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УДК 539.3
Влияние магнитного поля, начального напряжения, вращения и неоднородности на напряжения в ортотропном материале
K.S. Al-Basyouni, S.R. Mahmoud
Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия
В статье исследовано влияние магнитного поля, начального напряжения, вращения и неоднородности на радиальное смещение и соответствующие напряжения в ортотропном материале. Получено аналитическое решение уравнения эластодинамики в терминах смещения. Изменение напряжений, смещения и возмущений магнитного поля представлено графически. Проведено сравнение результатов с предыдущими данными, полученными при отсутствии магнитного поля, начального напряжения, вращения и неоднородности.
Ключевые слова: магнитное поле, начальное напряжение, вращение, неоднородный, ортотропный
DOI 10.24412/1683-805X-2021-1-88-95
Effect of the magnetic field, initial stress, rotation, and nonhomogeneity on stresses in orthotropic material
K.S. Al-Basyouni1 and S.R. Mahmoud2
1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia 2 GRC Department, Faculty of Applied Studies, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
In this article, the effect of the magnetic field, initial stress, rotation, and nonhomogeneity on the radial displacement and the corresponding stresses in orthotropic material is investigated. The analytical solution for the elastodynamic equation is solved in terms of displacements. The variation of stresses, displacement, and perturbation magnetic field have been shown graphically. Comparisons are made with the previous results in the absence of the magnetic field, the initial stress, the rotation, and nonhomogeneity.
Keywords: magnetic field, initial stress, rotation, nonhomogeneous, orthotropic
1. Introduction
Growing attention is being devoted to electromagnetic due to its many engineering applications in the fields of geophysical physics, magnetic structural elements, plasma physics, magnetic storage elements and the corresponding measurement techniques of magnetoelasticity as described elsewhere [1-3]. The interaction of electromagnetic fields with the motion of a deformable solid is receiving much attention by many researchers. Among many important problems considered in such studies, elastic wave propagation problems in the presence of a magnetic field were investigated. Large use has been made of elastic materials, especially in aerospace industries. It is thus of
considerable practical interest to investigate the elastodynamic behavior of such materials due to the effect of suddenly applied surface pressures. In addition to aerospace industries, spherical structures may also be used in submarines, nuclear reactors, and chemical plants. The elastodynamic response of anisotropic spheres is a fundamental problem of renewed contemporary interest. Mahmoud et al. [4, 5] investigated the effect of the rotation on plane vibrations in a transversely isotropic infinite hollow cylinder and effect of the rotation on wave motion through a cylindrical bore in a micropolar porous cubic crystal. Mahmoud [6] presented an analytical solution for electrostatic potential on wave propagation modeling in human long
© Al-Basyouni K.S., Mahmoud S.R., 2021
wet bones. Abd-Alla et al. [7] and Mahmoud [8] investigated effect of nonhomogeneity, magnetic field and gravity field on Rayleigh waves in an initially stressed elastic half-space of orthotopic material subject to rotation and they studied influence of the rotation and gravity field on Stonely waves in a nonho-mogeneous orthotopic elastic medium. Abd-Alla and Mahmoud [9, 10] investigated the magneto-thermo-elastic problem in rotating nonhomogeneous ortho-tropic hollow cylindrical under the hyperbolic heat conduction model and problem of radial vibrations in the nonhomogeneity isotropic cylinder under the effect of initial stress and magnetic field. Mahmoud [11] investigated wave propagation in hollow cylindrical poroelastic dry bones. Mofakhamia et al. [12] investigated finite cylinder vibrations with different end conditions at the boundary. The hollow spheres are frequently encountered in engineering industries, and the corresponding free vibration problem has become one of the basic problems in elastodynamics. The method has been widely used for static and free vibration analysis of beams [13]. Free vibration analysis of functionally graded curved panels was carried out using a higher-order formulation that has been investigated by Pradyumna and Bandyopadhyay [14]. Mahmoud [15] and Abd-Alla et al. [16, 17] studied the propagation of S-wave in a nonhomogeneous anisotropic incompressible and initially stressed medium under the influence of gravity field and effect of the rotation on a nonhomogeneous infinite cylinder of orthotropic material and studied the effect of the nonhomogeneity on wave propagation on orthotopic elastic media. Sofi-yev and Karaca [18] presented the vibration and buckling of laminated nonhomogeneous orthotropic conical shells subjected to external pressure. Addou et al. [19] investigated influences of porosity on dynamic response of FG plates resting on Winkler/Paster-nak/Kerr foundation using quasi 3D HSDT. Sahla et al. [20] studied the free vibration response of angle-ply laminated composite and soft core sandwich plates. Static and dynamic analysis of laminated composite and sandwich plates and shells is investigated by Allam et al. [21] based on a generalized 4-un-known refined theory. Bourada et al. [22] examined the buckling and free vibrational behaviors of SW-CNT reinforced concrete beam resting on WinklerPasternak elastic foundations. Argatov [23] investigated the approximate solution of the axisymmetric contact problem for an elastic sphere. Huang and Ho [24] discussed the analytical solution for vibrations of a polarly orthotopic Mindlin sectorial plate with simply supported radial edges. Mahmoud [25] studied
shear waves in a magneto-elastic half-space of initially stressed a nonhomogeneous anisotropic material under the influence of rotation and considered the effect of nonhomogeneity and rotation on an infinite generalized thermoelastic diffusion medium with a spherical cavity subject to the magnetic field and initial stress. Bahrami et al. [26] investigated the wave propagation technique for free vibration analysis of annular circular and sectorial membranes. Towfighi and Kundu [27] investigated elastic wave propagation in anisotropic spherical curved plates. Abd-Alla and Mahmoud [28] studied the analytical solution of wave propagation in nonhomogeneous orthotopic rotating elastic media. The radially nonhomogeneous axisym-metric problem is studied by Theotokoglou and Stam-pouloglou [29]. Chapra [30] studied the numerical methods with MATLAB for engineering and science. Stavsky and Greenberg [31] studied radial vibrations of orthotopic laminated hollow spheres.
In the present work, the equation of elastodynamic problem for orthotopic nonhomogeneous hollow sphere subject to a magnetic field, initial stress and rotation is solved in terms of displacement potentials. The analytical solution for the elastodynamic equation is shown in detail for different cases by figures. The results indicate that the effect of the magnetic field, initial stress, rotation and nonhomogeneity on radial displacement, the corresponding stresses, and perturbation magnetic field are very pronounced.
2. Formulation of the problem
Consider a hollow sphere placed initially in an axial magnetic field. For a spherically orthotopic elastic medium, the spherical coordinates (r, 9, 9) are helpful with r radial 9 colatitudinal and 9 meridional. The basic equations of the spherical orthotropic coincident with the origin. The only the radial displacement Ur = Ur(r, t) is a function of r and t only, the circumferential displacement U9 moreover, the longitudinal displacement U9, which are independent of 9 and 9. The dynamical equation in the r direction is given by
8xr
dr
2 1 1
Xrr T99 x
99
Me (J x H)r
xfí xU)r =p
8 2ur 8t2
(1)
where Q = (0, 0, Q), (Q x Q x U)r is a component of the centripetal acceleration in the radial direction r, due to the time-varying motion only, and p is the density of the material of the sphere and Q is the rotation, f = ^e(J x H)r is defined as a component of Lorentz's
force in the radial direction r, which may be written from the governing Maxwell equations:
ah
J = curlh, - ^e — = curle, divh = 0,
at
h = curl (U x H) = ( 0, 0, -1H0 d(rUr) I r dr
= 10,0, - Ho \ÔU + U
ôr r
= H| 'ôUr + 2Ur >=-H01 IT+—
(2)
fr = Це (J XH) = ^eH0
2 ô (ÔUr + 2Ur
ôr I ôr r J'
G, =
ôUr 2Ur
ôr
r
Ur
Tee = (ci2 + P) + (c22 + P)— + (c23 + PY- , ôr r r (3)
тфф = C13
ôUr Ur U,
+ c23 + C33
ôr r r
= ц>2m, i, j = 1,2,3,
(5)
xrr = r(-1+2m)
xl (2p* +a12 +a13)Ur + r(p* +an)
ôUr ôr
_ = r (—1+2m)
tûû — r
xl (2p* +a22 +a23)Ur + r(p* +a12)
_ = r (—1+2m) 1фф '
x\ (2p +a23 +a33)Ur + r(p +a13)
ôUr ôr
ôUr ôr
Substituting from Eqs. (6) and (5) into Eq. (4), then one obtains:
—1+m
r
[(-2 + 4m) p* + a12 + 2ma12 + a13 + 2ma
43
-a22 -2a23 -a33]Ur +r
ru1p0Q
V
dr r j
where crr is the magnetic stress, h is the perturbed magnetic field over the primary magnetic field, E is the electric intensity, J is the electric current density, is the magnetic permeability, H(0, 0, H0) is the constant primary magnetic field, and U = (Ur(r, t), 0, 0) is the displacement vector,
V = (cn + P) + (Ci2 + P) ^ + (C13 + P) ^,
rp0-
ô 2U,
ôt
ôu
2 + 2[(1 + m)(p* +an) + H 0^0] "
ôr
+r ( p* +an +H 0^0)
ô2U ^ ôr 2
= 0.
(7)
Tr^ = Tr9 = T99 = 0
Substituting from Eqs. (2) into Eq. (1), one gets:
arrr 2 -1 -1
— + — Irr —^09 _Tmm
or r r r
, a(aur 2U \ o a2ur ,A.
+^H[if+u J+pQ ur=p~U. (4)
Let us characterize the elastic constants c-, the density p of nonhomogeneous material in the form
Cj = a-r2m, p = p0r2m, P = p*r2m,
In the next part, the analytical solution for the radial vibration of an elastic spherical body of ortho-tropic material is studied.
3. Solution of the problem
In this part, the analytical solution of the above problem for a spherical region of inner radius a and outer radius b with different boundary conditions, by taking the harmonic vibrations. The solution of Eq. (7) has the following form:
Ur (r, t) = ui(r(8) where ю is the natural frequency of the vibrations, t is the time.
Substituting from Eq. (8) into Eq. (7), one gets:
e—Шг—1+m
[(-2 + 4m) p* + a12 + 2ma12 + a13
+ 2ma0 ^^22 2a 23 ^^33
■r 2p0(ra + Q2)]u(r)
where ay and p0 are the values of ctj and p in the homogeneous case, respectively, and m is the nonho-mogeneous parameter,
-[2r (1 + m)((p* +an) + 2rH 0V0] ^
ôr
î * TT2 \ d u ^
+ r (p +an +H0 Ц0)-1
dr2
J.
= 0,
(9)
where u1(r) is given in terms of m due to nonhomogeneity of material. Then one have
—1+m
a1r 2p0(ra +Q2)u1(r )
du,
2. Y
d2u
+ ra3~T2
v dr dr2 j
= 0,
(10)
where
a1 = (-2 + 4m) p* + a12 + 2ma12 + a13 + 2ma13 -a22,
a 2 = 2(1 + m)( p* +an) + H QVq,
(11)
a3 = p* +a11 + H 0M0
Equations (11) are called spherical Bessel's equations [32], and its general solution is known in the form:
1
= r2 2(p*+au
u1 = r
(
C1J1/2+t
/a^ p0 V ra2 + Q2
A
I
p* +an +H 0^0
+ C2Y1/2+n
ja1yl p0V ra2 +Q2 yjp* +an +H 02^0
(12)
Substituting from the above Eq. (12) into Eq. (8), one obtains the components of the displacements:
U = r 2 2(p*+a11+H0^0)
CJ
1/2+n
ja1yj p0V ra2 +Q2
t
p* +an +H 0^0
+ C2Y1/2+n
/a1A/ p0 V ra2 +Q2
p* +a11 +H 0^0
(13)
From Eq. (13), one get strain components in the form:
x [C1 J_V2+ n (k2r) + C2 Y-1/2+n (k2r)]e
-12+
x (p* +an + H02^0)'
-12
(14)
e00 = r
= r-1/2-n
[J*n (V) + C2V n (V )]e-itra ,(15)
U,
ew = eee =■
(16)
Substituting from Eq. (13) into Eqs. (5), one obtain the stresses:
t,, = (p* +an + H 02^0)_V2 r _1/2+2m_n x [^far (p* +a11 ra2 +Q2
x J-1/2 +n (V )C1 +(2 p*
+ a12 +a13) x (p* +a11 + H0V0)1/2 J1/2+ n (k2r)C1
+ C2yja1r (p* +an)^/p0\/ ra2 +Q2
x Y_V2+n (k2r) + C2(2 p* + a12 +a13)
x (p* +an + H02^0)V2Y-12+n(V)]e_iira, (17) Tee = ( p* +an + H 2m0>_v2 r _V2+2m_n
x [^/a"r(p* + a12)^/pQ\/ra2 +Q2
x J-1/2+n (k2r)C1 + (2p* + a22 + a23)
x (p* +a11 + H 02^0)V2 J 1/2+n (k2r )C1
+ C2*Ja1r (p* + a12)^pQV ra2 +Q2
x Y_1/2+n (k2r) + C2 (2p* +a22 +a23)
x (p* +an + H02^0)V2YV2+n (V)*^, (18) Tw = (p* +an + H 2^0)_12 r-1/2+2m-n
x [^/ar(p* +a13)^/p0Vra2 +Q2 x J-12+ n(k2r)C1 + (2p* +a23 + a33)
x (p* +an + hq2^2 J1/2+n(V)C1 + C2^fa1r (p* + a13)^/pQV ra2 + Q2
x Y1/2+n (k2r) + C2 (2p* + a23 +a33)
x (p* +an + H02^0)V2YV2+n (V)]e-itra. (19)
Substituting from Eq. (13) into Eqs. (2), the component of perturbed magnetic field h^ over the primary magnetic field and the magnetic stress crr are in the form
h^=-H 0( p* +an +H 02^0)-V2 x [^/arra2 +Q2 rV2-n x (J^ (k2r) + C2Y-12+n (k2r))e-itra ]
- H,r-x'2-n (J+n (V) + C2Y/2+n (k2r))e-itra, (20)
where
a0
n =
2(p* +an + H(2^0 )' 2
.,a1 Jp0 V ra2 +Q2 k2 = v, 1VF0 (21)
= e
+ C
p +an +H0 ^0 itra H 0V -1l2-n (-a2 + 3( p* +an + Hq2^)) JV2+n (V )C1
p* +an + H q2^q
/a1 r^/ pq V ra2 +Q2 J^2+n kr )C1
p* +an +H q2^q
(-a2 + 3( p* +a11 + H02^0))YV2+n (k2r) p* +an +H q2^q
a
2
a
2
X
- C
№2 +Q 2 >12 + n ( k 2 r )
* T T-2
p +a11 + H (Iq
where C1, C2 are arbitrary constants and jn(k2r) and yn(k2r) denote to spherical Bessel's function of the
(22)
Yn+1/2( k2 r ),
k2 is a constant.
We can determine the constants from the boundary conditions:
Ur ( r, t ) = 0 at r = a,
- ■ (23)
Trr (r, t) + arr (r, t) = -pe-mt at r = b,
where p is a constant, then from Eqs. (16), (17) and
(20) we have:
C1 = Pb - mVp* +a11 + HoV 0 Yn+1/2( k2 a )
Jn+1/2( k2 a ) >n-1/2( k2b)
+
:+1/2 ' n + m
X >n+12 (k2b ) - >n+1/2 (k2b )d9
y
C2 = -Pb - ^ VP*+aÛ^H(2(Ô7n+12( k2 a )
(24)
Jn+12(k2a) ( >n-1/2 (k2b) + n + m
V
b n-1
X >n+12 (k2b ) - >n+V2 (k2b )d9
= Jn-V2(k2b) + Cl n m Jn+1/2(k2b).
Substituting from Eqs. (24) into Eqs. (13), (17)-(20) then we have the corresponding radial displacement, radial stress, hoop stress, and perturbation magnetic field in a radial nonhomogeneous material.
4. Discussion and numerical results
The numerical results have been obtained graphi-
cally to display the distribution of displacements, stresses, and perturbation magnetic field through the radial direction of the inhomogeneous orthotropic hol-
low sphere. The elastic constants may be obtained from [30, 32, 33] may be taken as an example: a11 =
0.134, a12 = 0.101, a13 = 0.099, a22 = 0.674, a23 = 0.151, a33 = 0.297, with the above values of the elastic constants and b = 3 cm, a = 1 cm. Numerical calculations are carried out for the displacement and the stress components along the r-direction at different values of the rotation in the cases for nonhomogeneous material, orthotropic material. Figures 1-4 depict the variation of displacement, radial stress, hoop stress and perturbation magnetic field along the radial direction of the inhomogeneous hollow sphere with different values of the inhomogeneity exponent m and constant magnetic field H0. It is seen easily from all figures that the radial displacement satisfies the mechanical boundary conditions. Figure 1 shows the variation of radial displacement, with increasing r, in case of non-homogeneity m = 0.5 at different values for magnetic field H0 = 0.3 x 103, 0.8 x 103, 1.3 x 103, 1.8 x 103 as in
Fig. 1. Variation of the radial displacement U versus the radius r at different values for magnetic field H0 (a), rotation Q (b), initial stress p (c), Q = 0.8 (a, c), p = 1.2 (a, b), H0 = 1.3 x 103 (b, c), m = 0.5
1.0 1.5 2.0 2.5 r
-p* = 0.4
—....../ = 1.2
-.-p* = 1.6
Va. ___________
1.0 1.5 2.0 2.5 r
Fig. 2. Variation of the radial stress t,, versus the radius r at different values for magnetic field H0 (a), rotation Q (b), initial stress p (c), Q = 0.8 (a, c), p = 1.2 (a, b), H0 = 1.3 x 103 (b, c), m = 0.5
Fig. 1, a, different values of initial stress p* = 0.4, 0.8, 1.2, 1.6 as in Fig. 1, b and at different values of rotation Q = 0.3, 0.8, 1.3, 1.8 as in Fig. 1, c. The radial displacement satisfied the boundary conditions in Fig. 1 in the case an orthotropic inhomogeneous hollow sphere.
Figure 2 shows the variation of radial stress versus the radius r in case of nonhomogeneity m = 0.5 at different values for magnetic field Hq (Fig. 2, a), initial stress p (Fig. 2, b) and rotation Q (Fig. 2, c).
Figure 3 shows the variation of hoop stress versus the radius r in case of nonhomogeneity m = 0.5 at different values for magnetic field Hq (Fig. 3, a), initial stressp (Fig. 3, b) and rotation Q (Fig. 3, c).
Figure 4 shows the variation of perturbation magnetic field versus the radius r in case of nonhomogeneity m = 0.5 at different values for magnetic field Hq
-300-400-1-,-,-,-
1.0 1.5 2.0 2.5 r
-200- ^ —" -300-400-1-,-,-,-
1.0 1.5 2.0 2.5 r
Fig. 3. Variation of the hoop stress t ee versus the radius r at different values for magnetic field H0 (a), rotation Q (b), initial stress p (c), Q = 0.8 (a, c), p = 1.2 (a, b), H0 = 1.3 x 103 (b, c), m = 0.5
(Fig. 4, a), initial stress p* (Fig. 4, b) and rotation Q (Fig. 4, c). It is evident that magnetic field and rotation have a significant influence, more than the influence of initial stress on displacement, stresses, and perturbation magnetic field. The influence of the magnetic field, initial stress, rotation and the nonhomogeneity on radial displacement, stresses and perturbation magnetic field is very pronounced. These results are specific for the example considered, but other examples may have different trends because of the dependence of the results on the mechanical of the material. The influence of the nonhomogeneity and orthotropic properties of the material is pronounced. The results in this paper compared with previous results, in the absence of magnetic field, initial stress, rotation, and nonhomogeneity. These results are specific for the example considered; one more cases may have dif-
Fig. 4. Variation of the perturbation magnetic field hф versus the radius r at different values for magnetic field H0 (a), rotation Q (b), initial stress p (c), Q = 0.8 (a, c), p = 1.2 (a, b), H0 = 1.3 x 103 (b, c), m = 0.5
ferent trends because of the dependence of the results on the mechanical properties of the material as is displayed in [34-36], that have more applications in scientific and technical disciplines and materials science.
5. Conclusion
In the present study, the exact solutions for radial displacement, stresses and perturbation magnetic field of orthotropic hollow sphere subjected to the magnetic field, initial stress, rotation, and the nonhomogeneity are obtained. The distribution of displacement, stress, and perturbation of the magnetic field are drawn and discussed in detail for various effects. The present technique is applicable to other homogeneous material. The numerical results are obtained and represented graphically. The results indicate that the effect
of the magnetic field, initial stress, rotation and non-homogeneity on radial displacement, stresses, and perturbation magnetic field are pronounced. An improvement of the present formulation will be considered in the future work to consider other type of structures and materials.
Funding
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. 130-11-D1441. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
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Received 11.11.2020, revised 28.01.2021, accepted 01.02.2021
Сведения об авторах
Khalil Salem Al-Basyouni, Dr., King Abdulaziz University, Saudi Arabia, [email protected]
Samy Refahy Mahmoud, Prof., King Abdulaziz University, Saudi Arabia, [email protected], [email protected]
