**2. Theory**

## **2.1 Local defect resonance (LDR)**

The local defect resonance (LDR) is a phenomenon that occurs mainly due to presence of a defect in any structure. Whenever, a structure consists of damage, the local stiffness and mass of the structure at defect location changes. This change leads to a new parameter known as the effective stiffness and effective mass of the structure over the defect area. According to Solodov et al. [21], when the plate with defect is excited with a frequency that matches with the frequency of the defect due to its effective stiffness and effective mass, resonance occurs. This phenomenon of resonance at the defect area is termed as local defect resonance. The LDR phenomenon leads to high amplitude of vibration at the defect location while the rest of the plate remains at negligible amplitudes. Exciting a plate with one of its LDR

*Defect Detection in Delaminated Glass-Fibre/Epoxy Composite Plates Using Local Defect… DOI: http://dx.doi.org/10.5772/intechopen.101178*

frequency will lead to clapping and rubbing action at the corresponding defect. This clapping and rubbing action will further generate a local heat over the damage area. The analytical relation for detection of LDR frequency is given by Solodov et al. [21], in case of a flat bottom hole, which can be found analogous to a case of delamination in composites.

**Figure 1** shows a schematic of four layered composite with central delamination. The analytical relation for calculating LDR frequency is based on parameters such as diameter of delamination (*a*) and residual thickness (*t*). The effective stiffness (*Ke*) of the structure at defect location is given by [26]

$$K\_{\varepsilon} = \frac{64\pi Et^3}{a^2(1-\nu^2)}\tag{1}$$

Where *E* is modulus of elasticity and *ν* is Poisson's ratio. Subsequently, the effective mass of delamination is given as [26].

$$M\_{\epsilon} = \frac{9}{20} \pi \rho t a^2 \tag{2}$$

Finally, the analytical expression for calculating the LDR frequency of any defect in form of delamination is obtained by substituting Eqs. (**1**) and (**2**) in natural frequency relation of the structure, and is expressed as [26]

$$f\_{LDR} = \frac{6.4t}{a^2} \sqrt{\frac{E}{12\rho(1-\nu^2)}}\tag{3}$$

Where *ρ* is mass density of the material.

## **2.2 LDR based vibro-thermography**

The excitation of a structure having delamination with the LDR frequency corresponding to its damage leads to a rise in temperature at the defect location, as discussed previously. The heat generated at the defect location due to clapping and rubbing phenomena at the defect interface propagates away from the defect layer towards the surface of the structure. These thermal signatures are captured by using IR imaging and the technique is called as LDR based vibro-thermography. In this chapter, the LDR based vibro-thermography will be discussed by carrying out a

**Figure 1.** *Schematic of a four layered composite structure with circular delamination at the central layer.*

explicit coupled temperature-displacement analysis. The nonlinear wave propagation analysis is carried out on glass fibre reinforced polymer (GFRP) composite in order to obtain the thermal response of the structure at the defect interface. The numerical model considered must fulfil the equilibrium condition in the deformed structure to obtain the solution of forces and displacements at specific nodes. The momentum equation based on principal of virtual displacement can be used for translation and rotational motion of the body under harmonic loading, and is given as [25].

$$\int\_{\xi} \delta d\ \sigma d\xi + \int\_{\xi} \delta v \ \rho \dot{v} d\xi + \int\_{\xi} \delta v \ B d\xi - \int\_{\emptyset} \delta v \ \mathcal{S}\_t \, d\rho = \mathbf{0} \tag{4}$$

where *ξ* is current domain, *φ* is boundary of the body, *δd* is the virtual displacement, *σ* is stress, *δv* is virtual velocity, *B* is the body force vector and *St* is the surface traction. These terms represent the internal forces on the system, inertia force, body force and surface traction in the system, respectively.

The Courant-Friedrichs-Levy condition is satisfied while carrying out the explicit temperature-displacement analysis that requires integrating through small time increments [27]. The central-difference and forward-difference are required to be stable and hence a limit of time increment is chosen as

$$
\Delta t \le \min.\left(\frac{2}{\alpha\_{\max}}, \frac{2}{\lambda\_{\max}}\right) \tag{5}
$$

Where *ω*max is highest natural frequency of the structure and *λ*max is largest eigenvalue for the solution. The time increment, Δ*t* must be calculated such that the wavelength does not exceed more than a single element edge length. Therefore, the size of element and time increment during the temperature-displacement analysis is a critical factor that needs to be taken care of. Otherwise, higher time increment will lead to high amplitude oscillation of the time history variables and an unstable solution.

During the coupled temperature-displacement analysis, internal stress and strain are developed due to harmonic excitations from the incident wave which further results in temperature rise at the defect location. The transient heat transfer occurring at the defect surface is computed using the following equation [25].

$$
\rho \mathbf{C}\_p \frac{\partial T}{\partial t} = \nabla \phi + H\_\mathbf{g} \tag{6}
$$

Where *Cp* is specific heat capacity,*T* is time dependent temperature field, *ϕ* is heat flux vector per unit volume and *Hg* is total internal heat generated per unit volume. From the Fourier's heat conduction law, the time dependent temperature during each time step can be calculated which is discretised and expressed as [25].

$$T\_{n+1} = \left(\frac{C\_n}{\Delta t} + \chi\_n\right)^{-1} \left(F\_n + T\_n \frac{C\_n}{\Delta t}\right) \tag{7}$$

Where *n* is the analysis step (*n* = 1, 2, 3, … ), *C* is heat capacity matrix, *γ* is conductivity matrix and *F* is thermal force vector. The temperature rise at the defect location due to internal heat when the plate is excited at LDR frequency can be evaluated using the above equation. Therefore, vibro-thermography can be performed on a structure by incorporating LDR excitations that lead to higher amplitude of vibration at defect area causing a temperature rise.

*Defect Detection in Delaminated Glass-Fibre/Epoxy Composite Plates Using Local Defect… DOI: http://dx.doi.org/10.5772/intechopen.101178*
