# Identification of the Anisotropic Elastic Parameters of NiCrAlY Coating by Combining Nanoindentation and Finite Element Method

Authors: Jingyu Zhai; Yugang Chen; Xinyuan Song; Hongchun Wu; Qingkai Han
Journal: Shock and Vibration
 (2019)

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## Abstract

For vibration damping, coatings are prepared on surface of the structures (substrates), which constitute the coating-substrate composite structures. Elastic parameters of the coating are indispensable for the vibration and damping analysis of the composite structure. Due to the small scale of coating thickness and elastic difference compared with the substrate, the identification results are inevitably influenced by the existence of substrate. Moreover, resulting from the preparation process, elastic properties of hard coating often exhibit anisotropic properties. All the above factors bring about the difficulties of accurate identification. In this study, a method for identifying anisotropic elastic parameters of hard coatings considering substrate effect is proposed, by combining nanoindentation and finite element analysis. Based on the identification results, finite element models are established to analyze the vibration characteristics of the coating-substrate composite structure, which verify the rationality of the anisotropic elastic parameters for vibration analysis. The studies in this paper are significant to more accurately identify the mechanical parameters for establishing the dynamic model. Moreover, they lay the foundation for further optimization design of hard coating damping.


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## Body

## 1. Introduction

In recent years, hard coatings applied as functional coatings on thin plate have shown significant damping effects [1, 2] and are being widely studied as a new type of passive damping method. In the hard coating damping system, the elastic modulus of the hard coating directly affects the stiffness and the damping effect of the coating structure. Therefore, it is significant to accurately measure the elastic modulus of the hard coating for the modeling and vibration absorption designing of the coating-substrate composite structures [3, 4].The testing methods for mechanical properties of materials mainly include the stretching method [5], ultrasonic method [6], indentation method [7], vibration method [8], and special equipment such as thermomechanical dynamics analyzer measurement [9]. Hard coatings are brittle and generally coated on a substrate, which renders it difficult to be made into a single piece for testing. Therefore, when testing the mechanical properties of hard coatings, it is usually performed on composite structures composed of coatings and substrates [10]. For the small size and elastic difference of the hard coating compared with the substrate, the identification results are inevitably influenced by the existence of the substrate, which is called the substrate effect. The existence of the substrate effect makes these traditional measuring methods unable to accurately reflect the mechanical parameters of the coating.In addition, due to the test methods and sample size limits, hard coatings are often reduced to isotropic linear materials when testing their mechanical properties. For example, the American Society for Testing and Materials (ASTM) gave a standard test method [11] for the mechanical properties of coating materials, and the premise of the test is to assume that the properties of the coating are isotropic and linear. Several recent studies [12–15] on the mechanical properties of hard coatings have also equated coatings with isotropic materials. However, the microstructure feature of the hard coating resulting from the special preparation process (such as the layered structure prepared by air plasma spraying [16] and the columnar structure prepared by physical vapor deposition [17]) makes the elastic modulus of the hard coating show a significant difference in horizontal and vertical directions in the coating surface [16–22]. Because the vibration response of the coating-substrate composite structure contains complex modes such as bending and torsion, the anisotropy of the elastic modulus will affect the vibration characteristics [23]; the isotropic elastic modulus is difficult to accurately predict the mechanical behavior of the coating-substrate composite structure in vibration and the damping effect. In order to accurately predict the vibration characteristics of the coating-substrate composite structures and to evaluate the damping effect, it is necessary to develop a method being capable of accurately identifying the anisotropic elastic modulus of the coating under consideration of the substrate effect.The nanoindentation is a technique for evaluating the mechanical properties of materials by loading the indenter on the surface of the materials, in which the indentation depth and loading force can be controlled and be recorded simultaneously [24]. Compared with the traditional indentation method, the depth of the nanoindentation is small enough to nanoscale, and the displacement resolution achieves 1 nm, which is suitable for testing mechanical properties of coating materials [25]. In the nanoindentation system, the degree of elastic deformation and plastic deformation of the material is expressed in the load-displacement curve. The elastic deformation can measure the stiffness and elastic modulus of the material, and the plastic deformation can measure the hardness of the material. For the isotropic bulk materials, Oliver and Pharr [26] proposed a method based on the maximum loading force and the slope of the unloading curve to identify the elastic modulus and hardness of the material, which laid the foundation for the nanoindentation test. However, the data obtained from nanoindentation experiments cannot directly reflect the anisotropic mechanical parameters of the sample piece. Even though some recent studies have proposed methods to extract anisotropic mechanical parameters by nanoindentation [27] or other equipment [6], there are still limitations in use and inevitable effects of substrate.In this study, the anisotropic mechanical parameters of NiCrAlY hard coating that considered substrate effect was studied. Based on the nanoindentation test, finite element method (FEM), and analytical method, an identification method for anisotropic elastic properties of the hard coating considering the substrate effect was established. Finally, the identified parameters are imputed to the finite element model to analyze the vibration characteristics of the hard coating-substrate composite structure. The finite element results were compared with the test results to prove the reasonability of the method proposed in this paper.


## 2. Description of Anisotropic Material Model and Coating Indentation

### 2.1. Anisotropic Material Model of Hard Coating

For the general anisotropic elastic body, in the case of small deformation, the expression of the elastic coefficient matrix is(1)c=c11c12c13c14c15c16c21c22c23c24c25c26c31c32c33c34c35c36c41c42c43c44c45c46c51c52c53c54c55c56c61c62c63c64c65c66,where cmn are called elastic constants and are a total of 36. For extremely anisotropic materials, it proves that cmn = cnm; thus, there are only 21 independent elastic constants.If the elastic body has three elastic planes of symmetry, it is called an orthotropic elastic body where the elastic coefficient matrix can be reduced to(2)c=c11c12c13000c22c23000c33000c4400symmc550c66.There are only 9 independent elastic constants of the orthotropic elastic body. When the coordinate axis directions are consistent with the main directions of elasticity, the normal stress is only related to the normal strain and the shear stress is only related to the shear strain. Therefore, there is no coupling between the tension and shear, nor the shear stress and shear strain in different planes.Based on the orthotropy, if the elasticity is symmetric about an axis that is normal to a plane of isotropy and within this plane, the elasticity is the same in all directions; this material is known as a transversely isotropic material. Thex1-x2 plane is taken as the isotropic plane. When involving the x1 or x2 directions, the subscript of elastic coefficients can be indiscriminate, that is, c11 = c22, c13 = c23, c44 = c55, and c66 = 1/2c11−c12. In this way, there are only five independent elastic constants, and the elasticity matrix is(3)c=c11c12c13000c11c13000c33000c4400symmc44012c11−c12.Due to the microstructure depending on the preparation technology, the anisotropy of the hard coating is often expressed as transversely isotropic, and the mechanical properties are the same in the plane parallel to the substrate (as shown in Figure1); the mechanical properties in the direction vertical to the surface of the substrate are quite different from those in the isotropic plane [19–22].Figure 1
The coating-substrate composite structure.In engineering, the elasticity matrix is often expressed as(4)S=1ET−vTET−vLTEL000−vTET1ET−vLTEL000−vTLET−vTLET1EL0000001GT0000001GL0000001GL,where the subscript “L” indicates longitudinal and the subscript “T” indicates transverse, as shown in Figure 2. Poisson’s ratios vLT and vTL show the following relation:(5)vTLET=vLTEL.Figure 2
The stress distribution at different indentation depths.The shear modulus in the transverse plane can be expressed as(6)GT=ET21+vT.Since Poisson’s ratio has less influence on the calculation results, we supposevT = 0.3 which is Poisson’s ratio in the transverse plane, and the other two Poisson’s ratios have the following constraints [28]:(7)vTL+vLT=2vT.According to [29], the longitudinal shear modulus can be approximated to(8)GL=ET+EL/221+vTL+vLT/2.OnlyET and EL are needed to solve. According to the study of Vlassak et al. [30, 31], the indentation modulus of anisotropic materials is a weighted average of the modulus of elasticity in each direction. The indentation modulus can be expressed as(9)Em=αET+βEL,where α and β are weight coefficients and α + β = 1. In previous study, it was generally considered that α = β = 1/2 [29].


### 2.2. Substrate Effect in Indentation of Thin Coating and Dimensionless Analysis

During the indentation process of the bulk material, when the rigid conical indenter is pressed into the bulk material, the relationship between the loading force and the indentation depth can be expressed as [32](10)F=Cth2,where F is the loading force, h is the indentation depth, and Ct is the scaling factor, which is related to the elastic mechanical parameters of the base material.The coating-substrate composite structure is composed of two materials with different mechanical properties. When the indenter is pressed into the composite structure, the state of stress and strain in the coating near the indenter will be affected by the substrate. Figure2 shows the stress distribution near the indenter and interface at different indentation depths through the finite element simulation. When the indentation depth exceeds a certain range, the strain produced by the indentation has extended to the substrate. As the depth of indentation increases, the curve gained by the indentation test actually contains the properties of both the coating and the substrate, which is called the substrate effect.Due to the substrate effect, during the loading process of the indentation for the coating-substrate composite structure, the relationship between the loading force and the depth of indentation is different from the quadratic relationship for the bulk material. The exponential function can be used to amend the formula [32](11)F=Fmaxhhmx,where the index of the load curve x quantitatively reflects the effect of the substrate on the stress state of the coating during the indentation loading, that is, the extent of the substrate effect.The two important parameters from the indentation curve, the maximum loading force and the index of the load curve, are related to the elastic mechanical parameters and the geometrical parameters of the coating-substrate composite structure. The maximum loading force can be expressed by the functional relation of the elastic mechanical parameters with the geometric parameters of the composite structure, which is(12)Fmax=ϕET,EL,ES,hc,hm,where ET and EL are the transverse and longitudinal elastic modulus of the coating, respectively, and ES is the elastic modulus of the substrate. hc and hm are the coating thickness and the maximum depth of the indentation.Consistent with the maximum loading force, the index of the load curvex can also be expressed as the following functional relation:(13)x=φET,EL,ES,hc,hm.The elastic modulus and maximum depth of the indentation of the base material are selected as the fundamental quantity. Π theorem is applied to nondimensionalize equations (12) and (13), and the following equations can be obtained:(14)FmaxEShm2=ϕETES,ELES,hchm,x=φETES,ELES,hchm.In the numerical simulation of this study, fixed maximum depth of the indentation (equal to that in the indentation test) is taken, and equations (12) and (13) can be expressed as(15)FmaxEShm2=ϕETES,ELES,(16)x=φETES,ELES.From the above equations, as long as the maximum loading forceFmax and the index of the loading curve x at a fixed indentation depth are obtained and then combined with numerical analysis for back analysis, ET and EL of the anisotropic coating can be obtained by solving the equations.


## 2.1. Anisotropic Material Model of Hard Coating

For the general anisotropic elastic body, in the case of small deformation, the expression of the elastic coefficient matrix is(1)c=c11c12c13c14c15c16c21c22c23c24c25c26c31c32c33c34c35c36c41c42c43c44c45c46c51c52c53c54c55c56c61c62c63c64c65c66,where cmn are called elastic constants and are a total of 36. For extremely anisotropic materials, it proves that cmn = cnm; thus, there are only 21 independent elastic constants.If the elastic body has three elastic planes of symmetry, it is called an orthotropic elastic body where the elastic coefficient matrix can be reduced to(2)c=c11c12c13000c22c23000c33000c4400symmc550c66.There are only 9 independent elastic constants of the orthotropic elastic body. When the coordinate axis directions are consistent with the main directions of elasticity, the normal stress is only related to the normal strain and the shear stress is only related to the shear strain. Therefore, there is no coupling between the tension and shear, nor the shear stress and shear strain in different planes.Based on the orthotropy, if the elasticity is symmetric about an axis that is normal to a plane of isotropy and within this plane, the elasticity is the same in all directions; this material is known as a transversely isotropic material. Thex1-x2 plane is taken as the isotropic plane. When involving the x1 or x2 directions, the subscript of elastic coefficients can be indiscriminate, that is, c11 = c22, c13 = c23, c44 = c55, and c66 = 1/2c11−c12. In this way, there are only five independent elastic constants, and the elasticity matrix is(3)c=c11c12c13000c11c13000c33000c4400symmc44012c11−c12.Due to the microstructure depending on the preparation technology, the anisotropy of the hard coating is often expressed as transversely isotropic, and the mechanical properties are the same in the plane parallel to the substrate (as shown in Figure1); the mechanical properties in the direction vertical to the surface of the substrate are quite different from those in the isotropic plane [19–22].Figure 1
The coating-substrate composite structure.In engineering, the elasticity matrix is often expressed as(4)S=1ET−vTET−vLTEL000−vTET1ET−vLTEL000−vTLET−vTLET1EL0000001GT0000001GL0000001GL,where the subscript “L” indicates longitudinal and the subscript “T” indicates transverse, as shown in Figure 2. Poisson’s ratios vLT and vTL show the following relation:(5)vTLET=vLTEL.Figure 2
The stress distribution at different indentation depths.The shear modulus in the transverse plane can be expressed as(6)GT=ET21+vT.Since Poisson’s ratio has less influence on the calculation results, we supposevT = 0.3 which is Poisson’s ratio in the transverse plane, and the other two Poisson’s ratios have the following constraints [28]:(7)vTL+vLT=2vT.According to [29], the longitudinal shear modulus can be approximated to(8)GL=ET+EL/221+vTL+vLT/2.OnlyET and EL are needed to solve. According to the study of Vlassak et al. [30, 31], the indentation modulus of anisotropic materials is a weighted average of the modulus of elasticity in each direction. The indentation modulus can be expressed as(9)Em=αET+βEL,where α and β are weight coefficients and α + β = 1. In previous study, it was generally considered that α = β = 1/2 [29].


## 2.2. Substrate Effect in Indentation of Thin Coating and Dimensionless Analysis

During the indentation process of the bulk material, when the rigid conical indenter is pressed into the bulk material, the relationship between the loading force and the indentation depth can be expressed as [32](10)F=Cth2,where F is the loading force, h is the indentation depth, and Ct is the scaling factor, which is related to the elastic mechanical parameters of the base material.The coating-substrate composite structure is composed of two materials with different mechanical properties. When the indenter is pressed into the composite structure, the state of stress and strain in the coating near the indenter will be affected by the substrate. Figure2 shows the stress distribution near the indenter and interface at different indentation depths through the finite element simulation. When the indentation depth exceeds a certain range, the strain produced by the indentation has extended to the substrate. As the depth of indentation increases, the curve gained by the indentation test actually contains the properties of both the coating and the substrate, which is called the substrate effect.Due to the substrate effect, during the loading process of the indentation for the coating-substrate composite structure, the relationship between the loading force and the depth of indentation is different from the quadratic relationship for the bulk material. The exponential function can be used to amend the formula [32](11)F=Fmaxhhmx,where the index of the load curve x quantitatively reflects the effect of the substrate on the stress state of the coating during the indentation loading, that is, the extent of the substrate effect.The two important parameters from the indentation curve, the maximum loading force and the index of the load curve, are related to the elastic mechanical parameters and the geometrical parameters of the coating-substrate composite structure. The maximum loading force can be expressed by the functional relation of the elastic mechanical parameters with the geometric parameters of the composite structure, which is(12)Fmax=ϕET,EL,ES,hc,hm,where ET and EL are the transverse and longitudinal elastic modulus of the coating, respectively, and ES is the elastic modulus of the substrate. hc and hm are the coating thickness and the maximum depth of the indentation.Consistent with the maximum loading force, the index of the load curvex can also be expressed as the following functional relation:(13)x=φET,EL,ES,hc,hm.The elastic modulus and maximum depth of the indentation of the base material are selected as the fundamental quantity. Π theorem is applied to nondimensionalize equations (12) and (13), and the following equations can be obtained:(14)FmaxEShm2=ϕETES,ELES,hchm,x=φETES,ELES,hchm.In the numerical simulation of this study, fixed maximum depth of the indentation (equal to that in the indentation test) is taken, and equations (12) and (13) can be expressed as(15)FmaxEShm2=ϕETES,ELES,(16)x=φETES,ELES.From the above equations, as long as the maximum loading forceFmax and the index of the loading curve x at a fixed indentation depth are obtained and then combined with numerical analysis for back analysis, ET and EL of the anisotropic coating can be obtained by solving the equations.


## 3. Simulation of the Nanoindentation by FEM

The finite element model of the indenter and coating-substrate composite structure was established by ABAQUS to simulate the nanoindentation process, which was used to obtain the expression of the dimensionless equation.The indenter used in the nanoindentation test was a Berkovich indenter, which could be replaced by a conical indenter with a half-cone angle of 70.3° in the finite element model [33]. The material of Berkovich indenter is diamond with a modulus of elasticity of 1140 GPa, which is of a very high magnitude to the pressed test piece. So, the Berkovich indenter can be equivalent to a rigid body and set to Analytical Rigid in ABAQUS. The process of which the conical indenter presses into the test piece has typical axisymmetric feature, so an axisymmetric two-dimensional model can be used to simulate the loading process of nanoindentation. In order to ensure the sufficient computational accuracy and save the computational time, a gradient mesh is used to divide the body according to Saint-Venant’s principle. For the coating, a dense mesh is used near the indenter, and a sparse mesh is used in the area far from the indenter; for the substrate, the dense mesh is used near the vertical region of the indenter and near the interface, and others use the sparse mesh (as shown in Figure 3). A downward displacement load is applied to the reference point on the indenter to simulate the process of pressing the indenter into the coating-substrate composite structure. The load-displacement curve of the finite element simulation can be drawn by extracting the displacement of the indenter and the reaction force acting on the indenter for each load step.Figure 3
The model of the indenter and coating-substrate composite structure.According to the adaptability requirements of the inverse analysis, when fitting a dimensionless equation, choosing a suitable material parameter range can improve the accuracy of the fitting result. The coating studied in this paper is metal-based composite coatings, and the substrate is metal. Since the elastic modulus of them is generally in a range of tens to hundreds GPa, we setET/ES and EL/ES from 0.3 to 1.5 which basically covers the range of modulus of the metal-based composite coating. First, we fixed ET/ES as 0.3 and selected EL/ES as 0.3, 0.5, 0.7, 1, 1.2 and 1.5, respectively and calculated the loading curve of the indentation for each ET/ES. Then, we changed the ET/ES to 0.5, 0.7, 1, 1.2, and 1.5 and repeated the above process. Figure 4 shows the dimensionless load-displacement curves obtained from the simulations of different ET/ES and EL/ES. The dimensionless maximum loading force Fmax/Eshm2 and index of loading curve x under different situations were extracted from Figure 4 for the further analysis, which are represented in Sections 3.1 and 3.2, respectively.Figure 4
Loading versus depth curves of differentET/ES and EL/ES. (a) EL/ES = 0.3. (b) EL/ES = 0.5. (c) EL/ES = 0.7. (d) EL/ES = 1. (e) EL/ES = 1.2. (f) EL/ES = 1.5.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.1. Maximum Loading Force

A three-dimensional diagram of the relationship betweenFmax/EShm2, EL/ES, and ET/ES is shown in Figure 5. When ET/ES is fixed, Fmax/EShm2 and EL/ES show a nonlinear relationship. The dimensionless maximum loading force Fmax/EShm2 is increased with the increase of EL/ES. It can be seen that the dimensionless maximum loading force Fmax/EShm2 is more sensitive to EL/ES.Figure 5
The three-dimensional diagram of the relationship betweenFmax/EShm2, EL/ES, and ET/ES.According to the curves in Figure5, the relationship between Fmax/EShm2 and EL/ES can be fitted to a quadratic curve. Figure 6 shows the quadratic fit curve for ET/ES = 1.5. As can be seen from the figure, the quadratic fit curve describes the relationship between Fmax/EShm2 and EL/ES accurately. Thus, equation (15) can be expressed as(17)FmaxEShm2=AELES2+BELES+C.Figure 6
Quadratic curve fitting ofFmax/EShm2 and EL/ES.Obviously,A, B, and C are related to ET/ES.Corresponding to different values ofET/ES, 6 groups of A, B, and C are calculated, which are shown as the data points in Figure 7.Figure 7
Data fitting ofA, B, and C. (a) The coefficient A. (b) The coefficient B. (c) The coefficient C.
(a)
(b)
(c)According to the relationship betweenA, B, and C, the parameters A, B, and C against ET/ES can be fitted as(18)A=a1ETES2+a2ETES+a3,B=a4ETES2+a5ETES+a6,C=a7ETES3+a8ETES2+a9ETES+a10.LetET/ES = u and EL/ES = v. The dimensionless maximum loading force can be expressed as(19)FmaxEShm2==a1u2+a2u+a3v2+a4u2+a5u+a6v+a7u3+a8u2+a9u+a10,where the coefficients are shown in Table 1.Table 1
The coefficients in equation (19).
Coefficient
Value
a
1
0.1665
a
2
−0.5912
a
3
−0.0826
a
4
−0.5017
a
5
1.947
a
6
0.5461
a
7
0.1183
a
8
−0.3821
a
9
0.4326
a
10
0.1969


### 3.2. Index of Loading Curve

The indexes of each loading curve in Figure4 are fitted, and a three-dimensional diagram of the relationship between x, EL/ES, and ET/ES is shown in Figure 8.Figure 8
The three-dimensional diagram of the relationship betweenx, EL/ES, and ET/ES.Figure8 shows that the index x is increased as EL/ES increases and is decreased as ET/ES increases. Similar to the fitting procedure of Fmax/EShm2=φu,v, the fitting expression of B is(20)x=b1u4+b2u3+b3u2+b4u+b5v2+b6u4+b7u3+b8u2+b9u+b10v+b11u4+b12u3+b13u2+b14u+b15,where the coefficients are shown in Table 2.Table 2
The coefficients in equation (20).
Coefficient
Value
b
1
0.08735
b
2
−0.3086
b
3
0.3596
b
4
−0.1521
b
5
−0.01096
b
6
−0.1812
b
7
0.6484
b
8
−0.7641
b
9
0.3028
b
10
0.04343
b
11
0.1391
b
12
−0.5382
b
13
0.7501
b
14
−0.4917
b
15
2.216


## 3.1. Maximum Loading Force

A three-dimensional diagram of the relationship betweenFmax/EShm2, EL/ES, and ET/ES is shown in Figure 5. When ET/ES is fixed, Fmax/EShm2 and EL/ES show a nonlinear relationship. The dimensionless maximum loading force Fmax/EShm2 is increased with the increase of EL/ES. It can be seen that the dimensionless maximum loading force Fmax/EShm2 is more sensitive to EL/ES.Figure 5
The three-dimensional diagram of the relationship betweenFmax/EShm2, EL/ES, and ET/ES.According to the curves in Figure5, the relationship between Fmax/EShm2 and EL/ES can be fitted to a quadratic curve. Figure 6 shows the quadratic fit curve for ET/ES = 1.5. As can be seen from the figure, the quadratic fit curve describes the relationship between Fmax/EShm2 and EL/ES accurately. Thus, equation (15) can be expressed as(17)FmaxEShm2=AELES2+BELES+C.Figure 6
Quadratic curve fitting ofFmax/EShm2 and EL/ES.Obviously,A, B, and C are related to ET/ES.Corresponding to different values ofET/ES, 6 groups of A, B, and C are calculated, which are shown as the data points in Figure 7.Figure 7
Data fitting ofA, B, and C. (a) The coefficient A. (b) The coefficient B. (c) The coefficient C.
(a)
(b)
(c)According to the relationship betweenA, B, and C, the parameters A, B, and C against ET/ES can be fitted as(18)A=a1ETES2+a2ETES+a3,B=a4ETES2+a5ETES+a6,C=a7ETES3+a8ETES2+a9ETES+a10.LetET/ES = u and EL/ES = v. The dimensionless maximum loading force can be expressed as(19)FmaxEShm2==a1u2+a2u+a3v2+a4u2+a5u+a6v+a7u3+a8u2+a9u+a10,where the coefficients are shown in Table 1.Table 1
The coefficients in equation (19).
Coefficient
Value
a
1
0.1665
a
2
−0.5912
a
3
−0.0826
a
4
−0.5017
a
5
1.947
a
6
0.5461
a
7
0.1183
a
8
−0.3821
a
9
0.4326
a
10
0.1969


## 3.2. Index of Loading Curve

The indexes of each loading curve in Figure4 are fitted, and a three-dimensional diagram of the relationship between x, EL/ES, and ET/ES is shown in Figure 8.Figure 8
The three-dimensional diagram of the relationship betweenx, EL/ES, and ET/ES.Figure8 shows that the index x is increased as EL/ES increases and is decreased as ET/ES increases. Similar to the fitting procedure of Fmax/EShm2=φu,v, the fitting expression of B is(20)x=b1u4+b2u3+b3u2+b4u+b5v2+b6u4+b7u3+b8u2+b9u+b10v+b11u4+b12u3+b13u2+b14u+b15,where the coefficients are shown in Table 2.Table 2
The coefficients in equation (20).
Coefficient
Value
b
1
0.08735
b
2
−0.3086
b
3
0.3596
b
4
−0.1521
b
5
−0.01096
b
6
−0.1812
b
7
0.6484
b
8
−0.7641
b
9
0.3028
b
10
0.04343
b
11
0.1391
b
12
−0.5382
b
13
0.7501
b
14
−0.4917
b
15
2.216


## 4. Nanoindentation Testing and Elastic Parameters Determination

### 4.1. Nanoindentation Testing

The matrix adopts 304 stainless-steel beam, and its material parameters are density 8 × 103 kg/m3, elastic modulus E = 193 GPa, and Poisson’s ratio v = 0.3. The surface of steel beam is polished to ensure a good surface quality and conductivity. The geometrical size of the stainless-steel beam is 60 × 10 × 1.8 mm. The coating target is Ni75Cr19Al4Y powder. Using the FMA 90/80 magnetic-filtered cathodic arc ion coating machine in Northeastern University, the as-prepared coating thickness is 20 μm by using the electron beam physical vapor deposition (EB-PVD) technique. Figure 9 shows the stainless-steel beam before and after applying NiCrAlY.Figure 9
The bare and coated stainless-steel beam.The surface state of the coating, especially the surface roughness of the coating, has a noticeable effect on the accuracy of nanoindentation measurement. According to the requirement of nanoindentation test, the surface roughness of the coating surface should meet the conditions ofRa ≤ h/20 (h is indentation depth) to ensure that the measuring error of indentation depth is less than 5%.Nanoindentation experiments were performed by the Swiss CSM Nanoscratch Tester. Its main technical parameters are as follows: load range 0–500 mn, load resolution 40 nN, maximum pressure depth 200μm, and displacement resolution 0.004 nm. The testing process consists of three phases: loading phase, load-holding phase (eliminating the effect of creep in unloading process), and unloading phase. The maximum loading force of indenter is 20 mN, the loading and unloading speeds are 10 mN/min, and the pause time is 10 s.The nanoindentation method is very sensitive to the microstructure of the coating material, and the measured results can only reflect the local mechanical properties of the coating. Randomly select 5 points on the specimen for testing and continuously record the change of loading force and indenter displacement. According to the continuous displacement and load recorded during the indentation experiment, the load-displacement curves of 5 measuring points are plotted as shown in Figure10.Figure 10
The experimental load versus depth curves of coated beam.From the load-displacement curve, under the same maximum loading force, the maximum depth of indentation is relatively discrete, and the indentation modulus is quite different. The reason is that the nanoindentation test is a method which depends on the local phase of the coating. Even for the same material coating, due to the subtle differences of the process of spraying, it is difficult to have a similar surface roughness, so the indentation test results have a strong discreteness. Three close results are chosen as valid values for averaging, and the averageFmax = 20.2 mN and average hm = 256.3 nm. The loading phase of the averaged load-displacement curve is fitted to get that x = 2.076.According to Oliver and Pharr’s research [26], the elastic modulus of the pressed test piece can be solved by the following equation:(21)1Em=1−v2E+1−vi2Ei,where Em is the indentation modulus (also called simplified Young’s modulus); E and v are Young’s modulus and Poisson’s ratio of the test piece, respectively; and Ei and vi are Young’s modulus and Poisson’s ratio of the indenter, respectively. For adamantine indenter, Ei = 1141 GPa and vi = 0.07.The indentation modulusEm in equation (21) is calculated by the following equation for load-displacement curve [34]:(22)S=dPdh=2πEmA,where S = dP/dh is the slope of the upper part of the experimental unloading curve of nanoindentation, that is, the stiffness value measured; Em is the indentation modulus; and A is the area of the elastic contact between the indenter and the test piece.When the geometrical shape of the indenter is known, the contact area is a function of the indentation contact depth. For an ideal Berkovich indenter, the contact areaA can be given by the following equation:(23)A=24.56hco2,where hco is the contact indentation depth, and the value can be obtained by using the following equation:(24)hco=hm−εFmS,where ε is a parameter related to the shape of the indenter; for Berkovich indenter in this experiment, ε = 0.76. After calculating, the indentation modulus Em=S/2π/A=157.19GPa.


### 4.2. Elastic Parameters Determination

The maximum loading forceFmax, load curve index x, and indentation modulus Em extracted from the curve of indentation obtained from the nanoindentation experiment are solved by substituting into the dimensionless equations (23) and (24).The identification of coating material parameters belongs to the inverse analysis; the inverse analysis usually has the feature of nonlinearity, heavy calculation burden, and ill-posed problem, and the solution of coating material parameters is particularly strong in dependence on input data. In order to ensure the suitability of the obtained material parameter results,Em = (ET + EL)/2 will be used as the constraint equation while calculating.Because of the inconvenience of solving dimensionless equation directly, we defined an objective function to find the optimal solution. Define the objective function as(25)e=1−ϕu,vFmax/EShm2+1−φu,vx.Looking foru, v which makes the value of the objective function e the smallest can obtain the desired solution. Substituting (u + v)/2 = Em/ES into equation (25) to reduce the equation to single variable u, we can get the objective function e vs u curve, as shown in Figure 11. The curve shows that when u = 0.69, the objective function e has a minimum value. Therefore, the identification results of NiCrAlY transverse isotropic mechanical parameters are ET = 133.17 GPa and EL = 181.21 GPa, and then according to the equations (5)–(8), GT = 51.22 GPa, GL = 60.45 GPa, vTL = 0.254, and vLT = 0.346 are obtained.Figure 11
Search for the optimal solution.


## 4.1. Nanoindentation Testing

The matrix adopts 304 stainless-steel beam, and its material parameters are density 8 × 103 kg/m3, elastic modulus E = 193 GPa, and Poisson’s ratio v = 0.3. The surface of steel beam is polished to ensure a good surface quality and conductivity. The geometrical size of the stainless-steel beam is 60 × 10 × 1.8 mm. The coating target is Ni75Cr19Al4Y powder. Using the FMA 90/80 magnetic-filtered cathodic arc ion coating machine in Northeastern University, the as-prepared coating thickness is 20 μm by using the electron beam physical vapor deposition (EB-PVD) technique. Figure 9 shows the stainless-steel beam before and after applying NiCrAlY.Figure 9
The bare and coated stainless-steel beam.The surface state of the coating, especially the surface roughness of the coating, has a noticeable effect on the accuracy of nanoindentation measurement. According to the requirement of nanoindentation test, the surface roughness of the coating surface should meet the conditions ofRa ≤ h/20 (h is indentation depth) to ensure that the measuring error of indentation depth is less than 5%.Nanoindentation experiments were performed by the Swiss CSM Nanoscratch Tester. Its main technical parameters are as follows: load range 0–500 mn, load resolution 40 nN, maximum pressure depth 200μm, and displacement resolution 0.004 nm. The testing process consists of three phases: loading phase, load-holding phase (eliminating the effect of creep in unloading process), and unloading phase. The maximum loading force of indenter is 20 mN, the loading and unloading speeds are 10 mN/min, and the pause time is 10 s.The nanoindentation method is very sensitive to the microstructure of the coating material, and the measured results can only reflect the local mechanical properties of the coating. Randomly select 5 points on the specimen for testing and continuously record the change of loading force and indenter displacement. According to the continuous displacement and load recorded during the indentation experiment, the load-displacement curves of 5 measuring points are plotted as shown in Figure10.Figure 10
The experimental load versus depth curves of coated beam.From the load-displacement curve, under the same maximum loading force, the maximum depth of indentation is relatively discrete, and the indentation modulus is quite different. The reason is that the nanoindentation test is a method which depends on the local phase of the coating. Even for the same material coating, due to the subtle differences of the process of spraying, it is difficult to have a similar surface roughness, so the indentation test results have a strong discreteness. Three close results are chosen as valid values for averaging, and the averageFmax = 20.2 mN and average hm = 256.3 nm. The loading phase of the averaged load-displacement curve is fitted to get that x = 2.076.According to Oliver and Pharr’s research [26], the elastic modulus of the pressed test piece can be solved by the following equation:(21)1Em=1−v2E+1−vi2Ei,where Em is the indentation modulus (also called simplified Young’s modulus); E and v are Young’s modulus and Poisson’s ratio of the test piece, respectively; and Ei and vi are Young’s modulus and Poisson’s ratio of the indenter, respectively. For adamantine indenter, Ei = 1141 GPa and vi = 0.07.The indentation modulusEm in equation (21) is calculated by the following equation for load-displacement curve [34]:(22)S=dPdh=2πEmA,where S = dP/dh is the slope of the upper part of the experimental unloading curve of nanoindentation, that is, the stiffness value measured; Em is the indentation modulus; and A is the area of the elastic contact between the indenter and the test piece.When the geometrical shape of the indenter is known, the contact area is a function of the indentation contact depth. For an ideal Berkovich indenter, the contact areaA can be given by the following equation:(23)A=24.56hco2,where hco is the contact indentation depth, and the value can be obtained by using the following equation:(24)hco=hm−εFmS,where ε is a parameter related to the shape of the indenter; for Berkovich indenter in this experiment, ε = 0.76. After calculating, the indentation modulus Em=S/2π/A=157.19GPa.


## 4.2. Elastic Parameters Determination

The maximum loading forceFmax, load curve index x, and indentation modulus Em extracted from the curve of indentation obtained from the nanoindentation experiment are solved by substituting into the dimensionless equations (23) and (24).The identification of coating material parameters belongs to the inverse analysis; the inverse analysis usually has the feature of nonlinearity, heavy calculation burden, and ill-posed problem, and the solution of coating material parameters is particularly strong in dependence on input data. In order to ensure the suitability of the obtained material parameter results,Em = (ET + EL)/2 will be used as the constraint equation while calculating.Because of the inconvenience of solving dimensionless equation directly, we defined an objective function to find the optimal solution. Define the objective function as(25)e=1−ϕu,vFmax/EShm2+1−φu,vx.Looking foru, v which makes the value of the objective function e the smallest can obtain the desired solution. Substituting (u + v)/2 = Em/ES into equation (25) to reduce the equation to single variable u, we can get the objective function e vs u curve, as shown in Figure 11. The curve shows that when u = 0.69, the objective function e has a minimum value. Therefore, the identification results of NiCrAlY transverse isotropic mechanical parameters are ET = 133.17 GPa and EL = 181.21 GPa, and then according to the equations (5)–(8), GT = 51.22 GPa, GL = 60.45 GPa, vTL = 0.254, and vLT = 0.346 are obtained.Figure 11
Search for the optimal solution.


## 5. Vibration Experiment for Validation of the Anisotropic Elastic Parameters

### 5.1. Parameters of Composite Structure

Taking the titanium alloy cantilever plate as substrate (Figure12(a)), the EB-PVD method is used to apply NiCrAlY coating on one side of its surface (Figure 12(b)). Vibration characteristics of titanium alloy cantilever plate with NiCrAlY coating were studied. The material of the plate is Ti-6Al-4V, and the elastic modulus is 113.8 GPa, Poisson’s ratio is 0.3, and the density is 4200 kg m−3 [35]. The length, width, and thickness of the plate are 122 mm, 110 mm, and 1.5 mm, separately. The thickness of the coating is 15 μm. From the mass change of the titanium alloy plate before and after the coating weighted by using the pallet balance and the volume of the coating which is the product of thickness, length, and width of the coating, the density of the NiCrAlY coating can be obtained as 2840.7 kg·m−3. Natural frequencies of the coating-plate composite structure are measured by sweeping frequency on the vibrating table [36].Figure 12
Titanium alloy thin plate and coated plate. (a) Titanium alloy thin plate. (b) Coated plate with NiCrAlY.
(a)
(b)


### 5.2. Modeling of the Plate with Coating

The titanium Alloy plate adopts Solid-95 unit which is suitable for block structure, and the hard coating adopts shell-181 unit which is suitable for thin shell structure, as shown in Figure13. The input parameters of NiCrAlY coating material in ANSYS are EX = EY = 133.17 GPa, EZ = 181.21 GPa, GXZ = GYZ = 51.22 GPa, GXY = 60.45 GPa, vXY = 0.346, and vXZ = vYZ = 0.254.Figure 13
Finite element model of cantilever coating-plate composite structure.The coating prepared by EB-PVD is tightly bonded to the substrate. The bond is in good condition with no obvious interfacial layer, and there is an interfused phenomenon between the coating and the substrate interface. Therefore, this study does not take the interface between the coating and the substrate into consideration, that is, assuming that the displacement and stress between the coating and the substrate interface are continuous. The coating is bonded to the substrate by merging mesh nodes on relevant surfaces in ANSYS.The degree of freedom in all directions of the bottom plate is restrained to simulate the cantilever condition. The finite element model eventually established of the coating-plate composite structure is shown in Figure13. According to the finite element model, the block Lanczos method is used in ANSYS to analyze the first 10 natural frequencies and modal modes of the coating-plate composite structure. For comparison, the equivalent model of isotropic materials with E = 165.76 GPa measured by the nanoindentation experiment is calculated.


### 5.3. Comparison of the Natural Frequencies

In order to compare the results of the dynamic test with calculated results of coating-plate composite structure to verify the correctness of the material parameters, we should first confirm that the calculated and tested results of the titanium alloy plate are correct. Table3 shows the first 8-order natural frequencies obtained from the finite element and test of titanium alloy plate without coatings. From Table 3, except the 7th order, the difference between the finite element and experimental test results is less than 2%, indicating that both the experimental test and the finite element calculation results are correct and credible.Table 3
Natural frequencies of the bare plate.
Mode order
1
2
3
4
5
6
7
8
Experiment (Hz)
85.75
227.88
538.25
776
811.63
1443.75
1494.25
1765.25
FEM (Hz)
85.708
225.48
528.36
789.13
802.74
1459
1528.2
1753.2
Error (%)
0.049
1.053
1.837
1.692
1.095
1.056
2.272
0.683Table4 shows the first 8 natural frequencies of the coated plate and the errors/difference ratios between the experimental test results and finite element results calculated by both isotropic parameters and anisotropic parameters. For a better comparison, the differences are plotted in Figure 14.Table 4
Natural frequencies of the coated plate using isotropic and anisotropic parameters.
Mode order
1
2
3
4
5
6
7
8
Experiment (Hz)
87.75
227.75
538
777.25
814
1463.25
1526
1772.25
Isotropic (Hz)
87.453
230.17
539.80
804.69
820.16
1491.2
1563.4
1793.9
Error (%)
0.34
1.06
0.33
3.53
0.76
1.91
2.45
1.22
Anisotropic (Hz)
86.890
228.61
535.65
800.02
813.80
1479.0
1549.1
1777.2
Error (%)
0.98
0.38
0.44
2.93
0.025
1.08
1.51
0.28Figure 14
The errors of the natural frequencies calculated by isotropic and anisotropic parameters.Figure14 shows that both the errors of the results calculated by isotropic and anisotropic parameters are below 2%. Moreover, the natural frequency results calculated by using anisotropic material parameters are close to the results obtained by the experimental test in higher-order parts (4∼8 orders).


## 5.1. Parameters of Composite Structure

Taking the titanium alloy cantilever plate as substrate (Figure12(a)), the EB-PVD method is used to apply NiCrAlY coating on one side of its surface (Figure 12(b)). Vibration characteristics of titanium alloy cantilever plate with NiCrAlY coating were studied. The material of the plate is Ti-6Al-4V, and the elastic modulus is 113.8 GPa, Poisson’s ratio is 0.3, and the density is 4200 kg m−3 [35]. The length, width, and thickness of the plate are 122 mm, 110 mm, and 1.5 mm, separately. The thickness of the coating is 15 μm. From the mass change of the titanium alloy plate before and after the coating weighted by using the pallet balance and the volume of the coating which is the product of thickness, length, and width of the coating, the density of the NiCrAlY coating can be obtained as 2840.7 kg·m−3. Natural frequencies of the coating-plate composite structure are measured by sweeping frequency on the vibrating table [36].Figure 12
Titanium alloy thin plate and coated plate. (a) Titanium alloy thin plate. (b) Coated plate with NiCrAlY.
(a)
(b)


## 5.2. Modeling of the Plate with Coating

The titanium Alloy plate adopts Solid-95 unit which is suitable for block structure, and the hard coating adopts shell-181 unit which is suitable for thin shell structure, as shown in Figure13. The input parameters of NiCrAlY coating material in ANSYS are EX = EY = 133.17 GPa, EZ = 181.21 GPa, GXZ = GYZ = 51.22 GPa, GXY = 60.45 GPa, vXY = 0.346, and vXZ = vYZ = 0.254.Figure 13
Finite element model of cantilever coating-plate composite structure.The coating prepared by EB-PVD is tightly bonded to the substrate. The bond is in good condition with no obvious interfacial layer, and there is an interfused phenomenon between the coating and the substrate interface. Therefore, this study does not take the interface between the coating and the substrate into consideration, that is, assuming that the displacement and stress between the coating and the substrate interface are continuous. The coating is bonded to the substrate by merging mesh nodes on relevant surfaces in ANSYS.The degree of freedom in all directions of the bottom plate is restrained to simulate the cantilever condition. The finite element model eventually established of the coating-plate composite structure is shown in Figure13. According to the finite element model, the block Lanczos method is used in ANSYS to analyze the first 10 natural frequencies and modal modes of the coating-plate composite structure. For comparison, the equivalent model of isotropic materials with E = 165.76 GPa measured by the nanoindentation experiment is calculated.


## 5.3. Comparison of the Natural Frequencies

In order to compare the results of the dynamic test with calculated results of coating-plate composite structure to verify the correctness of the material parameters, we should first confirm that the calculated and tested results of the titanium alloy plate are correct. Table3 shows the first 8-order natural frequencies obtained from the finite element and test of titanium alloy plate without coatings. From Table 3, except the 7th order, the difference between the finite element and experimental test results is less than 2%, indicating that both the experimental test and the finite element calculation results are correct and credible.Table 3
Natural frequencies of the bare plate.
Mode order
1
2
3
4
5
6
7
8
Experiment (Hz)
85.75
227.88
538.25
776
811.63
1443.75
1494.25
1765.25
FEM (Hz)
85.708
225.48
528.36
789.13
802.74
1459
1528.2
1753.2
Error (%)
0.049
1.053
1.837
1.692
1.095
1.056
2.272
0.683Table4 shows the first 8 natural frequencies of the coated plate and the errors/difference ratios between the experimental test results and finite element results calculated by both isotropic parameters and anisotropic parameters. For a better comparison, the differences are plotted in Figure 14.Table 4
Natural frequencies of the coated plate using isotropic and anisotropic parameters.
Mode order
1
2
3
4
5
6
7
8
Experiment (Hz)
87.75
227.75
538
777.25
814
1463.25
1526
1772.25
Isotropic (Hz)
87.453
230.17
539.80
804.69
820.16
1491.2
1563.4
1793.9
Error (%)
0.34
1.06
0.33
3.53
0.76
1.91
2.45
1.22
Anisotropic (Hz)
86.890
228.61
535.65
800.02
813.80
1479.0
1549.1
1777.2
Error (%)
0.98
0.38
0.44
2.93
0.025
1.08
1.51
0.28Figure 14
The errors of the natural frequencies calculated by isotropic and anisotropic parameters.Figure14 shows that both the errors of the results calculated by isotropic and anisotropic parameters are below 2%. Moreover, the natural frequency results calculated by using anisotropic material parameters are close to the results obtained by the experimental test in higher-order parts (4∼8 orders).


## 6. Conclusion

In this paper, we proposed a method to identify the anisotropic elastic parameters of NiCrAlY coating. Based on the anisotropic elastic constitutive relation, the constitutive model of NiCrAlY coating was simplified to be described by two independent mechanical constantsET and EL. The expression of the dimensionless equation of indentation parameters and elastic properties of the coating was obtained by finite element simulations. According to the results obtained from the nanoindentation test, the process of solving the elastic parameters of the coating was transformed into the form of finding the optimal solution of the objective function. Then, mechanical properties ET and EL of the coating were obtained.According to the elastic parameters, the finite element model of the coating-substrate composite structure was established by ANSYS and natural characteristics of the cantilever titanium plate coated with NiCrAlY coating were analyzed. The comparative result shows that the results calculated by anisotropic elastic parameters are more close to the experimental results, especially in higher modes. The method proposed in this paper can also be used for other coating materials. It has important research significance for vibration reduction analysis and active design of hard coating.



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*Source: 9034750-2019-06-02.xml*