Interfacial cracks analysis of functionally graded materials using Voronoi cell finite element method

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Procedia Engineering 00 (2011) 000–000 Procedia Engineering 31 (2012) 1125 – 1130

Procedia Engineering www.elsevier.com/locate/procedia

International Conference on Advances in Computational Modeling and Simulation

Interfacial cracks analysis of functionally graded materials using Voronoi cell finite element method Guangjie Zhanga, Ran Guoa* a

Faculty of Civil Engineering and Architecture, Kunming University of Science and Technology, Kunming 650500, China

Abstract The behaviour of a functionally graded material with interface cracks was studied using Voronoi element analysis. A comparison with the mode which does not take the interface cracks into consideration was made. The results show that the mechanical properties of functionally gradient materials are influenced by the particles’ size, topological structure, and interfacial deboning strength. With the interface cracking the overall stiffness of functionally gradient materials is gradually reduced, as a consequence, the reduction of material property mismatch at the heterogeneous interface is not obvious, because the matrix and reinforced particles cannot work together. It is verified that the interface cracks can be reduced or eliminated by the adjustment of the particles’ size and the increasement of the interface deboning strength.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Keywords: functionally graded materials; Interfacial crack; Voronoi cell finite element model;

1. Introduction Functionally gradient materials (FGMS) are designable composite materials, which consider the variety of different materials medium along the space of the macroscopic length scale. Via reinforcements’ functionally gradient distribution, it can connect two different kinds of materials which achieves different uses together perfectly. It has been recognized for quite some time that FGMS can reduce the sharp stress in the interface of the bi-materials which can lead to significant improvement in damage resistance and mechanical durability. Therefore, FGMS are of great interest in disciplines as

* Corresponding author. Tel.:+86-13759177126; fax: +86-871-3801140. E-mail address: [email protected]

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.1152

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diverse as civil infrastructure, aerospace propulsion, microelectronics, biomechanics, nuclear power, and nanotechnology [1] However, from the macroscopically point of view FGMS are heterogeneous because of the variation of the matrix and inclusion at volume fractions, and from the microscopically point of view FGMS which are made up of matrixes and inclusions, are also heterogeneous. So it is very difficult or time consuming to simulate FGMS’s topological structure with traditional finite element method. While the detailed knowledge of the distribution of stress and the redistribution of the stress reduction mechanism when the interfacial crack was generated is lacking. And this will affect the design of functionally gradient materials. Another question is how the FGMS work when the inclusions’ size or topological structure or interfacial debonds strength varies. From the viewpoint of above it is very important to analyze the mechanics performance of functionally gradient material with another numerical analysis method. In 1991 Ghosh proposed a Voronoi cell finite element method (VCFEM) [2].It based on the crosssectional scanning topology graph of the material. So it can describe the microscopically structure perfectly. Since then Ghosh and co-workers in a series of papers [3-4] have calculated the effective material properties of particle reinforced composite material via VCFEM. In 2001 R. Guo proposed an improved VCFEM [5-6] which can consider the effect of the interface cracks. In this study all the analysis are based on this method. Four models were analyzed in order to simulate the behaviour of a functionally graded material with interface cracks, and how the overall stiffness of functionally gradient materials are influenced by the particle size, topological structure, and interfacial deboning strength. 2. VCFEM considering interfacial deboning crack In the VCFEM method each cell represents an element of the microstructure, which includes a particulate with its matrix neighbourhood. A new concept of an interfacial decohesion cell element is shown in Fig. 1. (a).

Fig. 1. (a) VCFEM with interfacial decohesion; (b) [7] schematic representation of FGM

In an incremental formulation to account for the onset and growth of the debonding at matrix-inclusion interfaces, an element complementary energy function can be expressed as follows: mc  element   B  ,   d     : d    e

e



 t   t   u  u  d   





te

m

b

m

e

      ne  u  u  d  

( m   m   c   c )  nc   ub  ub  d  

  m   n m   u m  u m  d   

c



c

  c   nc   uc  uc  d 

(1)

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Where, ub , uc and um are the displacement increments of the interface and B  ,   is the increment of complimentary energy density in an element. Then the stress in the matrix and heterogeneity phases can be individually described to accommodate stress jumps across the interface. The expressions may be assumed in the form as m  Pmm (in m ) c  Pc c (in c )

(2)

Compatible displacement increments on the element boundary as well as on the interface are generated by interpolation in terms of generalized nodal values as:





m

m

  L q  ,on 

b b b u  L q e , on e ; u  L q , on b ;

u  L q m

 , on 

m

c

; u 

c

c

c

(3)

mc Then set the first variations of  element with respect to the stress coefficients m and c to zero respectively, results in the weak forms of the kinematic relations. At last set the first variation of the total energy functional (1) with respect to qe , qb , qm and qc to zero respectively, results in the weak form of the traction reciprocity conditions, for more details about the derivation of the formula please see the paper[6].

3. Voronoi tessellation of functionally gradient materials As shown in Figure 1. (b), the particles’ density and size are uniform along the thickness direction while they are changing. At both ends is mostly one material, the fractions of another kind of material is extremely low, and the fractions of materials increase gradually from one end to the interface.

Figure. 2. (a) Delaney and Voronoi tessellation; (b) Voronoi tessellation of FGMS

Based on functionally gradient material cross-sectional SEM graphs, we can get the particles’ locations and its size in x, y direction. Then according to the coordinates of points and boundary constraints a Constrained Delaunay Tessellation can be achieved, the vertexes of the triangle are the positions of the particles, and then sequentially connected the centers of neighbour triangles’ circum circle a voronoi tessellation can be made, an example is shown in figure 2. (a). About the algorithm please refer to paper [8], in this paper, an adjustment was made in order to consider the gradient of inclusions. Because the voronoi tessellation can reflect the inclusion influence domain (matrix closer to the inclusion region will have more obvious effect on the interfacial mechanical).It is a perfect way to describe the microstructure of the FGMS. The specify voronoi tessellation of functionally gradient materials is shown in Figure 2. (b).

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4. Numerical examples 4.1. Example 1 Detailed information about the validation of VCFEM’s accuracy can be found in literature [9]. In the first example, I take the SiC/Al as functionally gradient material, the volume fraction ratio of SiC was 25%, 40%, 55%, 70% specific distribution as shown in figure 5.

Figure 3 section schematic of SiC / Al functionally gradient materials

In this model, the Al was Al2024-T6. The elastic modulus EAl = 72Gpa; the Poisson's ratio V Al = 0.33; the yield stress was 70MPa; tangent modulus E t = 14.5Gpa; followed J2 flow theory; the elastic modulus of SiC ESiC = 430Gpa; Poisson's ratio V SiC = 0.25; the yield stress of SiC was 1.0E8; tangent modulus was 5.0e4Mpa.The width of the specimen was 4mm; the thickness was 8mm, each gradient layer thickness was 2mm. About 4700 inclusions were modeled, the boundary conditions of the model were constrained as flows: the right hands of this specimen were constrained in Ux, Uy direction, and the left hands of the specimen were enforced 0.001 uniform displacement. Analysis type was plane stress analysis; the normal stress leading interfacial crack was 30Mpa. In order to evaluate the influence of the interface cracks, a comparison with the model which does not take the interface cracks into consideration was made. The curve of displacement and reaction force was shown in figure 6.

Figure 4. (a) curve of displacement and reaction force; (b) The effect of interface bond strength

In figure 4. (a), the results show that the overall stiffness of functionally gradient materials is gradually reduced when the interfaces are cracked. Compared with the model which does not take the interface cracks into consideration the overall stiffness is reduced largely. In figure 4. (b), the results show that the overall stiffness of the specimen is increasing with the enhancement of interface bond strength. In this test, we can see that when the interface bond strength is between 10-35Mpa the overall stiffness of the specimen is increased slowly or can't even reach the yield strength. In this case the sic particles are a kind

Guangjie Ran Guo / Procedia Engineering (2012) 1125 – 1130 Guangjie Zhang Zhang and et al./ Procedia Engineering 00 (2011)31000–000

of damage to the al matrixes, because they cannot work together perfectly. With interface bond strength increasing for example between 35-70Mpa, the overall stiffness of the specimen is increased obviously. And the yield strength is higher than 90Mpa. So in this situation the spacemen can work very well. And the gradient performance of the functionally gradient materials is not obviously affected by the interfacial crack. 4.2. Example 2 In this model, the size and the topological of inclusions were varied, so we can estimate their influence on the overall behavior of the functionally gradient materials, and how the interfacial cracks were generated. All the parameters of specimen are exactly the same with the example1, while the size and their direction angles of sic particles are varied, and the normal stress of particle debonding was increase to 70Mpa to ensure that the cracks will be generated in the yield stage. So we can figure out in what kind of situation increasing the interface bond strength will avoid the interfacial crack obviously. The effect of size of inclusions was shown in figure 8, and the effect of inclusion direction angle was shown in figure 9.

Figure 5. (a) The effect of SiC’ volume ratio; (b) The effect of inclusions’ direction angle

In figure 5. (a), we can see that when the volume ratio of SiC is varied from 0.2 to 0.9 the elastic modulus is increased from 1.10e5 to 1.20e5, that means in elastic stage to improve SiC volume content will increase the elastic modulus, thereby improving stiffness. While in the yield stage the tangent modulus decreases as the SiC content gradually increasing. Because in this situation, a large number of interface cracks were generated, so SiC and al matrix cannot work cooperatively. Thus the more volume content of SiC the more interface damage, and the stiffness degradation is more serious, so the stiffness of this part of the gradient layer relative to other regions is reduced more obviously. In figure 5. (b), the variations of SiCs’ direction angles have no obvious effect on the elastic modulus, while in the yield stage, the tangent modulus decreased with the increase of direction angle which is between SiC axis and the horizontal axis. Because when the angle increased the stress concentration will increase, and the number of interface cracks will gradually increase, then the stiffness degradation is more serious. 5. Conclusions This new kind of VCFEM method can predict functionally gradient material particles’ interfacial crack accurately. The particles’ size, topological structure, and interfacial deboning strength will influence FMES’ mechanical behavior. With the interface cracking the overall stiffness of functionally gradient materials is gradually reduced. When we design functionally material, we must improve the interface strength of the inclusion, make the size of inclusion and the topological structure moderate.

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6. Acknowledgements This research was supported by the National Natural Science Foundation of China (Nos. 11072092/ A020317), the financial supports from it are gratefully acknowledged. The first author wishes to thank professor Ran Guo for the useful help during the computer implementation of the algorithm. References [1] Suresh S. Graded materials for resistance to contact deformation and damage. Science 2001;292:2447–2451. [2] Ghosh S, Mukhopadhyay S.N Material based finite element analysis of heterogeneous media involving dirichlet tessellations. Computer Methods in Applied Mechanics and Engineering 1993;104:211–247. [3] Ghosh S and Moorthy S. Particle fracture simulation in non-uniform microstructures of metal-matrix composites. Acta Materialia 1998;46:965–982. [4] Ghosh S, Lee K, Raghavan P. A multi-level computational model for multi-scale damage analysis in composite and porous materials. international Journal of Solids and Structutres 2001;38: 2335–2385. [5] Guo R, Shi HJ, Yao ZH. Numerical simulation of thermo-mechanical fatigue properties for particulate reinforced composites. Acta Mechanica Sinica 2005;21(2):160–168 [6] Guo R, Shi HJ, Yao ZH. The modeling of interfacial debonding crack in particle reinforced composites using Voronoi cell finite element method. Comput Mech 2003;32(1-2):52–59 [7] Rahman S, Chakraborty A. A stochastic micromechanical model for elastic properties of functionally graded materials. Mechanics of Materials 39 (2007) 548–563. [8] Cesari F, Furgiuele F, Martini D. An automatic mesh generator for random composites, XII ADM International Conference, Grand Hotel Rimini Italy Sept 5th-7th; 2001,P. 18–24 [9] Guo R, Shi HJ, Yao ZH. The analysis of interface damage evolution of composite materials. Beijing Mechanical Institute Annual Meeting 2002:225–250.