Research Article | Open Access
Mechanics of Static Slip and Energy Dissipation in Sandwich Structures: Case of Homogeneous Elastic Beams in Transverse Magnetic Fields
Mechanics of static slip and energy dissipation in sandwich structures with respect to two-layer homogeneous elastic beams in a transverse magnetic field is presented. The mathematical physics problem derives from nonuniform contact conditions of press sandwich layers or joints. On this theory, equations governing the stresses and the deflection profile are derived. By restricting analysis to the case of cantilever architecture, closed form polynomial expressions are computed for the deflection, interfacial slip, slip, and strain energies of the system. In particular, the effects of magnetoelasticity and interfacial pressure gradient on these properties are demonstrated for design analysis and engineering applications. In addition, explicit mathematical equations couched in magnetoelasticity and pressure gradient polynomial kernels with fractional coefficients for critical values of pressure for which no slip occurs at the tip and the optimum clamping pressure for optimal slip energy dissipation are derived. It is also shown for special cases that recent results in literature are recoverable from the theory reported in this paper.
Investigation into the vibration and magnetoelastic stability of ferromagnetic flexible structures, beams, beam-plates, plates, and shells is abound in literature. Concerning these theoretical analyses or experimental studies, comprehensive reviews of trends are reported in [1–18]. For theoretical analyses, the effect of eddy current in the ferromagnetic material was neglected. Lee  in contrast to earlier investigators studied the dynamic stability with magnetic damping arising from eddy current and derived an explicit expression for the destabilizing effect. For experimental investigation, Moon and Pao  were credited with the pioneering work on magnetoelastic buckling of a ferromagnetic thin plate in transverse magnetic fields. Their results showed that a ferromagnetic plate buckles and loses its stability when the magnetic intensity approaches a critical value that is functionally related to the geometric ratio of length to plate thickness via a 3/2 power law. Based on these findings, a mathematical problem was contrived as the magnetic body coupled model to predict the experimental phenomenon of magnetoelastic instability and critical magnetic field. Additional experiment later showed that the natural frequency of a beam-plate decreased with increasing magnetic field intensity and becomes near to zero, as the field attains a critical value, which causes the same beam-plate to buckle statically. Following the emergence of large discrepancy between the theoretical predictions and experimental results of Moon and Pao in , research attentions were further devoted to the study of magnetoelastic stability and buckling problems.
Some investigators, Wallerstein and Peach , Miya et al. , Peach et al. , and so forth, directed their attentions to finding satisfactory explanations for these discrepancies. Notwithstanding the significance of previous findings, Lee  investigated the dynamic stability of electrically conducting beam-plates in transverse magnetic fields via a concise theory of flexural vibration of magnetoelastic plates immersed in transverse magnetic fields. Similarly, in the 1990s, additional theories were developed for the study of magnetoelastic buckling and bending of ferromagnetic plates in transverse and/or oblique magnetic fields via a generalized variational principle of magnetoelasticity by Zhou et al. , Zhou and Zheng , and Zhou and Miya . In 2002, the experimental results of Wang et al.  confirmed the theoretical predictions in the 1990s.
Following renewed interest in magnetoelasticity and its applications in engineering systems, namely, magnetic storage elements, magnetic structural devices, geophysical physics, and plasma physics, attentions were directed on the study of magnetothermodynamic stress and perturbation of magnetic field vector in both solid and orthotropic thermoelastic cylinders. In this regard, Wang et al.  employed finite integral transforms to examine theoretically the magnetothermoelastic waves and perturbation of the magnetic field vector produced by thermal shock in a solid conducting cylinder. In this study, closed forms expressions were derived for magnetothermodynamic stress and perturbation response of an axial magnetic field vector in a solid cylinder. Comprehensively, Wang et al.  investigated the magnetothermoelastic responses and perturbation of the magnetic field vector in a conducting orthotropic thermoelastic cylinder subjected to thermal shock using finite Hankel integral transform, whilst Librescu et al.  and Wang et al.  studied the effect of magnetothermoelasticity of ferromagnetic conducting plates under excitations theoretically.
In related development, Wang and Dai  investigated the dynamic responses of piezoelectric hollow cylinders in axial magnetic field. This study led to the development of a concise analytical solution to reveal the interaction between mechanical and electromagnetoelastics responses of piezoelectric hollow cylinders subjected to arbitrary mechanical loadings and electric potential shock. An interpolation method was employed to solve the resulting Volterra integral equation of the second kind, arising from interaction between different physical fields. Furthermore, closed forms results were derived for dynamic stresses, electric-displacements and electric-potentials as well as perturbation responses using finite and Laplace integral transforms.
Meanwhile, the problem of stability loss and free vibration of electromagnetically conducting plate conveying an electric current in magnetic field environment was investigated by Hansanyan et al. . Following the theoretical models in the 1990s, Wang and Lee  considered the magnetic damping effect induced by the eddy current and its effect on dynamic stability. Application of these structures is receiving significant attentions in magnetic propulsion devices for space transport and exploration. From experimental investigations, ferromagnetic flexible structures are usually subjected to magnetic forces arising from the coupling or mutual influence of the magnetization and magnetic fields.
Following recent advances in the mechanics of sandwich layered elastic structures, in an environment of nonuniform interface pressure by Damisa et al. [19, 20], Olunloyo et al. , Olunloyo et al. , and Osheku and Damisa  investigated the flexural vibration of a two-layer magnetoelastic beam in a transverse magnetic field. In their study, equations of mathematical physics governing the stresses and the structural vibration were derived via laminated beam theory employing Newtonian form of Cauchy’s stress equations.
Although the study was restricted to the case of cantilever structure, the effects of magnetoelasticity, material conductivity, and interfacial pressure gradient on the system response were computed in the form of polynomial expression via Laplace and finite Fourier integral transforms.
The study also shows that each mode of vibration was governed by a two-dimensional family of natural frequencies. The natures of the closed forms expressions for the natural frequencies indicate that the oscillation ceases when the two become simultaneously zero. In both theory and experiment, this is the required condition for static or quasistatic buckling of any layered elastic structure in a transverse magnetic field.
In fact, it is an indication that with suitable geometric parameters and matching transverse magnetic field, critical damping can be enhanced. For special and limit cases, recent theoretical and experimental results were validated. Following the increasing significant of studying both theoretically and experimentally the required condition for static or quasistatic buckling of any layered elastic structure in a transverse magnetic field, this study is devoted to the comprehensive investigation of the characteristics of statically loaded homogenous two-layer sandwich magnetoelastic cantilever structure in a transverse magnetic field.
This paper is organized as follows. Section 1 introduces the problem under investigation within a general context. In Section 2, the essential analytical mechanics leading to the mathematical physics problem with additional specialized static boundary values ordinary differential equations are presented. In Section 3, formal analysis of the problem of interest using finite Fourier integral transform is discussed. Section 4 is concerned with the analysis of static slip, whilst in Section 5 the energy dissipation ability and damping capacity of the structure are analysed. In Section 6, simulated results are discussed. Finally, the paper ends with conclusion in Section 7.
2. Formulation of the Governing Differential Equation Problem Definition
As illustrated in Figure 1(a), the problem here is to examine analytically the effect of the pressure gradient on the damping properties of a statically loaded two-layer magnetoelastic beams clamped together in an environment of nonuniform pressure.
2.1. Underlying Assumptions
A two-layer elastic structure is subject to a transverse magnetic field. For the contrived structure, the upper and the lower layers are assumed to be perfectly press fit surfaces of homogenous magnetoelastic beams. The contact conditions between the mating layers as itemized in Damisa et al.  hold, namely;(i)there is continuity of stress distributions at the interface to sufficiently hold the separate layers together both in the pre- and postslip conditions;(ii)the static deflection of each beam is small compared with the span;(iii)during bending, the magnetoelastic structure has (upper and lower) layers such that each has its neutral plane which may not necessarily coincide with its geometric mid plane of the resultant structure. These neutral planes are located at and , where is a function of as illustrated in Figure 1(b);(iv)the approximations involved in the forgoing beam theory are such that the field variables are linear and are expressible in terms of the derivatives of the transverse static deflection and is taken to be same for both layers.
By defining and as displacements along and , respectively, the following relations hold from the classical theory of elasticity, namely, Equation (2) can be evaluated as Now, ; , following assumption (iii) above, we can rewrite (3) as to obtain the following expression: where is the point of initiation of interfacial slip in the upper layer.
Similarly, the following expression holds for the lower layer as and evaluated to obtain the following expression: where admits same definition in the lower layer.
From classical theory of elasticity, the in-plane bending stress for the upper layer takes the following form: On substituting (5), the foregoing becomes Now is a nonlinear term, whilst is the strain at the point of initiation of static slip.
Following assumption (iv) (linear theory), is negligible while at the fixed end. Consequently, Similarly for layer (2), we have
On substituting , (11) becomes Following assumption (iv) (linear theory), is negligible while at the fixed end.
Consequently, Next, we invoke the static form of the generalized Cauchy stress equation in the absence of body forces, namely, where is the stress tensor.
In the upper and lower halves, (14) admits the following forms: On substitution of (10), (13), we rewrite the above as Following Goodman and Klumpp , (16)-(17) must satisfy the following postslip boundary conditions along -plane, namely, From Lee , the in-plane shear stress arising from electromagnetic surface traction follows from the generalized Maxwell’s stress tensor defined as By enforcing small perturbation on the primary bias field due to the field- structure interaction, following Lee , the following expression ensues Here the field quantities in lower-case letters are assumed to be small magnetic and electric perturbation variables. Consequently, their products can be neglected. Under this circumstance, the Maxwell’s stress tensor in (19) reduces to the form Utilizing the relations in Lee , the in-plane and out-plane static frictional stresses are modified as Consequently, (16) can be integrated to obtain Similar expression can be derived for the lower layer as On substituting (23)-(24) into the second parts of (15a)-(15b) the generalized ordinary differential equation governing the static deflection of the sandwich magnetoelastic structure is. The following specialized ordinary differential equations can be formulated from the foregoing as follows.
Case 1. For the case of a beam-plate of thickness , (25) becomes By dropping the variable , the formulated equation governing the static deflection takes the form
Case 2. For the case of a beam-plate of thickness , (25) becomes By dropping the variable , the formulated equation governing the static deflection takes the form On the other hand, we can rewrite (26b) to obtain the following specialized ordinary differential equations.
Case 3. A two-layer sandwich homogenous magnetoelastic beam-plate of thickness with non-uniform pressure at the interface. For such a problem, the formulated equation governing the static deflection takes the form
Case 4. A two-layer sandwich homogenous magnetoelastic beam-plate of thickness with non-uniform pressure at the interface. For such a problem, the formulated equation governing the static deflection takes the form
3. Analysis of Static Deflection
The generalized governing differential equation for the static deflection of each layer takes the following form: For the cantilever architecture under investigation, the usual boundary conditions hold as follows: in conjunction with the generalized end condition reported in Damisa et al. , namely, By limiting our investigation to linear interface pressure profile, we obtainEquation (30) takes the form The solution to the above is sorted via the Fourier finite sine transform namely, Equation (33) in the Fourier transform plane takes the formFollowing Damisa et al. , the bending moment and static deflection in the Fourier transform plane are computed asThe Fourier inversion of the above yieldsUtilizing binomial expansion, (37) is rewritten as The semiinfinite series in (38) can be converted to spatial polynomials via the following closed form Fourier series representations: Consequently, we can write (39) in the formwhere Imposing the condition in (40), the deflection at the end of the sandwich magnetoelastic cantilever structure isSubstitution of the above with rearrangement giveswhere the following nondimensionalized parameters have been introduced as For the special case in (43), the transverse deflection at uniform pressure is which for the case of agrees with the results in Damisa et al. .
4. Analysis of Static Slip
As shown in Figure 1(e), during bending, each half of the layered elastic structure has its own neutral plane that does not necessarily coincide with the geometric mid plane through the interface because of the frictional stresses. For the sandwich structure, the geometrical description of the gross interfacial slip is defined in Figure 1(e). In view of the foregoing, the expressions for the displacements of the two adjacent opposite points follow from Taylor series approximation as: which for the case of first order theory reduce to the forms For this problem, and must be zero at the fixed end. Hence, the relative static slip at the interface of the elastic structure is given by Following Goodman and Klumpp , (49) becomes where is a dummy axial spatial variable of integration across the interface, and , denote the origin of the transverse spatial variable for each layer; where subscripts 1 and 2 refer to the upper and lower laminates.
On substituting (23)-(24) into the first parts of (15a)-(15b), the derived corresponding spatial bending stresses are, namely, This gives (50) as and on introducing the following nondimensionalized parameters: Equation (52) simplifies to the formwhere On setting in (54), the static slip at uniform pressure isThe case agrees with the result in . The foregoing expression allows us to compute the maximum slip at the tip of the magnetoelastic structure in the context of the pressure gradient as On setting in (57), the maximum static slip at the tip for the case of uniform pressure isThis suggests that in the presence of coulomb friction and interfacial pressure , there are critical values of pressure for which no slip occurs at the tip such as whereThe case gives the expression for critical values of pressure for which no slip occurs at the tip as