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.

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 length , width , and thickness 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 where , are the coordinates along the thickness () and length () of the FGM beam as shown in Figure 1. denotes the effective material properties such as Young’s modulus, Poisson’s ratio, and thermal expansion efficient along coordinate for the FGM beam through its thickness direction or coordinate for the FGM beam along its axial direction. , denote the corresponding material properties of ceramics and metal, respectively. 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]:where are the coefficients of temperature 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):where the updating parameter vector is composed of material properties and volume fraction index of FGMs to be identified, that is, . The vector is the analytic modal parameters (modal frequencies and mode shapes), that is, , which is a nonlinear function of the updating parameters vector . The vector is the actual modal parameters. The vectors , 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]:where subscript denotes the iteration step. The vectors , , are the means of the updating parameters, the analytical modal parameters, and the actual modal parameters. The vectors , , are the corresponding zero-mean random parts with the same variances of . , is a diagonal weighting matrix to allow for regularization of ill-posed sensitivity equations. is a transformation matrix and is the sensitivity matrix of modal parameters with respect to updating parameters. In (6), the actual modal data and updating parameters are assumed to be uncorrelated; that is, , , and the covariance matrix between the analytical modal parameters and the updating parameters as well as the covariance matrix of the analytic modal parameters can be evaluated by forward propagation of uncertainty.

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:where are boundary vectors of the truncated region. is the distance vector of the current iterative point from the nearest constraints, and the size of truncation region depends on the parameter . It is suggested that 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:where is the mean value of each effective material property, and is the corresponding zero-mean random part with the variance of . is the element of the updating parameter vector , and 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]: where , are the axial and transverse displacements at any point of the FGM beam in the ,   directions, respectively. , are those on the mid-plane. is the cross-sectional rotation about -axis.

By assuming the small deformation, the following strain field can be obtained:where and are the normal and shear strains.

The constitutive relation in thermal environment can be expressed aswhere and are the normal and shear stresses. and are the elastic coefficients, that is, ,  . are the effective Young’s elastic modulus, which is the function of for the FGM beam through its thickness direction, and the function of for the FGM beam along its axial direction. is the mean and is the corresponding zero-mean random part. Poisson’s ratio is assumed to be constant. is the thermal expansion efficient. 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 where is the shape function matrix and is the nodal displacement vector of beam element.

The strain and kinetic energies of the beam element arewhere is the volume of a beam element. are the effective Young’s modulus, which is the function of for the FGM beam through its thickness, and the function of for the FGM beam along its axial direction. is the mean and 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:where is the element stiffness matrix, is the element mass matrix, and is the thermal force. Each matrix is composed of the mean matrix and the zero-mean random part.

Recalling (8), the means , , and the zero-mean random parts , , can be computed, respectively, by where the matrices , ,   can be computed bywhere , , and are the matrices mainly related to the shape function matrix and .

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 indicating the damage severity for the th damaged element, effective Young’s modulus of the th damaged element can be expressed aswhere is the effective Young’s elastic modulus for the undamaged FGM beam and is for the damaged FGM beam.

Recalling (15), the stiffness matrix of the damaged element can be expressed as where is the damage factor of the th damaged element and is the stiffness element for the th 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. where is the global random stiffness matrix and is the global random mass matrix. is the global thermal force and is a transformation matrix that transforms the local coordinate of the th element to the global coordinate. is the number of undamaged elements and is the number of damaged elements.

5. Forward Uncertainty Propagation

The modal frequency and mode shape are governed by the following:

The modal frequency and mode shape can also be represented as the sum of its mean and a zero-mean random part: and further the random part of the normalized mode shape can be written as where ,   are the small coefficients to be determined. Substituting (22) into (21), we have

Using the orthogonality of mode shapes, the random part of the modal frequency and the coefficients can be obtained by premultiplying (25) by and , respectively,

Recalling the orthogonality condition and using (20) and (22), the coefficient can be obtained by

Neglecting the high-order terms, the random parts of modal parameters in (22) can be obtained by the Taylor’s expansion:

Using (20), (26)–(28), we have the following equation for the solution of the unknown and by equating the coefficients of each :

Further, the variances of the modal frequency and normalized mode shape as well as the covariance between the modal parameters and the random material parameters can be obtained:where , , are the standard deviations of the random material parameters, that is, constituent material properties and volume fraction index, and is the correlation coefficient of random parameters , .

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 Table 1. 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.

In 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 the element with a reduction in the effective Young’s elastic modulus by 10%, that is, . 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. Figure 2 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 () compared to the undamaged case and Figure 3 shows those for the FGM beam along the axial direction (), 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 parameters to be identified, which are chosen as the updating parameters of stochastic model updating.

Step 2. Compute the mean vector and covariance matrix of the actual modal data.