## Abstract

In this paper, the inverses of both the elastic constants and the thermal expansion coefficients of anisotropic materials are used to decouple thermo-mechanical deformation using the virtual fields method (VFM). First, a method for inversing and decoupling the thermo-mechanical deformation field under three-point bending loading is proposed based on the VFM. Second, thermo-mechanical coupled deformation of the three-point bending specimen is simulated by means of the finite-element method, and the inverse of the thermo-mechanical elastic constants are used, which agree well with the reference value. Finally, the influences of the noise level on the accuracy of the thermo-mechanical constants from the VFM are discussed. The results will provide a new method for characterizing thermo-mechanical elastic constants and decoupling the thermo-mechanical deformation of anisotropic materials under high-temperature environments.

## 1. Introduction

Both hot structures [1] and high-temperature materials [2–4] are widely used in many engineering fields, such as hypersonic re-entry vehicles, rocket combustion chambers, turbine engines and so on. Owing to the complexity of high-temperature environments and high-temperature deformations, the measurement of high-temperature material parameters and deformations is still a difficult problem to solve when designing and evaluating the strength and lifespan of hot structures. How to characterize the thermo-mechanical deformation field of anisotropic materials is not only an important but also a novel scientific problem.

The virtual fields method (VFM) [5,6] is a tool for identification based on the principle of virtual work. Grediac *et al.* [7] proposed special virtual fields to determine the orthotropic material parameters using the VFM, and studied the inversion of elastic deformation for the unnotched Iosipescu test [8] and the bending of anisotropic plates [9]. As for the sensitivity analysis of the VFM, Avril *et al*. [10,11] added Gaussian white noise to the actual strain fields simulated by the finite-element method (FEM) for studying the sensitivity of the VFM to the noisy data, which indicated that the optimal virtual field is the ‘maximum-likelihood solution’ with minimal uncertainty. Rossi & Pierron [12] set a cost function to evaluate the influence of the noise for different simulated deformations of the unnotched Iosipescu specimen, and found the best combinations between the length and the material orientation, which were verified using the unnotched Iosipescu experiment [13]. For different constitutive models of engineering materials, Promma *et al*. [14] used a multi-axial mechanical test to identify the constitutive parameters of a Mooney model suitable for hyperelastic materials using the VFM. Palmieri *et al*. [15] estimated the material-dependent parameters with hyperelastic constitutive laws of the Ogden model and second-order Mooney–Rivlin model under planar tension tests by means of the VFM.

High-temperature anisotropic composites have more and more applications in aerospace engineering structures. The corresponding full-field deformation under high temperature can be measured by means of experimental solid mechanics technology, such as high-temperature electronic speckle pattern interferometry [16], high-temperature digital image correlation [17], high-temperature gratings [18] and so on. In fact, the full-field deformations in high-temperature experiments are thermo-mechanically coupled. Furthermore, the traditional measurements of anisotropic thermo-mechanical constants under high temperature need a series of complex experimental conditions. There is a lack of a simple and effective method to determine all the elastic constants and the thermal expansion coefficients of anisotropic composites using one experiment.

In this paper, a new method based on VFM is proposed to inverse and decouple the thermo-mechanical deformation field in anisotropic materials, which has potential applications in decoupling and obtaining thermo-mechanical constants under high temperature.

## 2. Theory of the virtual fields method

### (a) Principle of the virtual fields method

A specimen with constant thickness of any shape subjected to in-plane thermo-mechanical loading is shown in figure 1. Here, *V* and *S* are the volume and the external surface area of the specimen, respectively. *S*_{u} and *S*_{f} are the displacement boundary and the loading boundary of the external surface area, respectively. *u*_{i} (*i*=*x*,*y*) is the displacement over the displacement boundary. *T*_{i} (*i*=*x*,*y*) is the force per unit area over the loading boundary. Based on the principle of virtual work, the static equilibrium equation without body force is
2.1where (*i*=*x*,*y*) is an imagined differentiable displacement field (virtual displacement field), which is kinematically admissible, and over the displacement boundary. (*i*=*x*,*y*,*s*) is an imagined strain field (virtual strain field), which is derived from . *σ*_{i} (*i*=*x*,*y*,*s*) can be expressed based on the constitutive equations as follows:
2.2where *Q*_{xx}, *Q*_{yy}, *Q*_{xy} and *Q*_{s} are the coefficients of the stiffness matrix. *α*_{x} and *α*_{y} are the thermal expansion coefficients in the *x*-direction and the *y*-direction, respectively. *ε*_{i} (*i*=*x*,*y*,*s*) is the true strain field. Δ*T*(*x*,*y*) is the temperature increment in the coordinate (*x*,*y*). Then, equation (2.1) can be expressed as
2.3where *e* is the thickness of the specimen.

### (b) Construction of the special virtual fields

The three-point bending test is a type of high-temperature experiment. The special virtual fields are constructed for three-point bending loading (figure 2). Here, *h* is the height of the specimen, *L* is the span of the three-point bending specimen and the applied load is *F*. The displacement boundary is: (i) *u*_{By}=0 at position B in the *y*-direction; (ii) *u*_{Cy}=0 at position C in the *y*-direction. The loading boundary is: *T*_{Ay}=−*F* at position A in the *y*-direction. The temperature field is assumed to be uniform. Then the temperature increment is the constant (Δ*T*(*x*,*y*)=Δ*T*_{0}).

The objective of this study is to identify the thermo-mechanical elastic constants *Q*_{xx}, *Q*_{yy}, *Q*_{xy}, *Q*_{s}, *α*_{x} and *α*_{y}. Six unknown parameters need six kinds of special virtual fields to set up six equations for obtaining the thermo-mechanical elastic constants. The criteria to design the virtual fields are: (i) all virtual fields satisfy the displacement boundary: *u*_{By}=0 and *u*_{Cy}=0; (ii) each of the six equations has a unique solution; and (iii) the forms of the virtual fields should be as simple as possible to reduce the amount of computation. Figure 3 shows the six kinds of special virtual fields.

#### (i) First virtual field

The first virtual field (figure 3*a*) is expressed as follows: , over A_{1} and A_{2}, which represents the uniform compression in the *y*-direction. The virtual displacements at positions A, B and C in the *y*-direction are *u**_{Ay}=−*h*, *u**_{By}=0 and *u**_{Cy}=0, respectively, which satisfy the displacement boundary. The associated virtual strain fields are , and over A_{1} and A_{2}. Using the first virtual field, equation (2.3) can be expressed as
2.4

#### (ii) Second virtual field

The second virtual field (figure 3*b*) is expressed as follows: , over A_{1} and A_{2}, which represents the non-uniform compression in the *y*-direction. The virtual displacements at positions A, B and C in the *y*-direction are *u**_{Ay}=−*h*^{2}, *u**_{By}=0 and , respectively, which satisfy the displacement boundary. The associated virtual strain fields are , and over A_{1} and A_{2}. Using the second virtual field, equation (2.3) can be expressed as
2.5

#### (iii) Third virtual field

The third virtual field (figure 3*c*) is expressed as follows: , over A_{1} and A_{2}, which represents the uniform tension in the *x*-direction. The virtual displacements at positions A, B and C in the *y*-direction are *u**_{Ay}=0, *u**_{By}=0 and *u**_{Cy}=0, respectively, which satisfy the displacement boundary. The associated virtual strain fields are , and over A_{1} and A_{2}. Using the third virtual field, equation (2.3) can be expressed as
2.6

#### (iv) Fourth virtual field

The fourth virtual field (figure 3*d*) is expressed as follows: , over A_{1} and A_{2}, which represents the non-uniform tension in the *x*-direction. The virtual displacements at positions A, B and C in the *y*-direction are *u**_{Ay}=0, *u**_{By}=0 and *u**_{Cy}=0, respectively, which satisfy the displacement boundary. The associated virtual strain fields are , and over A_{1} and A_{2}. Using the fourth virtual field, equation (2.3) can be expressed as
2.7

#### (v) Fifth virtual field

The fifth virtual field (figure 3*e*) is expressed as follows: , over A_{1}; , over A_{2}, which represents the coupled deformation in the *x*-direction and *y*-direction. The virtual displacements at positions A, B and C in the *y*-direction are *u**_{Ay}=0, *u**_{By}=0 and *u**_{Cy}=0, respectively, which satisfy the displacement boundary. The associated virtual strain fields are , and over A_{1}; , and over A_{2}. Using the fifth virtual field, equation (2.3) can be expressed as
2.8

#### (vi) Sixth virtual field

The sixth virtual field (figure 3*f*) is expressed as follows: , over A_{1}; , over A_{2}, which represents the shear deformation. The virtual displacements at positions A, B and C in the *y*-direction are , *u**_{By}=0 and *u**_{Cy}=0, respectively, which satisfy the displacement boundary. The associated virtual strain fields are , and over A_{1}; , and over A_{2}. Using the sixth virtual field, equation (2.3) can be expressed as
2.9

### (c) Inversion and decoupling

Six unknown parameters (*Q*_{xx}, *Q*_{yy}, *Q*_{xy}, *Q*_{s}, *α*_{x} and *α*_{y}) can be obtained by solving equations (2.4)–(2.9), which can be expressed in the following discretized form:
2.10a
2.10b
2.10c
2.10d
2.10e
2.10f

Based on the six parameters (*Q*_{xx}, *Q*_{yy}, *Q*_{xy}, *Q*_{s}, *α*_{x} and *α*_{y}), the thermo-mechanical coupling displacement field (*u*_{x} and *u*_{y}) can be decoupled into the thermal displacement field ( and ) and the mechanical displacement field ( and ). The thermal displacement field is
2.11aand
2.11bAnd the mechanical displacement field is
2.12aand
2.12bThe related elastic constants such as the elastic moduli (*E*_{x} and *E*_{y}) in the *x*- and *y*-directions, Poisson's ratio (*ν*_{xy}) and the shear modulus (*G*) can be calculated as follows:
2.13

## 3. Simulated three-point bending deformation

A composite of C and SiC is a commonly used thermal protection material. Here, a three-point bending beam of the C/SiC composite with 30×10×1 (*L*×*h*×*e*) is considered. The elastic moduli (*E*_{x} and *E*_{y}) of the C/SiC composite in the *x*- and *y*-directions are 140 and 13 GPa [19], respectively. Poisson's ratio (*ν*_{xy}) of the C/SiC composite is 0.37 [19]. The shear modulus (*G*) of the C/SiC composite is 36 GPa. The thermal expansion coefficients (*α*_{x} and *α*_{y}) in the *x*- and *y*-directions are 1.14×10^{−6} and 2.35×10^{−6} K^{−1} [19], respectively.

For the three-point bending beam in figure 2, the displacement of positions B and C in the *y*-direction is zero, and the initial temperature is 298 K. Also, a point loading of 10 N is applied on position A, while a constant temperature loading of 308 K is applied to the whole three-point bending beam. A two-dimensional planar model with a four-node bilinear plane stress quadrilateral element is set up to simulate the three-point bending deformation under mechanical and thermal coupled loadings using the software ABAQUS. The calculated strain fields for the three-point bending beam with 7500 elements and 0.2 mm step length are shown in figure 4. In view of the discrete strain field in the data processing of the VFM, each element coordinate is selected as the average value of the nodal coordinates of that element. And each element strain is the average value of the integration point's strain of that element.

## 4. Results and discussions

Based on the strain fields obtained by means of the FEM, the thermo-mechanical deformation fields are inversed and decoupled using the VFM. Here, the influences of the mesh and the noise on the accuracy of the VFM are discussed.

### (a) Inversion

Based on the strain fields obtained in §3, six unknown parameters (*E*_{x}, *E*_{y}, *υ*_{xy}, *G*, *α*_{x} and *α*_{y}) can be solved from equations (2.10*a*–*f*) and (2.13). Table 1 shows the comparison of the six parameters between the reference values and the inversion values from the strain fields with different mesh densities. It is clear that the influence of the mesh density on Poisson's ratio (*ν*_{xy}), the shear modulus (*G*) and the thermal expansion coefficients (*α*_{x} and *α*_{y}) in the *x*- and *y*-directions is minimal, and that the influence of the mesh density on the elastic moduli (*E*_{x} and *E*_{y}) in the *x*- and *y*-directions is dominant. Furthermore, if the step length is smaller than 0.2 mm (and the element number is larger than 7500), the influence of the mesh on *E*_{x} and *E*_{y} can be ignored.

### (b) Decoupling displacement field

Figure 5*a*,*b* shows the thermo-mechanical decoupled displacement field directly obtained by the FEM. The thermal expansion coefficients (*α*_{x} and *α*_{y}) in the *x*- and *y*-directions are determined in §4.1. Thus, the thermo-mechanical displacement field can be decoupled by equations (2.11) and (2.12). Figure 5*c*,*d* shows the decoupling thermal displacement field in the *x*- and *y*-directions. Figure 5*e*,*f* shows the decoupling mechanical displacement field in the *x*- and *y*-directions, which shows good agreement with the typical three-point bending displacement fields.

### (c) Noise effect

It is known that experimental error in the strain field is inevitable. Here, a white Gaussian noise is introduced into the simulated three-point bending deformation. *N*_{x}(*x*,*y*), *N*_{y}(*x*,*y*) and *N*_{s}(*x*,*y*) are white Gaussian noises, which follow the normal distribution with an average of zero and a variance of 1. Then the measured strain can be defined as the superposition between the actual values (, and and the noise,
4.1a
4.1b
and
4.1cwhere *μ* is a strictly positive real number, which represents the random variability intensity of the measured strain. When the temperature increment is the constant (Δ*T*(*x*,*y*)=Δ*T*_{0}), and equation (4.1*a*–*c*) is put into equation (2.3), the static equilibrium equation without body force can be expressed as
4.2Using the special virtual fields in §2.2, six unknown parameters (*E*_{x}, *E*_{y}, *υ*_{xy}, *G*, *α*_{x} and *α*_{y}) can be determined using equations (2.10*a*–*f*) and (2.13). The results of the inversion of the thermo-mechanical elastic constants with the noise are shown in table 2. Each case is simulated five times. The standard deviation of *E*_{x} and *E*_{y} changing with *μ* is shown in figure 6*a*. The influence of the noise on the elastic moduli (*E*_{x} and *E*_{y}) in the *x*- and *y*-directions is dominant. Also, the influence of the noise on *E*_{y} is greater than that of *E*_{x}. The standard deviation of *ν*_{xy} and *G* changing with *μ* is shown in figure 6*b*. The influences of the noise on *ν*_{xy} and *G* are much smaller than those on *E*_{x} and *E*_{y}. Figure 6*c* shows the influence of the noise on the thermal expansion coefficients (*α*_{x} and *α*_{y}) in the *x*- and *y*-directions, and shows that the influence of the noise on *α*_{x} is minimal, and the influence of the noise on *α*_{y} is dominant. It shows that if the *μ* does not exceed 1×10^{−5.5} (approx. the magnitude of 5 μ*ε*), the error of this inversion method will be smaller than 10 per cent.

## 5. Conclusions

In this paper, the inverses of both the elastic constants and the thermal expansion coefficients of anisotropic materials are used to decouple the thermo-mechanical deformation field using the VFM. Some important conclusions are as follows:

— a new method for obtaining the inverses of the thermo-mechanical constants and decoupling the thermo-mechanical deformation fields of anisotropic materials is proposed based on the VFM, which has potential applications in thermo-mechanical parameter measurements and full-field deformation analysis under a high-temperature environment. In this paper, six special virtual fields are selected to simulate the three-point bending deformation;

— the strain fields of the simulated three-point bending deformation under high temperature are extracted by the FEM. The inverses of both the elastic constants and the thermal expansion coefficients are used and the thermo-mechanical deformation fields are decoupled, showing good agreement with the reference values;

— the influence of the noise on the elastic moduli in the

*x*- and*y*-directions is dominant. The influence of the noise on*α*_{x}is minimal, and the influence of the noise on*α*_{y}is dominant; and— if the full-field temperature (Δ

*T*(*x*,*y*)) for anisotropic materials can be obtained under the gradient temperature field, the theoretical analysis in this paper is also applicable to the inverse of the thermo-mechanical elastic constants. For gradient materials or temperature-dependent materials under a gradient temperature field, the inverses of many more unknown parameters are needed, which requires corresponding numbers of virtual fields to solve the unknown parameters.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant nos 91016007, 11227801 and 91216301) and the Specialized Research Fund for the Doctoral Programme of Higher Education (grant no. 20110002110073).

- Received February 22, 2013.
- Accepted March 19, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.