# Dynamic Effective Properties of Lead-Reinforced Epoxy Resin Composites with Differential Method

Authors: Haitao Zhu; Yongping Wan
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

In this paper, the quasi-P wave propagation in epoxy/lead composites is studied by the differential method with polarization stress and polarization momentum. The corresponding differential equations are numerically solved by the fourth-order Runge–Kutta method, and the dynamic effective modulus and density are obtained. Correctness of the results obtained by the differential method is verified by comparing with the results obtained by the self-consistent method as well as experiments. Results show that the dynamic effective modulusL1133 and L3333 increase with the increase of aspect ratio δ, whereas L1111 and L1122 decrease with the increase of δ. The phase velocity ratio of the quasi-P wave propagation in the composites tends to 1 when the dimensionless wave number is large, and the dynamic effective density and modulus obtained are not isotropic when δ≠1. It is believed that the differential method can be a choice for solving dynamic effective properties of heterogeneous composites.


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## Body

## 1. Introduction

In daily life, epoxy/lead composites have attracted more and more attention and been applied in many fields [1–3] since epoxy resins have the advantages of being green and degradable. The actual working environment of these composites is complex and changeable, especially in the case of nondestructive testing carried out on the composites with wave propagation. When damage degree of the composites is appraised during this working process, we need to know the dynamic effective properties of the composites.The study of dynamic effective properties of composites mainly includes two parts: experiments and theoretical solutions. In the experimental aspect, Tauchert and Guzelsu tested the propagation of P-wave and shear wave in spherical particle-reinforced boron-epoxy composite [4]. Kinra tested the propagation of P-wave in the spherical and randomly distributed glass spheres/epoxy, lead spheres/epoxy, and steel spheres/PMMA composites, respectively [5]. Wang and Yuan tested the propagation of higher-order Lamb waves in composite plates [6].In terms of theory, the dynamic constitutive relationship is constructed by ensemble average analysis with periodicity assumption by considering periodic composites and Bloch wave. Willis used Fourier series expansion and ensemble average to derive Green’s function and the analytical dynamic effective modulus and density for one-dimensional periodic elastic composite [7]. Nemat-Nasser used the dynamic equivalent inclusion method to construct the dynamic effective constitutive relation in the case of periodic layered elastic composite by introducing eigenstress and eigenvelocity [8]. Srivastava and Nemat-Nasser derived the dynamic constitutive relation for the case of cuboid inclusion embedded in the cube matrix of pure elastic unit cell by introducing the eigenstrain and eigenmomentum with the dynamic equivalent inclusion method [9]. Sigalas and Soukoulis analyzed and studied the propagation of elastic waves in periodic layered composite by the transfer matrix method [10]. Hou et al. analyzed and studied the transmission spectra of elastic wave propagation in two-dimensional lead-epoxy composite [11, 12]. Sigalas and Economou used the wave function expansion method to analyze and calculate the bandgap characteristics of elastic waves propagating in periodic phononic crystals [13]. Yan and Wang calculated the band gap structure and fluid-solid coupling system of one-dimensional periodic gold-epoxy composites by wavelet analysis [14]. García-Pablos et al. firstly analyzed and studied the frequency band structure of elastic wave propagation in composites by the using finite-difference time-domain method, and verified the calculated results by comparing with experimental results [15]. Kafesaki and Economou used the multiple scattering method to analyze and calculate the frequency band of elastic wave propagation in three-dimensional periodic composites [16]. Kohn et al. used the variational method to analyze and study the dispersion relations of harmonic wave propagation in layered periodic elastic composites [17]. Hussein analyzed the calculation of phononic crystal, photonic crystal, and electronic band structure, respectively, by means of the reduced Bloch mode expansion [18].Another theoretical method is based on the representative volume element (RVE) with the homogenization method, where the inclusion is randomly distributed in the composites. Tsinopoulos et al. solved the longitudinal wave and shear wave propagation in the random particle-reinforced composite with the iterative effective medium approximation method [19]. Verbis et al. used this method to analyze and solve the P wave, SH wave, and SV wave propagation in two-dimensional composite [20]. Kanaun and Levin used effective medium method to analyze and solve the case of axial shear wave propagation in infinite cylinder fiber composite [21]. Kim et al. studied the dynamic effective properties of longitudinal and shear wave propagation in spherical particle-reinforced two-phased composites with asymptotic expansion of the potential function and the self-consistent method [22]. Sabina et al. generalized the static self-consistent method to the dynamic situation by introducing polarization stress and polarization momentum and analyzed the dynamic effective properties of spherical particle-reinforced two-phased composites with P wave and S wave propagated in the composites [23] and quasi-longitudinal wave and transverse wave in spherical particle-reinforced composites and hole problems [24, 25]. Bussink et al. analyzed and studied the dynamic effective properties of longitudinal wave and shear wave propagation in two-dimensional lead-reinforced epoxy [26].On the other hand, the differential method is usually used to solve the static problems. Studies have proved the efficiency of the differential method [27–29]. In this paper, we study the possibility of analyzing the dynamic effective properties of two-phase particulate reinforced composites using the differential method. The quasi-P wave propagation in epoxy/lead composites is considered. By introducing the polarization stress and polarization momentum proposed by Willis [30], the differential method is generalized to analyze and solve the dynamic effective properties. The composite is assumed to be uniform macroscopically, while on the microscopic level, the inhomogeneities, with the same shape, size, and orientation as well as material properties, can be translationally but not rotationally scattered in the epoxy. The work of this paper is as follows. In Section 2, the materials and basic equations are introduced. Then, by introducing the scattering problem of approximate solution, the differential method is generalized to dynamic situation and differential equations are given, where the specific forms of differential equations are shown in appendix B. In Section 3, by the use of the fourth-order Runge–Kutta method, the differential equations are numerically solved for different aspect ratios. Phase velocity ratios obtained by the differential method are compared to the self-consistent method and experimental results for verification. Also, dynamic effective modulus and density obtained by the differential method are compared to the results obtained by the self-consistent method. In Section 4, conclusions of this paper are given.


## 2. Materials and Methods

For the sake of simplicity, quasi-P wave propagating in a two-phased linear elastic composite material is considered. The wave propagates along the negative direction parallel to the axisx3, and the material’s principal axes of properties are fixed and only translational. The composite structure is shown in Figure 1, where the spheroid is inclusion and outside the spheroid is matrix.Figure 1
Illustration of inclusions.Spheroidal inclusions satisfy the following equation:(1)Ω=x:x12+x22+x32δ2<a2,where δ is the aspect ratio and a is the radius.


### 2.1. Basic Equations

Generally, the motion equation of the linear elastic material is(2)divσ=p˙.The constitutive equation is(3)σ=Le,p=ρv,where σ denotes stress, L denotes modulus, e denotes strain, p denotes momentum, ρ denotes density, v denotes velocity, and the dot above the momentum p denotes derivative with respect to time.The relation between velocityv and displacement u is(4)eij=12ui,j+uj,i,vi=u˙i.Due to the linearity of equations (2) and (3), considering the ensemble average of equation (2), we have(5)divσ=p˙,where · represents ensemble average. Considering the two-phased composite material, where the modulus of the matrix is L2, density is ρ2, strain is e2, volume fraction of matrix is f2, the modulus of inclusion is L1, density is ρ1, strain is e1, and volume fraction of inclusion is f1, we have(6)e=f1e1+f2e2,(7)v=f1v1+f2v2,(8)σ=f1σ1+f2σ2=f1L1e1+f2L2e2,(9)p=f1p1+f2p2=f1ρ1v1+f2ρ2v2,(10)f1+f2=1.By eliminatinge2 and v2 through (6) and (7), (8) and (9) can be rewritten as(11)σ=L2e+f1L1−L2e1,(12)p=ρ2v+f1ρ1−ρ2v1,with(13)•rx,t=1Ωr∫Ur•x,t;x′dx′,where Ur represents the region x′:x−x′∈Ωr, Ωr represents the volume of Ωr, r represents the r-th phase material, x is the spatial coordinate, t is time, and x′ is the center of the inclusion.


### 2.2. Single Inclusion Scattering Problem

The propagation of plane harmonic wave in composite materials is considered and has the form [24]:(14)u0x,t=mexpikn⋅x−ωt,where u0 denotes physical quantities such as stress σ, strain e, density ρ, and velocity v, m is the amplitude of these physical quantities, i=−1, k is the wave number, n is the unit directional vector of wave propagation, and ω is the circular frequency.At the same time, polarization stressτ and polarization momentum π are introduced with the form [30]:(15)τ=Lr−L0e,(16)π=ρr−ρ0v,where r refers to r-th phase material, r=1 represents inclusion, r=2 represents matrix, L0 represents modulus of reference medium, and ρ0 represents density of the reference medium. For the convenience of analysis, let the modulus and density of the reference medium be the same as the matrix material. Then, it can be known from (15) and (16) that the polarization stress and polarization momentum only exist in the inclusion but not in the matrix.Then, stressσ and momentum p can be obtained from (15) and (16) with the following form:(17)σ=L0e+τ,(18)p=ρ0v+π.Substituting (17) and (18) into (2) and taking divτ−π˙ as the body force term, we have(19)u=u0+G∗divτ−π˙=u0−S∗τ−M∗π,where G is the elastodynamic Green’s function, S and M are the quantities related to Green’s function, and ∗ represents convolution operation. Specific expression of elastodynamic Green’s function can be seen from equation (A.1) in Appendix A.According to the study in [24] and in combination with (15) and (16), we have(20)L1−L0−1τ+Sx∗τ+Mx∗π=e0,(21)ρ1−ρ0−1π+St∗τ+Mt∗π=v0,where Sx, Mx, St, and Mt are the quantities related to Green’s function [24]. It can be verified that M¯x=0 and S¯t=0 because the inclusion has a center of symmetry, where the bar above Mx and St represents the volume average over the inclusion.From (20) and (21), we have(22)τ=L1−L0−1+S¯x−1e¯0,(23)π=ρ1−ρ0−1+M¯t−1v¯0,where the specific expressions of S¯x and M¯t can be seen from equations (A.6) and (A.7) in Appendix A, respectively.In combination with (15), (16), (22), and (23), we have(24)e=I+S¯xL1−L0−1e¯0,(25)v=I+M¯tρ1−ρ0−1v¯0.Assuming that the center of the inclusion is atx′ and e¯0=e0x′hkn, we obtain(26)e1x=hknI+S¯xL1−L0−11Ω∫Ωe0x′dx′,(27)v1x=hknI+M¯tρ1−ρ0−11Ω∫Ωv0x′dx′,with(28)hkn=1Ω∫Ωexpikn⋅xdx.Due toe0x′expikn⋅x=e0xexpikn⋅x′, e0x′=e0xexp−ikn⋅x−x′ is obtained, then,(29)e1x=hknI+S¯xL1−L0−11Ω∫Ωe0x′dx′=hknI+S¯xL1−L0−11Ω∫Ωe0xexp−ikn⋅x−x′dx′=hknI+S¯xL1−L0−11Ω∫Ωe0xexpikn⋅x′−xdx′−x=hknhknI+S¯xL1−L0−1e0x.Also, similarly,(30)v1x=hknhknI+M¯tρ1−ρ0−1v0x.Substituting (29) and (30) into (11) and (12), we have(31)σ=L2e+f1L1−L2e1=L2e+f1L1−L2hknhknI+S¯xL1−L0−1e0,(32)p=ρ2v+f1ρ1−ρ2v1=ρ2v+f1ρ1−ρ2hknhknI+M¯tρ1−ρ0−1v0.LetL¯ and ρ¯ represent the overall effective modulus and the overall effective density of the material, respectively. From equations (31) and (32), we have(33)σ=L¯e=L2e+f1L1−L2hknhknI+S¯xL1−L0−1e0,(34)p=ρ¯v=ρ2v+f1ρ1−ρ2hknhknI+M¯tρ1−ρ0−1v0.


### 2.3. Differential Method

For the two-phased heterogeneous composite material, it is assumed that the volume fraction of inclusion isf1 at some time. When the infinitesimal element of volume dV is removed and the same volume of inclusion is added, the volume fraction of inclusion is f1+df1. Then, the volume of inclusion in the new composite is(35)f1V+dV−f1dV=f1+df1V.Equation (35) can be simplified as [31](36)dVV=df11−f1.When the volume fraction of inclusion isf1 in composite materials at some time, let the modulus and density of the composite be L¯f1 and ρ¯f1, respectively. Then, let the composite at this time be as matrix, namely, L2=L¯f1 and ρ2=ρ¯f1. After adding the volume dV of inclusion, let the overall effective modulus be L¯f1+df1 and density be ρ¯f1+df1. By replacing L2, ρ2, L¯, and ρ¯ in equations (33) and (34) by L¯f1, ρ¯f1, L¯f1+df1, and ρ¯f1+df1, respectively, equations (33) and (34) can be rewritten as(37)L¯f1+df1−L¯f1e=dVVL1−L¯f1hknhknI+S¯xf1L1−L¯f1−1e0,(38)ρ¯f1+df1−ρ¯f1v=dVVhknhknρ1−ρ¯f1I+M¯tf1ρ1−ρ¯f1−1v0,WhendV⟶0[32–34], we have(39)e=e01+ΟdVV,(40)v=v01+ΟdVV.In combination with (36), (39), and (40), equations (37) and (38) can be rearranged as(41)dL¯f1df1=11−f1hknhknL1−L¯f1I+S¯xf1L1−L¯f1−1,(42)dρ¯f1df1=11−f1hknhknρ1−ρ¯f1I+M¯tf1ρ1−ρ¯f1−1.The initial conditions of the corresponding expressions of (37) and (38) are, respectively,(43)L¯f1|f1=0=L2,(44)ρ¯f1|f1=0=ρ2.


## 2.1. Basic Equations

Generally, the motion equation of the linear elastic material is(2)divσ=p˙.The constitutive equation is(3)σ=Le,p=ρv,where σ denotes stress, L denotes modulus, e denotes strain, p denotes momentum, ρ denotes density, v denotes velocity, and the dot above the momentum p denotes derivative with respect to time.The relation between velocityv and displacement u is(4)eij=12ui,j+uj,i,vi=u˙i.Due to the linearity of equations (2) and (3), considering the ensemble average of equation (2), we have(5)divσ=p˙,where · represents ensemble average. Considering the two-phased composite material, where the modulus of the matrix is L2, density is ρ2, strain is e2, volume fraction of matrix is f2, the modulus of inclusion is L1, density is ρ1, strain is e1, and volume fraction of inclusion is f1, we have(6)e=f1e1+f2e2,(7)v=f1v1+f2v2,(8)σ=f1σ1+f2σ2=f1L1e1+f2L2e2,(9)p=f1p1+f2p2=f1ρ1v1+f2ρ2v2,(10)f1+f2=1.By eliminatinge2 and v2 through (6) and (7), (8) and (9) can be rewritten as(11)σ=L2e+f1L1−L2e1,(12)p=ρ2v+f1ρ1−ρ2v1,with(13)•rx,t=1Ωr∫Ur•x,t;x′dx′,where Ur represents the region x′:x−x′∈Ωr, Ωr represents the volume of Ωr, r represents the r-th phase material, x is the spatial coordinate, t is time, and x′ is the center of the inclusion.


## 2.2. Single Inclusion Scattering Problem

The propagation of plane harmonic wave in composite materials is considered and has the form [24]:(14)u0x,t=mexpikn⋅x−ωt,where u0 denotes physical quantities such as stress σ, strain e, density ρ, and velocity v, m is the amplitude of these physical quantities, i=−1, k is the wave number, n is the unit directional vector of wave propagation, and ω is the circular frequency.At the same time, polarization stressτ and polarization momentum π are introduced with the form [30]:(15)τ=Lr−L0e,(16)π=ρr−ρ0v,where r refers to r-th phase material, r=1 represents inclusion, r=2 represents matrix, L0 represents modulus of reference medium, and ρ0 represents density of the reference medium. For the convenience of analysis, let the modulus and density of the reference medium be the same as the matrix material. Then, it can be known from (15) and (16) that the polarization stress and polarization momentum only exist in the inclusion but not in the matrix.Then, stressσ and momentum p can be obtained from (15) and (16) with the following form:(17)σ=L0e+τ,(18)p=ρ0v+π.Substituting (17) and (18) into (2) and taking divτ−π˙ as the body force term, we have(19)u=u0+G∗divτ−π˙=u0−S∗τ−M∗π,where G is the elastodynamic Green’s function, S and M are the quantities related to Green’s function, and ∗ represents convolution operation. Specific expression of elastodynamic Green’s function can be seen from equation (A.1) in Appendix A.According to the study in [24] and in combination with (15) and (16), we have(20)L1−L0−1τ+Sx∗τ+Mx∗π=e0,(21)ρ1−ρ0−1π+St∗τ+Mt∗π=v0,where Sx, Mx, St, and Mt are the quantities related to Green’s function [24]. It can be verified that M¯x=0 and S¯t=0 because the inclusion has a center of symmetry, where the bar above Mx and St represents the volume average over the inclusion.From (20) and (21), we have(22)τ=L1−L0−1+S¯x−1e¯0,(23)π=ρ1−ρ0−1+M¯t−1v¯0,where the specific expressions of S¯x and M¯t can be seen from equations (A.6) and (A.7) in Appendix A, respectively.In combination with (15), (16), (22), and (23), we have(24)e=I+S¯xL1−L0−1e¯0,(25)v=I+M¯tρ1−ρ0−1v¯0.Assuming that the center of the inclusion is atx′ and e¯0=e0x′hkn, we obtain(26)e1x=hknI+S¯xL1−L0−11Ω∫Ωe0x′dx′,(27)v1x=hknI+M¯tρ1−ρ0−11Ω∫Ωv0x′dx′,with(28)hkn=1Ω∫Ωexpikn⋅xdx.Due toe0x′expikn⋅x=e0xexpikn⋅x′, e0x′=e0xexp−ikn⋅x−x′ is obtained, then,(29)e1x=hknI+S¯xL1−L0−11Ω∫Ωe0x′dx′=hknI+S¯xL1−L0−11Ω∫Ωe0xexp−ikn⋅x−x′dx′=hknI+S¯xL1−L0−11Ω∫Ωe0xexpikn⋅x′−xdx′−x=hknhknI+S¯xL1−L0−1e0x.Also, similarly,(30)v1x=hknhknI+M¯tρ1−ρ0−1v0x.Substituting (29) and (30) into (11) and (12), we have(31)σ=L2e+f1L1−L2e1=L2e+f1L1−L2hknhknI+S¯xL1−L0−1e0,(32)p=ρ2v+f1ρ1−ρ2v1=ρ2v+f1ρ1−ρ2hknhknI+M¯tρ1−ρ0−1v0.LetL¯ and ρ¯ represent the overall effective modulus and the overall effective density of the material, respectively. From equations (31) and (32), we have(33)σ=L¯e=L2e+f1L1−L2hknhknI+S¯xL1−L0−1e0,(34)p=ρ¯v=ρ2v+f1ρ1−ρ2hknhknI+M¯tρ1−ρ0−1v0.


## 2.3. Differential Method

For the two-phased heterogeneous composite material, it is assumed that the volume fraction of inclusion isf1 at some time. When the infinitesimal element of volume dV is removed and the same volume of inclusion is added, the volume fraction of inclusion is f1+df1. Then, the volume of inclusion in the new composite is(35)f1V+dV−f1dV=f1+df1V.Equation (35) can be simplified as [31](36)dVV=df11−f1.When the volume fraction of inclusion isf1 in composite materials at some time, let the modulus and density of the composite be L¯f1 and ρ¯f1, respectively. Then, let the composite at this time be as matrix, namely, L2=L¯f1 and ρ2=ρ¯f1. After adding the volume dV of inclusion, let the overall effective modulus be L¯f1+df1 and density be ρ¯f1+df1. By replacing L2, ρ2, L¯, and ρ¯ in equations (33) and (34) by L¯f1, ρ¯f1, L¯f1+df1, and ρ¯f1+df1, respectively, equations (33) and (34) can be rewritten as(37)L¯f1+df1−L¯f1e=dVVL1−L¯f1hknhknI+S¯xf1L1−L¯f1−1e0,(38)ρ¯f1+df1−ρ¯f1v=dVVhknhknρ1−ρ¯f1I+M¯tf1ρ1−ρ¯f1−1v0,WhendV⟶0[32–34], we have(39)e=e01+ΟdVV,(40)v=v01+ΟdVV.In combination with (36), (39), and (40), equations (37) and (38) can be rearranged as(41)dL¯f1df1=11−f1hknhknL1−L¯f1I+S¯xf1L1−L¯f1−1,(42)dρ¯f1df1=11−f1hknhknρ1−ρ¯f1I+M¯tf1ρ1−ρ¯f1−1.The initial conditions of the corresponding expressions of (37) and (38) are, respectively,(43)L¯f1|f1=0=L2,(44)ρ¯f1|f1=0=ρ2.


## 3. Results and Discussion

For the convenience of obtaining numerical results for quantitative analysis, equations (41)∼(44) are numerically solved by the fourth-order Runge–Kutta method [35]. Specific formations of equations (41)∼(44) can be seen from equations (B.24) to (B.25) in Appendix B. The iteration step length h is 0.001, and the calculation formula of phase velocity can be seen from equation (B.13). Material properties are same with the matrix when calculating the phase velocity of the matrix material, and material properties are taken as dynamic effective properties when calculating phase velocity of the composite. The matrix is epoxy, and inclusion is lead. Material’s properties are shown in Table 1 [5]:Table 1
Elastic modulus and density of the matrix and inclusion. The units of modulus and density areGPa and 103Kg/m3, respectively.
MaterialsρL1111L1122L1133L3333L2323L1212Epoxy1.2028.3774.9154.9158.3771.7311.731Lead11.355.459838.759838.759855.45988.358.35


### 3.1. Verification of the Differential Method

Forδ=1, quasi-P wave propagation is calculated by the differential method and self-consistent method for inclusion volume fractions of 5.4% and 15.9%, respectively. The ratio of the phase velocity in the composite and matrix is compared with the experimental results and shown in Figures 2 and 3 below. For δ=0.2, quasi-P wave propagation is obtained by the differential method and self-consistent method, respectively, for inclusion volume fraction of 1%. The ratio of the phase velocity in the composite and matrix is shown in Figure 4.Figure 2
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM), and experimental results (ER) in the case of quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% andδ=1.Figure 3
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM), and experimental results (ER) in the case of quasi-P wave propagates in the composite material for inclusion volume fraction of 15.9% andδ=1.Figure 4
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM) when quasi-P wave propagates in the composite material for inclusion volume fraction of 1% andδ=0.2.As can be seen from Figure2, when the inclusion volume fraction is 5.4%, the ratio of the phase velocity calculated by the differential method is in good agreement with the results obtained by experiments as well as the self-consistent method. Figure 3 shows that when the inclusion volume fraction is 15.9% and the dimensionless wave number ωa/c1 is at 0.3–0.5, the ratio of the phase velocity obtained by the differential method is larger than the results obtained by the self-consistent method. This is mainly due to the influence of resonance area included in the range of 0.3–0.5, and in other areas, the results obtained by the differential method are in good agreement with the self-consistent method and experiments. As can be seen from Figure 4, when δ=0.2 and the inclusion volume fraction is 1%, the results obtained by the differential method are in good agreement with the self-consistent method. Figures 2–4 also verify the correctness of results obtained by the differential method.


### 3.2. Comparisons of Different Forms of Inclusion with the Differential Method and Self-Consistent Method

Whenδ=0.5 and δ=5, the differential method and self-consistent method are used to calculate the phase velocity ratio of quasi-P wave propagation in the composite material for inclusion volume fraction of 5.4% and 15.9%, respectively, and the comparisons of the phase velocity ratio are shown in Figures 5 and 6.Figure 5
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM) when quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% and 15.9% andδ=0.5.Figure 6
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM) when quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% and 15.9% andδ=5.Figures5 and 6 show that when the inclusion volume fraction is 5.4%, results obtained by the differential method are in good agreement with the self-consistent method. Phase velocity ratio near the resonance zone obtained by the differential method is larger than the self-consistent method when δ=0.5, and phase velocity ratio near the resonance zone obtained by the self-consistent method is larger than the differential method when δ=5.Whenδ=0.5, 1, and 5, the differential method is used to calculate the phase velocity ratio of quasi-P wave propagation in the composite material with inclusion volume fraction of 5.4% and 15.9%, respectively. Results are shown in Figures 7 and 8.Figure 7
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method when quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% andδ=0.5, 1, and 5, respectively.Figure 8
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method when quasi-P wave propagates in the composite material for inclusion volume fraction of 15.9% andδ=0.5, 1, and 5, respectively.It can be seen from Figure8 that when the inclusion volume fraction is 15.9%, the peak value of the phase velocity ratio becomes small as the value of δ increases.Figures2–8 show that when the dimensionless wave number ωa/c1 is large, the value of the ratio of the phase velocity in the composite and matrix tends to 1. This is because when ωa/c1 takes a larger value, according to equation (A.14) in Appendix A, we have hknhkn⟶0, and this leads to the right side of equations (41) and (42) tends to zero, namely, the dynamic effective properties of the composite tend to be same as matrix.


### 3.3. Dynamic Effective Properties of the Composites with the Differential Method and Self-Consistent Method

The differential method and self-consistent method are used to calculate the dynamic effective modulus and density of quasi-P wave propagation in the composite material for inclusion volume fractions of 5.4% and 15.9% whenδ=0.5, 1, and 5, respectively. The dynamic effective density ρ11 and ρ33 and modulus L1111, L1122, L1133, and L3333 are shown in Figures 9 and 10. For the convenience of analysis and discussion, the real part of the obtained results is taken as the dynamic effective density and modulus.Figure 9
When quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% andδ=0.5, 1, and 5, the dynamic effective density ρ11 and ρ33 and dynamic effective modulus L1111, L1122, L1133, and L3333 are obtained by the differential method (DM) and self-consistent method (SCM), respectively.
(a)(b)(c)(d)(e)(f)Figure 10
When quasi-P wave propagates in the composite material for inclusion volume fraction of 15.9% andδ=0.5, 1, and 5, the dynamic effective density ρ11 and ρ33 and dynamic effective modulus L1111, L1122, L1133, and L3333 are obtained by the differential method (DM) and self-consistent method (SCM), respectively.
(a)(b)(c)(d)(e)(f)According to Figures9 and 10, the dynamic effective densities obtained by the differential method are larger than those obtained by the self-consistent method near the resonance zone when δ=0.5 and 1, and this means the results obtained by the differential method are more accurate.According to Figures9 and 10, the dynamic effective modulus L1111 and L1122 decreases with the increase of δ. From Figures 9 and 10, the dynamic effective modulus L1133 and L3333 increases with δ increases when ωa/c1 is small. This is because the cross-section geometry of inclusion increases as δ increases in parallel to the direction of x3. Also, when ωa/c1 is small, especially when ωa/c1⟶0, which can be thought as quasi-static condition, the results are in line with the actual case at this time.As can be seen from Figures9 and 10, when δ=1, the matrix and inclusion are isotropic materials, and the dynamic effective density and modulus are also isotropic. When δ≠1, although the matrix and inclusion are isotropic, the dynamic effective density and modulus are not isotropic. This is because when δ=1, the inclusions are spherical particles with highly symmetric of geometry, but when δ≠1, the shape of the inclusion is rotating ellipsoid (δ>1) and flat (δ<1), respectively, and the geometry of the inclusions is not symmetrical. Therefore, even if the matrix and inclusion are isotropic, the dynamic effective density and modulus obtained are not isotropic when δ≠1. Furthermore, the dynamic effective density and modulus tend to be same for different δ when the dimensionless wave number ωa/c1 is large enough.


## 3.1. Verification of the Differential Method

Forδ=1, quasi-P wave propagation is calculated by the differential method and self-consistent method for inclusion volume fractions of 5.4% and 15.9%, respectively. The ratio of the phase velocity in the composite and matrix is compared with the experimental results and shown in Figures 2 and 3 below. For δ=0.2, quasi-P wave propagation is obtained by the differential method and self-consistent method, respectively, for inclusion volume fraction of 1%. The ratio of the phase velocity in the composite and matrix is shown in Figure 4.Figure 2
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM), and experimental results (ER) in the case of quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% andδ=1.Figure 3
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM), and experimental results (ER) in the case of quasi-P wave propagates in the composite material for inclusion volume fraction of 15.9% andδ=1.Figure 4
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM) when quasi-P wave propagates in the composite material for inclusion volume fraction of 1% andδ=0.2.As can be seen from Figure2, when the inclusion volume fraction is 5.4%, the ratio of the phase velocity calculated by the differential method is in good agreement with the results obtained by experiments as well as the self-consistent method. Figure 3 shows that when the inclusion volume fraction is 15.9% and the dimensionless wave number ωa/c1 is at 0.3–0.5, the ratio of the phase velocity obtained by the differential method is larger than the results obtained by the self-consistent method. This is mainly due to the influence of resonance area included in the range of 0.3–0.5, and in other areas, the results obtained by the differential method are in good agreement with the self-consistent method and experiments. As can be seen from Figure 4, when δ=0.2 and the inclusion volume fraction is 1%, the results obtained by the differential method are in good agreement with the self-consistent method. Figures 2–4 also verify the correctness of results obtained by the differential method.


## 3.2. Comparisons of Different Forms of Inclusion with the Differential Method and Self-Consistent Method

Whenδ=0.5 and δ=5, the differential method and self-consistent method are used to calculate the phase velocity ratio of quasi-P wave propagation in the composite material for inclusion volume fraction of 5.4% and 15.9%, respectively, and the comparisons of the phase velocity ratio are shown in Figures 5 and 6.Figure 5
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM) when quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% and 15.9% andδ=0.5.Figure 6
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method (DM) and self-consistent method (SCM) when quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% and 15.9% andδ=5.Figures5 and 6 show that when the inclusion volume fraction is 5.4%, results obtained by the differential method are in good agreement with the self-consistent method. Phase velocity ratio near the resonance zone obtained by the differential method is larger than the self-consistent method when δ=0.5, and phase velocity ratio near the resonance zone obtained by the self-consistent method is larger than the differential method when δ=5.Whenδ=0.5, 1, and 5, the differential method is used to calculate the phase velocity ratio of quasi-P wave propagation in the composite material with inclusion volume fraction of 5.4% and 15.9%, respectively. Results are shown in Figures 7 and 8.Figure 7
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method when quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% andδ=0.5, 1, and 5, respectively.Figure 8
Comparison of ratios of the phase velocity in the composite material and matrix obtained by the differential method when quasi-P wave propagates in the composite material for inclusion volume fraction of 15.9% andδ=0.5, 1, and 5, respectively.It can be seen from Figure8 that when the inclusion volume fraction is 15.9%, the peak value of the phase velocity ratio becomes small as the value of δ increases.Figures2–8 show that when the dimensionless wave number ωa/c1 is large, the value of the ratio of the phase velocity in the composite and matrix tends to 1. This is because when ωa/c1 takes a larger value, according to equation (A.14) in Appendix A, we have hknhkn⟶0, and this leads to the right side of equations (41) and (42) tends to zero, namely, the dynamic effective properties of the composite tend to be same as matrix.


## 3.3. Dynamic Effective Properties of the Composites with the Differential Method and Self-Consistent Method

The differential method and self-consistent method are used to calculate the dynamic effective modulus and density of quasi-P wave propagation in the composite material for inclusion volume fractions of 5.4% and 15.9% whenδ=0.5, 1, and 5, respectively. The dynamic effective density ρ11 and ρ33 and modulus L1111, L1122, L1133, and L3333 are shown in Figures 9 and 10. For the convenience of analysis and discussion, the real part of the obtained results is taken as the dynamic effective density and modulus.Figure 9
When quasi-P wave propagates in the composite material for inclusion volume fraction of 5.4% andδ=0.5, 1, and 5, the dynamic effective density ρ11 and ρ33 and dynamic effective modulus L1111, L1122, L1133, and L3333 are obtained by the differential method (DM) and self-consistent method (SCM), respectively.
(a)(b)(c)(d)(e)(f)Figure 10
When quasi-P wave propagates in the composite material for inclusion volume fraction of 15.9% andδ=0.5, 1, and 5, the dynamic effective density ρ11 and ρ33 and dynamic effective modulus L1111, L1122, L1133, and L3333 are obtained by the differential method (DM) and self-consistent method (SCM), respectively.
(a)(b)(c)(d)(e)(f)According to Figures9 and 10, the dynamic effective densities obtained by the differential method are larger than those obtained by the self-consistent method near the resonance zone when δ=0.5 and 1, and this means the results obtained by the differential method are more accurate.According to Figures9 and 10, the dynamic effective modulus L1111 and L1122 decreases with the increase of δ. From Figures 9 and 10, the dynamic effective modulus L1133 and L3333 increases with δ increases when ωa/c1 is small. This is because the cross-section geometry of inclusion increases as δ increases in parallel to the direction of x3. Also, when ωa/c1 is small, especially when ωa/c1⟶0, which can be thought as quasi-static condition, the results are in line with the actual case at this time.As can be seen from Figures9 and 10, when δ=1, the matrix and inclusion are isotropic materials, and the dynamic effective density and modulus are also isotropic. When δ≠1, although the matrix and inclusion are isotropic, the dynamic effective density and modulus are not isotropic. This is because when δ=1, the inclusions are spherical particles with highly symmetric of geometry, but when δ≠1, the shape of the inclusion is rotating ellipsoid (δ>1) and flat (δ<1), respectively, and the geometry of the inclusions is not symmetrical. Therefore, even if the matrix and inclusion are isotropic, the dynamic effective density and modulus obtained are not isotropic when δ≠1. Furthermore, the dynamic effective density and modulus tend to be same for different δ when the dimensionless wave number ωa/c1 is large enough.


## 4. Conclusions

In this paper, the dynamic effective properties of a two-phased composite with quasi-P wave propagation are solved by the differential method, and the results are compared with the experimental and theoretical results. From the results, we have as follows.(1)
The results obtained by the differential method are compared with the self-consistent method and experimental results, and the correctness of the results obtained by the differential method is verified.(2)
Whenδ=0.5 and δ=1, the peak value of the phase velocity obtained by the differential method is slightly higher than the value obtained by the self-consistent method near the resonance region, which is more accurate.(3)
When the dimensionless wave numberωa/c1 is large, the phase velocity ratio of the quasi-P wave propagation in the composites tends to 1.(4)
Whenδ≠1, even if the inclusion material and the matrix material are isotropic, the dynamic effective density and modulus obtained are not isotropic, which is mainly due to the geometric shape of the inclusion.(5)
The dynamic effective modulusL1111 and L1122 decreases with the increase of δ, and L1133 and L3333 increases with the increase of δ.



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*Source: 3807351-2022-02-12.xml*