Mechanics Research Communications 92 (2018) 15–20
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3D Path independent integral for thermoelastic and magnetostriction problem Awani Bhushan, S.K. Panda∗, P.K. Singh, P. Kartheek, R. Kumar, Y. Mittal Department of Mechanical Engineering, IIT (BHU), Varanasi 221005, India
a r t i c l e
i n f o
Article history: Received 13 November 2017 Revised 13 June 2018 Accepted 19 June 2018 Available online 28 June 2018 Keywords: 3D J-integral Contour integral Magnetostriction Thermoelastic Thermo-magnetic
a b s t r a c t In this work, a universal path independent integral Ju that represents the energy release rate or flux during crack extension in a homogeneous and isotropic material, has been derived for straight crack subjected to multiple loads. This version of integral includes the effect of elastic strain, Eigen strain, thermal strain, body forces and magnetic strain. Particularly focus is made on thermo-mechanical and magnetic strains. The 3D J-integral expression represents the energy release rate in the combined thermoelastic field with magnetostriction characteristics. This integral is applicable to the real field problems concerning medium in an electric field where the multiphysics of heat generation and magnetic field are involved. 3D finite element analysis of edge cracked magnetostrictive specimen has been carried out to demonstrate the efficacy of the integral expressions. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction The phenomenon of magnetostriction is very common in ferromagnetic materials, characterized by the changes in geometrical shape due to alignment of dipole, when such materials are subjected to magnetic fields [1]. The extensive use of ferromagnetic and magneto-ceramic materials in practical engineering, such as nuclear reactors components, aerospace components, electrical motors, transducers etc. where magnetic field is dominant, has given impetus to the magnetostriction problems [2]. Steel is widely used as a soft ferromagnetic material in nuclear reactors, where its magnetostrictive performance has been an important aspect of the structural analysis in the design of a nuclear reactor [3–5]. Further in the last few decades, many new types of functional ferromagnetic materials such as magnetostrictive composites [6,7], magnetostrictive alloy of iron and rare earth elements [8–10] and ferromagnetic shape memory alloys [11–14] has been emerged and studied for engineering applications. This addition of high magnetic field or magneto elasticity along with mechanical stresses under static or cyclic loading leads to development of micro cracks eventually resulting in failure of structure. Thus, it becomes increasingly important to explore crack parameters under external magnetic field over thermo-mechanical behavior. The authors are unaware of such a formulation, which might have wide application in current carrying medium, as the possibility of heat gener-
∗
Corresponding author. E-mail address:
[email protected] (S.K. Panda).
https://doi.org/10.1016/j.mechrescom.2018.06.005 0093-6413/© 2018 Elsevier Ltd. All rights reserved.
ation coupled with magnetostriction is of relevance to the crack progression behaviour in such cases. Estimation of fracture parameter for the cracked body is challenging task in fracture mechanics. Rice proposed the approach to identify a path independent line integral surrounding a class of notch tips in two-dimensional deformation fields which is addressed as J-integral [15]. For several years this J-integral is used for study of linear or non-linear crack behavior, but it is not valid when deformation is not reversible. Further the extension of the J-integral by Rice, is presented by Kishimoto [16], taking into account of the existence of a fracture process region and the effects of plastic deformations, body forces, thermal strains and inertia of material. However, the 2D J-integral in true sense is not applicable to three-dimensional problems due to non-consideration of out-ofplane stress and strain fields. Bui and Kishimoto [16,17] studied the three dimensional extension of the J-integral as a fracture criterion with point-wise calculation of J along the crack front. Numerical evaluation of J-integral has been preceded with the corresponding introduction of Domain Integrals (integrals over element volumes) and Contour Integrals (line integrals and area integrals) methods. Selection of the approach depends on element meshing and problem specification, though each approach leads to identical numerical values. The more extensive discussion of Domain and Contour Integrals for 2-D and 3-D applications was explained in R. H. Dodds [18–20]. In this manuscript, path independent integral expression for three dimensional straight crack has been derived. Further magnetic effect is considered and magnetostriction (responsible for crack extension due to magneto elasticity) is used as a term for
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A. Bhushan et al. / Mechanics Research Communications 92 (2018) 15–20
where, σ ij , Fi , ρ and ui are stress-tensor, body force per unit volume, density of solid and displacement respectively where the first and second dots specify the first or second time derivative of displacement. Multiplying u˙ i on both sides of Eq. (1) and integrating over the body volume V
Nomenclature
σ ij Fi
ρ
ui V Ti
εij S
1 , 2 , c A1 , A2 l Jku
εiej εi∗j εimj
We
α ij θ M
λs Ms
Stress Body forces per unit volume Density Displacement Volume of the body Traction vector Strain tensor Surface Contours Area inside the contours Length of the crack Energy release rate during crack extension Elastic strain component Eigen strain component Magnetic strain component Elastic strain energy density function Coefficient of thermal expansion Temperature increment from the natural state Magnetization Saturated magnetostriction Saturated magnetization
V
ρ u¨ i u˙ i dV
V
(2)
We know that
(σi j u˙ i ), j = σi j, j u˙ i + σi j u˙ i, j
(3)
Traction on a small differential element on the contour surface can be expressed as
Ti = σi j n j
(4)
where, nj is the outward normal vector. Eq. (3) and Eq. (4) is substituted into expanded integral Eq. (2) and strain derivatives ε˙ i j is used for the displacement derivative. Now using Gauss divergence theorem to convert volume integral to surface integral, Eq. (2) can be derived as
S
magnetic strain along with initial, elastic and thermal strains. Thus a new path independent expression for energy release rate is proposed. This energy release rate expression is further extended to three-dimensional straight crack propagation by using contour integral approach and the new expression is also proposed. 2. Formulation of path independent integral Fig. 1 shows the schematic of a plate containing a crack for the purpose of quantifying path independency in 3D domain. Now, the crack tip is assumed to virtually move an infinitesimally small distance from the fixed frame at O to moving frame at O1 . The direction of X2 and x2 are perpendicular to the crack surface corresponding to O and O1 respectively. Two contour paths are chosen, the first one being the outer contour noted by path 1 curves and the second contour 2 can be any arbitrary contour surrounding the crack surfaces. The region enclosed by these two contours is A1 and the area bounded by the crack plane and the second contour isA2 . Now stating the equilibrium equations for a stressed continuum of volume V subjected to arbitrary traction T and body forces F:
σi j, j + Fi = ρ u¨ i
(σi j, j + Fi ) u˙ i dV =
(1)
Ti u˙ i dS +
V
Fi u˙ i dV =
V
ρ u¨ i ui dV +
V
σi j ε˙ i j dV
(5)
For an infinitesimal virtual crack extension, the energy release rate can be evaluated from Eq. (5) for a differential change dl of the propagating crack as similar to time derivatives. This is expressed as
1 +c
Ti
+
d ui d + dl
A1
Fi
d ui dA = dl
A1
ρ u¨ i
d ui dA dl
dε σi j i j dA + Ju dl A1
(6)
where J u is the rate of change of energy of material in the fracture process region for name say a generalized universal integral, be it unimodular or bimodular. We can introduce zero integral terms with reference to contour 2 with an integral evaluated along the contour path and opposite the contour path by adding and sub du tracting the term Ti dli d to Eq. (6) and after rearrangement 2 the expressions for energy release rate J u is given as
Ju =
1 +c −2
Ti
d ui d ui d + Ti + dl dl 2 A1
du (Fi − ρ u¨ i ) i − dl
du σi j i dl
u
J =
A1
+
2
Ti
d εi j dA σi j
,j
(7)
dl
d ui + (Fi − ρ u¨ i ) − dl
d ui d dl
dε σi j i j dl
dA
(8)
Using Eq. (1), the first term of area integral in Eq. (8) vanishes and the integral may be rewritten as
Ju =
2
Ti
d ui d dl
(9)
Fig. 2(a) and 2(b) delineates the crack propagation along a plane in 2D boundary and propagation in any arbitrary spatial 3D domain respectively from crack tip O (X1 ,X2 , X3 ) to O1 (x1 ,x2 , x3 ). The angles are measured from the rectangular (X1 ,X2 , X3 ) axes to the (x1 ,x2 , x3 ) axes. With reference to Fig. 2, transformation equations from the fixed frame O − X1 ,X2 to moving frame O1 − x1 ,x2 for the infinitesimal crack extension can be expressed as Fig. 1. Configuration of crack tip { 1 (arbitrary curve surrounding area A1 ), c (curve along the crack surface), A2 (fracture process region), 2 (boundary of A2 )} around a region of infinitesimal thickness enclosing the crack front.
x1 = X1 cos θ0 + X2 sin θ0 − l x2 = −X1 sin θ0 + X2 cos θ0
(10)
A. Bhushan et al. / Mechanics Research Communications 92 (2018) 15–20
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Then, Eq. (16) is simplified to
∂ ui J =− Ti d ∂ x1 2
u
(17)
We obtain from Eq. (13), (14) & (15)
∂ ui ∂ ui ∂ ui = cos θ0 + sin θ0 ∂ x1 ∂ X1 ∂ X2
(18)
Substituting Eq. (18) in Eq. (17), the integral equation becomes
∂ ui ∂ ui J =− Ti cos θ0 + sin θ0 d ∂ X1 ∂ X2 2
u
(19)
For any arbitrary orientation θ 0 of the propagating crack front, the Ju can be resolved as
J u = J1u cos θ0 + J2u sin θ0
(20) Jku
Taking as single notation where k = 1, 2 correspond to respective coordinate axes, Eq. (17) can be modified using Eq. (20) as
Jku = −
Ti
2
∂ ui d ∂ Xk
(21)
However, for verification of that the above integral to be path independent, let us consider another integral surrounding the crack path and expressed as
J¯ku = −
Ti
1 +c
∂ ui d + Mk ( A ) ∂ Xk
(22)
where, Mk (A) are the terms determined when the area A1 surrounded by 1 , 2 and c is specified. Now, if both the integral Jku and J¯ku are different, then we can write
J¯ku − Jku = − Fig. 2. Representation of propagation of crack tip from O to O1 in (a) 2D boundary and (b) 3D domain.
J¯ku − Jku = −
1 +c
X2 = x1 sin θ0 + x2 cos θ 0 + l sin θ0
(11)
Similarly, displacements for the fixed frame O − X1 ,X2 is given by
ui (X1 , X2 , l ) = ui (x1 cos θ0 + x2 sin θ0 − l, −x1 sin θ0 + x2 cos θ0 , l ) (12) d ui = dl
∂ ui ∂ ui ∂ xi ∂ ui ∂ ui ∂ x1 ∂ ui ∂ x2 + = + + ∂l ∂ xi ∂ l ∂l ∂ x1 δ l ∂ x2 ∂ l
∂ ui ∂ x1 ∂ ui ∂ x2 =− ; =0 ∂ x1 ∂ l ∂ x1 ∂l d ui = dl
∂ ui ∂ ui − ∂l ∂ x1
(14)
2
Ti
∂ ui ∂ ui − d ∂l ∂ x1
Ju -integral
∂ ui Mk ( A ) = Ti d = ∂ Xk t J¯ku = Jku =
∂ ui d ∂ Xk
∂ ui d + Mk ( A ) ∂ Xk
(23) (24) (25)
σi j
A1
∂ ui ∂ Xk
∂ ui dA σi j ∂ Xk , j A
, j dA −
1 +c
Ti
∂ ui d ∂ Xk
(26)
(27)
Using Eq. (1), we can write Eq. (27) as
A1
{(ρ u¨ i − Fi )
−
Here, is the crack driving force or also known as the energy release rate during crack extension. We assume that the fracture process region does not depend upon load conditions or upon geometry of body or crack. Hence, the process region is assumed to be constant in dimensions and moving along with the same speed as the crack tip, and hence, ∂ ui ∂ l = 0 holds in 2 .
2
Ti
∂ ui d + Mk ( A ) ∂ Xk
(15)
(16)
Ti
Ti
t = 1 + c − 2 denotes the contour which surrounds the area A1 . For path independence J¯ku = Jku and hence from Eq. (25)
Jku =
Now substitution of Eq. (19) into Eq. (13) gives,
Ju =
t
Therefore,
(13)
∂ ui d + Mk ( A ) + ∂ Xk
1 +c −2
J¯ku − Jku = −
X1 = x1 cos θ0 − x2 cos θ0 + l cos θ0
Ti
1 + c
Ti
∂ εi j ∂ ui + ( σi j )}dA ∂ Xk ∂ Xk
∂ ui d ( k = 1, 2 ) ∂ Xk
(28)
Now we decompose ε ij asεiej , εi∗j , εimj
εi j =εiej + εi∗j + εimj εiej
(29)
Where, Elastic Strain Eigen Strain Componentεimj Magnetic Strain Component We take elastic strain energy density function, We (εiej ) which does not explicitly depend on X1 .
∂ W e = σi j ∂εiej
Componentεi∗j
(30)
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A. Bhushan et al. / Mechanics Research Communications 92 (2018) 15–20
We take thermal strain εi∗j = αi j θ . Where α ij denote the coefficient of thermal expansion and θ the temperature increment from the natural state. From Magnetostriction Model [1],
λ (t, x ) =
∞
γi M2i (t, x )
M 2 = 2 M M0 − M0 2
(31)
i=0
ε
m ij
3 λs = M2 (t, x ) 2 Ms 2
A1
Jku =
+
(33)
∂ ui (ρ u¨ i − Fi ) dA − ∂ Xk A1 ∂ εi j σi j + dA ∂ Xk A1
1 + c
∂ ui d ∂ Xk
Ti
(34)
e ∂εi j ∂εi∗j ∂εimj ∂ εi j dA = σi j + + dA σi j ∂ Xk ∂ Xk ∂ Xk ∂ Xk A1 A1
(35)
As the fracture process zone is very small
A1
σi j
∂ εi j dA = ∂ Xk
1 + c − 2
W e nk d =
1 + c
W e nk d (36)
∂ε ∗ ∂θ σi j i j dA = αi j σi j dA ∂ Xk ∂ Xk A1 A1
∂ε m 3λs ∂ M2 σi j i j dA = σi j dA ∂ Xk A1 A1 Ms 2 ∂ Xk 3λs ∂M = σi j M dA ∂ Xk A1 Ms 2
(37)
Jku = J1u cos θ0 + J2u sin θ0
u Jk
3D
(38)
∂ ui ∂ ui Jku = { W e nk − Ti } d + (ρ u¨ i − Fi ) dA ∂ X ∂ Xk 1 + c A1 k 3λs σi j ∂θ ∂M + αi j σi j dA + M dA (39) ∂ Xk ∂ Xk A1 A1 Ms 2
⇒ (Jku )2D +
M = M0 + m
3λs σi j M0 A1
Ms 2
∂ ui dA ∂ Xk
∂M dA ∂ Xk (48)
(40) (41)
( M − M0 ) 2 = m2
(42)
M 2 + M0 2 − 2 M M0 = m2
(43)
M 2 + M0 2 − 2 M M0 ≈ 0
(44)
(49)
At
∂ u (J ) n dA ∂ X3 k 2D 3
(50)
(51)
All the term except 1st term will vanish in above equation as variation of other terms in the direction n3 is constant. So, we can write above equation as
At
∂ u (J ) n dA = ∂ X3 k 2D 3
u Jk
3D
=
1 + c
∂ W e nk n3 dA ∂ X3 At ∂ ∂u − σi j n j i n3 dA ∂ Xk At ∂ X3
{ W e nk − Ti
∂ ui } d ∂ Xk
∂ W e nk n3 dA ∂ X3 ∂ ∂u − σi j n j i n3 dA ∂ Xk At ∂ X3 ∂ ui + (ρ u¨ i − Fi ) dA (i, j, k = 1, 2, 3) ∂ Xk At +
where, M0 is Biasing Magnetic Level.
∼ = J1u cos θ0
∂ u ∂ ∂ ui (J ) dA = {W e nk − Ti }n dA ∂ X3 k 2D ∂ X3 At ∂ Xk 3 ∂ ∂ ui + (ρ u¨ i − Fi ) dA ∂ X3 At ∂ Xk ∂θ + αi j σi j dA ∂ X1 At 3λs σ i j M0 ∂ M + dA ∂ Xk At Ms 2
Considering linear model:
εimj
(ρ u¨ i − Fi )
At
We also consider area of fracture process region is diminishing and contour integral [16] end = 0 Combining all the quantities Eq. (34) is expressed as
3 λs = M2 2 Ms 2
A1
As θ 0 will be very small and so 2nd term will vanish. Hence, k will be equal to 1. (Jku )2D can be taken as constant through the thickness dη. Now let At = A1 + A2 and the two faces of this At area is At + and At − . As normal to these two faces are parallel and in opposite directions, the sum of the two area integral is given by the X3 derivative of the integrand multiplied by dη. Division of both sides by dη gives the pointwise value for the integral and further sum with integrand yields J - integral in 3-D.
∂θ dA + ∂ X1 ( k = 1, 2 )
αi j σi j
A1 Jku 2D
∂ ui } d + ∂ Xk
(47)
Here k will be equal to 1 as we assume crack propagation variation is very small in other directions. And so we can write Jku as
Consider,
{ W e nk − Ti
Jku = ( )
Now, Eq. (28) can be rewritten as
1 + c
3 λs M2 2 Ms 2
(46)
∂εimj 3 λs ∂M dA = σ i j ( 2 M0 ) dA 2 ∂ Xk ∂ Xk A1 2 Ms 3M0 λs σi j ∂ M = dA ∂ Xk A1 Ms 2
Ms → Saturated Magnetization Hence,
Jku =
σij
(32)
λs → Saturated Magnetostriction
εi j = εimj + αi j θ +
3 λs ( 2 M M0 − M0 2 ) 2 Ms 2
εimj =
By assuming or approximating the value of λ to second order,
(45)
A1
(52)
A. Bhushan et al. / Mechanics Research Communications 92 (2018) 15–20
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Table 1 Properties of magnetostrictive material. Description
Value
Description
Value
Young’s modulus (E) Density (ρ ) Poisson’s ratio (ν )
60 × 109 Pa 7870 kg/m3 0.3
Saturated Magnetostriction (λs ) Saturation magnetization (Ms ) Effective domain density (a)
2 × 10−4 15 × 105 A/m 70 0 0 A/m
Table 2 The of (Jku )3D in value at saturate magnetization. Part of Integral Taken (Jku )3D
(Jku )3D /(Jku )3D average Contour 1
Contour 2
Contour 3
Contour 4
0.9745
1.0058
1.0121
0.9974
∂ ui e 1+c {W n1 − Ti ∂ X1 }d − At (σi3 ui,3 ),3 dA1
+
+
At
αi j σi j
∂θ dA + ∂ X1
3 λ s σi j
A1
Ms
2
M ∂∂XM dA k
3λs σ i j M0 Ms 2
At
∂M dA ∂ Xk (53)
Again,
u Jk
3D
= J1u cos θ0 + J2u cos + J3u cos ∼ = J1u cos θ0
After, taking k = 1
Jk
u 3D
=
W e n1 − Ti
1 + c
∂ ui ∂ X1
d +
(54) Fig. 3. Integrating contours and applied boundary condition in the magnetic field.
At
∂ W e n1 n3 d A ∂ X3
∂ ui (ρ u¨ i − Fi ) dA ∂ X1 At ∂ ∂ ui − n dA σi j n j ∂ X1 3 At ∂ X3 +
+
At
αi j σi j
∂θ dA + ∂ X1
3λs σ i j M0 At
Ms 2
∂M dA ∂ X1 (55)
Here, in the above expression, second term will be zero as dot product of normal vector (n1 ∗ n3 ) is zero and in the third term, j = 1,2 will vanish and only 3 contribute in the expression as follows:
Jk
∂ ui = W n1 − Ti d − (σi3 ui,3 ),3 dA1 ∂ X1 1 +c At ∂ ui + dA (ρ u¨ i − Fi ) ∂ X1 1 At ∂θ + αi j σi j d A1 (i, j = 1, 2, 3) ∂ X1 At 3λs σ i j M0 ∂ M + d A1 (56) ∂ X1 At Ms 2
u
e
3D
The applications of the integral (Jku )3D is in various real fields problems, where the pure magnetostriction applied on solid ferromagnetic cracked body. The interactive coupled field fracture problems arise with multi-turn coil windings around a cracked ferromagnet and nonlinear magnetostrictive transducer having straight crack. 4. Numerical results We have taken a numerical example of rectangular cracked bar under the magnetostriction. Three-dimensional finite element analyses have been carried out to solve the magnetoelastic multiphysics problem of crack characterization. The integral (Jku )3D is evaluated for the four different contours under magnetostrictive field induced due to applied current density as shown in Fig. 3.
Fig. 4. (a) Sectional view and (b) Mesh distribution of cracked specimen model with steel housing enclosing the drive coil.
The cracked specimen model contains the steel housing enclosing a drive coil as depicted in Fig. 4(a). The cracked magnetostrictive specimen inside the core has been subjected to the magnetic field induced by the current carrying drive coil. An air domain is enveloped around the steel housing to realistically model the magnetic flux path. The steel housing helps in minimizing flux leakage by creating a closed magnetic flux path. The current in the coil is considered to be DC for performing the analysis in stationary state. One end of the specimen is fixed to study its fracture behaviour. To specify the magnetic constitutive relation in steel housing, the nonlinear H (Magnetic field)∼B(Magnetic flux density) curve of soft iron material without losses has been incorporated in the numerical simulation. Anhysteretic magnetization model has been assumed for the magnetostrictive material, which is specified using the Langevin function [1,21,22] as follows.
M He = coth Ms a
−
a He
(57)
where, He is the effective magnetic field and a is domain density constant with dimensions of magnetic field. The results are obtained for the finite element model of rectangular bar of dimensions 50 × 6 × 6 mm3 , with an edge crack of 1.5 mm length. The material properties are specified in Table 1. Appropriate mesh model satisfying the patch test and convergence is obtained from some trial simulations. The chosen finite element model contains 92,210 tetrahedral mesh elements and 13,296 hex-
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A. Bhushan et al. / Mechanics Research Communications 92 (2018) 15–20
ahedral mesh elements. The tetrahedral elements are used to mesh coil, steel housing and air domains, whereas the hexahedral swept mesh is used to mesh the magnetostrictive cracked specimen. The path independence of integral has been proved numerically and for the four contours covering the analysis domain. The integral is estimated through contour integral method and shown in Table 2. The integral (Jku )3D for different applied current density has been estimated numerically and its values saturates at saturated magnetization. 5. Conclusion A new expression of conservation integral (Jku )3D for a straight crack has been proposed to have the physical meaning of energy release rate (both in two dimensional and three dimensional cases) for a homogeneous, isotropic material considering combined effects of inertia, thermal and magnetostriction effect. Numerical 3D finite element analyses have been conducted to evaluate the integral for a magnetostrictive edge crack specimen inside a current carrying coil enclosed in steel housing and surrounded by air domain. The path independence of the J-value has been proven numerically for induced magnetic load due to applied current density. The value of integral (Jku )3D and magnetostriction saturates at saturated magnetization. References [1] M.J. Dapino, R.C. Smith, A.B. Flatau, Structural magnetic strain model for magnetostrictive transducers, IEEE Trans. Magn. 36 (20 0 0) 545–556. [2] M.M. Carroll, The foundation of solid mechanics, Appl. Mech. Rev. 38 (1986) 1301–1308. [3] W.F.J. Brown, Magnetoelastic Interactions, Springer-Verlag, New York, 1966. [4] Y.H. Pao, Electromagnetic Force in Deformable Continua, 1978. [5] F.C. Moon, Magneto Solid Mechanics, Wiley, New York, 1984.
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