Results and discussion

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where a is applied ultrasonic amplitude of 0, 1.00, 2.00, 3.21µm, 𝜌_𝑚 is the mass density of materials, including 2700 and 8940 kg/𝑚³ for aluminum and copper respectively, and 𝜔 is circular frequency, which is expressed by the equation with frequency f of 60kHz.

𝜔 = 2πf (9) Combining Eqs. (1)-(9) could construct a constitutive model governing the relatioon between the flow stress σ and the normal plastic strain εincluding the ultrasonc acoustic softening effect. It can been seen that when the ultrasonic energy is increased, the value of 𝛾̇₀ is increased, resulting in an increase of the dynamic recovery factor 𝑘₂, thus the dislocation density and flow stress will be reduced, which is in accordance with the experimental results in chapter 2 and 3.

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Fig. 5-1 Stress-strain curves with oscillatory stress of pure aluminum without vibration with amplitude of 1.00µm.

Fig. 5-2. Stress-strain curves with oscillatory stress of pure copper without vibration with amplitude of 1.00, 2.00 and 3.21µm.

To model the real stress evolution by acoustic softning during compression, the experimental datas of stress-strain curves by acoustic sfotening were used to idenfy to unknow parameters in the proposed constitutive model. The steps to obtain the unknown parameters are as follows.

1. To obtain the evolution of dislocation density at each amplitude. Combining Eqs. (1-3, 6), the relationship between dislocation density and flow stress could be

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constructed, then combined with experimental data of stress-strain relations at different amplitude, the evolution of dislocation density by strain is obtained, as shown in Fig. 5-3. The constant parameters needed for the equations are refered from previous literatures, as shown in Table 5-1. In Fig. 5-3, it shows that dislocation densites increase with strain increasing, and acoustic softening significantly reduces the dislocation density with different amplitudes, which is according with the KAM values of EBSD results in chapter 4. By acoustic softening, mean KAM value of aluminum at top area is reduced from 0.490284 to 0.410087, and mean KAM values of copper at top and center areas are reduced from 0.853330 and 1.257880 to 0.499695 and 1.138760 respectively. These reductions of KAM value confirm the drop of dislocation density.

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Fig. 5-3 Dislocation density evolution by acoustic softening for different amplitudes on (a) pure aluminum and (b) pure copper.

2. To obtain the unknown parameters 𝑘₁and 𝑘₂ without ultrasonic vibration (a=0µm).

Eq. 4 indicates the relationship between dislocation density and strain, fitted with the dislocation density evolution results without ultrasonic vibration on specimens of pure aluminum and pure copper in Fig. 5-3, 𝑘₁and 𝑘₂ in Eq. 4 can be determined by curve fitting using the software of Auto2Fit 5.5,

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3. To obtain the unknown parameters 𝑘₂ at different amplitudes (a=1.00, 2.00, 3.21µm).

Since the ultrasonic acoustic softening mainly leads to dynamic recovery, the parameter 𝑘₁ for dislocation generation is assumed constant during ultrasonic assistance in this model. So 𝑘₂ should increase with amplitude increases for acoustic softening. By curve fitting with the dislocation density evolution results at different amplitudes in Fig. 5-3, 𝑘₂ can be determined, as shown in Table 5-1.

Table 5-1

Parameter values of 𝑘₁and 𝑘₂ obtained in the model

𝒌_𝟏 𝒌_𝟐

Aluminum No UV

13825 1.319841021

1.00µm 1.333871805

Copper No UV

37010

1.371876253

1.00µm 1.489502085

2.00µm 1.555558843

3.21µm 1.604442236

4. To obtain the unknown parameters for modification of the model. Combining Eqs.

(5, 7-9), the relation of 𝑘₂ and ultrasonic energy density can be determined. Using the values of 𝑘₂ in Table 5-1, the parameters of 𝑘₂₀, 𝜉, 𝛾̇₀₀ and 𝑚 can be obtained by curve fitting. Table 5-2 summarized the list of parameters obtained from references and determined by curving fitting from stress-strain curves.

Table 5-2

Parameter values used in the model

Parameter Value Equation

number Reference

From reference

𝛾̇₀

pre-exponential factor（/s）

10⁶ (1) [8]

μ

elastic shear modulus (MPa)

Copper:

4.60×10⁴ Al:

(3) [12], [5]

138 2.60×10⁴ b

length of Burgers vector (nm)

Copper:

0.256 Al:

0.286

(3) [13], [5]

p, q 1 (2) [8]

α 1/3 (3) [14]

n 4 (5) [11]

M Taylor factor 3.06 (6) [8]

𝑄_𝑑

activation energy for self-diffusion (kj/mol)

Copper:

211 Al:

144

(7) [15]

Experimentally determined

𝑘₁ dislocation storage coefficient

Copper:

37010 Al:

13825

(4)

𝑘₂₀

Copper:

3.51 Al:

1.99

(5)

𝛾̇₀₀

Copper:

4.66×10⁻⁴ Al:

9.75×10⁻⁵

(7)

𝜉

Copper:

2.28 Al:

4.45×10⁻²

(7)

m

Copper:

4.06×10⁻¹ Al:

5.01×10⁻³

(7)

These parameters were used in Eqs. (1)-(9) to calculate the values of flow stress for increasing strains. The calculated stresses versus strain were plotted and compared with the experimental compression results, as shown in Fig. 5-4 and Fig. 5-5.

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Fig. 5-4. Comparision between experimental and predicted stess-strain curves by acoustic softening on pure aluminum (a) without vibration and (b) without amplitude of 1.00µm.

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Fig. 5-5. Comparision between experimental and predicted stess-strain curves by acoustic softening on pure copper (a) without vibration, and with amplitude of (b) 1.00µm, (c)

2.00µm and (d) 3.21µm.

By comparing the experimental and predicted stress-strain curves from modeling, it can be seen that for both pure aluminum and pure copper, all curves fit well. The results of this modeling confirm that the mechanism of acoustic softening could be considered as dynamic recovery in plastic deformation, which leads to more annihilation of dislocations in metal.

In chapter 3, it is found that acoustic softening is more effective on aluminum than copper for the ratio of stress reduction. The reason is thought to be the higher lattice resistance to dislocations in copper, which can be related to the activation energy for self-diffusion or dislocation climb 𝑄_𝑑 in the modified constitutive model in this chapter. The 𝑄_𝑑 for copper, 211kj/mol, is larger than that for aluminum, 144kj/mol, confirming the higher resistance to dislocations in copper. Thus in the case of ultrasonic assistance, the relative ratio of ultrasonic energy density to activation energy for self-diffusion 𝐸 𝑄⁄ _𝑑 in Eq. (7) may be

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considered as a key parameter to determine the effect of acoustic softening on different materials. With larger E, the acoustic softening should be more significant, which can be confirmed by the results of copper at different amplitudes. But it has to be noticed that, E consists of amplitude a and frequency f. So both a and f should influence the effect of acoustic softening theoretically, although in this study only the effect of a is investigated. In the future, the effect of different ultrasonic frequency on acoustic softening should be investigated. For different material, with lower value of 𝑄_𝑑 should have a more significant effect of acoustic softening due to the lower lattice resistance to dislocations.

Compared to previous studies about numerical modeling on acoustic softening [4,5,13], the proposed constitutive model is a physically based model, which combines the construction of the model and observation of microstructure. So this model could provide a deeper understanding on the mechanism of ultrasonic effect on material deformation.

However, it has to be noticed that the effect of acoustic softening on dislocation density is in a dynamic balance of generation and annihilation. As for the EBSD results of acoustic softening in chapter 4, it is observed that dislocation density is reduced in annealed material but increased in as-received material. However, it possible that acoustic softening leads to promotion on both generation and annihilation of dislocation with different rate of promotion. For annealed material, the tendency of dislocation density is increasing, to inhibit this tendency, acoustic softening leads to higher promotion rate on dislocation generation than annihilation, which results in a decrease of dislocation density in the whole.

But for as-received material with large work hardening, the material has a tendency of decreasing dislocation density in static deformation without ultrasonic vibration. Once ultrasonic vibration works on the deformation, the original tendency of dislocation density is inhibited by higher promotion rate on dislocation generation than annihilation, so the dislocation density is increased finally.

In our constitutive model, there are some drawbacks. We just considered the acoustic softening effect on annealed material and the parameter 𝑘₁ for dislocation generation is assumed constant during ultrasonic assistance for simplification. As we discussed above, both dislocation generation 𝑘₁ and annihilation 𝑘₂ could be promoted in the real process,

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although the promotion rate on 𝑘₂ is higher than 𝑘₁. So in the future a more reliable model could be considered.