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International Scholarly Research NetworkISRN Mechanical EngineeringVolume 2012, Article ID 372019, 23 pagesdoi:10.5402/2012/372019

Research Article

Mechanics of Static Slip and Energy Dissipation inSandwich Structures: Case of Homogeneous Elastic Beams inTransverse Magnetic Fields

Charles A. Osheku

Centre for Space Transport and Propulsion, National Space Research and Development Agency,Federal Ministry of Science and Technology, FCT, Abuja, PMB 437, Nigeria

Correspondence should be addressed to Charles A. Osheku, [email protected]

Received 26 June 2012; Accepted 13 August 2012

Academic Editors: K. Abhary, J. Clayton, and K. Ismail

Copyright © 2012 Charles A. Osheku. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Mechanics of static slip and energy dissipation in sandwich structures with respect to two-layer homogeneous elastic beams in atransverse magnetic field is presented. The mathematical physics problem derives from nonuniform contact conditions of presssandwich layers or joints. On this theory, equations governing the stresses and the deflection profile are derived. By restrictinganalysis to the case of cantilever architecture, closed form polynomial expressions are computed for the deflection, interfacialslip, slip, and strain energies of the system. In particular, the effects of magnetoelasticity and interfacial pressure gradient onthese properties are demonstrated for design analysis and engineering applications. In addition, explicit mathematical equationscouched in magnetoelasticity and pressure gradient polynomial kernels with fractional coefficients for critical values of pressurefor which no slip occurs at the tip and the optimum clamping pressure for optimal slip energy dissipation are derived. It is alsoshown for special cases that recent results in literature are recoverable from the theory reported in this paper.

1. Introduction

Investigation into the vibration and magnetoelastic stabilityof ferromagnetic flexible structures, beams, beam-plates,plates, and shells is abound in literature. Concerning thesetheoretical analyses or experimental studies, comprehensivereviews of trends are reported in [1–18]. For theoreticalanalyses, the effect of eddy current in the ferromagneticmaterial was neglected. Lee [1] in contrast to earlier investi-gators studied the dynamic stability with magnetic dampingarising from eddy current and derived an explicit expressionfor the destabilizing effect. For experimental investigation,Moon and Pao [2] were credited with the pioneering workon magnetoelastic buckling of a ferromagnetic thin platein transverse magnetic fields. Their results showed thata ferromagnetic plate buckles and loses its stability whenthe magnetic intensity approaches a critical value that isfunctionally related to the geometric ratio of length to platethickness via a 3/2 power law. Based on these findings, amathematical problem was contrived as the magnetic body

coupled model to predict the experimental phenomenon ofmagnetoelastic instability and critical magnetic field. Addi-tional experiment later showed that the natural frequency ofa beam-plate decreased with increasing magnetic field inten-sity and becomes near to zero, as the field attains a criticalvalue, which causes the same beam-plate to buckle statically.Following the emergence of large discrepancy between thetheoretical predictions and experimental results of Moon andPao in [3], research attentions were further devoted to thestudy of magnetoelastic stability and buckling problems.

Some investigators, Wallerstein and Peach [4], Miya et al.[5], Peach et al. [6], and so forth, directed their attentionsto finding satisfactory explanations for these discrepancies.Notwithstanding the significance of previous findings, Lee[7] investigated the dynamic stability of electrically con-ducting beam-plates in transverse magnetic fields via aconcise theory of flexural vibration of magnetoelastic platesimmersed in transverse magnetic fields. Similarly, in the1990s, additional theories were developed for the study ofmagnetoelastic buckling and bending of ferromagnetic plates

2 ISRN Mechanical Engineering

in transverse and/or oblique magnetic fields via a generalizedvariational principle of magnetoelasticity by Zhou et al. [8],Zhou and Zheng [9], and Zhou and Miya [10]. In 2002,the experimental results of Wang et al. [11] confirmed thetheoretical predictions in the 1990s.

Following renewed interest in magnetoelasticity andits applications in engineering systems, namely, magneticstorage elements, magnetic structural devices, geophysicalphysics, and plasma physics, attentions were directed on thestudy of magnetothermodynamic stress and perturbation ofmagnetic field vector in both solid and orthotropic thermoe-lastic cylinders. In this regard, Wang et al. [12] employedfinite integral transforms to examine theoretically the mag-netothermoelastic waves and perturbation of the magneticfield vector produced by thermal shock in a solid conduct-ing cylinder. In this study, closed forms expressions werederived for magnetothermodynamic stress and perturbationresponse of an axial magnetic field vector in a solid cylinder.Comprehensively, Wang et al. [13] investigated the magne-tothermoelastic responses and perturbation of the magneticfield vector in a conducting orthotropic thermoelastic cylin-der subjected to thermal shock using finite Hankel integraltransform, whilst Librescu et al. [14] and Wang et al. [15]studied the effect of magnetothermoelasticity of ferromag-netic conducting plates under excitations theoretically.

In related development, Wang and Dai [16] investigatedthe dynamic responses of piezoelectric hollow cylinders inaxial magnetic field. This study led to the development of aconcise analytical solution to reveal the interaction betweenmechanical and electromagnetoelastics responses of piezo-electric hollow cylinders subjected to arbitrary mechanicalloadings and electric potential shock. An interpolationmethod was employed to solve the resulting Volterra inte-gral equation of the second kind, arising from interac-tion between different physical fields. Furthermore, closedforms results were derived for dynamic stresses, electric-displacements and electric-potentials as well as perturbationresponses using finite and Laplace integral transforms.

Meanwhile, the problem of stability loss and free vibra-tion of electromagnetically conducting plate conveying anelectric current in magnetic field environment was inves-tigated by Hansanyan et al. [17]. Following the theoreticalmodels in the 1990s, Wang and Lee [18] considered themagnetic damping effect induced by the eddy current andits effect on dynamic stability. Application of these structuresis receiving significant attentions in magnetic propulsiondevices for space transport and exploration. From experi-mental investigations, ferromagnetic flexible structures areusually subjected to magnetic forces arising from the cou-pling or mutual influence of the magnetization and magneticfields.

Following recent advances in the mechanics of sandwichlayered elastic structures, in an environment of nonuniforminterface pressure by Damisa et al. [19, 20], Olunloyo et al.[21], Olunloyo et al. [22], and Osheku and Damisa [23]investigated the flexural vibration of a two-layer magnetoe-lastic beam in a transverse magnetic field. In their study,equations of mathematical physics governing the stresses andthe structural vibration were derived via laminated beam

theory employing Newtonian form of Cauchy’s stress equa-tions.

Although the study was restricted to the case of can-tilever structure, the effects of magnetoelasticity, materialconductivity, and interfacial pressure gradient on the systemresponse were computed in the form of polynomial expres-sion via Laplace and finite Fourier integral transforms.

The study also shows that each mode of vibration wasgoverned by a two-dimensional family of natural frequencies.The natures of the closed forms expressions for the naturalfrequencies indicate that the oscillation ceases when the twobecome simultaneously zero. In both theory and experiment,this is the required condition for static or quasistatic bucklingof any layered elastic structure in a transverse magnetic field.

In fact, it is an indication that with suitable geometricparameters and matching transverse magnetic field, criticaldamping can be enhanced. For special and limit cases,recent theoretical and experimental results were validated.Following the increasing significant of studying both the-oretically and experimentally the required condition forstatic or quasistatic buckling of any layered elastic structurein a transverse magnetic field, this study is devoted to thecomprehensive investigation of the characteristics of stati-cally loaded homogenous two-layer sandwich magnetoelasticcantilever structure in a transverse magnetic field.

This paper is organized as follows. Section 1 introducesthe problem under investigation within a general context. InSection 2, the essential analytical mechanics leading to themathematical physics problem with additional specializedstatic boundary values ordinary differential equations arepresented. In Section 3, formal analysis of the problem ofinterest using finite Fourier integral transform is discussed.Section 4 is concerned with the analysis of static slip, whilst inSection 5 the energy dissipation ability and damping capacityof the structure are analysed. In Section 6, simulated resultsare discussed. Finally , the paper ends with conclusion inSection 7.

2. Formulation of the Governing DifferentialEquation Problem Definition

As illustrated in Figure 1(a), the problem here is to examineanalytically the effect of the pressure gradient on the damp-ing properties of a statically loaded two-layer magnetoelasticbeams clamped together in an environment of nonuniformpressure.

2.1. Underlying Assumptions. A two-layer elastic structureis subject to a transverse magnetic field. For the contrivedstructure, the upper and the lower layers are assumed tobe perfectly press fit surfaces of homogenous magnetoelasticbeams. The contact conditions between the mating layers asitemized in Damisa et al. [20] hold, namely;

(i) there is continuity of stress distributions at the inter-face to sufficiently hold the separate layers togetherboth in the pre- and postslip conditions;

(ii) the static deflection of each beam is small comparedwith the span;

ISRN Mechanical Engineering 3

Transverse magnetic fieldz

F = F0

Magnetoelastic beam layer 1


h

h

x

P(x): interfacial pressure

(−)P(x): interfacial pressure

(a)

Upper neutral axis

Lower neutral axis

Z

Z

1

o

z1 = ψ(x)h

2

z2 = ψ(x)h

2

Z2

(b)


F = F0

h

xO+

Transverse magnetic field

z


(c)

F = F0

h

xO−


z

Transverse magnetic field


(d)

Neutral plane of layer

A1

w(x)

ΔU1

ΔU2

A2Neutral plane of layer

0 X

Z

A1

A2

F = F0



(e)

Figure 1: (a) Preslip geometry for the sandwich structure under static load. (b) conceptual description of the upper and lower layers neutralaxes. (c) upper layer postslip geometry under static load. (d) lower layer postslip geometry under static load. (e) mechanism of interfacialslip geometry.

(iii) during bending, the magnetoelastic structure has(upper and lower) layers such that each has its neutralplane which may not necessarily coincide with itsgeometric mid plane of the resultant structure. Theseneutral planes are located at z1 = ψ(x)(h/2) andz2 = −ψ(x)(h/2), where ψ(x) is a function of x asillustrated in Figure 1(b);

(iv) the approximations involved in the forgoing beamtheory are such that the field variables are linearand are expressible in terms of the derivatives of the

transverse static deflection W(x) and is taken to besame for both layers.

By defining u(x, z) and W(x) as displacements along xand z, respectively, the following relations hold from theclassical theory of elasticity, namely,

εx = du

dx; γxz =

(dW

dx+du

dz

)= 0, (1)

du

dz=−dW

dxsuch that:

∫ U1

U01

du

dzdz =

∫ U1

U01

du =−∫ Z0

Z1

dW

dxdz.

(2)


Equation (2) can be evaluated as

U1 −U01 = −∫ Z1

Z0

dW

dxdz = −(z0 − z1)

dW

dx. (3)

Now, ∀z0 = z; z1 = ψ(x)(h/2), following assumption (iii)

above, we can rewrite (3) as

U1 −U01 = −∫ Z1

Z0

dW

dxdz =− (z0 − z1)

dW

dx

=−(z − ψ(x)

h

2

)dW

dx,

(4)

to obtain the following expression:

U1 = −(z − ψ(x)

h

2

)dW

dx+U01(x), (5)

whereU01(x) is the point of initiation of interfacial slip in theupper layer.

Similarly, the following expression holds for the lowerlayer as

U2−U02 =−∫ Z\2

Z0

dW

dxdz =−(z0−z2)

dW

dx∀z=−ψ(x)

h

2(6)

and evaluated to obtain the following expression:

U2 = −(z + ψ(x)

h

2

)dW

dx+U02(x), (7)

where U02(x) admits same definition in the lower layer.

From classical theory of elasticity, the in-plane bendingstress for the upper layer takes the following form:

σx(x, z) = σx(x, z)1 = Eεx1 = EdU1

dx. (8)

On substituting (5), the foregoing becomes

σx(x, z)1

= EdU1

dx

= −E(d

dx

((z − ψ(x)

h

2

)dW

dx+U01(x)

))

=−E(d

dx

(z−ψ(x)

h

2

)dW

dx+(z−ψ(x)

h

2

)d2W

dx2+dU01(x)dx

)

=−E((−h

2

(dψ(x)dx

)dW

dx

)+(z−ψ(x)

h

2

)d2W

dx2+dU01(x)dx

).

(9)

Now −(h/2)(dψ(x)/dx)(dW/dx) is a nonlinear term, whilstdU01(x)/dx is the strain at the point of initiation of static slip.

Following assumption (iv) (linear theory), −(h/2)(dψ(x)/dx)(dW/dx) is negligible while dU01(x)/dx = 0 atthe fixed end. Consequently,

σx(x, z)1

= EdU1

dx

= −E(d

dx

((z − ψ(x)

h

2

)dW

dx+U01(x)

))

= −E(d

dx

(z − ψ(x)

h

2

)dW

dx

+(z − ψ(x)

h

2

)d2W

dx2+dU01(x)dx

)

= −E(z − ψ(x)

h

2

)d2W

dx2.

(10)

Similarly for layer (2), we have

σx(x, z) = σx(x, z)2 = Eεx2 = EdU2

dx. (11)

On substitutingU2 = −(z+ψ(x)(h/2))(∂W/∂x)+U02(x),(11) becomes

σx(x, z)2

= EdU2

dx

=−E(d

dx

((z+ψ(x)

h

2

)dW

dx+U02(x)

))

=−E(d

dx

(z+ψ(x)

h

2

)dW

dx+(z+ψ(x)

h

2

)d2W

dx2+dU02(x)dx

)

=−E((

h

2

(dψ(x)dx

)dW

dx

)+(z+ψ(x)

h

2

)d2W

dx2+dU02(x)dx

).

(12)


Following assumption (iv) (linear theory), (h/2)(dψ(x)/dx)×(dW/dx) is negligible while dU02(x)/dx = 0 at the fixed end.

Consequently,

σx(x, z)2

= EdU2

dx

= −E(d

dx

((z + ψ(x)

h

2

)dW

dx+U02(x)

))

= −E(d

dx

(z + ψ(x)

h

2

)dW

dx

+(z + ψ(x)

h

2

)d2W

dx2+dU02(x)dx

)

= −E(z + ψ(x)

h

2

)d2W

dx2.

(13)

Next, we invoke the static form of the generalized Cauchystress equation in the absence of body forces, namely,

∇ · �τ = 0, (14)

where �τ is the stress tensor.In the upper and lower halves, (14) admits the following

forms:

∂

∂xσ(x)1 +

∂

∂zτ(xz)1 = 0,

∂

∂zσ(z)1 +

∂

∂xτ(xz)1 = 0,

(15a)

∂

∂xσ(x)2 +

∂

∂zτ(xz)2 = 0,

∂

∂zσ(z)2 +

∂

∂xτ(xz)2 = 0.

(15b)

On substitution of (10), (13), we rewrite the above as

−Ezd3W

dx3+Eh

2d

dx

(ψ(x)

(d2W

dx2

))+dτ(xz)1

dz= 0, (16)

−Ezd3W

dx3− Eh

2d

dx

(ψ(x)

(d2W

dx2

))+dτ(xz)2

dz= 0. (17)

Following Goodman and Klumpp [24], (16)-(17) must sat-isfy the following postslip boundary conditions along xz-plane, namely,

σ(z)1(x, 0) = −p(x, 0); σ(z)2(x,−h) = p(x, 0);

τ(xz)1(x,h) = 0; τ(xz)2(x,−h) = 0;

τ2(xz)1(x, 0) = μ2σ2

(z)1(x, 0); τ2(xz)2(x, 0) = μ2σ2

(z)2(x, 0);

∫ h0σ(x)1(x, z)dz =

∫ h0zσ(x)1(x, z)dz = 0;

∫ 0

−hσ(x)2(x, z)dz =

∫ 0

−hσ(x)2(x, z)dz = 0.

(18)

From Lee [7], the in-plane shear stress arising from electro-magnetic surface traction follows from the generalized Max-well’s stress tensor τM defined as

τM = μM �H ⊗ �H + εm�Em ⊗ �Em

− 12

(μm �H · �H + εm�Em · �Em

)Is.

(19)

By enforcing small perturbation on the primary bias fielddue to the field- structure interaction, following Lee [7], thefollowing expression ensues

�H(x) = �H0 +�h1(x); �EM(x) = 0 +�e(x). (20)

Here the field quantities in lower-case letters are assumedto be small magnetic and electric perturbation variables.Consequently, their products can be neglected. Under thiscircumstance, the Maxwell’s stress tensor in (19) reduces tothe form

τM =(�B0 ⊗�h1 +�bm ⊗ �H0

)− 1

2

(�B0 ·�h1 +�bm · �H0

)Is.

(21)

Utilizing the relations in Lee [7], the in-plane and out-planestatic frictional stresses are modified as

τ(xz)1(x, 0) = μp(x, 0)− 2μ0

(1− μ0

μm

)B0

2 dW

dx,

τ(xz)2(x, 0) = −μp(x, 0)− 2μ0

(1− μ0

μm

)B0

2 dW

dx.

(22)

Consequently, (16) can be integrated to obtain

τ(xz)1(x,h)

=

⎛⎜⎜⎜⎝E(z2 − zh)

2d3W

dx3+

2μ0

(1− μ0

μm

)B0

2 (z − h)h

dW

dx

−μp(x, 0)(z − h)h

⎞⎟⎟⎟⎠.

(23)

Similar expression can be derived for the lower layer as

τ(xz)2(x,h)

=

⎛⎜⎜⎜⎝E(z2 + zh

)2

d3W

dx3+

2μ0

(1− μ0

μm

)B0

2 (z + h)h

dW

dx

+μp(x, 0)(z + h)

h

⎞⎟⎟⎟⎠.

(24)

On substituting (23)-(24) into the second parts of (15a)-(15b) the generalized ordinary differential equation gov-erning the static deflection of the sandwich magnetoelasticstructure is.

EId4W

dx4+bh

μ0

(1− μ0

μm

)B0

2 d2W

dx2= 1

2bhμ

dP

dx. (25)

The following specialized ordinary differential equations canbe formulated from the foregoing as follows.


Case 1. For the case of a beam-plate of thickness h, (25) be-comes

EI∗d4W

dx4+bh

μ0

(1− μ0

μm

)B0

2 d2W

dx2

= 12bhμ

dP

dx∀I∗ = bh3

12(1− υ2).

(26a)

By dropping the variable b, the formulated equation govern-ing the static deflection takes the form

Dd4W

dx4+h

μ0

(1− μ0

μm

)B0

2 d2W

dx2

= 12hμdP

dx∀D = Eh3

12(1− υ2).

(26b)

Case 2. For the case of a beam-plate of thickness 2h, (25)becomes

EI∗∗d4W

dx4+bh

μ0

(1− μ0

μm

)B0

2 d2W

dx2

= 12bhμ

dP

dx∀I∗∗ = b(2h)3

12(1− υ2)= 2bh3

3(1− υ2).

(27a)

By dropping the variable b, the formulated equation govern-ing the static deflection takes the form

D∗d4W

dx4+

2hμ0

(1− μ0

μm

)B0

2 d2W

dx2

= hμdP

dx∀D = 2Eh3

3(1− υ2).

(27b)

On the other hand, we can rewrite (26b) to obtain the fol-lowing specialized ordinary differential equations.

Case 3. A two-layer sandwich homogenous magnetoelasticbeam-plate of thickness h with non-uniform pressure atthe interface. For such a problem, the formulated equationgoverning the static deflection takes the form

EId4W

dx4+bh

μ0

(1− μ0

μm

)(1− υ2)B0

2 d2W

dx2

= 12bhμ

(1− υ2)dP

dx∀I = bh3

12.

(28)

Case 4. A two-layer sandwich homogenous magnetoelasticbeam-plate of thickness 2h with non-uniform pressure atthe interface. For such a problem, the formulated equationgoverning the static deflection takes the form

EI∗d4W

dx4+

2bhμ0

(1− μ0

μm

)(1− υ2)B0

2 d2W

dx2

= bhμ(1− υ2)dP

dx∀I∗ = 2Eh3

3(1− υ2).

(29)

3. Analysis of Static Deflection

The generalized governing differential equation for the staticdeflection of each layer takes the following form:

d4W

dx4+

12μ0

(1− μ0

μm

)B0

2

Eh2

d2W

dx2= α

dP

dx; ∀α = 6μ

Eh2.

(30)

For the cantilever architecture under investigation, the usualboundary conditions hold as follows:

W(0) = dW(0)dx

= d2W(L)dx2

= 0, (31)

in conjunction with the generalized end condition reportedin Damisa et al. [19], namely,

∫ h0τ(xz)1

(x)dz = F

2bat x = L. (32a)

By limiting our investigation to linear interface pressure pro-file, we obtain

p(x) = p0

(1 +

ε

Lx). (32b)

Equation (30) takes the form

d4W

dx4+

12μ0

(1− μ0

μm

)B0

2

Eh2

d2W

dx2= α

ε

Lp0. (33)

The solution to the above is sorted via the Fourier finite sinetransform namely,

[·]F =∫ L

0[·] sin

(nπx

L

)dx; [·] = 2

L

∞∑n=0

[·]F sin(nπx

L

).

(34)

Equation (33) in the Fourier transform plane takes the form

n4π4

L4WF(λn)− 12B0

2

Eh2

(1μ0

(1− μ0

μm

))n2π2

L2WF(λn)

=

⎛⎜⎜⎜⎝

(n3π3

L3(−1)n+1− 12B0

2

Eh2

(1μ0

(1− μ0

μm

))nπ

L(−1)n+1

)W(L)− nπ

LWxx(0)

αP0ε

(1+(−1)n+1

nπ

)⎞⎟⎟⎟⎠.

(35)


Following Damisa et al. [19], the bending moment and staticdeflection in the Fourier transform plane are computed as

Wxx(0) ={(

6FEbh3

− 6μP0

Eh2

(1 +

ε

2

))L +

12B02

Eh2

1μ0

(1− μ0

μm

)W(L)

},

WF(λn) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

(n3π3

L3(−1)n+1 − nπ

L(−1)n+1 12B0

2

Eh2

1μ0

(1− μ0

μm

))W(L)

−nπ

⎛⎜⎜⎝ 6FEbh3

−6μP0

(1 +

ε

2

)

Eh2

⎞⎟⎟⎠− nπ

L

12B02

Eh2

1μ0

(1− μ0

μm

)W(L) +

6μP0ε

Eh2

(1 + (−1)n+1

nπ

)

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(n4π4

L4− 12B0

2

Eh2

(1μ0

(1− μ0

μm

))n2π2

L2

) .

(36)

The Fourier inversion of the above yields

W(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2W(L)∞∑n=1

(−1)n+1 sinnπxnπ

− 24W(L)χ2

(1− μ0

μm

) ∞∑n=1

(−1)n+1

sinnπxn3π3

−12

⎛⎜⎜⎜⎜⎜⎜⎜⎝

L3

⎛⎜⎜⎝ F

Ebh3−μP0

(1 +

ε

2

)

Eh2

⎞⎟⎟⎠

∞∑n=1

sinnπxn3π3

+

(2χ2

(1− μ0

μm

)W(L)

) ∞∑n=1

((−1)n+1

n3π3

)sinnπx

⎞⎟⎟⎟⎟⎟⎟⎟⎠

+3L3μ

8Eh2P0ε

∞∑n=1

sin 2nπxn5π5

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(1− 12

(1− μ0

μm

)χ2

n2π2

) ∀χ = B02L2

μ0Eh2.

(37)

Utilizing binomial expansion, (37) is rewritten as

W(x) =(

1 + 12

(1− μ0

μm

)χ2

n2π2

)

×

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2W(L)∞∑n=1

(−1)n+1 sinnπxnπ

− 24W(L)χ2

(1− μ0

μm

) ∞∑n=1

(−1)n+1 sinnπxn3π3

−12

⎛⎜⎜⎜⎜⎜⎜⎜⎝

L3

⎛⎜⎜⎝ F

Ebh3−μP0

(1 +

ε

2

)

Eh2

⎞⎟⎟⎠

∞∑n=1

sinnπxn3π3

+2χ2

(1− μ0

μm

)W(L)

∞∑n=1

sinnπxn3π3

⎞⎟⎟⎟⎟⎟⎟⎟⎠

+3L3μ

8Eh2P0ε

∞∑n=1

sin 2nπxn5π5

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

.

(38)


The semiinfinite series in (38) can be converted to spatialpolynomials via the following closed form Fourier series rep-resentations:

x = 1π

∞∑n=1

(−1)n+1

nsinnπx,

∀0 < x < 1,

∞∑n=1

sinnxn3

= π2x

6− πx2

4+x3

12,

∀0 < x < 2,

∞∑n=1

sinnxn5

= π4x

90− π2x3

36+πx4

48− x5

240,

∀0 < x < 2,

∞∑n=1

sinnxn7

= 2π6x

405− π4x3

540+π2x5

720− πx6

1440+

x7

980,

∀0 < x < 2.

(39)

Consequently, we can write (39) in the form

W(x) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

2W(L)x − 12L3

((F

Ebh3− μP0(1 + ε/2)

Eh2

)Λ1

)+

3L3μ

8Eh2P0εΛ2 −W(L)

⎛⎝24χ2

(1− μ0

μm

)+ 15χ4

(1− μ0

μm

)2⎞⎠Λ2

−144L3χ2

(1− μ0

μm

)(F

Ebh3− μP0(1 + ε/2)

Eh2

)Λ3 +

9L3μ

2Eh2χ2

(1− μ0

μm

)P0ε Λ4

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

,

(40)

where

Λ1 =(x

6− x2

4+x3

12

);

Λ2 =(x

45− 2x3

9+x4

3− 2x5

5

);

Λ3 =(x

90− x3

36+x4

48− x5

240

);

Λ4 = 4x405

− 2x3

135+

2x5

45− 2x6

45+

64x7

245.

(41)

Imposing the condition dW/dx = 0 in (40), the deflection atthe end of the sandwich magnetoelastic cantilever structureis

W(L) =

⎛⎜⎜⎜⎜⎝

(F

Ebh3− μP0(1 + ε/2)

Eh2− μP0ε

240Eh2

)+ χ2

(1− μ0

μm

) (F

Ebh3− μP0

Eh2

)

+1350χ4

(1− μ0

μm

)2(F

Ebh3− μP0(1 + ε/2)

Eh2− 5μP0ε

26Eh2

)+

225χ6

(1− μ0

μm

)3 (F

Ebh3− μP0

Eh2

)

⎞⎟⎟⎟⎟⎠. (42)

Substitution of the above with rearrangement gives

W(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1− μP0

)(3x2 − x3) + μP0ε

(−3x2 +

1112x3 +

18x4 − 1

20x5)

+χ2

(1− μ0

μm

)((1− μP0

)(−28

13x3 + 11x4 +

515x5)

+ μP0ε(−214

45x3 − 83

15x4 +

26750

x5 − 15x6 +

288245

x8))

+χ4

(1− μ0

μm

)2((1− μP0

)(263x3 − 13x4 +

785x5)

+ μP0ε(−121

72x3 +

12148

x4 +12140

x5))

+χ6

(1− μ0

μm

)3((1− μP0

)(−35475

x3 − 17725

x4 +1107125

x5)

+ μP0ε(− 1

20x3 +

2013x4 − 24

13x5))

+χ8

(1− μ0

μm

)4((1− μP0

)(9775x3 − 97

50x4 +

296125

x5)

+ μP0ε(−1

6x3 − 20

13x4 +

3350x5))

+χ10

(1− μ0

μm

)5((1− μP0

)( 415x3 − 2

5x4 +

1225x5))

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

, (43)


where the following nondimensionalized parameters havebeen introduced as

W(x) = W(x)Ebh3

L3F; P0 = P0

(F/bh); x = x

L. (44)

For the special case ε→ 0 in (43), the transverse deflection atuniform pressure is

W(x)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1− μP0

)(3x2 − x3)

+χ2

(1− μ0

μm

)((1− μP0

)(−28

13x3 + 11x4 +

515x5))

+χ4

(1− μ0

μm

)2((1− μP0

)(263x3 − 13x4 +

785x5))

+χ6

(1− μ0

μm

)3((1− μP0

)(−35475

x3 − 17725

x4 +1107125

x5))

+χ8

(1− μ0

μm

)4((1− μP0

)(9775x3 − 97

50x4 +

296125

x5))

+χ10

(1− μ0

μm

)5((1− μP0

)( 415x3 − 2

5x4 +

1225x5))

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

,

(45)

which for the case of χ = 0 agrees with the results in Damisaet al. [19].

4. Analysis of Static Slip

As shown in Figure 1(e), during bending, each half of thelayered elastic structure has its own neutral plane thatdoes not necessarily coincide with the geometric mid planethrough the interface because of the frictional stresses. Forthe sandwich structure, the geometrical description of thegross interfacial slip is defined in Figure 1(e). In view ofthe foregoing, the expressions for the displacements of thetwo adjacent opposite points follow from Taylor seriesapproximation as:

ΔU1(x, z)

=(ΔU1(0, 0+) +

(z − ψ(x)

h

2

)dW(x)dx

+12

(z − ψ(x)

h

2

)2

×d2W(x)dx2

+16

(z − ψ(x)

h

2

)3 d3W(x)dx3

),

(46)

ΔU2(x, z)

=(ΔU2(0, 0−)+

(z + ψ(x)

h

2

)dW(x)dx

+12

(z+ψ(x)

h

2

)2

×d2W(x)dx2

+16

(z+ψ(x)

h

2

)3 d3W(x)dx3

),

(47)

which for the case of first order theory reduce to the forms

ΔU1(x, z) = ΔU1(0, 0+) +(z − ψ(x)

h

2

)dW(x)dx

,

ΔU2(x, z) = ΔU2(0, 0−) +(z + ψ(x)

h

2

)dW(x)dx

.

(48)

For this problem, Δu1(0, 0+) and Δu2(0, 0−) must be zero atthe fixed end. Hence, the relative static slip at the interface ofthe elastic structure is given by

ΔU(x, 0) = ΔU1(x, 0+)− ΔU2(x, 0−). (49)

Following Goodman and Klumpp [24], (49) becomes

ΔU(x, 0) = E−1∫ x

0

{(σx)1(ξ, 0+)− (σx)2(ξ, 0−)

}dξ, (50)

where ξ is a dummy axial spatial variable of integrationacross the interface, and 0+, 0− denote the origin of thetransverse spatial variable for each layer; where subscripts 1and 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,

(σx)1(x, z)

= −E2

(2z − h)d2W(x)dx2

+B0

2

μ0

(1− μ0

μm

)

× (W(x)−W(L)) + μP0(1 + εx)(x − L)h

,

(σx)2(x, z)

= −E2

(2z + h)d2W(x)dx2

− B02

μ0

(1− μ0

μm

)

× (W(x)−W(L))− μP0(1 + εx)(x − L)h

.

(51)

This gives (50) as

ΔU(x)

=∫ x

0

⎧⎨⎩d2W(x)

dξ2 + 2B0

2(

1− μ0

μm

)

×(W(ξ)−W(1)

)+ 2μP0

(1 + εξ

)(ξ − 1

)⎫⎬⎭dξ,

(52)

and on introducing the following nondimensionalized para-meters:

ΔU = ΔU(x) Ebh2

L2F; P0 = P0

(F/bh). (53)


Equation (52) simplifies to the form

ΔU =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4μP0 − 3

)(x2 − 2x

)+ μP0ε

(−6x +

7x2

4+

7x3

6− x4

4

)

+χ2

(1− μ0

μm

)((1− μP0

)Π1(x) + μP0εΠ2(x)

)+ χ4

(1− μ0

μm

)2((1− μP0

)Π3(x) + μP0εΠ4(x)

)

+χ6

(1− μ0

μm

)3((1− μP0

)Π5(x) + μP0εΠ6(x)

)+ χ8

(1− μ0

μm

)4((1− μP0

)Π7(x) + μP0εΠ8(x)

)

+χ10

(1− μ0

μm

)5((1− μP0

)Π9(x) + μP0εΠ10(x)

)+ χ12

(1− μ0

μm

)6(1− μP0

)Π11(x)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

, (54)

where

Π1(x) =(

2x − 8413x2 + 44x3 + 51x4

),

Π2(x) =(−241

120x − 214

15x2 − 332

15x3 +

6463240

x4

−4740x5 − 1

120x6 − 2304

245x7)

,

Π3(x) =(

128365

x+1

26x2 − 52x3− 1021

13x4 +

115x5 +

1710x6)

,

Π4(x) =(

4x − 12124

x2 +12112

x3 +5017360

x4

−8375x5 +

267300

x6 − 135x7 +

32245

x9)

,

Π5(x) =(

1695x − 354

25x2− 708

25x3 +

6967150

x4− 135x5 +

3915x6)

,

Π6(x) =(

2783720

x − 320x2 +

8013x3

−361333744

x4 +121240

x5 +121240

x6)

,

Π7(x) =(−368

125x − 97

25x2 − 194

25x3

+53350

x4 − 177125

x5 +1107750

x6)

,

Π8(x) =(−93

260x − 1

2x2 +

8013x3 − 53

16x4 +

413x5 − 4

13x6)

,

Π9(x) =(

1291750

x − 9725x2 − 194

25x3

+3649300

x4 +97

250x5 − 148

375x6)

,

Π10(x) =(−1019

975x − 1

2x2 +

8013x3 − 401

120x4− 4

13x5 +

11100

x6),

Π11(x) =(

2675x +

115x4 − 1

5x5 +

225x6).

(55)

On setting ε→ 0 in (54), the static slip at uniform pressure is

ΔU =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4μP0 − 3

)(x2 − 2x

)+ χ2

(1− μ0

μm

)(1− μP0

)Π1(x) + χ4

(1− μ0

μm

)2(1− μP0

)Π3(x)

+χ6

(1− μ0

μm

)3(1− μP0

)Π5(x) + χ8

(1− μ0

μm

)4(1− μP0

)Π7(x)

+χ10

(1− μ0

μm

)5(1− μP0

)Π9(x) + χ12

(1− μ0

μm

)6(1− μP0

)Π11(x)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (56)


The case χ = 0 agrees with the result in [19]. The foregoingexpression allows us to compute the maximum slip at the tip

of the magnetoelastic structure in the context of the pressuregradient as

ΔU =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3− 4μP0

)− 3.33μP0ε + χ2

(1− μ0

μm

)(91(

1− μP0

)− 24μP0ε

)

+χ4

(1− μ0

μm

)2(−107

(1− μP0

)− 22.9μP0ε

)+ χ6

(1− μ0

μm

)3(38(

1− μP0

)+ 1.23μP0ε

)

+χ8

(1− μ0

μm

)4(−3.86

(1− μP0

)+ 1.98μP0ε

)+ χ10

(1− μ0

μm

)5(2.24

(1− μP0

)+ 1.07μP0ε

)

−0.3χ12

(1− μ0

μm

)6(1− μP0

)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (57)

On setting ε → 0 in (57), the maximum static slip at the tipfor the case of uniform pressure is

ΔUmax(tip) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3− 4μP0

)+ 91χ2

(1− μ0

μm

)(1− μP0

)− 107χ4

(1− μ0

μm

)2(1− μP0

)

+38χ6

(1− μ0

μm

)3(1− μP0

)− 3.86χ8

(1− μ0

μm

)4(1− μP0

)

+2.24χ10

(1− μ0

μm

)5(1− μP0

)− 0.3χ12

(1− μ0

μm

)6(1− μP0

)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (58)

This suggests that in the presence of coulomb friction andinterfacial pressure P0, there are critical values of pressure forwhich no slip occurs at the tip such as

P0(critical) = 0.75μ−1Υ1

(χ,B0

)

Υ2

(χ,B0

) , (59)

where

Υ1

(χ,B0

)=

⎛⎜⎜⎜⎜⎜⎜⎝

1 +913χ2

(1− μ0

μm

)− 107

3B0

4(

1− μ0

μm

)2

+383χ6

(1− μ0

μm

)3

− 3.863

χ8

(1− μ0

μm

)4

+2.24

3χ10

(1− μ0

μm

)5

− 110χ12

(1− μ0

μm

)6

⎞⎟⎟⎟⎟⎟⎟⎠

,

Υ2

(χ,B0

)=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 +3.33

4+ χ2

(1− μ0

μm

)(914

+ 6ε)− χ4

(1− μ0

μm

)2(107

4− 22.9

4ε)

+χ6

(1− μ0

μm

)3(384− 1.23

4ε)− χ8

(1− μ0

μm

)4(3.86

4+

1.234

ε)

+χ10

(1− μ0

μm

)5(2.24

4− 1.07

4ε)− 3

10χ12

(1− μ0

μm

)5

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(60)


The case ε→ 0 gives the expression for critical values of pres-sure for which no slip occurs at the tip as

P0(critical) = 0.75μ−1Υ1

(χ,B0

)

Υ3

(χ,B0

) , (61)

where

Υ3

(χ,B0

)=

⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 +3.33

4+

914χ2

(1− μ0

μm

)− 107

4χ4

(1− μ0

μm

)2

+384χ6

(1− μ0

μm

)3

− 3.864

χ8

(1− μ0

μm

)4

+2.24

4χ10

(1− μ0

μm

)5

− 310χ12

(1− μ0

μm

)5

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(62)

which for the case of B0 = 0 agrees with the result in [19].

5. Energy Dissipation

The energy dissipated per static slip, following Damisa [25]is given by the relation

D = 4μb∫ L

0p(x)Δu(x)dx, (63)

which can also be expressed as

D = 4μp0

∫ 1

0

(1 +

ε

2

)Δudx ∀pav = p0

∫ 1

0(1 + εx)dx,

(64)

on substituting for Δu gives

D =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(8μp0 −

323μ2p0

2)

+ 4μp0ε −42130

μ2p02ε − 261

60μ2p0

2ε2

+χ2

(1− μ0

μm

)(80μp0 − 80μ2p0

2 + 20μp0ε − 69.2μ2p02ε − 29.2

2μ2p0

2ε2)

+χ4

(1− μ0

μm

)2(−72.8μp0 + 72.8μ2p0

2 − 36.4μp0ε + 14μ2p02ε− 11.2μ2p0

2ε2)

+χ6

(1− μ0

μm

)3(57.2μp0 − 57.2μ2p0

2 + 28.6μp0ε − 21.8μ2p02ε + 3.4μ2p0

2ε2)

+χ8

(1− μ0

μm

)4(−10.4μp0 + 10.4μ2p0

2 − 5.2μp0ε + 7.4μ2p02ε + 1.1μ2p0

2ε2)

+χ10

(1− μ0

μm

)5(0.28μp0 − 0.28μ2p0

2 + 0.14μp0ε − 0.46μ2p02ε + 0.3μ2p0

2ε2)

+χ12

(1− μ0

μm

)6(0.68μp0 − 0.68μ2p0

2 + 0.32μp0ε − 0.32μ2p02ε2)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (65)

where D = DEbh3/(L3F20 ) is the dimensionless static energy

dissipated.

5.1. Analysis of Optimum Clamping Pressure. The optimumclamping pressure can be found from the partial derivativeof the energy dissipated if we set

∂D

∂p0

= 0. (66)

Thus we can derive the general expression for the optimumclamping pressure as

popt = 0.375μ−1 Λ1

Λ2, (67)


where

Λ1 =

⎛⎜⎜⎜⎜⎝

1 +12ε +

52χ2

(1− μ0

μm

)(1 + 4ε) +

9120χ4

(1− μ0

μm

)2

(2− ε) +14340

χ6

(1− μ0

μm

)3

(2 + ε)

+1320χ8

(1− μ0

μm

)4

(−2− ε) +7

400χ10

(1− μ0

μm

)5

(2 + ε) +1712χ12

(1− μ0

μm

)6

(1 + ε)

⎞⎟⎟⎟⎟⎠,

Λ2 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 +421320

ε +261640

ε2 + χ2

(1− μ0

μm

)(152

+8125ε +

137100

ε2)

+χ4

(1− μ0

μm

)2(−683

100− 131

100ε +

2120ε2)

+ χ6

(1− μ0

μm

)3(10720

+5120ε − 8

25ε2)

+χ8

(1− μ0

μm

)4(−49

50− 69

100ε − 103

1000ε2)

+ χ10

(1− μ0

μm

)5(3

100+

431000

ε− 3100

ε2)

+χ12

(1− μ0

μm

)6(1625

+3

100ε)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (68)

In the limit as χ → 0, we recover the optimum clamping pre-ssure in [19] as

p0 = 0.375μ−1

{(1 +

12ε)(

1 +421320

ε +261640

ε2)−1

}. (69)

By letting ε→ 0, the optimum clamping pressure for the mag-netoelastic structure computed is

popt = 0.375μ−1 Λ3

Λ4, (70)

where

Λ3

=

⎛⎜⎜⎜⎝

1 +52χ2

(1− μ0

μm

)+

9110χ4

(1− μ0

μm

)2

+14320

χ6

(1− μ0

μm

)3

−1310χ8

(1− μ0

μm

)4

+7

200χ10

(1− μ0

μm

)5

+1712χ12

(1− μ0

μm

)6

⎞⎟⎟⎟⎠,

Λ4

=

⎛⎜⎜⎜⎝

1 +152χ2

(1− μ0

μm

)− 683

100χ4

(1− μ0

μm

)2

+10720

χ6

(1− μ0

μm

)3

−4950χ8

(1− μ0

μm

)4

+3

100χ10

(1− μ0

μm

)5

+1625χ12

(1− μ0

μm

)6

⎞⎟⎟⎟⎠,

(71)

and for the special case χ → 0, the optimum clamping pres-

sure reduce to the form p0 = 0.375μ−1 = 3/8μ in Damisa[25]. At this optimal pressure, the corresponding energydissipation is

Dmax = 32Ψ

×

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(2 + ε)−Ψ(

1 +421320

ε)−Ψ

261640

ε2

+χ2

(1− μ0

μm

)(20− 5ε −Ψ

(152

+51980

ε +219160

ε2))

+χ4

(1− μ0

μm

)2(− 91

10(2 + ε) + Ψ

(27340

+2116ε − 21

20ε2))

+χ6

(1− μ0

μm

)3(14320

(2 + ε)−Ψ(

42980

+327160

ε − 153320

ε2))

+χ8

(1− μ0

μm

)4(− 13

10(2 + ε)−Ψ

(3940− 13

10ε − 33

320ε2))

+χ10

(1− μ0

μm

)5(7

100(1 + 5ε)−Ψ

(21

800+

691600

ε − 9320

ε2))

+χ12

(1− μ0

μm

)6(17

100+

225ε −Ψ

(51

800+

3100

ε2))

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(72)

where

Ψ = M1

M2

M1

=

⎛⎜⎜⎜⎝

1 +52χ2

(1− μ0

μm

)+

9110χ4

(1− μ0

μm

)2

+14320

χ6

(1− μ0

μm

)3

−1310χ8

(1− μ0

μm

)4

+7

200χ10

(1− μ0

μm

)5

+1712χ12

(1− μ0

μm

)6

⎞⎟⎟⎟⎠,

M2

=

⎛⎜⎜⎝

1 +152χ2

(1− μ0

μm

)− 683

100χ4

(1− μ0

μm

)2

+10720

χ6

(1− μ0

μm

)3

− 4950χ8

(1− μ0

μm

)4

+3

100χ10

(1− μ0

μm

)5

+1625χ12

(1− μ0

μm

)6

⎞⎟⎟⎠M,

(73)

which indicates that even when linear pressure variation isadmitted, the maximum dissipated energy still remains inde-pendent of the coefficient of friction as reported in [19].


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ε = 0ε = −0.2ε = 0.2

ε = −0.6ε = 0.6

Axial length x

Stat

ic r

espo

nse

W

Figure 2: W versus x with different values of ε for the followingcase: χ = 0.05; μr = 0.6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

−0.5

−1

−1.5

−2

−2.5

−3

ε = 0ε = −0.2ε = 0.2

ε = −0.6ε = 0.6

Axial length x

Stat

ic r

espo

nse

W


Axial length x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ε = 0ε = −0.2ε = 0.2

ε = −0.6ε = 0.6

201816141210

86420

Stat

ic r

espo

nse

W


Table 1: Material and geometric parameters.

Definition Symbol Value

Magnetic permeability μ0 4π × 10−7 mho/mInterfacial pressure P 1× 109 Nm−2

GeometryLength L 0.1 m–5 mWidth b 0.3 mThickness h 0.001 m–0.4 m

Modulus of rigidity of materials E 1.2× 1011 Nm−2

Density of material ρ 2810 kg m−3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Axial length x

1.4

1.2

1

0.8

0.6

0.4

0.2

0

χ = 0.05χ = 0.15χ = 0.2

χ = 0.25χ = 0.3

Stat

ic r

espo

nse

W

Figure 5: W versus x with different values of χ for the followingcase: ε = 0; μr = 0.6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Axial length x

1.4

1.2

1

0.8

0.6

0.4

0.2

0

χ = 0.05χ = 0.15χ = 0.2

Stat

ic r

espo

nse

W

χ = 0.25χ = 0.3

Figure 6: W versus x with different values of χ for the followingcase: ε = 0.6; μr = 0.6; μP0 = μPopt.

5.2. Analysis of Damping Capacity of the Magnetoelastic Struc-ture. The damping capacity of any structure is a measure ofthe ratio of its slip energy dissipation to total strain energyunder any conditions. For this case, we shall first derive theexpression for the total strain energy following Damisa et al.[19]. It is a combination of the energy introduced by thebending moment as well as that stored from the deflectionof the free end. While the former can be evaluated from


the theorem of Castigliano, the later can be computed fromthe free end deflection theory. For this problem, we canderive the total strain energy expression as

X = X1 + X2, (74)

where

X1 =

⎛⎜⎜⎜⎜⎝

1− μp0(2 + ε) + μ2p02

(1 + ε +

ε2

4

)+ χ2

(1− μ0

μm

)(1− μp0(2 + ε)Wmax

)

+χ4

(1− μ0

μm

)2

Wmax2

⎞⎟⎟⎟⎟⎠,

∀Wmax =

⎛⎜⎜⎜⎜⎝

1 + χ2

(1− μ0

μm

)

+3

20χ4

(1− μ0

μm

)2

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎝

(1− μP0

(1 +

ε

2

)− μP0ε

240

)

+45χ2

(1− μ0

μm

)(1− μP0

(1 +

ε

2

)+μP0ε

576

)⎞⎟⎟⎟⎠,

Wmax2 = Γ

(χ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎝

(1− μP0(2 + ε) + μ2P0

2(

1 + ε +ε2

4

))

+

⎛⎝μP0ε

120− μ2P0

2

120

(1 +

ε

2

)+μ2P0

2ε2

2402

⎞⎠

⎞⎟⎟⎟⎟⎠

+85χ2

(1− μ0

μm

)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(1− μP0

(1 +

ε

2

)+μP0ε

576

)

+μP0

(1 +

ε

2

)− μ2P0

2(

1 + ε +ε2

4

)+μ2P0

2

576

(ε +

ε2

2

)

+

⎛⎝−μP0ε

240+μ2P0

2

240

(1 +

ε

2

)− μ2P0

2ε2

(240)(576)

⎞⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+1625χ4

(1− μ0

μm

)2

⎛⎜⎜⎜⎜⎝

(1− μP0(2 + ε) + μ2P0

2(

1 + ε +ε2

4

))

+

⎛⎝μP0ε

288− μ2P0

2

288

(1 +

ε

2

)⎞⎠ +

μ2P02ε2

(162)(362)

⎞⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(75)

where

Γ(χ)=⎛⎝1 + χ2

(1− μ0

μm

)+

320χ4

(1− μ0

μm

)2⎞⎠; (76)

while

X2 =Wmax2 = Γ

(χ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎝

(1− μP0(2 + ε) + μ2P0

2(

1 + ε +ε2

4

))

+

⎛⎝μP0ε

120− μ2P0

2

120

(1 +

ε

2

)+μ2P0

2ε2

2402

⎞⎠

⎞⎟⎟⎟⎟⎠

+85χ2

(1− μ0

μm

)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(1− μP0

(1 +

ε

2

)+μP0ε

576

)

+μP0

(1 +

ε

2

)− μ2P0

2(

1 + ε +ε2

4

)+μ2P0

2

576

(ε +

ε2

2

)

+

⎛⎝−μP0ε

240+μ2P0

2

240

(1 +

ε

2

)− μ2P0

2ε2

(240)(576)

⎞⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+1625χ4

(1− μ0

μm

)2

⎛⎜⎜⎜⎜⎜⎝

(1− μP0(2 + ε) + μ2P0

2(

1 + ε +ε2

4

))+

⎛⎝μP0ε

288− μ2P0

2

288

(1 +

ε

2

)⎞⎠ +

μ2P02ε2

(162)(362)

⎞⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (77)


For this problem, the damping capacity following Damisaet al. [19] is

Φ = D

X, (78)

whilst the maximum damping capacity can be computed as

Φmax = Dmax

Xmax, (79)

where

Xmax = X1 + X2,

X1 =

⎛⎜⎜⎜⎜⎝

1− 38Ψ(2 + ε) +

964

Ψ2

(1 + ε +

ε2

4

)+ χ2

(1− μ0

μm

)(1− 3

8Ψ(2 + ε)Wmax

)

+χ4

(1− μ0

μm

)2

W2

max

⎞⎟⎟⎟⎟⎠,

∀Wmax =

⎛⎜⎜⎜⎜⎝

1 + χ2

(1− μ0

μm

)

+3

20χ4

(1− μ0

μm

)2

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎝

(1− 3

8Ψ(

1 +ε

2

)− Ψε

640

)

+45χ2

(1− μ0

μm

)(1− 3

8Ψ(

1 +ε

2

)+

Ψε(8)(192)

)⎞⎟⎟⎟⎠

Wmax2 = Γ

(χ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎝

(1− 3

8Ψ(2 + ε) +

964

Ψ2

(1 + ε +

ε2

4

))

+

(Ψε

320− 3Ψ2

(40)(64)

(1 +

ε

2

)+

Ψ2ε2

(102)(84)

)⎞⎟⎟⎟⎠

+85χ2

(1− μ0

μm

)⎛⎜⎜⎜⎜⎜⎜⎜⎝

(1− 3

8Ψ(

1 +ε

2

)+

Ψε(8)(192)

)

+38Ψ(

1 +ε

2

)− 9

64Ψ2

(1 + ε +

ε2

4

)+

Ψ2

642

(ε +

ε2

2

)

+

(− Ψε

640+

Ψ2

(102)(84)

(1 +

ε

2

)− Ψ2ε2

(642)(240)

)

⎞⎟⎟⎟⎟⎟⎟⎟⎠

+1625χ4

(1− μ0

μm

)2

⎛⎜⎜⎜⎝

(1− 3

8Ψ(2 + ε) +

964

Ψ2

(1 + ε +

ε2

4

))

+

(Ψε

768− Ψ2

211

(1 +

ε

2

))+

Ψ2ε2

(223)(32)

⎞⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(80)

while

X2 = Wmax2. (81)

6. Analysis of Results

In this paper, static slip and energy dissipation in two-layer sandwich homogeneous elastic beams in a transversemagnetic field is studied. The problem physics derives fromenergy dissipation via contact frictional stresses in press fitjoints. In the contrived problem, a constant tip force promptsthe two-layer cantilever elastic beams in an environmentof uniform transverse magnetic field. Simulation studiesemployed the characteristic values listed in Table 1. (Figures2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14) illustrate the effect ofmagnetoelasticity, relative permeability, and pressure gradi-ent at optimum interface clamping pressure. In Figures 2–7, atip force in an environment of weak magnetic field (χ = 0.05)prompts the structure. We displayed in the Figure 2 resultfor homogeneous laminates of relative permeability μr=0.6.

As can be seen, the deflection in the absence of pressuregradient (ε = 0) is lower and higher than the cases of negativeand positive pressure gradients. Such a response pattern isexpected ideally.

The rigidity or stiffness of the structure from the fixedend should increase with progressive increment in thetightening torque axially. It is conceivable to expect higherdeflection for the case ε=−0.6 than for ε=−0.2. Intuitively,a converse pattern should be expected in the reversed orderas correctly noted in the figure. Furthermore, for ε =−0.6, there is a tendency for buckling to occur with axialprogressive decrement in the tightening torque close to thefree end, whilst for ε = −0.2, buckling is expected to occurat the free end. In Figure 3, the magnetic field (χ = 0.5)intensity is relatively higher. Under the same parametricvariables, we note a reversion in the pattern of deflection andtendency for buckling in the neighborhood of the free endirrespective of the value of ε.

The effect of much higher magnetic field intensity (χ =1.5) on the deflection response is illustrated in Figure 4.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Axial length x

χ = 0.05χ = 0.15χ = 0.2

χ = 0.25χ = 0.3

2.5

2

1.5

1

0.5

0

Stat

ic r

espo

nse

W

Figure 7: W versus x with different values of χ for the followingcase: ε = −0.6; μr = 0.6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Axial length x

Stat

ic r

espo

nseW

χ = 0.05χ = 0.15χ = 0.2

χ = 0.25χ = 0.3

2.5

2

3.5

3

1.5

1

0.5

0

Figure 8: W versus x with different values of χ for the followingcase: ε = −0.6; μr = 6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Axial length x

χ = 0.05χ = 0.15χ = 0.2

χ = 0.25χ = 0.3

2.5

2

1.5

1

0.5

0

Stat

ic r

espo

nseW

Figure 9: W versus x with different values of χ for the followingcase: ε = 0.6; μr = 6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Axial length x

100

80

60

40

20

0

−20

−40

χ = 1.5χ = 2χ = 2.5

χ = 3χ = 3.5

Stat

ic r

espo

nseW

Figure 10: W versus x with different values of χ for the followingcase: ε = 0; μr = 6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ε = 0ε = −0.2ε = 0.2

ε = −0.6ε = 0.6

0

Axial length x

6

5

4

3

2

1

Stat

ic r

espo

nse

W

Figure 11: W versus x with different values of ε for the followingcase: χ = 0.5; μr = 6; μP0 = μPopt.

ε = 0

ε = −0.2ε = 0.2 ε = −0.6

ε = 0.6

1110

987654321

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Magnetic field χ

Stat

ic r

espo

nse

W

Figure 12: W versus χ with different values of ε for the followingcase: x = 1; μr = 6; μP0 = μPopt.


Nature of curves indicated identical trajectories with higherdeflection and symmetric ordering with respect to the roleof the pressure gradient. In Figure 5, we illustrate the effectof varying the magnetic field intensity for the case ofuniform clamping pressure on the static deflection profile.In the environment of lower magnetic field intensity χ,the trajectory is uniform but as field intensity increases,proportional decrements to respective optimum points arenoted. Next, we display in Figures 6 and 7 the effects ofpressure gradients on the deflection pattern in the samemagnetic field environment. For the case ε = 0.6, thepattern is the same from the fixed end to the middle ofthe structure. Beyond this point, the effect of the fieldintensity becomes noticeable with higher deflection in theweak magnetic environment. We also note identical patternfrom the fixed end to the middle of the structure for thecase ε = −0.6. Beyond this point, the effect of the fieldintensity becomes noticeable. Natures of curves indicatehigher deflection with respective optimum points in thestrong magnetic environment. Next, we show in Figures 8and 9 the effect of increasing the relative permeability on thestatic deflection.

As shown in Figures 8 and 9, the effect of increasingthe field intensity is noticeable beyond the middle of thestructure. Nevertheless, for the case ε = −0.6, a bisegmentedasymmetrical deflection appeared in the neighborhood ofthe free end of the structure compared to the case ε = 0.6where the pattern of deflection is progressively monotonic.Profiles for the case ε = 0 (uniform pressure) in much higherfield intensity are illustrated in Figure 10. In contrast toresults in Figures 8 and 9, we note a very visible bisegmentedasymmetrical deflection slightly away from the fixed pointto the free end of the structure. Next, we study the staticresponse of the structure at optimum clamping pressure forthe case displayed in Figure 11. Nature of curves indicatethat the deflections for negative pressure gradient are higherthan for the case of uniform pressure and positive gradientas reported in [19].

The cumulative effect of pressure gradient and fieldintensity on the mid structure and tip deflections are demon-strated in Figures 12 and 13. The picture in Figure 12 indi-cates a three-region segmentation except for the case ε = 0.6as modulated by the field intensity. In the first region, thefield intensity lies within the range 0 < χ ≤ 0.2. Here, thedeflections are ordered in consonant with the forms andvalues of pressure gradient ε with each profile, indicating amonotonic variable that is approaching a local maximum.

In the second region, the field intensity lies within therange 0.5 ≤ χ ≤ 1. Here, each profile is also a monotonicvariable approaching a local minimum, and in the lastregion, the field intensity lies within the range 1 ≤ χ ≤ 1.2and the respective tip deflection is progressively monotonic.Next, the deflection pattern in the middle of the structureis displayed in Figure 13. In contrast to tip deflection, arelatively stable structure is noted in the range 0 ≤ χ ≤0.4 and beyond, and a segmented distortional responsecharacterized the structure. To what extent is the optimumclamping pressure influenced by the pressure gradient and

ε = 0

ε = −0.2ε = 0.2 ε = −0.6

ε = 0.6

5

4

3

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Magnetic field χ

1.6 1.8 2

1

0

−1

−2

Stat

ic r

espo

nse

W

Figure 13: W versus χ with different values of ε for the followingcase: x = 0.5; μr = 6; μP0 = μPopt.

Figure 14: μP0 versus χ with different values of ε for the followingcase: μr = 6.

the magnetic field strength? Such modulating effect is dem-onstrated in Figure 14.

In the range 0 ≤ χ ≤ 1.5, a cyclic propagation ensued andbeyond, and the interface pressure becomes stable and higherfor positive pressure gradient. The profiles of interfacialstatic slip are shown in Figures 15–21. Effect of pressuregradient on the tip static slip as a function of the magneticfield intensity is displayed in Figure 15. Family of curvesindicate a three-region profile. In the first region, the slipfor positive pressure gradient is higher than the cases foruniform pressure and negative pressure gradient. As themagnetic intensity increases to χ = 0.2, the interfacial slipbecomes constant irrespective of the nature of the pressuregradient. Beyond this point, the second region begins and areversal in the pressure gradient becomes apparent.

In this domain, each curve indicates a monotonic pro-gressing slip that attains a maximum value subject to com-mon critical magnetic field intensity. Next, we display inFigures 16, 17, and 18 gross interfacial slip profiles for


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ε = 0

ε = −0.2ε = 0.2 ε = −0.6

ε = 0.6

Magnetic field χ

201816141210

86420

Stat

ic s

lipΔU

Figure 15: ΔU versus χ with different values of ε for the followingcase: x = 1; μr = 6.

χ = 0χ = 0.1χ = 0.2

χ = 0.3χ = 0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2.5

2

3

1.5

1

0.5

0

Stat

ic s

lipΔU

Axial length x

Figure 16: ΔU versus x with different values of χ for the followingcase; ε = 0; μr = 6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Stat

ic s

lipΔU

3

2.5

2

1.5

1

0.5

0

χ = 0

χ = 0.1χ = 0.2

χ = 0.3χ = 0.4

Axial length x

Figure 17: ΔU versus x with different values of χ for the followingcase: ε = 0.6; μr = 6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3

2.5

2

1.5

1

0.5

0

χ = 0

χ = 0.1χ = 0.2

χ = 0.3χ = 0.4

Axial length x

Stat

ic s

lipΔU

Figure 18: ΔU versus x with different values of χ for the followingcase: ε = −0.6; μr = 6; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 102468

101214161820

μr = 2μr = 3μr = 4

μr = 5

μr = 6

Axial length x

Stat

ic s

lipΔU

Figure 19: ΔU versus x with different values of μr for the followingcase: ε = 0; χ = 1.5; μP0 = μPopt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

20

μr = 2μr = 3μr = 4

μr = 5

μr = 6

Axial length x

Stat

ic s

lipΔU

Figure 20: ΔU versus x with different values of μr for the followingcase: ε = 0.6; χ = 1.5; μP0 = μPopt.


35

30

25

20

15

10

5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

μr = 2μr = 3μr = 4

μr = 5

μr = 6

Axial length x

Stat

ic s

lipΔU

Figure 21: ΔU versus x with different values of μr for the followingcase: ε = −0.6; χ = 1.5; μP0 = μPopt.

0 0.2 0.4 0.6 0.8 1 1.20

0.51

1.52

2.53

3.54

4.55

5.5

χ = 0χ = 0.1

χ = 0.2

χ = 0.3χ = 0.4χ = 0.5

Interface pressure, μP0

En

ergy

dis

sipa

tion

D

Figure 22: D versus μP0 with different values of χ for the followingcase: ε = 0; μr = 6.

the cases of uniform pressure, positive- and negative-pressure gradients, respectively. As noted in [19], gross slipfor negative pressure gradients is higher compared to thecases of uniform pressure and positive pressure gradients.

For the special case, χ = 0, the profiles replicated theresults in [19] for the same values of ε. Irrespective of theform of ε, the gross slip is proportional to the magnetic inten-sity and admits approximate linear profile when compared tothe case χ = 0. The gross slip for uniform optimum pressureas influenced by different relative permeability values in anenvironment of strong magnetic field defined by χ = 1.5 isshown in Figure 19. In general, interfacial slip increases asthe permeability increases. Apart from the curve typified byμr = 2, which rises progressively to an optimum value, therest profiles admit cyclic variation with local maxima andminima values.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

χ = 0χ = 0.1

χ = 0.2

χ = 0.3χ = 0.4χ = 0.5


En

ergy

dis

sipa

tion

D

Figure 23: D versus μP0 with different values of χ for the followingcase: ε = 0.6; μr = 6.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8


En

ergy

dis

sipa

tion

D

χ = 0χ = 0.1

χ = 0.2

χ = 0.3χ = 0.4χ = 0.5

Figure 24: X versus μP0 with different values of χ for the followingcase: ε = 0.6; μr = 6.

To what extent are the profiles modified for the cases ofpositive and negative pressure gradients? Such modificationor effects are displayed in Figures 20 and 21 respectively. InFigure 20, profiles admit the same noncyclic configurationpattern that are proportional to the permeability valuesacross the structures, and in Figure 21 slip profiles areproportionally parabolic curves. Energy dissipation abilityas influenced by the magnetic field intensity and interfacialpressure are shown in Figures 22–24. Natures of profilesindicate a three-region regime. As shown in the plottedcurves, the profiles are parabolic in the first region withlocal maxima as reported in [19, 20] and here, the inter-facial pressure values are restricted in consonance withthe form of pressure gradient. The same observationsare noted in Damisa [25], where energy dissipation ismaximum at respective optimum clamping pressure defined


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

χ = 0χ = 0.1χ = 0.2

χ = 0.3χ = 0.4


4

3.5

3

2.5

2

1.5

1

0.5

0

−0.5

Stra

in e

ner

gyX

Figure 25: X versus μP0 with different values of χ for the followingcase: ε = 0; μr = 6.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

χ = 0χ = 0.1

χ = 0.2

χ = 0.3χ = 0.4


10

8

6

4

2

0

Stra

in e

ner

gyX

−2

Figure 26: D versus μP0 with different values of χ for the followingcase: ε = 0.6; μr = 6.

by corresponding critical transition into the second region.Such is anticipated via (72). In the second region, the valuesof the energy dissipation quanta are reversed in accordancewith the respective magnetic field intensity prior to enteringthe third region. In this region, the dissipation quanta areproportional to the magnetic field intensities. The sametrends are noted for energy dissipation profiles for the casesof positive and negative pressure gradients as shown inFigures 23 and 24.

As expected, the magnetic environment has an increasingeffect on the energy dissipation mechanism as shown inFigures 22–24. Nevertheless, as correctly noted in [19, 20],dissipation is more with negative pressure gradient thanfor the uniform pressure and positive pressure gradient.The strain energy profiles as influenced by the magnetic field

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

χ = 0χ = 0.1

χ = 0.2

χ = 0.3χ = 0.4

3.5

3

2.5

2

1.5

1

0.5

0

−0.5


Stra

in e

ner

gyX

Figure 27: X versus μP0 with different values of χ for the followingcase: ε = −0.6; μr = 6.

intensity and pressure gradient parameter are illustrated inFigures 25, 26, and 27. For the case of uniform pressure,symmetric parabolic curves proportional to the magneticintensity ensued as shown in Figure 25, whilst skewedasymmetric parabolas are displayed for the positive andnegative pressure gradients.

7. Conclusion

A well-posed mathematical physics problem on the mechan-ics of interfacial slip and energy dissipations mechanismwith a two-layer sandwich homogenous elastic beam in atransverse magnetic field is presented. By employing oper-ational methods, closed form polynomial expressions arederived for the responses defined in the body of the paper.In particular, the effects of magnetoelasticity and interfacialpressure gradient are demonstrated for design analysis andengineering applications. For special and limit cases, recenttheoretical and experimental results are validated from thetheory reported in the paper.

Nomenclature

b: Width of laminated beamB0: Magnetic flux densityd/dx: Differential operatorE: Electric field intensityE: Modulus of rigidityF: Applied end force amplitudeh: Depth of laminated beamH : Magnetic field intensityI : Moment of inertiaL: Length of laminated beamP: Clamping pressure at the interface of the

laminated beamst: Time coordinateu, ν: Velocity


u, ν: AccelerationU1: Displacement of the lower laminateU2: Displacement of the upper laminateW : Dynamic responseWF : Dynamic response in Laplace transform

planeW : Dynamic response in Fourier transform

planeWF : Dynamic response in Fourier-Laplace

transform planeWF: Transverse response in Fourier planex: space coordinate along the beam interfacez: Space coordinate perpendicular to the

beam interface⊗: Tensor Productε : Pressure gradientμ: Dry friction coefficientμm: Permeability of the mediumμr : Relative permeability of the mediumχ: Normalized magnetic field intensityεx1: Axial strain in layer-1εx2: Axial strain in layer-2γxz: Angular strain�h(x): Disturbed magnetic field vector�H(x): Undisturbed magnetic field vector�e(x): Disturbed electric field vector�bm: Disturbed magnetic induction vectorτM : Maxwell stress tensorIs: Identity stress tensor.

References

[1] J. S. Lee, “Destabilizing effects of magnetic dampimg in platestrip,” Journal of Engineering Mechanics, vol. 118, no. 1, pp.161–173, 1992.

[2] F. C. Moon and Y. H. Pao, “Magnetoelastic buckling of a thinplate,” Journal of Applied Mechanics, vol. 35, pp. 53–58, 1968.

[3] F. C. Moon and Y. H. Pao, “Vibration and dynamic instabilityof a beam-plate in a transverse magnetic field,” Journal of Ap-plied Mechanics, vol. 36, pp. 92–100, 1969.

[4] D. V. Wallerstein and M. O. Peach, “Magnetoelastic buckling ofbeams and thin plates of magnetically soft material,” Journal ofApplied Mechanics, vol. 39, pp. 451–455, 1972.

[5] K. Miya, T. Takagi, and Y. Ando, “Finite element analysis ofmagnetoelastic buckling of a ferromagnectic beam-plate,” Jou-rnal of Applied Mechanics, vol. 47, no. 2, pp. 377–382, 1980.

[6] M. O. Peach, N. S. Christopherson, J. M. Dalrymple, and G. L.Viegelahn, “Magnetoelastic buckling: why theory and experi-ment disagree,” Experimental Mechanics, vol. 28, no. 1, pp. 65–69, 1988.

[7] J. S. Lee, “Dynamic stability of conducting beam-plates in tra-nsverse magnetic fields,” Journal of Engineering Mechanics, vol.122, no. 2, pp. 89–94, 1996.

[8] Y. H. Zhou, X. J. Zheng, and K. Miya, “Magnetoelastic bendingand snapping of ferromagnetic plates in oblique magneticfields,” Fusion Engineering and Design, vol. 30, no. 4, pp. 325–337, 1995.

[9] Y. H. Zhou and X. Zheng, “A general expression of magneticforce for soft ferromagnetic plates in complex magnetic fields,”

International Journal of Engineering Science, vol. 35, no. 15, pp.1405–1417, 1997.

[10] Y. H. Zhou and K. Miya, “A theoretical prediction of naturalfrequency of a ferromagnetic beam plate with low susceptibil-ity in an in-plane magnetic field,” Journal of Applied Mechan-ics, vol. 65, no. 1, pp. 121–126, 1998.

[11] X. Wang, G. Lu, and S. R. Guillow, “Magnetothermodynamicstress and perturbation of magnetic field vector in a solid cyli-nder,” Journal of Thermal Stresses, vol. 25, no. 10, pp. 909–926,2002.

[12] X. Wang, J. S. Lee, and X. Zheng, “Magneto-thermo-elasticinstability of ferromagnetic plates in thermal and magneticfields,” International Journal of Solids and Structures, vol. 40,no. 22, pp. 6125–6142, 2003.

[13] X. Wang and H. L. Dai, “Magnetothermodynamic stressand perturbation of magnetic field vector in an orthotropicthermoelastic cylinder,” International Journal of EngineeringScience, vol. 42, no. 5-6, pp. 539–556, 2004.

[14] L. Librescu, D. Hasanyan, Z. Qin, and D. R. Ambur, “Nonlin-ear magnetothermoelasticity of anisotropic plates immersedin a magnetic field,” Journal of Thermal Stresses, vol. 26, no.11-12, pp. 1277–1304, 2003.

[15] X. Wang, J. S. Lee, and X. Zheng, “Magneto-thermo-elasticinstability of ferromagnetic plates in thermal and magneticfields,” International Journal of Solids and Structures, vol. 40,no. 22, pp. 6125–6142, 2003.

[16] X. Wang and H. L. Dai, “Magnetothermodynamic stressand perturbation of magnetic field vector in an orthotropicthermoelastic cylinder,” International Journal of EngineeringScience, vol. 42, no. 5-6, pp. 539–556, 2004.

[17] D. J. Hasanyan, L. Librescu, and D. R. Ambur, “A few results onthe foundation of the theory and behavior of nonlinear mag-netoelastic plates carrying an electrical current,” InternationalJournal of Engineering Science, vol. 42, no. 15-16, pp. 1547–1572, 2004.

[18] X. Wang and J. S. Lee, “Dynamic stability of ferromagneticbeam-plates with magnetoelastic interaction and magneticdamping in transverse magnetic fields,” Journal of EngineeringMechanics, vol. 132, no. 4, pp. 422–428, 2006.

[19] O. Damisa, V. O. S. Olunloyo, C. A. Osheku, and A. A.Oyediran, “Static analysis of slip damping with clampedlaminated beams,” European Journal of Scientific Research, vol.17, no. 4, pp. 455–475, 2007.

[20] O. Damisa, V. O. S. Olunloyo, C. A. Osheku, and A. A.Oyediran, “Dynamic analysis of slip damping in clampedlayered beams with non-uniform pressure distribution at theinterface,” Journal of Sound and Vibration, vol. 309, no. 3–5,pp. 349–374, 2008.

[21] V. O. S. Olunloyo, C. A. Osheku, and O. Damisa, “Vibrationdamping in structures with layered viscoelastic beam-plate,”Journal of Vibration and Acoustics, Transactions of the ASME,vol. 130, no. 6, Article ID 061002, 2008.

[22] V. O. S. Olunloyo, O. Damisa, C. A. Osheku, and A. A. Oye-diran, “Analysis of the effects of laminate depth and materialproperties on the damping associated with layered structuresin a pressurized environment,” Transactions of the CanadianSociety for Mechanical Engineering, vol. 34, no. 2, pp. 165–196,2010.

[23] C. A. Osheku and O. Damisa, “Vibration of conducting two-layer sandwich homogeneous elastic beams in transverse mag-netic fields part I,” ASCE Journal of Aerospace EngineeringASENG-245. In press.


[24] L. E. Goodman and J. H. Klumpp, “Analysis of slip dampingwith reference to turbine blade vibration,” Journal of AppliedMechanics, vol. 23, pp. 421–429, 1956.

[25] O. Damisa, “Slip damping of Timoshenko beam close to reso-nance,” Journal of Engineering Research, vol. 11, no. 1-2, pp.13–26, 2003.

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