**1. Introduction**

It is widely known that residual stresses are created by almost every material process and harmful to a variety of structures and devices [1]. Yet, the problem is far from being solved. Although a number of techniques have been developed to evaluate residual stresses, destructively [2–6] or nondestructively [7–20], have been developed to evaluate residual stresses, there is no single method applicable to general cases for accurate evaluations. The fundamental complexity of the problem lies in the fact that residual stresses are locked into the material and therefore hidden from observance. Unless the locking mechanism is removed, the residual stress is not visible from the outside. Under such a situation, it is especially difficult to diagnose residual stresses nondestructively. Techniques classified as nondestructive methods normally use diffractometry [9, 10, 19] or acoustic probing [7, 8, 12–16]. The techniques classified as diffractometry detect the atomic rearrangement due to residual stresses from a shift in the diffraction angle using X-ray, neutron, synchrotron, or similar radiation. The techniques classified as acoustic probing detect residual stresses from the change in the acoustic impedance due to

stress-induced alternation of the elastic constant. Both are well-established methods but have limitations. Diffractometry is applicable to detection of atomic rearrangement within the penetration depth of the radiation used. Neutron and synchrotron radiations can have reasonable penetration depths of tens of mm, but a large facility is required, and access is most often an issue. X-ray diffraction instruments are relatively more accessible, but the typical penetration depth is in the range of tens of *μ*m [19]. The acoustic waves can penetrate much further, but the coupling of the acoustic signal from the emitter to the specimen is sensitive. According to our experience, the conditions of the coupling medium (usually distilled water) such as the layer thickness and the total amount can affect the measurement. Slight shifts in the contact locations seem to cause a considerable change in the measurement as well. It is not clear if such changes are due to errors associated with the contact or actual spatial variation of the residual stress. In either case, the measurement is essentially pointwise, and therefore these methods are timeconsuming if applied to a certain area of the object. These issues are especially significant for analysis of welding-induced residual stress because by nature compressive and tensile residual stresses can alternate with high spatial frequency.

integer multiple of *λ*. Under this condition, the detector (not shown in this figure)

*(a) Bragg diffraction. (b) Change in crystallographic plane orientation and spaces due to residual stress. (c) Coordinate system αβγ affixed to specimen surface where the γ-axis is normal to specimen surface and α and β are axes of principal stress. Coordinate system xyz is affixed to grain where the* z*-axis is normal to lattice plane. The* z*-axis is rotated around the* y*-axis, which is in αβ-plane.* S*<sup>ϕ</sup> denotes direction of in-plane strain εϕ, which*

ð Þ *<sup>ε</sup>*<sup>Ψ</sup> *zz* <sup>¼</sup> *<sup>d</sup>*<sup>Ψ</sup> � *<sup>d</sup>*<sup>0</sup>

Here ð Þ *ε*<sup>Ψ</sup> *zz* is the normal strain along the *z*-axis in the coordinate system affixed to the grain, as **Figure 1c** illustrates. In this coordinate system, the lattice plane is parallel to the *xy*-plane and perpendicular to the *z*-axis. The *αβγ* coordinate system is affixed to the specimen where the *γ*-axis is normal to the specimen surface and *α*and *β*-axes are the principal axes. The local stress has angle *ϕ* to the *α*-axis. Through the proper coordinate transformation, we can express this strain using the in-plane

> *<sup>ϕ</sup>* <sup>þ</sup> *εαβ* sin 2*<sup>ϕ</sup>* <sup>þ</sup> *εββ* sin <sup>2</sup> *ϕ* sin <sup>2</sup>

The first part on the right-hand side of Eq. (3) enclosed by the parenthesis is the in-plane strain in the direction of *S<sup>ϕ</sup>* (*εϕ* in **Figure 1**). Eq. (3) indicates that *εϕ* is

> *<sup>ϕ</sup>* <sup>þ</sup> *εαβ* sin 2*<sup>ϕ</sup>* <sup>þ</sup> *εββ* sin <sup>2</sup> *<sup>ϕ</sup>* <sup>∝</sup> *<sup>∂</sup>d*<sup>Ψ</sup>

is proportional to *εzz*. These altogether indicate that the in-plane strain *εϕ* can be

Since the in-plane strain *εϕ* is connected with the in-plane stress *σϕ* with an elastic constant, Eq. (4) indicates that *σϕ* can be evaluated as the slope of *d*<sup>Ψ</sup> vs.

Ψ plot. In the present context, *σϕ* is the in-plane residual stress of interest.

Experimentally, we can evaluate *σϕ* by changing angle Ψ and plotting the

determined from the corresponding shift in the diffraction angle as discussed above. When the angle Ψ is varied for this plot, the slope (4) remains unchanged.

Ψ

*∂* sin <sup>2</sup>

Ψ. The deformed lattice distance *d*<sup>Ψ</sup> can be

<sup>Ψ</sup> *:* (4)

<sup>Ψ</sup> <sup>þ</sup> *εγα* cos *<sup>ϕ</sup>* sin 2<sup>Ψ</sup> <sup>þ</sup> *εβγ* sin *<sup>ϕ</sup>* sin 2Ψ*:* (3)

Ψ plot. Eq. (2) tells us the deformed lattice distance *d*<sup>Ψ</sup>

*d*0

(2)

The XRD for residual stress analysis exploits the fact that an in-plane stress alters the distance between the neighboring lattice planes via Poisson's effect and thereby shifts the diffraction angle from the nominal (unstressed) value. **Figure 1b** illustrates the situation where an in-plane tensile stress increases the lattice distance in a grain whose lattice plane is oriented at angle Ψ to a line normal to the surface. The resultant change in the lattice distance Δ*d*<sup>Ψ</sup> ¼ *d*<sup>Ψ</sup> � *d*<sup>0</sup> from the initial distance *d*<sup>0</sup>

records a maximum when it is oriented at the angle of diffraction.

causes the strain.

*makes angle ϕ from the α-axis in αβ-plane.*

*Opto-Acoustic Technique for Residual Stress Analysis DOI: http://dx.doi.org/10.5772/intechopen.90299*

**Figure 1.**

strain parallel to the specimen surface as.

given as the slope of *εzz*- sin <sup>2</sup>

*εϕ* <sup>¼</sup> *<sup>∂</sup>*ð Þ *<sup>ε</sup>*<sup>Ψ</sup> *zz ∂* sin <sup>2</sup>

sin <sup>2</sup>

**11**

given by the following differential:

<sup>Ψ</sup> <sup>¼</sup> *εαα* cos <sup>2</sup>

corresponding *d*<sup>Ψ</sup> as a function of sin <sup>2</sup>

ð Þ *<sup>ε</sup>*<sup>Ψ</sup> *zz* <sup>¼</sup> *εαα* cos <sup>2</sup>

<sup>þ</sup>*εγγ* cos <sup>2</sup>

Given the above situation, our approach is to employ multiple methods. We are especially interested in combining an optical technique capable of full-field analysis and an acoustic technique. It should be noted that the optical methods, including the diffractometry discussed above, measure a change in displacement (strain), whereas the acoustic methods detect a change in elastic constant. With an appropriate combination of optical and acoustic techniques, we can essentially obtain both elastic constant and strain information. In principle, through the knowledge of strain with the elastic constant, we can estimate the stress. This is contrastive to the application of diffractometry by itself. In that case, even if the method provides us with accurate strain data, the residual stress heavily depends on the elastic constant to be used. The use of the nominal value (the residual stress free value) to estimate the residual stress from the measured residual strain is somewhat contradictory. The aim of this article is to discuss the application of an optical technique known as the electronic speckle pattern interferometry and acoustic methods for assessment of residual stress induced by welding. After briefly reviewing several techniques widely used for residual stress analysis, the results of our research are presented and discussed. The presented combined method is still in the process of development. It is our intention to introduce the techniques to the readers hoping that the information is useful to them. Future directions of the research are discussed as well.

### **2. Techniques to evaluate residual stresses**

#### **2.1 X-ray diffractometry (XRD)**

In this section the principle of X-ray diffractometry (XRD) for residual stress analysis is described briefly. A more detailed description about the technique can be found elsewhere [21]. **Figure 1a** illustrates that Bragg's law relates the angle of diffraction *θ* and the lattice distance as follows:

$$n\lambda = 2d\sin\theta\tag{1}$$

Here *n* is an integer and *λ* is the wavelength of the X-ray. Eq. (1) represents the condition of constructive interference that takes place when the path difference between the reflections from the top lattice plane and the second lattice plane is an *Opto-Acoustic Technique for Residual Stress Analysis DOI: http://dx.doi.org/10.5772/intechopen.90299*

**Figure 1.**

stress-induced alternation of the elastic constant. Both are well-established methods

Given the above situation, our approach is to employ multiple methods. We are especially interested in combining an optical technique capable of full-field analysis and an acoustic technique. It should be noted that the optical methods, including the diffractometry discussed above, measure a change in displacement (strain), whereas the acoustic methods detect a change in elastic constant. With an appropriate combination of optical and acoustic techniques, we can essentially obtain both elastic constant and strain information. In principle, through the knowledge of strain with the elastic constant, we can estimate the stress. This is contrastive to the application of diffractometry by itself. In that case, even if the method provides us with accurate strain data, the residual stress heavily depends on the elastic constant to be used. The use of the nominal value (the residual stress free value) to estimate the residual stress from the measured residual strain is somewhat contradictory. The aim of this article is to discuss the application of an optical technique known as the electronic speckle pattern interferometry and acoustic methods for assessment of residual stress induced by welding. After briefly reviewing several techniques widely used for residual stress analysis, the results of our research are presented and discussed. The presented combined method is still in the process of development. It is our intention to introduce the techniques to the readers hoping that the information is useful to them. Future directions of the research are discussed as well.

In this section the principle of X-ray diffractometry (XRD) for residual stress analysis is described briefly. A more detailed description about the technique can be found elsewhere [21]. **Figure 1a** illustrates that Bragg's law relates the angle of

Here *n* is an integer and *λ* is the wavelength of the X-ray. Eq. (1) represents the condition of constructive interference that takes place when the path difference between the reflections from the top lattice plane and the second lattice plane is an

*nλ* ¼ 2*d* sin *θ* (1)

**2. Techniques to evaluate residual stresses**

diffraction *θ* and the lattice distance as follows:

**2.1 X-ray diffractometry (XRD)**

**10**

rearrangement within the penetration depth of the radiation used. Neutron and synchrotron radiations can have reasonable penetration depths of tens of mm, but a large facility is required, and access is most often an issue. X-ray diffraction instruments are relatively more accessible, but the typical penetration depth is in the range of tens of *μ*m [19]. The acoustic waves can penetrate much further, but the coupling of the acoustic signal from the emitter to the specimen is sensitive. According to our experience, the conditions of the coupling medium (usually distilled water) such as the layer thickness and the total amount can affect the measurement. Slight shifts in the contact locations seem to cause a considerable change in the measurement as well. It is not clear if such changes are due to errors associated with the contact or actual spatial variation of the residual stress. In either case, the measurement is essentially pointwise, and therefore these methods are timeconsuming if applied to a certain area of the object. These issues are especially significant for analysis of welding-induced residual stress because by nature compressive and tensile residual stresses can alternate with high spatial frequency.

but have limitations. Diffractometry is applicable to detection of atomic

*New Challenges in Residual Stress Measurements and Evaluation*

*(a) Bragg diffraction. (b) Change in crystallographic plane orientation and spaces due to residual stress. (c) Coordinate system αβγ affixed to specimen surface where the γ-axis is normal to specimen surface and α and β are axes of principal stress. Coordinate system xyz is affixed to grain where the* z*-axis is normal to lattice plane. The* z*-axis is rotated around the* y*-axis, which is in αβ-plane.* S*<sup>ϕ</sup> denotes direction of in-plane strain εϕ, which makes angle ϕ from the α-axis in αβ-plane.*

integer multiple of *λ*. Under this condition, the detector (not shown in this figure) records a maximum when it is oriented at the angle of diffraction.

The XRD for residual stress analysis exploits the fact that an in-plane stress alters the distance between the neighboring lattice planes via Poisson's effect and thereby shifts the diffraction angle from the nominal (unstressed) value. **Figure 1b** illustrates the situation where an in-plane tensile stress increases the lattice distance in a grain whose lattice plane is oriented at angle Ψ to a line normal to the surface. The resultant change in the lattice distance Δ*d*<sup>Ψ</sup> ¼ *d*<sup>Ψ</sup> � *d*<sup>0</sup> from the initial distance *d*<sup>0</sup> causes the strain.

$$(\varepsilon\Psi)\_{\text{xx}} = \frac{d\_{\Psi} - d\_0}{d\_0} \tag{2}$$

Here ð Þ *ε*<sup>Ψ</sup> *zz* is the normal strain along the *z*-axis in the coordinate system affixed to the grain, as **Figure 1c** illustrates. In this coordinate system, the lattice plane is parallel to the *xy*-plane and perpendicular to the *z*-axis. The *αβγ* coordinate system is affixed to the specimen where the *γ*-axis is normal to the specimen surface and *α*and *β*-axes are the principal axes. The local stress has angle *ϕ* to the *α*-axis. Through the proper coordinate transformation, we can express this strain using the in-plane strain parallel to the specimen surface as.

$$\begin{split} (\epsilon \mu)\_{xx} &= \left(\varepsilon\_{aa}\cos^2\phi + \varepsilon\_{a\beta}\sin2\phi + \varepsilon\_{\beta\beta}\sin^2\phi\right)\sin^2\Psi \\ &+ \varepsilon\_{\gamma\gamma}\cos^2\Psi + \varepsilon\_{\gamma a}\cos\phi\sin2\Psi + \varepsilon\_{\beta\gamma}\sin\phi\sin2\Psi. \end{split} \tag{3}$$

The first part on the right-hand side of Eq. (3) enclosed by the parenthesis is the in-plane strain in the direction of *S<sup>ϕ</sup>* (*εϕ* in **Figure 1**). Eq. (3) indicates that *εϕ* is given as the slope of *εzz*- sin <sup>2</sup> Ψ plot. Eq. (2) tells us the deformed lattice distance *d*<sup>Ψ</sup> is proportional to *εzz*. These altogether indicate that the in-plane strain *εϕ* can be given by the following differential:

$$\varepsilon\_{\phi} = \frac{\partial (\varepsilon \boldsymbol{\mu})\_{\text{zz}}}{\partial (\sin^{2} \boldsymbol{\Psi})} = \left(\varepsilon\_{a\text{a}} \cos^{2} \phi + \varepsilon\_{a\beta} \sin 2\phi + \varepsilon\_{\beta\beta} \sin^{2} \phi\right) \propto \frac{\partial d\boldsymbol{\mu}}{\partial (\sin^{2} \boldsymbol{\Psi})}.\tag{4}$$

Since the in-plane strain *εϕ* is connected with the in-plane stress *σϕ* with an elastic constant, Eq. (4) indicates that *σϕ* can be evaluated as the slope of *d*<sup>Ψ</sup> vs. sin <sup>2</sup> Ψ plot. In the present context, *σϕ* is the in-plane residual stress of interest. Experimentally, we can evaluate *σϕ* by changing angle Ψ and plotting the corresponding *d*<sup>Ψ</sup> as a function of sin <sup>2</sup> Ψ. The deformed lattice distance *d*<sup>Ψ</sup> can be determined from the corresponding shift in the diffraction angle as discussed above. When the angle Ψ is varied for this plot, the slope (4) remains unchanged.

**Figure 2.** *(a) Interatomic potential energy curve. (b) Slope of potential energy curve.*

So, *d*Ψ- sin <sup>2</sup> Ψ graph is a linear plot. Since the twice of the diffraction angle 2*θ* is the quantity directly measured in this type of experiment, often 2*<sup>θ</sup>* � sin <sup>2</sup> Ψ graph is used to evaluate *σϕ*. The constant of proportionality *K* is called the stress constant:

$$
\sigma\_{\phi} = \text{KM} \tag{5}
$$

where *C*ð Þ *<sup>n</sup>* is the *nth*-order coefficient of the strain energy. (The exponent used for *C* indicates the order of the strain energy. Since the elastic constant is the slope of the strain energy, i.e., its differential, *<sup>n</sup>* appears on the *<sup>n</sup>* � <sup>1</sup>*th* term in this expression. It appears on the *nth* term in the potential energy expression.) Under

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>E</sup>*<sup>0</sup> <sup>þ</sup> *<sup>C</sup>*ð Þ<sup>3</sup> *<sup>ε</sup> ρ*

> > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>0</sup> are the acoustic velocity in a residually stressed specimen and an

*C*ð Þ<sup>3</sup> *εres E*0

≃1 þ 1 2

*rel* � <sup>1</sup> � � (9)

*C*ð Þ<sup>3</sup> *εres E*0

1 þ

(7)

(8)

this condition, the acoustic velocity can be expressed as follows:

*Opto-Acoustic Technique for Residual Stress Analysis DOI: http://dx.doi.org/10.5772/intechopen.90299*

*vac* <sup>¼</sup>

ity can be expressed with the residual strain *εres* and the TOEC as follows:

*<sup>ε</sup>res* <sup>¼</sup> <sup>2</sup>*E*<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>E</sup>*<sup>0</sup> <sup>þ</sup> *<sup>C</sup>*ð Þ<sup>3</sup> *<sup>ε</sup>res E*0

s

use of the nonlinear elastic modulus expression (6).

devices used for the acoustoelasticity measurement.

*(a) Typical contact acoustic transducer arrangement. (b) Working principle of SAM.*

*vac rel* <sup>¼</sup> *vac res vac* 0 ¼

*res* and *vac*

solving Eq. (8) for *εres* as.

Here *vac*

**Figure 3.**

**13**

s

Here *ρ* is the density, and up to the third order of the higher-order terms in Eq. (6) is considered. The coefficient *C*ð Þ<sup>3</sup> is called the third-order elastic constant (TOEC). When a residual stress causes the nonlinearity, the relative acoustic veloc-

¼

unstressed specimen of the same material, respectively. By measuring these velocities and knowing the value of the TOEC, we can evaluate the residual strain by

*<sup>C</sup>*ð Þ<sup>3</sup> *<sup>v</sup>ac*

Once *εres* is found, the corresponding residual stress can be evaluated with the

**2.3 Contact acoustic transducer and scanning acoustic microscope (SAM)**

A contact acoustic transducer and scanning acoustic microscope are typical

**Figure 3a** illustrates a typical contact acoustic transducer arrangement. The transducer placed on the specimen surface through a coupling medium (typically distilled water) sends a pulsed longitudinal or shear acoustic wave. The signal goes

s

Here *σϕ* is the residual stress, *<sup>M</sup>* is the slope of the 2*<sup>θ</sup>* � sin <sup>2</sup> Ψ plot. *K* depends on the wavelength (i.e., the X-ray source line) and the lattice plane used for diffraction. As an example, for aluminum alloy 5083 with the use of Cr-K*α* line for the X-ray source and the aluminum's [4 2 2] lattice plane for diffraction, *K* = �168.80 MPa/°.

#### **2.2 Acoustoelasticity**

Acoustoelasticity [14–17] evaluates residual stresses based on a change in the acoustic velocity from the nominal value. Residual stresses cause the strain so large that the elastic coefficient is altered from the nominal value. **Figure 2** illustrates the situation schematically. The strain energy curve is steeper on the short-range side of the equilibrium position (where the strain is null) than the long-range side. Consequently, the region of tensile residual stress makes the acoustic velocity lower than the nominal value, and the region of compressive residual stress makes the acoustic velocity higher. Acoustic velocity is proportional to the ratio of the elastic modulus to the density. Thus, through measurement of acoustic velocity at each point of the specimen and scanning through the entire specimen, it is possible to map out the residual stress distribution. Typically, a contact acoustic transducer is used for acoustic velocity measurement.

For quantitative analyses, the lowest order of the nonlinear terms in the elastic modulus is used. As **Figure 2** indicates, the strain energy curve is quadratic around the equilibrium (the bottom of the well). Being the first-order spatial derivative of the energy, the stress is proportional to the strain; hence, the elastic coefficient (the stress divided by the strain) is a constant. When a residual stress shifts the strain from the equilibrium, the strain energy curve is no more quadratic at that point. Hence, the elastic coefficient *E* becomes a function of strain and can be expressed as a polynomial expansion around the nominal value (*E*0):

$$E = E\_0 + C^{(3)}\varepsilon + \frac{1}{2}C^{(4)}\varepsilon^2 + \dots + \frac{1}{(n-2)!}C^{(n)}\varepsilon^{(n-2)} + \dotsb \tag{6}$$

#### *Opto-Acoustic Technique for Residual Stress Analysis DOI: http://dx.doi.org/10.5772/intechopen.90299*

where *C*ð Þ *<sup>n</sup>* is the *nth*-order coefficient of the strain energy. (The exponent used for *C* indicates the order of the strain energy. Since the elastic constant is the slope of the strain energy, i.e., its differential, *<sup>n</sup>* appears on the *<sup>n</sup>* � <sup>1</sup>*th* term in this expression. It appears on the *nth* term in the potential energy expression.) Under this condition, the acoustic velocity can be expressed as follows:

$$
\nu^{ac} = \sqrt{\frac{E\_0 + C^{(3)}\varepsilon}{\rho}}\tag{7}
$$

Here *ρ* is the density, and up to the third order of the higher-order terms in Eq. (6) is considered. The coefficient *C*ð Þ<sup>3</sup> is called the third-order elastic constant (TOEC). When a residual stress causes the nonlinearity, the relative acoustic velocity can be expressed with the residual strain *εres* and the TOEC as follows:

$$\boldsymbol{\upsilon}\_{\rm rel}^{\rm ac} = \frac{\boldsymbol{\upsilon}\_{\rm res}^{\rm ac}}{\boldsymbol{\upsilon}\_{0}^{\rm ac}} = \sqrt{\frac{E\_0 + C^{(3)}\boldsymbol{\varepsilon}\_{\rm res}}{E\_0}} = \sqrt{\mathbf{1} + \frac{C^{(3)}\boldsymbol{\varepsilon}\_{\rm res}}{E\_0}} \simeq \mathbf{1} + \frac{\mathbf{1}}{2} \frac{C^{(3)}\boldsymbol{\varepsilon}\_{\rm res}}{E\_0} \tag{8}$$

Here *vac res* and *vac* <sup>0</sup> are the acoustic velocity in a residually stressed specimen and an unstressed specimen of the same material, respectively. By measuring these velocities and knowing the value of the TOEC, we can evaluate the residual strain by solving Eq. (8) for *εres* as.

$$
\varepsilon\_{res} = \frac{2E\_0}{C^{(3)}} \left( v\_{rel}^{ac} - \mathbf{1} \right) \tag{9}
$$

Once *εres* is found, the corresponding residual stress can be evaluated with the use of the nonlinear elastic modulus expression (6).

#### **2.3 Contact acoustic transducer and scanning acoustic microscope (SAM)**

A contact acoustic transducer and scanning acoustic microscope are typical devices used for the acoustoelasticity measurement.

**Figure 3a** illustrates a typical contact acoustic transducer arrangement. The transducer placed on the specimen surface through a coupling medium (typically distilled water) sends a pulsed longitudinal or shear acoustic wave. The signal goes

**Figure 3.** *(a) Typical contact acoustic transducer arrangement. (b) Working principle of SAM.*

So, *d*Ψ- sin <sup>2</sup>

**Figure 2.**

*K* = �168.80 MPa/°.

**2.2 Acoustoelasticity**

acoustic velocity measurement.

**12**

a polynomial expansion around the nominal value (*E*0):

1 2

*<sup>C</sup>*ð Þ <sup>4</sup> *<sup>ε</sup>*<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup>

1 ð Þ *n* � 2 !

*<sup>C</sup>*ð Þ *<sup>n</sup> <sup>ε</sup>*ð Þ *<sup>n</sup>*�<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> (6)

*<sup>E</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>0</sup> <sup>þ</sup> *<sup>C</sup>*ð Þ<sup>3</sup> *<sup>ε</sup>* <sup>þ</sup>

Ψ graph is a linear plot. Since the twice of the diffraction angle 2*θ* is the

*σϕ* ¼ *KM* (5)

Ψ graph is

Ψ plot. *K* depends on

quantity directly measured in this type of experiment, often 2*<sup>θ</sup>* � sin <sup>2</sup>

*(a) Interatomic potential energy curve. (b) Slope of potential energy curve.*

*New Challenges in Residual Stress Measurements and Evaluation*

Here *σϕ* is the residual stress, *<sup>M</sup>* is the slope of the 2*<sup>θ</sup>* � sin <sup>2</sup>

used to evaluate *σϕ*. The constant of proportionality *K* is called the stress constant:

the wavelength (i.e., the X-ray source line) and the lattice plane used for diffraction. As an example, for aluminum alloy 5083 with the use of Cr-K*α* line for the X-ray source and the aluminum's [4 2 2] lattice plane for diffraction,

Acoustoelasticity [14–17] evaluates residual stresses based on a change in the acoustic velocity from the nominal value. Residual stresses cause the strain so large that the elastic coefficient is altered from the nominal value. **Figure 2** illustrates the situation schematically. The strain energy curve is steeper on the short-range side of the equilibrium position (where the strain is null) than the long-range side. Consequently, the region of tensile residual stress makes the acoustic velocity lower than the nominal value, and the region of compressive residual stress makes the acoustic velocity higher. Acoustic velocity is proportional to the ratio of the elastic modulus to the density. Thus, through measurement of acoustic velocity at each point of the specimen and scanning through the entire specimen, it is possible to map out the residual stress distribution. Typically, a contact acoustic transducer is used for

For quantitative analyses, the lowest order of the nonlinear terms in the elastic modulus is used. As **Figure 2** indicates, the strain energy curve is quadratic around the equilibrium (the bottom of the well). Being the first-order spatial derivative of the energy, the stress is proportional to the strain; hence, the elastic coefficient (the stress divided by the strain) is a constant. When a residual stress shifts the strain from the equilibrium, the strain energy curve is no more quadratic at that point. Hence, the elastic coefficient *E* becomes a function of strain and can be expressed as through the specimen, is reflected at the bottom surface, and returns to the transducer. The received signal shows two peaks as the insert in **Figure 3a** illustrates. The first peak represents the input signal, and the second represents the returning signal. From the time of flight, the acoustic velocity inside the specimen can be evaluated for each polarization (longitudinal or shear polarization) of the acoustic signal. Since the acoustic velocity is proportional to the square root of the elastic constant for the given polarization, the elastic constant can be estimated.

**Figure 3b** illustrates a typical SAM setup. Details of its operation principle can be found in a number of references [22, 23]. In short, it works as follows. The acoustic lens sends the incident acoustic wave to the specimen at an angle higher than the critical angle. Consider the two acoustic paths labeled #1 and #2 in the figure. The former represents the acoustic wave incident to the specimen surface and specularly reflected off the surface. The latter represents the acoustic reradiation at the liquid–solid interface due to the surface acoustic wave generated by the incident wave. These two acoustic waves interfere with each other. The acoustic lens is then moved toward the specimen. As this happens, the voltage signal from the transducer (not shown in the figure) placed on top of the lens undergoes a series of crests and troughs (as the path difference goes through constructive and destructive interference). This oscillatory voltage pattern is called the V(z) curve [22]. The insert in **Figure 3b** is a sample V(z) curve. Since the frequency is fixed at the acoustic source, the acoustic path length (the path length in the unit of the wavelength) over AB depends on the phase velocity of the surface wave. Thus, the interval of these peaks (Δ*z*) is related to the velocity of the surface acoustic wave relative to the acoustic velocity in the coupling water as follows.

$$V\_s = \frac{V\_w}{\sqrt{1 - \left(1 - \frac{V\_w}{2\Delta x} \cdot f\right)^2}}\tag{10}$$

are formed in the image plane of a digital camera that captures images of the specimen. When the specimen is deformed by the tensile machine, the optical path length of one interferometric path increases and the other decreases. This changes the phase of each of all the speckles formed by the respective interfering beams. The digital camera takes images as the tensile machine keeps applying the load. The image captured at a certain time step is subtracted from the one captured at a different time step. The subtracted image exhibits an interferometric fringe pattern because the speckles undergo the phase change corresponding to the time difference between the two time steps. In those regions on the specimen where the speckles undergo a phase change corresponding to an integer multiple of 2*π*, the intensity of the interferometric images taken at the first and second time steps is the

*Opto-Acoustic Technique for Residual Stress Analysis DOI: http://dx.doi.org/10.5772/intechopen.90299*

same. Consequently, dark fringes are formed, as illustrated by the inserts in

In this arrangement the tensile machine applies a tensile load to the specimen so that the interference fringe patterns can be formed at least in three time steps. The applied load is kept as small as possible so that it does not relax the residual stress on the specimen. Since each of the fringe patterns contains the contours of displacement that the specimen undergoes in the duration between the first and second images are captured, the physical information contained by the fringe pattern represents the velocity. So, by subtracting two fringe patterns obtained by subtraction consecutively, the resultant frame contains acceleration of the points on the specimen surface. With the algorithm described below, it is possible to diagnose the status of residual stress through analysis of these frames containing the acceleration

**Figure 4b** illustrates the principle of operation. Consider that a certain part of the specimen has a compressive residual stress. As the top right part of this figure illustrates, the situation can be modeled by a mass connected to a compressed spring. If an external agent applies a tensile load (the load opposite to the compression), the mass returns to the equilibrium (represented by a dashed line in the figure) with acceleration in the same direction as the applied load. If the residual stress is tensile, the same external load displaces the mass away from the equilibrium. Hence, in this case, the acceleration of the mass is opposite to the applied load. This mechanism can be summarized as follows. "If the residual stress and applied load are of the same type (tensile or compression), the acceleration is opposite to the applied load. Otherwise, the acceleration is in the same direction as the applied load." The situation is the same when the applied load is compressive, as illustrated

By applying this algorithm to diagnose the type of residual stress at a given point

representing acceleration, it is possible to map out the type of residual stress on all

It is also possible to estimate the elastic modulus at all points. As mentioned above, the fringe pattern resulting from subtraction of the interferometric image taken at one time step from another time step represents the displacement occurring in the time difference between the two time steps. By evaluating the displacement at all coordinate points (e.g., via interpolation between dark fringes) and dividing them by the pixel interval, it is possible to map out strain as a full-field two-dimensional data. By assuming that all the points on the specimen are under equilibrium when the tensile machine applies the load, the stress can be evaluated by dividing the applied load by the cross-sectional area of the specimen. This procedure yields a map of relative elastic modulus over the specimen. If the thirdorder elastic constant of the material is known, the elastic modulus can be calibrated by performing an acoustic velocity measurement at several points of the specimen.

and combining it with the abovementioned formation of fringe pattern

**Figure 4a-2, a-3**.

information.

in the lower part of **Figure 4b**.

points of the specimen.

**15**

where *Vs* is the surface acoustic wave velocity, *Vw* is the acoustic velocity in water, and *f* is the acoustic frequency. The elastic modulus of the near-surface region of the specimen can be characterized from *Vs*:

#### **2.4 Electronic speckle pattern interferometry (ESPI)**

**Figure 4a** illustrates a typical electronic speckle pattern interferometry (ESPI) setup. A laser beam is split into two beams that constitute interferometric paths. The two beams are expanded and recombined on the surface of the specimen attached to the tensile machine. As superposition of coherent light beams, speckles

#### **Figure 4.**

*(a) Typical ESPI setup. (a-1) Optical configuration. (a-2) Sample dark fringes. (a-3) Another sample dark fringes. (b) Principle of operation.*

#### *Opto-Acoustic Technique for Residual Stress Analysis DOI: http://dx.doi.org/10.5772/intechopen.90299*

through the specimen, is reflected at the bottom surface, and returns to the transducer. The received signal shows two peaks as the insert in **Figure 3a** illustrates. The first peak represents the input signal, and the second represents the returning signal. From the time of flight, the acoustic velocity inside the specimen can be evaluated for each polarization (longitudinal or shear polarization) of the acoustic signal. Since the acoustic velocity is proportional to the square root of the elastic constant for the given polarization, the elastic constant can be estimated.

*New Challenges in Residual Stress Measurements and Evaluation*

**Figure 3b** illustrates a typical SAM setup. Details of its operation principle can be found in a number of references [22, 23]. In short, it works as follows. The acoustic lens sends the incident acoustic wave to the specimen at an angle higher than the critical angle. Consider the two acoustic paths labeled #1 and #2 in the figure. The former represents the acoustic wave incident to the specimen surface and specularly reflected off the surface. The latter represents the acoustic reradiation at the liquid–solid interface due to the surface acoustic wave generated by the incident wave. These two acoustic waves interfere with each other. The acoustic lens is then moved toward the specimen. As this happens, the voltage signal from the transducer (not shown in the figure) placed on top of the lens undergoes a series

of crests and troughs (as the path difference goes through constructive and destructive interference). This oscillatory voltage pattern is called the V(z) curve [22]. The insert in **Figure 3b** is a sample V(z) curve. Since the frequency is fixed at the acoustic source, the acoustic path length (the path length in the unit of the wavelength) over AB depends on the phase velocity of the surface wave. Thus, the interval of these peaks (Δ*z*) is related to the velocity of the surface acoustic wave

> *Vs* <sup>¼</sup> *Vw* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>1</sup> � *Vw*

where *Vs* is the surface acoustic wave velocity, *Vw* is the acoustic velocity in water, and *f* is the acoustic frequency. The elastic modulus of the near-surface

**Figure 4a** illustrates a typical electronic speckle pattern interferometry (ESPI) setup. A laser beam is split into two beams that constitute interferometric paths. The two beams are expanded and recombined on the surface of the specimen attached to the tensile machine. As superposition of coherent light beams, speckles

*(a) Typical ESPI setup. (a-1) Optical configuration. (a-2) Sample dark fringes. (a-3) Another sample dark*

<sup>2</sup>Δ*<sup>z</sup>* � *<sup>f</sup>* � �<sup>2</sup> <sup>q</sup> (10)

relative to the acoustic velocity in the coupling water as follows.

region of the specimen can be characterized from *Vs*:

**Figure 4.**

**14**

*fringes. (b) Principle of operation.*

**2.4 Electronic speckle pattern interferometry (ESPI)**

are formed in the image plane of a digital camera that captures images of the specimen. When the specimen is deformed by the tensile machine, the optical path length of one interferometric path increases and the other decreases. This changes the phase of each of all the speckles formed by the respective interfering beams. The digital camera takes images as the tensile machine keeps applying the load. The image captured at a certain time step is subtracted from the one captured at a different time step. The subtracted image exhibits an interferometric fringe pattern because the speckles undergo the phase change corresponding to the time difference between the two time steps. In those regions on the specimen where the speckles undergo a phase change corresponding to an integer multiple of 2*π*, the intensity of the interferometric images taken at the first and second time steps is the same. Consequently, dark fringes are formed, as illustrated by the inserts in **Figure 4a-2, a-3**.

In this arrangement the tensile machine applies a tensile load to the specimen so that the interference fringe patterns can be formed at least in three time steps. The applied load is kept as small as possible so that it does not relax the residual stress on the specimen. Since each of the fringe patterns contains the contours of displacement that the specimen undergoes in the duration between the first and second images are captured, the physical information contained by the fringe pattern represents the velocity. So, by subtracting two fringe patterns obtained by subtraction consecutively, the resultant frame contains acceleration of the points on the specimen surface. With the algorithm described below, it is possible to diagnose the status of residual stress through analysis of these frames containing the acceleration information.

**Figure 4b** illustrates the principle of operation. Consider that a certain part of the specimen has a compressive residual stress. As the top right part of this figure illustrates, the situation can be modeled by a mass connected to a compressed spring. If an external agent applies a tensile load (the load opposite to the compression), the mass returns to the equilibrium (represented by a dashed line in the figure) with acceleration in the same direction as the applied load. If the residual stress is tensile, the same external load displaces the mass away from the equilibrium. Hence, in this case, the acceleration of the mass is opposite to the applied load. This mechanism can be summarized as follows. "If the residual stress and applied load are of the same type (tensile or compression), the acceleration is opposite to the applied load. Otherwise, the acceleration is in the same direction as the applied load." The situation is the same when the applied load is compressive, as illustrated in the lower part of **Figure 4b**.

By applying this algorithm to diagnose the type of residual stress at a given point and combining it with the abovementioned formation of fringe pattern representing acceleration, it is possible to map out the type of residual stress on all points of the specimen.

It is also possible to estimate the elastic modulus at all points. As mentioned above, the fringe pattern resulting from subtraction of the interferometric image taken at one time step from another time step represents the displacement occurring in the time difference between the two time steps. By evaluating the displacement at all coordinate points (e.g., via interpolation between dark fringes) and dividing them by the pixel interval, it is possible to map out strain as a full-field two-dimensional data. By assuming that all the points on the specimen are under equilibrium when the tensile machine applies the load, the stress can be evaluated by dividing the applied load by the cross-sectional area of the specimen. This procedure yields a map of relative elastic modulus over the specimen. If the thirdorder elastic constant of the material is known, the elastic modulus can be calibrated by performing an acoustic velocity measurement at several points of the specimen.

This yields a map of absolute elastic modulus, and from this data the residual strain can be evaluated with Eq. (6).

Once the residual strain is found, the residual stress can be estimated as follows:

$$
\sigma\_{\rm res} = \left( E\_0 + C^{(3)} \varepsilon\_{\rm res} \right) \varepsilon\_{\rm res} \tag{11}
$$

Δ*C*<sup>12</sup> Δ*C*<sup>22</sup> Δ*C*<sup>32</sup> Δ*C*<sup>42</sup> Δ*C*<sup>52</sup> Δ*C*<sup>62</sup>

Δ*C*<sup>13</sup> Δ*C*<sup>23</sup> Δ*C*<sup>33</sup> Δ*C*<sup>43</sup> Δ*C*<sup>53</sup> Δ*C*<sup>63</sup>

Δ*C*<sup>14</sup> Δ*C*<sup>24</sup> Δ*C*<sup>34</sup> Δ*C*<sup>44</sup> Δ*C*<sup>54</sup> Δ*C*<sup>64</sup>

Δ*C*<sup>15</sup> Δ*C*<sup>25</sup> Δ*C*<sup>35</sup> Δ*C*<sup>45</sup> Δ*C*<sup>55</sup> Δ*C*<sup>65</sup>

Δ*C*<sup>16</sup> Δ*C*<sup>26</sup> Δ*C*<sup>36</sup> Δ*C*<sup>46</sup> Δ*C*<sup>56</sup> Δ*C*<sup>66</sup> 1

0

*Opto-Acoustic Technique for Residual Stress Analysis DOI: http://dx.doi.org/10.5772/intechopen.90299*

BBBBBBBBBB@

0

BBBBBBBBBB@

0

BBBBBBBBBB@

0

BBBBBBBBBB@

0

BBBBBBBB@

*C*<sup>112</sup> *C*<sup>112</sup> *C*<sup>123</sup> 000 *C*<sup>112</sup> *C*<sup>111</sup> *C*<sup>112</sup> 000 *C*<sup>123</sup> *C*<sup>112</sup> *C*<sup>112</sup> 000 000 *C*<sup>155</sup> 0 0 0000 *C*<sup>144</sup> 0 00000 *C*<sup>155</sup>

*C*<sup>112</sup> *C*<sup>123</sup> *C*<sup>112</sup> 000 *C*<sup>123</sup> *C*<sup>112</sup> *C*<sup>112</sup> 000 *C*<sup>112</sup> *C*<sup>112</sup> *C*<sup>111</sup> 000 000 *C*<sup>155</sup> 0 0 0000 *C*<sup>144</sup> 0 00000 *C*<sup>155</sup>

000 *C*<sup>144</sup> 0 0 000 *C*<sup>155</sup> 0 0 000 *C*<sup>155</sup> 0 0 *C*<sup>144</sup> *C*<sup>155</sup> *C*<sup>155</sup> 000 0000 0 *C*<sup>456</sup> 0000 *C*<sup>456</sup> 0

0000 *C*<sup>155</sup> 0 0000 *C*<sup>144</sup> 0 0000 *C*<sup>155</sup> 0 00000 *C*<sup>456</sup> *C*<sup>155</sup> *C*<sup>144</sup> *C*<sup>155</sup> 000 000 *C*<sup>456</sup> 0 0

000 0 0 *C*<sup>144</sup> 000 0 0 *C*<sup>155</sup> 000 0 0 *C*<sup>155</sup> 000 0 *C*<sup>456</sup> 0 000 *C*<sup>456</sup> 0 0 *C*<sup>155</sup> *C*<sup>155</sup> *C*<sup>144</sup> 000

The isotropic TOE tensor is described by three linearly independent elements [24]. Choosing *C*123, *C*144, and *C*<sup>456</sup> to be the three independent elements, we can use the following relations for complete expression of the third-order

1

0

*ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 1

CCCCCCCCCCA

(14)

(15)

(16)

(17)

(18)

CCCCCCCCCCA

1

0

*ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 1

CCCCCCCCCCA

CCCCCCCCCCA

1

0

*ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 1

CCCCCCCCCCA

CCCCCCCCCCA

1

0

*ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 1

CCCCCCCCCCA

CCCCCCCCCCA

1

0

BBBBBBBB@

*ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 1

CCCCCCCCA

CCCCCCCCA

*C*<sup>111</sup> ¼ *C*<sup>123</sup> þ 6*C*<sup>144</sup> þ 8*C*<sup>456</sup> (19)

*C*<sup>112</sup> ¼ *C*<sup>123</sup> þ 2*C*<sup>144</sup> (20) *C*<sup>155</sup> ¼ *C*<sup>144</sup> þ 2*C*<sup>456</sup> (21)

BBBBBBBBBB@

BBBBBBBBBB@

BBBBBBBBBB@

BBBBBBBBBB@

CCCCCCCCCCA ¼

1

CCCCCCCCCCA ¼

1

CCCCCCCCCCA ¼

1

CCCCCCCCCCA ¼

1

CCCCCCCCA ¼

0

BBBBBBBBBB@

0

BBBBBBBBBB@

0

BBBBBBBBBB@

0

BBBBBBBBBB@

0

BBBBBBBB@

coefficient:

**17**

#### **2.5 Finite element modeling (FEM)**

Accurate numerical modeling of residual stress is extremely difficult. However, finite element modeling is useful because it can provide us with some insight and qualitative analysis when combined with experiment. This section discusses frameworks of such modeling in conjunction with the TOEC algorithm. For simplicity, the discussion here is limited to an isotropic case [24].

With the third-order term included, the constitutive relation can be expressed as follows:

$$
\begin{aligned}
\begin{pmatrix}\sigma\_{1}\\\sigma\_{2}\\\sigma\_{3}\\\sigma\_{4}\\\sigma\_{5}\\\sigma\_{6}\\\sigma\_{6}\\\sigma\_{7}\end{pmatrix} &= \begin{pmatrix}\begin{pmatrix}C\_{11} & C\_{12} & C\_{13} & C\_{14} & C\_{15} & C\_{16} \\\\\begin{pmatrix}C\_{21} & C\_{22} & C\_{23} & C\_{24} & C\_{25} & C\_{26}\\\\\vdots &C\_{31} & C\_{32} & C\_{33} & C\_{34} & C\_{35} & C\_{36}\\\\\ CA\_{41} & C\_{42} & C\_{43} & C\_{44} & C\_{45} & C\_{46}\\\\\vdots & C\_{51} & C\_{52} & C\_{53} & C\_{54} & C\_{55} & C\_{56}\\\\\end{pmatrix} \\\\ &+ \begin{pmatrix}\Delta C\_{11} & \Delta C\_{12} & \Delta C\_{13} & \Delta C\_{14} & \Delta C\_{15} & \Delta C\_{16}\\\\\Delta C\_{21} & \Delta C\_{22} & \Delta C\_{23} & \Delta C\_{24} & \Delta C\_{25} & \Delta C\_{26}\\\\\Delta C\_{31} & \Delta C\_{32} & \Delta C\_{33} & \Delta C\_{34} & \Delta C\_{35} & \Delta C\_{36}\\\\\Delta C\_{41} & \Delta C\_{42} & \Delta C\_{43} & \Delta C\_{44} & \Delta C\_{45} & \Delta C\_{46}\\\\\hline\end{pmatrix} \begin{pmatrix}e\_{1} \\\\e\_{2} \\\\e\_{3} \\\\\varepsilon\_{4} \\\\\varepsilon\_{5} \\\\\varepsilon\_{6} \\\end{pmatrix} \end{aligned} \tag{12}$$

Here the first term inside the bracket on the right-hand side is the second-order elastic coefficient, and the second term is the third-order elastic coefficient. In the case of an isotropic material, the third-order elastic coefficient can be expressed as follows [25]:

$$
\begin{pmatrix}
\Delta C\_{11} \\
\Delta C\_{21} \\
\Delta C\_{31} \\
\Delta C\_{31} \\
\Delta C\_{41} \\
\Delta C\_{51} \\
\Delta C\_{61}
\end{pmatrix} = \begin{pmatrix}
\mathbf{C}\_{111} & \mathbf{C}\_{112} & \mathbf{C}\_{112} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{C}\_{112} & \mathbf{C}\_{112} & \mathbf{C}\_{123} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{C}\_{112} & \mathbf{C}\_{123} & \mathbf{C}\_{112} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{144} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{155} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{155}
\end{pmatrix} \begin{pmatrix}
\varepsilon\_{1} \\
\varepsilon\_{2} \\
\varepsilon\_{3} \\
\varepsilon\_{4} \\
\varepsilon\_{5} \\
\varepsilon\_{6}
\end{pmatrix} \tag{13}
$$

*Opto-Acoustic Technique for Residual Stress Analysis DOI: http://dx.doi.org/10.5772/intechopen.90299*

This yields a map of absolute elastic modulus, and from this data the residual strain

*<sup>σ</sup>res* <sup>¼</sup> *<sup>E</sup>*<sup>0</sup> <sup>þ</sup> *<sup>C</sup>*ð Þ<sup>3</sup> *<sup>ε</sup>res* � �

Once the residual strain is found, the residual stress can be estimated as follows:

Accurate numerical modeling of residual stress is extremely difficult. However, finite element modeling is useful because it can provide us with some insight and qualitative analysis when combined with experiment. This section discusses frameworks of such modeling in conjunction with the TOEC algorithm. For simplicity,

With the third-order term included, the constitutive relation can be expressed as

1

CCCCCCCCCCCCCCCA

1

9

0

*ε*1

1

(12)

*ε*2

*ε*3

*ε*4

CCCCCCCCCCCCCCCA

BBBBBBBBBBBBBBB@

*ε*5

*ε*6

>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>;

CCCCCCCCCCCCCCCA

1

0

*ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6

1

CCCCCCCCCCCA

(13)

CCCCCCCCCCCA

BBBBBBBBBBB@

*εres* (11)

can be evaluated with Eq. (6).

follows:

0

*σ*1

1

CCCCCCCCCCCCCCCA ¼

*σ*2

*σ*3

*σ*4

BBBBBBBBBBBBBBB@

*σ*5

*σ*6

follows [25]:

Δ*C*<sup>11</sup> Δ*C*<sup>21</sup> Δ*C*<sup>31</sup> Δ*C*<sup>41</sup> Δ*C*<sup>51</sup> Δ*C*<sup>61</sup> 1

0

BBBBBBBBBBB@

CCCCCCCCCCCA ¼

0

BBBBBBBBBBB@

**16**

**2.5 Finite element modeling (FEM)**

0

8

>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>:

þ

BBBBBBBBBBBBBBB@

BBBBBBBBBBBBBBB@

0

the discussion here is limited to an isotropic case [24].

*New Challenges in Residual Stress Measurements and Evaluation*

*C*<sup>11</sup> *C*<sup>12</sup> *C*<sup>13</sup> *C*<sup>14</sup> *C*<sup>15</sup> *C*<sup>16</sup>

*C*<sup>21</sup> *C*<sup>22</sup> *C*<sup>23</sup> *C*<sup>24</sup> *C*<sup>25</sup> *C*<sup>26</sup>

*C*<sup>31</sup> *C*<sup>32</sup> *C*<sup>33</sup> *C*<sup>34</sup> *C*<sup>35</sup> *C*<sup>36</sup>

*C*<sup>41</sup> *C*<sup>42</sup> *C*<sup>43</sup> *C*<sup>44</sup> *C*<sup>45</sup> *C*<sup>46</sup>

*C*<sup>51</sup> *C*<sup>52</sup> *C*<sup>53</sup> *C*<sup>54</sup> *C*<sup>55</sup> *C*<sup>56</sup>

*C*<sup>61</sup> *C*<sup>62</sup> *C*<sup>63</sup> *C*<sup>64</sup> *C*<sup>65</sup> *C*<sup>66</sup>

Δ*C*<sup>11</sup> Δ*C*<sup>12</sup> Δ*C*<sup>13</sup> Δ*C*<sup>14</sup> Δ*C*<sup>15</sup> Δ*C*<sup>16</sup>

Δ*C*<sup>21</sup> Δ*C*<sup>22</sup> Δ*C*<sup>23</sup> Δ*C*<sup>24</sup> Δ*C*<sup>25</sup> Δ*C*<sup>26</sup>

Δ*C*<sup>31</sup> Δ*C*<sup>32</sup> Δ*C*<sup>33</sup> Δ*C*<sup>34</sup> Δ*C*<sup>35</sup> Δ*C*<sup>36</sup>

Δ*C*<sup>41</sup> Δ*C*<sup>42</sup> Δ*C*<sup>43</sup> Δ*C*<sup>44</sup> Δ*C*<sup>45</sup> Δ*C*<sup>46</sup>

Δ*C*<sup>51</sup> Δ*C*<sup>52</sup> Δ*C*<sup>53</sup> Δ*C*<sup>54</sup> Δ*C*<sup>55</sup> Δ*C*<sup>56</sup>

Δ*C*<sup>61</sup> Δ*C*<sup>62</sup> Δ*C*<sup>63</sup> Δ*C*<sup>64</sup> Δ*C*<sup>65</sup> Δ*C*<sup>16</sup>

*C*<sup>111</sup> *C*<sup>112</sup> *C*<sup>112</sup> 000 *C*<sup>112</sup> *C*<sup>112</sup> *C*<sup>123</sup> 000 *C*<sup>112</sup> *C*<sup>123</sup> *C*<sup>112</sup> 000 000 *C*<sup>144</sup> 0 0 0000 *C*<sup>155</sup> 0 00000 *C*<sup>155</sup>

Here the first term inside the bracket on the right-hand side is the second-order elastic coefficient, and the second term is the third-order elastic coefficient. In the case of an isotropic material, the third-order elastic coefficient can be expressed as

Δ*C*<sup>12</sup> Δ*C*<sup>22</sup> Δ*C*<sup>32</sup> Δ*C*<sup>42</sup> Δ*C*<sup>52</sup> Δ*C*<sup>62</sup> 0 BBBBBBBBBB@ 1 CCCCCCCCCCA ¼ *C*<sup>112</sup> *C*<sup>112</sup> *C*<sup>123</sup> 000 *C*<sup>112</sup> *C*<sup>111</sup> *C*<sup>112</sup> 000 *C*<sup>123</sup> *C*<sup>112</sup> *C*<sup>112</sup> 000 000 *C*<sup>155</sup> 0 0 0000 *C*<sup>144</sup> 0 00000 *C*<sup>155</sup> 0 BBBBBBBBBB@ 1 CCCCCCCCCCA *ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 0 BBBBBBBBBB@ 1 CCCCCCCCCCA (14) Δ*C*<sup>13</sup> Δ*C*<sup>23</sup> Δ*C*<sup>33</sup> Δ*C*<sup>43</sup> Δ*C*<sup>53</sup> Δ*C*<sup>63</sup> 0 BBBBBBBBBB@ 1 CCCCCCCCCCA ¼ *C*<sup>112</sup> *C*<sup>123</sup> *C*<sup>112</sup> 000 *C*<sup>123</sup> *C*<sup>112</sup> *C*<sup>112</sup> 000 *C*<sup>112</sup> *C*<sup>112</sup> *C*<sup>111</sup> 000 000 *C*<sup>155</sup> 0 0 0000 *C*<sup>144</sup> 0 00000 *C*<sup>155</sup> 0 BBBBBBBBBB@ 1 CCCCCCCCCCA *ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 0 BBBBBBBBBB@ 1 CCCCCCCCCCA (15) Δ*C*<sup>14</sup> Δ*C*<sup>24</sup> Δ*C*<sup>34</sup> Δ*C*<sup>44</sup> Δ*C*<sup>54</sup> Δ*C*<sup>64</sup> 0 BBBBBBBBBB@ 1 CCCCCCCCCCA ¼ 000 *C*<sup>144</sup> 0 0 000 *C*<sup>155</sup> 0 0 000 *C*<sup>155</sup> 0 0 *C*<sup>144</sup> *C*<sup>155</sup> *C*<sup>155</sup> 000 0000 0 *C*<sup>456</sup> 0000 *C*<sup>456</sup> 0 0 BBBBBBBBBB@ 1 CCCCCCCCCCA *ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 0 BBBBBBBBBB@ 1 CCCCCCCCCCA (16) Δ*C*<sup>15</sup> Δ*C*<sup>25</sup> Δ*C*<sup>35</sup> Δ*C*<sup>45</sup> Δ*C*<sup>55</sup> Δ*C*<sup>65</sup> 0 BBBBBBBBBB@ 1 CCCCCCCCCCA ¼ 0000 *C*<sup>155</sup> 0 0000 *C*<sup>144</sup> 0 0000 *C*<sup>155</sup> 0 00000 *C*<sup>456</sup> *C*<sup>155</sup> *C*<sup>144</sup> *C*<sup>155</sup> 000 000 *C*<sup>456</sup> 0 0 0 BBBBBBBBBB@ 1 CCCCCCCCCCA *ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 0 BBBBBBBBBB@ 1 CCCCCCCCCCA (17) Δ*C*<sup>16</sup> Δ*C*<sup>26</sup> Δ*C*<sup>36</sup> Δ*C*<sup>46</sup> Δ*C*<sup>56</sup> Δ*C*<sup>66</sup> 0 BBBBBBBB@ 1 CCCCCCCCA ¼ 000 0 0 *C*<sup>144</sup> 000 0 0 *C*<sup>155</sup> 000 0 0 *C*<sup>155</sup> 000 0 *C*<sup>456</sup> 0 000 *C*<sup>456</sup> 0 0 *C*<sup>155</sup> *C*<sup>155</sup> *C*<sup>144</sup> 000 0 BBBBBBBB@ 1 CCCCCCCCA *ε*1 *ε*2 *ε*3 *ε*4 *ε*5 *ε*6 0 BBBBBBBB@ 1 CCCCCCCCA (18)

The isotropic TOE tensor is described by three linearly independent elements [24]. Choosing *C*123, *C*144, and *C*<sup>456</sup> to be the three independent elements, we can use the following relations for complete expression of the third-order coefficient:

$$\mathbf{C\_{111}} = \mathbf{C\_{123}} + \mathbf{6C\_{144}} + \mathbf{8C\_{456}} \tag{19}$$

$$\mathbf{C\_{112}} = \mathbf{C\_{123}} + \mathbf{2C\_{144}} \tag{20}$$

$$\mathbf{C\_{1\ $5}} = \mathbf{C\_{144}} + \mathbf{2C\_{4\$ 6}} \tag{21}$$

By substituting Eqs. (19)–(21) into Eqs. (13)–(18), we can find the third-order effect for each stress tensor component.

To compare with acoustoelastic measurement, it is necessary to express the effect of the inclusion of the third-order elastic coefficient in the corresponding acoustic velocity. Assuming that the density is unaffected by the inclusion of the third-order effect, the relative acoustic velocity can be expressed as follows:

$$\frac{w\_{ij}^{(3)}}{w\_{ij}^{(2)}} = \frac{\sqrt{\left(\mathbf{c}\_{i\bar{\jmath}} + \Delta \mathbf{C}\_{i\bar{\jmath}}\right)/\rho}}{\sqrt{\mathbf{C}\_{i\bar{\jmath}}/\rho}} = \sqrt{\frac{\left(\mathbf{c}\_{i\bar{\jmath} + \Delta \mathbf{C}\_{i\bar{\jmath}}\right)}}{\mathbf{C}\_{i\bar{\jmath}}}}\tag{22}$$

Here *i*, *j* ¼ 1 … 6, and *v* ð Þ3 *ij* and *v* ð Þ2 *ij* denote the acoustic velocity of the corresponding mode with and without the third-order effect.
