Ultrasonic Basic Equations

1. Acoustic softening and residual hardening in aluminum: Modeling
and experiments
Zhehe Yao a,b
, Gap-Yong Kim a,⇑
, Zhihua Wang a
, LeAnn Faidley a
, Qingze Zou c
, Deqing Mei b
,
Zichen Chen b
a
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
b
The State Key Lab of Fluid Power Transmission and Control, Department of Mechanical Engineering, Zhejiang University, Hangzhou 310027, PR China
c
Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854, USA
a r t i c l e i n f o
Article history:
Received 25 December 2011
Received in final revised form 10 June 2012
Available online 23 June 2012
Keywords:
A. Acoustics
A. Cutting and forming
B. Crystal plasticity
B. Metallic material
a b s t r a c t
It is known that high-frequency vibration affects metal plasticity during metal forming and
bonding operations. Metal plasticity is significantly affected by the acoustic field leading to
acoustic softening and acoustic residual hardening. In this study, a modeling framework for
the acoustic plasticity was proposed based on the crystal plasticity theory. The acoustic
softening and acoustic residual hardening effects were modeled based on the thermal acti-
vation theory and dislocation evolution theory, respectively. To validate the developed
model, vibration-assisted upsetting tests were conducted using pure aluminum specimens.
Results showed that the stress decrease due to the acoustic softening was proportional to
the vibration amplitude. Moreover, the acoustic residual hardening effect was influenced
by the vibration amplitude and duration. The unified acoustic plasticity model accurately
captured the acoustic softening and hardening in aluminum. The predicted stress–strain
response of the vibration-assisted upsetting agreed well with the experimental results.
The findings confirmed the significant effects of high-frequency vibration on metal plastic-
ity and provided a basis to understand the underlying mechanisms of vibration-assisted
forming.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
High-frequency vibration has been used for various processes, such as vibration assisted forming (Daud et al., 2007;
Ngaile and Bunget, 2011; Siddiq and El Sayed, 2012b), ultrasonic welding (Matsuoka, 1998; Tsujino et al., 2002), etc. The
thermal effect of the high-frequency vibration plays a significant role in several cases like ultrasonic welding (Kim et al.,
2011; Zhang et al., 2010). On the other hand, the high-frequency vibration can remarkably soften the metallic materials
without significant heating, which is usually referred to as acoustic softening (Langenecker, 1966; Siddiq and El Sayed,
2011; Yao et al., 2012). The acoustic softening is considered to be more efficient than plasticity originating from the thermal
softening (Langenecker, 1966). Even though the acoustic softening has been observed and studied for decades (Dawson et al.,
1970; Eaves et al., 1975; Langenecker, 1966; Winsper et al., 1970), the underlying mechanism is still not so clear. From the
viewpoint of dislocation theory, one explanation for acoustic softening is that the preferential absorption of acoustic energy
by lattice imperfections like dislocations or grain boundaries reduces the critical resolved shear stress (Langenecker, 1966;
Lum et al., 2009).
0749-6419/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijplas.2012.06.003
⇑ Corresponding author. Tel.: +1 515 294 6938; fax: +1 515 294 3261.
E-mail address: gykim@iastate.edu (G.-Y. Kim).
International Journal of Plasticity 39 (2012) 75–87
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2. Besides the acoustic softening effect during the high-frequency vibration mentioned above, a residual hardening effect
was detected in various studies (Gindin et al., 1972; Langenecker, 1966; Tyapunina et al., 1982; Westmacott and Langeneck-
er, 1965). Unlike the acoustic softening, which only exists temporarily while the vibration is applied, the residual hardening
effect is retained after stopping the vibration. If sufficiently large ultrasonic intensity is applied during the vibration, hard-
ening occurs and can be detected after the vibration is stopped. The acoustic residual hardening is mainly attributed to the
increase of the dislocation density (Tyapunina et al., 1982), which is due to the dislocation multiplication by the ultrasonic
irradiation (Langenecker, 1966). On the other hand, acoustic residual softening effect for metals after the ultrasound was also
reported in ultrasonic bonding processes for thin metallic wires (Huang et al., 2009; Lum et al., 2009), which was explained
by the reduction of dislocation density due to dynamic annealing. In recent study by Siu et al. (2011), indentation experi-
ments performed on aluminum samples simultaneously excited by ultrasound revealed that the subgrain formation was
extensively enhanced during deformation, and suggested that the enhancement of dipole annihilation induced by ultrasound
caused the softening effect. Siu and Ngan (2011) also carried out dislocation dynamics simulations under different conditions
of combined oscillatory and quasi-static stresses to investigate the acoustic residual softening and hardening effects.
Recently, several studies have revisited acoustic plasticity in metallic materials. Rusinko (2011) developed an analytical
model which introduced a new term, ultrasonic defect intensity, into the synthetic theory of plastic deformation. This
model may describe the ultrasonic softening during plastic straining and the ultrasonic hardening without static loading,
but did not consider the acoustic residual softening or the acoustic residual hardening after ultrasonic-assisted plastic
deformation. Siddiq and El Sayed (2011, 2012a) proposed a phenomenological crystal plasticity model to account for
acoustic softening effects based on the level of ultrasonic intensity supplied to the single and polycrystalline metals. In
their work, the stress reduction by acoustic softening was considered to be proportional to the acoustic intensity based
on the experimental results by Langenecker (1966). However, a recent study conducted by Huang et al. (2009) showed
that the stress reduction by acoustic softening in the copper compression experiments was proportional to the vibration
amplitude rather than the intensity (acoustic intensity is proportional to the square of the amplitude). There still seems no
such model that can completely explain the effects of high-frequency vibration on metallic plasticity observed in various
experiments.
In this study, a framework for an acoustic plasticity was developed to describe the effects of high-frequency vibration on
metal plasticity based on the crystal plasticity theory. To validate the model, high-frequency vibration experiments were
conducted during upsetting of an aluminum material. The acoustic softening during the vibration and the acoustic residual
hardening after the vibration were measured. The agreement between the model prediction and the experimental results
were discussed.
2. Acoustic plasticity framework
2.1. General framework of crystal plasticity
For decades, several constitutive models have been proposed to describe the metal plasticity. For example, Kocks (1987)
developed the crystal plasticity framework based on the micromechanism of the crystal plastic deformation to predict the
mechanical behavior of the material. Kalidindi (1998) proposed a mathematical description to capture the four-stage strain-
hardening behavior of low stacking fault energy polycrystalline face-centered cubic (FCC) metallic alloys that deform plas-
tically by both slip and twinning mechanisms. Bai and Wierzbicki (2008) postulated a general form of asymmetric metal
plasticity, considering both the pressure sensitivity and the Lode dependence. Barlat et al. (2002) modeled the stress–strain
behavior by using a simple dislocation model, which used three variables to characterize the dislocation population: the
average forest and mobile dislocation densities, and the average dislocation mean free path. Wu (2002) proposed a simple
anisotropic theory of plasticity including the concept of combined isotropic-kinematic hardening. Brünig and Obrecht (1998)
considered a rate-independent formulation that deviated from the classical Schmid-rule. In addition, Brünig (1999, 2001)
proposed a large strain elastic-rate-independent plastic macroscopic model taking into account the hydrostatic stress sen-
sitivity of metals and irreversible dilatant deformation behavior observed in experiments. Lademo et al. (1999), Stoughton
(2002), Stoughton and Yoon (2004, 2006, 2008, 2009), and Cvitanic et al. (2008) used non-associated flow rule models for
metal deformation to obtain a more accurate description of the anisotropic response. Liu and Khan (2012) proposed a phe-
nomenological constitutive model based on experiments to predict the mechanical behavior of alloy materials over wide
ranges of strain rate and temperature.
Among these models, the crystal plasticity framework, which includes the thermal activation model and the dislocation
kinetics model, clearly reflects the physical processes during the plastic deformation and is capable of accurately simulating
the stress–strain behavior. Therefore, it has been widely accepted by the academic community (Kocks, 1987; Krausz and
Krausz, 1996; Kuo et al., 2005; Messerschmidt, 2010; Siddiq and El Sayed, 2011). In this study, the acoustic plasticity frame-
work has been developed based on the crystal plasticity. Among various governing equations of the crystal plasticity (Krausz
and Krausz, 1996), only those used in this study are briefly introduced.
The Taylor model (Frost and Ashby, 1982; Taylor, 1938) establishes the relation between the polycrystal uniaxial stress/
strain and the single crystal shear stress/strain simply by the Taylor factor (M), which is expressed as:
M ¼ r=s ¼ c=e; ð1Þ
76 Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87

3. where r and e are the flow stress and the normal plastic strain, respectively; and s and c are the critical resolved shear stress
and the shear strain in the active slip system. M is assumed constant in this work.
The plastic deformation of metals at low homologous temperatures is mostly dislocation motions and the interactions
between dislocations and obstacles (Kuo et al., 2005). Besides the applied stress, the obstacles can be overcome by the assis-
tance of thermal energy (Messerschmidt, 2010). Therefore, the flow stress of metals at constant strain rate is significantly
affected by the temperature, which can be described by the thermal activation model (Kocks, 1987; Messerschmidt,
2010). One commonly used thermal activation model is in the form of Arrhenius equation (Kocks, 1987), which is expressed
as:
_
cp ¼ _
c0exp
DG
kT
; ð2Þ
where _
cp is the shear plastic strain rate; _
c0 is the pre-exponential factor (also known as frequency factor); k is the Boltzmann
constant; T is the Kelvin temperature; and DG is the Gibbs free-energy of activation for overcoming an obstacle, which de-
pends on the shape, strength, and distribution of obstacles, as well as the applied stress. For short range obstacles, a general
equation of DG for various types of obstacle distribution (Frost and Ashby, 1982; Kocks, 1987) has been suggested as:
DG ¼ DF 1
s
^
s
p
q
; ð3Þ
where DF is the total free energy (or the activation energy) required to surmount the obstacle without the aid from external
stress; and p and q are the obstacle distribution parameters with the range of 0 p 6 1 and 1 6 q 6 2. In this study, p and q
are assumed as p = q = 1 for simplification, which are commonly adopted values (Frost and Ashby, 1982; Kuo and Chu, 2005;
Kuo et al., 2005). ^
s is a material property named mechanical threshold, which can be considered as the shear strength of a
metal at absolute zero (0 K). The mechanical threshold ^
s depends on the dislocation density q, which is usually expressed as
(Barlat et al., 2002; Krausz and Krausz, 1996):
^
s ¼ s0 þ lab
ffiffiffiffi
q
p
; ð4Þ
where s0 is the friction stress; l is the elastic shear modulus; a is a coefficient close to 1/3; b is the length of the Burgers
vector.
The dislocation density evolves during the plastic deformation. One of the typical forms of the dislocation evolution law
(Krausz and Krausz, 1996) is:
dq
dc
¼ k1
ffiffiffiffi
q
p
k2q; ð5Þ
where the dislocation storage coefficient k1 is regarded as a constant; and the dynamic recovery coefficient k2 depends on the
plastic strain rate _
e and the temperature (Krausz and Krausz, 1996). k2 can be expressed as:
k2 ¼ k20
_
e
_
e0
1=n
; ð6Þ
where k20 is a constant; and _
e0 and n are temperature-dependent parameters. In this study, _
e0 and n are assumed constant
since there is no significant temperature variance in the deformation process. Based on Eqs. (4) and (5), ^
s depends on the
dislocation evolution in the plastic deformation.
Eqs. (1)–(6) provide a constitutive model governing the relation between the flow stress r and the normal plastic strain e.
Several material-dependent parameters, i.e., s0, k1, k20, _
e0 and n, in the model may be determined from experiments.
2.2. Modeling of acoustic softening based on thermal activation theory
Applying high-frequency vibration during the plastic deformation may lead to a stress reduction. Combining Eqs. (2) and
(3), s can be expressed as (Frost and Ashby, 1982):
s ¼ ^
s½1 kTlnð_
c0=_
cpÞ=DF: ð7Þ
The high-frequency vibration may affect T, _
c0 and/or DF, or directly generate an acoustic pressure pv to aid the plastic
deformation. Based on the previous study conducted by the authors, no significant temperature rise was observed during
the high-frequency vibration-assisted upsetting experiments (Yao et al., 2012). Hence, the acoustic softening observed in this
study did not involve thermal softening mechanism. The acoustic pressure leads to an effect of stress superposition (or added
stress effect) (Malygin, 2000; Schinke and Malmberg, 1987; Yao et al., 2010), which does not soften the material and cannot
completely account for the stress reduction induced by the high-frequency vibration based on the previous study (Yao et al.,
2012). Consequently, the acoustic softening effect is considered to be mainly attributed to the changes in _
c0 and/or DF due to
the high-frequency vibration.
Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87 77

4. After non-dimensionalizing and substituting kTlnð_
c0=_
cpÞ=DF with W, Eq. (7) converts to:
k ¼ s=^
s ¼ 1 W; ð8Þ
where the non-dimensional stress ratio k only depends on W. For the condition with no aid from other energy, the stress
ratio k0 equals 1 W0. The stress ratio during vibration can be expressed as kv = 1 Wv=1 (W0 + DWv), where subscript
v indicates variable under vibration, and subscript 0 indicates variable without vibration or other forms of excitation energy.
Therefore, the Wv consists of value without vibration (W0) and net-change due to vibration (DWv). The changes of _
c0 and/or
DF due to the high-frequency vibration are reflected in DWv, and it can be found that Dk ¼ kV k0 ¼ DWV . From previous
studies (Huang et al., 2009; Langenecker, 1966; Siddiq and El Sayed, 2011; Siddiq and El Sayed, 2012b), the stress reduction
due to acoustic softening mainly depended on the vibration magnitude. As mentioned in Section 1, the stress reduction by
acoustic softening was observed to be proportional to the acoustic energy density (or acoustic intensity) in the studies by
Siddiq and El Sayed (2011, 2012a) and Langenecker (1966), but was proportional to the vibration amplitude (or square root
of acoustic energy density) in the study by Huang et al. (2009). To provide a generic form to accommodate these discrepan-
cies, an exponential expression was used in the model. The change of the non-dimensional stress ratio, k, induced by the
acoustic softening effect is modeled as Dk ¼ bðE=xÞm
. The negative sign indicates that the involved acoustic energy leads
to a stress decrease. E is the acoustic energy density; b and m are parameters to be found by experiments. m may vary rang-
ing from 0.5 to 1 based on relations found in previous studies (Huang et al., 2009; Langenecker, 1966; Siddiq and El Sayed,
2011; Siddiq and El Sayed, 2012b). x is a term to non-dimensionalize E. Non-dimensionalizing E with ^
s scales the input vibra-
tion energy density with the mechanical threshold of a given material, which leads to:
Dk ¼ kV k0 ¼ DWv ¼ bðE=^
sÞm
: ð9Þ
Eq. (9) provides a relation between the normalized input acoustic energy density E and the net-change in stress ratio
(DWv). Physically, the change of W induced by the high-frequency vibration leads to the acoustic softening effect and the
decrease of the forming stress during the vibration.
The framework proposed above provides a simplified model to explain the acoustic softening based on the thermal acti-
vation theory. Besides the acoustic energy, the electric energy has also been proved to have the capability to soften the met-
als (Conrad, 2000; Dzialo et al., 2010), which has been used in the electricity-assisted metal forming. The effect of electric
field on plastic deformation in metals can also be explained by the proposed framework.
2.3. Modeling of acoustic residual effects based on dislocation evolution theory
The observed acoustic residual hardening or softening effect is attributed to the change of the dislocation density, and there-
fore, results in a residual effect. Reflected in Eq. (5), k1 and k2 may change due to the acoustic field, leading to the change of the
dislocation evolution.Inthisstudy,itisassumed that the additional changes inEq. (5) induced by thevibration are allequivalently
included in the changes of k1 and k2. Then, the dislocation evolution process exposed in the acoustic field can be expressed as:
dq
dc
¼ k1ð1 þ gk1Þ
ffiffiffiffi
q
p
k2ð1 þ gk2Þq; ð10Þ
where the non-dimensional parameters gk1 and gk2 are the change ratios of k1 and k2 due to the exposure in the acoustic
field, respectively. Physically, gk1 and gk2 relate to the additional dislocation multiplication and annihilation induced by
the high-frequency vibration, respectively. gk1 and gk2 may depend on material properties, acoustic field parameters, and
the duration exposed in the acoustic field. In this study, acoustic residual hardening was dominant, which had also been
the case in prior studies by Langenecker (1966) and Tyapunina et al. (1982). Therefore, the acoustic residual hardening
was assumed to be related to gk1, while gk2 was neglected in this study.
In acoustic residual hardening, the dislocation density cannot increase infinitely, and therefore, saturation occurs after a
long exposure in the acoustic field. This saturation phenomenon was experimentally detected in previous studies (Pesloa,
1984; Tyapunina et al., 1982). To describe this saturation behavior in the form of S-shape, a logistic function, which is a typ-
ical growth function applied in various fields (Gallagher, 2011; Thornley et al., 2007; Tsoularis and Wallace, 2002), has been
adopted. This phenomenological model is expressed as gk1 ¼ KP0=½P0 þ ðK P0ÞertV for a vibration duration of tv, where P0
and K are the initial and the limiting (saturation) value of gk1, respectively. r is the growth rate, which is affected by the mag-
nitude of the acoustic field. For simplification, the growth rate r is assumed to be proportional to the vibration amplitude n
with a rate of u. With these substitutions, Eq. (10) becomes:
dq
dc
¼ k1 1 þ
K
1 þ ðk=P0 1Þe/nV
ffiffiffiffi
q
p
k2q: ð11Þ
The dependence of Eq. (11) on the vibration duration tv indicates that this model is history-dependent rather than con-
dition-dependent.
Eq. (11) provides a means to phenomenologically describe the additional increase of dislocation density due to the acous-
tic residual hardening. Besides the acoustic residual hardening observed in this study, the proposed framework can also cap-
ture the acoustic residual softening through gk2. When acoustic residual softening occurs (Lum et al., 2009), the dislocation
density may decrease indicating that gk2 is dominant and should not be neglected.
78 Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87

5. 3. Experimental setup
To investigate the metal plasticity under high-frequency vibration, an experimental setup was developed for vibration-
assisted upsetting (compression) tests as shown in Fig. 1. A DC motor (071-300-0058, Bison) controls the compression mo-
tion, while a magnetostrictive (Terfenol-D) transducer (CU-18A, Etrema Products Inc.) generates high-frequency oscillation
applied to the specimen. The stress–strain curves in the compression tests are obtained from the force sensor (9133B21, Kis-
tler) and the laser displacement sensor (optoNCDT 1401, Micro-Epsilon). The details of the setup can be found in the earlier
publications by the authors (Yao et al., 2011, 2012; Yao et al., 2010).
The vibration generated by the magnetostrictive transducer is amplified by a titanium horn, which is directly used as the
compression punch. The longitudinal and transverse vibration at the horn tip was measured by an inductive displacement
sensor (SMU-9000, resolution 0.1 lm, Kaman), which is connected to a DSP lock-in amplifier (SR830, Stanford Research Sys-
tems) to capture the oscillation amplitude. With the excitation frequency of 9.6 kHz, there is significant transverse vibration
at the horn tip, which was chosen as the working frequency in this study. The relation between the transverse oscillation
amplitude at the horn tip and the input voltage amplitude applied to the transducer is plotted in Fig. 2, which displays a near
linear relationship.
Commercially pure aluminum (Al 1100, McMaster-Carr) in annealed condition was used in the upsetting tests. The
dimension of each sample is 2.032 mm in diameter and 2 mm in length. The material after annealing usually has a random
texture or a very weak texture (Azari et al., 2004; Doherty, 1997; Liu et al., 2008; Liu and Morris, 2005; Poorganji et al., 2010).
Therefore, it was assumed M = 3.06, a typical value for polycrystalline FCC metals, which had been used in numerous studies
(Frost and Ashby, 1982; Haasen, 1996; Kassner, 2004; Lian and Chen, 1991; Stoller and Zinkle, 2000).
It was very difficult to directly measure the oscillation in the samples during vibration-assisted compression, so the mag-
nitude of the acoustic field in the sample was estimated based on the vibration at the horn tip. The sound power transmitted
coefficient from the titanium horn to the aluminum specimen can be expressed as at = 4qAlcAlqTicTi/(qAlcAl + qTicTi)2
(Frederick, 1965), where qAl and qTi are the material densities for aluminum and titanium, respectively; and cAl and cTi
are the transverse wave speeds for aluminum and titanium, respectively. The transverse wave speed was calculated based
on Ci ¼
ffiffiffiffiffiffiffiffiffiffiffi
ffi
Gi=qi
p
, where G is the shear modulus and the subscript i is the material. Then, at was calculated as 0.9355 based on
material properties. Consequently, the sound energy density in the sample can be expressed as (Frederick, 1965; Pierce,
2007):
E ¼ n2
A1x2
qA1 ¼ n2
Tix2
qTiat; ð12Þ
where nTi is the vibration amplitude at the titanium horn tip shown in Fig. 2; nAl is the vibration amplitude in the specimen;
and x is the excitation angular frequency. Based on Fig. 2 and Eq. (12), the sound energy density E for various input voltage
can be calculated, and the results are shown in Fig. 3.
In addition, the acoustic pressure in the aluminum specimen can be expressed as pAl = qAlcAlxnAl = (EGAl)1/2
(Frederick,
1965). When the input voltage amplitude was 60 V, the acoustic energy density in the sample was around 125 J/m3
from
Fig. 3. Consequently, the acoustic pressure in the aluminum specimen pAl was around 1.8 MPa when the input voltage ampli-
tude was 60 V.
Fig. 1. Experimental setup for vibration-assisted upsetting tests.
Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87 79

6. 4. Experimental results and model development
This section discusses the experimental results and modeling prediction based on the parameters determined from the
experiments. While details of the parameters and predicted results are discussed in sub-sections, a general calculation pro-
cedure is described as the following: (i) identify the basic parameters by curve-fitting between the model and the experi-
ments without vibration; (ii) identify the parameters related to the acoustic softening and hardening; and (iii) simulate
the stress–strain curves with vibration and compare with the experimental results.
4.1. Basic experimental results for acoustic plasticity
Based on the developed experimental setup, upsetting tests were conducted with the aid of the high-frequency vibration.
The vibration was turn on and off during each upsetting experiment to clearly show the effects of the high-frequency vibra-
tion on metal plasticity. Significant acoustic softening effect and acoustic residual hardening effect were detected in various
experiments, among which several typical experimental results are illustrated in Fig. 4. The input voltage amplitude applied
to the transducer was 60 V. The flow stress during the vibration oscillates in the stress–strain curve, however, only the aver-
age stress is shown to avoid overlapping of curves in Fig. 4.
As shown in Fig. 4, Line 1 is a typical stress–strain curve from upsetting without vibration. For the process denoted as Line
2, the flow stress was significantly reduced as soon as the high-frequency vibration started. On the other hand, the flow
stress recovered to even higher than Line 1 after the vibration had stopped, displaying an acoustic residual hardening. In
the process, the temporary acoustic softening effect emerges immediately when vibration starts but also disappears after
vibration stops. Although the acoustic residual hardening effect is detected after stopping the vibration, the change of the
dislocation density should gradually accumulate while the vibration is applied. It is reasonable to assume that this hardening
effect is negligible or small at the beginning of the vibration, but results in noticeable amount after a long vibration duration
as observed in Line 2 in Fig. 4. This hardening effect during the vibration, however, is obscured by the relatively large acoustic
softening effect. In order to validate this assumption, experiments with short duration of vibration were conducted with a
typical result shown as Line 3. Compared with the duration of 8 s in Line 2, the vibration duration of 2 s in Line 3 did not
display acoustic residual hardening, i.e., there was no noticeable stress increase once the vibration had stopped. Too short
of a vibration duration cannot lead to significant acoustic residual hardening effect. It should also be noted that the stress
Fig. 2. The transverse vibration amplitude at the horn tip at different input voltages.
Fig. 3. The acoustic energy density in the compression sample for different input voltage.
80 Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87

7. decrease (acoustic softening) during the vibration in Line 3 is larger than the stress decrease in Line 2 at the same strain. The
gap between Line 2 and Line 3 during the vibration comes from the acoustic hardening present during vibration. These
experimental phenomena support the above assumption that the additional increase of dislocation density due to the acous-
tic hardening gradually accumulates during the vibration.
Experiments were performed at different strain rates to verify whether the acoustic residual hardening effect was depen-
dent on the duration of the vibration (time-dependent) or only the strain during the vibration (strain-dependent). A typical
experimental result is shown as Line 4 in Fig. 4. The strain rate in Line 4 is double the one in Line 2, while the vibration dura-
tion in Line 4 is half of Line 2. Therefore, the stress–strain curve during the vibration in Line 4 closely followed that of Line 2,
but the total vibration duration applied was different. The results showed that the stress increased once the vibration
stopped in Line 4 but was less than the stress increase in Line 2. The observation indicates that the residual strain hardening
response increased with longer vibration duration even the strain was the same. The dislocation evolution law expressed as
Eq. (5) only considers strain dependency. However, the acoustic residual hardening effect is time-dependent rather than
strain-dependent, which is supported by the experiments shown in Fig. 4.
4.2. Experimental results and model development for acoustic softening
Acoustic softening effect definitely depends on the vibration magnitude. Based on the developed setup, the vibration
magnitude can be varied by changing the input voltage amplitude supplied to the transducer as shown in Fig. 3. Experiments
with various input voltage amplitudes were conducted to investigate the influence of vibration magnitude on the acoustic
softening. Typical stress–strain curves of compression tests with and without high-frequency vibration for various vibration
magnitudes are shown in Fig. 5. The acoustic softening effect due to the high-frequency vibration significantly increased with
increasing vibration amplitude. In addition, there was a sharp overshoot at the start of the vibration when the input voltage
amplitude was 60 V, which can be observed in Fig. 4 as well. This overshoot is considered to be an artifact caused by the
control system, which did not show up when the vibration amplitude was small. The acoustic energy density E in the sample
can be obtained based on the input voltage amplitude as plotted in Fig. 3. For various magnitudes of the acoustic field, the
stress decrease due to the acoustic softening at the onset of vibration was measured from the stress–strain curves.
To apply the developed modeling framework in Section 2, several parameters introduced in Section 2.1 were identified
based on upsetting experiments without vibration. The identified parameters are listed in Table 1. When the vibration is ap-
plied, the dependence of the stress decrease on the magnitude of the acoustic field is expressed by Eq. (9), where the param-
eters m and b need to be determined from experiments. Combining Eqs. (1) and (7), it can be obtained that:
^
s ¼ r=½M MkTlnð_
c0=_
cpÞ=DF; ð13Þ
^
s at each strain can be calculated based on Eq. (13), parameter values in Table 1, and the flow stress r obtained from
experiments. For example, the flow stress r in the experiments was 94.8 MPa at the onset of the vibration in Fig. 4. s
was 31 MPa, and ^
s was 39.4 MPa based on Eq. (1) and (13). The decrease of the non-dimensional stress ratio |Dk| is then
obtained based on Eq. (8). The dependence of |Dk| on E is plotted in Fig. 6. The experimental results in Fig. 6 show that
|Dk| is nearly proportional to
ffiffiffiffiffiffiffiffi
E=^
s
p
, i.e., the stress decrease due to acoustic softening is almost proportional to the vibration
amplitude, which agrees with the experimental results by Huang et al. (2009). Therefore, the exponential factor m in Eq. (9)
is considered to be 0.5 for the aluminum in this study. By using the linear model expressed as Eq. (9), b was 207.9 from curve
Fig. 4. Stress–strain curves of typical compression tests with and without high-frequency vibration (v is the punch speed; and tH is the vibration duration).
Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87 81

8. fitting. At the onset of the vibration, the reduction of s (about 15 MPa) was much larger than the applied acoustic stress of
1.8 MPa calculated in Section 3. This supports that the stress reduction in this study cannot be solely explained by the stress
superposition or added stress effect.
4.3. Experimental results and model development for acoustic residual hardening
Acoustic residual hardening accumulates over time and is time-dependent based on the results in Fig. 4. This is further
confirmed by additional experiments performed for various vibration durations at the same input voltage amplitude of 60 V.
The stress increase due to acoustic residual hardening Drh was measured from the stress–strain curves when the vibration is
stopped. The dependence of Drh on vibration duration tv is plot in Fig. 7. When the vibration duration was less than 2 s, no
significant acoustic residual hardening effect was detected. The rate of hardening increased significantly when the vibration
duration was around 3 s, and then decreased around 6 s. The overall trend of curve is S-shaped, which agrees well with the
assumption in Section 2.3.
Acoustic residual hardening is also affected by the magnitude of the acoustic field. Experiments with various input volt-
age amplitudes under the same vibration duration of 8 s were conducted to investigate the influence of vibration magnitude
Fig. 5. Stress–strain curves of compression tests with and without high-frequency vibration for various vibration magnitude (U is the input voltage
amplitude applied to the transducer).
Table 1
Parameter values used in the model and experiments.
Parameters Values Sources
Parameters obtained from references Taylor factor M 3.06 (Haasen, 1996)
Pre-exponential factor (s1
) _
c0 106
(Frost and Ashby, 1982)
Coefficient in Eq. (4) a 1/3 (Barlat et al., 2002)
Burgers vector length (nm) b 0.286 (François et al., 1998)
Elastic shear modulus (GPa) l 26 (Cubberly et al., 1979)
Yield strength (MPa) Y 34 (Kaufman, 1999)
Activation energy DF 0.5lb3
(Frost and Ashby, 1982)
Working parameters Kelvin temperature (K) T 300
Punch speed (mm/s) v 0.085
Vibration frequency (kHz) f 9.6
Parameters identified from experiments Friction stress (MPa) s0 0.1
Coefficient in Eq. (5) k1 1.396 108
Coefficient in Eq. (6) k20 40.93
Coefficient in Eq. (6) _
e0 3.212 103
Coefficient in Eq. (6) n 1.610
Coefficient in Eq. (9) b 207.9
Coefficient in Eq. (9) m 0.5
Coefficient in Eq. (11) P0 5.521 103
Coefficient in Eq. (11) K 8.339 102
Coefficient in Eq. (11) u 4.170 105
82 Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87

9. on the acoustic residual hardening. After measuring Drh for various magnitudes of the acoustic field, the dependence of Drh
on the acoustic energy density E is plotted in Fig. 8. The results also displayed a S-shape dependence and existence of a sat-
uration value for the acoustic residual hardening.
Eq. (1) through Eq. (6) provide a constitutive relation for a regular plastic deformation in metallic materials without vibra-
tion. By replacing Eq. (5) with Eq. (11), the constitutive relation can account for the acoustic residual hardening predicting
the stress increase during and after vibration. Based on the experimental results in Fig. 7 and 8, the parameters P0, K and u in
the model is optimized to fit the experimental results, which are listed in Table 1. The fitted curves with the parameter val-
ues in Table 1 are provided in Fig. 7 and 8 and shows a reasonable agreement with the experimental values.
4.4. Prediction of stress–strain curves based on the developed model
The stress–strain curve of a typical experiment without vibration is shown as Line 1 in Fig. 9(a) (the same curve as Line 1
in Fig. 4). This stress–strain curve can be simulated by the model governed by Eq. (1) through Eq. (6). Only the plastic behav-
ior is considered in this study, while the minor elastic deformation is neglected. The initial condition in the numerical cal-
culation is the yield state with the flow stress r equaling the yield strength Y. Table 1 summarizes all the parameter values
used in the model and their sources. The five parameters, s0, k1, k20, _
e0 and n, in the model were determined by curve fitting.
The predicted stress–strain curve based on the model described in Section 2.1 is plot as Line 2 in Fig. 9(a), which shows a
good agreement with the experimental result (Line 1).
The stressstrain curves in the vibration-assisted upsetting tests with various vibration amplitudes and durations are
shown as Line 3 (the same curve as Line 2 in Fig. 4), Line 5 (the same curve as Line 3 in Fig. 4) and Line 7 in Fig. 9(a).
The upsetting curve during the initial segment without vibration can be simulated by the model governed by Eq. (1) through
Eq. (6). Once the vibration starts, the acoustic softening effect is included in the model by replacing Eqs. (2) and (3) with
Eqs. (7)–(9), and the acoustic residual hardening is incorporated into the simulation by substituting Eq. (5) with Eq. (11).
Thus, Eqs. (1), (4), (6), (7), (8), (9), and (11) govern the stress–strain relation during the high-frequency vibration. After
the vibration stopped, the acoustic softening effect disappeared. Thus, Eqs. (7)–(9) are switched back to Eqs. (2) and (3).
For the acoustic residual hardening, the dislocation evolution is still governed by Eq. (11) due to the residual effect.
Fig. 6. The dependence of the nondimensional stress ratio decrease on the acoustic energy density in the specimen (Dk is the nondimensional stress ratio,
defined as Eq. (8) and (9); and E is the acoustic energy density in the specimen).
Fig. 7. The dependence of the stress increase due to acoustic residual hardening on the vibration duration (Drh is the stress increase due to acoustic
residual hardening when vibration stopped; and tv is the duration of the vibration applied).
Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87 83

10. gk1 was unchanged once the vibration stopped, since the vibration duration no longer increased. Therefore, Eqs. (1)–(4), (6),
and (11) govern the stress–strain relation after the vibration stopped.
As shown in Fig. 9(a), there were ramp segments at recovery after stopping the vibration, which were also detected in
other studies (Daud et al., 2006, 2007). This nearly linear response in transition included the elastic behavior of the actual
material and the testing system. Since the proposed model only describes the material plastic behavior, this transition
behavior including elasticity cannot be simulated solely by the proposed model. In the simulation process, this transition
Fig. 9. Prediction of the stress–strain curves in the vibration-assisted upsetting processes based on the developed model: (a) comparison of the stress–
strain curves between model prediction and experimental results, and (b) prediction of the dislocation density in the vibration-assisted upsetting processes
(ER denotes an experimental result; PR denotes an predicted result; tH is the vibration duration; and U is the input voltage amplitude applied to the
transducer).
Fig. 8. The dependence of the stress increase due to acoustic residual hardening on the acoustic energy density in the specimen (Drh is the stress increase
due to acoustic residual hardening after vibration stops; and E is the acoustic energy density in the specimen).
84 Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87

11. segment was simulated with an equivalent elastic portion. When the elastic stress increased to the predicted plastic stress, it
was replaced with the proposed plastic model. The elastic constant in the recovering curves was adjusted to match the
experimental results in Fig. 9(a), and the equivalent elastic modulus was 0.7 GPa in the simulation.
Combining the calculation processes described above, the predicted stress–strain curves for the vibration-assisted upset-
ting processes are shown as Lines 4, 6 and 8 in Fig. 9(a). A good agreement was found between the predicted curves and the
experimental results as shown by Line 3, 5 and 7. As discussed in Sections 4.2 and 4.3, the behavior of the stress–strain curve
during vibration is affected by both acoustic softening and hardening, which has been accurately captured by the proposed
model. In addition, the evolution of the dislocation density during the upsetting processes were predicted and plotted in
Fig. 9(b). Since the temporary acoustic softening effect cannot be explained by changes of the dislocation density in this
study, the dislocation density evolution only reflects the influence of acoustic residual hardening.
5. Discussion
In this study, the effects of high-frequency vibration on the stress–strain response of aluminum were investigated. The
observed acoustic softening effect was temporary only appearing during the application of the high-frequency vibration
and vanishing as soon as the vibration had stopped. Therefore, the acoustic softening cannot be simply explained by the
change of dislocation density. On the contrary, acoustic residual softening as mentioned in the prior studies (Huang et al.,
2009; Lum et al., 2009; Siu et al., 2011) is a residual effect induced by the high-frequency vibration. This is related to the
change of dislocation density. Consequently, the mechanisms of acoustic softening and acoustic residual softening are dif-
ferent. Moreover, the acoustic softening cannot be solely explained by the stress superposition (added stress effect) (Malygin,
2000; Schinke and Malmberg, 1987; Yao et al., 2010) or heating effect based on authors’ previous studies (Yao et al., 2012).
Regarding the crystal plasticity model of Eq. (7), the acoustic softening is related to the changes in the frequency factor _
c0
and/or the activation energy DF rather than the change of the dislocation density. _
c0 is related to the attempt frequency for
dislocation motion (for softening, it is increased), and DF is physically related to the energy barrier for dislocation motion (for
softening, it is lowered). The additional energy applied in the form of vibration helps to increase the attempt frequency and/
or lower the energy barrier for the dislocation motion. To the best of the authors’ knowledge, it is the first time that the
acoustic softening during high-frequency vibration is interpreted as such. A similar approach may be found in electric energy
assisted metal forming (Conrad, 2000; Dzialo et al., 2010), where the role of _
c0 and DF has been discussed in explaining the
softening effect by the applied current. This potentially suggests that the effects of various forms of energy such as acoustic
and electric energy in metal deformation can be uniformly interpreted using Arrhenius equation.
From the observations in this study and several prior studies, either acoustic residual hardening (Gindin et al., 1972;
Langenecker, 1966; Tyapunina et al., 1982; Westmacott and Langenecker, 1965) or softening (Huang et al., 2009; Lum
et al., 2009; Siu and Ngan, 2011; Siu et al., 2011) could occur in vibration-assisted metal forming. It is speculated that the
initial dislocation density of the material may be a factor contributing to these different outcomes (hardening or softening).
The acoustic residual hardening often occurs in well-annealed metals with low initial dislocation density, while work-
hardened metals are apt to experience acoustic residual softening (Huang et al., 2009; Lum et al., 2009; Siu and Ngan,
2011). This may be the reason why acoustic residual hardening was experimentally detected in this study, since fully an-
nealed aluminum was used. In reality, both k1 and k2 may change during vibration. However, it is difficult to precisely mea-
sure evolution of k1 and k2 during vibration due to the combined effects of both acoustic softening and hardening during the
vibration as described in Section 4.1. Therefore, the change of k2 is neglected in this study.
Measuring of dislocation densities with and without vibration (Pesloa, 1984; Tyapunina et al., 1982) confirms that the
acoustic residual hardening is caused by the additional dislocation multiplication; however, the exact physics behind these
effects are still unclear. In this study, the effects of high-frequency vibration on the dislocation multiplication are phenom-
enologically included in the change of k1, which is history-dependent. Physically, k1 is related to the ratio of the mean free
path travelled by the dislocations to the average obstacle spacing. k1 statistically depends on the material microstructures,
which may have been altered by the high-frequency vibration. The studies (Siu et al., 2011; Westmacott and Langenecker,
1965) that demonstrated significant subgrain formation induced by ultrasound provides a proof for the microstructural
change due to the vibration. If it is true that the change of k1 is related to the change of the substructure, k1 should remain
unchanged right after stopping the vibration since the substructure cannot revert back instantaneously. Nevertheless, the
substructure may revert back to the original structure after some strain, which implies that k1 should switch back to its ori-
ginal value. This certainly is a possibility, but further investigation is needed. In this study, however, such phenomenon has
not been clearly observed due to the limitation of the experimental setup.
The limit of the input voltage amplitude of the transducer (CU-18A, Etrema Products Inc.) is 60 V. Therefore, experiments
with extremely high vibration amplitude could not be conducted. For acoustic residual hardening, a near-saturation behavior
is observed in Fig. 7 and Fig. 8. The stress decrease due to the acoustic softening, however, remains almost linearly dependent
on the vibration amplitude within the current testing limit. This linear dependence certainly will not be valid if the stress
decrease calculated based on the model is larger than the static stress. The limit of the acoustic softening effect is to deform
the material with very low applied stress (close to zero), which has been observed by prior study (Langenecker, 1966).
Quantitatively, based on the model developed in Section 2.2, kV ¼ k0 þ Dk must be larger than zero. From Eqs. (8) and (9),
the application range of the model expressed in Eq. (9) can be described as:
Z. Yao et al. / International Journal of Plasticity 39 (2012) 75–87 85

12. E 6
s ^
sðm1Þ
b
1=m
: ð14Þ
Namely, the dependence of the stress decrease due to acoustic softening on the vibration amplitude is no longer linear
when E approaches s ^
sðm1Þ
=b
1=m
.
Besides the softening and hardening effects, the vibration may influence friction involved in a metal forming operation.
For the case of upsetting, the influence of friction on reduction of flow stress proved to be insignificant compared with the
acoustic softening and hardening effects as noted in the authors’ previous study (Yao et al., 2012).
6. Conclusions
The acoustic softening and the acoustic residual hardening in the vibration-assisted plastic deformation of aluminum
were investigated in this study. A unified acoustic plasticity model based on the crystal plasticity theory was developed
to accurately capture the acoustic softening and hardening phenomena. The acoustic softening was modeled based on ther-
mal activation theory. The frequency factor _
c0 and the activation energy DF are considered to be significantly affected by the
applied acoustic field, leading to the decrease of the flow stress during vibration. The acoustic residual hardening was mod-
eled based on dislocation evolution theory. The multiplication speed of dislocation increases with increase in exposure time
to the acoustic field, resulting in the increase of the flow stress. To validate the proposed model, high-frequency vibration-
assisted upsetting tests were performed for pure aluminum. Significant acoustic softening effect during the vibration and
acoustic residual hardening were detected in the experiments. The flow stress decrease due to the acoustic softening during
the vibration was found to be nearly proportional to the vibration amplitude. The acoustic residual hardening was affected
by the vibration duration and magnitude. The acoustic residual hardening was found to be time-dependent rather
than strain-dependent. The stress increase after the vibration due to acoustic residual hardening had a S-shaped increase
over the vibration duration. Based on the proposed model, the stress–strain curve of the aluminum specimen under
high-frequency vibration can be accurately predicted. The model helps to understand the underlying mechanisms of
high-frequency vibration in vibration-assisted metal forming or bonding processes.
Acknowledgements
The authors greatly appreciate the financial support from the United States National Science Foundation (CMMI-
0800353). We also acknowledge the support from National Natural Science Foundation of China (Grant No. 50930005,
51175460), Zhejiang Provincial Natural Science Foundation of China (Grant No. Z1090373), and China Scholarship Council
for the international collaborative research opportunity.
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