Numerical investigation of the stress field near a crack normal to ceramic-metal interface

Journal of Mechanical Science and Technology 25 (2) (2011) 309~315

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-010-1209-3

Numerical investigation of the stress field near a crack normal

to ceramic–metal interface† Liviu Marsavina1,*, Tomasz Sadowski2 and Nicolae Faur1

1Faculty of Mechanical Engineering, University Politehnica of Timisoara, Blvd. M. Viteazu, No. 1 Timisoara 300222, Romania 2Faculty of Civil and Sanitary Engineering, Lublin University of Technology, Nadbystrzycka 40 str., 20-618 Lublin, Poland

(Manuscript Received March 8, 2010; Revised July 9, 2010; Accepted October 4, 2010)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Ceramic–metal interfaces are often present in composite materials. The presence of cracks has a major impact on the reliability of ad-

vanced materials, such as fiber or particle reinforced ceramic composites, ceramic interfaces and laminated ceramics. The understanding of the failure mechanisms is very important, as is as the estimation of fracture parameters at the tip of the crack approaching an interface and crack propagation path. A cracked sandwich plate loaded with axial uniform normal stress was numerically investigated using plane strain Finite Element Analysis. The numerical results for the singularity orders were compared with the analytical solution. The influ-ences of the material combination and crack length on the radial and circumferential stresses and displacement distributions were investi-gated. The Stress Intensity Factors were determined based on numerical results using a displacement extrapolation method. The results for the non-dimensional stress intensity factors show that at lower crack lengths the influence of material mismatch is lower, but this influence increases with increasing crack length.

Keywords: Ceramic-metal interface; Crack; Stress field; Stress intensity factor ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Ceramic-metal composites often appear in engineering ap-plications. The presence of cracks has a major impact on the reliability of advanced materials, like fiber or particle rein-forced ceramic composites, ceramic interfaces and laminated ceramics. Normal cracks to a bi-material interface could ap-pear in crystalline composites (Fig. 1(a)), in fiber-matrix com-posites (Fig. 1(b)) or in layered composites (Fig. 1(c)).

Different researchers have investigated the interaction between an interface and a crack perpendicular or inclined to bi-material interface. Zak and Williams [1] showed that the stress field singularity at the tip of a crack perpendicular to an interface or terminating at the interface is of order r-λ, where λ is the real part of the eigenvalue and depends on the elastic properties of the bi-material. Cook and Erdogan [2] used the Mellin transform method to derive the govern-ing equation of a finite crack perpendicular to the interface and obtained the stress intensity factors (SIFs). Erdogan and Biricikoglu [3] solved the problem of two bounded half planes with a crack going through the interface. Bogy [4]

investigated the stress singularity of an infinite crack termi-nating at the interface with an arbitrary angle. Wang and Chen [5] used photoelasticity to determine the stress distri-bution and the stress intensity factors of a crack perpendicu-lar to the interface. Lin and Mar [6], Ahmad [7] and Tan et al. [8] used finite element to analyze cracks perpendicular to bi-material in a finite elastic body. Chen [9] used the body force method to determine the stress intensity factors for a normal crack terminating at a bi-material interface. Chen et al. [10] used the dislocation simulation approach in order to investigate the crack tip parameters for a crack perpendicu-lar to an interface of a finite solid. He and Hutchinson [11] also considered cracks approaching the interface at oblique angles. Chang and Xu [12] presented the singular stress field and the stress intensity factors solution for an inclined crack terminating at a bi-material interface. A theoretical description of the stress singularity at an inclined crack terminating at an anisotropic bi-material interface was pro-posed by Lin and Sung [13]. Wang and Stahle [14], using a dislocation approach, presented the complete solution of the stress field ahead of a crack approaching a bi-material inter-face. They calculated the stress intensity factor solutions and the T-stress. Liu et al. [15] determined the mixed mode stress intensity factors for a bi-material interface crack in the infinite strip configuration and in the case where both

† This paper was recommended for publication in revised form by Associate EditorYoungseog Lee

*Corresponding author. Tel.: +40 256 403577, Fax.: +40 256 403523 E-mail address: [email protected]

© KSME & Springer 2011

310 L. Marsavina et al. / Journal of Mechanical Science and Technology 25 (2) (2011) 309~315

phases are fully anisotropic. Marsavina and Sadowski [16] and Marsavina et al. [17] used the finite element analysis to investigate the interaction between a crack and an interface for inclined and normal crack to an interface. They investi-gated the effects of the elastic properties of the two bounded materials and the crack deflection at the interface using the energy release rate.

Most of the previous studies provide the singularity or-ders and stress intensity factors for some particular cases of shear modulus ratio (G1/G2), corresponding to aluminum - epoxy [8-10, 18]; boron – epoxy [8-10] and aluminum – steel [18], and there are no results regarding ceramic – metal bi-material systems. The present study numerically investigates the asymptotic stress field (radial and circum-ferential variation of stresses and displacements) and the stress intensity factors at the tip of a normal crack terminat-ing on a ceramic – metal interface. The influence of materi-als properties mismatch and the influence of the crack length on the non-dimensional stress intensity factors are presented.

2. Asymptotic stress field for a crack normal to a bi-material interface

Chen [9] provides the analytical solution of the asymp-totic stress field for a crack normal to a bi-material interface, Fig. 2, using the eigenfunction expansion with complex function theory:

( )

2

2 '( ) '( )

2 2 "( ) '( )

2 ( ) '( ) ( )

rr

irr r

ir

z z

i e z z z

G u iu e z z z z

θθ

θθθ θ

θθ

σ σ φ φ

σ σ τ φ ψ

κφ φ ψ−

⎡ ⎤+ = +⎣ ⎦⎡ ⎤− + = +⎣ ⎦

⎡ ⎤+ = − −⎣ ⎦

(1)

where (r, θ) are the polar coordinates of a system with the origin at the crack tip (Fig. 2).

1i −= φ (z), ψ (z) are complex functions assumed in the following form:

1 2

1 2

( )

( ) , 1,2,3.k k k

k k k

z a z a z

z b z b z k

λ λ

λ λ

φ

ψ

= +

= + = (2)

For a crack normal to an interface, λ is the double root

of the following eigenequation:

2 2 2 22 ( )( 1) (1 )cos( ) 0λ α β β α β β πλ⎡ ⎤− + − + + − =⎣ ⎦ (3) with α and β the Dundur’s bi-material parameters [19],

( ) ( )( ) ( )

1 2 2 1

1 2 2 1

1 11 1

G GG G

κ κα

κ κ+ − +

=+ + +

(4a)

and

( ) ( )( ) ( )

1 2 2 1

1 2 2 1

1 11 1

G GG G

κ κβ

κ κ− − −

=+ + +

(4b)

where Gi and νi are the shear modulus and Poisson’s ratio of material i (i = 1, 2), ii 43 ν−=κ for plane strain and

( ) ( )iii 1/3 ν+ν−=κ for plane stress. The boundary conditions that should be satisfied are

listed below. - The stress-free boundary conditions at the crack sur-

faces:

( ) ( )( ) ( )

,1 ,1

,3 ,3

,0 0 ,0 0

,2 0 ,2 0r

r

r r

r rθθ θ

θθ θ

σ σ

σ π σ π

= =

= = (5)

- The continuity conditions at the interface:

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )

,1 ,2

,1 ,2

,2 ,3

,2 ,3

,1 ,2

,1 ,2

,2 ,3

,2 ,3

, / 2 , / 2

, / 2 , / 2

, / 2 , / 2

, / 2 , / 2

, / 2 , / 2

, / 2 , / 2

, / 2 , / 2

, / 2 , / 2

rr rr

rr rr

r r

r r

u r u r

u r u r

u r u r

u r u r

r r

r r

r r

r r

θθ θθ

θθ θθ

θθ θθ

θ θ

θθ θθ

θ θ

π π

π π

π π

π π

σ π σ π

σ π σ π

σ π σ π

σ π σ π

=

=

− = −

− = −

=

=

− = −

− = −

(6)

1

1

( )

( )k k

k k

z a z

z b z

λ

λ

φ

ψ

=

=

(a) (b) (c) Fig. 1. Cracks normal to interfaces in composite materials.

Fig. 2. A crack normal to an interface.

L. Marsavina et al. / Journal of Mechanical Science and Technology 25 (2) (2011) 309~315 311

The general expressions for displacements and stresses could be expressed by the following terms in regard to symmetric (I) and skew symmetric (II) modes:

( )( ) ( )I IIij I ij II iju K g K g rλθ θ= + (7)

( ) 1

1( ) ( )I IIij I ij II ijK f K f

rλσ θ θ −= + (8)

where (r, θ) represents the polar coordinates, λ is the stress singularity, KI and KII are the stress intensity factors for mode I and II respectively, and functions ( ), ( ),I II

ij ijg gθ θ ( ), ( )I II

ij ijf fθ θ are specified in [9]. Using complex potential equations, Chen [9] describes

the stress and displacement field at the tip of a crack normal to interface. 3. Asymptotic stress field for a crack normal to a ce-

ramic-metal interface

3.1 Problem description

A completely cracked sandwich layer, remote loaded by uniform normal stress σ was considered for the analysis, results a mode I load at the crack tip (Fig. 3).

The material combinations were between Al2O3 and dif-ferent metals (Fe, Cu, Al, Mg) with the properties indicated in Table 1. The Dundur’s parameters for the considered combinations are shown in Table 2. 3.2 FE model

To assess the numerical results, the plane strain Finite Element Analysis was performed using ABAQUS finite

element package [20]. Based on the symmetry, a quarter of the model was examined and plane strain conditions were considered. A representative mesh for the FE model is shown in Fig. 4 for the case when crack length was equal to the remaining ligament (a/W=0.5). Eight node bi-quadratic plane strain quadrilateral (CPE8) elements were used to model a part of the crack tip where a ring (36 elements) of 6 node quadratic plane strain triangle (CPE6) elements was placed. A highly focused mesh (bias = 10) was used at the crack tip. Different crack length to width ratios of model were considered : a/W = 0.2, 0.33, 0.5, 0.67 and 0.8.

The symmetric boundary conditions were applied in the symmetry axes and a uniform constant stress, σ =1 MPa was applied. 3.3 Radial variation of the asymptotic stress field

Logarithmic variations of the stresses σθθ in the radial di-rection θ = 450 are plotted in Fig. 5(a) for the case when a/W = 0.5. The stress values are normalized to the applied stress σ and the distance is normalized to the crack length a. The slope of the obtained lines, representing (1-λ), increases with increasing ratio of G1/G2. For comparison G1/G2=1 represents the homogeneous case. Fig. 5(b) presents the radial variation of σθθ/σ for different crack lengths and for material combination with G1/G2=10.06. As we expected, the crack length does not influence the singularity order.

Table 1. Material properties.

Material E [MPa]

ν [-]

G [MPa]

σys [MPa]

Ceramic Al2O3

400000

0.22

327869.0

-

Metal Fe Cu Al Mg

200000 110000 70000 44000

0.29 0.35 0.30 0.35

155038.7 81481.5 53846.2 32592.6

100 33.3 50 21

Fig. 3. Cracked sandwich plate.

Table 2. Material combinations and Dundurs parameters.

Material 1/ Material 2 G1/G2 α β

Mg/Al2O3 0.099 -0.959 -0.787

Al/Al2O3 0.164 -0.757 -0.691

Cu/Al2O3 0.249 -0.561 -0.541

Fe/Al2O3 0.473 -0.301 -0.316

Al2O3/ Al2O3 1.000 0.000 0.000

Al2O3/Fe 2.115 0.301 0.316

Al2O3/Cu 4.024 0.561 0.541

Al2O3/Al 6.089 0.757 0.691

Al2O3/Mg 10.060 0.959 0.787

Fig. 4. The FE model and detail at the crack tip for a/W=0.5 (1980 CPE8 elements and 36 CPE6 elements connected in 6197 nodes).

312 L. Marsavina et al. / Journal of Mechanical Science and Technology 25 (2) (2011) 309~315

Fig. 6 presents a comparison of the singularity orders ob-tained analytically from Eq. (3) and numerically from the slope of the plots (for radial directions with θ = 450) in Fig. 5. The singularity order decreases with increases in the G1/G2 ratio. It can be observed that the relative errors be-tween analytical and numerical results are in the limit of finite element analysis (± 4.5 %).

3.4 Circumferential variation of the asymptotic stress field

Fig. 7 presents the circumferential distributions of dis-placements ur and uθ for the case a/W=0.5 and radius r = 1 mm (r/a = 0.02). The conditions of displacements continu-ity on the interface are fulfilled. It could be observed that higher displacements were obtained for the bi-material sys-tems compared to the homogeneous case (G1/G2=1). Maxi-mum displacement ur occurs for an angle of 1730 for the homogeneous case G1/G2=1 and for G1/G2 < 1. For G1/G2 > 1, the angle corresponding to maximum value of displace-ment ur decreases from 1650 (G1/G2=2.115) to 1440 (G1/G2=10.06). The angle corresponding to the maximum value of the displacement uθ decreases from 900 for G1/G2=0.099 to approximately 63.50 for G1/G2=10.06.

Comparisons of the circumferential stress distributions (σrr, σθθ and σrθ) at integration points, normalized to the applied load σ at a radius r = 0.5 mm (r/a = 0.01), are plot-ted in Fig. 8 for the case when a/W=0.5. In comparing the distributions for bi-material and homogeneous cases, rele-vant differences could be seen. A jump at the interface for

(a) Influence of properties mismatch (a/W = 0.5)

(b) Influence of a/W (G1/G2 = 10.06)

Fig. 5. Logarithmic stress distributions of σθθ (θ = 450).

Fig. 6. Singularity orders.

(a) Displacement ur

(b) Displacement uθ

Fig. 7. Circumferential displacement distributions at radius r = 1 mm(r/a=0.02) for a/W=0.5.

L. Marsavina et al. / Journal of Mechanical Science and Technology 25 (2) (2011) 309~315 313

bi-material cases appears for σrr/σ, with maximum values in material 1. The maximum value of the radial stress was obtained for material combination G1/G2=10.06 and de-creases with decreasing G1/G2 ratio. The variations of the tangential stress σrθ/σ highlight that higher values are ob-

tained for the material combination with G1/G2=10.06, and for a particular angle θ, the ratio σrθ/σ decreases with de-creasing G1/G2.

It can be observed that the stress free conditions at the crack edges σθθ = σrθ = 0 for all material combinations, and the continuity on the interface conditions for circumferen-tial stress σθθ and tangential stress σrθ are fulfilled. 4. Evaluation of stress intensity factors (SIF’s) at the

tip of a crack normal to a bi-material interface

The displacement extrapolation technique was used by Yilan & Hua [21] to extract the SIFs for a crack normal to interface using the moiré interferometry technique. They express the SIFs starting from the asymptotic displacement solution Eq. (7), and, for example, for Mode I the SIF KI could be found:

( )0

( , )lim y

I Iry

u rK

r gλ

θθ→

= (9)

where uy (r,θ) represents the displacements on the y axis collected on a particular angle (θ =450 for our determina-tions) and )(gI

y θ is a non dimensional function expressed in Cartesian coordinates, [9].

Taking into account the dimensions of the SIFs, KI [MPa m1-λ] is difficult to compare between results for different material combinations and usually a non-dimensional SIF is used. For example, in [9, 22]:

1I

IKfa λσ −= . (10)

The results for the non-dimensional stress intensity fac-

tors fI are shown in Fig. 9(a) for two different crack lengths (a/W = 0.2 and 0.8) versus ratio G1/G2 at logarithmic scale. From Fig. 9(a) it can be observed at lower crack lengths that the influence of material mismatch is lower, but this influ-ence increases as crack length increases. Fig. 9(b) presents the results of the non-dimensional SIFs fI versus crack length a/W for different material combination G1/G2. It can be seen that fI increases with crack length and for a particu-lar value of the crack length, a/W, non-dimensional SIF is higher for material combination G1/G2 = 0.099, and de-creases with increasing G1/G2 ratio.

SIF solutions for this particular geometry could not be found in the literature. However, in order to compare to those published in Murakami [22], where the load is applied on the crack faces, non-dimensional SIFs are plotted in Fig. 10. Taking into account that the results from Ref. [22] are provided for an infinite geometry (W→∞) and the material Poisson ratios were ν1=0.35 and ν2=0.22, the difference for the non-dimensional SIFs are between 0.8% for the homo-geneous case (G1/G2 = 1) to a maximum of 7.2% for G1/G2 = 10.06.

(a) Circumferential distribution of σrr

(b) Circumferential distribution of σθθ

(c) Circumferential distribution of σrθ

Fig. 8. Circumferential stress distributions at radius r = 0.5 mm(r/a=0.01) for a/W=0.5.

314 L. Marsavina et al. / Journal of Mechanical Science and Technology 25 (2) (2011) 309~315

5. Concluding remarks

Finite element analysis under plane strain conditions has been developed for determining the singularity order and the asymptotic stress field at the tip of a crack normal to a ceramic-metal interface for a cracked sandwich layer sub-jected to remote load σ. The ceramic material was consid-ered Al2O3 in addition to metallic materials: Fe, Cu, Al and Mg resulting in nine combinations. For each material com-bination, the following five crack lengths were considered: a/W = 0.2, 0.33, 0.5, 0.67 and 0.8.

The linear relationship between Log (σθθ/σ) and Log (r/a) on a radial direction corresponding to θ = 450 allows the determination of the singularity order, λ. The singularity order decreases from 0.635 for G1/G2 = 0.099 with increas-ing the G1/G2 ratio to 0.275 for G1/G2 = 10.06. For the nine considered material combinations good agreement was ob-tained between the analytical and numerical singularity orders (maximum difference being 4.5% for G1/G2 = 10.06).

The circumferential plots of displacements and stresses at integration points show the influence of mate-rial mismatch on the displacements ur and uθ, respec-tively on the σrr/σ, σθθ/σ, and σrθ/σ, distributions.

Using the displacement extrapolation method the

stress intensity factors were determined. The results for the non-dimensional stress intensity factors show that at lower crack lengths the influence of material mismatch is lower but this influence increases with increasing crack length. The non-dimensional SIF increases with crack length and for a particular value of the crack length a/W, the non-dimensional SIF is higher for material combina-tion G1/G2 = 0.099 and decreases as the G1/G2 ratio in-creases.

Acknowledgment

This work was supported by the European Reintegration Grant PERG03-GA-2008-230991: Reliability of Interfaces on Composite Materials and European Union Seventh Framework Programme (FP7/2007–2013), FP7-REGPOT – 2009 –1, under grant agreement No: 245479.

Nomenclature------------------------------------------------------------------------

a : Half length of crack fI : Non-dimensional stress intensity factor KI : Mode I stress intensity factor r : Polar radius ur, uθ : Displacements in polar coordinates W : Half width of model α, β : Dundur’s parameters θ : Polar angle λ : Singularity order G1, G2 : Shear modulus ν1, ν2 : Poisson ratios σ : Applied load σrr, σrr, σrr : Stress components in polar coordinates

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Fig. 10. Comparison of non-dimensional SIF’s.

(a) Effect of materials combinations (G1/G2)

(b) Effect of crack length (a/W)

Fig. 9. Non-dimensional SIF’s variations with crack length and G1/G2.

L. Marsavina et al. / Journal of Mechanical Science and Technology 25 (2) (2011) 309~315 315

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Liviu Marsavina received his MEng and PhD degrees in Mechanical Engineering from Politehnica University of Timisoara, Romania, in 1988 and 1998, respectively. He works as a postd-octoral researcher at Loughborough University, UK (2000), research asso-ciate at University of

Sheffield, UK (2001-2002) and experienced researcher at Lublin University of Technology, Poland (2007-2008). Dr. Marsavina is currently a Professor at the Faculty of Mechanical Engineering, from Politehnica University of Timisoara, Romania. He serves as an editorial board member of International Journal of Structural Integrity. Dr. Marsavina’s research interests include mechanics of materials, fracture mechanics, and finite element analysis.