**5. Constitutive model for thixotropic plastic forming of composites**

On the basis of analysis of behavior of thixotropic plastic deformation of composites in compression process, its constitutive model is established. Then the model parameters are determined using the multiple nonlinear regression method.

Based on the experimental analysis of axial compression of composites in semi-solid state, there is a certain non-liner relationship among stress σ and strain rate *<sup>z</sup>* , strain *<sup>z</sup>* , temperature T, liquid phase rate *Lf* , as well as the volume fraction of reinforcement *pf* [Yan & Wang, 2011]. At the same time Hong Yan present the constitutive relationship of semisolid magnesium alloy as follow [Yan & Zhou, 2006]:

$$
\sigma \propto \exp(1/T) \dot{\varepsilon}^{a\_1} \varepsilon^{a\_2} (1 - \beta f\_L)^{a\_3} \tag{3}
$$

Where σ – stress, ε – strain, – strain rate, T – temperature, β – geometric parameters(β=1.5), fL – liquid phase rate.

In the study of deformation behavior of composites under high strain rate [Bao & Lin, 1996] and [Li & Ramesh, 2000] found that the influence of volume fraction of reinforcement on the mechanical behavior of the material was present as following:

$$
\sigma(f\_p) = \sigma(\varepsilon, \dot{\varepsilon}) \cdot \lg(f\_p) \cdot \left[1 + (a\dot{\varepsilon})^m f\_p\right] \tag{4}
$$

Where σ- stress, ε- strain, - strain rate, (,) - function of strain and strain rate, fpvolume fraction of reinforcement.

So the constitutive model of thixotropic plastic deformation of composites reinforced with particles is proposed.

Study on Thixotropic Plastic Forming of Magnesium Matrix Composites 269

Mean Square are 413.512 and 59.073 respectively. Residual in Sum of Squares and Residual in Mean Square are 421.492 and 0.451 respectively. The test statistic observations F = 130.902. The concomitant probability p is approximately 0. The linear relationship between

Model Sum of Squares df Mean Square F Sig. 1 Regression 413.512 7 59.073 130.902 0.000(a)

Table 5 shows the regression coefficient analysis. As can be seen from the table and the estimated value of the test results, the corresponding variable regression coefficient A0=- 8.27366, A1=50.158,A2=-296.555,A3 =14253.359,A4 =-0.053,A5=0.242,A6 =2.316,A7 =-0.505. The concomitant probability p is 0, whose regression is a significant. From comparison of regression coefficients, those indicate that the constitutive model has a significant meaning. The sensitivity coefficient m A5=0.242 of strain rate resulted from regression is good close to

1 (Constant) -8.27366 1.818 -9.728 0.000 X1 50.158 7.917 1.303 6.335 0.000 X2 -296.555 53.608 -0.934 -5.532 0.000 X3 14253.359 1402.791 0.342 10.161 0.000 X4 -0.053 0.012 -0.108 -4.601 0.000 X5 0.242 0.045 0.474 5.382 0.000 X6 2.316 0.553 0.143 4.186 0.000 X7 -0.505 0.463 -0.135 -1.091 0.000

The analysis of the regression equation is a meaningful, and the following relation is got.

<sup>2</sup> 50 158

*L p*

*f f*

exp(-8.27366+ . -296.555 14253.359 / )

*p p*

Equation (11) is a constitutive relationship of thixotropic plastic forming of SiCp/AZ61

*ff T*

(11)

From the inverse transform of equations (9) and (10), equation (6) becomes:

0.242 2.316 4 0.242 -0.505

[ ] [ ( .1 10 ) ]

1 12

 

1 2 34567 *y X X XXXXX* 8 27366 50 158 296 555 14253 359 0 053 0 242 2 316 0 505 . . . ..... (10)

Standardized

Coefficients t Sig.


 

Coefficients

B Std. Error Beta

variables x and y is significant, which create a linear model.

 Total 835.005 941 a Predictors: (Constant),x7,x6,x4,x2,x3,x5,x1 b Dependent Variable:y

Model Unstandardized

Table 4. ANOVA (b)

the replaced m value.

a Dependent Variable:y Table 5. Coefficients (a)

composites.

Residual 421.492 934 0.451

$$\sigma = \exp(d \;/\ T) \cdot \varepsilon^{\mu} \cdot \dot{\varepsilon}^{m} \cdot \left[1 - \beta f\_{L}\right]^{a\_{1}} \cdot g(f\_{p}) \cdot \left[1 + (\alpha \dot{\varepsilon})^{m} f\_{p}\right]^{a\_{2}} \tag{5}$$

Under assumption of <sup>2</sup> ( ) *<sup>p</sup> <sup>p</sup> a bf cf <sup>p</sup> gf e* the constitutive model is established in the following form.

$$\sigma = \exp(a + bf\_p + cf\_p^{\;\;2} + d \;/\; T) \cdot \varepsilon^n \cdot \dot{\varepsilon}^m \cdot [1 - \beta f\_L]^{a\_1} \cdot [1 + (a\dot{\varepsilon})^m f\_p]^{a\_2} \tag{6}$$

Where a, b, c, d, a1, a2 - constant, n—strain hardening index, m—strain rate sensitivity index,β—constant (β=1.5), α - correction coefficient , fp - volume fraction of reinforcement, fL - liquid phase rate. 1 1-K ( ), *M L L M T T <sup>f</sup> T T* TM - the melting point of pure metal, TL - liquidus temperature of alloy, k - balance coefficient.

The parameters in proposed constitutive model were determined by the multiple nonlinear regression method. The nonlinear equation is transformed into linear one using legarithms for Esq.(3).

$$\ln \sigma = a + bf\_p + cf\_p^{\;\;\;\;\;\sigma} + d \;/\; T + n \ln \varepsilon + m \ln \dot{\varepsilon} + a\_1 \ln(1 - \beta f\_L) + a\_2 \ln[1 + (a \dot{\varepsilon})^m f\_p] \tag{7}$$

Where

$$\begin{aligned} y &= \ln \sigma, X\_1 = f\_p, X\_2 = f\_p \ ^2, X\_3 = 1/\ T, X\_4 = \ln \bar{\sigma}, X\_5 = \ln \bar{\varepsilon}, X\_6 = \ln(1 - \beta f\_L), \\ X\_7 &= \ln[1 + (a\bar{x})^m f\_p] \\ A\_0 &= a\_r A\_1 = b\_r A\_2 = c\_r A\_3 = d\_r A\_4 = n\_r A\_5 = m\_r A\_6 = a\_1, A\_7 = a\_2 \end{aligned} \tag{8}$$

Esq.(12) is changed as follow

$$y = A\_0 + A\_1 X\_1 + A\_2 X\_2 + A\_3 X\_3 + A\_4 X\_4 + A\_5 X\_5 + A\_6 X\_6 + A\_7 X\_7 \tag{9}$$

Table 3 shows the common statistic values. The correlation coefficient R = 0.974, determination coefficient <sup>2</sup> *R* = 0.949, the adjustment determination coefficient <sup>2</sup> *R* = 0.931, the Std. Error of the Estimate S =0.0670. As the equation has a number of explained variables, the determination should be based on the adjustment determination coefficient <sup>2</sup> *R* . As can be seen from the output that <sup>2</sup> *R* is close to 1, the fit degree is high. So the representativeness of proposed constitutive model is strong.


a Predictors:(Constant), x7,x6,x4,x2,x3,x5,x1 b Dependent Variable:y

Table 3. Model Summary

The analysis is listed in Table 4. The significant test of regression equation is based on the table. Total in Sum of Squares is 835.005, Regression in Sum of Squares and Regression in Mean Square are 413.512 and 59.073 respectively. Residual in Sum of Squares and Residual in Mean Square are 421.492 and 0.451 respectively. The test statistic observations F = 130.902. The concomitant probability p is approximately 0. The linear relationship between variables x and y is significant, which create a linear model.


a Predictors: (Constant),x7,x6,x4,x2,x3,x5,x1 b Dependent Variable:y

Table 4. ANOVA (b)

268 Mechanical Engineering

 *d T f gf*

 

1

1-K ( ), *M L*

*M T T <sup>f</sup> T T*

2

*L*

temperature of alloy, k - balance coefficient.

( ) *<sup>p</sup> <sup>p</sup> a bf cf*

Under assumption of <sup>2</sup>

fL - liquid phase rate.

7

Esq.(12) is changed as follow

Table 3. Model Summary

1

*X f*

ln[ ( ) ]

*m p*

for Esq.(3).

Where

form.

1 1 1 2 exp( / ) [ ] ( ) [ ( ) ] *n m a a <sup>m</sup>*

<sup>2</sup> 1 1 1 2 exp( /) [ ] [ ( ) ] *n m a a <sup>m</sup> p p L p*

Where a, b, c, d, a1, a2 - constant, n—strain hardening index, m—strain rate sensitivity index,β—constant (β=1.5), α - correction coefficient , fp - volume fraction of reinforcement,

The parameters in proposed constitutive model were determined by the multiple nonlinear regression method. The nonlinear equation is transformed into linear one using legarithms

1 2 ln / ln ln ln( ) ln[ ( ) ] 1 1 *<sup>m</sup>*

12 3 4 5 6

*y X f X f X TX X X f*

Table 3 shows the common statistic values. The correlation coefficient R = 0.974, determination coefficient <sup>2</sup> *R* = 0.949, the adjustment determination coefficient <sup>2</sup> *R* = 0.931, the Std. Error of the Estimate S =0.0670. As the equation has a number of explained variables, the determination should be based on the adjustment determination coefficient <sup>2</sup> *R* . As can be seen from the output that <sup>2</sup> *R* is close to 1, the fit degree is high. So

Model R R Square Adjusted R Square Error of the Estimate 1 0.974(a) 0.949 0.931 0.0671449560

The analysis is listed in Table 4. The significant test of regression equation is based on the table. Total in Sum of Squares is 835.005, Regression in Sum of Squares and Regression in

 *a bf cf d T n m a f a*

0 1 2 3 4 5 6 17 2

,,, , , , ,

*A aA bA cA dA nA mA a A a*

2

the representativeness of proposed constitutive model is strong.

a Predictors:(Constant), x7,x6,x4,x2,x3,x5,x1 b Dependent Variable:y

*p p L p*

*p p L*

 

0 11 2 2 33 4 4 5 5 6 6 7 7 *y A AX A X AX A X AX AX A X* (9)

ln , , , / , ln , ln , ln( ),

 *a bf cf d T*

*Lp p*

 *f*

TM - the melting point of pure metal, TL - liquidus

 

1 1

(8)

*<sup>p</sup> gf e* the constitutive model is established in the following

*f* (5)

*f* (6)

*f* (7)

Table 5 shows the regression coefficient analysis. As can be seen from the table and the estimated value of the test results, the corresponding variable regression coefficient A0=- 8.27366, A1=50.158,A2=-296.555,A3 =14253.359,A4 =-0.053,A5=0.242,A6 =2.316,A7 =-0.505. The concomitant probability p is 0, whose regression is a significant. From comparison of regression coefficients, those indicate that the constitutive model has a significant meaning. The sensitivity coefficient m A5=0.242 of strain rate resulted from regression is good close to the replaced m value.


a Dependent Variable:y

Table 5. Coefficients (a)

The analysis of the regression equation is a meaningful, and the following relation is got.

$$y = -8.27366 + 50.158X\_1 - 296.5\% \text{X}\_2 + 142\text{S}3.3\\$ \text{9X}\_3 - 0.05\text{S}X\_4 + 0.242X\_5 + 2.316X\_6 - 0.50\text{S}X\_7 \text{ (10)}$$

From the inverse transform of equations (9) and (10), equation (6) becomes:

$$\begin{aligned} \sigma &= \exp(-8.27366 + 50.158 f\_p \text{-} 296.555 f\_p \text{\*} + 14253.359 \text{ /} \text{T}) \cdot \varepsilon^{-0.053} \cdot \\ \varepsilon^{0.242} \cdot \left[ \text{l} - \beta f\_L \right]^{2.316} \cdot \left[ \text{l} + \text{(2.1} \times 10^4 \text{\textdegree s})^{0.242} f\_p \text{]}^{0.505} \end{aligned} \tag{11}$$

Equation (11) is a constitutive relationship of thixotropic plastic forming of SiCp/AZ61 composites.

Study on Thixotropic Plastic Forming of Magnesium Matrix Composites 271

*f f T*

For establishing material modal of SiCp/AZ61 composite in forging, true stress-strain curves at various temperature and strain rates were performed by mean of isothermal

In this study, the workpiece is formed by the close-forge method. The experiment set-up was shown in Fig.19. Fig.20 shows the workpiece, whose structure and flow character are complicated. Comparisons between forging and thixo-forging of the workpiece will be done and predicted in advance using numerical simulation. This is an effective method to instruct

The same simulated parameters are used to analyze the differences of mechanics properties and flow rule between forging and thixo-forging processes. The materials are normal and semi-solid SiCp/AZ61 composite respectively. Environment temperature is 20℃, warm-up temperature of the die is 320℃. The friction model is constant shearing stress model, whose coefficient is 0.25. Billet size is ø50×18.5mm, which is meshed to 50000 tetrahedron elements.

where σ is the stress; ε the strain; z .

. 

0.242 1

Volume fraction of SiC particle; *Lf* is liquid volume fraction

application of semi-solid forming technology into its practice production.

 -0.053

compression experiments.

Stroke of up-die is 14mm.

Fig. 19. Experiment set-up

Fig. 20. SiCp/AZ61 composite workpiece

<sup>2</sup> exp( . . 8 27366 50 158 296 55 14253 359 . . /) *p p*

*fL* 2.316 <sup>4</sup> [ (. ) 1 2 1 10

the strain rate; T temperature; β constant(β=1.5); *pf* is

0.242 ] *pf* -0.505 (12)

Fig.18 is the real stress test - a true strain curves and regression curve of the results of the comparison, Solid line is the experimental curve, dotted line is the calculation of one. The results calculated by multiple non-linear regression method are good agreement with experimental ones. So the proposed constitutive model has the higher forecast precision and practical significance.

(c) 9vol.% SiCP/AZ61,530℃

Fig. 18. A comparison between true strain—stress curves of the test and regression curves
