Generalized Differential Quadrature Research Articles

Thermal Vibration of Thick FGM Conical Shells by Using Third-Order Shear Deformation Theory.

A time-dependent third-order shear deformation theory (TSDT) approach on the displacements of thick functionally graded material (FGM) conical shells under dynamic thermal vibration is studied. Dynamic equations of motion with TSDT for thick FGM conical shells are applied directly with the partial derivative of variable R*θ in the curve coordinates (x, θ, z) instead of y in the Cartesian coordinates (x, y, z) for thick FGM plates, where R* is the middle-surface radius at any point on conical shells. The generalized differential quadrature (GDQ) numerical method is used to solve the dynamic differential equations in equilibrium matrix forms under thermal loads. It is the novelty of the current study to identify the parametric effects of shear correction coefficient, environment temperature, TSDT model, and FGM power law index on the displacements and stresses in the thick conical shells only subjected to sinusoidal heating loads. The physical parts with values on the length-to-thickness ratio equals 5, and 10 FGMs can be used in an area of an airplane engine that usually operates near more than 1000 K of temperatures when the thermal stress is considered and affected. The important findings of the presented study are listed as follows. The values of normal stress are in decreasing tendencies with time in cases when the coefficient c1 equals 0.925925/mm2 in TSDT and length-to-thickness ratio equals 5. The shear stress values in x plane z direction on the minor middle-surface radius (r) equals the major middle-surface radius (R) over 8 and length-to-thickness ratio equals to 5 can withstand T = 1000 K of pressure.

Free vibration analysis of laminated anisotropic doubly-curved shell structures reinforced with three-phase polymer/CNT/fiber material

The present work investigates the vibrational response of laminated anisotropic doubly-curved shell structures reinforced with Carbon Nanotubes (CNTs) short fibers. The fundamental equations are derived from a curvilinear reference system of principal coordinates, employing the Equivalent Single Layer (ESL) methodology. Furthermore, a general variation in laminate thickness is considered. The unknown field variable is described using higher order theories through the unified formulation, taking into account a general interpolation of the unknown variables with Lagrange polynomials across the physical domain. The Hamiltonian Principle is adopted for the derivation of the dynamic equilibrium equations, which arediscretized numerically by using the Generalized Differential Quadrature (GDQ) method. In addition, an efficient isogeometric NURBS-based mapping is adopted for the distortion of the physical domain. The boundary conditions of the differential problem are modelled through a general distribution of linear elastic springs along the lateral surfaces of the three-dimensional doubly-curved solid. The theory is then applied to study the vibrational modes of structures with different curvatures and lamination schemes, taking into account a general distribution of CNTs along the thickness direction. The homogenized properties of CNTs layers are obtained from the Mori-Tanaka procedure, which accounts for the agglomeration effects of the dispersed nanofibers within the matrix. Unlike previous studies focusing on doubly-curved shells reinforced with composite materials and CNTs, the present formulation enables to derive the vibration characteristics of structures made of generally anisotropic materials with general orientations. Furthermore, the calibration of the parameters for the linear elastic springs’ distribution leads to general external constraints within a single element, thus reducing the computational cost of the problem.

Frequency Regulation in Nanocomposite Sandwich Joined Conical-Conical Shells with Open Cell Foam Partially Supported by Pasternak Elastic Foundation Using GDQ

This study investigated the natural vibration of a sandwich conical-conical shell that is partially supported by Winkler–Pasternak foundations. The sandwich system comprises an open-cell foam (OCF) core and laminated composite face sheets reinforced with graphene platelets using functionally graded models (FG-GPLRC). To account for through-the-thickness shear deformations and rotary inertias, the first-order theory of shells is combined with Donnell-type kinematic assumptions. Hamilton’s principle is utilized to establish the general motion equations and related boundary and continuity conditions. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature (GDQ) approach. An eigenvalue problem is formulated to analyze the corresponding mode shapes and vibration frequencies for the shell ends and continuity criteria under various boundary conditions. Parametric experiments are conducted for sandwich conical-conical shells partially supported by Winkler–Pasternak foundations. This study examined the impact of multiple factors on the frequency of a joined shell structure, including the total thickness, FG-GPL face sheet thickness, various boundary conditions, the length of each cone segment, and the Winkler and shear elastic foundation. Notably, this research also analyzed the effects of a partial elastic foundation on the structure’s frequency for the first time.

Inherent resonance of carbon and graphene-based nanocomposite coupled single-span arch beams

In recent decades, there has been a significant rise in the utilization of composite materials for various engineering applications. These advanced materials offer the potential to improve the mechanical properties and vibration characteristics of structural components. This particular study is dedicated to enhancing the vibration performance of coupled curved-curved beams that feature a linear elastic mid-layer, achieved through the incorporation of carbon nanotubes (CNTs), graphene nanoplates (GNPs), and graphene oxide powder (GOPs). The governing equations of the system are solved using the generalized differential quadrature (GDQ) method. While previous research primarily focused on the use of CNTs to enhance the vibration behavior of coupled-curved beams, this study delves into the utilization of multiple nanofillers for this purpose. An essential aspect of modeling composite materials lies in determining their equivalent mechanical properties. This research undertakes a comparison between the rule of mixture (RoM) and Halpin-Tsai methods for calculating these properties, revealing that frequencies derived from the RoM method are higher than those obtained through the Halpin-Tsai approach. Additionally, the study highlights that systems incorporating GNPs demonstrate higher frequencies at lower nanofiller volumes, with CNTs and GOPs following in ranking. However, this hierarchy shifts at higher nanofiller volumes. The arrangement of nanofillers within the system is influenced by its boundary conditions, with the curvature of the bottom beam playing a significant role in affecting vibration behavior. Increasing the radius of the bottom beam (R2) leads to higher system frequencies, which subsequently decrease with higher R2 values.

Adjusting dynamic and damping performance in fiber-reinforced magnetorheological elastomer composite conical shells subjected to compressive loads

This study explores the dynamic behavior of axially loaded conical shells composed of magnetorheological elastomer composites (MRECs). The investigation focuses on analyzing the impact of compressive loads on the frequencies and loss factors of these composites. The MRECs consist of laminates reinforced with fibers, enabling tunable viscoelastic properties through the utilization of magnetic fields. The equivalent mechanical characteristics of the MREC are calculated using the modified Voigt and Halpin–Tsai micromechanical rules. By employing the first-order shear deformation theory and Sanders-based strains, the kinematics of the conical shell are accurately modeled. A semi-analytical analysis, combining trigonometric expansion and the generalized differential quadrature (GDQ) methods, is employed to solve the system's highly coupled partial differential equations. Two consecutive analyses are performed. Firstly, the critical buckling load of the MREC conical shell is obtained through a stability analysis. Subsequently, the frequencies and loss factors of the shell in the pre-buckling zone are computed. Parametric analyses are conducted to study the impact of key factors, including magnetic field strength, compressive load magnitude, composite properties, geometric parameters, and boundary conditions. The proposed methodology is rigorously validated through meticulous accuracy and convergence analysis. The outcomes provide valuable insights into the dynamic response of axially loaded magnetorheological elastomer composite conical shells, highlighting their potential in adjustable dynamic and damping systems.

Buckling and post-buckling of anisogrid lattice-core sandwich plates with nanocomposite skins

This research offers an in-depth examination of the buckling and post-buckling responses observed in sandwich plates distinguished by the presence of an anisogrid lattice core and face layers composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC). To model this structure, a global continuous model for the lattice core is employed, while the displacement field is approximated using the First-Order Shear Deformation Theory (FSDT). Geometrically nonlinear strain relations based on the von-Kármán assumptions are incorporated to accurately capture the structural response. The homogenization of the FG-CNTRC skins is achieved through the modified rule of mixture, enabling estimations of effective material properties. To separate pre-buckling and post-buckling phases, we implement the adjacent-equilibrium criterion. Solving the linear (pertaining to buckling) and nonlinear (related to post-buckling) equations is based on the Generalized Differential Quadrature (GDQ) technique alongside an iterative method grounded in the displacement control strategy. Our analysis explores the influence of core and skin characteristics on the stability of these sandwich plates. By examining critical buckling thresholds, post-buckling equilibrium paths, and the underlying mechanisms driving these responses, we contribute insights into the structural stability and performance of such composite structures. This study not only enhances our understanding of buckling behavior in anisogrid lattice core sandwich plates but also holds promise for diverse engineering applications where structural stability is paramount.

Nonlinear dynamic response of a temperature-dependent FGM spherical shell under various boundary conditions and thermal shocks: Examination of dynamic snap-through

In this study, a comprehensive exploration of the nonlinear dynamic snap-through phenomenon observed in shallow spherical shells composed of functionally graded materials (FGMs) is examined. The analysis encompasses the interplay of thermomechanical characteristics within metal and ceramic phases, taking into account their temperature-dependent behaviors. To address this problem, the transient heat conduction equation in a nonlinear form, accounting for the temperature dependence factor is derived. To solve this complexity, the Generalized Differential Quadrature (GDQ) method is employed, coupled with the Crank-Nicolson time integrating scheme, leveraging an iterative approach for precision. The formulation of nonlinear dynamic motion equations is based on Hamilton's principle and incorporates the utilization of nonlinear strain-displacement equations and uncoupled thermoelasticity. In the spatial domain, GDQ proves invaluable in solving these nonlinear, coupled equations. In the temporal domain, the Newmark procedure in conjunction with the Newton-Raphson iterative method are utilized to navigate through dynamic equations, yielding insights into shell's behavior over time. In the initial stages of the investigation, we validate our spherical shell formulation and methodologies by comparing shell's response with existing, simpler works. This step ensures the reliability and accuracy of our model. The primary objective is to assess the presence and characteristics of the thermal snap-through phenomenon within the shell under these diverse conditions.

Equivalent layer-wise theory for the hygro-thermo-magneto-electro-elastic analysis of laminated curved shells

The paper presents a multifield formulation involving five different physical problems under the equilibrium thermodynamic conditions for laminated doubly-curved shell structures. More specifically, the study focuses on the coupling between the mechanical elasticity and the thermo-hygrometric problem, while also considering the magneto-electricity of the solid. The configuration variables are described with a generalized formulation based on the Equivalent Layer Wise (ELW) approach, taking into account higher order polynomial interpolations along the thickness direction. The fundamental relations are derived from the Master Balance principle and solved using the Navier's method. Furthermore, the three-dimensional response of the doubly-curved shell solid in terms of primary and secondary variables is recovered from the two-dimensional solution with a methodology based on the three-dimensional multifield balance equations and the Generalized Differential Quadrature (GDQ) numerical technique. Some examples are then presented in which panels of different curvatures and lamination schemes are investigated. The results are compared with success to those coming from three-dimensional numerical models developed with a commercial software. It is shown that the present analytical solution is a valid tool for modelling multifield problems for the evaluation of the response of doubly-curved shells under generalized external actions and pre-determined values of the configuration variables.

Study on the Fundamental Frequency and Dynamic Mode of Traveling Wave Vibration of Rotating PJCS

The vibrations of rotating joined conical–conical shells with classical supported conditions have been studied extensively. As a matter of fact, in some cases, these classical boundary conditions cannot exactly model actual situations. Moreover, theoretical frameworks on them are still limited. This research aims to investigate the fundamental frequencies and dynamic mode shapes of the traveling wave of the rotating porous metal material joined conical–conical thin shells (PJCS) with elastic supports. By utilizing artificial spring technology, arbitrary elastic supported boundary conditions and classical boundary conditions are achieved efficiently. A new dynamic model has been formulated with the help of the first-order shear deformation theory (FSDT) and Hamilton’s principle. By employing the generalized differential quadrature (GDQ) method along with stress boundary conditions and generalized eigenvalues, various factors such as porosity, semi-vertex angles and stiffness are analyzed for their impact on the fundamental frequencies of forward wave (FW), backward wave (BW) and mode shapes. The presented results are validated through the convergence and comparison studies from literatures. The interesting and novel results indicate that the in-plane displacement constraints have the most significant impact on the critical speed, while the lateral displacement constraint has the least effect. The vibrations are more easily excited for the part with a larger half vertex angle. Rotating PJCS with Type 1 has the biggest critical rotating speed.

Analysis of hot band effects on the vibration and stability of a rotating FGM heavy vehicle disc brake

In this paper, vibration and stability analysis of a rotating Functionally Graded Material (FGM) disc brake considering the hot band phenomena are studied. In order to model the hot band phenomena, the disc brake is divided into two sections, the inner disc and the outer ring. The latter is where thermal and mechanical loads are applied. Based on Hamilton’s total potential energy principle and applying first-order shear deformation theory (FSDT), the governing equations are derived. Furthermore, the derived equations of motion are solved via the Generalized Differential Quadrature (GDQ) method. The results reveal that using a functionally graded material would improve the mechanical and thermal characteristics of a disc brake, most notably higher buckling temperature and smaller deflection in a disc brake. Moreover, using a functionally graded material for a disc brake will result in increasing the natural frequency of the disc, that is to say, the buckling will occur at higher rotational speeds. Following this further, the effects of the radius of loading annular ring, where thermal and mechanical loads are applied, as well as gradation index and mechanical and thermal loading, are also studied. The results show there is a non-linear relation between the radius of the loading annular ring and deflection as well as critical buckling temperature. Furthermore, a comparison of results with those available in the literature shows good agreement.

On the Importance of the Recovery Procedure in the Semi-Analytical Solution for the Static Analysis of Curved Laminated Panels: Comparison with 3D Finite Elements.

The manuscript presents an efficient semi-analytical solution with three-dimensional capabilities for the evaluation of the static response of laminated curved structures subjected to general external loads. A two-dimensional model is presented based on the Equivalent Single Layer (ESL) approach, where the displacement field components are described with a generalized formulation based on a higher-order expansion along the thickness direction. The fundamental equations are derived from the Hamiltonian principle, and the solution is found by means of Navier's approach. Then, an efficient recovery procedure, derived from the three-dimensional elasticity equations and based on the Generalized Differential Quadrature (GDQ) method, is adopted for the derivation of the three-dimensional solution. Some examples of investigation are presented, where the numerical predictions of refined three-dimensional Finite-Element-based models are matched with a high level of accuracy. The model is validated for both straight and curved panels, taking into account different lamination schemes and load shapes. Furthermore, it is shown that the numerical solution to the elasticity problem in the recovery procedure is determining and accurately predicting the three-dimensional static response of the doubly-curved shell solid.

Advanced Thick FGM Plates-Cylindrical Shells in Linear Unsteady Supersonic Flow

Introduction: Thick functionally graded (FG) plates-cylindrical shells with third-order shear deformation theory (TSDT) of displacement models under thermal vibration in linear unsteady supersonic flow are investigated by using the generalized differential quadrature (GDQ) method. Method: The nonlinear coefficient term of TSDT models and the advanced nonlinear shear correction coefficient expression are used to study the thermal vibration of thick FG plates and cylindrical shells in linear unsteady supersonic flow. At the intersection of FG plates-cylindrical shells, the displacements, stress resultants, and moment resultants could be reasonably assumed in the continuity condition for the differential volume in linear unsteady supersonic flow, respectively. Result: The dynamic equilibrium differential equations of thick FG plates and cylindrical shells in linear unsteady supersonic flow, respectively in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads consist of the linear supersonic aerodynamic pressure and inertia terms can be presented. Two parametric effects including environment temperature and functionally graded material (FGM) power law index on the thermal stress and center displacement of thick FG plates-cylindrical shells in linear unsteady supersonic flow are investigated. Conclusion: The linear unsteady supersonic flow over the outer surface of thick FG platescylindrical shells provides the additional aerodynamic effect on the values of displacements and stresses.

Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions

This study investigates the sensitivity of dynamic properties in coupled curved beams reinforced with carbon nanotubes (CNTs) to thermal variations. Temperature-dependent (TD) mechanical properties are considered for poly methyl methacrylate (PMMA) to be strengthened with single-walled CNTs (SWCNTs), employing the basic rule of mixture to define the equivalent mechanical properties of nanocomposites. The governing equations of motion are derived using a first-order shear deformation theory (FSDT) and Hamilton’s principle, accounting for elastic interfaces modeled using elastic springs. A meshfree solution method based on a generalized differential quadrature (GDQ) approach is employed to discretize the eigenvalue problem and to obtain the frequency response of the structure. The proposed numerical procedure’s accuracy is verified against predictions in the literature for homogeneous structural cases under a fixed environmental temperature. The systematic investigation assesses the impact of various geometric and material properties, including curvature, boundary conditions, interfacial stiffness, and CNT distribution patterns, on the vibrational behavior.

Efficient analysis on buckling of FG-CNT reinforced composite joined conical–cylindrical laminated shells based on GDQ method under multiple loading conditions

This work presents an efficient analysis of buckling for functionally graded carbon nanotube reinforced composite (FG-CNTRC) joined conical–cylindrical laminated shells with lateral pressure and axial compressive load considering thermal effect. The governing equilibrium equation of the joined conical–cylindrical shell (CO-CYL) is determined using the first-order shear deformation theory (FSDT), Donnell kinematic assumptions, von-Kármán geometrical nonlinearity, and considering thermal effect. A semi-analytical solution, generalized differential quadrature (GDQ) method, and artificial springs are applied during the solution procedure. The comparison studies have demonstrated a remarkable 97% reduction in running time compared to the finite element method (FEM).

Artificial neural networks coupled with numerical approach for the stability prediction of non-uniform functionally graded microscale cylindrical structures

Functionally graded materials (FGMs) have garnered significant attention in research due to their potential for creating innovative materials with diverse applications. This study focuses on investigating the behavior of steel-concrete functionally graded structures under axial loading, specifically exploring nonlinear buckling. The research considers cylindrical structures with simply-supported boundary conditions and analyzes four different configurations with varying thickness. The governing equations are derived using the modified couple stress theory, incorporating assumptions of nonlinear von-Kármán theory coupled with classical beam theory. Numerical solutions are obtained using the general differential quadrature (GDQ) method. A notable aspect of this work is the use of Artificial Neural Networks (ANN) to predict nonlinear buckling forces, greatly enhancing the accuracy and efficiency of the predictions. This approach offers a novel method for analyzing concrete structures. The results are validated by comparing them with data from other published works, using dimensionless parameters for consistent comparisons. The findings provide a comprehensive analysis of various parameters, including wall thickness function and porosity, on the buckling stability of the cylindrical structure. Both numerical and ANN approaches are employed to report the effects. The study concludes that the uniform thickness cylinder exhibits the highest resistance to axial loading, resulting in buckling.

Aero-Hygro-Thermoelastic size-dependent analysis of NCMF-reinforced GNPs sector microplates located between piezoelectric patches in supersonic flow considering surface stress effects

Lately, metal foams have been widely used in various engineering structures due to their exceptional mechanical properties, including their lightweight. However, scientists have proposed adding nano-fillers like graphene nanoplatelets (GNPs) to overcome the issue of low stiffness caused by increased porosity. Hence, aerodynamic stability and vibrational behaviors of nanocomposite metal foams (NCMF)-reinforced GNPs sector microplate with piezoelectric/metallic face sheets are provided in this work. The microstructure is sited on Kerr elastic substrate and also is in a hygro-thermo-electrical media, subjected to supersonic flow based on first-order piston’s theory (FPT). Gaussian random field (GRF), Halpin-Tsai, and extended rule of mixture (ERM) in conjunction with Gurtin-Morduch, modified strain gradient theories, and Hamilton’s principle are used to derive the governing equations and linked boundary conditions. Generalized differential quadrature (GDQ) scheme is chosen to attain the results. By validating the results’ precision, the effects of different parameters on the natural frequencies (NFs) and critical aerodynamic pressures (CAPs) are observed. It is perceived that adding GNPs enhances both NFs and CAPs significantly, and increasing porosity up to seventy percent, leads the results to drop. Various industries use simpler models to evaluate the microstructure, therefore the present model can pave the way for future uses and push the boundaries of knowledge in these industries based on the results it produces.

A variational quantum algorithm-based numerical method for solving potential and Stokes flows

This paper presents a numerical method based on the variational quantum algorithm to solve potential and Stokes flow problems. In this method, the governing equations for potential and Stokes flows can be respectively written in the form of Laplace's equation and Stokes equations using velocity potential, stream function and vorticity formulations. Then the finite difference method and the generalized differential quadrature (GDQ) method are applied to discretize the governing equations. For the prescribed boundary conditions, the corresponding linear systems of equations can be obtained. These linear systems are solved by using the variational quantum linear solver (VQLS), which resolves the potential and Stokes flow problems equivalently. To the best of authors' knowledge, this is the first study that incorporates the GDQ method which is inherently a high-order discretization method with the VQLS algorithm. Since the GDQ method can utilize much fewer grid points than the finite difference method to approximate derivatives with a higher order of accuracy, the size of the input matrix for the VQLS algorithm can be smaller. In this way, the computational cost may be saved. The performance of the present method is comprehensively assessed by two representative examples, namely, the potential flow around a circular cylinder and Stokes flow in a lid-driven cavity. Numerical results validate the applicability and accuracy of the present VQLS-based method. Furthermore, its time complexity is evaluated by the heuristic scaling, which demonstrates that the present method scales efficiently in the number of qubits n and the precision ε. This work brings quantum computing to the field of computational fluid dynamics. By virtue of quantum advantage over classical methods, promising advances in solving large-scale fluid mechanics problems of engineering interest may be prompted.

Higher order theories for the modal analysis of anisotropic doubly-curved shells with a three-dimensional variation of the material properties

The manuscript investigates the dynamic properties of doubly-curved shell structures laminated with innovative materials employing the Generalized Differential Quadrature (GDQ) method. A higher order generalized expression of the displacement field variable is assumed following the Equivalent Single Layer (ESL) approach. The geometrical description of the structures follows the differential geometry basics, whereas arbitrarily-shaped domains are distorted by means of generalized isogeometric blending functions. A three-dimensional generally anisotropic elastic constitutive behaviour is assumed within each layer of laminates, and a smooth variation of the material properties is implemented. In particular, a two-dimensional distribution alongside the physical domain is considered, whereas a through-the-thickness dispersion of the material properties is associated to each layer, taking into account both polynomial and non-polynomial analytical expressions. The fundamental equations are derived from the Hamiltonian principle, and they are solved directly in strong form via the GDQ method taking into account a non-uniform discrete computational grid. After some preliminary validating examples, some parametric investigations are performed systematically, showing the influence of the variation of the material properties on the modal response of the structures characterized by different lamination schemes, curvatures and boundary conditions. The proposed model allows for the dynamic analysis of the structures of different curvatures characterized by very complicated dispersions of the material properties following a simple and accurate methodology.

Self-weight and thermal bifurcation analysis of functionally graded vertical annular plates

The present paper thoroughly investigates the influence of structural weight on the thermal buckling of standing circular and annular plates. The structure is composed of symmetrically functionally graded materials (FGM) in terms of its thickness. The first-order theory of shear deformation along with nonlinear strain–displacement equations has been employed to derive the governing equations of the FG plate. First, the asymmetric pre-buckling path of the structure under the influence of weight and heat is calculated by a semi-analytical method. The plate's stability relations are extracted using the adjacent equilibrium criterion and virtual work principle. Then, the stability equations are analyzed by the 2D generalized differential quadrature (GDQ) method. Parametric results are presented based on the analysis of the FGM power-law index, geometric ratios, and edge constraints on the buckling of the vertical plate. The study reveals the critical power-law index and thickness at which the structure becomes unstable, shedding light on the previously neglected calculation of weight-induced buckling in vertical structures. Also, the effect of weight on the buckling mode shapes of heavy plates and disturbing its symmetry is demonstrated. Additionally, this study presents the first investigation into the effect of weight on the thermal stability of annular FG plates.

An Explicit Modeling for the Pseudoelastic Response of Porous SMA Thick-Walled Cylinders under Internal and External Pressure

Shape-memory alloy (SMA) structures, either dense or porous, are widely used because of their functional characteristics arising from shape-memory effect and pseudoelasticity. Hence, the precise modeling of these materials is essential due to design issues. In this study, the solution algorithm based on an explicit phenomenological constitutive model is investigated to analyze pressurized cylinders. The proposed method is extracted following return mapping scheme to find the evolution of phase transformation in these structures. The formulation is validated by comparing uniaxial tension–compression curves during loading, unloading and thermal actuation against published literature. By the aid of this model, assuming elastic and phase transformation strains, a comprehensive study is presented to inspect stress, strain and displacement fields along the thickness of internally and externally pressurized thick-walled cylinders. To this end, the governing nonlinear equation in elastic–inelastic zones is discretized by the efficient method of generalized differential quadrature (GDQ) and the resultant relations are solved numerically. The influences of internal and external pressure on stress distribution along the cylinder wall during loading–unloading for different temperatures including phase transformation and saturation states are discussed. Also, the closed-form solution for critical pressure at the onset of the phase transformation is studied.