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Fracture test of nanocomposite ceramics under ultrasonic vibration based on nonlocal theory
Ceramics International 45 (2019) 20945–20953
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Fracture test of nanocomposite ceramics under ultrasonic vibration based on nonlocal theory
T
Jinglin Tong*, Peng Chen, Junshuai Zhao, Bo Zhao School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, 45400, Henan, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Nonlocal theory Ultrasonic vibration Three-point bending Crack propagation Nanocomposite ceramics
Based on the non-local theory, the dispersion characteristics of ultrasonic vibration in three-point bending are analyzed, and the effects of ultrasonic frequency and amplitude on the dispersion characteristics are obtained through theoretical analysis. Three-point bending tests were carried out using an ultrasonic vibration system and specimens, and the test process as well as the workpiece after the test were observed and analyzed. The effects of ultrasonic vibration on fracture properties and crack growth mechanism of ceramics under different loading conditions were verified by experiments. The introduction of ultrasonic vibration changes the internal stresses of the ceramic fracture, affects the hardness of the workpiece, and also causes the scattering of particles, which leads to the intensification of ultrasonic attenuation and phase velocity dispersion. SEM and XRD analysis showed that with the increase in ultrasonic frequency and amplitude, the transgranular fracture was more obvious, tearing edges and dimples increased, and microvoids increased in size and number. Ultrasonic vibration can cause ZrO2 to undergo t-m phase transformation under stress induction and absorb the strain, thus greatly improving the mechanical properties of nanocomposite ceramics. Due to the toughening mechanism of the inherent phase transformation, the material has improved plastic mechanical properties, and it is easier to achieve ductile domain processing.
1. Introduction Ceramic materials are widely used in aviation and other high-end technology fields. Because of their high strength, high hardness, high temperature resistance, wear resistance, corrosion resistance, chemical stability, and good biocompatibility, they have broad application prospects in aerospace, aviation industry, machinery, petroleum industry, and nuclear industry [1]. However, because of their brittleness and sensitivity to internal defects, ceramic materials show sudden and catastrophic fracture without warning. Therefore, the reliability of ceramic components is poor, which is a bottleneck that affects the popularization and application of ceramic materials [2]. With the development of nanotechnology, nano-sized ceramic particles were introduced into a micron-sized ceramic matrix as a dispersed phase, and nano-composite ceramics were prepared, which achieved an obvious strengthening and toughening effect [3]. Nanocomposite ceramics are structural ceramics with excellent high temperature mechanical properties. In recent years, many scholars at home and abroad have studied nanocomposite ceramic materials. Soh A K et al. [4] studied the effect of nanoparticles on the toughness of nanocomposite ceramics. The results showed that with the increase of the volume fraction of nanoparticles, the
*
transgranular fracture would increase, and the aggregation of nanoparticles would lead to the decrease in the strength and toughness of nanocomposite ceramics. Zhao Bo et al. [5] analyzed the brittleness, toughness, and material removal mechanism of nano-composite ceramics, established the formula for critical toughness and grinding depth of nano-composite ceramics, and carried out grinding experiments under ultrasonic and ordinary processes. It was found that the critical grinding depth of nano-composite ceramics under ordinary grinding was 12 μm, and under ultrasonic grinding was 20 μm. Based on impulse theory and fracture mechanics, Wu Y et al. [6] analyzed the contact model of two-dimensional ultrasonic vibration of abrasive particles and the workpiece. The experiment shows that toughness grinding of ceramics can be realized only when the grinding depth is less than the critical grinding depth and the appropriate grinding parameters were determined. Yunteng Wang et al. [7] established a thermo-mechanical coupling bond-based peridynamic model to study the thermal fracturing behavior of ceramic nuclear pellets, which overcomes the difference in typical time scales between thermal and mechanical systems, and can accurately predict the thermal crack morphology. Radial cracks occur when power increases and circumferential cracks occur when power decreases. In addition to the advantages of ordinary ceramics,
Corresponding author. E-mail address: [email protected] (J. Tong).
https://doi.org/10.1016/j.ceramint.2019.07.084 Received 15 May 2019; Received in revised form 5 July 2019; Accepted 8 July 2019 Available online 09 July 2019 0272-8842/ © 2019 Published by Elsevier Ltd.
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Nomenclature
u (x , t ) w (x , t ) εxx σxx Q M ρ S E EI e0 a ωn N A
wave number circle frequency of the wave phase velocity group velocity classic Lame coefficient Poisson's ratio of the material phase velocity of the shear wave in the classical theory internal stress of vibration speed of sound frequency of ultrasonic bending strength the maximum load measured span of the specimen width of the specimen height of the specimen
kn ω Cp Cg μ δ C1 σ c f σf P L0 b h
axial displacements of any point lateral displacements of any point bending strain axial stress on the section axial force bending moment density of the material cross-sectional area Young's modulus of the material bending stiffness of the beam non-local scale parameter circular frequency of the nth sampling point the number of sampling points amplitude of ultrasonic
nano-ceramics have greatly improved their mechanical properties and reliability, but their plastic toughness is still very low and their hardness is high. It is difficult to achieve high precision, high efficiency, and high reliability in their processing [8,9]. In the study of solid mechanics, Eringen et al. [10,11] proposed the theory of nonlocal linear elasticity and applied it widely. This nonlocal theory is based on the theory of lattice dynamics and phonon scattering. It assumes that the interaction between atoms in a material is a longrange force and that the stress state at a point in an object is related not only to the strain at that point, but also to the strain at other points in the entire object. The theory of nonlocal elasticity describes the relative effect of the stress at given locations on the strain states at other locations. This type of constitutive relation involves the integral of the entire body and includes nonlocal kernel functions. This belongs forms a part of generalized continuum mechanics, which is anextension and development of classical continuum mechanics. This theory has contributed to several advancements in research related to fracture [12], dislocation [13], crack initiation and propagation [14], wave propagation and so on. Reddy J. N. et al. [15] deduced the non-local theoretical equations of motion, and gave the analytical solutions for bending, vibration, and buckling, revealing the influence of the nonlocal theory on deflection, buckling load, and natural frequency. Huang W. G et al. [16] and others used this theory to study the stability of compressive rods and the axial vibration of elastic rods under three boundary conditions, and deduced the nonlocal theoretical solutions of critical pressure and natural frequency. Bian P.Y et al. [17] established the non-local constitutive model under ultrasonic vibration and studied the grinding of nanocomposite ceramics. The results showed that the grinding force could be greatly reduced by applying ultrasonic vibration, the grinding force increased with the increase in frequency, and the surface quality was improved. To aid the study of precise and highefficiency processing of hard and brittle materials and realize the ductile domain processing of hard and brittle materials, this study presents a non-local constitutive model of ultrasonic vibrating nanocomposite ceramics under three-point bending based on the non-local theory and fracture mechanics. The effect of ultrasonic vibration on the fracture, crack propagation, and microscopic characteristics of nanocomposite ceramics was studied by three-point bending. The effects of ultrasonic vibration on the mechanical properties of nanocomposite ceramics were analyzed by SEM and XRD observations on the fracture of three-point bending specimens.
Fig. 1. Stress synthesis diagram of nano-beam. ∂w (x , t ) ∂x
⎧ u x (x , z , t ) = u ( x , t ) − z ⎪ u y (x , z , t ) = 0 ⎨ ⎪ u ( x , z , t ) = w (x , t ) z ⎩
(1)
2. Nonlocal wave dispersion characteristics of three-point bending in ultrasound vibration of nanocomposite ceramic rods 2.1. Constitutive equation of Euler-Bernoulli beam with nonlocal motion Based on the Euler -Bernoulli beam theory, Fig. 1 axial and lateral displacement fields at any point in the rod are as follows [18]:where, u (x , t ) and w (x , t ) are the axial and lateral displacements of any point in the rod along the central axis. Bending strain is given by:
∂u x ∂u ∂ 2w = −z 2 ∂x ∂x ∂x
εxx =
(2)
According to the stress relationships, the Euler beam-Bernoulli theory of motion equation can be stated as follows:
⎧
∂Q ∂x
= ρS
∂2u ∂x 2
⎨ ∂2M = ρS ∂2w ∂t 2 ⎩ ∂x 2 Q=
∫ σxx dS S
(3)
M=
∫ zσxx dS S
(4)
where, σxx is the axial stress on the section, Q is the axial force, and M is the bending moment. In contrast with the linear algebraic local stress-strain equation, the non-local constitutive relation introduces the differential stress-strain relation. Assuming that the beam is homogeneous and isotropic, the stress-strain relationship based on the Euler-Bernoulli beam theory is expressed by the non-local constitutive relation, and the non-local constitutive relation of the nano-beam is as follows:
σxx − (e0 a)2
∂2σxx = Eεxx ∂x 2
(5)
From the non-local constitutive relation and the proposed equation of motion, according to the generalized displacement condition, the 20946
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moment equation can be expressed as follows:
M = −EI
∂ 2w ∂x 2
+ (e0 a)2ρS
∂ 2w (6)
∂t 2
The non-local motion equation of the Euler-Bernoulli beam is:
EI
∂ 4w ∂ 2w ∂ 4w + ρS 2 − ρS (e0 a)2 2 2 = 0 4 ∂x ∂t ∂x ∂t
(7)
where, ρ is the density, S is the cross-sectional area, EI is the bending stiffness of the beam, and e0 a is the non-local scale parameter. It can be seen that if the internal length a is zero, then equation (7) reduces to the local Euler-Bernoulli beam equation. 2.2. Wave dispersion characteristics The wave characteristic parameters of nano-beams are the wave number and wave velocity. The time variable can be eliminated from the nonlocal beam equation (7) of the partial differential equation by using Fourier transform. The transverse displacement is expressed as follows [19]: N
w(x , t ) =
∑ wˆ (x , ωn) e−iωt
(8)
n=1
where, ωn is the circular frequency of the nth sampling point, and N is the number of sampling points. N should be large enough to achieve better resolution for both high and low frequency response. By introducing equation (8) into equation (7), we obtain: N
d4wˆ
∑ ⎡⎢EI dx 4
n=1
⎣
+ ρSωn2 wˆ − ρSωn2 (e0 a)2
d 2w ⎤ −iωt e =0 dx 2 ⎥ ⎦
(9)
The equation must satisfy every n, so that it can be transformed into a general differential equation with a single parameter x:
EI
d4wˆ d 2w + ρSωn2 ⎡wˆ − (e0 a)2 2 ⎤ = 0 ⎢ dx 4 dx ⎥ ⎣ ⎦
wˆ (x ) =
Ae−ikn t
(10) (11)
where, A is the ultrasonic amplitude, kn is the wave number. Since the amplitude A is not zero, the wave number kn is obtained by solving the features by equations (11) and (10) (only the real part is taken).
kn =
ρSω2 (e0 a)2 +
2.3. Standard non-local modulus under ultrasonic vibration According to the above formula, the curve of the standard non-local modulus varying with frequency can be obtained as follows [20]:
‾l (k ) = cp2/ c12 =
2ρIω (1 + δ ) ρSω (e0
a)2
+
ρS (4EI + ρSω2 (e0 a) 4)
where, C1 is the phase velocity of the shear wave in the classical theory, C12 = 2μ/ ρ . ρ is the density, 2μ is the classic Lame's coefficient, μ = E /2(1 + δ ) . E and δ are the Young's modulus and Poisson's ratio of the material, respectively. The ZTA nano-composite ceramics used in the experiment are large particles tightly bound by nano-ZrO2 and Al2O3, with a diameter of about 20 μm. Therefore, the intrinsic length of the specimens is approximately 2 × 10−5 m, e0 = 0.3, e0 a = 6 μm , E = 377 Gpa, and ρ = 4880 kg/m3 . The interaction of external ultrasonic wavelength with Al2O3 and ZrO2 clusters results in wave dispersion and stress attenuation at a certain point. Fig. 4 shows that the wave dispersion phenomenon can be observed in the frequency band from 20 kHz to 100 kHz on the scale of 2 × 10−5 m intrinsic length of the test material, and it is aggravated with the increase in frequency. 2.4. Effect of ultrasonic vibration on fracture properties of nanocomposite ceramics
(12)
2EI
ω = ω 2EI / ρSω2 (e0 a)2 + kn
ρSω2 (4EI + ρSω2 (e0 a) 4)
Ultrasonic vibrations of different frequencies and amplitudes will produce different vibrational stresses inside the material, which will lead to changes in the mechanical properties of the material. The wave equation transmitted by the ultrasonic wave in the material shows [21]:
2πf ⎞ x ·sin2πft u (x , t ) = A (x ) sin (ωt ) = Acos ⎛ ⎝ c ⎠ The internal stress of vibration is:
(13)
If the group velocity is defined as Cg = Real (∂ωn / ∂kn ) , the wave group velocity according to the non-local theory is:
Cg =
(15)
ρSω2 (4EI + ρSω2 (e0 a) 4)
where, the wave number kn is a function of the non-local scale parameter e0 a , the wave circle frequency ω and the material parameters of the rod. The phase velocity is defined as Cp = Real (ω/ kn ) , which varies with the change in ω. This is different from the classical elasticity theory. Therefore, the speed changes with the frequency, showing a strong dispersion phenomenon.
Cp =
30 kHz. The non-linear change in the wave number indicates that the wave is dispersive, and the shape of the wave changes during wave propagation. However, the wave number of the bending mode starts propagating from zero frequency, which indicates that the mode can propagate at any excitation frequency, and there is no cut-off frequency. In the results of non-local elasticity (e0a = 10 μm), the wave number exhibits many of the same characteristics as the local elastic wave. Therefore, the different slopes of the two curves indicate that the change in group velocity is different. The wave number calculated from the nonlocal elasticity theory is higher than that from the classical elasticity theory, as shown in Fig. 2. It can be seen from Fig. 3 that the phase velocity under the non-local theory is smaller than the phase velocity computed using the local elastic theory. Under the non-local theory, the phase velocity tends to zero with the increase in frequency. Therefore, the frequency of ultrasonic vibration grinding should not be too high, and the recommended frequency should not exceed 60 kHz.
EIkn3 − ρSω2 (e0 a)2kn ρSω (1 + (e0 a)2kn2)
(14)
The Euler-Bernoulli model is used to analyze the dispersion characteristics of beams (cross-sectional area b × h = 5 × 2.5). The frequency spectrum curve (wavenumber vs. frequency) and dispersion curve (phase velocity vs. frequency) are shown in Fig. 2 and Fig. 3. In the classical elastic (e0 a = 0 nm) solution, the number of bending modes varies nonlinearly with respect to frequency in the range of 20947
Fig. 2. Theoretical local and non-local frequency spectrum curves.
(16)
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propagation during the fracture process. Five specimens are used in each group, and the specimens should be finished before the test. The ultrasonic three-point bending test was carried out on an Instron-10 universal material testing machine in the United States. In order to ensure consistency, the span is 20 mm and the loading rate is 0.3 mm/ min. The bending strength σf of the non-cut ceramic material can be expressed by the following equation:
σf =
3PL0 2bh2
(20)
where, P is the maximum load measured; L0 is the span of the specimen, the test is 30 mm; b is the width of the specimen, h is the height of the specimen. The variation of elastic strain distribution caused by the centralized load and short load span are considered. The ultrasonic three-point bending device is shown in Fig. 6.
Fig. 3. Theoretical local and non-local dispersion curves.
4. Results and discussion In order to study the effect of ultrasonic frequency on the fracture properties of ceramics, different horn shapes with similar amplitude at different frequencies were selected, such as a 20 kHz conical horn (f = 20.565 kHz, A1 = 14.7 μm), 28 kHz exponential horn (f = 27.46 kHz, A2 = 14.9 μm), and 35 kHz catenary horn (f = 35.367 kHz, A3 = 15 μm). 4.1. Dynamic observation of three-point bending fracture process under ultrasonic vibration Fig. 4. Relationship between nonlocal standard kernel function and frequencies.
σ = εE = uˆ (x , t ) E = −AE
2πf 2πf ⎞ x sin2πft sin ⎛ c ⎝ c ⎠
(17)
The maximum internal stress is:
σmax = AE
2πf = 2πfA Eρ c
(18)
where, uˆ (x , t ) is the derivative of u (x , t ) , f is the ultrasonic frequency, c is the speed of sound, c = E / ρ ; ρ is the material density. From the above formula, it can be seen that the internal stress of vibration is proportional to the frequency and amplitude. Therefore, the introduction of different ultrasonic vibrations changes the internal stress of vibration, which will affect the hardness of the workpiece. 3. Experimental materials and equipment The Al2O3 powder selected in this experiment is industrial 99αAl2O3 having an original crystal grain diameter of 200–500 μm, and a slurry particle size of approximately 2.5 μm. The ZrO2 powder used was a ZrO2 (3Y) powder (having an average particle diameter of 100–1000 nm, with 2% Y2O3 as a stabilizer), and an additive such as Si and MgCO3. The material used is ZTA (Al2O3–ZrO2 (n)), in which the volume content of ZrO2 is 15%. The nanocomposite ceramic blank is made by a static pressure process and sintered at a high temperature of 1640 °C, with an average particle size of 500–1000 nm. The specimens selected for the bending test are 36 mm × 5 mm × 2.5 mm (l × b × h). The size of the three-point bending specimen satisfies the resonance conditions of the ultrasonic frequencies of 20 kHz, 28 kHz, and 35 kHz. The surface roughness Ra of the sample was controlled within 0.1 μm, and there should be no damage or crack. The ultrasonic three-point bending performance test of nanocomposite ceramics is shown in Fig. 5. The purpose of the experiment is to study the effect of frequency and amplitude of ultrasonic vibration on the fracture properties of ceramic materials and the mechanism of crack
For the online observation of crack propagation, a high-speed camera system was used in this experiment. The effect of ultrasonic vibration is illustrated using images corresponding to an ultrasonic frequency of f = 35 kHz and amplitude A = 15 μm. The dynamic images of crack propagation are shown in Fig. 7. It can be seen from Fig. 7 that the fracture of ceramic materials under ultrasonic vibration is actually a slow crack propagation process. In the initial stage of loading, there is no crack formation because the upper surface is subjected to compressive stress and the lower surface is subjected to tensile stress. With the effect of ultrasonic vibration on the repeated loading of the ceramic surface, there is a slight collapse of the surface. When the loading is continued for a period of time to reach the critical state, that is, when the transverse tensile stress at the crack tip is exactly equal to the bonding strength of the material, cracks will appear on the lower surface of the loading point, as shown in Fig. 7 (b). According to the non-local theory, the crack breaks the stress balance of the entire ceramic material. Due to the long-range force between microparticles in the material, the stress redistribution around the material will occur in order to maintain the internal stress balance. The interaction between these particles will cause wave scattering, which will lead to stress attenuation. Cracks will continue to propagate under ultrasonic vibration, and will follow the path of the smaller bonding force, which will lead to the deflection and bifurcation of secondary cracks [22]. Subsequently, smaller cracks begin to coalescence and larger cracks appear to balance the overall internal stress of the material. When the cracks accumulate to a certain extent, sudden fracture will occur. According to the non-local theory, when the ultrasonic wave
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Fig. 5. Schematic diagram of three-point bending test.
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Fig. 6. Ultrasonic three-point bending device diagram.
propagates in the ceramic material, the wave velocity decreases and the speed of crack propagation decreases. When the applied ultrasonic energy interacts with the internal micro-structure of materials, it consumes energy and produces heat, which changes the micro-structure and stress state of materials. The crack propagation path in brittle and hard materials is closely related to temperature, which changes the crack propagation behavior during the fracture process [23,24].
4.2. Fracture morphology analysis of three-point bending The fracture morphology of the fractured specimens was observed by scanning electron microscopy (SEM) as shown in Fig. 8 and Fig. 9. The fracture morphology of ZTA ceramics with three-point bending is examined using secondary electron diffraction (SED) ( × 1000 enlargement), and the partially enlarged view is the back scatter electron diffraction (BSED) ( × 3000 enlargement). It can be seen from Fig. 8 that Fig. 8 (a) is the fracture morphology under normal loading, and
Fig. 7. Dynamic crack growth process in three-point bending. 20949
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Fig. 8. Fracture morphology of three-point bending at different frequencies.
Fig. 8 (b), (c), (d) are the fracture morphologies under different ultrasonic frequencies with similar amplitude. In the figure, the black and coarse grains are A12O3, while the white and fine grains are ZrO2. The ZrO2 distributed at grain boundaries aggregate and coarsen (agglomerates) at trigeminal grain boundaries due to the pushing effect of the Al2O3 growth. Some fine ZrO2 particles are encapsulated by the A12O3 matrix and become “inner crystals”. The presence of ZrO2 in the inner crystal of the A12O3 results in abundant dislocation morphologies. Because of the randomness of grain size and distribution, the residual stress field is very complex, which leads to the complexity and diversification of dislocations. The intersection, accumulation, and entanglement of these dislocations will increase the strength of A12O3 in varying degrees and give rise to a strengthening effect. From the fracture morphology of Fig. 8 (a), it can be seen that under normal loading, there are more complete grains and voids in the fracture. The grain boundaries of Al2O3 and ZrO2 grains are relatively complete, which indicates that intergranular fracture mainly occurs. The intergranular fracture that occurs in three-point bending under normal loading (most of the intergranular fractures are brittle fractures) refer to fractures that occur in the low energy absorption process along the grain boundaries of different orientations [25]. Since the energy of the grain boundary is higher than that of the grain interior, the second phase particles precipitate easily at the grain boundary, which makes the mechanical, physical, and chemical properties of the grain boundary or adjacent region worse than that of the interior of the grain. According to the principle of minimum fracture energy consumption, the crack propagation path always follows the surface with the weakest atomic bonding force, and hence, intergranular fracture as shown in Fig. 8 (a) is obtained. Due to the effect of ultrasonic vibration, at the interface between the second phase particles and the matrix, visible micro-voids are generated by the aggregation of adjacent micro-cracks;
the voids grow and increase, resulting in transgranular fracture (transgranular fracture is generally ductile fracture). This phenomenon is due to the change in the microstructure of the material by ultrasonic vibration, which enhances the toughness of the material and changes the fracture mode. With the increase in frequency, it can be seen that the transgranular fracture phenomenon becomes more obvious and the microcracks on the grains increase. A large number of wrinkles are found on Al2O3 grains at the fracture surfaces of 28 kHz and 35 kHz. According to the non-local theory of wave propagation, with the increase in ultrasonic frequency, the dispersion phenomenon becomes more pronounced, and the interaction between the ultrasonic wave and the internal structure of the ceramic material becomes more obvious. The greater the number of surface microcracks the more complex the crack propagation path is. From the processing point of view, this phenomenon makes it easier to form a surface that is similar in structure to wool felt fiber, thus improving the surface accuracy. It can be seen from Fig. 9 that, at the same frequency, with the increase in amplitude, intergranular-transgranular mixed fracture occurs, and the crack propagation path is broken randomly and tortuously along the crystal plane. It can be seen from the SED picture that the cross-section is uneven and honeycomb-shaped, and there are a small number of shallow dimples and tearing edges. This means that, when the material is damaged, the crack no longer spreads completely along the grain boundary, but a part passes through the middle of the grain [26]. Generally, the propagation of cracks follows the principle of minimum energy. Transgranular fracture occurs in ZTA ceramics, which indicates that grain boundary is strengthened and cracks are forced to penetrate the grains, thus improving material properties. As the amplitude increases, the propagation speed of the ultrasonic wave in the material increases. From the non-local mechanics, the ultrasonic wave exhibits a dispersion phenomenon in the material. The higher the
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Fig. 9. Fracture morphology of three-point bending specimen under different amplitudes.
speed, the stronger the interaction between the ultrasonic wave and the long-range force of the internal structure of the material, which leads to the attenuation of the required fracture stress. This attenuation is greater than the toughening effect induced by the microscopic change. Therefore, under the action of ultrasonic vibration, the fracture stress is reduced [27].
4.3. Three-point bending fracture XRD diffraction analysis XRD scanning was performed on the fracture surface of specimens with the same amplitude and different frequencies. A Bruker
D8ADVANCE x-ray diffractometer was used for the analysis. It can be seen from Fig. 10 and Table 1 that the nanocomplex ZTA ceramics undergo different degrees of martensite transformation of t-ZrO2 particles to m-ZrO2 particles during fracture at different frequencies. When the material is subjected to ultrasonic vibration, a large number of nanoparticles are randomly distributed within the crystal, which is manifested as a diversity of stresses. Under the action of the stress field at the crack tip, the t-ZrO2 particles dispersed in the matrix will produce a 3%–5% volume expansion during martensitic transformation, and the tensile stress will induce the transformation of t-ZrO2 particles to mZrO2 particles. A stress-induced phase transition occurs in the matrix,
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Fig. 10. Three-point bending fracture XRD diffraction. Table 1 Chemical composition of nanocomposite ceramics after XRD test. Chemical composition
Al2O3 t-ZrO2 m-ZrO2
0 kHz
20 kHz
28 kHz
35 kHz
wt%
wt%
wt%
wt%
42 48.3 5.2
42.1 46.7 6.8
41.9 44.7 8.8
41.8 42.9 10.6
which results in a dispersed distribution of microcracks. During the expansion of the main crack tip, secondary deflection and bifurcation of the transgranular crack occur (the crack changes the direction of extension) and hinders crack propagation. This increases the energy consumption of crack propagation, lengthens the path and difficulty of crack propagation, and increases the fracture surface energy [28]. The transgranular fracture registers a significant increase in toughness and the phase change layer produces a relatively high residual compressive stress, generating a toughening effect, thereby improving the fracture toughness and strength of the test piece. In addition to generating a new fracture surface to absorb energy, it also absorbs energy due to the volume effect at the time of phase change (3%–5% volume expansion). Stress-induced tissue transformation consumes the applied stress. At the same time, compressive stress is generated on the crack due to the volume expansion of the particles that
underwent phase change, which hinders the crack propagation. Secondary interface and microcracks are produced in the grain. The growth of these microcracks themselves also plays a role in dispersing the energy of the main crack tip. This leads to the differentiation of grains and the appearance of local weak links, so that the cracks blocked at the grain boundary can enter the grain, thus inhibiting the rapid growth of main cracks and improving the plasticity of materials [29]. This shows that the mechanical properties of ceramics are greatly improved by ultrasonic vibration, and t-m phase transformation can occur under stress induction to absorb strain, so that the ceramics have better plastic properties. It can also be seen from the results in Fig. 10 that the martensite transformation rate increases with the increase in frequency. The t-m phase transition rate at 35 kHz is higher than that at other frequencies. According to the mechanism of phase transformation toughening, the higher the t-m phase transformation rate of ZTA nanocomposite ceramics, the stronger the plastic properties of ceramics, which indicates that it is easier to achieve ductile processing under ultrasonic excitation. 5. Conclusion Based on the non-local theory, the effects of ultrasonic vibration on wave dispersion characteristics and non-local modulus are analyzed and compared with the results obtained using the local theory. Threepoint bending test in the presence of ultrasonic vibration was carried
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out using different ultrasonic frequencies and amplitudes. SEM and XRD observations were carried out on the fracture surface of the workpiece. The effects of ultrasonic vibration on the fracture properties and crack propagation mechanism of the material were analyzed. The following conclusions were drawn: (1) Under the non-local theory, due to the interaction between the ultrasonic vibration and ceramic microstructures, the fracture stress of ultrasonic vibration decreases with the increase in frequency and amplitude, and is less than the flexural strength under ordinary loading. Different ultrasonic vibration characteristics change the internal stress of material vibration and cause particle scattering, which leads to the intensification of the phase velocity dispersion. (2) From the SEM images of the fracture surface of three-point bending specimens, it can be found that under ultrasonic vibration, the fracture surface shows obvious grain sections, and a large number of cracks appear on the grain, which indicates that the material shows transgranular and intergranular mixed fracture. With the increase in ultrasonic frequency and amplitude, the transgranular fracture of the fracture surface becomes more obvious, and the number of surface microcracks, complexity of the crack propagation path, and the interaction between ultrasonic energy and the internal structure of ceramic materials all increase. (3) It was found by fracture XRD that ultrasonic vibration can induce the t-m phase transformation of ceramic materials under stress, which absorbs strain and causes the materials to have good plasticity. The intrinsic phase transformation toughening mechanism makes it easier for materials to be processed in the ductile domain, which greatly improves the mechanical properties of nanocomposite ceramics.
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Acknowledgments [22]
This research was supported financially by the Henan Natural Science Foundation (162300410120) and National Joint Fund (U1604255).
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Contents lists available at ScienceDirect
Ceramics International journal homepage: www.elsevier.com/locate/ceramint
Fracture test of nanocomposite ceramics under ultrasonic vibration based on nonlocal theory
T
Jinglin Tong*, Peng Chen, Junshuai Zhao, Bo Zhao School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, 45400, Henan, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Nonlocal theory Ultrasonic vibration Three-point bending Crack propagation Nanocomposite ceramics
Based on the non-local theory, the dispersion characteristics of ultrasonic vibration in three-point bending are analyzed, and the effects of ultrasonic frequency and amplitude on the dispersion characteristics are obtained through theoretical analysis. Three-point bending tests were carried out using an ultrasonic vibration system and specimens, and the test process as well as the workpiece after the test were observed and analyzed. The effects of ultrasonic vibration on fracture properties and crack growth mechanism of ceramics under different loading conditions were verified by experiments. The introduction of ultrasonic vibration changes the internal stresses of the ceramic fracture, affects the hardness of the workpiece, and also causes the scattering of particles, which leads to the intensification of ultrasonic attenuation and phase velocity dispersion. SEM and XRD analysis showed that with the increase in ultrasonic frequency and amplitude, the transgranular fracture was more obvious, tearing edges and dimples increased, and microvoids increased in size and number. Ultrasonic vibration can cause ZrO2 to undergo t-m phase transformation under stress induction and absorb the strain, thus greatly improving the mechanical properties of nanocomposite ceramics. Due to the toughening mechanism of the inherent phase transformation, the material has improved plastic mechanical properties, and it is easier to achieve ductile domain processing.
1. Introduction Ceramic materials are widely used in aviation and other high-end technology fields. Because of their high strength, high hardness, high temperature resistance, wear resistance, corrosion resistance, chemical stability, and good biocompatibility, they have broad application prospects in aerospace, aviation industry, machinery, petroleum industry, and nuclear industry [1]. However, because of their brittleness and sensitivity to internal defects, ceramic materials show sudden and catastrophic fracture without warning. Therefore, the reliability of ceramic components is poor, which is a bottleneck that affects the popularization and application of ceramic materials [2]. With the development of nanotechnology, nano-sized ceramic particles were introduced into a micron-sized ceramic matrix as a dispersed phase, and nano-composite ceramics were prepared, which achieved an obvious strengthening and toughening effect [3]. Nanocomposite ceramics are structural ceramics with excellent high temperature mechanical properties. In recent years, many scholars at home and abroad have studied nanocomposite ceramic materials. Soh A K et al. [4] studied the effect of nanoparticles on the toughness of nanocomposite ceramics. The results showed that with the increase of the volume fraction of nanoparticles, the
*
transgranular fracture would increase, and the aggregation of nanoparticles would lead to the decrease in the strength and toughness of nanocomposite ceramics. Zhao Bo et al. [5] analyzed the brittleness, toughness, and material removal mechanism of nano-composite ceramics, established the formula for critical toughness and grinding depth of nano-composite ceramics, and carried out grinding experiments under ultrasonic and ordinary processes. It was found that the critical grinding depth of nano-composite ceramics under ordinary grinding was 12 μm, and under ultrasonic grinding was 20 μm. Based on impulse theory and fracture mechanics, Wu Y et al. [6] analyzed the contact model of two-dimensional ultrasonic vibration of abrasive particles and the workpiece. The experiment shows that toughness grinding of ceramics can be realized only when the grinding depth is less than the critical grinding depth and the appropriate grinding parameters were determined. Yunteng Wang et al. [7] established a thermo-mechanical coupling bond-based peridynamic model to study the thermal fracturing behavior of ceramic nuclear pellets, which overcomes the difference in typical time scales between thermal and mechanical systems, and can accurately predict the thermal crack morphology. Radial cracks occur when power increases and circumferential cracks occur when power decreases. In addition to the advantages of ordinary ceramics,
Corresponding author. E-mail address: [email protected] (J. Tong).
https://doi.org/10.1016/j.ceramint.2019.07.084 Received 15 May 2019; Received in revised form 5 July 2019; Accepted 8 July 2019 Available online 09 July 2019 0272-8842/ © 2019 Published by Elsevier Ltd.
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Nomenclature
u (x , t ) w (x , t ) εxx σxx Q M ρ S E EI e0 a ωn N A
wave number circle frequency of the wave phase velocity group velocity classic Lame coefficient Poisson's ratio of the material phase velocity of the shear wave in the classical theory internal stress of vibration speed of sound frequency of ultrasonic bending strength the maximum load measured span of the specimen width of the specimen height of the specimen
kn ω Cp Cg μ δ C1 σ c f σf P L0 b h
axial displacements of any point lateral displacements of any point bending strain axial stress on the section axial force bending moment density of the material cross-sectional area Young's modulus of the material bending stiffness of the beam non-local scale parameter circular frequency of the nth sampling point the number of sampling points amplitude of ultrasonic
nano-ceramics have greatly improved their mechanical properties and reliability, but their plastic toughness is still very low and their hardness is high. It is difficult to achieve high precision, high efficiency, and high reliability in their processing [8,9]. In the study of solid mechanics, Eringen et al. [10,11] proposed the theory of nonlocal linear elasticity and applied it widely. This nonlocal theory is based on the theory of lattice dynamics and phonon scattering. It assumes that the interaction between atoms in a material is a longrange force and that the stress state at a point in an object is related not only to the strain at that point, but also to the strain at other points in the entire object. The theory of nonlocal elasticity describes the relative effect of the stress at given locations on the strain states at other locations. This type of constitutive relation involves the integral of the entire body and includes nonlocal kernel functions. This belongs forms a part of generalized continuum mechanics, which is anextension and development of classical continuum mechanics. This theory has contributed to several advancements in research related to fracture [12], dislocation [13], crack initiation and propagation [14], wave propagation and so on. Reddy J. N. et al. [15] deduced the non-local theoretical equations of motion, and gave the analytical solutions for bending, vibration, and buckling, revealing the influence of the nonlocal theory on deflection, buckling load, and natural frequency. Huang W. G et al. [16] and others used this theory to study the stability of compressive rods and the axial vibration of elastic rods under three boundary conditions, and deduced the nonlocal theoretical solutions of critical pressure and natural frequency. Bian P.Y et al. [17] established the non-local constitutive model under ultrasonic vibration and studied the grinding of nanocomposite ceramics. The results showed that the grinding force could be greatly reduced by applying ultrasonic vibration, the grinding force increased with the increase in frequency, and the surface quality was improved. To aid the study of precise and highefficiency processing of hard and brittle materials and realize the ductile domain processing of hard and brittle materials, this study presents a non-local constitutive model of ultrasonic vibrating nanocomposite ceramics under three-point bending based on the non-local theory and fracture mechanics. The effect of ultrasonic vibration on the fracture, crack propagation, and microscopic characteristics of nanocomposite ceramics was studied by three-point bending. The effects of ultrasonic vibration on the mechanical properties of nanocomposite ceramics were analyzed by SEM and XRD observations on the fracture of three-point bending specimens.
Fig. 1. Stress synthesis diagram of nano-beam. ∂w (x , t ) ∂x
⎧ u x (x , z , t ) = u ( x , t ) − z ⎪ u y (x , z , t ) = 0 ⎨ ⎪ u ( x , z , t ) = w (x , t ) z ⎩
(1)
2. Nonlocal wave dispersion characteristics of three-point bending in ultrasound vibration of nanocomposite ceramic rods 2.1. Constitutive equation of Euler-Bernoulli beam with nonlocal motion Based on the Euler -Bernoulli beam theory, Fig. 1 axial and lateral displacement fields at any point in the rod are as follows [18]:where, u (x , t ) and w (x , t ) are the axial and lateral displacements of any point in the rod along the central axis. Bending strain is given by:
∂u x ∂u ∂ 2w = −z 2 ∂x ∂x ∂x
εxx =
(2)
According to the stress relationships, the Euler beam-Bernoulli theory of motion equation can be stated as follows:
⎧
∂Q ∂x
= ρS
∂2u ∂x 2
⎨ ∂2M = ρS ∂2w ∂t 2 ⎩ ∂x 2 Q=
∫ σxx dS S
(3)
M=
∫ zσxx dS S
(4)
where, σxx is the axial stress on the section, Q is the axial force, and M is the bending moment. In contrast with the linear algebraic local stress-strain equation, the non-local constitutive relation introduces the differential stress-strain relation. Assuming that the beam is homogeneous and isotropic, the stress-strain relationship based on the Euler-Bernoulli beam theory is expressed by the non-local constitutive relation, and the non-local constitutive relation of the nano-beam is as follows:
σxx − (e0 a)2
∂2σxx = Eεxx ∂x 2
(5)
From the non-local constitutive relation and the proposed equation of motion, according to the generalized displacement condition, the 20946
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moment equation can be expressed as follows:
M = −EI
∂ 2w ∂x 2
+ (e0 a)2ρS
∂ 2w (6)
∂t 2
The non-local motion equation of the Euler-Bernoulli beam is:
EI
∂ 4w ∂ 2w ∂ 4w + ρS 2 − ρS (e0 a)2 2 2 = 0 4 ∂x ∂t ∂x ∂t
(7)
where, ρ is the density, S is the cross-sectional area, EI is the bending stiffness of the beam, and e0 a is the non-local scale parameter. It can be seen that if the internal length a is zero, then equation (7) reduces to the local Euler-Bernoulli beam equation. 2.2. Wave dispersion characteristics The wave characteristic parameters of nano-beams are the wave number and wave velocity. The time variable can be eliminated from the nonlocal beam equation (7) of the partial differential equation by using Fourier transform. The transverse displacement is expressed as follows [19]: N
w(x , t ) =
∑ wˆ (x , ωn) e−iωt
(8)
n=1
where, ωn is the circular frequency of the nth sampling point, and N is the number of sampling points. N should be large enough to achieve better resolution for both high and low frequency response. By introducing equation (8) into equation (7), we obtain: N
d4wˆ
∑ ⎡⎢EI dx 4
n=1
⎣
+ ρSωn2 wˆ − ρSωn2 (e0 a)2
d 2w ⎤ −iωt e =0 dx 2 ⎥ ⎦
(9)
The equation must satisfy every n, so that it can be transformed into a general differential equation with a single parameter x:
EI
d4wˆ d 2w + ρSωn2 ⎡wˆ − (e0 a)2 2 ⎤ = 0 ⎢ dx 4 dx ⎥ ⎣ ⎦
wˆ (x ) =
Ae−ikn t
(10) (11)
where, A is the ultrasonic amplitude, kn is the wave number. Since the amplitude A is not zero, the wave number kn is obtained by solving the features by equations (11) and (10) (only the real part is taken).
kn =
ρSω2 (e0 a)2 +
2.3. Standard non-local modulus under ultrasonic vibration According to the above formula, the curve of the standard non-local modulus varying with frequency can be obtained as follows [20]:
‾l (k ) = cp2/ c12 =
2ρIω (1 + δ ) ρSω (e0
a)2
+
ρS (4EI + ρSω2 (e0 a) 4)
where, C1 is the phase velocity of the shear wave in the classical theory, C12 = 2μ/ ρ . ρ is the density, 2μ is the classic Lame's coefficient, μ = E /2(1 + δ ) . E and δ are the Young's modulus and Poisson's ratio of the material, respectively. The ZTA nano-composite ceramics used in the experiment are large particles tightly bound by nano-ZrO2 and Al2O3, with a diameter of about 20 μm. Therefore, the intrinsic length of the specimens is approximately 2 × 10−5 m, e0 = 0.3, e0 a = 6 μm , E = 377 Gpa, and ρ = 4880 kg/m3 . The interaction of external ultrasonic wavelength with Al2O3 and ZrO2 clusters results in wave dispersion and stress attenuation at a certain point. Fig. 4 shows that the wave dispersion phenomenon can be observed in the frequency band from 20 kHz to 100 kHz on the scale of 2 × 10−5 m intrinsic length of the test material, and it is aggravated with the increase in frequency. 2.4. Effect of ultrasonic vibration on fracture properties of nanocomposite ceramics
(12)
2EI
ω = ω 2EI / ρSω2 (e0 a)2 + kn
ρSω2 (4EI + ρSω2 (e0 a) 4)
Ultrasonic vibrations of different frequencies and amplitudes will produce different vibrational stresses inside the material, which will lead to changes in the mechanical properties of the material. The wave equation transmitted by the ultrasonic wave in the material shows [21]:
2πf ⎞ x ·sin2πft u (x , t ) = A (x ) sin (ωt ) = Acos ⎛ ⎝ c ⎠ The internal stress of vibration is:
(13)
If the group velocity is defined as Cg = Real (∂ωn / ∂kn ) , the wave group velocity according to the non-local theory is:
Cg =
(15)
ρSω2 (4EI + ρSω2 (e0 a) 4)
where, the wave number kn is a function of the non-local scale parameter e0 a , the wave circle frequency ω and the material parameters of the rod. The phase velocity is defined as Cp = Real (ω/ kn ) , which varies with the change in ω. This is different from the classical elasticity theory. Therefore, the speed changes with the frequency, showing a strong dispersion phenomenon.
Cp =
30 kHz. The non-linear change in the wave number indicates that the wave is dispersive, and the shape of the wave changes during wave propagation. However, the wave number of the bending mode starts propagating from zero frequency, which indicates that the mode can propagate at any excitation frequency, and there is no cut-off frequency. In the results of non-local elasticity (e0a = 10 μm), the wave number exhibits many of the same characteristics as the local elastic wave. Therefore, the different slopes of the two curves indicate that the change in group velocity is different. The wave number calculated from the nonlocal elasticity theory is higher than that from the classical elasticity theory, as shown in Fig. 2. It can be seen from Fig. 3 that the phase velocity under the non-local theory is smaller than the phase velocity computed using the local elastic theory. Under the non-local theory, the phase velocity tends to zero with the increase in frequency. Therefore, the frequency of ultrasonic vibration grinding should not be too high, and the recommended frequency should not exceed 60 kHz.
EIkn3 − ρSω2 (e0 a)2kn ρSω (1 + (e0 a)2kn2)
(14)
The Euler-Bernoulli model is used to analyze the dispersion characteristics of beams (cross-sectional area b × h = 5 × 2.5). The frequency spectrum curve (wavenumber vs. frequency) and dispersion curve (phase velocity vs. frequency) are shown in Fig. 2 and Fig. 3. In the classical elastic (e0 a = 0 nm) solution, the number of bending modes varies nonlinearly with respect to frequency in the range of 20947
Fig. 2. Theoretical local and non-local frequency spectrum curves.
(16)
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propagation during the fracture process. Five specimens are used in each group, and the specimens should be finished before the test. The ultrasonic three-point bending test was carried out on an Instron-10 universal material testing machine in the United States. In order to ensure consistency, the span is 20 mm and the loading rate is 0.3 mm/ min. The bending strength σf of the non-cut ceramic material can be expressed by the following equation:
σf =
3PL0 2bh2
(20)
where, P is the maximum load measured; L0 is the span of the specimen, the test is 30 mm; b is the width of the specimen, h is the height of the specimen. The variation of elastic strain distribution caused by the centralized load and short load span are considered. The ultrasonic three-point bending device is shown in Fig. 6.
Fig. 3. Theoretical local and non-local dispersion curves.
4. Results and discussion In order to study the effect of ultrasonic frequency on the fracture properties of ceramics, different horn shapes with similar amplitude at different frequencies were selected, such as a 20 kHz conical horn (f = 20.565 kHz, A1 = 14.7 μm), 28 kHz exponential horn (f = 27.46 kHz, A2 = 14.9 μm), and 35 kHz catenary horn (f = 35.367 kHz, A3 = 15 μm). 4.1. Dynamic observation of three-point bending fracture process under ultrasonic vibration Fig. 4. Relationship between nonlocal standard kernel function and frequencies.
σ = εE = uˆ (x , t ) E = −AE
2πf 2πf ⎞ x sin2πft sin ⎛ c ⎝ c ⎠
(17)
The maximum internal stress is:
σmax = AE
2πf = 2πfA Eρ c
(18)
where, uˆ (x , t ) is the derivative of u (x , t ) , f is the ultrasonic frequency, c is the speed of sound, c = E / ρ ; ρ is the material density. From the above formula, it can be seen that the internal stress of vibration is proportional to the frequency and amplitude. Therefore, the introduction of different ultrasonic vibrations changes the internal stress of vibration, which will affect the hardness of the workpiece. 3. Experimental materials and equipment The Al2O3 powder selected in this experiment is industrial 99αAl2O3 having an original crystal grain diameter of 200–500 μm, and a slurry particle size of approximately 2.5 μm. The ZrO2 powder used was a ZrO2 (3Y) powder (having an average particle diameter of 100–1000 nm, with 2% Y2O3 as a stabilizer), and an additive such as Si and MgCO3. The material used is ZTA (Al2O3–ZrO2 (n)), in which the volume content of ZrO2 is 15%. The nanocomposite ceramic blank is made by a static pressure process and sintered at a high temperature of 1640 °C, with an average particle size of 500–1000 nm. The specimens selected for the bending test are 36 mm × 5 mm × 2.5 mm (l × b × h). The size of the three-point bending specimen satisfies the resonance conditions of the ultrasonic frequencies of 20 kHz, 28 kHz, and 35 kHz. The surface roughness Ra of the sample was controlled within 0.1 μm, and there should be no damage or crack. The ultrasonic three-point bending performance test of nanocomposite ceramics is shown in Fig. 5. The purpose of the experiment is to study the effect of frequency and amplitude of ultrasonic vibration on the fracture properties of ceramic materials and the mechanism of crack
For the online observation of crack propagation, a high-speed camera system was used in this experiment. The effect of ultrasonic vibration is illustrated using images corresponding to an ultrasonic frequency of f = 35 kHz and amplitude A = 15 μm. The dynamic images of crack propagation are shown in Fig. 7. It can be seen from Fig. 7 that the fracture of ceramic materials under ultrasonic vibration is actually a slow crack propagation process. In the initial stage of loading, there is no crack formation because the upper surface is subjected to compressive stress and the lower surface is subjected to tensile stress. With the effect of ultrasonic vibration on the repeated loading of the ceramic surface, there is a slight collapse of the surface. When the loading is continued for a period of time to reach the critical state, that is, when the transverse tensile stress at the crack tip is exactly equal to the bonding strength of the material, cracks will appear on the lower surface of the loading point, as shown in Fig. 7 (b). According to the non-local theory, the crack breaks the stress balance of the entire ceramic material. Due to the long-range force between microparticles in the material, the stress redistribution around the material will occur in order to maintain the internal stress balance. The interaction between these particles will cause wave scattering, which will lead to stress attenuation. Cracks will continue to propagate under ultrasonic vibration, and will follow the path of the smaller bonding force, which will lead to the deflection and bifurcation of secondary cracks [22]. Subsequently, smaller cracks begin to coalescence and larger cracks appear to balance the overall internal stress of the material. When the cracks accumulate to a certain extent, sudden fracture will occur. According to the non-local theory, when the ultrasonic wave
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Fig. 5. Schematic diagram of three-point bending test.
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Fig. 6. Ultrasonic three-point bending device diagram.
propagates in the ceramic material, the wave velocity decreases and the speed of crack propagation decreases. When the applied ultrasonic energy interacts with the internal micro-structure of materials, it consumes energy and produces heat, which changes the micro-structure and stress state of materials. The crack propagation path in brittle and hard materials is closely related to temperature, which changes the crack propagation behavior during the fracture process [23,24].
4.2. Fracture morphology analysis of three-point bending The fracture morphology of the fractured specimens was observed by scanning electron microscopy (SEM) as shown in Fig. 8 and Fig. 9. The fracture morphology of ZTA ceramics with three-point bending is examined using secondary electron diffraction (SED) ( × 1000 enlargement), and the partially enlarged view is the back scatter electron diffraction (BSED) ( × 3000 enlargement). It can be seen from Fig. 8 that Fig. 8 (a) is the fracture morphology under normal loading, and
Fig. 7. Dynamic crack growth process in three-point bending. 20949
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Fig. 8. Fracture morphology of three-point bending at different frequencies.
Fig. 8 (b), (c), (d) are the fracture morphologies under different ultrasonic frequencies with similar amplitude. In the figure, the black and coarse grains are A12O3, while the white and fine grains are ZrO2. The ZrO2 distributed at grain boundaries aggregate and coarsen (agglomerates) at trigeminal grain boundaries due to the pushing effect of the Al2O3 growth. Some fine ZrO2 particles are encapsulated by the A12O3 matrix and become “inner crystals”. The presence of ZrO2 in the inner crystal of the A12O3 results in abundant dislocation morphologies. Because of the randomness of grain size and distribution, the residual stress field is very complex, which leads to the complexity and diversification of dislocations. The intersection, accumulation, and entanglement of these dislocations will increase the strength of A12O3 in varying degrees and give rise to a strengthening effect. From the fracture morphology of Fig. 8 (a), it can be seen that under normal loading, there are more complete grains and voids in the fracture. The grain boundaries of Al2O3 and ZrO2 grains are relatively complete, which indicates that intergranular fracture mainly occurs. The intergranular fracture that occurs in three-point bending under normal loading (most of the intergranular fractures are brittle fractures) refer to fractures that occur in the low energy absorption process along the grain boundaries of different orientations [25]. Since the energy of the grain boundary is higher than that of the grain interior, the second phase particles precipitate easily at the grain boundary, which makes the mechanical, physical, and chemical properties of the grain boundary or adjacent region worse than that of the interior of the grain. According to the principle of minimum fracture energy consumption, the crack propagation path always follows the surface with the weakest atomic bonding force, and hence, intergranular fracture as shown in Fig. 8 (a) is obtained. Due to the effect of ultrasonic vibration, at the interface between the second phase particles and the matrix, visible micro-voids are generated by the aggregation of adjacent micro-cracks;
the voids grow and increase, resulting in transgranular fracture (transgranular fracture is generally ductile fracture). This phenomenon is due to the change in the microstructure of the material by ultrasonic vibration, which enhances the toughness of the material and changes the fracture mode. With the increase in frequency, it can be seen that the transgranular fracture phenomenon becomes more obvious and the microcracks on the grains increase. A large number of wrinkles are found on Al2O3 grains at the fracture surfaces of 28 kHz and 35 kHz. According to the non-local theory of wave propagation, with the increase in ultrasonic frequency, the dispersion phenomenon becomes more pronounced, and the interaction between the ultrasonic wave and the internal structure of the ceramic material becomes more obvious. The greater the number of surface microcracks the more complex the crack propagation path is. From the processing point of view, this phenomenon makes it easier to form a surface that is similar in structure to wool felt fiber, thus improving the surface accuracy. It can be seen from Fig. 9 that, at the same frequency, with the increase in amplitude, intergranular-transgranular mixed fracture occurs, and the crack propagation path is broken randomly and tortuously along the crystal plane. It can be seen from the SED picture that the cross-section is uneven and honeycomb-shaped, and there are a small number of shallow dimples and tearing edges. This means that, when the material is damaged, the crack no longer spreads completely along the grain boundary, but a part passes through the middle of the grain [26]. Generally, the propagation of cracks follows the principle of minimum energy. Transgranular fracture occurs in ZTA ceramics, which indicates that grain boundary is strengthened and cracks are forced to penetrate the grains, thus improving material properties. As the amplitude increases, the propagation speed of the ultrasonic wave in the material increases. From the non-local mechanics, the ultrasonic wave exhibits a dispersion phenomenon in the material. The higher the
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Fig. 9. Fracture morphology of three-point bending specimen under different amplitudes.
speed, the stronger the interaction between the ultrasonic wave and the long-range force of the internal structure of the material, which leads to the attenuation of the required fracture stress. This attenuation is greater than the toughening effect induced by the microscopic change. Therefore, under the action of ultrasonic vibration, the fracture stress is reduced [27].
4.3. Three-point bending fracture XRD diffraction analysis XRD scanning was performed on the fracture surface of specimens with the same amplitude and different frequencies. A Bruker
D8ADVANCE x-ray diffractometer was used for the analysis. It can be seen from Fig. 10 and Table 1 that the nanocomplex ZTA ceramics undergo different degrees of martensite transformation of t-ZrO2 particles to m-ZrO2 particles during fracture at different frequencies. When the material is subjected to ultrasonic vibration, a large number of nanoparticles are randomly distributed within the crystal, which is manifested as a diversity of stresses. Under the action of the stress field at the crack tip, the t-ZrO2 particles dispersed in the matrix will produce a 3%–5% volume expansion during martensitic transformation, and the tensile stress will induce the transformation of t-ZrO2 particles to mZrO2 particles. A stress-induced phase transition occurs in the matrix,
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Fig. 10. Three-point bending fracture XRD diffraction. Table 1 Chemical composition of nanocomposite ceramics after XRD test. Chemical composition
Al2O3 t-ZrO2 m-ZrO2
0 kHz
20 kHz
28 kHz
35 kHz
wt%
wt%
wt%
wt%
42 48.3 5.2
42.1 46.7 6.8
41.9 44.7 8.8
41.8 42.9 10.6
which results in a dispersed distribution of microcracks. During the expansion of the main crack tip, secondary deflection and bifurcation of the transgranular crack occur (the crack changes the direction of extension) and hinders crack propagation. This increases the energy consumption of crack propagation, lengthens the path and difficulty of crack propagation, and increases the fracture surface energy [28]. The transgranular fracture registers a significant increase in toughness and the phase change layer produces a relatively high residual compressive stress, generating a toughening effect, thereby improving the fracture toughness and strength of the test piece. In addition to generating a new fracture surface to absorb energy, it also absorbs energy due to the volume effect at the time of phase change (3%–5% volume expansion). Stress-induced tissue transformation consumes the applied stress. At the same time, compressive stress is generated on the crack due to the volume expansion of the particles that
underwent phase change, which hinders the crack propagation. Secondary interface and microcracks are produced in the grain. The growth of these microcracks themselves also plays a role in dispersing the energy of the main crack tip. This leads to the differentiation of grains and the appearance of local weak links, so that the cracks blocked at the grain boundary can enter the grain, thus inhibiting the rapid growth of main cracks and improving the plasticity of materials [29]. This shows that the mechanical properties of ceramics are greatly improved by ultrasonic vibration, and t-m phase transformation can occur under stress induction to absorb strain, so that the ceramics have better plastic properties. It can also be seen from the results in Fig. 10 that the martensite transformation rate increases with the increase in frequency. The t-m phase transition rate at 35 kHz is higher than that at other frequencies. According to the mechanism of phase transformation toughening, the higher the t-m phase transformation rate of ZTA nanocomposite ceramics, the stronger the plastic properties of ceramics, which indicates that it is easier to achieve ductile processing under ultrasonic excitation. 5. Conclusion Based on the non-local theory, the effects of ultrasonic vibration on wave dispersion characteristics and non-local modulus are analyzed and compared with the results obtained using the local theory. Threepoint bending test in the presence of ultrasonic vibration was carried
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out using different ultrasonic frequencies and amplitudes. SEM and XRD observations were carried out on the fracture surface of the workpiece. The effects of ultrasonic vibration on the fracture properties and crack propagation mechanism of the material were analyzed. The following conclusions were drawn: (1) Under the non-local theory, due to the interaction between the ultrasonic vibration and ceramic microstructures, the fracture stress of ultrasonic vibration decreases with the increase in frequency and amplitude, and is less than the flexural strength under ordinary loading. Different ultrasonic vibration characteristics change the internal stress of material vibration and cause particle scattering, which leads to the intensification of the phase velocity dispersion. (2) From the SEM images of the fracture surface of three-point bending specimens, it can be found that under ultrasonic vibration, the fracture surface shows obvious grain sections, and a large number of cracks appear on the grain, which indicates that the material shows transgranular and intergranular mixed fracture. With the increase in ultrasonic frequency and amplitude, the transgranular fracture of the fracture surface becomes more obvious, and the number of surface microcracks, complexity of the crack propagation path, and the interaction between ultrasonic energy and the internal structure of ceramic materials all increase. (3) It was found by fracture XRD that ultrasonic vibration can induce the t-m phase transformation of ceramic materials under stress, which absorbs strain and causes the materials to have good plasticity. The intrinsic phase transformation toughening mechanism makes it easier for materials to be processed in the ductile domain, which greatly improves the mechanical properties of nanocomposite ceramics.
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Acknowledgments [22]
This research was supported financially by the Henan Natural Science Foundation (162300410120) and National Joint Fund (U1604255).
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