The plastic energy dissipation in metal matrix composites during cyclic loading

Transcription

1 Computational Materials Science 15 (1999) 96±100 The plastic energy dissipation in metal matrix composites during cyclic loading D. Xu a, S. Schmauder b, * a Department of Engineering Mechanics, Xi'an Jiaotong University, Xi'an , Shaanxi Province, People's Republic of China b Staatliche Materialprufungsanstalt (MPA), Universitat Stuttgart, Pfa enwaldring 32, D Stuttgart, Germany Received 19 November 1998; accepted 28 January 1999 Abstract Plastic deformation of materials is a major source of energy dissipation during external loading. In metal matrix composites (MMCs), local plastic strain may arise even if the overall external load is below the yield stress of the matrix because of the di erence of elastic modulus between the reinforcement and the matrix. In this paper, the nite element method combined with cell modelling is employed to discuss the energy dissipation of MMCs under cyclic loading for this case. The in uence of the elastic modulus, the shape and the volume fraction of reinforcement on the energy dissipation was investigated numerically. The relation between external loading and energy dissipation was clari ed. It was found that the energy dissipation became constant after two or three cycles and showed a strong dependence on loading amplitude and the volume fraction as well as the elastic modulus of inclusions. However, the energy dissipated is hardly in uenced by the shape of the inclusions. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Metal matrix composites (MMCs); Plastic energy dissipation; Cell modelling; FEM in uencing factors 1. Introduction There is a great amount of literature discussing the damping of metal matrix composites (MMCs) as a kind of energy dissipation during external cyclic loading [1±5]. Based on their own and other people's results, Zhang and colleagues [6] summarised the mechanisms which are applied to explain the damping in MMCs from a microscopic view. Furthermore, most people have noticed that the stress-induced martensitic transformation in shape memory alloys can be an e ective energy absorber [7,8]. But, plasticity known as a main * Corresponding author. Tel.: ; fax: ; schmauder@mpa.uni-stuttgart.de source of energy dissipation is seemingly neglected, although Baker [9] derived an expression for log decrement of vibration amplitude by treating energy absorption processes as essentially the same plastic ow phenomena as those in an unreinforced matrix. Since there exists the di erence of the elastic moduli between the matrix and the reinforced particles, the local plasticity can be generated even if the loading level of overall material is not high enough to produce plasticity in the unreinforced matrix. The local plastic deformation in MMCs usually starts in the area adjacent to the reinforcement [10,11]. Accordingly, it may be supposed that the properties of the reinforcement including elastic moduli and shape can a ect the local plasticity and then the energy dissipation of /99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S ( 9 9 )

2 D. Xu, S. Schmauder / Computational Materials Science 15 (1999) 96± MMCs. Another factor which can in uence the energy dissipation is the loading level. If the load is too small, local plasticity will not be expected to occur in MMCs. The next step which should be taken is to investigate how the loading, the elastic modulus and the shape of reinforced particles in- uence the energy dissipation in MMCs. This is the object of this paper. equivalent plastic strain and n the power of strain hardening. The matrix is also assumed to have the same initial yield stress for both tensile and compressive loading. The energy dissipation ability of the composite material is evaluated by the ratio of the dissipated work by the material DW to the maximum of elastic energy stored in the cyclic loading process, and we use w to represent the ratio 2. Calculation model w ˆ DW W ; 2 The MMC studied here is particle-reinforced and it is ideal: the reinforcing particles are assumed to be uniformly dispersed in the matrix and there are no sliding and debonding occurring on the particle±matrix interfaces during the loading process. Thus, the nite element method combined with cell modelling can be employed to evaluate the plastic energy dissipation. The sketch of the cell which is axisymmetric is shown in Fig. 1. Because of its symmetry, only one quarter of the cell is taken for calculation. The load is applied along the axis Z. Owing to its high yield stress, the particle will keep elastic as for real MMCs during the whole loading process. The matrix is assumed to be elasto-plastic and obeys the von Mises yield criterion. When the matrix yields, the isotropic power hardening model is assumed to be valid and the ow stress is written as follows [12], where DW can be written as I DW ˆ R de; 3 which is the area surrounded by the cyclic stress± strain curve. W is usually expressed as follows, W ˆ Z e 0 R de; 4 where e 0 is the strain maximum during the loading process (as shown in Fig. 2) while R and E are the macroscopic average stress and strain of the cell (shown in Fig. 1). The loading process is quasistatic because what we study here is elasto-plastic deformation. r f ˆ r 0 H e p n ; 1 where r 0 is the initial yield stress of the matrix, H the coe cient of hardening, e p the von Mises Fig. 1. Sketch of the cell model for the calculation of energy dissipation. Fig. 2. Sketch of the calculation for DW.

3 98 D. Xu, S. Schmauder / Computational Materials Science 15 (1999) 96± Numerical results and discussion In our calculation, we take Al as the matrix and SiC as the reinforcing particles. The parameters of these two kinds of material are listed below [6,12]: E m ˆ 69 GPa; m m ˆ 0:3; r 0 ˆ 43 MPa; n ˆ 0:3; H ˆ 137:2 MPa; E r ˆ 440 GPa; m r ˆ 0:25; where E is the Young's modulus and m the Poisson's ratio. The subscript m and r stand for matrix and reinforcing particles, respectively. In the following, the in uences of loading amplitude, mechanical properties and shape of particles as well as number of cyclic loadings on the ratio w are discussed Loading amplitude Three kinds of displacement loading are taken to apply on the composite cell and their strain amplitudes are , and , respectively. The results are shown in Fig. 3, which is a plot showing the relation of w and the reinforcement volume fraction. The ratio w is seen to increase with the volume fraction of SiC and it is very sensitive to the loading amplitude. We also noticed that when the strain amplitude is very small, say in our calculation model, there is almost no energy dissipation. This phenomenon suggests that there exists a threshold for the occurrence of energy dissipation in MMCs. Only when the loading crosses the threshold, energy dissipation will take place Modulus of particles If there were no reinforcing particles in the metal, the material would become pure Al and energy dissipation would not occur under the loading level below the yield stress as chosen in our calculation. It is reasonable to suppose that it is the di erence of mechanical properties existing between reinforcing particles and matrix that makes energy dissipation due to plastic deformation in microstructure possible. Consequently, we can also suppose that the ratio w will change if we change the SiC particle to another kind of hard particle. First, we change SiC to Al 2 O 3 and then to TiC. The results are shown in Fig. 4. The Young's modulus of Al 2 O 3 is taken as E ˆ 380 GPa and Fig. 3. The energy dissipation increases with the loading amplitude. Fig. 4. The in uence of the reinforcing particles' elastic modulus and volume fraction on the ratio w.

4 D. Xu, S. Schmauder / Computational Materials Science 15 (1999) 96± Fig. 5. Sketch of the cell models of spherical, ``double cone without tips'' particle and circular cylinder particles. that of TiC is E ˆ 300 GPa. Fig. 4 suggests that the higher Young's modulus induces higher w-values Particle shape Three kinds of shapes of the particles are considered here and their schematic sketch is presented in Fig. 5. The results with respect to w are shown in Fig. 6. From Fig. 6, it can be seen that the double cones without tips' particle produces the highest ratio w because `stress concentrations' appear at all corners of these kinds of particles Number of cycles The values of the above w are obtained in the rst cycle. In the following, we analyse whether the value of w alters with the number of cycles. Fig. 7 shows the results of two kinds of materials whose volume fractions of reinforcement are 20% (solid curve) and 40% (dotted curve), respectively, and the load is a four cycle load. The strain amplitude is We can only distinguish the rst cycle and the second one in this guration which means that the results of Fig. 6. The in uence of the particle's shape on the ratio w. Fig. 7. In uence of loading cycles and particle volume fraction on ratio w.

5 100 D. Xu, S. Schmauder / Computational Materials Science 15 (1999) 96±100 Table 1 The value of w in the rst four cycles Volume fraction (%) Cycle 1 Cycle 2 Cycle 3 Cycle cycle 3 and cycle 4 are almost the same. The values of w in this group of calculations are listed in Table 1. From Table 1, we can notice that the value of the ratio does not change much from cycle 1 to cycle 4 and becomes steady after 4 cycles of loading. 4. Conclusion 1. The ratio w is sensitive to the loading level. There exists a threshold of loading level above which the higher loading level and the higher volume fraction of reinforcements will generate a higher ratio of dissipated energy w. 2. The mechanical properties and the volume fraction of reinforcing particles can a ect the ratio w: a higher ratio w can be induced by higher Young's modulus and higher volume fractions of reinforcing particles. 3. The shape of particles can a ect the ratio w as well, but the in uence is not as signi cant as that of loading level and Young's modulus of reinforcing particles. 4. The energy dissipation reaches its steady state value after several (typically 4) cycles. The differences between the value of ratio w in the rst cycle and its steady state value is negligibly small. References [1] C.W. Bert, Composite Material: A survey of the damping capacity of ber-reinforced composites, Damping Application in Vibration Control, AMD-Vol. 38, American Society of Mechanical Engineers, New York, 1978, pp. 53± 63. [2] R.F. Gibson, P. Plumkett, Dynamic mechanical behaviour of ber-reinforced composites: Measurement and analysis, J. Comp. Mater. 10 (1976) 325±341. [3] S.A. Suarez, R.F. Gibson, Improved impulse frequency response techniques for measurement of dynamic mechanical properties of composite materials, Exp. Tech. 8 (1984) 19±25. [4] A. Wolfenden, C.K. Frisby, K.J. Heritage, S.S. Vinson, R.C. Knight, Internal friction and dynamic modulus of metal matrix composites and advanced alloys, J. de Physique C8 (1978) 337. [5] T. Mori, M. Koda, R. Monzen, Particle blocking in grain boundary sliding and associated internal friction, Acta Metall. 31 (1983) 275±283. [6] J. Zhang, R.J. Perez, C.R. Wong, E.J. Lavernia, E ects of secondary phases on the damping behavior of metals, alloys and metal matrix composites, Mater. Sci. Eng. R13 (1994) 325±390. [7] A. Wolfenden, Internal friction in nickel near the Curie temperature, Script. Metall. 12 (1978) 103±109. [8] R. De Batist, Mechanical energy dissipation related with martensitic transformation process, M3D: mechanics and mechanisms of material damping, in: V.K. Kinra, A. Wolfenden (Eds.), American Society for Testing and Materials, Philadelphia, 1992, pp. 45±59. [9] A.A. Baker, The fatigue of ber reinforced aluminium, J. Mater. Sci. 3 (1968) 412±423. [10] J.R. Brockenbrough, S. Suresh, H.A. Wienecke, Deformation of metal matrix composites with continuous bers: Geometrical e ects of ber distribution and shape, Acta Metall. Mater. 39 (1991) 735±751. [11] M. Dong, S. Schmauder, Modelling of metal matrix composites by a self-consistent embedded cell model, Acta Mater. 44 (1996) 2465±2478. [12] J. Weizhi, Z. Shixi, The Mechanical Properties of Engineering Materials, Aerospace and Aeronautics University, Beijing, 1991 (in Chinese).