**2.3 Material properties**

The Young's modulus for the specimens is calculated from Eq. 1, the calibrated experimental setup is used to generate Acoustic Emission (AE) on to the test specimen. AE velocities travelling in the material are determining from the instrument EPOCH 4 PLUS [22, 23].


E1 <sup>¼</sup> <sup>V</sup><sup>2</sup>

*Histogram representation of % amplitude of waveform at constant gain.*

*Material properties arrived from experimental setup.*

*DOI: http://dx.doi.org/10.5772/intechopen.86134*

E2 <sup>¼</sup> V2

model and loss less finite element model respectively.

**3.1 Lamb wave model for laminated composite plate**

presented in **Table 2**.

**Table 2.**

**Figure 4.**

**3. Methodology**

shown in **Figure 5**.

**155**

<sup>T</sup> ρð Þ 1 þ ν<sup>12</sup> ð Þ 1 � 2ν<sup>12</sup> 1 � ν<sup>12</sup>

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material…*

<sup>L</sup> ρð Þ 1 þ ν<sup>21</sup> ð Þ 1 � 2ν<sup>21</sup> 1 � ν<sup>21</sup>

where VL is longitudinal velocity of AE and VT is transverse velocity of AE traveling in the material respectively, ρ is material density and *υ*<sup>12</sup> is Poisson's ratio. The material properties of specimens arrived from the experimental setup is

In this work a hybrid method has been proposed for identify change in damping capacity of a material using combined finite element and Lamb wave method. The process diagram of the hybrid method for the dynamic mechanical analysis is

The group velocity (*cgn* ), modal frequency (fn) are determined from Lamb wave

Lamb waves can be of two groups, symmetric and anti-symmetric, these waves propagate independently of the other and boundary conditions of the wave equation are being satisfied by both of them for this problem. Actuating frequency relating the velocity of Lamb wave propagation has been derived in the following section. Dispersion curves of Lamb wave in a particular material, which plot the

(1)

**Table 1.** *EPOCH 4 PLUS parameters.*

**Figure 3.** *Optimal driving frequency selection for different materials.*

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material… DOI: http://dx.doi.org/10.5772/intechopen.86134*

#### **Figure 4.**

The Panametrics-NDT™ EPOCH 4 PLUS is used to generate acoustic waves and

parameters were arrived through calibration of the device using editable parameters

The parameters have been varied one by one and arrived to a conclusion of optimal driving frequency for different specimens. **Figure 3** shows the optimal driving frequency of different materials calibrated through ultrasonic pulse generator test setup. With the help fitted peak value and percentage amplitude at constant gain 55 (db) Lamb waves generation frequency is identified for different materials. **Figure 4** shows the histogram representation of % amplitude of the waveform with a bin range of 0–820 kHz of pulser [21, 22]. The most effective range of frequencies to generate Lamb waves is 140–420 kHz. The frequency range to generate Lamb waves for GFRP is 170–190 kHz where as it is 260–280 kHz for

The Young's modulus for the specimens is calculated from Eq. 1, the calibrated

Device Unit

Angle Thickness Waveform Rang

Rectification Offset

experimental setup is used to generate Acoustic Emission (AE) on to the test specimen. AE velocities travelling in the material are determining from the

> Broad band Low pass High pass By pass

the device is equipped with four channels. Optimal Lamb wave propagation

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

as shown in **Table 1**.

**2.3 Material properties**

**Editable parameters** Pulser Mode

*EPOCH 4 PLUS parameters.*

**Table 1.**

**Figure 3.**

**154**

*Optimal driving frequency selection for different materials.*

Energy Wave Type Frequency

instrument EPOCH 4 PLUS [22, 23].

CFRP and for Aluminum (Al) it is 360–380 kHz.

Receiver Gain

*Histogram representation of % amplitude of waveform at constant gain.*


#### **Table 2.** *Material properties arrived from experimental setup.*

$$\begin{aligned} \mathbf{E}\_1 &= \frac{\mathbf{V}\_\mathrm{T}^2 \rho (\mathbf{1} + \nu\_{12})(\mathbf{1} - 2\nu\_{12})}{\mathbf{1} - \nu\_{12}} \\ \mathbf{E}\_2 &= \frac{\mathbf{V}\_\mathrm{L}^2 \rho (\mathbf{1} + \nu\_{21})(\mathbf{1} - 2\nu\_{21})}{\mathbf{1} - \nu\_{21}} \end{aligned} \tag{1}$$

where VL is longitudinal velocity of AE and VT is transverse velocity of AE traveling in the material respectively, ρ is material density and *υ*<sup>12</sup> is Poisson's ratio. The material properties of specimens arrived from the experimental setup is presented in **Table 2**.

### **3. Methodology**

In this work a hybrid method has been proposed for identify change in damping capacity of a material using combined finite element and Lamb wave method. The process diagram of the hybrid method for the dynamic mechanical analysis is shown in **Figure 5**.

The group velocity (*cgn* ), modal frequency (fn) are determined from Lamb wave model and loss less finite element model respectively.

#### **3.1 Lamb wave model for laminated composite plate**

Lamb waves can be of two groups, symmetric and anti-symmetric, these waves propagate independently of the other and boundary conditions of the wave equation are being satisfied by both of them for this problem. Actuating frequency relating the velocity of Lamb wave propagation has been derived in the following section. Dispersion curves of Lamb wave in a particular material, which plot the

*Q*<sup>11</sup> ¼ *E*1*=*ð Þ 1 � *υ*12*υ*<sup>21</sup> *Q*<sup>22</sup> ¼ *E*2*=*ð Þ 1 � *υ*12*υ*<sup>21</sup> *Q*<sup>12</sup> ¼ *υ*12*E*1*=*ð Þ 1 � *υ*12*υ*<sup>21</sup>

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material…*

where *E*<sup>1</sup> Young's moduli in the longitudinal and *E*<sup>2</sup> Young's moduli in the transverse directions. The major and minor Poisson's ratios represented by *ν*<sup>12</sup> and *ν*<sup>21</sup> respectively. The relation between Poisson's ratios in Eq. (4) is given by:

> *<sup>υ</sup>*<sup>21</sup> <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> *E*1

> > *Q*<sup>0</sup> *ij* � � *k*

where plate thickness is represented by "*h*" and "*k*" *represents* each individual

<sup>11</sup> <sup>¼</sup> *<sup>m</sup>*<sup>4</sup>*Q*<sup>11</sup> <sup>þ</sup> *<sup>n</sup>*<sup>4</sup>*Q*<sup>22</sup> <sup>þ</sup> <sup>2</sup>*m*<sup>2</sup>*n*<sup>2</sup>*Q*<sup>12</sup> <sup>þ</sup> <sup>4</sup>*m*<sup>2</sup>*n*<sup>2</sup>*Q*<sup>66</sup>

<sup>22</sup> <sup>¼</sup> *<sup>n</sup>*<sup>4</sup>*Q*<sup>11</sup> <sup>þ</sup> *<sup>m</sup>*<sup>4</sup>*Q*<sup>22</sup> <sup>þ</sup> <sup>2</sup>*m*<sup>2</sup>*n*<sup>2</sup>*Q*<sup>12</sup> <sup>þ</sup> <sup>4</sup>*m*<sup>2</sup>*n*<sup>2</sup>*Q*<sup>66</sup>

<sup>12</sup> <sup>¼</sup> *<sup>m</sup>*<sup>2</sup>*n*<sup>2</sup>*Q*<sup>11</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup>*n*<sup>2</sup>*Q*<sup>22</sup> <sup>þ</sup> *<sup>m</sup>*<sup>4</sup> <sup>þ</sup> *<sup>n</sup>*<sup>4</sup> ð Þ*Q*<sup>12</sup> � <sup>4</sup>*m*<sup>2</sup>*n*<sup>2</sup>*Q*<sup>66</sup>

where *m* = cos(θ) and *n* = sin(θ), the angle θ is taken positive for counterclockwise rotation and it is considered from the primed (laminate) axes to the unprimed

<sup>11</sup> � �

<sup>22</sup> � �

<sup>12</sup> � �

0 deg ¼ *Q*<sup>11</sup> *Q*<sup>0</sup>

0 deg ¼ *Q*<sup>22</sup> *Q*<sup>0</sup>

0 deg ¼ *Q*<sup>12</sup> *Q*<sup>0</sup>

The extensional plate mode velocity is related to the in-plane stiffness of a composite [14]. *A*<sup>11</sup> and *A*<sup>22</sup> are the stiffnesses propagating in the 0° and 90° directions respectively. The relation between extensional plate mode velocity and

for 0°direction *ct* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

for 90°direction *cl* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The inplane stiffnesses *A*<sup>11</sup> and *A*<sup>22</sup> are calculated using Eqs. (4)–(8) by substituting engineering stiffnesses of the composite. The extensional plate mode velocities are substituted into Eq. (2) and it is solved numerically for phase velocity in MathematicaTH. Phase velocity (*cphase*) is the dependent variable being solved for the independent variable being iteratively supplied is the frequency-thickness product, where *ω* is the driving frequency in radians. Group velocity dispersion

*∂cphase*

*<sup>∂</sup><sup>k</sup> <sup>k</sup>* <sup>¼</sup> *cphase* <sup>1</sup> � *<sup>f</sup> cphase :*

*∂cphase ∂f*

curve, which are derived from the phase velocity curve using Eq. (11):

*cgroup* ¼ *cphase* þ

where *f* is the frequency in Hz.

**157**

*ij* for the 0° and 90° laminas are given by

90 deg ¼ *Q*<sup>22</sup>

90 deg ¼ *Q*<sup>11</sup>

90 deg ¼ *Q*<sup>12</sup>

*Aij* ¼

lamina. The transformed stiffness coefficients *Q*<sup>0</sup>

*Q*0 <sup>11</sup> � �

*Q*0 <sup>22</sup> � �

*Q*0 <sup>12</sup> � �

*Q*0

*DOI: http://dx.doi.org/10.5772/intechopen.86134*

*Q*0

*Q*0

(individual lamina) axes. *Q*<sup>0</sup>

stiffness is given by:

ð*<sup>h</sup>=*<sup>2</sup> �*h=*2

*A*<sup>11</sup> and *A*22, are in-plane stiffnesses of plate and these are obtained by integrating the *Qij* across the thickness of the plate [21]. These stiffness values are given as:

(4)

(7)

(8)

(11)

*υ*<sup>12</sup> (5)

*dz, i, j* ¼ 1*,* 2*,* (6)

*<sup>A</sup>*11*=ρ<sup>h</sup>* <sup>p</sup> (9)

*<sup>A</sup>*22*=ρ<sup>h</sup>* <sup>p</sup> (10)

*ij* are defined as

**Figure 5.** *Methodology chart for dynamic mechanical analysis by hybrid method.*

phase and group velocities versus the excitation frequency given by Dalton et al. [24]. The anti-symmetric Lamb wave solution formulated as seen in Eq. 2:

$$\frac{\tan\left(qh\right)}{\tan\left(ph\right)} = \frac{\left(k^2 + q^2\right)^2}{4k^2 qp} \tag{2}$$

where *<sup>p</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *c*2 *l* � *<sup>k</sup>*<sup>2</sup> *, q*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *c*2 *t* � *<sup>k</sup>*<sup>2</sup> *, and k* <sup>¼</sup> *<sup>ω</sup> cphase* individual laminate stress-strain relationship is given by

$$
\begin{bmatrix} \sigma\_1\\ \sigma\_2\\ \sigma\_6 \end{bmatrix} = \begin{bmatrix} Q\_{11} & Q\_{12} & \mathbf{0} \\ Q\_{12} & Q\_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & Q\_{66} \end{bmatrix} \begin{bmatrix} \varepsilon\_1\\ \varepsilon\_2\\ \gamma\_6 \end{bmatrix} \tag{3}
$$

where *σ* is normal stress, τ represent shear stress, *ε* is normal strain and *ϒ* represent the shear strain. Reduced stiffness components *Qij* are defined in terms of the engineering constants as

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material… DOI: http://dx.doi.org/10.5772/intechopen.86134*

$$\begin{aligned} Q\_{11} &= E\_1 / (\mathbf{1} - \nu\_{12} \nu\_{21}) \\ Q\_{22} &= E\_2 / (\mathbf{1} - \nu\_{12} \nu\_{21}) \\ Q\_{12} &= \nu\_{12} E\_1 / (\mathbf{1} - \nu\_{12} \nu\_{21}) \end{aligned} \tag{4}$$

where *E*<sup>1</sup> Young's moduli in the longitudinal and *E*<sup>2</sup> Young's moduli in the transverse directions. The major and minor Poisson's ratios represented by *ν*<sup>12</sup> and *ν*<sup>21</sup> respectively. The relation between Poisson's ratios in Eq. (4) is given by:

$$
v\_{21} = \frac{E\_2}{E\_1} v\_{12} \tag{5}$$

*A*<sup>11</sup> and *A*22, are in-plane stiffnesses of plate and these are obtained by integrating the *Qij* across the thickness of the plate [21]. These stiffness values are given as:

$$A\_{\vec{\eta}} = \int\_{-h/2}^{h/2} \left( Q'\_{\vec{\eta}} \right)\_k dz, i, j = 1, 2,\tag{6}$$

where plate thickness is represented by "*h*" and "*k*" *represents* each individual lamina. The transformed stiffness coefficients *Q*<sup>0</sup> *ij* are defined as

$$\begin{aligned} Q'\_{11} &= m^4 Q\_{11} + n^4 Q\_{22} + 2m^2 n^2 Q\_{12} + 4m^2 n^2 Q\_{\delta\delta} \\ Q'\_{22} &= n^4 Q\_{11} + m^4 Q\_{22} + 2m^2 n^2 Q\_{12} + 4m^2 n^2 Q\_{\delta\delta} \\ Q'\_{12} &= m^2 n^2 Q\_{11} + m^2 n^2 Q\_{22} + (m^4 + n^4) Q\_{12} - 4m^2 n^2 Q\_{\delta\delta} \end{aligned} \tag{7}$$

where *m* = cos(θ) and *n* = sin(θ), the angle θ is taken positive for counterclockwise rotation and it is considered from the primed (laminate) axes to the unprimed (individual lamina) axes. *Q*<sup>0</sup> *ij* for the 0° and 90° laminas are given by

$$\begin{aligned} \left( \left( \mathbf{Q}\_{11}' \right)\_{0 \text{ deg}} = \mathbf{Q}\_{11} \left( \mathbf{Q}\_{11}' \right)\_{90 \text{ deg}} = \mathbf{Q}\_{22} \\ \left( \mathbf{Q}\_{22}' \right)\_{0 \text{ deg}} = \mathbf{Q}\_{22} \left( \mathbf{Q}\_{22}' \right)\_{90 \text{ deg}} = \mathbf{Q}\_{11} \\ \left( \mathbf{Q}\_{12}' \right)\_{0 \text{ deg}} = \mathbf{Q}\_{12} \left( \mathbf{Q}\_{12}' \right)\_{90 \text{ deg}} = \mathbf{Q}\_{12} \end{aligned} \tag{8}$$

The extensional plate mode velocity is related to the in-plane stiffness of a composite [14]. *A*<sup>11</sup> and *A*<sup>22</sup> are the stiffnesses propagating in the 0° and 90° directions respectively. The relation between extensional plate mode velocity and stiffness is given by:

$$\text{for } 0^{\circ} \text{direction } c\_t = \sqrt{A\_{11}/\rho h} \tag{9}$$

$$\text{for } 90^{\circ} \text{direction } \, c\_{l} = \sqrt{A\_{22}/\rho h} \tag{10}$$

The inplane stiffnesses *A*<sup>11</sup> and *A*<sup>22</sup> are calculated using Eqs. (4)–(8) by substituting engineering stiffnesses of the composite. The extensional plate mode velocities are substituted into Eq. (2) and it is solved numerically for phase velocity in MathematicaTH. Phase velocity (*cphase*) is the dependent variable being solved for the independent variable being iteratively supplied is the frequency-thickness product, where *ω* is the driving frequency in radians. Group velocity dispersion curve, which are derived from the phase velocity curve using Eq. (11):

$$c\_{group} = c\_{phase} + \frac{\partial c\_{phase}}{\partial k}k = \frac{c\_{phase}}{1 - \frac{f}{c\_{phase}} \cdot \frac{\partial c\_{phase}}{\partial f}} \tag{11}$$

where *f* is the frequency in Hz.

phase and group velocities versus the excitation frequency given by Dalton et al. [24]. The anti-symmetric Lamb wave solution formulated as seen in Eq. 2:

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

tan ð Þ *ph* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> � �<sup>2</sup> 4*k*<sup>2</sup> *qp*

*, and k* <sup>¼</sup> *<sup>ω</sup>*

where *σ* is normal stress, τ represent shear stress, *ε* is normal strain and *ϒ* represent the shear strain. Reduced stiffness components *Qij* are defined in terms of

*Q*<sup>11</sup> *Q*<sup>12</sup> 0 *Q*<sup>12</sup> *Q*<sup>22</sup> 0 0 0 *Q*<sup>66</sup>

*cphase*

3 7 5

2 6 4

*ε*1 *ε*2 *γ*6 3 7

<sup>5</sup> (3)

(2)

tan ð Þ *qh*

where *<sup>p</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>

**Figure 5.**

*c*2 *l* � *<sup>k</sup>*<sup>2</sup>

the engineering constants as

**156**

*, q*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *c*2 *t* � *<sup>k</sup>*<sup>2</sup>

*Methodology chart for dynamic mechanical analysis by hybrid method.*

*σ*1 *σ*2 *τ*6

2 6 4

individual laminate stress-strain relationship is given by

2 6 4

### **3.2 Finite element model for free vibration of a laminated composite plate**

Natural frequency is the phenomenon that occurs with oscillatory motion at certain frequencies known as characteristic values, and it follows well defined deformation pattern known as mode shapes or characteristic modes. The study free vibration is important in finding the dynamic response of elastic structures. It is assumed that the external force vector *P* ! to be zero and the harmonic displacement as:

$$
\overrightarrow{Q} = \overrightarrow{\underline{Q}} \cdot \overrightarrow{\varepsilon}^{\rm{tot}} \tag{12}
$$

In order to develop shape functions two different interpolations are used one interpolation within the *xy*-plane and the other in the *z*-axis. For the *xy*-plane interpolation, shape function *Ni*(*x,y*) are used where subscript *i* varies depending on the number of nodes on the *xy*-plane. Shape function *Hj*(*z*) is used for interpolation along the *z*-axis, where subscript *j* varies depending on the number of nodes along the plate thickness. Since two inplane displacement are functions of *x*, *y*, and *z*, both shape functions are used while the shape functions *Ni*(*x,y*) was used for transverse displacement. The mapping of *ξ, η*-plane onto *xy*-plane and *ζ*-axis to zaxis, was done using isoparametric element and the three displacements are

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material…*

*<sup>u</sup>* <sup>¼</sup> <sup>X</sup> *N*<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.86134*

*<sup>v</sup>* <sup>¼</sup> <sup>X</sup> *N*<sup>1</sup>

*i*¼1

*i*¼1

X *N*<sup>2</sup>

*Ni*ð Þ *ξ; η Hj*ð Þ*ζ uij* (18)

*Ni*ð Þ *ξ; η Hj*ð Þ*ζ vij* (19)

*Ni*ð Þ *ξ; η wi* (20)

*j*¼1

X *N*<sup>2</sup>

*j*¼1

*i*¼1

where *N*<sup>1</sup> represents the number of nodes in *xy*-plane (*ξ, η*-plane) and *N*<sup>2</sup> represents the number of nodes in *z*-axis (*ζ*-axis). The first subscript for *u* and *v* denotes the node numbering in terms of *xy*-plane (*ξ, η*-plane) and the second subscript indicates the node numbering in terms of *z*-axis (*ζ*-axis). Four-node quadrilateral shape function is considered for the *xy*-plane (*ξ, η*-plane) interpolation, i.e., *N*<sup>1</sup> = 4 and *N*<sup>2</sup> = 2 that is linear shape function which is considered for the *z*-axis (*ζ*-axis) interpolation. Nodal displacement *ui*<sup>1</sup> and *vi*<sup>1</sup> are displacement on the bottom surface of the plate element and *ui*<sup>2</sup> and *vi*<sup>2</sup> are displacement on the top surface. As seen in Eqs. (18)–(20), there is no rotational degree of freedom for the present plate bending element were as both bending strain energy and transverse shear strain

The relation between bending strains and transverse shear strain with respect to

*∂ ∂x*

*∂ ∂y*

*∂ ∂z*

where f g *ε<sup>b</sup>* and f g *ε<sup>s</sup>* are the bending strain and transverse shear strain respec-

Substitution of Eqs. (18)–(20), into the Eqs. (25) and (26), with *N*<sup>1</sup> = 4 and

<sup>0</sup> *<sup>∂</sup> ∂z*

<sup>0</sup> *<sup>∂</sup> ∂y* 0

> *∂ ∂x* 0

<sup>0</sup> *<sup>∂</sup> ∂x*

*∂ ∂y*

0 0

*u v w*

*u v w*

f g *<sup>ε</sup><sup>s</sup>* <sup>¼</sup> ½ � *Bb de* f g (23)

9 >=

>;

9 >=

>;

(21)

(22)

8 ><

>:

8 ><

>:

*<sup>w</sup>* <sup>¼</sup> <sup>X</sup> *N*<sup>1</sup>

expressed as

energy are included.

*N*<sup>2</sup> = 2 gives:

**159**

displacements is given by:

f g *ε<sup>b</sup>* ¼

*εx εy γxy*

*γx z* � �

tively. The normal strain along the plate thickness *ε<sup>z</sup>* is not considered.

¼

9 >= >; ¼

8 ><

>:

f g *<sup>ε</sup><sup>s</sup>* <sup>¼</sup> *<sup>γ</sup>y z*

and the free vibration is given by:

$$\left[ [k] - \alpha^2 [\mathcal{M}] \right] \stackrel{\rightarrow}{\underline{Q}} = \stackrel{\rightarrow}{\mathcal{O}} \tag{13}$$

where *Q* ! is displacement amplitude, *Q* ! eigen vector and *ω* denotes the natural frequency of vibration. Eq. (12) is a linear algebraic eigenvalue problem where neither [*k*] nor [*M*] is a function of the circular frequency ð Þ *ω* , *Q* ! is nonzero solution therefore the determinant of coefficient matrix ½ �� *<sup>k</sup> <sup>ω</sup>*<sup>2</sup>½ � *<sup>M</sup>* is zero, i.e.,

$$[k] - \alpha^2 [\mathbf{M}] = \mathbf{0} \tag{14}$$

where [*k*] is stiffness matrix and [*M*] is mass matrix, which are derived through finite element formulation.

**Figure 6** shows the plate bending formulation where *x*, *y*, and *z* describes the global coordinate of the plate whereas *u*, *v*, and *w* are the displacements, *h* represents plate thickness. The *xy* plane is parallel to the midsurface plane prior to deflection. The displacements in the plate at any point is expressed as

$$
\mu = \mu(x, y, z) \tag{15}
$$

$$\boldsymbol{v} = \boldsymbol{v}(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}) \tag{16}$$

$$w = w(x, y, z)\tag{17}$$

The plane displacement *u* and *v* vary through the plate thickness as well as with in the *xy*-plane while the transverse displacement *w* remains constant through the plate thickness.

**Figure 6.** *Plate element with displacement degrees of freedom.*

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material… DOI: http://dx.doi.org/10.5772/intechopen.86134*

In order to develop shape functions two different interpolations are used one interpolation within the *xy*-plane and the other in the *z*-axis. For the *xy*-plane interpolation, shape function *Ni*(*x,y*) are used where subscript *i* varies depending on the number of nodes on the *xy*-plane. Shape function *Hj*(*z*) is used for interpolation along the *z*-axis, where subscript *j* varies depending on the number of nodes along the plate thickness. Since two inplane displacement are functions of *x*, *y*, and *z*, both shape functions are used while the shape functions *Ni*(*x,y*) was used for transverse displacement. The mapping of *ξ, η*-plane onto *xy*-plane and *ζ*-axis to zaxis, was done using isoparametric element and the three displacements are expressed as

$$\mu = \sum\_{i=1}^{N\_1} \sum\_{j=1}^{N\_2} N\_i(\xi, \eta) H\_j(\zeta) u\_{\vec{\eta}} \tag{18}$$

$$\nu = \sum\_{i=1}^{N\_1} \sum\_{j=1}^{N\_2} N\_i(\xi, \eta) H\_j(\zeta) \nu\_{ij} \tag{19}$$

$$w = \sum\_{i=1}^{N\_1} N\_i(\xi, \eta) w\_i \tag{20}$$

where *N*<sup>1</sup> represents the number of nodes in *xy*-plane (*ξ, η*-plane) and *N*<sup>2</sup> represents the number of nodes in *z*-axis (*ζ*-axis). The first subscript for *u* and *v* denotes the node numbering in terms of *xy*-plane (*ξ, η*-plane) and the second subscript indicates the node numbering in terms of *z*-axis (*ζ*-axis). Four-node quadrilateral shape function is considered for the *xy*-plane (*ξ, η*-plane) interpolation, i.e., *N*<sup>1</sup> = 4 and *N*<sup>2</sup> = 2 that is linear shape function which is considered for the *z*-axis (*ζ*-axis) interpolation. Nodal displacement *ui*<sup>1</sup> and *vi*<sup>1</sup> are displacement on the bottom surface of the plate element and *ui*<sup>2</sup> and *vi*<sup>2</sup> are displacement on the top surface. As seen in Eqs. (18)–(20), there is no rotational degree of freedom for the present plate bending element were as both bending strain energy and transverse shear strain energy are included.

The relation between bending strains and transverse shear strain with respect to displacements is given by:

$$\begin{aligned} \{e\_b\} = \left\{ \begin{array}{c} e\_\chi\\ e\_\chi\\ \chi\_{xy} \end{array} \right\} = \begin{bmatrix} \frac{\partial}{\partial x} & 0 & 0\\ 0 & \frac{\partial}{\partial y} & 0\\ \frac{\partial}{\partial y} & \frac{\partial}{\partial \chi} & 0 \end{bmatrix} \begin{Bmatrix} u\\ v\\ w \end{Bmatrix} \end{aligned} \tag{21}$$

$$\{e\_i\} = \begin{Bmatrix} \chi\_{yz} \\ \chi\_{xz} \end{Bmatrix} = \begin{bmatrix} \frac{\partial}{\partial x} & 0 & \frac{\partial}{\partial x} \\ 0 & \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{bmatrix} \begin{Bmatrix} u \\ v \\ w \end{Bmatrix} \tag{22}$$

where f g *ε<sup>b</sup>* and f g *ε<sup>s</sup>* are the bending strain and transverse shear strain respectively. The normal strain along the plate thickness *ε<sup>z</sup>* is not considered.

Substitution of Eqs. (18)–(20), into the Eqs. (25) and (26), with *N*<sup>1</sup> = 4 and *N*<sup>2</sup> = 2 gives:

$$\{\varepsilon\_{\mathfrak{t}}\} = [\mathcal{B}\_{\mathfrak{b}}] \{d^{\mathfrak{t}}\} \tag{23}$$

**3.2 Finite element model for free vibration of a laminated composite plate**

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

is assumed that the external force vector *P*

and the free vibration is given by:

is displacement amplitude, *Q*

neither [*k*] nor [*M*] is a function of the circular frequency ð Þ *ω* , *Q*

therefore the determinant of coefficient matrix ½ �� *<sup>k</sup> <sup>ω</sup>*<sup>2</sup>½ � *<sup>M</sup>* is zero, i.e.,

½ �� *<sup>k</sup> <sup>ω</sup>*<sup>2</sup>

displacement as:

where *Q* !

plate thickness.

**Figure 6.**

**158**

*Plate element with displacement degrees of freedom.*

finite element formulation.

Natural frequency is the phenomenon that occurs with oscillatory motion at certain frequencies known as characteristic values, and it follows well defined deformation pattern known as mode shapes or characteristic modes. The study free vibration is important in finding the dynamic response of elastic structures. It

!

! ¼*O* !

*Q* ! ¼ *Q* ! *:e*

½ �� *<sup>k</sup> <sup>ω</sup>*<sup>2</sup> ½ � *<sup>M</sup> <sup>Q</sup>*

!

where [*k*] is stiffness matrix and [*M*] is mass matrix, which are derived through

**Figure 6** shows the plate bending formulation where *x*, *y*, and *z* describes the global coordinate of the plate whereas *u*, *v*, and *w* are the displacements, *h* represents plate thickness. The *xy* plane is parallel to the midsurface plane prior to deflection. The displacements in the plate at any point is expressed as

The plane displacement *u* and *v* vary through the plate thickness as well as with in the *xy*-plane while the transverse displacement *w* remains constant through the

frequency of vibration. Eq. (12) is a linear algebraic eigenvalue problem where

to be zero and the harmonic

*<sup>i</sup>ω<sup>t</sup>* (12)

eigen vector and *ω* denotes the natural

!

½ �¼ *M* 0 (14)

*u* ¼ *u x*ð Þ *; y; z* (15) *v* ¼ *v x*ð Þ *; y; z* (16) *w* ¼ *w x*ð Þ *; y; z* (17)

(13)

is nonzero solution

where

$$\begin{bmatrix} \mathbf{B}\_{b} \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} \mathbf{B}\_{b1} \end{bmatrix} & \begin{bmatrix} \mathbf{B}\_{b2} \end{bmatrix} & \begin{bmatrix} \mathbf{B}\_{b3} \end{bmatrix} & \begin{bmatrix} \mathbf{B}\_{b4} \end{bmatrix} \end{bmatrix} \tag{24}$$

In which

And

(42) because of the reciprocal relation

*DOI: http://dx.doi.org/10.5772/intechopen.86134*

*<sup>k</sup>*ð Þ*<sup>e</sup>* h i <sup>¼</sup>

where Ω represents the plate domain. Similarly the mass matrix is given by

for damaged specimen.

**3.3 Bandwidth method**

**161**

ð Ω*e* ½ � *Bb* *<sup>D</sup>*<sup>11</sup> <sup>¼</sup> *<sup>E</sup>*<sup>1</sup>

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material…*

*<sup>D</sup>*<sup>12</sup> <sup>¼</sup> *<sup>E</sup>*1*υ*<sup>21</sup>

*<sup>D</sup>*<sup>22</sup> <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

*Ds* ½ �¼ *<sup>G</sup>*<sup>13</sup> <sup>0</sup>

*υ*<sup>12</sup> *E*1

*<sup>T</sup>*½ � *Db* ½ � *Bb <sup>∂</sup><sup>Ω</sup>* <sup>þ</sup>

The stiffness matrix of the element *<sup>k</sup>*ð Þ*<sup>e</sup>* h i is expressed as:

*<sup>M</sup>*ð Þ*<sup>e</sup>* h i <sup>¼</sup> *<sup>ρ</sup>A t*

9

where *A* is area of the element, *t* is the element thickness and *ρ* density of material. Natural frequencies are arrived for composite plates from the lossless finite element formulation. The MathematicaTH software has been used to compute the Eigen values using the inputs taken from the experimental data discussed in the previous sections. The analytical model developed was correlated ANSYS model. Block Lancozs method was used to carry out modal analysis in ANSYS. The analysis was done for 30 subsets and shell-190 has been used as meshing element. **Figures 7** and **8** shows the first and twentieth mode of natural frequency of the undamaged specimen, i.e., GFRP and CFRP respectively and similarly **Figures 9** and **10** shows

The damping parameters in FRP composites are based on the energy dissipation mechanism. Vibrational parameters such as frequency and amplitude are used to

Here, the longitudinal direction is represented with 1 and transverse direction is represented with 2 for the FRP composite. Further *Ei* and *Gij* are the elastic modulus and shear modulus respectively, whereas υ*ij* is Poisson's ratio for strain in the *j*direction. Five independent material properties will be considered for Eqs. (37)–

> <sup>¼</sup> *<sup>υ</sup>*<sup>21</sup> *E*2

> > ð Ω*e*

1 � *υ*12*υ*<sup>21</sup>

1 � *υ*12*υ*<sup>21</sup>

1 � *ν*12*ν*<sup>21</sup>

*D*<sup>33</sup> ¼ *G*<sup>12</sup> (41)

<sup>0</sup> *<sup>G</sup>*<sup>12</sup> � � (42)

*Bs* ½ �*<sup>T</sup> Ds* ½ � *Bs* ½ � *<sup>∂</sup><sup>Ω</sup>* (44)

(38)

(39)

(40)

(43)

(45)

$$\begin{aligned} \begin{bmatrix} B\_b \end{bmatrix} = \begin{bmatrix} H\_1 \frac{\partial N\_i}{\partial \mathbf{x}} & \mathbf{0} & H\_2 \frac{\partial N\_i}{\partial \mathbf{x}} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & H\_1 \frac{\partial N\_i}{\partial \mathbf{y}} & \mathbf{0} & H\_2 \frac{\partial N\_i}{\partial \mathbf{y}} & \mathbf{0} \\ H\_1 \frac{\partial N\_i}{\partial \mathbf{y}} & H\_1 \frac{\partial N\_i}{\partial \mathbf{x}} & H\_2 \frac{\partial N\_i}{\partial \mathbf{y}} & H\_2 \frac{\partial N\_i}{\partial \mathbf{x}} & \mathbf{0} \end{bmatrix} \end{aligned} \tag{25}$$

$$\begin{array}{ccccc}\{\boldsymbol{d}^{\boldsymbol{e}}\} = \left\{ \begin{Bmatrix} \boldsymbol{d}\_{1}^{\boldsymbol{e}} \end{Bmatrix} & \begin{Bmatrix} \boldsymbol{d}\_{2}^{\boldsymbol{e}} \end{Bmatrix} & \begin{Bmatrix} \boldsymbol{d}\_{1}^{\boldsymbol{e}} \end{Bmatrix} & \begin{Bmatrix} \boldsymbol{d}\_{2}^{\boldsymbol{e}} \end{Bmatrix} & \begin{Bmatrix} \boldsymbol{d}\_{2}^{\boldsymbol{e}} \end{Bmatrix} \right\}^{T} \\ & & . \end{array} \tag{26}$$

$$\left\{d\_i^{\epsilon}\right\} = \left\{u\_{i1} \quad v\_{i1} \quad u\_{i2} \quad v\_{i2} \quad w\_i\right\} \tag{27}$$

$$\{\varepsilon\_{\varepsilon}\} = [B\_{\varepsilon}] \{d^{\varepsilon}\} \tag{28}$$

where

$$\begin{bmatrix} \mathbf{B}\_{\mathfrak{t}} \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} \mathbf{B}\_{\mathfrak{t}1} \end{bmatrix} & \begin{bmatrix} \mathbf{B}\_{\mathfrak{t}2} \end{bmatrix} & \begin{bmatrix} \mathbf{B}\_{\mathfrak{t}3} \end{bmatrix} & \begin{bmatrix} \mathbf{B}\_{\mathfrak{t}4} \end{bmatrix} \end{bmatrix} \tag{29}$$

$$\begin{aligned} \begin{bmatrix} B\_{i\end{bmatrix} \end{bmatrix} = \begin{bmatrix} N\_i \frac{\partial H\_1}{\partial \mathbf{z}} & \mathbf{0} & N\_i \frac{\partial H\_2}{\partial \mathbf{z}} & \mathbf{0} & \frac{\partial N\_i}{\partial \mathbf{x}}\\ \mathbf{0} & N\_i \frac{\partial H\_1}{\partial \mathbf{z}} & \mathbf{0} & N\_i \frac{\partial H\_2}{\partial \mathbf{z}} & \frac{\partial H\_2}{\partial \mathbf{y}} \end{bmatrix} \end{aligned} \tag{30}$$

The constitutive equation is

$$\{\sigma\_b\} = [D\_b] \{\varepsilon\_b\} \tag{31}$$

$$\begin{Bmatrix} \sigma\_b \end{Bmatrix} = \begin{Bmatrix} \sigma\_\mathbf{x} & \sigma\_\mathbf{y} & \tau\_\mathbf{xy} \end{Bmatrix}^T \tag{32}$$

$$[D\_b] = \frac{E}{\mathbf{1} - \nu^2} \begin{bmatrix} \mathbf{1} & \nu & \mathbf{0} \\ \nu & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \frac{\mathbf{1} - \nu}{2} \end{bmatrix} \tag{33}$$

For the bending components

$$\{\sigma\_{\mathfrak{s}}\} = [D\_{\mathfrak{s}}] \{\mathfrak{e}\_{\mathfrak{s}}\} \tag{34}$$

where

$$\{\sigma\_{\mathbf{s}}\} = \begin{Bmatrix} \mathfrak{r}\_{\mathbf{y}x} & \mathfrak{r}\_{\mathbf{x}x} \end{Bmatrix}^T \tag{35}$$

$$[D\_s] = \frac{E}{2(1+\nu)} \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \end{bmatrix} \tag{36}$$

where Eq. (33) is for the plane stress condition for the plate bending theory and for a FRP composite, is given by

$$\begin{bmatrix} D\_b \end{bmatrix} = \begin{bmatrix} D\_{11} & D\_{12} & \mathbf{0} \\ D\_{12} & D\_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & D\_{33} \end{bmatrix} \tag{37}$$

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material… DOI: http://dx.doi.org/10.5772/intechopen.86134*

In which

where

where

where

**160**

½ �¼ *Bb*

*Bsi* ½ �¼

The constitutive equation is

For the bending components

for a FRP composite, is given by

*H*<sup>1</sup> *∂Ni ∂x*

*H*<sup>1</sup> *∂Ni ∂y*

*<sup>d</sup><sup>e</sup>* f g <sup>¼</sup> *<sup>d</sup><sup>e</sup>*

*de i*

*Ni ∂H*<sup>1</sup> *∂z*

0 *Ni*

½ �¼ *Db*

0 *H*<sup>1</sup>

*H*<sup>1</sup> *∂Ni ∂x*

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

1 � � *de*

½ �¼ *Bb Bb*<sup>1</sup> ½ � ½ � ½ � *Bb*<sup>2</sup> ½ � *Bb*<sup>3</sup> ½ � *Bb*<sup>4</sup> (24)

0 0

0

(25)

(30)

(33)

(36)

0

*∂Ni ∂y*

*∂Ni ∂x*

1 � � *de*

*∂H*<sup>2</sup> *∂z*

1 *υ* 0 *υ* 1 0

> 1 � *υ* 2

1 0 0 1 � �

> 3 7

0 0

0 *Ni*

0 *H*<sup>2</sup>

*H*<sup>2</sup> *∂Ni ∂x*

2 � � � � *<sup>T</sup>* (26)

f g *<sup>ε</sup><sup>s</sup>* <sup>¼</sup> *Bs* ½ � *de* f g (28)

<sup>0</sup> *<sup>∂</sup>Ni ∂x*

*∂H*<sup>2</sup> *∂y*

*∂H*<sup>2</sup> *∂z*

f g *σ<sup>b</sup>* ¼ ½ � *Db* f g *ε<sup>b</sup>* (31)

f g *σ<sup>s</sup>* ¼ *Ds* ½ �f g *ε<sup>s</sup>* (34)

� �*<sup>T</sup>* (35)

<sup>5</sup> (37)

� �*<sup>T</sup>* (32)

� � <sup>¼</sup> f g *ui*<sup>1</sup> *vi*<sup>1</sup> *ui*<sup>2</sup> *vi*<sup>2</sup> *wi* (27)

*Bs* ½ �¼ ½ � ½ � *Bs*<sup>1</sup> ½ � *Bs*<sup>2</sup> ½ � *Bs*<sup>3</sup> ½ � *Bs*<sup>4</sup> (29)

0 *H*<sup>2</sup>

2 � � *d<sup>e</sup>*

0 *Ni*

*∂H*<sup>1</sup> *∂z*

f g *σ<sup>b</sup>* ¼ *σ<sup>x</sup> σ<sup>y</sup> τxy*

f g *σ<sup>s</sup>* ¼ *τy z τx z*

2 1ð Þ þ *υ*

where Eq. (33) is for the plane stress condition for the plate bending theory and

*D*<sup>11</sup> *D*<sup>12</sup> 0 *D*<sup>12</sup> *D*<sup>22</sup> 0 0 0 *D*<sup>33</sup>

*Ds* ½ �¼ *<sup>E</sup>*

2 6 4

½ �¼ *Db*

*E* 1 � *υ*<sup>2</sup> *H*<sup>2</sup> *∂Ni ∂y*

*∂Ni ∂y*

$$D\_{11} = \frac{E\_1}{1 - \nu\_{12}\nu\_{21}}\tag{38}$$

$$D\_{12} = \frac{E\_1 \nu\_{21}}{1 - \nu\_{12}\nu\_{21}}\tag{39}$$

$$D\_{22} = \frac{E\_2}{1 - \nu\_{12}\nu\_{21}}\tag{40}$$

$$D\_{33} = G\_{12} \tag{41}$$

And

$$\begin{bmatrix} \mathbf{D}\_{\mathbf{s}} \end{bmatrix} = \begin{bmatrix} \mathbf{G}\_{13} & \mathbf{0} \\ \mathbf{0} & \mathbf{G}\_{12} \end{bmatrix} \tag{42}$$

Here, the longitudinal direction is represented with 1 and transverse direction is represented with 2 for the FRP composite. Further *Ei* and *Gij* are the elastic modulus and shear modulus respectively, whereas υ*ij* is Poisson's ratio for strain in the *j*direction. Five independent material properties will be considered for Eqs. (37)– (42) because of the reciprocal relation

$$\frac{\upsilon\_{12}}{E\_1} = \frac{\upsilon\_{21}}{E\_2} \tag{43}$$

The stiffness matrix of the element *<sup>k</sup>*ð Þ*<sup>e</sup>* h i is expressed as:

$$\mathbb{E}\left[\boldsymbol{k}^{(\epsilon)}\right] = \int\_{\Omega'} \left[\boldsymbol{B}\_b\right]^T \left[\boldsymbol{D}\_b\right] \left[\boldsymbol{B}\_b\right] \partial \Omega + \int\_{\Omega'} \left[\boldsymbol{B}\_s\right]^T \left[\boldsymbol{D}\_s\right] \left[\boldsymbol{B}\_t\right] \partial \Omega \tag{44}$$

where Ω represents the plate domain. Similarly the mass matrix is given by

$$
\begin{bmatrix} M^{(\epsilon)} \end{bmatrix} = \frac{\rho A t}{9} \begin{bmatrix} 4 & 2 & 1 & 2 \\ 2 & 4 & 2 & 1 \\ 1 & 2 & 4 & 2 \\ 2 & 1 & 2 & 4 \end{bmatrix} \tag{45}
$$

where *A* is area of the element, *t* is the element thickness and *ρ* density of material. Natural frequencies are arrived for composite plates from the lossless finite element formulation. The MathematicaTH software has been used to compute the Eigen values using the inputs taken from the experimental data discussed in the previous sections. The analytical model developed was correlated ANSYS model. Block Lancozs method was used to carry out modal analysis in ANSYS. The analysis was done for 30 subsets and shell-190 has been used as meshing element. **Figures 7** and **8** shows the first and twentieth mode of natural frequency of the undamaged specimen, i.e., GFRP and CFRP respectively and similarly **Figures 9** and **10** shows for damaged specimen.

#### **3.3 Bandwidth method**

The damping parameters in FRP composites are based on the energy dissipation mechanism. Vibrational parameters such as frequency and amplitude are used to

*<sup>ζ</sup>* <sup>¼</sup> <sup>Δ</sup>*<sup>ω</sup>* 2*ω<sup>r</sup>*

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material…*

*<sup>ζ</sup><sup>i</sup>* <sup>¼</sup> <sup>1</sup> 2 *Δω<sup>i</sup> ωi*

The equation of motion of a system with viscous damping, when the excitation

Eq. (48) represents the forced vibration of a damped system and the resulting motion occurs at the forcing frequency *ω*. The damping coefficient *c* is greater than zero, leads to change in the phase between the force and resulting motion. The phase change is termed as phase angle δ which is a function of the frequency ratio *ω/ω<sup>r</sup>* and for several values of the fraction of critical damping *ζ*, given by [25].

*r*

*<sup>δ</sup>* <sup>¼</sup> *tan* �**<sup>1</sup> <sup>2</sup>***ζ ω*ð Þ *<sup>=</sup>ω<sup>r</sup>* **<sup>3</sup> 1** � *ω***<sup>2</sup>***=ω***<sup>2</sup>**

*mx::* <sup>þ</sup> *cx:* <sup>þ</sup> *kx* <sup>¼</sup> *Fo* sin*ω<sup>t</sup>* (48)

<sup>þ</sup> ð Þ **<sup>2</sup>***ζω=ω<sup>r</sup>* **<sup>2</sup>** (49)

for ith mode damping ratio is given by

*DOI: http://dx.doi.org/10.5772/intechopen.86134*

**Figure 10.**

**Figure 11.**

**163**

is a force *F* ¼ *Fo* sin*ωt* applied to the system, is given by

*Damaged CFRP specimen's first and twentieth mode of natural frequency.*

*Response curve showing bandwidth at half-power point.*

(46)

(47)

**Figure 7.** *Undamaged GFRP specimen's first and twentieth mode of natural frequency.*

#### **Figure 8.**

*Undamaged CFRP specimen's first and twentieth mode of natural frequency.*

**Figure 9.**

*Damaged GFRP specimen's first and twentieth mode of natural frequency.*

determine the dynamic characteristics of a system. The best practice to study vibrational parameters is with nondestructive evaluation. Damping characteristics of a system can be determined by the maximum response, i.e., the response at the resonance frequency as indicated by the maximum value of *Rv*. **Figure 11** illustrates the Bandwidth method of damping measurement where, damping in a system is indicated by the sharpness or width of the response curve in the vicinity of a resonance frequency *ω*r, designating the width as a frequency increment (i.e., Δ*ω*=Δ*ω*2–*ω*1) measured at the "half-power point" (i.e., at a value (*R=* ffiffi 2 <sup>p</sup> )) and the damping ratio ζ can be estimated by using band width in the relation given by

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material… DOI: http://dx.doi.org/10.5772/intechopen.86134*

$$
\zeta = \frac{\Delta o}{2a\_r} \tag{46}
$$

for ith mode damping ratio is given by

$$\zeta\_i = \frac{1}{2} \frac{\Delta o\_i}{o\_i} \tag{47}$$

The equation of motion of a system with viscous damping, when the excitation is a force *F* ¼ *Fo* sin*ωt* applied to the system, is given by

$$-m\mathbf{x}^{\cdot\cdot} + c\mathbf{x}^{\cdot} + k\mathbf{x} = F\_{\boldsymbol{\theta}}\sin\boldsymbol{at}\tag{48}$$

Eq. (48) represents the forced vibration of a damped system and the resulting motion occurs at the forcing frequency *ω*. The damping coefficient *c* is greater than zero, leads to change in the phase between the force and resulting motion. The phase change is termed as phase angle δ which is a function of the frequency ratio *ω/ω<sup>r</sup>* and for several values of the fraction of critical damping *ζ*, given by [25].

$$\delta = \tan^{-1} \frac{\mathbf{2\xi}(\boldsymbol{\omega}/\boldsymbol{\omega}\_r)^3}{\mathbf{1} - \left(\boldsymbol{\omega}^2/\boldsymbol{\omega}\_r^2\right) + \left(\mathbf{2\xi}\boldsymbol{\omega}/\boldsymbol{\omega}\_r\right)^2} \tag{49}$$

**Figure 10.** *Damaged CFRP specimen's first and twentieth mode of natural frequency.*

**Figure 11.** *Response curve showing bandwidth at half-power point.*

determine the dynamic characteristics of a system. The best practice to study vibrational parameters is with nondestructive evaluation. Damping characteristics of a system can be determined by the maximum response, i.e., the response at the resonance frequency as indicated by the maximum value of *Rv*. **Figure 11** illustrates the Bandwidth method of damping measurement where, damping in a system is indicated by the sharpness or width of the response curve in the vicinity of a resonance frequency *ω*r, designating the width as a frequency increment (i.e., Δ*ω*=Δ*ω*2–*ω*1) measured at the "half-power point" (i.e., at a value (*R=* ffiffi

*Undamaged CFRP specimen's first and twentieth mode of natural frequency.*

*Undamaged GFRP specimen's first and twentieth mode of natural frequency.*

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

*Damaged GFRP specimen's first and twentieth mode of natural frequency.*

**Figure 8.**

**Figure 7.**

**Figure 9.**

**162**

damping ratio ζ can be estimated by using band width in the relation given by

2 <sup>p</sup> )) and the

## **4. Results and discussion**

The phenomenon of change in modal parameters has been used to identify the damage in the specimens. The damage in the specimen is identified by change in damping capacity with respect to undamaged specimen. The first order Lamb wave equation is used to determine the storage modulus. Lamb wave propagating is quite complex to understand, i.e., an increase in modulus slightly speeds the wave velocity. An increase in the density would have the opposite effect slowing wave velocity, as it appears in all the same terms as the modulus but on the reciprocal side of the divisor.

The AE velocities of the specimens were arrived experimentally using ultrasonic pulse generator test setup and the engineering constants, Young's modulus and poison's ratio were calculated. The engineering constants are substituted in Lamb wave model discussed in previous sections for finding dispersion characteristics shown in **Figure 12**. The same material properties are used for finite element model to determine natural frequencies.

The group velocity (*cgn* ) at natural frequency (*f <sup>n</sup>*) and thickness (*h*) is substituted in Eq. (50) to determine the phase shift and thus finding material damping capacity ð *Tan δ*). Dynamic mechanical analysis can be carried out using the same procedure by getting the *Eo* value from group velocity dispersion at iteratively supplied frequencies.

$$\delta = 2\pi f\_n h / c\_{\text{g.}} \tag{50}$$

$$E\_o = \mathfrak{c}\_{\mathfrak{g}\_u}^2 \rho \tag{51}$$

fast Fourier transform (FFT) which yield a single peak from the calibrated optimal driving frequency, however for a few finite cycles, the FFT appears as a Gaussian curve. The response curve of the undamaged and damaged specimens being tested

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material…*

The Lamb wave dispersion curves have been obtained from the iterative supply of the frequency using MathematicaTH code. The group velocity dispersion curve of the specimens used in this research is shown in **Figure 12**. The group velocities and

**Table 3** shows the damping capacities of the undamaged specimen in compari-

for damping capacity using bandwidth method is shown in **Figure 13**.

damping capacity at various mode of interest.

*DOI: http://dx.doi.org/10.5772/intechopen.86134*

*Response curve of GFRP and CFRP showing bandwidth.*

*Damping capacity of undamaged test specimens.*

*Damping capacity of damaged test specimens.*

**Figure 13.**

**Table 3.**

**Table 4.**

**165**

the natural frequencies obtained from modal analysis are used to determine

son at critical modes similarly for damaged specimen it has been reported in **Table 4**. It is observed that the natural frequencies of the damaged specimen fell

down and the damping capacities have increased slightly with respect to undamaged specimens. **Figure 14** shows the damping capacities and dynamic

Damping is the term used in vibration and noise analysis to describe any mechanism whereby mechanical energy in the system is dissipated. The damping properties of so-called damping materials, such as elastomeric materials, are usually temperature and frequency dependent, so the experimental determination of damping material properties requires a long and repeating process.

In dynamic mechanical analysis damping measurements is done in temperature sweep mode whereas in this work frequency sweep mode is used. In the present work damping measurements were carried out using combined finite element and Lamb wave method and the results were compared with bandwidth method.

The modal analysis was carried out using developed finite element model and it was correlated with ANSYS. The waveform from the instrument is processed through virtual controlling software and the continuous waveform is subjected to

**Figure 12.** *Lamb wave dispersion curves of CFRP and GFRP.*

### *Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material… DOI: http://dx.doi.org/10.5772/intechopen.86134*

fast Fourier transform (FFT) which yield a single peak from the calibrated optimal driving frequency, however for a few finite cycles, the FFT appears as a Gaussian curve. The response curve of the undamaged and damaged specimens being tested for damping capacity using bandwidth method is shown in **Figure 13**.

The Lamb wave dispersion curves have been obtained from the iterative supply of the frequency using MathematicaTH code. The group velocity dispersion curve of the specimens used in this research is shown in **Figure 12**. The group velocities and the natural frequencies obtained from modal analysis are used to determine damping capacity at various mode of interest.

**Table 3** shows the damping capacities of the undamaged specimen in comparison at critical modes similarly for damaged specimen it has been reported in **Table 4**. It is observed that the natural frequencies of the damaged specimen fell down and the damping capacities have increased slightly with respect to undamaged specimens. **Figure 14** shows the damping capacities and dynamic

**Figure 13.** *Response curve of GFRP and CFRP showing bandwidth.*


#### **Table 3.**

**4. Results and discussion**

to determine natural frequencies.

iteratively supplied frequencies.

divisor.

**Figure 12.**

**164**

*Lamb wave dispersion curves of CFRP and GFRP.*

The phenomenon of change in modal parameters has been used to identify the damage in the specimens. The damage in the specimen is identified by change in damping capacity with respect to undamaged specimen. The first order Lamb wave equation is used to determine the storage modulus. Lamb wave propagating is quite complex to understand, i.e., an increase in modulus slightly speeds the wave velocity. An increase in the density would have the opposite effect slowing wave velocity, as it appears in all the same terms as the modulus but on the reciprocal side of the

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

The AE velocities of the specimens were arrived experimentally using ultrasonic

*δ* ¼ 2*πf <sup>n</sup>h=cgn* (50)

*:ρ* (51)

pulse generator test setup and the engineering constants, Young's modulus and poison's ratio were calculated. The engineering constants are substituted in Lamb wave model discussed in previous sections for finding dispersion characteristics shown in **Figure 12**. The same material properties are used for finite element model

The group velocity (*cgn* ) at natural frequency (*f <sup>n</sup>*) and thickness (*h*) is substituted in Eq. (50) to determine the phase shift and thus finding material damping capacity ð *Tan δ*). Dynamic mechanical analysis can be carried out using the same procedure by getting the *Eo* value from group velocity dispersion at

> *Eo* ¼ *c* 2 *gn*

damping material properties requires a long and repeating process.

Damping is the term used in vibration and noise analysis to describe any mechanism whereby mechanical energy in the system is dissipated. The damping properties of so-called damping materials, such as elastomeric materials, are usually temperature and frequency dependent, so the experimental determination of

In dynamic mechanical analysis damping measurements is done in temperature sweep mode whereas in this work frequency sweep mode is used. In the present work damping measurements were carried out using combined finite element and Lamb wave method and the results were compared with bandwidth method.

The modal analysis was carried out using developed finite element model and it

was correlated with ANSYS. The waveform from the instrument is processed through virtual controlling software and the continuous waveform is subjected to

*Damping capacity of undamaged test specimens.*


#### **Table 4.**

*Damping capacity of damaged test specimens.*

#### **Figure 14.**

*Damping capacity and dynamic storage modulus for CFRP and GFRP.*

storage modulus of the tested specimens with respect to their natural frequencies. The material GFRP and CFRP exhibits similar damping property to a certain range of frequency, and in between 2 and 8 kHz GFRP has better damping property among the two and at higher range of frequencies CFRP is found to be good in damping characteristics.

### **5. Conclusions**

Dynamic mechanical analysis is a technique used to study and characterize damping behavior of materials. It is most useful for studying the viscoelastic behavior of polymers. The tests were conducted on polymer composites CFRP and GFRP laminates in their undamaged and damaged state. A hybrid method has been explored in this work and the materials have been characterized for damping parameters at their mode frequencies. The change in the modal parameters (i.e., natural frequencies and damping capacity) can be used to identify and assesses the health of the structures. It is very advantageous method to obtain damping characteristics of the materials at higher frequency and at relatively low amplitudes.

**Author details**

Beera Satish Ben<sup>1</sup>

**167**

\* and Beera Avinash Ben<sup>2</sup>

2 Avanthi Institute of Engineering and Technology, Visakhapatnam, India

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material…*

*DOI: http://dx.doi.org/10.5772/intechopen.86134*

1 National Institute of Technology, Warangal, India

\*Address all correspondence to: satishben@nitw.ac.in

provided the original work is properly cited.

*Damage Identification and Assessment Using Lamb Wave Propagation Parameters and Material… DOI: http://dx.doi.org/10.5772/intechopen.86134*
