Investigating the Elastic Response of Smart Cylinders Under Asymmetric Loading

This paper investigates the hygrothermal-magneto-elastic response of functionally graded piezomagnetic (FGPM) cylinders under asymmetric loading. The cylinders are supported by a Winkler-type elastic foundation, and their properties vary with the radius according to a power-law function. By solving 2D equations of Fickian diffusion and Fourier relations, the distribution of asymmetric moisture concentration and temperature field is determined. Incorporating constitutive equations into mechanical and magnetic equilibrium equations yields three second-order partial differential equations. The equations are solved using the separation of variables and complex Fourier series. Simulation results demonstrate the influence of hygrothermal loading, magnetic field, elastic foundation, and material inhomogeneity on the cylinder's response.


Introduction
Piezomagnetic materials, possessing both piezoelectric and magnetic characteristics, have garnered considerable attention due to their potential applications in aerospace, automotive, and biomedical industries for sensing and actuating purposes.Functionally graded materials (FGMs), characterized by spatially varying compositions enabling tailored mechanical, thermal, and electromagnetic properties, have also gained prominence.The combination of piezoelectric and FGM materials has led to the emergence of smart structures with exceptional properties.To effectively employ these smart structures in real-world scenarios, comprehending their response under diverse loading conditions is imperative.Researchers have extensively investigated the diverse loading-induced behaviors of smart structures composed of piezoelectric and functionally graded materials (Sui et al., 2023;Yang et al., 2021).Dini et al. examined the effects of strain gradients and flexoelectricity on thermal stresses in micro-cylinders made of functionally graded piezoelectric materials (Dini et al., 2020;Hosseini et al., 2017).Galic and Horgan provided an exact solution for the axisymmetric problem of a piezoelectric cylinder subjected to internal pressure and electric loading (Galic & Horgan, 2003).Das et al. investigated stress and displacement fields in a FGM thick-walled sphere under constant interior and exterior pressure, using analytical and numerical models that account for radial variations in material properties (Das et al., 2023).Dai and Wang studied the thermo-electro-elastic behavior of a piezoelectric cylinder subjected to various loading conditions (Dai & Wang, 2006).Additionally, Khoshgoftar et al. analyzed the thermo-piezo-electric response of FG cylinders under thermal and mechanical loads (Khoshgoftar et al., 2009).Dini et al. presented analytical solutions for the magneto-thermo-elastic behavior of cylindrical and spherical pressure vessels, as well as three-layered sandwich disks made of FGMs, considering internal heat generation and convective boundary conditions (Dini et al., 2019;Hosseini & Dini, 2015;Nematollahi et al., 2019).Several studies have investigated the behavior of FG and piezoelectric materials under various physical loadings (Fazelzadeh et al., 2011;Hosseini et al., 2019;Hosseini & Fazelzadeh, 2010;Hosseini et al., 2011;Rastgoo et al., 2017;Zandi-Baghche-Maryam et al., 2022).Chen et al. explored the 3D free vibration of FGPM cylinders containing compressible fluid and derived frequency equations based on 3D piezoelasticity exact equations using the state-space formulation (Chen et al., 2004).Jabbari et al. presented a general solution for the 3D thermal and mechanical stresses in FG cylinders, assuming power-law material properties and employing Bessel functions and Fourier series to solve the Navier equations (Jabbari et al., 2007).Shao et al. provided an exact solution for the thermomechanical behavior of FG cylinders under thermal and mechanical loadings using the Laplace transform method and complex Fourier series (Shao et al., 2008).Li et al. (2023) investigated the multi-field coupling of FG materials by analyzing the thermal distribution, displacement, strain, and stress of rotating FG cylinders or circular disks subjected to a uniform constant magnetic field (Li et al., 2023).Allam et al. provided an analytical solution for the interaction of electric potentials, electric displacement, elastic deformations, and hygrothermal effects in hollow and solid cylinders made of functionally graded piezoelectric material (Allam et al., 2014).Dini and Abolbashari investigated the hygrothermo-electroelastic response of cylinders composed of FGPM materials under non-axisymmetric thermal, mechanical, and electrical loadings (Dini & Abolbashari, 2016).Sayman conducted a stress analysis for multi-layered composite cylinders subjected to hygrothermal loading, considering both free-end and fixed-end boundary conditions (Sayman, 2005).
The behaviors of smart structures composed of piezoelectric and functionally graded materials have been extensively studied under various loading conditions.However, the specific investigation of the hygrothermal and electromagnetic loading effects on functionally graded piezomagnetic cylinders, considering the presence of an elastic foundation, remains limited.This manuscript aims to provide a comprehensive analysis of the behavior of such cylinders under hygrothermal and electromagnetic loading, taking into account the influence of an elastic foundation.The analysis focuses on asymmetric loading conditions to enhance the realism and applicability of the study to real-world scenarios.

Theoretical Formulation
The current investigation examines a FG piezomagnetic cylinder, which is subjected to mechanical, magnetic, and hygrothermal loads.The cylinder's inner and outer radii are denoted as  and , respectively, and it is supported by a Winkler elastic foundation with a stiffness coefficient of   at its outer surface.The material behavior of the piezomagnetic FG cylinder is modeled using linear constitutive equations that account for the effects of hygrothermal, mechanical, and magnetic fields, as expressed in Eq. ( 1) (Arani et al., 2010;Saadatfar & Aghaie-Khafri, 2014).
The equilibrium equations of a piezomagnetic cylinder, which take into account both mechanical and magnetic effects, can be expressed in terms of the radial and circumferential directions as follows: Equation ( 4) describes the mechanical, magnetic, and hygrothermal properties of the piezomagnetic cylinder, which are characterized by different non-homogeneity constants depending on the specific property under consideration (Akbarzadeh & Chen, 2013).
where  1 ,  2 , and  3 are the non-homogeneity constants of the material.Substituting Eqs. ( 1) and ( 4) into Eq.( 3) results in three second-order partial differential equations that are coupled.

Heat Transfer and Moisture Diffusion Problem
The Fourier heat transfer equation in steady-state, without any internal heat generation, for a 2D problem in cylindrical coordinates is expressed below along with the general boundary conditions (Sih G.C., 1986).
where   and   expressed by Eq. ( 4) are the thermal conductivity coefficients.  represents the constant thermal parameters associated with the conduction and convection coefficients.Functions  1 () and  2 () are given functions in the inner and outer radius, respectively.Simplifying Eq. ( 5) yields: The steady-state 2D equation governing Fickian moisture diffusion in a functionally graded piezomagnetic cylinder can be expressed in cylindrical coordinates, along with the general boundary conditions, as follows (Sih G.C., 1986): In Eq. ( 7),   and   are the moisture diffusion, and vary according to Eq. ( 4) along the cylinder thickness.  represents the constant parameters related to moisture, and ℎ 1 () and ℎ 2 () are given functions at the inner and outer radii, respectively.To solve Eqs.(6, 7), the complex Fourier series method can be employed.By solving Eqs.(1-7) we can obtained, displacments, stresses and magnetic induction components

Numerical results
This Fig. 2 illustrates the impact of hygrothermal loading on the distribution of displacement, stresses, and magnetic potential at  =  3 ⁄ , assuming   = 10 9 / 3 and  1 =  2 =  3 =  = 0.5.Fig. 2a indicates that the radial stresses decrease as the hygrothermal loading increases.In Fig. 2b, it can be realized that an increase in hygrothermal loading results in a decrease and increase in circumferential stress at

Conclusion
In this study, hygrothermal stresses in thick-walled cylinders made of functionally graded piezomagnetic materials were investigated considering elastic foundation and asymmetric loadings.The mechanical, hygrothermal, electrical, and magnetic properties were assumed to be a function of the radius of the cylinder according to the power law.The distributions of temperature and moisture concentration were computed using 2D equations of Fourier heat transfer and Fickian moisture diffusion.The coupled partial differential equations were solved using the method of separation of variables and the complex Fourier series.Numerical simulations were obtained for a certain angle to investigate the effects of hygrothermal loading, elastic foundation, and non-homogeneity constant.Based on the results and analysis of the diagrams, the following conclusions were made in this study: • The non-homogeneity constant has significant effects on the distribution of stresses, magnetic potential, temperature, and moisture concentration in the thick-walled cylinder.In addition, when  1 =  2 =  3 = , increasing the non-homogeneity constant in the range of  > 0 leads to a decrease in the absolute values of radial stress, and magnetic potential, while the decrement of non-homogeneity constant in the range of  < 0 increases the above-mentioned parameters.
• Hygrothermal loading has remarkable effects on the distribution of stresses, and magnetic potential.