2013 simulia regional user meeting - · pdf file2013 simulia regional user meeting modeling...

2013 SIMULIA Regional User Meeting

Modeling Creep Fracture for Ultra-High Temperature Ceramics Using ABAQUS UEL and UMAT

Chi-Hua Yu1, Chang-Wei Huang2, Chuin-Shan Chen1, Yanfei Gao3 and Chun-Hway Hsueh4,5

1Department of Civil Engineering, National Taiwan University, Taipei, Taiwan 2 Department of Civil Engineering, Chung Yuan Christian University, Chung Li, Taiwan

3 Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA 4 Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA

5 Department of Materials Science and Engineering, National Taiwan University, Taipei, Taiwan ABSTRACT A rate-dependent cohesive model based on a micromechanical model of grain boundary sliding and cavity nucleation and growth was developed. The model was implemented in ABAQUS to study creep fracture of polycrystalline ceramics at high temperatures. User-defined element (UEL) subroutine for grain boundary cohesive model and user-defined material (UMAT) subroutine for grain creep in ABAQUS were developed. The influence of nucleation and diffusion was explored. We found that the nucleation rate controls the cavity spacing while the diffusion increases the cavity growth rate. Grain boundary heterogeneity effects of polycrystalline ceramics were also studied. We showed that the stress concentration occurs in the presence of grain boundary heterogeneities which, in turn, could result in relatively weak and strong surfaces at the grain boundaries. Keywords: rate-dependent cohesive model, creep fracture, grain boundary nucleation, cavity growth. 1. Introduction Intergranular fracture occurs when high strength ceramic materials suffer from creep deformation at high service temperatures. Fracture of ceramics at high temperatures by creep can be classified into two categories: slow crack growth and creep fracture. Slow crack growth is a direct extension of a microcrack by diffusion of atoms through the crack tip. On the other hand, creep fracture is a process of nucleation, growth, and coalescence of cavities along the grain boundary, and it is often a localized and inhomogeneous manner. Generally, fracture of ceramics at high temperatures exhibits a transition from slow crack growth to creep fracture as the temperature increases [1, 2]. This transition also depends on stresses, such that low stresses are in favor of creep fracture [1, 2]. However, in these two mechanisms, surface diffusion along the crack surfaces and grain boundary diffusion to remove atoms from the crack tip are involved. While the surface diffusion is driven by the curvature of the crack surfaces, grain boundary diffusion is driven by the stress distribution along the grain boundary [3].

Cohesive interface models have been used widely in the numerical simulation of void nucleation and fracture. The aim of this

paper is to develop a robust user-defined rate-dependent cohesive element in ABAQUS to simulate the intergranular creep fracture problems. In the following, a physical-based constitutive model of the intergranular creep fracture is presented in Section 2. The explicit form of the Jacobian matrix (i.e., the tangential stiffness matrix for nonlinear problems) for the grain boundary and the corresponding stress update form are derived in Section 3. The kernel of the computational framework and the ABAQUS implementation are also described in this section. In Section 4, two numerical examples are presented. The first example is a simple bicrystal model to verify the accuracy and stability of the implementation. The second example is a more realistic polycrystal model to study the creep fracture of the ZrB2-SiC composites. Finally, concluding remarks are given in Section 5. 2. Intergranular creep fracture

A constitutive model accounting for elasticity and creep in the grains, the cavity nucleation, growth and coalescence on the grain boundaries, and grain boundary sliding is briefly presented below. 2.1. Grain deformation

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The material behaviors within the grains are assumed to be deformed by a power law creep [4]. All the grains are treated as homogeneous and isotropic materials. The total strain rate tensor is the sum of the elastic and the creep strain-rate tensors, such that

e crij ij ij

e crij ij ijij ijij

eeij (1)

The creep strain-rate components can be calculated from the power law creep

00

3 exp ( )2

ijcr neij

e

S QkT

((03 exp2

ij expcr 3 ijij

Sijcr jij

Q (2)

where Q is the activation energy, k is the Boltzmann constant, T is the absolute temperature, 0 and 00 are the reference stress and strain rate, respectively, n is the creep exponent, ijS is the deviatoric stress tensor, and e is the von Mises stress given by

32e ij ijS S (3)

2.2. Grain boundary sliding

For grain boundary sliding, it is assumed to be governed by Newtonian viscous flow, such that [5]

sb

u wsu ws (4)

where susu is the relative sliding velocity of adjacent grains resulting from the shear stress on the grain boundary, w and b are the

thickness and the viscosity of the grain boundary, respectively. Because b/w always appears as one term in the analysis, it can be treated as an independent material parameter to simplify the analysis. 2.3. Grain boundary cavitation

1(a) 1(b) Fig. (1a) Geometry of cavities in the spherical-caps shape; Fig. (1b) Smeared-out representation of grain boundary cavitation in

terms of un

To simplify the cavitation problem from

complex nucleation phenomena to a manageable manner in a numerical modeling, a smeared-out representation of the grain boundary cavitation is adopted [6]. The distribution of cavities on each grain boundary is replaced by a continuous separation as shown in Fig. 1. The cavity on the grain boundary can generally be characterized by its radius a, void half-spacing b, and spherical-caps shape parameter h that is determined by the cavity tip angle (see Fig. 1(a)). The cavity volume can be expressed as

34 ( )3

V a h (5)

where the spherical-caps shape parameter is defined by

1 1[(1 cos ) cos ]2( )

sinh (6)

The value of has a typical value of 75° during the cavity growth.

The smeared-out representative separation of grain boundaries, as shown in Fig. 1(b), is described by

2nVub

(7)

As a result, the rate of change of the normal separation is

2 3

2n

V Vbub bV Vb2 b

2nVub2

V (8)

The rate of void spacing change bb is related to the strain rate of adjacent grains and the cavity nucleation, such that [9]

1 1( )2 2I II

b Nb Nb 1 N) N1)

N)

2)) N (9)

where II and IIII are in-plane principal logarithmic strain rates at the grain boundary and N is the cavity density, which is defined as the density per unit undeformed grain boundary to describe the nucleation of new cavities. 2.4. Cavity nucleation

A phenomenological model for cavity nucleation is adopted [7]. The nucleation rate of grain boundary can be expressed by

2

0

( ) 0crnn e nN F forN F (F ( cr (10)

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where Fn is a material parameter and 0 is a stress normalized factor. 2.5. Cavity growth

Cavity growth results from two kinds of mechanisms: diffusion of atoms which diffuse from cavity surfaces to the grain boundary layer and creep deformation of the surrounding grains. Hence, the volumetric cavity growth rate can be written as

1 2V V V2V V V2VV1 (11)

where 1V1V1 and 2V2V2 represent the contributions of atom diffusion and creep, respectively. They are [8]:

1(1 )

41 1ln( ) (3 )(1 )

2

n sfV D

f ff

4V D1 4 (12a)

3

23

2 ( )[ ] , 1

2 ( )[ ] , 1

c nm me n n

e e

c n m me n n

e e

a h

V

a h

3c 3 ( )[(( )[)[( )[3 ( )[)[( )[3

2V23c 3 ( )[( )[( )[( )[)[3

(12b)

with 2 2

max ,1.5

a afb a L

(13a)

13

ece

L D ccec (13b)

32n n

(13c)

2

( 1)( 0.4319)n

n nn

(13d)

where L is a stress and temperature dependent length scale factor introduced by Needleman and Rice [9] and D is the grain boundary diffusion parameter. It should be noted that the effective stress e and the mean stress

m are obtained from the adjacent grains instead of at the cavity. The sintering stress

s is usually neglected owning to its small value. It has been concluded that the cavity growth is dominated by creep for large values of a/L, while the atom diffusion is more important for small values of a/L (e.g., a/L < 0.1). It implies that the growth rate of void radius aa is dominated by the free atom diffusion when the effective creep strain rate is small.

3. Finite element implementation

A user-defined element subroutine (UEL) in ABAQUS has been implemented for the grain boundary cavitation and shear sliding [10]. As mentioned in the pervious section, the stress state and effective creep rate of adjacent grains are also needed for the constitutive model of grain boundary element. To obtain these data, a user-defined material subroutine (UMAT) of the power law creep with isotropic elasticity is implemented. To establish a communication between grain elements and grain boundary elements, a prescribed order of grain and grain element is introduced to reduce the computing time in searching the common block between all grain elements.

Fig. 2 Two-dimensional linear cohesive element

A weak interface in solids can be

described by a cohesive model. There are many ways to implement cohesive laws in ABAQUS, and the most versatile one is the development of cohesive elements. The cohesive element implemented herein is made up of two linear line elements (as shown in Fig. 2), which connect the faces of adjacent plane elements during the fracture process. The two lines of interface elements initially lie together in the unstressed state with zero thickness and separate as the adjacent elements deform. The relative displacements of the nodes in the cohesive element are obtained from the displacement of grain boundary elements in both the normal and the tangential directions. In tangential sliding, the stress update form can be derived from Eq. (4), that is

bsu

w su (14)

The related material tangent is b

su w ts tsu wws

(15)

where t is the incremental time step in the

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time integration scheme. In the normal direction, the normal stress

is related to the normal displacement which is governed by the cavity nucleation, growth, and coalescence on the grain boundary. Substituting Eqs. (9), (10), (12) and (13) into Eq. (8), the normal displacement rate can be expressed as a function of the normal stress in a quadratic form, such that

2n n nu A B Cn nu An

2A nu A 2A (16) where

2 20

1

cne

FVANb

ce (17a)

2

1 41 1 ln( ) (3 )(1 )

2

DBb f f

f

(17b)

22 2 ( )

I IIV VC

b b2V2 V ) (17c)

It is worth noting that A, B, and C signify, respectively, the contributions of cavity nucleation, atom diffusion, and creep to the normal separation.

The material tangent with respect to the incremental normal displacement can be derived from Eq. (16):

1(2 )

n

n nu A B t (18)

Combining Eqs. (15) and (18), the material Jacobian (tangent stiffness) matrix can be obtained, such that

1 0(2 )

0

nn n

s

uA B tu

w t

(19)

All components in the material Jacobian matrix are derived in an explicit integration scheme and are implemented in the user subroutine UEL in ABAQUS. Since the creep information of the grain element should be gathered from adjacent grain elements for the grain boundary element, it is necessary to access the information of state variables, including the stress state and the effective creep strain rate. A user-defined material (UMAT) for power law creep is implemented accordingly. By a special ordering of elements and a common block, all the information can be accessed effectively.

4. Numerical examples All the parameters in simulation are normalized by the initial grain facet length Ri, Young’s Modulus of grain element E and the applied strain rate appapp . The facet length is set to 1 m at initial, Young’s modulus is 500 GPa and Poisson’s ration is 0.129 for grain elements. The reference stress 0 is 0.001E, the reference strain rate 00 is 1.0 and the order n is 5 for power law creep in grain elements. For a grain boundary element, the initial void radius a is set to be 30.67 10 iR and the half void spacing b is 0.67Ri. The initial reference density of the undeformed grain boundary NR is 1 iR , and the initial density of the undeformed grain boundary Ni is 40NR. The material parameter Fn in a nucleation law is set to 45.4 10 RN for brittle material, and the reference stress 0 is 0.01E. The viscosity is normalized by the applied strain rate appapp and Young’s modulus E:

* 35 10b app

E (20)

The normalized diffusion parameter D* is determined by

* 53 10

app i

DEDR3 10

app iRi

(21)

For all the time scale, t is normalized by applied strain rate *

appt t app and all the length scale are normalized by Ri. 4.2 Model verification: a bi-crystal model

The first example is a simple bi-crystal that consists of two square gains with a facet length of Ri and a single grain boundary as depicted in Fig. 3 [11, 12]. This example is conducted to verify the accuracy of current model. Specifically, the two grains are modelled by the plane strain element (CPE4 in ABAQUS) and the grain boundary is modelled using the rate-dependent cohesive element developed herein. The bi-crystal model is loaded by a constant displacement distributed along the top and bottom edges, while the transverse degree of freedom (in the

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x-direction) of each node is constrained. The applied strain rate is set to meet the stead state creep strain rate. The initial normalized time step t* is set at 10-9 and the following time steps are adjusted automatically by ABAQUS to meet the desirable convergence with efficiency.

Fig. 3 Bi-crystal with simple boundary value problem

To verify the accuracy, a direct integration form updated by an analytical solution has been computed via a MATLAB code and the results are compared with those obtained from the finite element analysis.

Figures 4 (b)-4(d) show the relationship of the void radius a, void spacing b, and a/b as functions of time. In Fig. 4(b), the increase of the void radius, a, with time represents the growth of the void on the grain boundary. It indicates that the cavity growth remains almost the same for different nucleation rates before fracture. On the other hand, the void spacing, b, decreases faster with time which indicates more cavity nucleation occurs before fracture.

Fig. 4(a) Normalized traction-separation relation with different nucleation rates.

Fig. 4(b) Normalized void radius a v.s. Normalized time t* with different nucleation rates.

Fig 4(c) Normalized void spacing b* v.s. normalized time t* with different nucleation rates.

Fig 4(d) a/b ratio v.s normalized time t* with different nucleation rates.

We noted that b attains a minimum and remains almost constantly at the final stage of fracture. This is because the nucleation rate is also influenced by effective creep rate and when the normal traction reaches its limit, the effective strain rate dropped quickly. Finally, the ratio of a to b increases monotonically and reaches the upper limit of 1.0 which represents complete separation of the grain

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boundary. 4.2. Polycrystalline ZrB2-SiC composites

A polycrystal model is conducted to study the creep fracture of the zirconium diboride (ZrB2) and silicon carbide (SiC) composite. A configuration of polycrystalline grains is shown in Fig. 6. For the sake of convergence, the facet length of each grain boundaries is equally discretized with 500 nm cohesive elements. Each grain is constructed by numbers of three-node triangle plane strain elements (CPE3 element in ABAQUS). In our study, each grain boundary element is attached to two adjacent grain elements. Material properties adopted are the same as those used in the previous bi-crystal example. The model is subjected to a constant tension of 75 MPa, while the horizontal degree of freedom at the right and the left boundaries is fixed.

Fig. 6 Polycrystalline structure

Because of the composites of ceramics,

such as the SiC-ZrB2, there exists different properties on the grain boundary due to impurities [13]. The cavity nucleation ability is only allowed at the grain boundaries between ZrB2 and SiC grains. Figure 7 shows the von-Mises stress contour and crack pattern for polycrystalline grains at high temperatures. We observe that there exists a severe stress concentration at the triple junction between ZrB2 and SiC grains. This stress heterogeneity phenomenon agrees with the experimental observation [14]. This phenomenon is a direct consequence of the nucleation properties assigned at the ZrB2-SiC grain boundaries.

Fig. 7 Stress contour at t*=0.1325 10-4

5. Conclusions

A robust rate-dependent cohesive element has been developed and implemented in ABAQUS. The model presented here accounts for the fracture mechanisms in microstructures, i.e., viscous grain boundary sliding, the nucleation of grain boundary cavities, their growth by grain boundary diffusion and local creep, and link up of microcracks to form macrocracks. The bicrystal verification confirms the robustness and stabilities of the implementation. The method is also used to study the creep fracture of ultra-high temperature ceramics composites. The results indicate there exists a stress heterogeneity due to a different grain boundary properties. This presence of heterogeneity predicted by the simulation agrees with experimental observations. Acknowledgements The research was performed under the finical support by National Science Council, Taiwan. The authors would like to express their gratitude to Simutech Solution Corp. Taiwan for providing computational support. We are also grateful to the National Center for High-Performance Computing for providing the computational resources. Finally, we would also like to express our sincere gratitude to Marc Bird, Professors Ken White and P. Sharma (University of Houston) and P. F. Becher (University of Tennessee) for their enlightening comments and discussions. References [1]G. D. Quinn, "Fracture mechanism maps for advanced structural ceramics - Part 1 Methodology and hot-pressed silicon nitride

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results," Journal of Materials Science, vol. 25, pp. 4361-4376, 1990. [2]G. D. Quinn and W. R. Braue, "Fracture mechanism maps for advanced structural ceramics - Part 2 Sintered silicon nitride," Journal of Materials Science, vol. 25, pp. 4377-4392, 1990. [3] C. H. Hsueh and A. G. Evans, "Overview 14 Creep fracture in ceramic polycrystals-II. effects of inhomogeneity on creep rupture," Acta Metallurgica, vol. 29, pp. 1907-1917, 1981. [4] A. F. Bower, Aplied Mechanics of Solids: Taylor & Francis Group, LLC, 2010. [5] M. F. Ashby, "Boundary Defects, and Atomistic Aspects of Boundary Sliding and Diffusional Creep," Surface Science, vol. 31, pp. 498-&, 1972. [6] J. R. Rice, "Constraints on the Diffusive Cavitation of Isolated Grain-Boundary Facets in Creeping Polycrystals," Acta Metallurgica, vol. 29, pp. 675-681, 1981. [7] P. Onck and E. Van Der Giessen, "Growth of an initially sharp crack by grain boundary cavitation," Journal of the Mechanics and Physics of Solids, vol. 47, pp. 99-139, 1998. [8] E. Van Der Giessen and V. Tvergaard, "Development of final creep failure in polycrystalline aggregates," Acta Metallurgica Et Materialia, vol. 42, pp. 959-973, 1994. [9] A. Needleman and J. R. Rice, "Plastic creep flow effects in the diffusive cavitation of grain boundaries," Acta Metallurgica, vol. 28, pp. 1315-1332, 1980. [10] C.-H. Yu, C.-W. Huang, C.-S. Chen, Y. Gao, and C.-H. Hsueh, "Effects of Grain Boundary Heterogeneities on Creep Fracture of Ultra-High Temperature Ceramics," presented at the 6th European Congress on Computational Methods in Applied Sciences and Engineering, Vienna, Austria, 2012. [11] Simulia, "ABAQUS," 6.10 ed. Pawtucket, Rhode Island, 2010. [12] Y. F. Gao and A. F. Bower, "A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces," Modelling and Simulation in Materials Science and Engineering, vol. 12, pp. 453-463, May 2004. [13] C.-H. Yu, C.-W. Huang, C.-S. Chen, Y.

Gao, and C.-H. Hsueh, "A micromechanics study of competing mechanisms for creep fracture of zirconium diboride polycrystals," Journal of the European Ceramic Society, vol. 33, pp. 1625-1637, 2013. [14] M. W. Bird, R. P. Aune, A. F. Thomas, P. F. Becher, and K. W. White, "Temperature-dependent mechanical and long crack behavior of zirconium diboride-silicon carbide composite," Journal of the European Ceramic Society, 2012.

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