# Statistical Identification of Parameters for Damaged FGM Structures with Material Uncertainties in Thermal Environment

Authors: Yalan Xu; Yu Qian; Kongming Guo
Journal: Complexity
 (2018)

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## Abstract

Considering that the statistic numerical characteristics are often required in the probability-based damage identification and safety assessment of functionally graded material (FGM) structures, an stochastic model updating-based inverse computational method to identify the second-order statistics (means and variances) of material properties as well as distribution of constituents for damaged FGM structures with material uncertainties is presented by using measurable modal parameters of structures. The region truncation-based optimization method is employed to simplify the computational process in stochastic model updating. In order to implement the forward propagation of uncertainties required in the stochastic model updating and avoid large error resulting in the nonconvergence of the iteration process, an algorithm is proposed to compute the covariance between the modal parameters and the identified parameters for damaged FGM structures. The proposed method is illustrated by a numerically simulated damaged FGM beam with continuous spatial variation of material properties and verified by comparing with the Monte-Carlo simulation (MCS) method. The influences of the levels and sources of measured data uncertainties as well as the boundary conditions on the identification results are investigated. The numerical simulation results show the efficiency and effectiveness of the presented method for the identification of material parameter variability by using the measurable modal parameters of damaged FGM structures.


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## Body

## 1. Introduction

Functionally graded materials (FGMs) have found wide application in modern industries including aerospace engineering, military application, and mechanical engineering, due to resistant high temperature gradient and strong mechanical performance [1], and a great deal of research has already been done into both deterministic [2–6] and stochastic [7–12] mechanical behavior of FGM structures. However, damage such as crack and fracture often occurs in FGM structures as most of FGM structures work in harsh environment such as high temperature gradient and corrosion. Damage problems in FGM structures have been widely investigated in literatures [13–19].As is well-known, accurate and reliable mechanical behavior analysis of FGM structures with initial damage is vital for the early condition assessment and damage prognosis in order to guarantee the safe performance and prevent the possible disastrous failure of FGM structures [20, 21]. What is more, the reliable analysis of overall mechanical behavior of any FGM structure, undamaged or damaged, relies on a precise knowledge of the material properties and, especially, the distribution of constituents. Traditionally, mechanical properties and volume fraction distribution of heterogeneous materials can be determined by starting from indentation tests [22, 23]. Owing to the inhomogeneous nature of FGMs, however, experimental characterization of materials constants and volume distribution is cumbersome and time-consuming since a large number of property parameters need to be determined, and nondestructive techniques have been developed to evaluate the material property parameters and the volume fraction index for FGMs by utilizing complex relationships between the structural behavior and the material properties. Liu et al. [24] suggested a progressive neural network process for characterizing the material properties of functionally graded material (FGM) plate by using elastic wave. Han and Liu [25] presented an inverse method for determining the material properties and distribution in the thickness direction of FGM plates by using uniform crossover microgenetic algorithm. Rahmani et al. [26] presented a regularized finite element model updating for identification of elastic constitutive parameters identification of 2D composites from full-field measured displacement data, in which mechanical constraints are used as regularization factors in the optimization algorithm. Sun et al. [27] developed a strategy to identify the temperature-dependent properties of a thermoelastic structure in thermal environment taking into time-varying material properties and thermal stresses account. Mishra and Chakraborty [28, 29] dealt with a modal analysis based inverse identification of material properties of fiber reinforced plastics composite plates with rotational flexibility at boundaries and panels having elastically restrained boundary from the experimental modal testing using finite element model updating.All of the above studies focus on the identification of deterministic quantities of material constants or volume distribution of composite materials and structures. In reality, however, both constituent material properties and volume distribution in FGMs present inherent fluctuation due to the typicality and technical variety in the manufacturing and fabrication of FGMs and environmental temperature [30, 31]. From this point of view, therefore, the inherent uncertainties need to be incorporated in parameter identification, which is an important issue in mechanical behavior analysis and safety evaluation of FGM structures, especially in the early stage of damage.In the present work, considering that the statistic numerical characteristics, for example, second-order statistics (means and variances) of material properties, are often required in evaluating the mechanical behavior and safety of FGM structures, an inverse computational procedure is presented to identify the second-order statistics (means and variances) of material properties and volume fraction index of FGM structures with initial damage by using stochastic finite element model updating [32], which is implemented by minimizing the differences between the analytical and actual structural modal parameters [33] easily obtained by measuring the structural response signal. It is worth noting that, as an inverse problem, both the efficient optimization method with good convergence and the forward uncertainty propagation with relatively small errors are two key issues for the statistical identification of FGM parameters based on stochastic model updating. Instead of utilizing the trust region method with expensive computation, the region truncation-based optimization method has been proposed to simplify the computational process without harming the convergence of the iteration process in this work. On the other hand, the forward uncertainty propagation needs to be carried out in every iteration step, in which not only the means and variances of the dynamic characteristics [7–12] but also the covariance between the dynamic characteristics and the identification parameters needs to be computed, and the computation of first-order derivatives is involved. Finite difference approximation is traditionally used for obtaining the involved derivatives for structures with a large number of degrees, and, however, it is obviously not suitable to be utilized in computing the derivatives with respect to volume fraction index for FGM structures due to large errors which will accumulate in every iteration step and lead to the nonconvergences of the results, since the effective material properties of FGMs are assumed as power functions of volume fraction index. On account of the fact mentioned above, an algorithm is developed for computing the first derivative of dynamic characteristics with respect to random variables, which is employed to compute the covariance between the dynamic characteristics and the identification parameters. In addition, in consideration of the difficult of random experiment test for the damaged FGM structure with enough samples required for the random analysis, the actual modal data will be obtained by numerical simulation method in this study.


## 2. Volume Distribution of Material Constituents in FGM Beam

An FGM rectangular beam composed of ceramics (top surface/left surface) and metal (bottom surface/right surface) having the lengthl, width b, and thickness h is considered in this wok. The effective material properties of the beam are assumed to vary continuously through its thickness direction or along its axial direction complying with power law distribution [11] and can be, respectively, expressed as (1)Pz=Pc-Pmzh+12n+Pm,-h2≤z≤h2,0≤n<∞,Px=Pc-Pm1-xln+Pm,0≤x≤l,0≤n<∞,where z, x are the coordinates along the thickness (h) and length (l) of the FGM beam as shown in Figure 1. P∙ denotes the effective material properties such as Young’s modulus, Poisson’s ratio, and thermal expansion efficient along z coordinate for the FGM beam through its thickness direction or x coordinate for the FGM beam along its axial direction. Pc, Pm denote the corresponding material properties of ceramics and metal, respectively. n denotes the volume fraction index.Figure 1
Configuration of an FGM beam.Each material property of ceramics and metal can be expressed as a function of temperature [30]:(2)P=P0P-1T-1+1+P1T+P2T2+P3T3,where P0,P-1,P1,P2,P3 are the coefficients of temperature T and change with the constituent materials as well as the temperature.


## 3. Stochastic Model Updating

### 3.1. Updating the Means and Variances of Material Parameters

The model updating can often be posed as a minimization problem of the objective function which is a sum of square difference between the analytic and actual data, typically dynamic characteristics (modal frequencies and mode shapes):(3)minθfθ=zakθ-zm2,θ_≤θ≤θ-,where the updating parameter vector θ is composed of material properties and volume fraction index of FGMs to be identified, that is, θ=PcTPmTnT. The vector zak is the analytic modal parameters (modal frequencies and mode shapes), that is, zak=fakTφakTT, which is a nonlinear function of the updating parameters vector θ. The vector zm=fmTφmTT is the actual modal parameters. The vectors θ_, θ- denote the upper and lower bounds on the updating parameters.In the stochastic model updating, the parameters can be updated by the following iterative expression [32]:(4)
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### 3.2. Optimization Algorithm by Region Truncation

Region truncation-based method is employed in the optimization process of model updating, since the widely used trust region method is less straightforward and more expensive in computational though it is powerful and reliable in convergence [34]. In region truncation-based optimization algorithm, the constrained minimization problem (3) with the constraints is converted to the unconstrained problem and the updating parameters are limited within a controlled region in each iteration step. Thus, (5) can be replaced with the following iterative process:(7)θ-k+1=θ-k+τ¯k,Δθk≥τ¯k,θ-k+STθkWεSθk-1+WϑSTθkz-ak-z-mτ_k≤Δθk≤τ¯k,θ-k+τ_k,Δθk≤τ_k,Δθ-k+1=θ-k-θ-k-1,τ-k=βκk,τ_k=-βκk,β∈0,1,κk=min⁡θ--θ-k,θ_-θ-k,where τ-k,τ_k are boundary vectors of the truncated region. κk is the distance vector of the current iterative point from the nearest constraints, and the size of truncation region depends on the parameter β. It is suggested that β should be set small for a large sensitivity of modal parameters to the updating parameters and large for a small sensitivity.


## 3.1. Updating the Means and Variances of Material Parameters

The model updating can often be posed as a minimization problem of the objective function which is a sum of square difference between the analytic and actual data, typically dynamic characteristics (modal frequencies and mode shapes):(3)minθfθ=zakθ-zm2,θ_≤θ≤θ-,where the updating parameter vector θ is composed of material properties and volume fraction index of FGMs to be identified, that is, θ=PcTPmTnT. The vector zak is the analytic modal parameters (modal frequencies and mode shapes), that is, zak=fakTφakTT, which is a nonlinear function of the updating parameters vector θ. The vector zm=fmTφmTT is the actual modal parameters. The vectors θ_, θ- denote the upper and lower bounds on the updating parameters.In the stochastic model updating, the parameters can be updated by the following iterative expression [32]:(4)
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## 3.2. Optimization Algorithm by Region Truncation

Region truncation-based method is employed in the optimization process of model updating, since the widely used trust region method is less straightforward and more expensive in computational though it is powerful and reliable in convergence [34]. In region truncation-based optimization algorithm, the constrained minimization problem (3) with the constraints is converted to the unconstrained problem and the updating parameters are limited within a controlled region in each iteration step. Thus, (5) can be replaced with the following iterative process:(7)θ-k+1=θ-k+τ¯k,Δθk≥τ¯k,θ-k+STθkWεSθk-1+WϑSTθkz-ak-z-mτ_k≤Δθk≤τ¯k,θ-k+τ_k,Δθk≤τ_k,Δθ-k+1=θ-k-θ-k-1,τ-k=βκk,τ_k=-βκk,β∈0,1,κk=min⁡θ--θ-k,θ_-θ-k,where τ-k,τ_k are boundary vectors of the truncated region. κk is the distance vector of the current iterative point from the nearest constraints, and the size of truncation region depends on the parameter β. It is suggested that β should be set small for a large sensitivity of modal parameters to the updating parameters and large for a small sensitivity.


## 4. Finite Element Model for Damaged FGM Structures

### 4.1. Random Effective Material Properties

Recalling (1), the uncertainty in the effective material properties of an FGM comes from the uncertainty in both material parameters and volume fraction of constituents. Taking into account the low variability in both the physical properties (such as the Young modulus and mass density of each constituent material) and volume fraction index, and neglecting high-order terms, the effective material properties for the FGM beam can be expanded by Taylor’s series:(8)P·=P-·+P~·,P~·=∑i=1l∂P-·∂θiθ~i,where P-(·) is the mean value of each effective material property, and P~· is the corresponding zero-mean random part with the variance of σP·2. θi is the ith element of the updating parameter vector θ, and θ~i is the corresponding zero-mean random part.


### 4.2. Finite Element Model of Undamaged FGM Beam

According to the third-order shear deformation theory, the displacement field of the FGM beam can be expressed as follows [11, 12]: (9)ux,z,t=u0x,t+zϕxx,t-43zh2ϕxx,t+∂w0x,t∂x,wx,t=w0x,t,where u, v are the axial and transverse displacements at any point of the FGM beam in the x,  z directions, respectively. u0, w0 are those on the mid-plane. ϕx is the cross-sectional rotation about x-axis.By assuming the small deformation, the following strain field can be obtained:(10)εxx,z,t=∂u0∂x+z∂ϕx∂x-43zh2∂ϕx∂x+∂2w0∂x2,γxzx,z,t=ϕx+∂w0∂x-4zh2ϕx+∂w0∂x,where εx and γxz are the normal and shear strains.The constitutive relation in thermal environment can be expressed as(11)σxx,z,tτxzx,z,t=Q1100Q55εxγxz-10αx,z,TΔTx,z,where σx and τxz are the normal and shear stresses. Q11 and Q55 are the elastic coefficients, that is, Q11=Ex,z,T/1-ν2,  Q55=Ex,z,T/21+ν. Ex,z,T=E-x,z,T+E~x,z,T are the effective Young’s elastic modulus, which is the function of z for the FGM beam through its thickness direction, and the function of x for the FGM beam along its axial direction. E-x,z,T is the mean and E~x,z,T is the corresponding zero-mean random part. Poisson’s ratio ν is assumed to be constant. αx,z,T is the thermal expansion efficient. ΔTx,z is the temperature change distribution for the FGM beam.The two-node shear deformable beam element with four degrees of freedom in each node is employed in the development of the finite element model. The displacement vector at the mid-plane of a beam element can be expressed as(12)u0w0∂w0∂xϕxT=Nqe,qe=u0e1,w0e1,∂w0e1∂x,ϕxe1,u0e2,w0e2,∂w0e2∂x,ϕxe2T,where N is the shape function matrix and qe is the nodal displacement vector of beam element.The strain and kinetic energies of the beam element are(13)U=∭Ve12εxQ11εx+γxzQ55γxz-εxQ11αΔTdxdydz,T=∭Ve12ρx,z,Tu˙2+w˙2dxdydz,where Ve is the volume of a beam element. ρx,z,T=ρ-x,z,T+ρ~x,z,T are the effective Young’s modulus, which is the function of z for the FGM beam through its thickness, and the function of x for the FGM beam along its axial direction. ρ-x,z,T is the mean and ρ~x,z,T is the corresponding zero-mean random part.The following stochastic finite element equation of vibration for the beam element in thermal environment can be obtained by using Hamilton’s principle:(14)Meq¨e+Keqe=Fe,where Ke=K-e+K~e is the element stiffness matrix, Me=M-e+M~e is the element mass matrix, and Fe=F-e+F~e is the thermal force. Each matrix is composed of the mean matrix and the zero-mean random part.Recalling (8), the means K-e, M-e, F-e and the zero-mean random parts K~e, M~e, F~e can be computed, respectively, by (15)K-e=∬E-x,z,TK^edxdz,M-e=∬ρ-x,z,TM^edxdz,F-e=∬E-x,z,TF^edxdz,(16)K~e=∬∑i=1l∂E-x,z,T∂θ~iθ~iK^edxdz,M~e=∬∑k=1l∂ρ-x,z∂θ~iθ~iM^edxdz,F~e=∬∑k=1l∂E-x,z∂θiθ~iF^edxdz,where the matrices K^e, M^e,  F^e can be computed by(17)Κ^e=b1-ν2BεTBε+BγTBγ,M^e=bDTD,F^e=b1-ν2BεTαx,z,TΔTx,z,Bε=10-c1z3z1-c1z2∂N∂x,Bγ=001-3c1z21-3c1z2N,D=10-c1z3z1-c1z20100N,where Bε, Bγ, and D are the matrices mainly related to the shape function matrix N and c1=4/3h2.


### 4.3. Finite Element Model of Damaged FGM Beam

For the damaged FGM beam, a local damage often leads to the reduction in the local stiffness parameter. In this section, damage is introduced by reducing effectiveness elastic moduli of the corresponding elements [16]. Introducing a damage factor λe indicating the damage severity for the eth damaged element, effective Young’s modulus of the eth damaged element can be expressed as(18)E-dx,z,T=E-x,z,T1-λe,where E-x,z,T is the effective Young’s elastic modulus for the undamaged FGM beam and E-dx,z,T is for the damaged FGM beam.Recalling (15), the stiffness matrix of the damaged element can be expressed as (19)Ke_d=Ke1-λe,where λe is the damage factor of the eth damaged element and Ke_d is the stiffness element for the eth damaged element.With (18) and (19), the element matrices in (14) can be assembled to obtain the global stochastic finite element equation of vibration for the damaged FGM beam. (20)Mq¨+Kq=F,K=K-+K~,M=M-+M~,F=F-+F~,K-=∑enuTeTK-eTe+∑end1-λeTeTK-eTe,M-=∑enu+ndTeTMeTe,F-=∑enuTeTF-e+∑end1-λeTeTF-e,K~=∑enuTeTK~eTe+∑end1-λeTeTK~eTe,M~=∑eneTeTM~eTe,F~=∑enuTeTF~e+∑end1-λeTeTF~e,where K is the global random stiffness matrix and M is the global random mass matrix. F is the global thermal force and Te is a transformation matrix that transforms the local coordinate of the eth element to the global coordinate. nu is the number of undamaged elements and nd is the number of damaged elements.


## 4.1. Random Effective Material Properties

Recalling (1), the uncertainty in the effective material properties of an FGM comes from the uncertainty in both material parameters and volume fraction of constituents. Taking into account the low variability in both the physical properties (such as the Young modulus and mass density of each constituent material) and volume fraction index, and neglecting high-order terms, the effective material properties for the FGM beam can be expanded by Taylor’s series:(8)P·=P-·+P~·,P~·=∑i=1l∂P-·∂θiθ~i,where P-(·) is the mean value of each effective material property, and P~· is the corresponding zero-mean random part with the variance of σP·2. θi is the ith element of the updating parameter vector θ, and θ~i is the corresponding zero-mean random part.


## 4.2. Finite Element Model of Undamaged FGM Beam

According to the third-order shear deformation theory, the displacement field of the FGM beam can be expressed as follows [11, 12]: (9)ux,z,t=u0x,t+zϕxx,t-43zh2ϕxx,t+∂w0x,t∂x,wx,t=w0x,t,where u, v are the axial and transverse displacements at any point of the FGM beam in the x,  z directions, respectively. u0, w0 are those on the mid-plane. ϕx is the cross-sectional rotation about x-axis.By assuming the small deformation, the following strain field can be obtained:(10)εxx,z,t=∂u0∂x+z∂ϕx∂x-43zh2∂ϕx∂x+∂2w0∂x2,γxzx,z,t=ϕx+∂w0∂x-4zh2ϕx+∂w0∂x,where εx and γxz are the normal and shear strains.The constitutive relation in thermal environment can be expressed as(11)σxx,z,tτxzx,z,t=Q1100Q55εxγxz-10αx,z,TΔTx,z,where σx and τxz are the normal and shear stresses. Q11 and Q55 are the elastic coefficients, that is, Q11=Ex,z,T/1-ν2,  Q55=Ex,z,T/21+ν. Ex,z,T=E-x,z,T+E~x,z,T are the effective Young’s elastic modulus, which is the function of z for the FGM beam through its thickness direction, and the function of x for the FGM beam along its axial direction. E-x,z,T is the mean and E~x,z,T is the corresponding zero-mean random part. Poisson’s ratio ν is assumed to be constant. αx,z,T is the thermal expansion efficient. ΔTx,z is the temperature change distribution for the FGM beam.The two-node shear deformable beam element with four degrees of freedom in each node is employed in the development of the finite element model. The displacement vector at the mid-plane of a beam element can be expressed as(12)u0w0∂w0∂xϕxT=Nqe,qe=u0e1,w0e1,∂w0e1∂x,ϕxe1,u0e2,w0e2,∂w0e2∂x,ϕxe2T,where N is the shape function matrix and qe is the nodal displacement vector of beam element.The strain and kinetic energies of the beam element are(13)U=∭Ve12εxQ11εx+γxzQ55γxz-εxQ11αΔTdxdydz,T=∭Ve12ρx,z,Tu˙2+w˙2dxdydz,where Ve is the volume of a beam element. ρx,z,T=ρ-x,z,T+ρ~x,z,T are the effective Young’s modulus, which is the function of z for the FGM beam through its thickness, and the function of x for the FGM beam along its axial direction. ρ-x,z,T is the mean and ρ~x,z,T is the corresponding zero-mean random part.The following stochastic finite element equation of vibration for the beam element in thermal environment can be obtained by using Hamilton’s principle:(14)Meq¨e+Keqe=Fe,where Ke=K-e+K~e is the element stiffness matrix, Me=M-e+M~e is the element mass matrix, and Fe=F-e+F~e is the thermal force. Each matrix is composed of the mean matrix and the zero-mean random part.Recalling (8), the means K-e, M-e, F-e and the zero-mean random parts K~e, M~e, F~e can be computed, respectively, by (15)K-e=∬E-x,z,TK^edxdz,M-e=∬ρ-x,z,TM^edxdz,F-e=∬E-x,z,TF^edxdz,(16)K~e=∬∑i=1l∂E-x,z,T∂θ~iθ~iK^edxdz,M~e=∬∑k=1l∂ρ-x,z∂θ~iθ~iM^edxdz,F~e=∬∑k=1l∂E-x,z∂θiθ~iF^edxdz,where the matrices K^e, M^e,  F^e can be computed by(17)Κ^e=b1-ν2BεTBε+BγTBγ,M^e=bDTD,F^e=b1-ν2BεTαx,z,TΔTx,z,Bε=10-c1z3z1-c1z2∂N∂x,Bγ=001-3c1z21-3c1z2N,D=10-c1z3z1-c1z20100N,where Bε, Bγ, and D are the matrices mainly related to the shape function matrix N and c1=4/3h2.


## 4.3. Finite Element Model of Damaged FGM Beam

For the damaged FGM beam, a local damage often leads to the reduction in the local stiffness parameter. In this section, damage is introduced by reducing effectiveness elastic moduli of the corresponding elements [16]. Introducing a damage factor λe indicating the damage severity for the eth damaged element, effective Young’s modulus of the eth damaged element can be expressed as(18)E-dx,z,T=E-x,z,T1-λe,where E-x,z,T is the effective Young’s elastic modulus for the undamaged FGM beam and E-dx,z,T is for the damaged FGM beam.Recalling (15), the stiffness matrix of the damaged element can be expressed as (19)Ke_d=Ke1-λe,where λe is the damage factor of the eth damaged element and Ke_d is the stiffness element for the eth damaged element.With (18) and (19), the element matrices in (14) can be assembled to obtain the global stochastic finite element equation of vibration for the damaged FGM beam. (20)Mq¨+Kq=F,K=K-+K~,M=M-+M~,F=F-+F~,K-=∑enuTeTK-eTe+∑end1-λeTeTK-eTe,M-=∑enu+ndTeTMeTe,F-=∑enuTeTF-e+∑end1-λeTeTF-e,K~=∑enuTeTK~eTe+∑end1-λeTeTK~eTe,M~=∑eneTeTM~eTe,F~=∑enuTeTF~e+∑end1-λeTeTF~e,where K is the global random stiffness matrix and M is the global random mass matrix. F is the global thermal force and Te is a transformation matrix that transforms the local coordinate of the eth element to the global coordinate. nu is the number of undamaged elements and nd is the number of damaged elements.


## 5. Forward Uncertainty Propagation

Thejth modal frequency fj and mode shape φj are governed by the following:(21)K-2πfj2Mφj=0,j=1,2,…,N.Thejth modal frequency and mode shape can also be represented as the sum of its mean and a zero-mean random part: (22)fj=f-j+f~j,φj=φ-j+φ~jand further the random part of the jth normalized mode shape can be written as (23)φ~j=∑r=1Nεrjφ-r,where εrj,  r=1,2,…,N are the small coefficients to be determined. Substituting (22) into (21), we have (24)K--2πf-j2M-φ-j=0,(25)K-∑r=1Nεrjφ-r+K~φ-j=2πf-j2M-∑r=1Nεrjφ-r+2πf-j2M~φ-j+8π2f-jf~jM-φ-j.Using the orthogonality of mode shapes, the random part of thejth modal frequency and the coefficients εiji≠j can be obtained by premultiplying (25) by φ-jT and φ-iT, respectively,(26)f~j=18π2f-jφ-jTK~-2πf-j2M~φ-j,εij=14π2f-j2-f-i2φ-jTK~-2πf-j2M~φ-j.Recalling the orthogonality conditionφjTMφj=1 and using (20) and (22), the coefficient εjj can be obtained by(27)εjj=-12φ-jTM~φ-j.Neglecting the high-order terms, the random parts of modal parameters in (22) can be obtained by the Taylor’s expansion:(28)f~j=∑i=1l∂f-j∂θiθ~i,φ~j=∑k=1l∂φ-jT∂θiθ~i.Using (20), (26)–(28), we have the following equation for the solution of the unknown ∂f-j/∂θi and ∂φ-jT/∂θi by equating the coefficients of each θ~i:(29)∑i=1l∂f-j∂θiθ~i=18π2f-jφ-jTK~-2πf-j2M~φ-j,∑i=1l∂φ-jT∂θiθ~i=-12φ-jTK~φ-jφ-j+∑i=1i≠jN14π2f-j2-f-i2φ-jTK~-2πf-j2M~φ-jφ-i,K~=∑enuTeT∬∑i=1l∂E-x,z,T∂θ~iθ~iK^edxdzTe+∑endTeT∬1-λe∑i=1l∂E-x,z,T∂θ~iθ~iK^edxdzTe,M~=∑e=1nu+ndTeT∬∑k=1l∂ρ-x,z∂θ~iθ~iM^edxdzTe.Further, the variances of thejth modal frequency and normalized mode shape as well as the covariance between the modal parameters and the random material parameters can be obtained:(30)Varfj=∑k=1l∑q=1l∂f-j∂θk∂f-j∂θqρkqσθ~kσθ~q,Varφj=∑k=1l∑q=1ldiag∂φj∂θk∂φjT∂θqρkqσθ~kσθ~q,Covfj,θi=∑k=1l∂f-j∂θkρkiσθ~kσθ~i,Covφj,θi=∑k=1l∂φj∂θkρkiσθ~kσθ~i,where σθ~k, σθ~q, σθ~i are the standard deviations of the random material parameters, that is, constituent material properties and volume fraction index, and ρkq is the correlation coefficient of random parameters θ~k, θ~q.


## 6. Numerical Simulation

In order to demonstrate the stochastic model updating-based parameter identification approach outlined in the previous sections for damaged FGM structures with random material properties in thermal environment, a damaged FGM beam is considered for the numerical simulation. The results obtained from the presented identification approach have been validated by comparing with the given values. The FGM is composed of ceramics (Si3N4) and metal (SUS304) with the mean values of material properties given in Table1. In the FGM, material properties are assumed to vary, either through its thickness direction or along its axial direction, according to power law distribution. Therefore, the bottom surface is pure metal and the top surface is pure ceramics for the FGM beam through its thickness direction, while the right side is pure metal and the left side is pure ceramics for the FGM beam along its axial direction. The FGM beam has the geometric parameters of 20 m in length, 1 m in thickness, and 0.8 m in width. In this section, Young’s elastic moduli of two materials and volume fraction index are chosen as the parameters to be identified, since the former may significantly change with the environmental temperature and the latter is hard to be determined by traditional experiments.Table 1
The following SuS304/Si3N4 type of FGM properties are used for computation [30].
Material Properties P 0 P - 1 P 1 P 2 P 3 Ceramics(Si3N4) E   ( P a ) 348.43 e + 9 0 - 3.070 e - 4 2.160 e - 7 - 8.964 e - 11 α   ( 1 / K ) 5.8723 e - 6 0 9.095 e - 4 0 0 ρ   ( k g / m 3 ) 2370 0 0 0 0  Metal(SUS304) E   ( P a ) 201.04 e + 9 0 3.079 e - 4 - 6.534 e - 7 0 α   ( 1 / K ) 12.330 e - 6 0 8.086 e - 4 0 0 ρ   ( k g / m 3 ) 8166 0 0 0 0In the finite element, the FGM beam was discretized into eight two-node shear deformable beam elements with four degrees of freedom in each node. It is assumed that the damage is located at the8th element with a reduction in the effective Young’s elastic modulus by 10%, that is, λ8=0.1. Analytically, modal analysis is first carried out to obtain the initial analytical modal parameters by using finite element model of the damaged FGM beam with initial material properties and distribution. Modal analysis is then again carried out on the damaged FGM beam with given material properties and distribution to obtain the actual modal parameters due to the difficulty of random experiment with large samples.The actual modal data is generated in a simulated way assuming that the uncertainty of modal data is caused by 5% COV (coefficient of variation, the ratio of the square root of the variance to the mean) in elastic moduli of two constituents as well as volume fraction index and 0.1% COV in measurement noise. Figure2 shows the scatter of the actual modal frequencies set compared to the initial modal frequencies in the temperature of 300 K for the damaged FGM beam through the thickness direction (n=2) compared to the undamaged case and Figure 3 shows those for the FGM beam along the axial direction (n=2), in which the clamped-free (CF) boundary condition is firstly considered.Figure 2
First three actual modal frequencies set (3000 samples) and initial modal frequencies (thickness).Figure 3
First three actual modal frequencies set (3000 samples) and initial modal frequencies (axial).The stochastic finite element model updating is conducted using the region truncation-based optimization algorithm, and, as a result, the means and variances of elastic moduli of two materials and volume fraction index are identified by the stochastic model updating-based method (SMUM).The statistical identification of material parameters for a damaged FGM structure can be implemented by the following steps.Step 1.
Develop the finite element model of the damaged FGM structure and determine the parametersθ to be identified, which are chosen as the updating parameters of stochastic model updating.Step 2.
Compute the mean vectorz-m and covariance matrix Covz~m,z~m of the actual modal data.Step 3.
Initialize the means and variances of the updating parametersθ.Step 4.
Obtain the mean vectorz-ak of the analytical modal data after carrying out the modal analysis, and determine the covariance matrices Covz~ak,θ~k and Covz~ak,z~ak using the forward uncertainty propagation (30).Step 5.
Update the mean valuesθ-k+1 and variances Cov(θ~k+1,θ~k+1) of the updating parameters using (7) and (6), respectively.Step 6.
Go to Step4 until zakθ-zm2 is small enough and the second-order statistics (means and variances) of the identified parameters can be obtained.The convergences of the mean values and variances of elastic moduli (Ec-elastic modulus of ceramics and Em-elastic modulus of metal of metal) and volume fraction index (n) for the damaged FGM beam with material properties varying through its thickness in the temperature of 300 K and 600 K are shown in Figure 4. The results show that, for different temperatures, the means and variances of elastic moduli of two constituents and volume fraction index can fast converge by the proposed method. It is also seen that the means of elastic moduli of two materials converge to different values for different temperatures since the elastic modulus of each material changes with the environmental temperature. The situation does not occur to volume fraction index, since volume fraction index is not affected by the temperature environment and depends on the manufacturing and fabrication process. The temperature has no obvious effect on the identification results of parameter variances.Figure 4
Iterative process of means and variances of material elastic moduli and volume fraction index.For the comparison of convergence between the trust region method and the proposed method, Figure5 gives the iteration process of mean values and variances of Ec, Em, and n for the damaged FGM beam with material properties varying through its thickness direction in the temperature of 600 K by two methods. It is seen that mean values and variances of Ec, Em, and n can converge faster by the proposed method than those by the trust region method. In fact, the CPU run time for the proposed method is about 6.356 seconds, and the CPU run time for the trust region method is about 13.251 seconds.Figure 5
Iterative process of means and variances ofEc, Em, and n for two methods (T=600 K).Compared to the Monte-Carlo simulation (MCS) method with 3000 samples, the mean values and COVs (coefficient of variation, the ratio of the square root of the variance to the mean) of elastic moduli of two materials and volume fraction index for both the damaged FGM beam through its thickness direction and the FGM beam along its axial direction with different volume fraction indices are shown in Tables2 and 3, respectively.Table 2
Means of material elastic moduli and volume fraction index for SMUM and MCS.
Temperature Parameters(given values) n = 0.5 n = 2 n = 5 n = 10 SMUM MCS SMUM MCS SMUM MCS SMUM MCS FGM beam through its thickness  T = 300 K E c  (3.22272e11) 3.2174 e 112 3.2236 e 11 3.2226 e + 11 3.2233 e 11 3.2163 e 11 3.2261 e 11 3.2079 e 11 3.2287 e 11 E m  (2.07788e11) 2.07768 e 11 2.0778 e 11 2.0731 e + 11 2.0783 e 11 2.0759 e 11 2.0789 e 11 2.0758 e 11 2.0750 e 11 n (0.5/2/5/10) 0.4978 0.5027 1.9910 1.9986 4.9583 5.0508 9.7925 9.9449  T = 600 K E c  (3.04610e11) 3.0388 e 11 3.0579 e 11 3.0465 e 11 3.0437 e 11 3.0396 e 11 3.0608 e 11 3.0265 e 11 3.0242 e 11 E m  (1.90890e11) 1.9059 e 11 1.9176 e 11 1.9084 e 11 1.9061 e 11 1.9027 e 11 1.9101 e 11 1.9008 e 11 1.9069 e 11 n  (0.5/2/5/10) 0.4985 0.5086 1.9943 2.0049 4.9412 5.0509 9.7276 9.7003  FGM along its axial direction  T = 300 K E c  (3.22272e11) 3.2177 e 11 3.2210 e 11 3.2158 e 11 3.2247 e 11 3.2139 e 11 3.2220 e 11 3.2125 e 11 3.2231 e 11 E m  (2.07788e11) 2.0832 e 11 2.0752 e 11 2.0775 e 11 2.0734 e 11 2.0759 e 11 2.0768 e 11 2.0769 e 11 2.0789 e 11 n  (0.5/2/5/10) 0.4986 0.5013 1.9965 1.9973 5.0012 4.9949 9.9779 10.0179  T = 600 K E c  (3.04610e11) 3.0386 e 11 3.0418 e 11 3.0396 e 11 3.0454 e 11 3.0383 e 11 3.0478 e 11 3.0489 e 11 3.0521 e 11 E m  (1.90890e11) 1.9149 e 11 1.8969 e 11 1.9125 e 11 1.9035 e 11 1.9069 e 11 1.9069 e 11 1.9105 e 11 1.9072 e 11 n  (0.5/2/5/10) 0.4987 0.5014 1.9989 1.9998 4.9951 5.0155 9.9955 10.0383Table 3
COVs of material elastic moduli and volume fraction index for SMUM and MCS.
Temperature Parameters(given values) n = 0.5 n = 2 n = 5 n = 10 SMUM MCS SMUM MCS SMUM MCS SMUM MCS FGM beam through its thickness  T = 300 K E c (5%) 6.9139% 6.3355% 6.1553% 6.0225% 5.8767% 5.4019% 8.4827% 4.1443% E m (5%) 8.4283% 7.4319% 6.2834% 5.3610% 5.7577% 5.0743% 5.9237% 5.1053% n (5%) 11.9278% 12.2767% 9..9623% 6.8676% 9.5556% 10.4554% 26.9936% 18.9727%  T = 600 K E c (5%) 7.3813% 6.4775% 6.4379% 5.9881% 6.2478% 5.2932% 8.9632% 8.8812% E m (5%) 9.0423% 7.2826% 6.8224% 5.3459% 6.3822% 5.2756% 6.5709% 6.5534% n (5%) 11.3814% 11.8691% 9.7212% 6.7731% 8.2936% 9.5587% 26.0417% 28.2942%  FGM beam along its axial direction  T = 300 K E c (5%) 5.6619% 5.0117% 5.6988% 5.1429% 6.2028% 5.6765% 6.4789% 6.0394% E m (5%) 5.9169% 5.8154% 5.9243% 5.0552% 5.5639% 5.0349% 5.0739% 5.1134% n (5%) 6.2618% 6.2209% 5.9821% 5.3645% 6.0643% 5.8934% 6.6918% 6.1129%  T = 600 K E c (5%) 5.9368% 4.9651% 6.0989% 5.3348% 6.5508% 5.5821% 6.7065% 5.9315% E m (5%) 6.5070% 6.5424% 6.2054% 5.4021% 6.0851% 5.1826% 6.0854% 5.0826% n (5%) 6.2779% 6.1971% 5.7743% 5.5224% 6.1823% 6.0234% 6.6069% 5.9734%It is seen from Tables2 and 3 that, for the mean values or the COVs, the identification results by the stochastic model updating-based method (SMUM) proposed in this work agree favorably with those from the MCS. It is also observed that, compared to the given values, the means of parameters can be identified with good accuracy in the case of different volume fraction indices, while the identification results of the COVs are obviously affected by the volume fraction index and present relatively large errors, especially in the case of relatively small volume fraction indices or relatively large volume fraction indices; that is, one of constituents plays an dominant role in the FGM. In fact, the identified COV of Ec will be far away from the given value when the volume fraction index is large enough (e.g., n=10) so that the FGM is rich in metal and few ceramics are in it. In contrast, the identified COV of Em will be far away from the given value when the volume fraction index is small enough (e.g., n=0.5) so that the FGM is rich in ceramics and few metal is in it. However, the situation does not happen to the FGM beam along its axial direction for these two cases (i.e., n=0.5 and n=10) and the volume fraction has no significant influence on the identification results of COVs. This can be explained that the material properties vary along the axial direction from the value of Em to the value of Ec much more gradually than those along the thickness since the length (8 m) of beam is much larger than its thickness (1 m).Figure6 gives the convergences of the mean values and variances of Ec, Em, and n for the FGM beam with material properties varying through its thickness direction in different cases of boundary conditions (T=600 K). Four boundary conditions of clamped-free (CF), clamped-clamped (CC), clamped-simply (CS), and simply-simply (SS) are considered. It is observed that the means and variances of elastic moduli of two constituents and volume fraction index can converge by the proposed method no matter what kind of boundary conditions, which means that the boundary condition has no influence on the convergence of iterative process. It is also seen that the boundary condition has more influences on the identification result of the variance than that of the mean, and this means that the identification of the dispersion (i.e., variance or COV) is more susceptible to the external disturbance compared to the identification of the mean.Figure 6
Iterative process of means and variances ofEc, Em, and n for different boundary conditions (T=600 k).In order to further investigate the influence of the boundary conditions on the means and COVs of identified parameters, the means and COVs of material elastic moduli and volume fraction index of the FGM beam through its thickness direction or along its axial direction (n=2) for different cases of boundary conditions are shown in Table 4. From Table 4, under the same case of parameter randomness and measurement noise, boundary conditions have no significant effects on the identified results of means and COVs of parameters with the exception of small fluctuation in values.Table 4
Means and COVs of material elastic moduli and volume fraction index for different boundary conditions.
Temperature Parameters(given means/COVs) Means COVs CF CC CS SS CF CC CS SS FGM beam through its thickness  T = 300 k E c   ( 3.22272 e 11 / 5 % ) 3.2226 e + 11 3.2244 e 11 3.2249 e + 11 3.2276 e + 11 6.1553 % 6.1149 % 6.0565 % 6.0745 % E m   ( 2.07788 e 115 % ) 2.0731 e + 11 2.07223 e 11 2.0734 e + 11 2.0733 e + 11 6.2834 % 6.6021 % 6.8732 % 6.7610 % n(2/5%) 1.9910 1.9954 1.9934 1.9899 9 . . 9623 % 9.4556 % 10.1517 % 9.4319 %  T = 600 k E c   ( 3.04610 e 11 / 5 % ) 3.0465 e 11 3.0511 e 11 3.0465 e 11 3.0510 e 11 6.4379 % 6.5151 % 6.2227 % 6.3908 % E m   ( 1.90890 e 11 / 5 % ) 1.9084 e 11 1.9032 e 11 1.9045 e 11 1.9026 e 11 6.8224 % 6.9642 % 7.47109 % 7.2721 % n(2/5%) 1.9943 1.9945 1.9934 1.9918 9.7212 % 9.2149 % 9.6028 % 9.1508 %  FGM beam along its axial direction  T = 300 k E c   ( 3.22272 e 11 / 5 % ) 3.2158 e 11 3.2064 e 11 3.2087 e + 11 3.1971 e + 11 5.6988 % 5.7067 % 6.0838 % 5.9943 % E m   ( 2.07788 e 11 / 5 % ) 2.0775 e 11 2.0754 e 11 2.0724 e + 11 2.0732 e + 11 5.9243 % 6.1247 % 6.0863 % 6.0629 % n(2/5%) 1.9965 1.9870 1.9845 1.9782 5.9821 % 6.1728 % 6.0602 % 5.9818 %  T = 600 k E c(3.04610e11/5%) 3.0396 e 11 3.0398 e 11 3.0356 e + 11 3.0298 e + 11 6.0989 % 5.9839 % 6.1146 % 5.9315 % E m   ( 1.90890 e 11 / 5 % ) 1.9125 e 11 1.9076 e 11 1.9081 e + 11 1.9054 e + 11 6.2054 % 6.6679 % 6.4078 % 6.5143 % n(2/5%) 1.9989 1.9845 1.9854 1.9790 5.7743 % 6.1718 % 6.0829 % 6.0728 %In order to investigate the influence of the uncertainty source of actual modal data on the means and COVs of identified parameters, the means and COVs of material elastic moduli and volume fraction index of the FGM beam through its thickness direction or along its axial direction (n=2) for different cases of uncertainty are shown in Tables 5 and 6. These cases include that the uncertainty of actual modal data is caused by the structural material (i.e., material elastic moduli and volume fraction index) randomness (SR) only, the measurement noise (MN) only, and the combination of two sources (both SR and MN). With structural material randomness (SR) and measurement noise (MN), four combinations of levels of uncertainty are considered. These combinations are (1) Level 1: 1% COV for SR (elastic moduli and volume fraction) and 0.1% for MN; (2) Level 2: 1% for SR and 0.5% for MN; (3) Level 3: 5% for SR and 0.1% for MN; (4) Level 4: 5% for SR and 0.5% for MN.Table 5
Means of material elastic moduli and volume fraction index for different cases of uncertainty.
Temperature Parameters(Given values) SR and MN SR only MN only Level 1(1% SR 0.1% MN) Level 2(1% SR 0.5% MN) Level 3(5% SR 0.1% MN) Level 4(5% SR 0.5% MN) 1% 5% 0.1% 0.5% FGM beam through its thickness  T = 300 k E c   ( 3.22272 e 11 ) 3.2223 e 11 3.2192 e 11 3.2226 e + 11 3.2256 e 11 3.2231 e 11 3.2197 e 11 3.2231 e 11 3.2276 e 11 E m   ( 2.07788 e 11 ) 2.0769 e 11 2.0779 e 11 2.0731 e + 11 2.0774 e 11 2.0780 e 11 2.0780 e 11 2.0798 e 11 2.0780 e 11 n   ( 2 ) 1.9982 1.9964 1.9910 1.9952 1.9996 1.9927 2.0018 2.0029  T = 600 k E c   ( 3.04610 e 11 ) 3.0465 e 11 3.0408 e 11 3.0465 e 11 3.0398 e 11 3.0461 e 11 3.0454 e 11 3.0455 e 11 3.0426 e 11 E m   ( 1.90890 e 11 ) 1.9087 e 11 1.9131 1.9084 e 11 1.9101 e 11 1.9088 e 11 1.9074 e 11 1.9079 e 11 1.9135 e 11 n   ( 2 ) 2.0002 2.0019 1.9943 1.9965 2.0001 1.9936 1.9986 2.0028  FGM beam along its axial direction  T = 300 k E c   ( 3.22272 e 11 ) 3.2224 e 11 3.2261 e 11 3.2158 e 11 3.2236 e 11 3.2226 e 11 3.2161 e 11 3.2247 e 11 3.2261 e 11 E m   ( 2.07788 e 11 ) 2.0778 e 11 2.0779 e 11 2.0775 e 11 2.0732 e 11 2.0786 e 11 2.0768 e 11 2.0787 e 11 2.0786 e 11 n   ( 2 ) 1.9981 2.0018 1.9965 1.9989 1.9998 1.9981 2.0019 2.0054  T = 600 k E c   ( 3.04610 e 11 ) 3.0462 e 11 3.0421 e 11 3.0396 e 11 30357 e 11 3.0458 e 11 3.0394 e 11 3.0467 e 11 3.0432 e 11 E m   ( 1.90890 e 11 ) 1.9093 e 11 1.9080 e 11 1.9125 e 11 1.9061 e 11 1.9095 e 11 1.9076 e 11 1.9087 e 11 1.9079 e 11 n   ( 2 ) 2.0001 1.9963 1.9989 1.9955 2.0001 1.9993 1.9999 1.9959Table 6
COVs of material elastic moduli and volume fraction index for different cases of uncertainty.
Temperature Parameters(given values) SR and MN SR only MN only Level 1(1% SR 0.1% MN) Level 2(1% SR 0.5% MN) Level 3(5% SR 0.1% MN) Level 4(5% SR 0.5% MN) 1% 5% 0.1% 0.5% FGM through its thickness  T = 300 k E c  (1%/5%) 3.0398% 14.0489% 6.1553% 15.3656% 1.0956% 5.5865% 2.8725% 14.2591% E m  (1%/5%) 2.1465% 9.2254% 6.2834% 11.5036% 1.1123% 5.9712% 1.7987% 9.8175% n  (1%/5%) 4.5745% 22.0164% 9..9623% 22.8086% 1.0342% 5.2633% 4.4465% 22.0228%  T = 600 K E c  (1%/5%) 3.2289% 15.0723% 6.4379% 16.0919% 1.1446% 5.6354% 2.9657% 15.0258% E m  (1%/5%) 2.2965% 9.9145% 6.8224% 11.6645% 1.1962% 6.2367% 1.9336% 9.8444% n  (1%/5%) 4.4755% 21.9856% 9.7212% 22.7138% 1.0331% 5.3643% 4.2429% 21.9586%  FGM beam along its axial direction  T = 300 K E c  (1%/5%) 2.0698% 8.6736% 5.6988% 10.1014% 1.1134% 5.3245% 1.7669% 8.6012% E m  (1%/5%) 1.2578% 3.2402% 5.9243% 6.2348% 1.0989% 5.3717% 0.6229% 3.0479% n  (1%/5%) 2.7756% 12.6854% 5.9821% 13.5611% 1.0334% 4.8631% 2.6207% 12.6327%  T = 600 K E c  (1%/5%) 2.1761% 9.5164% 6.0989% 11.0243% 1.1456% 5.6354% 1.8449% 9.4465% E m  (1%/5%) 1.3856% 3.6823% 6.2054% 6.91654% 1.1934% 5.9669% 0.6939% 3.4742% n  (1%/5%) 2.7799% 13.1743% 5.7743% 14.0605% 1.0299% 4.9781% 2.5838% 13.1392%The results from Tables5 and 6 show that the uncertainty does not bring too much changes in the means of identified parameters no matter what sources the uncertainty of actual modal data comes from, and, unlikely, the uncertainty contributes to the COV results, whose errors increase with the uncertainty level. Taking the FGM beam through its thickness direction with the environmental temperature of 300 K as an example, the COVs of identified parameters range from 2.1465% to 11.5036% for metal constituent elastic modulus, 3.0398% to 15.3656% for ceramics constituent modulus, and 4.5745% to 22.8086% for constituent volume fraction index when the level of uncertainty varies from Level 1 (1% SR, 0.1% MN) to Level 4 (5% SR, 0.5% MN). The same situation occurs to the FGM beam along its axial direction. It is worth noting that the accuracy of identified COVs is much more sensitive to the measurement noise, for example, ranging from 6.1553% to 15.3656% for Ec, 6.2834% to 11.5036% for Em, and 9.9623% to 22.8086% for n when the level of MN varies from 0.1% (Level 3) to 0.5% (Level 4) with the same SR of 5%. It is also observed that the influence of the measurement noise with a large value on the identified results is much more larger than that of structural material randomness although the identified COVs for each parameter obtained in the case of the combination of two uncertainty sources are higher than those obtained in the case of SR only or MN only; for example, the COVs is 15.3656% for Ec of the FGM through its thickness direction (T=300 K) in the case of 5% SR and 0.5% MN compared to 5.5865% in the case of 5% SR only and 14.2591% in the case of 0.5% MN only, which means that the identified results of COVs when both SR and MN are included are not the simple linear superposition of the results with respect to SR only or MN only concerned, and, in fact, the measurement noise has a pronounced effect on the accuracy of the identified results. It is also seen that, for the parameters with the smaller dispersion, the higher level of measurement noise can lead to the results with much larger errors, for example, 3.0398% COVs for Ec in the case of 1% SR and 0.1% MN (level 1) compared to 6.1553% in the case of 5% SR and 0.1% MN (Level 3), which means an error of 203.98% for Level 1 compared to an error of 23.11% for Level 3. In fact, in the case of a relatively small level of SR combined with a relatively large level of MN, the identified results of COVs could be totally annihilated by measurement noise.The same situation occurs to the case of FGM beam along the axial direction. Compared to the given value, Figures7, 8, and 9 show the mean values and COVs of Ec, Em, and n for the FGM beam along its axial direction under different cases and levels of uncertainty. For each identified parameter, the result shows that the mean can be identified with good accuracy in the presence of uncertainty, while the identified result of COV is susceptible to the measurement noise. In fact, the increase of measurement noise may result in a large error in the identified result of COV compared to the given value.Figure 7
Means and COVs of elastic moduli of ceramics (Ec) for FGM along axial direction with different uncertainties.Figure 8
Means and COVs of elastic moduli of metal (Em) for FGM along axial direction with different uncertainties.Figure 9
Means and COVs of volume fraction index (n) for FGM along axial direction with different uncertainties.Compared to the given probability density functions (PDFs), Figures10(a), 11(a), and 12(a) show the identified PDFs of Ec, Em, and n for the FGM beam (CF) through its thickness direction or along its axial direction with volume fraction index of n=2 in the thermal environment of 300 K and 600 K for the uncertainty level of 5% SR and 0.1% MN, in which A3 and A6 denote the given PDFs for 300 K and 600 K, respectively, PT3 and PT6 denote the identified PDFs for 300 K and 600 K, respectively, when the FGM beam through its thickness direction is considered, and PA3 and PA6 denote the identified PDFs for 300 K and 600 K, respectively, when the FGM beam along its axial direction is considered. It can be seen that the identified results for the FGM beam along its axial direction are more close to the given PDFs than those for the FGM beam through its thickness direction. Compared to the given PDFs, Figures 10(b), 11(b), and 12(b) show the identified PDFs of the FGM beam through its thickness direction in the thermal environment of 300 k with different cases of uncertainty, in which A3 denotes the given PDF, P5SR denotes the identified PDF with 5% SR (5% COVs in material elastic moduli and volume fraction index), P01MN and P05MN denote the identified PDFs for 0.1% MN (measurement noise) and 0.5% MN, respectively, and P5SR01MN denotes 5% SR and 0.1% MN, and P5SR05MN denotes 5% SR and 0.5% MN. It is obviously seen that the PDFs are far away from the given PDFs when MN reaches 0.5% no matter whether 5% SR is included, which also shows that the identified results of COVs are completely submerged in the measurement noise. The same situation occurs to the PDF results of all identified parameters.Figure 10
Identified PDFs of ceramics material elastic modulus for different cases.
(a) (b)Figure 11
Identified PDFs of metal material elastic modulus for different cases.
(a) (b)Figure 12
Identified PDFs of volume fraction index for different cases.
(a) (b)


## 7. Conclusions

Identification of the statistic numerical characteristics for constituent material properties and distribution is an important issue in mechanical behavior analysis and safety assessment of FGM structures, especially in the early stage of damage. The statistic numerical characteristics are identified for material elastic moduli and constituent volume fraction index of a random FGM beam with initial damage by using a stochastic finite element model updating-based inverse technique in this paper. A region truncation-based optimization algorithm is utilized for simplifying the optimization process and improving computational efficiency without impairing the convergence of iterative process, and a new algorithm is developed for computing the covariance between the modal parameters and the identification parameters for damaged FGM structures. The influence of structural material randomness and measurement noise on the accuracy of identification results has been investigated.The following conclusions can be drawn from this study:(1)
Compared to the given values and the results from MCS method, the presented inverse technique based on stochastic finite element model updating can be utilized as an alternative to identify the statistics of the material properties and volume fraction index for random damaged FGM structures, provided that the modal parameters of the corresponding FGM structures are be obtained from experimental test.(2)
Compared to the trust region method, the faster convergence of the iterative process can be guaranteed by the region truncation-based optimization. The elastic moduli of constituent materials converge to the different values in the case of different temperatures, while volume fraction indices not affected by the environment converge to the same value in the case of different temperatures.(3)
The volume distribution has more obvious influence on the accuracy of the dispersion (COVs) of identified parameters for the FGM with material properties varying quickly (through the thickness direction) compared to that for the FGM with material properties varying slowly (along the axial direction).(4)
The boundary conditions have no significant influence on the convergence of iterative process as well as the identified results of means and COVs of material elastic moduli and volume fraction index.(5)
The measurement noise plays a dominant role in the accuracy of COV identification results while the uncertainty does not bring too much change in the means of identified parameters no matter what sources the uncertainty comes from. The dispersion of material properties and volume fraction can be identified with relatively good accuracy in the presence of small measurement noise, while the COV identification results could be totally annihilated by a relatively large measurement noise.



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