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Stress–strain behavior in initial yield stage of short fiber reinforced metal matrix composite
Composites Science and Technology 62 (2002) 841–850 www.elsevier.com/locate/compscitech
Stress–strain behavior in initial yield stage of short fiber reinforced metal matrix composite X.D. Dinga,*, Z.H. Jiangb, J. Suna, J.S. Lianb, L. Xiaoa a
State Key Laboratory for Mechanical Behavior of Materials, School of Material Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China b The Key Laboratory of Automobile Materials, Ministry of Education, China, Jilin University, Changchun, 130025, PR China Received 18 May 2001; received in revised form 4 January 2002; accepted 11 January 2002
Abstract Based on the law of mixture and the large strain axisymmetric elasto-plastic finite element method, the analytical expressions for the stress strain response of short fiber reinforced metal matrix composite were derived and a new method of defining the yield behavior of short fiber reinforced metal matrix composite was proposed. The effects of the material parameters (fiber volume fraction, fiber aspect ratio, fiber end distance and matrix strain hardening coefficient) on the deformation behavior of the composite were also investigated. It was demonstrated that there is a close relationship between the stress strain partition parameter and the deformation behavior of the composite. The effect of the material parameters on the initial yield behavior can be revealed well by this method. The predicted elastic modulus and yield stress are in good agreement with the experiments. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Metal matrix composites; FEM
1. Introduction In recent years, there has been considerable interest in short fiber-reinforced metal matrix composites (MMCs) since they exhibit increased stiffness, strengths and good elevated temperature properties relative to the matrix material. However, since MMCs are microstructurally inhomogeneous, it is important to understand the consequences of such incompatibility in terms of the resultant stress–strain response of the system as a whole. For example, there is little advantage in reinforcing a matrix with the aim of improving its elastic stiffness if the strain range over which the composite behaves elastically becomes very small. With regard to the level of homogeneity, it can easily be shown that when a load is applied to a composite containing a stiff reinforcing fiber, high local stress gradients are generated in the matrix around the fiber [1–6]. It is thus not surprising that MMCs characteristically
* Corresponding author. E-mail address: [email protected] (X.D. Ding).
exhibit small deviations from ideal elastic behavior at strains far smaller than those required to produce bulk plastic flow in unreinforced matrix alloys. For a given system, this deviation will in turn influence the effective engineering modulus and the stress range over which it can be used practically without introducing unacceptable levels of non-recoverable deformation. In order to optimize composite mechanical properties, an accurate and reliable method is therefore desired to determine the overall composite properties pertain to the yield process, such as elastic modulus, proportion limit and yield strengths. It can be noted that in spite of considerable attempts to investigate the deformation and properties of MMCs, the investigations about the stress–strain behavior in initial yield stage are quite limited [7–11]. Historically, the difficulties involved in the definition of a yield point are well exemplified by the widespread adoption of a ‘‘proof stress’’ measured at a somewhat arbitrary plastic strain offset. The use of such a definition causes considerable confusion in MMCs, which have differing degree of inhomogeneity in their early flow behaviors. The problem lies in the fact that MMCs
0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00024-6
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exhibit a progressive transition from local plastic flow to bulk flow [1–6]. It is show that the demarcation between elastic and plastic behavior defined by the traditionally used 0.2% proof yield stress will lost the insight of flow behavior of MMCs in their initial yield and subsequence yield behaviors [9–11]. By comparison of the monotonic stress–strain curve and the corresponding load/unload stress–strain curve, Corbin et al. [7] defined the yield strength (or proportional limit) of particle reinforced metal matrix composite (PRMMC) as the stress at which a permanent plastic strain of 5105 is reached. Zhang et al. [8] use the stress at plastic strain relaxation of about 105 to denote elastic limit of PRMMC. By correlating the stress–strain response during the early stages of loading with evolution of matrix microstructure observed using TEM, Prangnell et al. [9] evaluated the yield behavior of MMCs. They found that the initial yield stress can be better defined in an inhomogenieous metal matrix material by an analysis of the first, second and third differentials of the initial stress–strain response than by the traditionally used 0.2% proof stress. Daymond and Withers [10] confirmed that the initial yield stress defined by Prangnell et al. [9] is really sensitive to the degree of inhomogeneity of the local stress by using finite element method (FEM). Shi et al. [11] investigated the matrix plastic flow in a whisker reinforced metal matrix composite under applied tensile load by using FEM, they found that the overall matrix plastic flow can be divided into several characteristic stages, and related to the global stress–strain relationship. It can be noted that the previous investigations have usually studied the yield behavior by examining the variation of the experimental stress–strain curves or by establishing some connection of these behaviors with the local stress field distributions in the matrix and the fiber using FEM. However, there is a lake of a theoretical connection between the deformation of the matrix and the fiber. Furthermore, because of the very rounded nature of the stress–strain curves of MMCs in the vicinity of yield, it is still very difficult and time consuming to determine the actual onset of plastic flow by experiment and finite element method. Therefore, in the present study, large strain axisymmetric elasto-plastic finite element method and the law of mixture were used to study the stress strain response of MMCs in initial yield stage under tensile loading. The analytical expressions for the stress strain response of the composite were derived and the method for determining the yield properties, i.e. proportion limit and yield strengths was proposed according to the variation of the stress–strain partition parameter. The effects of material parameters (fiber aspect ratio, fiber volume fraction, fiber end distance and matrix strain hardening coefficient) on the yield properties were also analyzed.
2. Analytical modeling 2.1. FEM model The modeled composite was assumed to consist of an array of hexagonal prism-shaped unit cells, each contains a longitudinally alighted fiber. In the finite element model, the unit cell and the fiber were approximated, respectively, as a cylinder of radius RC and length LC and a cylinder of radius of RF and length LF . The fiber– matrix interface bounding was assumed to be perfect. A schematic of the model unit cell is shown in Fig. 1 (a)– (b). The aspect ratios of the fiber and the unit cell are defined as: AF ¼ LF =RF , AC ¼ LC =RC , the fiber volume fraction and the fiber end distance are written as: VF ¼ ðR2F LF Þ=ðR2C LC Þ and K ¼ LC =LF . Fig. 1(d) illustrates the variation of the end spacing between the fibers with K for VF given and AF. Because of symmetry, only a quadrant of the unit cell [the positive quadrant of the RZ plane, see Fig. 1(c)] was considered. In the finite element analysis, a two dimensional axisymmetric approach was applied. The composite was loaded by an incremental application of a uniform displacement to the Z ¼ LC plane of the cell. To fulfil displacement continuity and symmetry requirements, the following boundary conditions were imposed [see Fig. 1(c)] [1,12] UZ ¼ 0 on Z ¼ 0 and uZ ¼ U on Z ¼ LC
ð1aÞ
UR ¼ 0 on R ¼ 0 and UR ¼ U on R ¼ RC
ð1bÞ
TR ¼ 0 on Z ¼ 0 and Z ¼ LC ; R 6¼ RC
ð1cÞ
TZ ¼ 0 on R ¼ 0 and R ¼ RC ; Z 6¼ LC
ð1dÞ
where U is a prescribed displacement, UR and UZ stand for the displacements in the and directions, respectively, TR and TZ are the applied tractions in the same directions and U is determined from the condition ð LC TR dZ ¼ 0
on
R ¼ RC
ð2Þ
0
The updated-Lagrangian–Jaumann (ULJ) formulation and the J2 F flow theory of elasto-plastic were used in the finite element analysis. In the ULJ formulation, the mean normal technique was used for the large strain elasto-plastic finite element solutions [13]. The numerical models were constructed using a four-crossed-triangle element [14] [see Fig. 1(c)]. The total numbers of the rectangle elements and the nodes are 980 and 2045, respectively. In this analysis, the fiber is assumed as a linear elastic material and the matrix is approximated as a linear elastic, plastic material exhibiting isotropic power law
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Fig. 1. (a) Schematic illustration of the fibers arrangement in the composite, (b) a circular unit cell representative, (c) FEM meshes in the unit cell, (d) the representative of the unit cell at different K values for VF=0.184 and AF=5.
hardening and obeying the Von Mises yield criterion. The material data were taken from a typical SiC whisker reinforced aluminum matrix composite (SiCw/ 6061Al T6) [2,4,23], i.e. the Young’s moduli and the Poisson’s ratios of the matrix and the fiber are EM =74 GPa, EF =483 GPa, M =0.33 and F =0.17; the yield strength and the strain hardening ratio of the matrix are MY =357 MPa and n ¼ 0:20: Furthermore, the effects of thermal residual stress were not considered in the present analysis, which will be reported elsewhere. 2.2. The law of mixture The law of mixture has been applied to describe the tensile stress–strain behaviors of dual-phase materials [15–18] and metal–ceramic graded composites [19–22]. For the composite studied, the law of mixture can be written as C ¼ ð1 VF ÞM þ VF F
ð3Þ
"C ¼ ð1 VF Þ"M þ VF "F
ð4Þ
F ¼
M ¼
NF 1 X F VFJ VFT J¼1 J NM 1 X M J VMJ
VMT
M F "M "F
where FJ ; M J are the average axis stresses obtained from the Gaussian integration points within each finite element of the fiber and the matrix, respectively; VFJ ; VMJ ; NF and NM are the volume and number of elements of the fiber and the matrix, respectively; VFT ; VMT are the volumes of the fiber and the matrix, respectively. The expressions for "F and "M have similar forms to those for F and M . The meanings of the law of mixture and the stress– strain partition parameter are shown in Fig. 2. Obviously, and correspond to the equal stress and equal
ð5Þ
In this analysis, the stress–strain relations of the matrix and the fiber can be calculated using a volume average approach as follows
ð6bÞ
J¼1
where C ; "C ; M ; "M ; F and "F are the axis stresses and strains of the composite, the matrix and the fiber, respectively. The relationship between the stresses and strains of the matrix and the fiber can be described by a parameter called generally as the stress–strain partition parameter q¼
ð6aÞ
Fig. 2. Schematics of the law of mixture.
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strain states respectively, 0 4 q 4 1 corresponds to a deformation state between the equal stress and equal strain states. The quantity F M can be defined as the load transfer stress TR ; which is a measurement of the stress transfer from the matrix to the fiber. The meaning of TR is also shown in Fig. 2. Combing Eqs. (3)–(5), the relationship between TR and can be expressed as TR ¼ qð"M "F Þ
ð8Þ
3. Dependence of stress–strain curves on q The q "C curves were calculated by using FEM and typical q "C curves are shown in Fig. 3. It can be seen that the q "C curves exhibit an initial plateau region at lower strain, then the q value decreases with increasing "C : The initial plateau region or the constant q value at lower strain reveals that the deformations of both matrix and fiber, hence the composite are elastic. This can be derived if assuming that C ¼ EC "C ; M ¼ EM "M and F ¼ EF "F ; where EC is the elastic modulus of the composite. Substituting these relations into Eqs. (3)–(5), we can obtain ð1 VF ÞEF EC þ VF EM EC EM EF q¼ ð1 VF ÞEM þ VF EF EC
ð10Þ
EF q EM M C ¼ ð1 VF Þ þ VF EM q EF
ð11Þ
ð7Þ
Thus, the composite stress can be given by C ¼ M þ VM TR ¼ M qVM ð"M "F Þ
1 q EF þ VF EC ¼ ð1 VF Þ q EM q EF VF EM þ VF EF q EM
ð9Þ
This equation indicates that q should be a constant independence of EC if both matrix and fiber deform elastically. Furthermore, two relations of EC and C to q can be obtained from Eqs. (3)–(5) and (9)
If we adopt the traditional assumption [2,15,16] that the composite yields when the average matrix stress reaches the yield strength of the matrix material MY ; i.e. M ¼ MY ; the yield strength of the composite CY can be expressed as EF q EM ð12Þ MY CY ¼ ð1 VF Þ þ VF EM q EF It should be pointed out that CY given by Eq. (12) should be an upper limit value of the yield strength of the composite. In fact, for a composite reinforced by an elastic fiber, the plastic yielding would occur in the immediate vicinity regions around the fiber at an applied stress below the yield strength of the matrix material MY : This effect has been reported in the previous investigations [1–4, 25, 26]. Thus, CY can be reexpressed as EF q EM 0 ð13Þ CY ¼ ð1 VF Þ þ VF MY EM q EF where s0 MY is the average matrix stress corresponding to the initial yield stress of the composite, which will be given by the following finite element calculation. At higher strain stage, the plastic deformation of the matrix will result in the decrease in the stress transfer effect. This has been implied by the decreased q values with increasing "C : If we assume that the deformation of the matrix follows the power law, i.e. M ¼ l"nM ; then, the elasto-plastic tangent modulus of the composite HC can be derived as dC ¼ HC ¼ d"C " n1 # VF lnð"C =ð1 VF ÞÞn1 q "C EF þ ln EF q 1 VF 1 VF ð14Þ
Fig. 3. The effect of the material parameters on q value.
Detailed derivations for this expression are given in the Appendix. This expression gives the dependence of the composite plastic response on q for given material parameters. The above equations indicate that q is an important parameter, the overall deformation behavior of the composite can be well described if one knows its varia-
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845
tion. In addition, it can be seen from Fig. 3 that the q "C curves depend greatly on the material parameters, such as AF ; VF ; k and n, although they do not change the shapes of the q "C curves. Among the four parameters, the effects of AF and VF are significant, while those of k and n are relatively small. The calculated stress–strain curves by the above expressions are shown in Fig. 4(a) for several typical cases. It is shown that the effects of AF ; k and n on C are consistent with their effects on q, but the effect of VF is opposite. This opposite effect can be explained well by Eq. (8). Although q decrease with increasing VF ; the contribution of TR to C ðVF TR Þ increases with increasing VF : As a result, C increase with increasing VF . The stress–strain curves of the composite calculated by FEM were compared with those from Eqs. (10), (13) and (14) by using the corresponding q values in Fig. 3. The results are shown in Fig. 4(b). It can be seen that the stress–strain curves predicted by Eqs. (10), (13) and (14) are in good agreement with those by FEM.
4. Transition of deformation behavior It can be clearly seen from Eq. (5) that q is very sensitive to the changes in the stress–strain relations of the matrix and the fiber. It should be possible to reveal the yield behaviors of the composite by examining
the varia tion of q and its second order derivatives, q00 ¼ d 2 q=d"2C : 00 Typical q "C curves and the corresponding C "C and q "C curves are shown in Fig. 5(a). In the calculations, to keep the precision and avoid the numerical noise of the calculation data, the strain range of the neighboring data points was kept as about 0.00004. Furthermore, a method of five data point smoothing was used to smooth the curves. The first and second order derivatives of were then calculated from the linear gradient between the neighboring data points of the smoothed curves. It should be noted that the choice in the number of data points for the smoothing influence the absolute values of the derivatives, but it does not change the trends we report here. It can be seen in Fig. 5(a) that there are two distinct points (points A and B) in the q "C curves. q keeps constant before point A, then declines. q decreases gradually after point B. From the q00 "C curves, it can be seen that there are two distinct extremum points corresponding to points A and B, respectively. Between points A and B, the q00 "C curves show sharper variation. After point C, q00 keeps nearly constant. To establish the connection of the above stress–strain behaviors with the development of the matrix plasticity, the contours of the Von Mises effective stresses in the matrix (regions E and F) and the fiber (region D) were calculated and are shown in Fig. 6 at the corresponding points A, B and C. As shown in Figs. 5(a) and 6, at
Fig. 4. (a) The effect of the material parameters on the stress–strain curves of the composite; (b) comparisons of the calculated stress– strain curves by Eqs. (10), (13) and (14) with those by FEM.
point A, the plastic region has formed in smaller region near the fiber end face (region F), the matrix in the fiber region (region E) is still elastic. At point B, most regions in the fiber end have entered the plastic state. In the strain stage between points A and B, the strain-hardening rate of the composite (the slope of the stress– strain curve) becomes small relative to that in strain stage before point A. At point C, the plastic deformation has occurred in the entire matrix including the fiber region and the fiber end region, the strain-hardening rate of the composite becomes markedly small. According to the results in Figs. 5(a) and 6, the composite properties in the initial yield stage can be well determined. It can be thought that point A is a strain point before which the overall deformation behavior is basically elastic although there exists very small local plastic region in the matrix near the fiber end face. This argument is supported by the results in Fig. 5(a) that the slope of the stress–strain curve keeps basically constant before point A. Therefore, phenomenally, the strain corresponding point A can be defined as the maximum elastic strain of the composite and the stress at this point can be defined as the proportional limit of the composite. The elastic modulus of the composite can be thus determined as the average value of the slope of the
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Fig. 5. The variations of the C "C , q "C and q00 "C curves with "C .
stress–strain curve before point A. Since significant plastic deformation has occurred in the fiber end region at point B, the stress at point B can be considered as the initial yield strength. Furthermore, the stress at point C can be defined as the final yield strength because the plastic deformation has occurred in the entire matrix. The effects of VF ; AF and K on the maximum elastic strain, the initial and final yield strains are shown in Fig. 5 (b)–(d). The corresponding matrix stresses and strains, i.e. the stress and strain at the points A1, B1 and C1 are also given in Fig. 5 (b)-(d), where the values of corresponding to points A, B and C are represented by 0 the dash lines. The value of MY in Eq. (13) can be given by the stress at point B1. As shown, the stresses and strains in the matrix, corresponding to the proportional limit and the initial yield stress, are smaller than the yield stress and strain of the matrix material (357 MPa and 0.00482). But the stress and strain in the matrix, corresponding to final yield stress, are larger than those of the matrix material. It can also be seen that the proportional limit decreases with increasing VF and increases with increasing AF and K. The initial yield stress decreases with increasing AF and VF and increases with
increasing K. The final yield stress increases with increasing AF ; VF and K. Fig. 6 also shows the contours of the Von Mises effective stress in the matrix and the fiber corresponding to points A, B and C defined by Fig. 5(b)–(d). From the comparison of Fig. 5(b)–(d) with Fig. 6, it can be seen that the plastic deformation in the fiber end region just begins at point A for all cases with different material parameters. At meantime, the matrix in the fiber end regions deforms plastically mainly at point B and the entire matrix deforms plastically at point C. The results above revealed that the present method can describe the yield process of the composite regardless the variations of the material parameters. Table 1 gives the comparisons of the elastic modulus and the proportion limit calculated by FEM with the published experiment data [6,23,24]. As shown, the calculated values are in good agreement with the experimental values. Fig. 7 shows the variations of the calculated proportion limit ðCP Þ; initial yield stress ðCIY Þ and the final yield stress ðCFY Þ with AF and VF for given k and n values. k and n were taken as K ¼ LC =LF ¼ RC =RF , n=0.2. Moreover, the published experiment data of the
X.D. Ding et al. / Composites Science and Technology 62 (2002) 841–850
Fig. 6. Contours of the Von Mises effective stress in the matrix and the fiber.
Fig. 7. The variations of the composite proportion limit and yield strength with AF and VF.
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Table 1 Comparison between the experimental and calculated proportion limit Material
VF
AF
4 1 4 1 4 2 2 2
EC (GPa)
CP (GPa)
Exp.
FEM
Exp.
120 81 88 106 119 81 102 114
119 80 90 104 119 80 98 116
348 193 220 241 269 – – –
FEM
SiCW/Al 6061 T6 [23] SiCd/Al 5456 [6] SiCd/Al 5456 SiCd/Al 5456 SiCd/Al 5456 SiCW/Al 6061 T6 [24] SiCW/Al 6061 T6 SiCW/Al 6061 T6
0.20 0.08 0.08 0.20 0.20 0.10 0.20 0.30
356 197 224 228 272 – – –
Al 6061 T6 Al 5456 Al 6061 T6
EM=74 GPa EF=483 GPa M =357 MPa EM=73 GPa EF=485 GPa M =241 MPa EM=68.8GPa EF=450 GPa M =280 MPa
yield stress ð0:2 Þ of SiCw/6061Al composites are also given in Fig. 7(b). As shown, CIY and CFY increase with increasing of AF and VF : The variation tendencies of CIY and CFY are similar to that of the yield stress CY calculated by Eq. (12). CY is higher than CIY ; but lower than CFY : This is consistent with the above argument that CY is an upper limit value of the yield strength of the composite. CIY is slightly higher than 0.2, but has the same variation tendency as 0.2. All of these comparisons verified the validity of the method proposed in the present study. Furthermore, it can be seen from Fig. 7 that the variation of CP is complex. CP first decreases and then increases with increasing AF and VF : This result is consistent with the results in works [8,28], in which a similar relationship between the proportional limit and the volume fraction of reinforcing particles were observed. This behavior results from two competitive effects of the local stress concentrations and the local plastic constraints introduced by the fibers. It is known that the load is transferred from the matrix to the fiber by the interface along the fiber and the fiber end face. As a consequence of this load transfer, the stress concentrations arise in the local region around the fiber and at the region near the fiber end. These matrix regions experience higher local stresses and yield first. Thus the proportional limit is lowered. Simultaneously, the presence of the elastic fiber restrains the matrix from plastic flow and tends to raise the proportional limit and initial yield stress. That is, there are competitive effects between local stress concentrations and local plastic constraints. The former is predominating when AF and VF are smaller and the latter become more determinative as AF and VF increase.
5. Conclusions (1) The analytical expressions for the stress strain response of the composite have been derived
based on the law of mixture. The expressions have predicted well the entire stress–strain curves of the composite with the given variations in the stress–strain partition parameter q by a large strain axisymmetric elasto-plastic finite element method. (2) A new method has been proposed to define the yield responses of the composite by examining the variations in and its second order derivations. The proportion limit, the elastic modulus, the initial yield stress and the final yield stress can be predicted well by using this method. (3) With the increases of the aspect ratio and the volume fraction, the proportion limit first decreases and then increases. This behavior is thought to be a consequence of the competitive effects of local stress concentrations and local plastic constraints due to the presence of the fiber. (4) The predicted elastic modulus and proportion limit by the derived analytical expressions and the finite element method exhibit good agreement with the published experimental results.
Acknowledgements This work was supported by the National Outstanding Young Investigator Grant of China. The financial support from National Natural Science Foundation of China has been also gratefully acknowledged. Valuable comments and kindly suggestions from the reviewer are sincerely appreciated too.
Appendix For the case that the fiber is elastic and the matrix is plastic, the differential of the composite stress C gives dC ¼
@C @F @C @M d"F þ d"M @F @"F @M @"M
ðA1Þ
The stress–strain relation of the matrix is characterized by ¼ EM "M 4 MY ðA2Þ ¼ l"nM > MY where EM ; l and are Young’s modulus, the strength constant and strain hardening exponent of the matrix, respectively. While the fiber is elastic, Hooke’s law can be used F ¼ EF "F
ðA3Þ
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849
where EF are Young’s modulus of the fiber. The differentials of (A2) and (A3) give
This is an explicit expression for the elasto-plastic tangent modulus of the composite.
@M ¼ M;M ¼ ln"n1 M @"M
ðA4Þ
References
@F ¼ F;F ¼ EF @"F
ðA5Þ
According to the law of mixture C ¼ ð1 VF ÞM þ VF F
ðA6Þ
"C ¼ ð1 VF Þ"M þ VF "F
ðA7Þ
we have @C ¼ 1 VF @M
ðA8Þ
@C ¼ VF @F
ðA9Þ
Introduction of Eqs. (A4)–(A5) and (A8)–(A9) into Eq. (A1) leads to dC ¼ EF VF d"F þ ð1 VF Þln"n1 M d"M
ðA10Þ
The definition of can be expressed as q¼
M F "M "F
ðA11Þ
The derivative of Eq. (A10) gives d"F ¼
ln"n1 M q d"M Eq
Combining Eqs. (A9) and (A11) gives ln"n1 n1 M q þ ð1 VF Þln"M d"M dC ¼ EF VF Eq
ðA12Þ
ðA13Þ
Since the fiber is elastic, i.e. "F 0; thus Eq. (A6) can be approximately written as "C ¼ ð1 VF Þ"M
ðA14Þ
So we have from Eqs. (A12) and (A13) " n1 # dC VF lnð"C =ð1VF ÞÞn1 q "C EF þln ¼ EF q d"C 1VF 1VF ðA15Þ
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[24] McDanels DL. Analysis of stress–strain, fracture and ductility behavior of aluminum matrix composites containing discontinuous silicon carbide reinforcements. Metall Trans A 1985; 16(6):1105–15. [25] Hutchinson JW. Elastic-plastic behavior of polycrystalline metals and composites. Proc Roy Soc Lond 1970;A319:247–72. [26] Berveiller M, Zaoui A. An extension of the self-consistent scheme to plastically-flowing polycrystals. J Mech Phys Solids 1978;26: 325–44. [27] Watanabe H, Ohori K, Takeuchi Y. Effects of hot extrusion and rolling on the tensile strength of SIC whisker reinforced aluminum alloy composites. Jpn Inst Light Metals 1988;38(10): 633–9. [28] Papazian JM, Adler PN. Tensile Properties of short fiber-reinforced SiC/Al composites: part I. Effects of matrix precipitates. Metall Trans A 1990;21(2):401–10.
Stress–strain behavior in initial yield stage of short fiber reinforced metal matrix composite X.D. Dinga,*, Z.H. Jiangb, J. Suna, J.S. Lianb, L. Xiaoa a
State Key Laboratory for Mechanical Behavior of Materials, School of Material Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China b The Key Laboratory of Automobile Materials, Ministry of Education, China, Jilin University, Changchun, 130025, PR China Received 18 May 2001; received in revised form 4 January 2002; accepted 11 January 2002
Abstract Based on the law of mixture and the large strain axisymmetric elasto-plastic finite element method, the analytical expressions for the stress strain response of short fiber reinforced metal matrix composite were derived and a new method of defining the yield behavior of short fiber reinforced metal matrix composite was proposed. The effects of the material parameters (fiber volume fraction, fiber aspect ratio, fiber end distance and matrix strain hardening coefficient) on the deformation behavior of the composite were also investigated. It was demonstrated that there is a close relationship between the stress strain partition parameter and the deformation behavior of the composite. The effect of the material parameters on the initial yield behavior can be revealed well by this method. The predicted elastic modulus and yield stress are in good agreement with the experiments. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Metal matrix composites; FEM
1. Introduction In recent years, there has been considerable interest in short fiber-reinforced metal matrix composites (MMCs) since they exhibit increased stiffness, strengths and good elevated temperature properties relative to the matrix material. However, since MMCs are microstructurally inhomogeneous, it is important to understand the consequences of such incompatibility in terms of the resultant stress–strain response of the system as a whole. For example, there is little advantage in reinforcing a matrix with the aim of improving its elastic stiffness if the strain range over which the composite behaves elastically becomes very small. With regard to the level of homogeneity, it can easily be shown that when a load is applied to a composite containing a stiff reinforcing fiber, high local stress gradients are generated in the matrix around the fiber [1–6]. It is thus not surprising that MMCs characteristically
* Corresponding author. E-mail address: [email protected] (X.D. Ding).
exhibit small deviations from ideal elastic behavior at strains far smaller than those required to produce bulk plastic flow in unreinforced matrix alloys. For a given system, this deviation will in turn influence the effective engineering modulus and the stress range over which it can be used practically without introducing unacceptable levels of non-recoverable deformation. In order to optimize composite mechanical properties, an accurate and reliable method is therefore desired to determine the overall composite properties pertain to the yield process, such as elastic modulus, proportion limit and yield strengths. It can be noted that in spite of considerable attempts to investigate the deformation and properties of MMCs, the investigations about the stress–strain behavior in initial yield stage are quite limited [7–11]. Historically, the difficulties involved in the definition of a yield point are well exemplified by the widespread adoption of a ‘‘proof stress’’ measured at a somewhat arbitrary plastic strain offset. The use of such a definition causes considerable confusion in MMCs, which have differing degree of inhomogeneity in their early flow behaviors. The problem lies in the fact that MMCs
0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00024-6
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exhibit a progressive transition from local plastic flow to bulk flow [1–6]. It is show that the demarcation between elastic and plastic behavior defined by the traditionally used 0.2% proof yield stress will lost the insight of flow behavior of MMCs in their initial yield and subsequence yield behaviors [9–11]. By comparison of the monotonic stress–strain curve and the corresponding load/unload stress–strain curve, Corbin et al. [7] defined the yield strength (or proportional limit) of particle reinforced metal matrix composite (PRMMC) as the stress at which a permanent plastic strain of 5105 is reached. Zhang et al. [8] use the stress at plastic strain relaxation of about 105 to denote elastic limit of PRMMC. By correlating the stress–strain response during the early stages of loading with evolution of matrix microstructure observed using TEM, Prangnell et al. [9] evaluated the yield behavior of MMCs. They found that the initial yield stress can be better defined in an inhomogenieous metal matrix material by an analysis of the first, second and third differentials of the initial stress–strain response than by the traditionally used 0.2% proof stress. Daymond and Withers [10] confirmed that the initial yield stress defined by Prangnell et al. [9] is really sensitive to the degree of inhomogeneity of the local stress by using finite element method (FEM). Shi et al. [11] investigated the matrix plastic flow in a whisker reinforced metal matrix composite under applied tensile load by using FEM, they found that the overall matrix plastic flow can be divided into several characteristic stages, and related to the global stress–strain relationship. It can be noted that the previous investigations have usually studied the yield behavior by examining the variation of the experimental stress–strain curves or by establishing some connection of these behaviors with the local stress field distributions in the matrix and the fiber using FEM. However, there is a lake of a theoretical connection between the deformation of the matrix and the fiber. Furthermore, because of the very rounded nature of the stress–strain curves of MMCs in the vicinity of yield, it is still very difficult and time consuming to determine the actual onset of plastic flow by experiment and finite element method. Therefore, in the present study, large strain axisymmetric elasto-plastic finite element method and the law of mixture were used to study the stress strain response of MMCs in initial yield stage under tensile loading. The analytical expressions for the stress strain response of the composite were derived and the method for determining the yield properties, i.e. proportion limit and yield strengths was proposed according to the variation of the stress–strain partition parameter. The effects of material parameters (fiber aspect ratio, fiber volume fraction, fiber end distance and matrix strain hardening coefficient) on the yield properties were also analyzed.
2. Analytical modeling 2.1. FEM model The modeled composite was assumed to consist of an array of hexagonal prism-shaped unit cells, each contains a longitudinally alighted fiber. In the finite element model, the unit cell and the fiber were approximated, respectively, as a cylinder of radius RC and length LC and a cylinder of radius of RF and length LF . The fiber– matrix interface bounding was assumed to be perfect. A schematic of the model unit cell is shown in Fig. 1 (a)– (b). The aspect ratios of the fiber and the unit cell are defined as: AF ¼ LF =RF , AC ¼ LC =RC , the fiber volume fraction and the fiber end distance are written as: VF ¼ ðR2F LF Þ=ðR2C LC Þ and K ¼ LC =LF . Fig. 1(d) illustrates the variation of the end spacing between the fibers with K for VF given and AF. Because of symmetry, only a quadrant of the unit cell [the positive quadrant of the RZ plane, see Fig. 1(c)] was considered. In the finite element analysis, a two dimensional axisymmetric approach was applied. The composite was loaded by an incremental application of a uniform displacement to the Z ¼ LC plane of the cell. To fulfil displacement continuity and symmetry requirements, the following boundary conditions were imposed [see Fig. 1(c)] [1,12] UZ ¼ 0 on Z ¼ 0 and uZ ¼ U on Z ¼ LC
ð1aÞ
UR ¼ 0 on R ¼ 0 and UR ¼ U on R ¼ RC
ð1bÞ
TR ¼ 0 on Z ¼ 0 and Z ¼ LC ; R 6¼ RC
ð1cÞ
TZ ¼ 0 on R ¼ 0 and R ¼ RC ; Z 6¼ LC
ð1dÞ
where U is a prescribed displacement, UR and UZ stand for the displacements in the and directions, respectively, TR and TZ are the applied tractions in the same directions and U is determined from the condition ð LC TR dZ ¼ 0
on
R ¼ RC
ð2Þ
0
The updated-Lagrangian–Jaumann (ULJ) formulation and the J2 F flow theory of elasto-plastic were used in the finite element analysis. In the ULJ formulation, the mean normal technique was used for the large strain elasto-plastic finite element solutions [13]. The numerical models were constructed using a four-crossed-triangle element [14] [see Fig. 1(c)]. The total numbers of the rectangle elements and the nodes are 980 and 2045, respectively. In this analysis, the fiber is assumed as a linear elastic material and the matrix is approximated as a linear elastic, plastic material exhibiting isotropic power law
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843
Fig. 1. (a) Schematic illustration of the fibers arrangement in the composite, (b) a circular unit cell representative, (c) FEM meshes in the unit cell, (d) the representative of the unit cell at different K values for VF=0.184 and AF=5.
hardening and obeying the Von Mises yield criterion. The material data were taken from a typical SiC whisker reinforced aluminum matrix composite (SiCw/ 6061Al T6) [2,4,23], i.e. the Young’s moduli and the Poisson’s ratios of the matrix and the fiber are EM =74 GPa, EF =483 GPa, M =0.33 and F =0.17; the yield strength and the strain hardening ratio of the matrix are MY =357 MPa and n ¼ 0:20: Furthermore, the effects of thermal residual stress were not considered in the present analysis, which will be reported elsewhere. 2.2. The law of mixture The law of mixture has been applied to describe the tensile stress–strain behaviors of dual-phase materials [15–18] and metal–ceramic graded composites [19–22]. For the composite studied, the law of mixture can be written as C ¼ ð1 VF ÞM þ VF F
ð3Þ
"C ¼ ð1 VF Þ"M þ VF "F
ð4Þ
F ¼
M ¼
NF 1 X F VFJ VFT J¼1 J NM 1 X M J VMJ
VMT
M F "M "F
where FJ ; M J are the average axis stresses obtained from the Gaussian integration points within each finite element of the fiber and the matrix, respectively; VFJ ; VMJ ; NF and NM are the volume and number of elements of the fiber and the matrix, respectively; VFT ; VMT are the volumes of the fiber and the matrix, respectively. The expressions for "F and "M have similar forms to those for F and M . The meanings of the law of mixture and the stress– strain partition parameter are shown in Fig. 2. Obviously, and correspond to the equal stress and equal
ð5Þ
In this analysis, the stress–strain relations of the matrix and the fiber can be calculated using a volume average approach as follows
ð6bÞ
J¼1
where C ; "C ; M ; "M ; F and "F are the axis stresses and strains of the composite, the matrix and the fiber, respectively. The relationship between the stresses and strains of the matrix and the fiber can be described by a parameter called generally as the stress–strain partition parameter q¼
ð6aÞ
Fig. 2. Schematics of the law of mixture.
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strain states respectively, 0 4 q 4 1 corresponds to a deformation state between the equal stress and equal strain states. The quantity F M can be defined as the load transfer stress TR ; which is a measurement of the stress transfer from the matrix to the fiber. The meaning of TR is also shown in Fig. 2. Combing Eqs. (3)–(5), the relationship between TR and can be expressed as TR ¼ qð"M "F Þ
ð8Þ
3. Dependence of stress–strain curves on q The q "C curves were calculated by using FEM and typical q "C curves are shown in Fig. 3. It can be seen that the q "C curves exhibit an initial plateau region at lower strain, then the q value decreases with increasing "C : The initial plateau region or the constant q value at lower strain reveals that the deformations of both matrix and fiber, hence the composite are elastic. This can be derived if assuming that C ¼ EC "C ; M ¼ EM "M and F ¼ EF "F ; where EC is the elastic modulus of the composite. Substituting these relations into Eqs. (3)–(5), we can obtain ð1 VF ÞEF EC þ VF EM EC EM EF q¼ ð1 VF ÞEM þ VF EF EC
ð10Þ
EF q EM M C ¼ ð1 VF Þ þ VF EM q EF
ð11Þ
ð7Þ
Thus, the composite stress can be given by C ¼ M þ VM TR ¼ M qVM ð"M "F Þ
1 q EF þ VF EC ¼ ð1 VF Þ q EM q EF VF EM þ VF EF q EM
ð9Þ
This equation indicates that q should be a constant independence of EC if both matrix and fiber deform elastically. Furthermore, two relations of EC and C to q can be obtained from Eqs. (3)–(5) and (9)
If we adopt the traditional assumption [2,15,16] that the composite yields when the average matrix stress reaches the yield strength of the matrix material MY ; i.e. M ¼ MY ; the yield strength of the composite CY can be expressed as EF q EM ð12Þ MY CY ¼ ð1 VF Þ þ VF EM q EF It should be pointed out that CY given by Eq. (12) should be an upper limit value of the yield strength of the composite. In fact, for a composite reinforced by an elastic fiber, the plastic yielding would occur in the immediate vicinity regions around the fiber at an applied stress below the yield strength of the matrix material MY : This effect has been reported in the previous investigations [1–4, 25, 26]. Thus, CY can be reexpressed as EF q EM 0 ð13Þ CY ¼ ð1 VF Þ þ VF MY EM q EF where s0 MY is the average matrix stress corresponding to the initial yield stress of the composite, which will be given by the following finite element calculation. At higher strain stage, the plastic deformation of the matrix will result in the decrease in the stress transfer effect. This has been implied by the decreased q values with increasing "C : If we assume that the deformation of the matrix follows the power law, i.e. M ¼ l"nM ; then, the elasto-plastic tangent modulus of the composite HC can be derived as dC ¼ HC ¼ d"C " n1 # VF lnð"C =ð1 VF ÞÞn1 q "C EF þ ln EF q 1 VF 1 VF ð14Þ
Fig. 3. The effect of the material parameters on q value.
Detailed derivations for this expression are given in the Appendix. This expression gives the dependence of the composite plastic response on q for given material parameters. The above equations indicate that q is an important parameter, the overall deformation behavior of the composite can be well described if one knows its varia-
X.D. Ding et al. / Composites Science and Technology 62 (2002) 841–850
845
tion. In addition, it can be seen from Fig. 3 that the q "C curves depend greatly on the material parameters, such as AF ; VF ; k and n, although they do not change the shapes of the q "C curves. Among the four parameters, the effects of AF and VF are significant, while those of k and n are relatively small. The calculated stress–strain curves by the above expressions are shown in Fig. 4(a) for several typical cases. It is shown that the effects of AF ; k and n on C are consistent with their effects on q, but the effect of VF is opposite. This opposite effect can be explained well by Eq. (8). Although q decrease with increasing VF ; the contribution of TR to C ðVF TR Þ increases with increasing VF : As a result, C increase with increasing VF . The stress–strain curves of the composite calculated by FEM were compared with those from Eqs. (10), (13) and (14) by using the corresponding q values in Fig. 3. The results are shown in Fig. 4(b). It can be seen that the stress–strain curves predicted by Eqs. (10), (13) and (14) are in good agreement with those by FEM.
4. Transition of deformation behavior It can be clearly seen from Eq. (5) that q is very sensitive to the changes in the stress–strain relations of the matrix and the fiber. It should be possible to reveal the yield behaviors of the composite by examining
the varia tion of q and its second order derivatives, q00 ¼ d 2 q=d"2C : 00 Typical q "C curves and the corresponding C "C and q "C curves are shown in Fig. 5(a). In the calculations, to keep the precision and avoid the numerical noise of the calculation data, the strain range of the neighboring data points was kept as about 0.00004. Furthermore, a method of five data point smoothing was used to smooth the curves. The first and second order derivatives of were then calculated from the linear gradient between the neighboring data points of the smoothed curves. It should be noted that the choice in the number of data points for the smoothing influence the absolute values of the derivatives, but it does not change the trends we report here. It can be seen in Fig. 5(a) that there are two distinct points (points A and B) in the q "C curves. q keeps constant before point A, then declines. q decreases gradually after point B. From the q00 "C curves, it can be seen that there are two distinct extremum points corresponding to points A and B, respectively. Between points A and B, the q00 "C curves show sharper variation. After point C, q00 keeps nearly constant. To establish the connection of the above stress–strain behaviors with the development of the matrix plasticity, the contours of the Von Mises effective stresses in the matrix (regions E and F) and the fiber (region D) were calculated and are shown in Fig. 6 at the corresponding points A, B and C. As shown in Figs. 5(a) and 6, at
Fig. 4. (a) The effect of the material parameters on the stress–strain curves of the composite; (b) comparisons of the calculated stress– strain curves by Eqs. (10), (13) and (14) with those by FEM.
point A, the plastic region has formed in smaller region near the fiber end face (region F), the matrix in the fiber region (region E) is still elastic. At point B, most regions in the fiber end have entered the plastic state. In the strain stage between points A and B, the strain-hardening rate of the composite (the slope of the stress– strain curve) becomes small relative to that in strain stage before point A. At point C, the plastic deformation has occurred in the entire matrix including the fiber region and the fiber end region, the strain-hardening rate of the composite becomes markedly small. According to the results in Figs. 5(a) and 6, the composite properties in the initial yield stage can be well determined. It can be thought that point A is a strain point before which the overall deformation behavior is basically elastic although there exists very small local plastic region in the matrix near the fiber end face. This argument is supported by the results in Fig. 5(a) that the slope of the stress–strain curve keeps basically constant before point A. Therefore, phenomenally, the strain corresponding point A can be defined as the maximum elastic strain of the composite and the stress at this point can be defined as the proportional limit of the composite. The elastic modulus of the composite can be thus determined as the average value of the slope of the
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Fig. 5. The variations of the C "C , q "C and q00 "C curves with "C .
stress–strain curve before point A. Since significant plastic deformation has occurred in the fiber end region at point B, the stress at point B can be considered as the initial yield strength. Furthermore, the stress at point C can be defined as the final yield strength because the plastic deformation has occurred in the entire matrix. The effects of VF ; AF and K on the maximum elastic strain, the initial and final yield strains are shown in Fig. 5 (b)–(d). The corresponding matrix stresses and strains, i.e. the stress and strain at the points A1, B1 and C1 are also given in Fig. 5 (b)-(d), where the values of corresponding to points A, B and C are represented by 0 the dash lines. The value of MY in Eq. (13) can be given by the stress at point B1. As shown, the stresses and strains in the matrix, corresponding to the proportional limit and the initial yield stress, are smaller than the yield stress and strain of the matrix material (357 MPa and 0.00482). But the stress and strain in the matrix, corresponding to final yield stress, are larger than those of the matrix material. It can also be seen that the proportional limit decreases with increasing VF and increases with increasing AF and K. The initial yield stress decreases with increasing AF and VF and increases with
increasing K. The final yield stress increases with increasing AF ; VF and K. Fig. 6 also shows the contours of the Von Mises effective stress in the matrix and the fiber corresponding to points A, B and C defined by Fig. 5(b)–(d). From the comparison of Fig. 5(b)–(d) with Fig. 6, it can be seen that the plastic deformation in the fiber end region just begins at point A for all cases with different material parameters. At meantime, the matrix in the fiber end regions deforms plastically mainly at point B and the entire matrix deforms plastically at point C. The results above revealed that the present method can describe the yield process of the composite regardless the variations of the material parameters. Table 1 gives the comparisons of the elastic modulus and the proportion limit calculated by FEM with the published experiment data [6,23,24]. As shown, the calculated values are in good agreement with the experimental values. Fig. 7 shows the variations of the calculated proportion limit ðCP Þ; initial yield stress ðCIY Þ and the final yield stress ðCFY Þ with AF and VF for given k and n values. k and n were taken as K ¼ LC =LF ¼ RC =RF , n=0.2. Moreover, the published experiment data of the
X.D. Ding et al. / Composites Science and Technology 62 (2002) 841–850
Fig. 6. Contours of the Von Mises effective stress in the matrix and the fiber.
Fig. 7. The variations of the composite proportion limit and yield strength with AF and VF.
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Table 1 Comparison between the experimental and calculated proportion limit Material
VF
AF
4 1 4 1 4 2 2 2
EC (GPa)
CP (GPa)
Exp.
FEM
Exp.
120 81 88 106 119 81 102 114
119 80 90 104 119 80 98 116
348 193 220 241 269 – – –
FEM
SiCW/Al 6061 T6 [23] SiCd/Al 5456 [6] SiCd/Al 5456 SiCd/Al 5456 SiCd/Al 5456 SiCW/Al 6061 T6 [24] SiCW/Al 6061 T6 SiCW/Al 6061 T6
0.20 0.08 0.08 0.20 0.20 0.10 0.20 0.30
356 197 224 228 272 – – –
Al 6061 T6 Al 5456 Al 6061 T6
EM=74 GPa EF=483 GPa M =357 MPa EM=73 GPa EF=485 GPa M =241 MPa EM=68.8GPa EF=450 GPa M =280 MPa
yield stress ð0:2 Þ of SiCw/6061Al composites are also given in Fig. 7(b). As shown, CIY and CFY increase with increasing of AF and VF : The variation tendencies of CIY and CFY are similar to that of the yield stress CY calculated by Eq. (12). CY is higher than CIY ; but lower than CFY : This is consistent with the above argument that CY is an upper limit value of the yield strength of the composite. CIY is slightly higher than 0.2, but has the same variation tendency as 0.2. All of these comparisons verified the validity of the method proposed in the present study. Furthermore, it can be seen from Fig. 7 that the variation of CP is complex. CP first decreases and then increases with increasing AF and VF : This result is consistent with the results in works [8,28], in which a similar relationship between the proportional limit and the volume fraction of reinforcing particles were observed. This behavior results from two competitive effects of the local stress concentrations and the local plastic constraints introduced by the fibers. It is known that the load is transferred from the matrix to the fiber by the interface along the fiber and the fiber end face. As a consequence of this load transfer, the stress concentrations arise in the local region around the fiber and at the region near the fiber end. These matrix regions experience higher local stresses and yield first. Thus the proportional limit is lowered. Simultaneously, the presence of the elastic fiber restrains the matrix from plastic flow and tends to raise the proportional limit and initial yield stress. That is, there are competitive effects between local stress concentrations and local plastic constraints. The former is predominating when AF and VF are smaller and the latter become more determinative as AF and VF increase.
5. Conclusions (1) The analytical expressions for the stress strain response of the composite have been derived
based on the law of mixture. The expressions have predicted well the entire stress–strain curves of the composite with the given variations in the stress–strain partition parameter q by a large strain axisymmetric elasto-plastic finite element method. (2) A new method has been proposed to define the yield responses of the composite by examining the variations in and its second order derivations. The proportion limit, the elastic modulus, the initial yield stress and the final yield stress can be predicted well by using this method. (3) With the increases of the aspect ratio and the volume fraction, the proportion limit first decreases and then increases. This behavior is thought to be a consequence of the competitive effects of local stress concentrations and local plastic constraints due to the presence of the fiber. (4) The predicted elastic modulus and proportion limit by the derived analytical expressions and the finite element method exhibit good agreement with the published experimental results.
Acknowledgements This work was supported by the National Outstanding Young Investigator Grant of China. The financial support from National Natural Science Foundation of China has been also gratefully acknowledged. Valuable comments and kindly suggestions from the reviewer are sincerely appreciated too.
Appendix For the case that the fiber is elastic and the matrix is plastic, the differential of the composite stress C gives dC ¼
@C @F @C @M d"F þ d"M @F @"F @M @"M
ðA1Þ
The stress–strain relation of the matrix is characterized by ¼ EM "M 4 MY ðA2Þ ¼ l"nM > MY where EM ; l and are Young’s modulus, the strength constant and strain hardening exponent of the matrix, respectively. While the fiber is elastic, Hooke’s law can be used F ¼ EF "F
ðA3Þ
X.D. Ding et al. / Composites Science and Technology 62 (2002) 841–850
849
where EF are Young’s modulus of the fiber. The differentials of (A2) and (A3) give
This is an explicit expression for the elasto-plastic tangent modulus of the composite.
@M ¼ M;M ¼ ln"n1 M @"M
ðA4Þ
References
@F ¼ F;F ¼ EF @"F
ðA5Þ
According to the law of mixture C ¼ ð1 VF ÞM þ VF F
ðA6Þ
"C ¼ ð1 VF Þ"M þ VF "F
ðA7Þ
we have @C ¼ 1 VF @M
ðA8Þ
@C ¼ VF @F
ðA9Þ
Introduction of Eqs. (A4)–(A5) and (A8)–(A9) into Eq. (A1) leads to dC ¼ EF VF d"F þ ð1 VF Þln"n1 M d"M
ðA10Þ
The definition of can be expressed as q¼
M F "M "F
ðA11Þ
The derivative of Eq. (A10) gives d"F ¼
ln"n1 M q d"M Eq
Combining Eqs. (A9) and (A11) gives ln"n1 n1 M q þ ð1 VF Þln"M d"M dC ¼ EF VF Eq
ðA12Þ
ðA13Þ
Since the fiber is elastic, i.e. "F 0; thus Eq. (A6) can be approximately written as "C ¼ ð1 VF Þ"M
ðA14Þ
So we have from Eqs. (A12) and (A13) " n1 # dC VF lnð"C =ð1VF ÞÞn1 q "C EF þln ¼ EF q d"C 1VF 1VF ðA15Þ
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