Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 1, pp. 143-159, Warsaw 2009 ANALYSIS OF A PENNY-SHAPED CRACK IN A MAGNETO-ELASTIC MEDIUM Bogdan Rogowski Technical University of Lodz, Chair of Mechanical of Materials, Łódź, Poland e-mail: brogowsk@p.lodz.pl The problem of a crack in a piezomagnetic material under magneto- mechanical loading is considered. The exact solution, obtained in this work, includes the unknown a priori normal component of the magnetic induction vector inside the crack. Several different physical assumptions associated with limited magnetic permeability of the crack are utilized to determine those unknownmagnetic inductions through the crack bo- undaries. Analytical formulae for the stress and magnetic induction in- tensity factors are derived. The effects of magnetic boundary conditions (limited permeability) at the crack surface on the basic parameters of fracture mechanics are analysed and some features of the solution are discussed. If the permeability of the medium inside the crack tends to zero or is very large, extreme results i.e. impermeable or permeable crack solutions are obtained. Key words: magnetoelasticity, limited permeable crack, stress and ma- gnetic intensity factors 1. Introduction Magneto-mechanical modelling of the piezomagnetic fracture is complicated by the fact that piezomagnetic materials exhibit magneto-elastic coupling be- haviour as well as anisotropy. The attractive property of piezomagnetic materials, that become strained when subjected to amagnetic field, is the underlying foundation for achieving numerous types of smart structures. When subjected to mechanical and ma- gnetical loads in service, piezomagnetic materials may fail prematurely due to their brittleness or due to the presence of defects or flaws produced during their manufacturing process. Therefore, it is important to study the fracture behaviour of piezomagnetic materials. 144 B. Rogowski Among theoretical studies on piezomagnetic bodies, magnetic permeable and impermeable conditions on crack faces are most commonly adopted. For permeable cracks, there is a nonzero magnetic field in the free space inside voids,while for impermeable cracks themagneticfield inside thevoids is always zero. In recent years, the study on magneto-electro-elastic materials with de- fects or crack has received considerable interest. Themagneto-elastic problem of straight cracks lying along the interface of two dissimilar soft ferromagne- tic materials subjected to a remote uniform magnetic induction was consi- dered by Lin and Lin (2002). The magneto-elastic coupling effect in an in- finite soft ferromagnetic material with a crack was also studied by Liang et al. (2002), where the nonlinear effect of magnetic field upon stress and the effect of deformed crack configuration were taken into consideration. Tho- se papers considered the coupling between magnetic and elastic fields. The electro-elastic field inside a piezoelectric material, where the limited elec- trical permeability inside the crack was taken into account, were conside- red, and closed form solutions were derived by Rogowski (2007). Rogow- ski (2008) discussed the limited electric boundary conditions on the crack faces in electro-elastic materials under transient thermal loading and also mechanic and electric (two cases) loadings, and closed form solutions were obtained. In this paper, a limited permeable crack model is considered. The effects of magnetic boundary conditions (limited permeability) at the crack surfaces on the fracture mechanics of piezomagnetic materials are analysed and some features of the solutions are discussed. In two limiting cases (infinitely large or zeromagnetic permeablities of themedium inside the crack) we can obtain the limiting solutions from the general results presented here. For piezoelectric materials, there are two kinds of ideal electric boundary conditions for the crack faces, that is, electrically impermeable crack and electrically permeable crack (Zhang et al., 2002). Although this paper is a generalisation to magnetoelasticity described by Zhang et al. (2002) who dealt with a piezoelectric medium in addition to different results regarding expressions for elastic and magnetic fields, there exists one distinct difference, namely the limited permeable crack boundary conditions are considered. The physical laws for piezomagnetic materials were explored by Nowacki (1983).Many theoretical problems can be found in the book byPurcell (1965) and Parkus (1972). Analysis of a penny-shaped crack... 145 2. Fundamental equations for piezomagnetic medium We consider an axi-symmetric problem. Assume that field variables are func- tions of r and z in the cylindrical coordinate system (r,θ,z). Constitutive equations for a piezomagnetic material polarised along the z direction subjec- ted to mechanical andmagnetic fields can be written as (σrr,σθθ,σzz,σrz,Br,Bz) ⊤ =C(εr,εθ,εz,εrz,Hr,Hz) ⊤ (2.1) where (·)⊤ denotes the transpose of a matrix and C=   c11 c12 c13 0 0 −q31 c12 c11 c13 0 0 −q31 c13 c13 c33 0 0 −q33 0 0 0 c44 −q15 0 0 0 0 q15 µ11 0 q31 q31 q33 0 0 µ33   (2.2) Here σij,Bi and Hi are stresses, components of the magnetic induction vec- tor and components of the magnetic field vector; cij, qij and µij are elastic constants, piezomagnetic constants andmagnetic permeabilities, respectively. The strain is related to the mechanical displacements ur, uz as follows εr = ∂ur ∂r εθ = ur r εz = ∂uz ∂z (2.3) εrz = ∂ur ∂z + ∂uz ∂r The equilibrium equations for stresses andmagnetic flux are ∂σrr ∂r + ∂σrz ∂z + σrr−σθθ r =0 ∂σzr ∂r + ∂σzz ∂z + σzr r =0 (2.4) ∂Br ∂r + Br r + ∂Bz ∂z =0 Here we neglect the body forces and magnetic sources in piezomagnetic ceramics. TheMaxwell equations in the quasi-static approximation are Hr =− ∂φ ∂r Hz =− ∂φ ∂z (2.5) where φ(r,z) is the magnetic potential. 146 B. Rogowski Substituting equations (2.1), (2.2) and (2.3) into equations (2.4) and using relations (2.5), we obtain the following equilibrium equations c11B1ur +c44D 2ur+(c13+ c44)D ∂uz ∂r +(q31+ q15)D ∂φ ∂r =0 c44B0uz + c33D 2uz +(c13+ c44)D ∂(rur) r∂r +q15B0φ+ q33D 2φ=0 (2.6) (q31+q15)D ∂(rur) r∂r +q15B0uz +q33D 2uz −µ11B0φ−µ33D2φ=0 where the following differential operators have been introduced Bk = ∂2 ∂r2 + 1 r ∂ ∂r − k r2 k=0,1 D= ∂ ∂z (2.7) The quasi-harmonic functions ϕi(r,z), such that ( ∂2 ∂r2 + 1 r ∂ ∂r + 1 λ2i ∂2 ∂z2 ) ϕi(r,z) = 0 i=1,2,3 (2.8) determine all field variables as follows ur(r,z) = 3∑ i=1 ai1λi ∂ϕi ∂r uz(r,z) = 3∑ i=1 1 λi ∂ϕi ∂z φ(r,z) =− 3∑ i=1 ai3 λi ∂ϕi ∂z σrr =− 3∑ i=1 ai4 λi ∂2ϕi ∂z2 − (c11− c12) ur r σzz = 3∑ i=1 ai4 λ3i ∂2ϕi ∂z2 σθθ =− 3∑ i=1 ai4 λi ∂2ϕi ∂z2 − (c11−c12) ∂ur ∂r σzr = 3∑ i=1 ai4 λi ∂2ϕi ∂r∂z Hr =− ∂φ ∂r = 3∑ i=1 ai3 λi ∂2ϕi ∂r∂z (2.9) Hz =− ∂φ ∂z = 3∑ i=1 ai3 λi ∂2ϕi ∂z2 Br = 3∑ i=1 ai5λi ∂2ϕi ∂r∂z Bz = 3∑ i=1 ai5 λi ∂2ϕi ∂z2 Analysis of a penny-shaped crack... 147 where ai1 = a1λ 2 i + b1 a2λ 4 i + b2λ 2 i + c2 ai3 = c13+ c44 q31+ q15 − c11− c44λ2i q31+q15 ai1 ai4 = q31c44λ 2 i +q15c11 q31+q15 ai1+ c44q31− c13q15 q31+q15 ai5 = q33µ11−q15µ33 µ11−µ33λ2i − q31µ11− q15µ33λ2i µ11−µ33λ2i ai1 (2.10) a1 = c33(q31+q15)− (c13+ c44)q33 a2 = c44q33 b1 = c13q15−c44q31 b2 =(c13+ c44)q31+ c13q15− c11q33 c2 = c11q15 and λ2i (i = 1,2,3) are the roots of the following cubic algebraic equation in λ2i a0λ 6+ b0λ 4+ c0λ 2+d0 =0 (2.11) with the coefficients defined by a0 = c44(c33µ33+q 2 33) b0 =(q31+q15)[2c13q33−c33(q31+q15)]+2c44q33q31− c11q233+ −µ11c33c44−µ33c2 c0 =2q15[c11q33− c13(q31+q15)]+c44q231+µ33c11c44+µ11c 2 (2.12) d0 =−c11(c44µ11+q215) c2 = c11c33− c13(c13+2c44) The roots of the above equation for a real material can be expressed for two cases: (a) +R1,−R1, +R2,−R2, +R3,−R3 (b) +R1,−R1,R2+iR3,R2− iR3,−R2+iR3,−R2− iR3 where R1,R2,R3 are positive real numbers and i= √ −1. 3. Formulation of the problem Consider a crack with a finite dimension in a transversely isotropic piezoma- gnetic solid under combinedmechanical (σ∞) andpuremagnetical loads (B∞ or H∞) applied at infinity (Fig.1). 148 B. Rogowski Fig. 1. A crack in a magnetoelastic medium and loading conditions To solve the crack problem in linear elastic solids, the superposition techni- que is usually used. Thus, we first solve the stress andmagnetic field problem without the cracks in the medium under magnetical and/or mechanical lo- ads. Then, we use equal and opposite stresses andmagnetic inductions as the crack surface tractions and solve the crack problem (the so called perturbation problem, Fig.2) σ∞ = { (1+q0)σ0− c̃1B∞ case I σ0−q3H∞ case II B∗ = { B∞ case I µ0σ0+µ3H∞ case II H∗ = { −q2σ0+ c̃3B∞ case I H∞ case II c̃3 = c11+ c12 q1 q2 = c̃3µ0 Fig. 2. Crack loading in the perturbation problem Thematerial parameters in the above solution are q1 =µ33(c11+ c12)+2q 2 31 q0 = c̃22 c̃20q1 = c̃2µ0 q1 µ0 = c̃2 c̃20 c̃2 = q33(c11+ c12)−2q31c13 c̃1 = c̃2 q1 (3.1) µ3 =µ33+ 2q231 c11+ c12 q3 = q33− 2c13 c11+ c12 q31 Analysis of a penny-shaped crack... 149 Note that σ0 is the uniform normal stress at zero magnetical loads. Employing the superposition principle, one arrives at an equivalent pro- blem with the loading σz = −σ∞, Bz = B0 −B∗ being applied on both surfaces of the crack. Inside the crack there is often air or vacuum, and the magnetic induction is usually considered constant under a uniform remote applied load. This unk- nown component is denoted by B0 and the following assumption is stated to determine B0 B0 =µaH c z (3.2) where µa is the magnetic permeability of the medium inside the crack and Hcz is the component of themagnetic field vector in the z-direction inside the crack. The quasi-harmonic function needed for the solution is ϕi(r,z) = ∞∫ 0 Ai(ξ)exp(−λiξz)J0(ξr) dξ (3.3) 4. Solution for a limited magnetically permeable crack problem The boundary conditions along the crack plane z=0 are stated as follows uz(r,0)= 0 φ(r,0)= 0 r­ a σzr(r,0)= 0 r­ 0 Bz(ρ,0)=B0−B∗ 0¬ r 0 case I ( 1−q3 H∞ σ0 ) K∗I > 0 case II (5.13) for the crack tip opening displacement to exist. Analysis of a penny-shaped crack... 155 Fig. 3. KB versus δ0/(aµa) for BaTiO3-CoFe2O4 composite and piezomagnetic CoFe2O4 The stress intensity factor vanishes if B∞ σ0 = 1+q0 c̃1 case I H∞ σ0 = 1 q3 case II (5.14) The right-hand sides of those equations are 5,303 · 10−6m/A and 3.049 · 10−6m/A (case I) and 0.033Am/N and 0.036Am/N (case II) for single ma- terial CoFe2O4 and composite, respectively. 5.2. The effect of crack opening displacement We assume that the magnetic field inside the crack can be found by Haz =− ϕ+−ϕ− u+z −u−z (5.15) Taking into account that B0 =µaH a z (5.16) one arrives at the magnetic condition B0uz(r)=−µaϕ(r) (5.17) along the crack region. 156 B. Rogowski Using the above obtained results for uz(r) and ϕ(r) on the crack surface, we obtain B0 =−µa m5σ∞+m(B0−B∗) m7σ∞+m6(B0−B∗) (5.18) This gives a quadratic equation with respect to B0 η1B 2 0 +η2B0+η3 =0 (5.19) where η1 =m6 η2 =m7σ∞−m6B∗+mµa (5.20) η3 =(m5σ∞−mB∗)µa For the two limiting cases, we obtain (a) µa =0, B imp 0 =0, since m7σ∞−m6B∗ 6=0 (b) µa →∞, Bperm0 =−(m5/m)σ∞+B∗ which are the solutions for impermeable and permeablemagnetic crack boun- dary conditions, respectively. For thismodel ofmagnetic boundary conditions, the magnetic induction intensity factors in the extreme cases are the same as given by equations (5.6) and (5.8), respectively. For µa →∞we obtain uz(r,0)= 2 π σ∞ m2 m̃ √ a2−r2 (permeable) (5.21) For µa → 0, we have uz(r,0)= 2 π m2 m̃ (m7σ∞−m6B∗) √ a2−r2 (impermeable) (5.22) Figure 4 shows one half of the crack opening displacement for two analysed materials and permeable or impermeable boundary conditions for σ∞ =10MPa and B ∗ =0.01N/(Am) (case I). Note that for both materials the crack opening displacement is large for the permeable boundary conditions in comparison to the impermeable case, but the difference is not visible. Analysis of a penny-shaped crack... 157 Fig. 4. The crack opening displacement for somemagnetic materials 6. Concluding remarks The following conclusions can be made based on the results obtained in the paper: (a) The stress intensity factor does not depend on the assuptions applied to the crack-face magnetic boundary condition assumptions. (b) The stress intensity factor depends on the applied mechanical and ma- gnetic loads and on thematerial constants (details are given in Section 5 and in Eq. (5.13)). (c) The stress intensity factor decreases with the magnetic field if the field is applied in the poling direction; in the opposite case KI increases. (d) Themagnetic induction intensity factor depends on the properties of the material and on the applied magnetic and mechanical loads, as shown by equations (5.12). (e) The magnetic permeability of air or vacuum inside the crack cannot be ignored while calculating the magnetic induction intensity factor. The effect of finite thickness of a very flat notch or of a crack opening displa- cement in a realistic structuremust be assessed. It can be seen that the function of permittivity andmaterial parameters f(·) describes the ratio of the normalmagnetic inductionwhich is stored inside the crack to the total normal magnetic induction B perm 0 which may be stored inside the crack. Hence, it can be said that the calculated magnetic induction in- tensity factor is an average magnetic induction intensity factor weighted by the function f(·). This situation exists for the notch solution model 158 B. Rogowski given by solution (5.11) and showngraphically in Fig.3. However, know- ledge of the notch thickness to length ratio is essential for obtaining the correct B0 in this model. Thus, the crack opening displacement model is more useful. The crack opening displacement is obtained explicitly in a closed-form, and this model may be applied to analysis of fracture of piezomagnetic materials in engineering applications. (f) Summing up, itmust be emphasised that the basic discrepancy exists in the field singularity for a crack and notch. For an elliptic hole, unlike a crack, the field has no singularity. Therefore, it should be noted that the field intensity factors presented in Section 5.1 are only valid for very flat notches (when the notch thickness-to-length ratio δ0/a is very small). One cannot find in the open literature a substantial experimental data for piezomagnetic materials about the applicability of permeable or impermeable boundary conditions and about the magnetic permeability inside a crack of a piezomagnetic material. But the comparison with experimental results is critical for the assessment of the most appropriate boundary conditions. The works in scientific laboratories must be underway and test results must be published. References 1. Aboudi J., 2001, Micromechanical analysis of fully coupled electro-magneto- thermo-elastic multiphase composites, Smart Materials and Structures, 10, 5, 867-877 2. HuK.Q.,LiG.Q., 2005,Constantmovingcrackproblems inamagnetoelectro- elastic material under anti-plane shear loading, Int. J. Solids Struct., 42, 2823-2835 3. Huang J., Chin Y, Hsien K.L., 1998, Magneto-electro-elastic Eshelby ten- sor for a pizoelectric-piezomagnetic composite reinforced ellipsoidal inclusions, J. Appl. Phys, 83, 5364-5370 4. Liang W., Fang D., Shen Y., Soh A.K., 2002, Nonlinear magnetoelastic coupling effects in a soft ferromagnetic material with a crack, International Journal of Solids and Structures, 39, 3997-4011 5. Lin C.B., Lin H.M., 2002, The magnetoelastic problem of cracks in bon- ded dissimilar materials, International Journal of Solids and Structures, 39, 2807-2826 Analysis of a penny-shaped crack... 159 6. Nowacki W., 1983, Efekty Elektromagnetyczne w Ciałach Stałych Odkształ- calnych, PWNWarszawa 7. Parkus H., 1972,Magneto-Thermoelasticity, Springer-Verlag, 1972 8. Purcell E.M., 1965, Electricity and Magnetism, Berkeley Physics Course vol. 2, McGraw-Hill Inc. 9. Rogowski B., 2007, The limited electrically permeable crack model in linear piezoelasticity, Int. J. Pressure Vessels Piping, 84, 572-581 10. Rogowski B., 2008, The transient thermo-electro-elastic fields in a piezoelec- tric plate with a crack, Int. J. Pressure Vessels Piping, 85, in press 11. Sih G.C., Song Z.F., 2003, Magnetic and electric poling effects associated with crack growth in BaTiO3-CoFe2O4 composite, Theoretical and Applied Fracture Mechanics, 39, 209-227 12. Zhang T.Y., Zhao M., Tong P., 2002, Fracture of piezoelectric ceramics, Advances in Applied Mechanics, 38, 147-289 Analiza kołowej szczeliny w ciele magnetosprężystym Streszczenie Rozpatrzono zagadnienie szczeliny w materiale piezomagnetycznym przy obcią- żeniumechanicznym imagnetycznym.Dokładne rozwiązanie, otrzymanew tej pracy, zawiera nieznaną a priori normalną składowąmagnetycznej indukcji wewnątrz szcze- liny. Fizyczne założenia, odnoszące się do ograniczonej magnetycznej przenikalności ośrodkawypełniającego szczelinę orazmagnetycznychwarunkówna brzegu szczeliny, prowadzą do wyznaczenia tej magnetycznej indukcji. Otrzymano analityczne wzory określające naprężeniowe i magnetyczne współczynniki intensywności typu I. Zbada- no wpływ magnetycznych warunków brzegowych na brzegu szczeliny na parametry mechaniki pękania i przedyskutowanopewnewłasności rozwiązań.Nieprzepuszczalny i przepuszczalny model szczeliny otrzymuje się jako przypadki graniczne. W pierw- szymmodelu uproszczonym indukcjamagnetycznaw szczelinie jest zawsze równa ze- ru. W drugimmodelu otrzymuje się różne wartości magnetycznej indukcji wewnątrz szczeliny, a tym samym współczynnika intensywności magnetycznej indukcji. Zależy to od warunków, jakie przyjmuje się na powierzchni szczeliny dla określenia w ośrod- ku szczeliny składowej wektora natężenia pola magnetycznego prostopadłej do jej brzegów. Manuscript received December 12, 2007; accepted for print April 25, 2008