_{3} + 1)}]},

(22)

D_{2} = \frac{t_{1} [(m_{3} + 1) D_{21} b^{- (m_{3} + 2)} - D_{22} b^{- (m_{3} + 1)}] - t_{2} [(m_{3} + 1) D_{11} a^{- (m_{3} + 2)} - D_{12} a^{- (m_{3} + 1)}]}{D_{12} [(m_{3} + 1) D_{21} b^{- (m_{3} + 2)} - D_{22} b^{- (m_{3} + 1)}] - D_{22} [(m_{3} + 1) D_{11} a^{- (m_{3} + 2)} - D_{12} a^{- (m_{3} + 1)}]},

(23)

Solution of governing displacement equation of FGM thick-walled spherical shell

By substituting equation (21) into equation (14) and doing some mathematical simplifications, the following nonhomogeneous ODE, namely, the Cauchy–Euler ODE governing the radial displacement is obtained

r^{2} \frac{d^{2} u}{d r^{2}} + A r \frac{d u}{d r} + B u = C_{1} r^{(m_{2} - m_{3})} + C_{2} r^{(m_{2} + 1)},

(24)

in which constant coefficients C₁ and C₂ are obtained as below

C_{1} = \frac{α_{0} (1 + ν) (m_{1} + m_{2} - m_{3} - 1) D_{1}}{1 - ν}, C_{2} = \frac{α_{0} (1 + ν) (m_{1} + m_{2}) D_{2}}{1 - ν},

(25)

The auxiliary relation corresponded to equation (24) is determined by the following quadratic relation

n^{2} + (A - 1) n + B = 0,

(26)

The discriminant value for the quadratic equation (26) is written as follows:

Δ = {(A - 1)}^{2} - 4 B = [({m_{1}}^{2} + 9) (1 - ν) + 2 m_{1} (1 - 5 ν)] / (1 - ν),

(27)

since for the pure metals like steel, mostly 0<ν < 0.3; therefore, Δ >0,

\forall

m₁, ν. So, there are two distinct real roots of n₁ and n₂ (i.e., n₁≠n₂) for relation (26) as follows:

n_{1, 2} = \frac{1 - A \pm \sqrt{Δ}}{2},

(28)

So, the homogenous solution of equation (24) is obtained as below

u_{h} (r) = L_{1} r^{n_{1}} + L_{2} r^{n_{2}},

(29)

where L₁ and L₂ are coefficients to be determined later applying the boundary conditions (BCs) of mechanical loading. The particular solution of equation (24) can be considered as

u_{p} (r) = L_{3} r^{(m_{2} - m_{3})} + L_{4} r^{(m_{2} + 1)},

(30)

Then, the solution of equation (24) is written as follows:

u (r) = u_{h} (r) + u_{p} (r),

(31)

Substituting equations (31) into (24), then the coefficients L₃ and L₄ are obtained as follows:

L_{3} = \frac{C_{1}}{(m_{2} - m_{3}) (m_{2} - m_{3} - 1) + A (m_{2} - m_{3}) + B}, L_{4} = \frac{C_{2}}{(m_{2} + 1) (m_{2} + A) + B},

(32)

Substituting equation (31) into the strain relations (9) and (10) and then putting the results into the equations (6) and (7), the stress components in terms of radial displacement are obtained

σ_{r} = \frac{E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)} [Q_{1} L_{1} r^{n_{1} - 1} + Q_{2} L_{2} r^{n_{2} - 1} + Q_{3} L_{3} r^{m_{2} - m_{3} - 1} + Q_{4} L_{4} r^{m_{2}}],

(33)

σ_{θ} = \frac{E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)} [G_{1} L_{1} r^{n_{1} - 1} + G_{2} L_{2} r^{n_{2} - 1} + G_{3} L_{3} r^{m_{2} - m_{3} - 1} + G_{4} L_{4} r^{m_{2}}],

(34)

in which

\begin{array}{l} Q_{1} = n_{1} (1 - ν) + 2 ν, \\ Q_{2} = n_{2} (1 - ν) + 2 ν, \\ Q_{3} = m_{2} (1 - ν) - m_{3} (1 - ν) + 2 ν - α_{0} (1 + ν) D_{1}, \\ Q_{4} = m_{2} (1 - ν) + 1 + ν - α_{0} (1 + ν) D_{2}, \\ G_{1} = 1 + n_{1} ν, \\ G_{2} = 1 + n_{2} ν, \\ G_{3} = 1 + (m_{2} - m_{3}) ν - α_{0} (1 + ν) D_{1}, \\ G_{4} = 1 + (m_{2} + 1) ν - α_{0} (1 + ν) D_{2}, \end{array}

(35)

The BCs of the hollow spherical vessel are expressed as

{σ_{r} |}_{r = a} = - P_{a},

(36)

{σ_{r} |}_{r = b} = - P_{b},

(37)

By applying the boundary conditions (36) and (37) into equation (33), coefficients L₁ and L₂ are obtained as follows:

L_{1} = \frac{- P_{a} b^{(n_{2} - 1)} A_{2} + P_{b} a^{(n_{2} - 1)} A_{1} - B_{2} - C_{4} - D_{4} - E_{2}}{Q_{1} A_{1} A_{2} (b^{(n_{2} - 1)} a^{(n_{1} - 1)} - a^{(n_{2} - 1)} b^{(n_{1} - 1)})},

(38)

L_{2} = \frac{- P_{a} b^{(n_{1} - 1)} A_{2} + P_{b} a^{(n_{1} - 1)} A_{1} - B_{1} - C_{3} - D_{3} - E_{1}}{Q_{2} A_{1} A_{2} (b^{(n_{1} - 1)} a^{(n_{2} - 1)} - a^{(n_{1} - 1)} b^{(n_{2} - 1)})},

(39)

where in equations (38) and (39) the following parameters are used

\begin{array}{l} A_{1} = \frac{E_{0} a^{m_{1}}}{(1 + ν) (1 - 2 ν)}, A_{2} = \frac{E_{0} b^{m_{1}}}{(1 + ν) (1 - 2 ν)}, \\ B_{1} = A_{1} A_{2} Q_{3} L_{3} a^{(m_{2} - m_{3} - 1)} {b^{(}}^{n_{1} - 1)}, \\ B_{2} = A_{1} A_{2} Q_{3} L_{3} a^{(m_{2} - m_{3} - 1)} b^{(n_{2} - 1)}, \\ C_{3} = A_{1} A_{2} Q_{4} L_{4} {a^{m}}^{2} b^{(n_{1} - 1)}, \\ C_{4} = A_{1} A_{2} Q_{4} L_{4} {a^{m}}^{2} b^{(n_{2} - 1)}, \\ D_{3} = - A_{1} A_{2} Q_{3} L_{3} b^{(m_{2} - m_{3} - 1)} a^{(n_{1} - 1)}, \\ D_{4} = - A_{1} A_{2} Q_{3} L_{3} b^{(m_{2} - m_{3} - 1)} a^{(n_{2} - 1)}, \\ E_{1} = - A_{1} A_{2} Q_{4} L_{4} {b^{m}}^{2} a^{(n_{1} - 1)}, \\ E_{2} = - A_{1} A_{2} Q_{4} L_{4} {b^{m}}^{2} a^{(n_{2} - 1)}, \end{array}

(40)

Yielding behavior of FGM thick-walled spherical shell

Based on Prantle–Reuss flow rule, the total strain increment is split into elastic and plastic strain increments as follows:

d ε_{i}^{T} = d ε_{i}^{e} + d ε_{i}^{p}, i = r, θ, φ

(41)

in which the superscripts e and p indicate quantities in the elastic and the plastic zones, respectively, and superscript T stands for total value of the quantity. According to Prantle–Reuss flow rule, the stress-strain increments relations is given by¹⁹

\frac{d ε_{r}^{p}}{σ_{r}^{'}} = \frac{d ε_{θ}^{p}}{σ_{θ}^{'}} = \frac{d ε_{φ}^{p}}{σ_{φ}^{'}} = d λ,

(42)

in which dλ is an instantaneous proportional constant (strain increment) with a positive value. Also, deviatoric stress

σ_{i}^{'}

and volumetric stress (hydrostatic or mean) σ_m are defined as below³³

\begin{array}{l} σ_{i}^{'} = σ_{i} - σ_{m}, i = r, θ, φ \\ σ_{m} = (σ_{r} + σ_{θ} + σ_{φ}) / 3 = (σ_{r} + 2 σ_{θ}) / 3, \end{array}

(43)

Thus, the stress-strain increments relations is rewritten as¹⁹

\frac{d ε_{r}^{p}}{(σ_{r} - \frac{1}{2} (σ_{θ} + σ_{φ}))} = \frac{d ε_{θ}^{p}}{(σ_{θ} - \frac{1}{2} (σ_{r} + σ_{φ}))} = \frac{d ε_{φ}^{p}}{(σ_{φ} - \frac{1}{2} (σ_{r} + σ_{θ}))} = \frac{2}{3} d λ,

(44)

The relation for von Mises stress σ_e (or equivalent stress) based on the principal stresses σ₁, σ₂, and σ₃ is written as

\bar{σ} = σ_{e} = \frac{\sqrt{2}}{2} \sqrt{{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{1} - σ_{3})}^{2}},

(45)

in which because of symmetry in spherical coordinates the radial, hoop, and meridional stress components are considered as principal stresses in equation (45), in which σ_φ(=σ_θ) > σ_r. So, von Mises yield criterion is written as

2 Y^{2} (r) = {(σ_{θ} - σ_{r})}^{2} + {(σ_{φ} - σ_{θ})}^{2} + {(σ_{φ} - σ_{r})}^{2},

(46)

by imposing the condition σ_φ = σ_θ in equation (46), von Mises yield criterion is obtained as follows:

Y (r_{y}) = σ_{θ} - σ_{r},

(47)

It is noted that in axisymmetric problems if circumferential stress becomes greater than radial stress then the yielding occurs, that is σ_θ > σ_r. Moreover, r_y is the specific yield onset radius. Now non-dimensional stress ${\bar{σ}}_{i}$ is defined as below

{\bar{σ}}_{i} = \frac{σ_{i}}{Y (a)} = \frac{σ_{i}}{Y_{0} a^{m_{4}}}, i = r, θ

(48)

In order to determine the yield onset radius, non-dimensional yield stress function Φ_Y is defined as below

Φ_{Y} = {(\frac{a}{r_{y}})}^{m_{4}} [{\bar{σ}}_{θ} (r_{y}) - {\bar{σ}}_{r} (r_{y})],

(49)

Substituting equations (33) and (34) into equation (48) and then putting the results into equation (49), the non-dimensional yield stress function Φ_Y to predict the yield onset value is rewritten as follows:

Φ_{Y} = {(\frac{1}{r_{y}})}^{m_{4}} \frac{1}{Y_{0}} [\frac{E_{0} r^{m_{1}}}{(1 + ν) (1 - 2 ν)}] [(G_{1} - Q_{1}) L_{1} {r_{y}}^{n_{1} - 1} + (G_{2} - Q_{2}) L_{2} {r_{y}}^{n_{2} - 1} + (G_{3} - Q_{3}) L_{3} {r_{y}}^{m_{2} - m_{3} - 1} + (G_{4} - Q_{4}) L_{4} {r_{y}}^{m_{2}}],

(50)

According to equation (50), it can be easily examined how can plastic zone spread and change in different temperature gradients of FGM with different inhomogeneity constants m_i (i = 1, 2, 3, and 4). To predict the yield onset, Φ_Y assigns values as follows:

Φ_{Y} : {\begin{cases} < 1 \to Elastic zone \\ = 1 \to Yield onset \\ > 1 \to Plastic zone \end{cases}

(51)

It is to be noted that when Φ_Y = 1 the plastic zone starts to spread radially from a specific yield onset radius in the shell wall. In other words, Φ_Y(r_y) = 1 indicates that at a threshold of thermoelasto-plastic behavior the plastic zone initially is launched at r = r_y. Also, for those radii of the spherical shell that are kept with Φ_Y ≤ 1 and Φ_Y ≥ 1 the corresponding layers in the shell wall are completely exposed to elastic and plastic zones, respectively.

Perfectly plastic analysis of FGM thick-walled spherical shell

By substituting equation (47) into equilibrium differential equation (11) consequently, radial and circumferential stresses in plastic zone are obtained

σ_{r} (r) = \frac{2 Y_{0}}{m_{4}} r^{m_{4}} + E_{3},

(52)

σ_{θ} (r) = \frac{2 Y_{0}}{m_{4}} (1 + m_{4} / 2) r^{m_{4}} + E_{3},

(53)

in which E₃ is a constant depended on BCs. Plastic deformation is supposed to be incompressible, that is, the sum of plastic strain components in the plastic zone are set to zero²⁰

ε_{r}^{p} + ε_{θ}^{p} + ε_{φ}^{p} = 0,

(54)

Therefore, according to the stress-strain relationship using equations (6) to (10), the sum of strain components is given by the following relation

ε_{r}^{T} + (1 + 2 ν) ε_{θ}^{T} = \frac{(1 + ν) (1 - 2 ν)}{E (r)} [σ_{θ} (r) + σ_{r} (r)] + 2 (1 + ν) α (r, T) T (r),

(55)

in which

ε_{r}^{T}

and

ε_{θ}^{T}

are total elastic radial and circumferential strains, respectively. Now, based on strain-displacement relations in equations (9) and (10), and with the help of equations (12) and (13) and then substituting the stress relations in the plastic zone in equations (52) and (53) into equation (55), finally summation of total strain components in equation (55) is converted in terms of radial displacement as follows:

\begin{array}{l} \frac{d u (r)}{d r} + (1 + 2 ν) \frac{u (r)}{r} = \frac{2 (1 + ν) (1 - 2 ν) (2 + m_{4} / 2) Y_{0}}{E_{0} m_{4}} r^{m_{4} - m_{1}} \\ + \frac{2 E_{3} (1 + ν) (1 - 2 ν)}{E_{0}} r^{- m_{1}} + 2 (1 + ν) α_{0} D_{1} (r^{m_{2} - m_{3} - 1} + r^{m_{2}}), \end{array}

(56)

It is noted that equation (56) is governing ODE of radial displacement of the shell whose solution results to the radial displacement of the shell in the perfectly plastic zone as follows:

u (r) = \frac{F_{1}}{r} + F_{2} r^{m_{2} - m_{3}} + F_{3} r^{m_{2} + 1} + F_{4} r^{m_{4} - m_{1} + 1} + F_{5} r^{1 - m_{1}},

(57)

In which constant coefficients are given by

\begin{array}{l} F_{2} = \frac{2 (1 + ν) α_{0} D_{1}}{m_{2} - m_{3} + 1}, F_{3} = \frac{2 (1 + ν) α_{0} D_{2}}{m_{2} + 2}, \\ F_{4} = \frac{2 (1 + ν) (1 - 2 ν) (2 + m_{4} / 2) Y_{0}}{E_{0} m_{4} (m_{4} - m_{1} + 2)}, F_{5} = \frac{2 E_{3} (1 + ν) (1 - 2 ν)}{E_{0} (2 - m_{1})}, \end{array}

(58)

in which constant coefficient F₁ is a constant depended on the boundary condition. When all power-law parameters of FGM are set zero, the yielding behavior is not dependent on radial direction of the spherical vessel. In this case, it is assumed that shell is made up of homogenous and isotropic material. In this condition, the governing relation (56) and radial and circumferential stresses of the shell in equations (52) and (53) become infinite. So, the perfectly plastic solution in equation (57) is not applicable and the solution is separately obtained in what follows. Note that the yield stress value is constant (Y(r) = Y₀ = const.) and by substituting von Mises yield criterion in equation (47) into the equilibrium differential equation (11) yields to

\frac{d σ_{r}}{d r} = \frac{2 Y_{0}}{r},

(59)

Then, the stress components in the plastic zone are obtained as below

σ_{r} (r) = 2 Y_{0} \ln (r) + E_{4},

(60)

σ_{θ} (r) = Y_{0} [1 + 2 \ln (r)] + E_{4},

(61)

in which E₄ is a constant depended on boundary conditions. Using equations (56), (60), and (61), the radial displacement u(r) is obtained as follows:

u (r) = \frac{F_{6}}{r} + F_{7} r \ln r + F_{8} r,

(62)

in which constant coefficients are

F_{7} = \frac{2 Y_{0} (1 + ν) (1 - 2 ν)}{E_{0}}, F_{8} = \frac{(1 + ν) (1 - 2 ν) (E_{4} - Y_{0} / 2)}{E_{0}} + (1 + ν) α_{0} (D_{1} + D_{2}),

(63)

in which constant coefficients E₄ and F₆ are depended on the boundary conditions.

Numerical results and discussions

In this section, based on a written code in MATLAB, at first, we examine the validity of the obtained numerical results by comparing them with the results reported in literature for special cases. Then, for FGM thick-walled hollow spheres with different material properties m_i (i = 1, 2, 3, 4) under internal pressure and thermal gradient the obtained results for the yield onset varaible, distribution of radial displacement and radial and circumferential stresses in elastic and plastic zones are evaluated. Moreover, a detailed parameter study on the numerical simulations is conducted to investigate the influence of different parameters on the thermoelasto-plastic behavior of hollow sphere.

Model validation and verification of numerical simulations

Case 1: An FGM thick-walled hollow sphere under internal pressure and applied steady thermal gradient between the inner and outer surfaces with the following specifications is considered ²: E(=E₀) = 200 GPa, ν = 0.3, α(=α₀) = 1.2 × 10⁻⁶ °C⁻¹, a = 1 m, b = 1.2 m, P_a = 50 MPa, P_b = 0, T_a = 10°C, T_b = 0, n(=m₁ = m₂ = m₃) = −2 and 3, and MATLAB code was run. In Figure 2, variation of the non-dimensional radial stress against radial distance of the thick-walled spherical shell is shown for n = −2 and 3 in this study and Ref. 2. It is observed that there is a complete accordance between the obtained results of this work and the ones in Ref. 2.

Case 2: In order to further study on the verification of obtained results, an FGM thick-walled spherical pressure vessel with geometry and mechanical specifications as²² ν = 0.3, Y₀ = 430 MPa, a/b = 0.6, P_b = 0 with m₁ = 1.1 and different m₄ values of m₄ = 0, −1, −2.13,026, −2, −3, −4, respectively, under non-dimensional internal pressure of P_a/Y(a) = −0.6131, P_a/Y(a) = −0.6131, P_a/Y(a) = −0.6131, P_a/Y(a) = −0.3932 and P_a/Y(a) = −0.2359 with no thermal gradient is used. Moreover, it is considered t₂ = 0, D₁₁ = 0°C⁻¹, D₁₂ = 1°C⁻¹, D₂₁ = 0°C⁻¹, D₂₂ = 1°C⁻¹. The MATLAB code was run and in Figure 3 variation of the yield variable Φ_Y versus non-dimensional radial distance r/b are plotted and compared with the results in Ref. 22. Numerical simulations show that there is a complete accordance between the obtained results of this work and the ones obtained in Ref. 22.

Thermoelasto-plastic analysis results for FGM thick-walled spherical shell under internal pressure and thermal gradient

In order to examine the role of temperature-dependent FGM parameters on the obtained thermoelasto-plastic analysis results of a hollow spherical vessel, Touloukian’s relation are used as follows³¹:

q_{j} = q_{0} (q_{- 1} T^{- 1} + 1 + q_{1} T^{1} + q_{2} T^{2} + q_{3} T^{3}),

(64)

in which T is temperature. In Table 1, the values of coefficients q_j corresponded to the modulus of elasticity E₀ and thermal expansion coefficient α₀ for stainless steel SUS304 are presented.³¹ In Table 2, values of elastic modulus E₀ and thermal expansion coefficient α₀ for SUS304 in different thermal gradient ΔT are listed.³¹

Table 1.

Values of coefficients q_j corresponded to the Touloukian’s relation for modulus of elasticity E₀ and thermal expansion coefficient α₀ for stainless steel SUS304.³¹

q _j	Elastic modulus E₀ (Pa)	Thermal expansion coefficient α₀ (°C⁻¹)
q ₀	201.04 × 10⁹	12.33 × 10⁻⁶
q _-1	0	0
q ₁	3.079 × 10⁻⁴	8.086 × 10⁻⁴
q ₂	−6.534 × 10⁻⁷	0
q ₃	0	0

Table 2.

Values of modulus of elasticity E₀ and thermal expansion coefficient α₀ in different thermal gradient for stainless steel SUS304.³¹

Temperature difference ΔT (°C)	modulus of elasticity E₀ × 10¹¹ (Pa)	Thermal expansion coefficient α₀ × 10⁻⁵ (°C⁻¹)
0	2.0815	1.5053
100	2.0585	1.6050
200	2.0092	1.7047
300	1.9337	1.8044
400	1.8319	1.9041
500	1.7088	2.0038
600	1.5494	2.1035

As an example in Figure 4, for an FGM hollow spherical vessel with a = 0.5 m, b = 1 m, t₂ = 0, D₁₁ = 0°C⁻¹, D₁₂ = 1°C⁻¹, D₂₁ = 0°C⁻¹, D₂₂ = 1°C⁻¹ the variation of temperature gradient ΔT(r) against the non-dimensional radial distance (r/b) with material property m₃ = −2.2 is illustrated.

Now effect of combination of internal pressure and thermal gradient on the yield onset radius, radial displacement and stress components of FGM hollow spherical vessels and how the plastic zones radially spread are studied in different examples in below.

In Figure 5(a), the variation of different non-dimensional mechanical and physical properties of the FGM including E/E(a), Y/Y(a) and α/α(a) for a hollow sphere with (m₁, m₂, m₃, m₄)=(2.2, −2.9, −2.2, −4)³¹ at temperature 600°C are shown. In Figure 5(b)–(e), the through-thickness variations of the non-dimensional yield onset variable Φ_Y, radial stress ${\bar{σ}}_{r}$ , circumferential stress ${\bar{σ}}_{θ}$ and radial displacement ${\bar{u}}_{r}$ (= $\frac{u E_{0}}{Y (a) b}$ ), respectively, with respect to temperature gradient ΔT along the non-dimensional radial distance (r/b) for this FGM hollow sphere with a/b = 0.5 under internal pressure of P_a = 8 × 10⁸ Pa are illustrated. Figure 5(b) shows the commencement of yield incidence. From Figure 5(b), it can be seen that by increasing the value of the temperature gradient ΔT the value of the yield variable Φ_Y decreases in radial direction from its minimum value at the inner surface up to r/b = 0.67 and then significantly increases afterward to its maximum value at the outer surface. From Figure 5(b), it can be observed that for the temperature gradient ΔT = 0°C the shell wall of hollow sphere is completely exposed to elastic zone, whereas for the temperature gradient ΔT=(100, 200, 300, 400, 500, and 600)°C for non-dimensional radii of r/b ≤ 0.87, r/b ≤ 0.82, r/b ≤ 0.77, r/b ≤ 0.76, r/b ≤ 0.75, and r/b ≤ 0.74, respectively, the shell wall of hollow sphere is maintained in the elastic zone and for the greater values of these radii the shell wall of hollow sphere is maintained in the plastic zone, respectively. Briefly, as illustrated, the temperature gradient influences the plastic onset radius and causes the plastic deformation launches to grow at different distance from inner surface. From Figure 5(c), it is seen that the maximum absolute value of non-dimensional radial stress ${\bar{σ}}_{r}$ in the inner radius decreases to the zero value in the outer radius of the hollow sphere. Also, it is observed that by increasing the temperature gradient ΔT, the absolute value of the non-dimensional radial stress ${\bar{σ}}_{r}$ increases in radial direction of the shell wall in which for ΔT=(300, 400, 500, and 600)°C the non-dimensional radial stress ${\bar{σ}}_{r}$ has its maximum absolute value in a specific interior radius of the shell wall. In other words, for ΔT=(300, 400, 500, and 600)°C the absolute value of the non-dimensional radial stress ${\bar{σ}}_{r}$ in the inner radius reaches to its maximum absolute value at the interior surface of the shell with increasing both the temperature gradient and non-dimensional radial distance and then decreases radially and reaches to the zero value in the outer radius. From this figure, it is noticed that for any specific temperature difference ΔT = T(a)−T(b) > 0, all the shell wall layers are exposed to the thermal gradient, and consequently, the hollow sphere tends to expand radially. Since the thermal expansion coefficient of the FGM decreases radially (m₂ < 0) in which the outer layers have smaller α values than the ones at the inner layers, the outer weaker metal layers with the smaller thermal expansion coefficient and also with the larger rigidity (m₁ > 0) have a great tendency to be stayed with the least expansion/elongation in radial direction. On the other hand, the inner richer metal layers with the higher thermal expansion coefficients and smaller rigidity tend to expanded much more. Therefore, due to this interaction between the shell wall layers, the inner layers will be more compressed; because the outer layers shrink more than the inner layers, increasing compressive radial stresses will be produced in all layers by increasing temperature gradient. From Figure 5(d), it is seen that by increasing the non-dimensional radial distance the absolute value of non-dimensional circumferential stress ${\bar{σ}}_{θ}$ increases up to r/b = 0.76 and then increases to the positive value at the outer surface. As can be seen in Figure 5(c) and (d), the temperature gradient influences the distribution of stresses noticeably. From Figure 5(e), it is seen that by increasing the temperature gradient, the non-dimensional radial displacement ${\bar{u}}_{r}$ increases, accordingly. Also, from this figure, it can be concluded that when the temperature gradient ΔT increases, the difference between every two successive results of ${\bar{u}}_{r}$ almost increases, accordingly.

Figure 6(a) shows the variation of different non-dimensional mechanical and physical properties of the FGM including E/E(a), Y/Y(a), and α/α(a) for a hollow sphere with (m₁, m₂, m₃, m₄)=(−2.9, −2.9, −2.9, −3.4) ³¹ at temperature 600°C. In Figure 6(b)–(e), the through-thickness variations of the non-dimensional yield onset variable Φ_Y, radial stress ${\bar{σ}}_{r}$ , circumferential stress ${\bar{σ}}_{θ}$ , and radial displacement ${\bar{u}}_{r}$ , respectively, with respect to temperature gradient ΔT along the non-dimensional radial distance (r/b) for this FGM hollow sphere with a/b = 0.5 under internal pressure of P_a = 2 × 10⁹ Pa are illustrated. From Figure 6(b), it can be seen that by increasing the value of the temperature difference ΔT (for ΔT >0) the value of the yield variable Φ_Y decreases up to r/b = 0.58 and then increases afterward. Also, by increasing the value of the non-dimensional radial distance r/b, the value of Φ_Y increases drastically for ΔT >0 in radial direction from its minimum value at the inner surface to the maximum value at the outer surface. As it is seen, increasing the thermal gradient results in the growth of plastic zone in outward direction. From Figure 6(c), it is seen that the non-dimensional radial stress ${\bar{σ}}_{r}$ at the inner radius is compressive and the maximum absolute vale of non-dimensional radial stress value decreases to the zero value at the outer radius. Also, it is observed that by increasing the temperature gradient ΔT, the absolute value of the non-dimensional radial stress ${\bar{σ}}_{r}$ increases in which for ΔT=(100, 200, 300, 400, 500, and 600)°C the maximum absolute value of the non-dimensional radial stress ${\bar{σ}}_{r}$ occurs in a specific interior radius through the shell wall. It is to be mentioned that for ΔT=(100, 200, 300, 400, 500, and 600)°C the absolute value of the non-dimensional radial stress ${\bar{σ}}_{r}$ in the inner radius increases to its maximum value at a specific interior surface of the shell with increasing both the temperature gradient and non-dimensional radial distance and then decreases in which reaches to the zero value at the outer radius. From this figure, it can be concluded that for any specific temperature difference ΔT >0, all the shell wall layers are exposed to the temperature gradient, and consequently, the hollow sphere tends to expand radially. Since the thermal expansion coefficient of the FGM decreases radially (m₂ < 0) in which the outer layers have lower values than the ones at the inner layers, the outer weaker metal layers with lower thermal expansion coefficients and also with the lesser rigidity (m₁ < 0) have a greater tendency to expand. Therefore, the inner layers will be compressed accordingly. Because the inner layers shrink more than the outer layers, more increasing compressive radial stresses will be produced in all layers compared to the depicted results in Figure 5(c). From Figure 6(d), it is seen that by increasing the non-dimensional radial distance, the non-dimensional circumferential stress ${\bar{σ}}_{θ}$ has a decreasing trend up to r/b = 0.58 and then increases to the positive value at the outer surface. From Figure 6(e), it is seen that by increasing the temperature gradient, the non-dimensional radial displacement ${\bar{u}}_{r}$ increases, accordingly. Also, from this figure, it can be concluded that when the temperature gradient ΔT increases, the difference between every two successive results of ${\bar{u}}_{r}$ almost increases, accordingly.

In Table 3 the obtained results of limit internal pressure P_y in which the yield incidence initiates for a thick-walled hollow sphere with a/b = 0.6 and with no thermal gradient with different FGM parameters made up of SUS304³⁰ are listed. As can be seen for the spherical pressure vessel, under internal pressure, the yielding region begins from the inside radius.

Table 3.

The calculated values for limit internal pressure P_y for a thick-walled hollow sphere with a/b = 0.6 and different FGM parameters without temperature gradient.

Yield onset location	P_y (MPa)	m ₁	m ₂	m ₃	m ₄
Inside radius	69.05	−1.1	3	3	2
Inside radius	39.15	−1.5	−1.4	−2.1	3
Inside radius	56.95	11	−1.4	−2.1	3
Inside radius	312.1	0.5	−0.5	−3	−0.5
Inside radius	187.25	0.5	−0.7	−3	0.5

In order to monitor the radial location of the commencement of the yielding in purely elastic state, the variation of non-dimensional yield variable Φ_Y, radial and hoop stresses, and radial displacement are illustrated in Figure 7–10, for a hollow spherical vessel with different FGM parameters. Figure 7 shows the variation of non-dimensional yield variable Φ_Y, the consequent radial and circumferential stresses and radial displacement for a spherical vessel with the ratio of inner to outer radius of the hollow sphere is taken a/b = 0.4 under internal pressure P_a/Y(a) = 0.73,372 ²² with respect to different temperature gradients for (m₁, m₂, m₃, m₄)=(0.9, −1.4, −1.5, −2.35,696). As it is seen in Figure 7(a) that yielding commences simultaneously at both inside and outside surfaces of the pressure vessel without temperature gradient since Φ_Y (0.4) = 1 and Φ_Y(1) = 1.

In Figure 8, the variation of non-dimensional yield variable Φ_Y, the consequent radial and circumferential stresses, and radial displacement for a spherical vessel with a/b = 0.4 under internal pressure P_a/Y(a) = 0.73,372²² with respect to different temperature gradients for (m₁, m₂, m₃, m₄)=(0.9, −1.4, −2.1, −1.8) are shown. From Figure 8(a), it is seen that the yielding begins at the inner surface of the pressure vessel without temperature gradient since Φ_Y(0.4) = 1. The matching radial and circumferential stresses and radial displacement are plotted in Figure 8(b)–(d) for different temperature gradients.

In Figure 9, the variation of non-dimensional yield variable Φ_Y, the consequent radial and circumferential stresses and radial displacement for a spherical vessel under internal pressure P_a/Y(a) = 0.488,908²² with respect to different temperature gradients for (m₁, m₂, m₃, m₄)=( 0.9, −1.4, −2.1, −2.8) are plotted. It is seen in the Figure 9(a) that yielding begins at the outer surface of the pressure vessel without temperature gradient since Φ_Y(1) = 1. The corresponding distributions of the radial and circumferential tresses and radial displacement are plotted in Figure 9(b)–(d) for different temperature gradients.

It is interesting to be mentioned that for the values of m₄ < 0, an unusual deformation behavior, that is yielding commencing at r = r_Y, where a<r_Y<b, may be observed. Figure 10 shows the variation of non-dimensional yield variable Φ_Y, the consequent stresses, and displacement for a spherical vessel with a/b = 0.3 under internal pressure P_a/Y(a) = 0.601,914²² with respect to different temperature gradients for (m₁, m₂, m₃, m₄)=(−0.25, −1.5, −2, −3.2). As can be noticed in Figure 10(a), the plastic flow begins at r/b = 0.76 of the pressure vessel without temperature gradient since Φ_Y(0.76) = 1.

To avoid only a classical discretization conducted for elasto-plastic analysis in this study, a natural framework based on NN solution is also employed utilizing efficient and automatically reliable account for uncertainties for inverse analysis. To do this, at the first step using analytical method in the present work an extensive tabulated data is gathered involving obtained results for yield onset variable of hollow spherical vessels composed of different FGMs under thermo-mechanical loading. Then, a computer code based on NN scheme was written to take care of 12 input variables of internal pressure P_a, temperature gradient ΔT, inside radius r_a, outside radius r_b, temperature-dependent parameters of modulus of elasticity E₀, thermal expansion coefficient α₀, thermal conductivity coefficient k₀, and yield stress Y₀, FGM parameters of m₁, m₂, m₃, and m₄ on the yield onset variable Φ_Y as output variable. The aim is reliable prediction of the value of yield onset variable under considered input parameters using trained data. Then, the NN based code was run and the results of trained neural network was used to predict output variable Φ_Y for a set of test data. In this basis, Figure 11 shows comparison of actual and scatted plot of predicted results for yield onset variable Φ_Y using NN solution for FGM hollow spheres with different material parameters under thermo-mechanical loading. As can be seen, the trained results using NN scheme are very close to the actual ones obtained using analytical method in this study. The simulation results shown in this figure reveal that the NN base is capable of quantifying equally well the influence of all uncertain input parameters with respect to the uncertain output parameter.

In Figure 12, a comparison between actual and scatted plot of predicted results for yield onset variable Φ_Y using analytical method and NN solution for FGM hollow spherical shells with different material parameters (m₁, m₂, m₃, m₄) based on tabulated listed rows of trained data is shown.

Figure 13 shows the error between actual and predicted results for yield variable Φ_Y using analytical and NN methods based on tabulated listed rows of trained data in this study. It is seen that the maximum difference between these two methods of solution is negligible with the value of 2.5%.

Using the trained data utilized based on neural network scheme, in Table 4, the obtained results for output variable Φ_Y for different set of input test data are listed. As can be seen, NN solution can accurately predict the output variable Φ_Y for these considered test data.

Table 4.

Predicted output results for Φ_Y using NN solution for different input test data.

r_a (m)	r_b (m)	P_a (GPa)	ΔT (°C)	E₀ (GPa)	α₀ × 10⁻⁵ (°C⁻¹)	Y₀ (MPa)	m ₁	m ₂	m ₃	m ₄	r (m)	Φ_Y, (Analytical)	Φ_Y, (NN)
0.5	1	50	200	200.92	1.7047	415	−4.23	8	8	−6	0.7	0.985	0.979
0.6	1	0	136	204.38	1.6409	430	0.5	−0.7	−3	0.5	0.6	1	0.998
0.3	1	12.196	0	200	1.5053	430	−0.25	0	0	−3.2	0.4	0.956	0.961

Conclusion

A thermoelasto-plastic behavior analysis of a thick-walled hollow sphere of FGM under internal pressure and thermal gradient loading was performed in steady-state conduction heat transfer condition. Temperature-dependent material properties of modulus of elasticity, thermal conductivity and thermal expansion coefficients along with yield stress were assumed to follow power-law functions in radius. The governing differential equation are derived analytically and then solved to determine through-thickness variation of the radial and circumferential stresses and radial displacement in elastic and perfectly plastic zones in the vessel wall to predict the onset radius of yielding based on von Mises yield criterion. Using NN solution, the influence of all uncertain input parameters with respect to uncertain output parameter of yield onset variable was utilized. The main outcome results are as follows:

(1) It was shown that when the yield onset variable is less than the one, the hollow FGM sphere remains in the elastic zone, and for those values equal or larger than one, the yield condition commences.

(2) The numerical results showed that for a combination of internal pressure and thermal gradient the plastic zone can commence from inside, outside radius or simultaneously in both radii or may be launched in interior surface of FGM thick-walled spherical shell depended on the applied thermo-mechanical loading and FGM parameters.

(3) It was observed that depended on the values of functionally gradient material properties for specific internal pressure, the direction of plastic zone growth could change in either inward or outward directions by varying the thermal gradient.

(4) It was observed that for an FGM with m₂ < 0 in which the outer layers have smaller thermal expansion coefficient values than the ones at the inner layers, the outer weaker metal layers with the smaller thermal expansion coefficient and also with the larger rigidity (m₁ > 0) have a great tendency to be stayed with the least expansion/elongation in radial direction. On the other hand, the inner richer metal layers with the higher thermal expansion coefficients and smaller rigidity tend to expanded much more. Therefore, due to this interaction between the shell wall layers, the inner layers will be more compressed; because the outer layers shrink more than the inner layers, increasing compressive radial stresses will be produced in all layers by increasing temperature gradient. More increasing compressive radial stresses can be found for an FGM with the values of m₁ and m₂ > 0.

(5) Numerical simulations using neural network solution showed that NN scheme is reliable and capable to accurately predict the value of yielding onset variable as the output parameter under the influence of all uncertain input parameters.

Footnotes

ORCID iD

Ahmad Mamandi

Data availability statement

The author confirms that the data supporting the findings of this study are available within the article.*

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