Eur. Phys. J. Special Topics 222, 1587–1595 (2013) © EDP Sciences, Springer-Verlag 2013 DOI: 10.1140/epjst/e2013-01947-3

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

Crack propagation in functionally graded strip under thermal shock I.V. Ivanov1,a , T. Sadowski2,b , and D. Pietras2 1 2

University of Ruse, Department of Engineering Mechanics, Studentska 8, Ruse 7017, Bulgaria Lublin University of Technology, Faculty of Civil Engineering and Architecture, Department of Solid Mechanics, 20-618 Lublin, Nadbystrzycka 36, Poland Received 4 June 2013 / Received in final form 2 August 2013 Published online 30 September 2013 Abstract. The thermal shock problem in a strip made of functionally graded composite with an interpenetrating network micro-structure of Al2 O3 and Al is analysed numerically. The material considered here could be used in brake disks or cylinder liners. In both applications it is subjected to thermal shock. The description of the position-dependent properties of the considered functionally graded material are based on experimental data. Continuous functions were constructed for the Young’s modulus, thermal expansion coefficient, thermal conductivity and thermal diffusivity and implemented as user-defined material properties in user-defined subroutines of the commercial finite element software ABAQUS™. The thermal stress and the residual stress of the manufacturing process distributions inside the strip are considered. The solution of the transient heat conduction problem for thermal shock is used for crack propagation simulation using the XFEM method. The crack length developed during the thermal shock is the criterion for crack resistance of the different graduation profiles as a step towards optimization of the composition gradient with respect to thermal shock sensitivity.

1 Introduction Ceramic materials are widely used in various engineering applications because of their excellent properties at high temperature and their superior corrosion and wear resistance. A major limitation of ceramics, however, is their inherent brittleness that can result in catastrophic failure under thermal shocks. To overcome this disadvantage, considerable efforts have been made to toughen ceramics by including another material. One solution is the concept of Functionally Graded Materials (FGM), which are composite materials with gradually changing volume fraction of the components. Functionally graded materials have better resistance to thermal shock cracking in comparison with layered composites [1] and homogeneous ceramics [2], as it is proved a b

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Fig. 1. Infinitely long and thick strip of FGM.

by experiments. In most of the cases, FGM are created at elevated temperature by some manufacturing process and then cooled to ambient temperature. The graded thermal expansion properties cause residual stress as a result of the FGM fabrication [3]. They usually work at higher temperature, but it is different than the stress free manufacturing temperature. The residual stress of manufacturing process could significantly influence crack resistance of FGM in thermal shocks. The thermal shock analysis of FGM is a challenging task. The functions of the material properties should be found or approximated, then they should be implemented in a mathematical or a numerical model. Finite Element (FE) method is a favourite method which can solve problems of continua. The utilisation of graded FEs is described in [4–6]. The fracture parameters based on contour integrals, however, are not strictly defined for FGM and residual stress, which leads to development of suitable methods to evaluate the fracture toughness [7–10]. Another way is to estimate the fracture toughness from the stress field at the crack tip using very dense mesh [11]. Some researchers seek analytical solution by space discretisation [12] or entire analytical solution [13]. Functionally graded composite of alumina(Al2 O3 )/aluminium(Al) is considered here with different composition profiles as a step toward optimization of composite material content for better thermal shock resistance. The residual stress of manufacturing process is included in the considerations. Extended Finite Element Method (XFEM) is used to simulate the crack propagation during transient thermal shock as a result of cooling. The crack propagation criterion is based on a modified Virtual Crack Closure Technique (VCCT) [14]. The XFEM is innovative method for crack propagation simulations, which is implemented in ABAQUS commercial software. The modified VCCT for crack growth decision is very suitable for FGM, since it is based on the local characteristic of the crack tip parameters. The demonstrated methodology in this work avoids most of the problems arising for FGM and their thermal residual stress.

2 Material properties A infinitely long and thick strip is considered here, which has width, b, and graded properties in the direction of x-axis as it is shown in Fig. 1. A edge crack with length, a, is supposed to exist for crack resistance analysis. The volume fraction of aluminium in the composite is given by the expression: vf = C0 + Cn ξ n

(1)

where ξ = x/b is dimensionless coordinate for the function of the property grade. For the composite Al2 O3 /Al considered here, C0 = 0.03 and Cn = 0.3. The thermal and elastic properties are experimentally determined in [15]. The composition profiles in dependence of the power degree, n, considered here are three: n = 1/3, n = 1, and

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Fig. 2. Volume fraction profiles of FGM.

n = 3. The diagrams of the volume fraction functions of the profiles can be seen in Fig. 2. The thermal properties of the FGM, which are developed as functions of the volume fraction of Al in the composite are the thermal conductivity, k, and the thermal diffusivity, κ: k = 37.71 + 363vf1.45 exp(−1.5vf )

N s.K

(2)

mm2 · (3) s The functions are programmed in the code of user subroutine UMATHT for the commercial software ABAQUS™. Since the software uses directly the specific heat of the material, cv , in order to calculate the thermal diffusivity, κ, the specific heat is provided by calculation: k (4) cv = ρκ where ρ is the mass density of the material, which is given as a constant value, ρ = 4 g/cm3 . The thermal expansion coefficient, α, and the Young’s modulus, E, of the FGM are approximated by third order polynomials as functions of the Al volume fraction: κ = 10.9 + 184.4vf1.5 exp(−2.5vf )

α = 281.24vf3 − 102.98vf2 + 15.112vf + 7.71 × 10−6 K−1

(5)

E = −1482.6vf3 + 1973vf2 − 1096.9vf + 398.42 GPa.

(6)

The Poisson’s ratio of the material is assumed constant, ν = 0.22, because its variation is small and it has negligible influence on the thermal stress [16]. The properties calculation are programmed in the code of user subroutine UMAT of ABAQUS™ commercial software.

3 Thermal shock temperature distribution The thermal shock simulated here is a sudden cooling. It is assumed that the whole sample is at 220 ◦ C temperature, which is the initial condition. The boundary conditions applied are as follows: T0 = Tξ=0 = 20 ◦ C,

T1 = Tξ=1 = 220 ◦ C,

ΔT = T1 − T0 = 200 ◦ C.

(7)

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b)

a)

c) Fig. 3. Dimensionless temperature distribution during the thermal shock: a) n = 1/3, b) n = 1, c) n = 3.

The thermal conduction equation governing the temperature distribution in time is:   ∂T ∂ ∂T k(x) = ρcv (x) · (8) ∂x ∂x ∂t The equation is solved by 2-D FE model of a sample with size b = 10 mm and length 100 mm. The discretisation of the elements reaches quad elements of size 0.25 mm, which provides enough accuracy of the solution. The terminal time of the problem solution, tt , is: tt = 0.1ts = 0.1

b2 κ0

(9)

where ts is the time to reach the steady state solution and κ0 is the thermal diffusivity at ξ = 0, when vf = v0 = C0 . Using steady state time, ts , it allows us to express the functions depending on time in dimensionless time: τ=

tκ0 t = 2 · ts b

(10)

The results for temperature distribution at different times for different composition profiles are given in Fig. 3 in dimensionless temperature: θ=

T − T1 · ΔT

(11)

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The curves are steeper on the left hand side of diagrams as the power degree, n, is higher. This could give higher gradient of the thermal stress there.

4 Thermal stress analysis The stress analysis can be done simultaneously with the heat transfer analysis or the temperature at nodes of FE model during the thermal shock calculations can be written in a file and then reproduced during the stress analysis. In both cases, we can add a step to the stress analysis to calculate the fabrication residual stress. It is assumed that the stress free state of the FGM considered here is at 620 ◦ C. When we want the residual stress to be accounted for, a step of cooling the sample from temperature of 620 ◦ C to 220 ◦ C is added before the thermal shock stress analysis step. The stress analysis is governed by plane stain constitutive law in an incremental solution and statically determinant boundary conditions. The thermal expansion strain increment is calculated from the temperature increment: dεT = αdT.

(12)

The plain strain constitutive law gives the stress increment from the mechanical strain increment, {dεm }: (13) {dσ} = [C]{dεm } where

⎡

1−ν ν ν E ⎢ ν 1−ν ν [C] = ⎣ ν ν 1−ν (1 + ν)(1 − 2ν) 0 0 0

⎤ 0 0 ⎥ 0 ⎦

(14)

1−2ν 2

is the material stiffness matrix and the engineering mechanical strain increment is obtained from the thermal expansion increment: {dεm } = [dεx − dεT

dεy − dεT

− dεT

dγxy ]T .

(15)

The stress increment has four components: {dσ} = [dσx

dσy

dσz

dτxy ]T .

(16)

The results of the stress analysis are given in Fig. 4 for the dimensionless stress component σy∗ . The dimensionless stress is obtained as follows: σy∗ =

σy (ξ, τ )(1 − ν) E0 α0 ΔT

(17)

where E0 and α0 are the Young’s modulus and the thermal expansion coefficient, respectively at ξ = 0, when vf = v0 = C0 . The analysis of diagrams with thermal shock stress distribution (see Fig. 4) shows that a great tensile stress is developed at the cooled surface, which can cause crack initiation and propagation. The tensile stress is as great as the power degree, n, of the composition profile is higher. The residual stress (at τ = 0) is significant and greater as n is greater, but compared with the maximum tensile stress is relatively lower as n is greater. The maximum tensile stress is developed in the very beginning of the thermal shock (at τ = 0.025) and then the stress calms down. The tensile zone is restricted by compressive zone, which could arrest a crack growth, but the size of those zones will change with the compliance developed by the crack propagation.

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a)

b)

c)

Fig. 4. Dimensionless stress distribution during the thermal shock: a) n = 1/3, b) n = 1, c) n = 3.

5 Crack resistance The crack resistance of the different composition profiles of Al2 O3 /Al FGM in a thermal shock was examined by simulation of crack propagation by XFEM. Initial edge crack with length of 0.32 mm in the direction perpendicular to the surface is created on the side of cooled surface. The crack is allowed to grow during the stress analysis. The criterion for crack growth in the XFEM simulations is the strain energy released rate in the Maximum Tangential Stress (MTS) direction. The critical strain energy released rate was given as GCI = GCII = GCIII = 40 N/m. The simulation of unstable crack propagation by XFEM is a challenging task. It requires some regularisation technique to be applied as viscous regularisation [14]. The investigation of crack propagation in FGM as a result of thermal shock in [17] shows that it is unstable and the crack path is difficult to be predicted. The results of our simulations also show unstable crack growth in curve paths as it can be seen in Fig. 5. The simulations are successfully done due to a very high viscous coefficient (0.6–1 s−1 ), which could slow down the development of the crack and could influence the crack length in great degree, since the crack propagates very fast in the very beginning of the thermal shock and then it is arrested. It is very difficult to find difference in the crack path length of composition profiles for n = 1/3 (Fig. 5(b)) and n = 1 (Fig. 5(c)). In order to avoid the influence of crack propagation instability on the assessment of composition profile crack resistance, the crack is forced to grow only in the direction tangential to the initial crack. As a result

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Fig. 5. Simulated crack propagation on MTS principle: a) initial crack, b) n = 1/3, b) n = 1, c) n = 3.

Fig. 6. Simulated crack propagation in tangential to the crack direction: a) initial crack, b) n = 1/3, b) n = 1, c) n = 3.

of the simulations, long stable crack growths are obtained, which can be seen in Fig. 6, using very small viscous coefficient of regularisation (0.02 s−1 ). The fracture toughness of FGM is dependent on the volume fraction of the composite components as well as on the temperature. It is possible to give such dependence of the critical strain energy released rate in the simulations. Since the reliable sources for such dependence are not found, a function of critical strain energy released rate in dependence of temperature is constructed as follows: GTC = G0C



T − T0 ΔT

0.075 (18)

where G0C = GCI = 40 N/m. The graphs of the function and its tabular linear interpolation encoded in the ABAQUS input file are given in Fig. 7. The fracture toughness could increase with approximately 20% within the temperature range of the thermal shock, accounting also for the Al content graded in the same direction as the temperature.

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Fig. 7. Temperature dependent critical strain energy released rate.

Fig. 8. Relative crack length developed for different composition profiles.

The straight crack growth has been allowed during the stress analysis under different conditions. Simulations of crack propagation are obtained when fabrication residual stress exists and when it does not exist. Two other cases are considered when the critical strain energy released rate is constant and when it is temperature dependent. The obtained relative crack lengths are depicted graphically in Fig. 8 in dependence of the composition profile. Comparing the results for crack lengths, one can conclude that the composition profile with lowest power degree, n, is the profile with the best crack resistance, the influence of the temperature is negligible, while the influence of the fabrication residual stress on the crack resistance is well pronounced.

6 Conclusions Thermal shock in a strip of FGM is modelled successfully by FE method. Graded property functions are coded in user subroutines and together with dense mesh they give very smooth solution for temperature and stress distribution. The crack resistance of different composition profiles is examined by simulations of the crack propagation by XFEM. The local fracture criterion used by this method avoids the difficulties of such analysis of FGM. The inherent instability of crack growth in FGM can be avoided by forcing the crack to propagate straight forward for the purposes of crack resistance assessment. The higher is the content of aluminium in alumina/aluminium composite with graded profile, the higher is the thermal shock crack resistance of the material. The influence of the temperature dependence of

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the fracture toughness of material is negligible on the crack resistance while the fabrication residual stress strongly reduces it. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013), FP7 - REGPOT - 2009 - 1, under grant agreement No. 245479. This work was also financially supported by Polish Ministry of Science and Higher Education within the statutory research number S/20/2013.

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