Stress concentration around a hyperboloidal notch under tension in a magnetoelectroelastic material

ZAMM · Z. Angew. Math. Mech. 84, No. 12, 818 – 824 (2004) / DOI 10.1002/zamm.200310134

Stress concentration around a hyperboloidal notch under tensionin a magnetoelectroelastic material

Xiao-Hong Wu, Ya-Peng Shen∗, and Xu Wang

The State Key Laboratory of Mechanical Structural Strength and Vibration, Xi’an Jiaotong University, 710049, P.R. China

Received 30 April 2003, revised 15 September 2003, accepted 8 October 2003Published online 1 December 2004

Key words stress concentration, potential functions approach, magnetoelectroelasticMSC (2000) 74B99, 74H35

A transversely isotropic magnetoelectroelastic medium made of piezoelectric BaTiO3 and magnetostrictive CoFe2O4 is boundedby a single sheeted hyperboloid of revolution which is traction, electric charges, and magnetic charges free. This solid is sub-jected to a finite tensile force and electric and magnetic charges at infinity. A closed-form solution based upon the potentialfunctions approach is obtained. The present solutions include all the solutions, such as piezoelectric, piezomagnetic, magne-tostrictive, and purely elastic solutions, as special cases. Numerical examples of the tensile stress at the narrowest section arepresented and show that the stress concentration is significantly affected by material property and applied electric charges, andnot appreciably influenced by magnetic charges.

c© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The magnetoelectroelastic materials which simultaneously possess piezoelectric, piezomagnetic, and magnetoelectric effectshave drawn much attention. These materials consisting of a piezoelectric phase and a piezomagnetic phase exhibit the magne-toelectric effect that is not presented in single-phase piezoelectric or piezomagnetic materials [1–3]. Van Run et al. [4] reportedthe fabrication of a BaTiO3-CoFe2O4 composite with a magnetoelectric coefficient two orders larger than that of Cr2O3, whichhas the highest magnetoelectric coefficient among single-phase materials known at that time. A micromechanics approach isdeveloped to analyze the average fields and effective moduli of heterogeneous magnetoelectroelastic solid that exhibit fullcoupling between stationary elastic, electric, and magnetic fields by Li and Dunn [5]. Although various inclusion, crack, orcavity related problems in magnetoelectroelastic media have been studied in recent years [6–9], no exact solution is availableto the external circumferential notch problem associated with stress concentration of magnetoelectroelastic media.

Chen [10] obtained the elasticity solution to the problem of a transversely isotropic medium externally bounded by ahyperboloid of revolution, subjected to axial tension at infinity. The corresponding problem in isotropic elasticity and also ofthe notch under other loading conditions was originally established by Neuber [11].

In this paper, the stress concentration of a magnetoelectroelastic hyperboloidal notch subjected to axial loading conditionsat infinity will be studied. The magnetoelectroelastic solid is transversely isotropic and the elastic symmetric axis is parallelto the geometric axis of the hyperboloidal surface. There is a prescribed tensile force, electric charges or magnetic chargesacting in the direction of this axis. The potential functions approach is employed to obtain a complete closed-form solution.Numerical examples of the tensile stress at the narrowest section are presented.

2 General solution of axisymmetric problems in magnetoelectroelastic media

The basis for potential solutions in magnetoelectroelastic solids is presented in the paper by Wang and Shen [12]. Cylindricalcoordinates (r, θ, z) with z axis parallel to the axis of symmetry of the physical property is introduced. The dimensionlessparameters λj (j = 1, 2, 3, 4) which are dependent on the 17 independent constants including five elastic constants, threepiezoelectric constants, three piezomagnetic constants, two electromagnetic constants, two dielectric permittivities, and twomagnetic permeabilities are characteristic distinct roots of the equation

uT (A − λB)−1 u = c11 − c44λ−1, (1)

where

u =

c13 + c44

e15 + e31

f15 + f31

, A =

c33 e33 f33

e33 −ε33 −g33f33 −g33 −µ33

, B =

c44 e15 f15

e15 −ε11 −g11f15 −g11 −µ11

. (2)

∗ Corresponding author, e-mail: [email protected]


ZAMM · Z. Angew. Math. Mech. 84, No. 12 (2004) / www.zamm-journal.org 819

The above equation is a quartic equation in λ.Constants kij (i = 1, 2, 3, j = 1, 2, 3, 4) dependent on material constants and characteristic roots λj can be expressed by

K = [k1 k2 k3 k4] = (A − λB)−1uv, (3)

where

kj =

k1j

k2j

k3j

(j = 1, 2, 3, 4), v = [λ1 λ2 λ3 λ4] . (4)

For the axisymmetric problem, four displacement functions Φj(r, zj), (j = 1, 2, 3, 4) are needed and satisfy, respectively,the following equations:

(∂2

∂r2+

∂

r∂r+

∂2

∂z2j

)Φj = 0, (j = 1, 2, 3, 4), (5)

where zj = z/√λj (j = 1, 2, 3, 4).

Displacements ur, w, electric potential ϕ, and magnetic potential ψ in the cylindrical coordinate system can be expressedconcisely in terms of the four potential functions as follows:

ur = JP,r, W = KH−1/2P,zj , (6)

where

J = [1 1 1 1], W =

w

ϕ

ψ

, P = [Φ1(r, z1) Φ2(r, z2) Φ3(r, z3) Φ4(r, z4)]

T,

H = diag [λ1 λ2 λ3 λ4] .

The components of stresses, electric displacements, and magnetic inductions are related to the potential functions throughthe following equations:

σrr =(2c66JH − c44J − IT

0 BK)H−1P,zjzj + 2c66JP,rr,

σθθ = −(c44J + IT

0 BK)H−1P,zjzj

− 2c66JP,rr,

σzz

Dz

Bz

= B

(I0J + K

)P,zjzj ,

τrz

Dr

Br

= B

(I0J + K

)H−1/2P,rzj , (7)

where I0 = [1 0 0]T .

3 Boundary conditions

The hyperboloidal notch surface is described by the equation

r2

a2 (1 − e2)− z2

a2e2= 1, (8)

where e ≺ 1, a is a length parameter, and e is a dimensionless parameter describing the curvature of the notch as shown inFig. 1. In describing stress concentration factors Neuber [11] used the ratio of the half width at the base of the notch (i.e. thenarrowest cross section) and the radius of curvature there. This ratio called R is given by

R =(1 − e2

)/e2. (9)

There are no external loads over this hyperboloidal surface, in addition, considering that the dielectric permittivities andmagnetic permeabilities of magnetoelectroelastic materials are three orders higher than the environment (e.g. air). The boundaryconditions over the hyperboloidal surface could be described as follows:

σzz

Dz

Bz

nz +

τrz

Dr

Br

nτ =

000

, (10)


820 X.-H. Wu and Y.-P. Shen et al.: Stress concentration in magnetoelectroelastic material

212 ea −

LdABA z =∫

QdADA z =∫

LdABA z =∫

QdADA z =∫

TdAA zz =∫ σ

PdAA zz =∫ σ

θ

Fig. 1 Deep hyperboloidal notch under magneto-electro-elastic tension.

σrrnr + τrznz = 0, (11)

where nr and nz are direction cosines of the normal at the free surface.The notch is loaded by a tensile force T , a pair of electric charges ±Q and magnetic flux L along the z axis. The narrowest

section of the notch has the radius of r = a√

1 − e2.

4 Analysis

Following the usual definition of spheroidal coordinates, functions pj(r, zj) and qj(r, zj) (j = 1, 2, 3, 4) are defined by

pj = cos ξj , qj = i sinh ηj , (12)

r = aj cosh ηj sin ξj , z = aj sinh ηj cos ξj , (13)

a2j = a2 (1 − e2 + e2/λj

), (14)

where i is the imaginary unit. It can be shown that the hyperboloid of one sheet defined by eq. (8) is represented by

pj = ρj , (15)

where

ρj = ae/aj

√λj . (16)

The ρj (j = 1, 2, 3, 4) are governed by the shape of the notch and the material property. After some algebraic manipulations,the left terms in eqs. (10) and (11) become

τrz

Dr

Br

+

σzz

Dz

Bz

nz

nr=

r

aeB(I0J + K

)M−1P,pjzj

, (17)

σrr + τrznz

nr= −

(c44J + IT

0 BK)N−1H−1P,qjzj − 2c66

rJP,r, (18)

where

M = diag(a1ch

2η1, a2ch2η2, a3ch

2η3, a4ch2η4),



N = diag (a1ρ1, a2ρ2, a3ρ3, a4ρ4) .

The external loads acting on the notch are tensile force T , electric charges Q, and magnetic charges L and act along the zaxis. It follows that over any arbitrary plane cross section, the resultant force, surface electric charger, and magnetic flux arealong the z direction. From the expression for σzz , Dz , and Bz in eq. (7), we have

T

Q

L

= 2π

∫ r(ρ)

0

σzz

Dz

Bz

rdr = −2πB

(I0J + K

)rP,r

∣∣∣r(ρj)

r=0, (19)

where r(ρ) denotes the radius of the hyperboloidal free surface. Since these resultant force, electric displacement flux, andmagnetic displacement flux are the same for any cross section, the term rP,r must be independent of qj . In selecting thepotential functions, we are guided by Chen’s work on the corresponding transversely isotropic problem, and the requirementthat the elastic, electric, and magnetic fields do not contain any singular point. We shall assume that

P = Sh,

S = diag(

− A1

1 − ρ1,− A2

1 − ρ2,− A3

1 − ρ3,− A4

1 − ρ4

),

h = [H1(r, z1) H2(r, z2) H3(r, z3) H4(r, z4)]T,

(20)

whereAj (j = 1, 2, 3, 4) are four unknown constants. The functionsHj(r, zj) will now take the form of a particular combinationof zero and first order Legendre functions:

Hj(r, zj) = ln r +Q0(pj) + P1(pj)Q1(qj). (21)

Partial differentiation with respect to r and zj reveals that

r∂Hj(r, zj)

∂r= 1 − pj (22)

and

∂Hj(r, zj)∂zj

=i

ajQ0(qj). (23)

Note that r∂Hj/∂r is independent of qj . Substituting eqs. (20) and (22) into eq. (19) leads to

B(I0J + K

)X = f ,

X = [A1, A2, A3, A4]T, f =

12π

[T,Q,L]T .(24)

The next step is to consider the traction, electric, magnetic free surface condition given by eqs. (10) and (11). Eq. (10)represents the surface condition in the axial direction. From the axisymmetric nature of the stress, electric, and magnetic fields,it can be argued on the basis of governing field equations that these conditions will be identically satisfied, if the resultantforce, surface electric charger, and magnetic flux in the axial direction for any plane section are constants (these are true fromeqs. (19) and (22)). It can be readily verified from eqs. (17) and (23) that eq. (10) is satisfied. What remains then is the tractionconditions in the radial directions represent by eq. (11). It is required that this condition be satisfied everywhere over thehyperboloidal surface.

After some manipulation with eqs. (18) and (23), we find that eq. (11) is satisfied if

[(c44J + IT

0 BK)GH−1 − 2c66J

]X = 0,

G = diag(

1 + ρ1

ρ1,1 + ρ2

ρ2,1 + ρ3

ρ3,1 + ρ4

ρ4

).

(25)

By means of eqs. (24) and (25), the constants Aj (j = 1, 2, 3, 4) can be determined.

X ={b(I0J + K

)+ Q

[(c44J + IT

0 BK)d − 2c66J

]}−1F, (26)



b =

c44 e15 f15 0e15 −ε11 −g11 0f15 −g11 −µ11 00 0 0 0

, Q =

0001

, F =

12π

T

Q

L

0

,

d = diag(

1 + ρ1

ρ1λ1,1 + ρ2

ρ2λ2,1 + ρ3

ρ3λ3,1 + ρ4

ρ4λ4

).

The elastic, electric, and magnetic fields can now be expressed in terms of the variables pj and qj . They are

ur = − 1rDX, D = diag

(1 − p1

1 − ρ1,1 − p2

1 − ρ2,1 − p3

1 − ρ3,1 − p4

1 − ρ4

), (27)

W = −iKEH−1/2X, E = diag(

Q0(q1)a1(1 − ρ1)

,Q0(q2)

a2(1 − ρ2),

Q0(q3)a3(1 − ρ3)

,Q0(q4)

a4(1 − ρ4)

)(28)

σrr = −((c44J + IT

0 BK)CH−1 − 2

c66r2

JR)X, (29)

C = diag

(p1

a21(1 − ρ1)

(p21 − q21

) , p2

a22(1 − ρ2)

(p22 − q22

) , p3

a23(1 − ρ3)

(p23 − q23

) , p4

a24(1 − ρ4)

(p24 − q24

)),

R = diag (1 − p1, 1 − p2, 1 − p3, 1 − p4) ,

σθθ =((

2c66JH − c44J − IT0 BK

)CH−1 + 2

c66r2

JR)X, (30)

σzz

Dz

Bz

= B

(I0J + K

)CX,

τzr

Dr

Br

= B

(I0J + K

)H−1/2CTX, (31)

T =1r

diag

(z1(1 − p2

1)

p21

,z2(1 − p2

2)

p22

,z3(1 − p2

3)

p23

,z4(1 − p2

4)

p24

).

For stress concentration calculations, one wishes to know the stress at the outside fiber of the narrowest section, z = 0,r = a

√1 − e2, i.e., qj = 0, pj = ρj . The longitudinal stress at that point is

σzz =(c44J + IT

0 BK)UX,

U = diag(

1a21ρ1(1 − ρ1)

,1

a22ρ2(1 − ρ2)

,1

a23ρ3(1 − ρ3)

,1

a24ρ4(1 − ρ4)

).

(32)

The average value of σzz at the narrowest section is

σaν =P

πa2 (1 − e2). (33)

Then we get the stress concentration factor

K =σc

σaν, (34)

where σc is the maximum value of σzz at the narrowest section.

5 Numerical result and discussion

The stress concentration factor has been computed by means of eq. (34) for a variety of different materials and notch shapes.When the notch is subjected to tensile force T only along the z axis, the stress concentration factors have been plotted in Fig. 2for five materials: CoFe2O4, BaTiO3, PZT-4 crystals, BaTiO3-CoFe2O4 fibrous composite (magnetoelectroelastic media), andEPZT-4. Note that EPZT-4 is a hypothetical non-piezoelectric transversely isotropic material that is assumed to have the sameelastic constants as that of PZT-4, and piezoelectric constants zero. The magnetoelectroelastic material contains 60% CoFe2O4

and 40% BaTiO3, and its effective moduli can be found in Li [13]. It is observed that the stress concentration factor can besubstantially affected by material properties. Fig. 2 shows that the stress concentration factor of magnetostrictive CoFe2O4 ishighest, and that of EPZT-4 higher than that of PZT-4.



0 5 10 15 20 25 301

2

3

4

5

6S

tres

sC

once

ntra

tion

Fac

tor

K

R (ratio of notch radius to radius of curvature)

BaTiO3-CoFe

2O

4

CoFe2O

4

BaTiO3

PZT-4EPZT-4

Fig. 2 Stress concentration factor for hyperboloidal notch.

0 100 200 300 400 500

-1

0

1

2

3

4

)C/N(100*Q

z=0,r=0

z=0,r=r0

Str

ess

Con

cent

ratio

nF

acto

rK

BaTiO3-CoFe

2O

3

BaTiO3

PZT-4BaTiO

3-CoFe

2O

3

BaTiO3

PZT-4

Fig. 3 The disturbance of electric charge Q on the longitudinal stressat the section z = 0.

-200 -100 0 100 200 300 400 5000.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

)m /A(L

z=0,r=0

z=0,r=r0

Str

ess

Con

cent

ratio

nF

acto

rK BaTiO

3-CoFe

2O

3

CoFe2O

3

BaTiO3-CoFe

2O

3

CoFe2O

3

Fig. 4 The disturbance of magnetic charge L on the longitudinalstress at the section z = 0.

Under tensile force T and electric charge Q along the z axis, the disturbances of charge Q = Q1010/T on the stressconcentration factor at center point r = z = 0 and at the outside fiber of the narrowest section z = 0, r = a

√1 − e2 for PZT-4,

BaTiO3, and BaTiO3-CoFe2O4 are given in Fig. 3 where the shape parameter R is equal to 10. It is shown that the electricchargeQ affects substantially the stress concentration factor of outside fiber of the narrowest section and insignificantly to thatof the central point. At outside fiber, such an effect is most significant for BaTiO3-CoFe2O4 material with the same magnitudeof chargeQ, and a higher magnitude implies a larger effect. Using this effect, we can diminish the stress concentration factor bychanging the value of the chargeQ. Under tensile force T and magnetic fluxL along the z axis, the disturbances of the magneticflux L = L1010/T on the stress concentration factor at center point r = z = 0 and at the outside fiber of the narrowest sectionz = 0, r = a

√1 − e2 for CoFe2O4 and BaTiO3-CoFe2O4, where the shape parameter R is 10 are given in Fig. 4. It suggests

that the effect on the stress concentration factor of outside fiber of the narrowest section by the magnetic flux L is obviouslyless than the electric charge Q while the same insignificantly to that of the central point. And negative magnetic flux L leads



0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

( )21 ear −

)C/N(5.3

)C/N(1

)C/N(0

===

Q

Q

Q

σ zz/σ

av

Fig. 5 Longitudinal stress along the r at the narrowest section z = 0for several Q.

to a lower stress concentration factor while the inverse charge leads to a higher one. From these figures, we can also see thatthe loads along the z axis have greater effect on the stress concentration factor of magnetoelectroelastic media among thesedifferent kinds of materials. Fig. 5 shows the values of σzz along the radius for different chargeQ for BaTiO3-CoFe2O4. It canbe seen that the biggest stress concentration factor is decreased and the location of the maximum stress is not the outside ofthe fiber when certain charge Q is applied to notch.

6 Conclusion

The elasticity solution to the problem of a magnetoelectroelastic hyperboloidal notch subjected to axial loading conditions atinfinity is obtained. The present solution includes all the solutions, such as piezoelectric, piezomagnetic, magnetostrictive, andpurely elastic solutions, as special cases. Numerical examples of the tensile stress at the narrowest section are presented andshow that the stress concentration is significantly affected by material property and applied electric charges, and not appreciablyinfluenced by magnetic charges.

Acknowledgements The authors would like to thank for the financial support by the National Natural Science Foundation of China(No. 50135030 and 0132010), and for the partial support by the Doctorate Foundation of the Xi’an Jiaotong University.

References

[1] G. Harshe, J. P. Dougherty, and R. E. Newnham, Theoretical modeling of multilayer magnetoelectric composites, Int. J. Appl. Elec-tromagn. Mater. 4, 145–159 (1993).

[2] C.W. Nan, Magnetoelectric effect in composites of piezoelectric and piezomagnetic phase, Phys. Rev. B 50, 6082–6088 (1994).[3] Y. Benveniste, Magnetoelectric effect in fibrous composites with piezoelectric and piezomagnetic phase, Phys. Rev. B 51, 16424–16427

(1995).[4] A. M. J. G. Van Run, D. R. Terrell, and J. H. Scholing, An in situ grown eutectic magnetoelectric composite material, J. Mater. Sci.

(UK) 9, 1710–1714 (1974).[5] J.Y. Li and M. L. Dunn, Anisotropic coupled-field inclusion and inhomogeneity problems, Philos. Mag. 77, 1341–1350 (1998).[6] J. H. Huang and W. S. Kuo, The analysis of piezoelectric/piezomagnetic composite materials containing an ellipsoidal inclusion, J.

Appl. Phys. 81(3), 1378–1386 (1997).[7] J.Y. Li and M. L. Dunn, Anisotropic coupled field inclusion and inhomogeneity problems, Philos. Mag. 77(5), 1341–1350 (1998).[8] J. X. Liu, X. G. Liu, and Y. B. Zhao, Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack,

Int. J. Eng. Sci. 39, 1405–1418 (2001).[9] J. H. Huang,Analytical predictions for the magneto-electric coupling in piezomagnetic materials reinforced by piezoelectric ellipsoidal

inclusions, Phys. Rev. B 58, 12–15 (1998).[10] W.T. Chen, Stresses concentration around a hyperboloidal notch under tension in a transversely isotropic material, ASME J. Appl.

Mech. 38, 185–189 (1971).[11] H. Neuber, Theory of Notch Stresses. English translation (Edwards, Ann Arbor, 1946).[12] X. Wang and Y. P. Shen, The general solution of three-dimensional problems in magnetoelectroelastic media, Int. J. Eng. Sci. 40,

1069–1080 (2002).[13] J.Y. Li and M. L. Dunn, Micromechanics of magnetoelectroelastic composite materials: Average fields and effective behavior, J. Intell.

Mater. Syst. Struct. 7, 404–416 (1998).


Documents

Stress concentration around a hyperboloidal notch under tension in a magnetoelectroelastic material