# Possibilities of Detecting Damage Due to Osmosis of GFRP Composites Used in Marine Applications
## 3 Modelling Vibrothermography NDT  
### Matematical Problem  
The principles of vibrothermography are illustrated in Figure 3.  
Theoretical numerical calculations can be divided into two steps with 3D solutions to the following problems:  
Figure 3: Vibrothermography principle [21]. * propagation of mechanical waves in the object and the thermal power generated by the defect,
* propagation of heat generated by the defects based on the results of the first stage. Below, a description of the steps for numerical calculations is presented. The presented algorithms were developed on the basis of the theoretical approach described in the publications [23; 24]. The thermomechanical model implements parallel numerical solutions for the propagation of thermal energy and mechanical waves. Figure 3 shows the mechanical stresses (normal and tangential) that act on each face of the elementary volume of the tested object. Along the two coordinates, the tangential stresses have two components. By adding up one normal component and two tangent components there are three stress components on each surface of the parallelepiped. For example, on the surface perpendicular to X we have one normal component, \(\sigma_{x}\), and two tangent components, \(\tau_{xy}\) and \(\tau_{xz}\) (the first subscript shows the coordinate axis which is parallel to the normal vector of the parallelepiped surface, and the second subscript indicates the coordinate axis which is parallel to the tangent stress component). As described in Figure 4, the mechanical stresses acting on each plane of a solid parallelepiped act as three components of stress. For example, on the plane perpendicular to the \(x\) coordinate, these are \(\sigma_{x}\), \(\tau_{xy}\), and \(\tau_{xz}\) stresses. The balance of forces along \(x\) can be described by the formula:  
\[(\sigma_{x+0}-\sigma_{x-0})\Delta y\Delta z+\big{(}\tau_{y+0,x}-\tau_{y-0,z} \big{)}\Delta x\Delta z+(\tau_{z+0,x}-\tau_{z-0,x})\Delta x\Delta y+X\Delta x \Delta y\Delta z=0 \tag{1}\]  
where \(\Delta x\), \(\Delta y\), and \(\Delta z\) are the parallelepiped dimensions and \(X\) is the projection of volume forces (if they are present) on the \(x\) axis. The subscripts \(x+0\) and \(x-0\) correspond to the maximum and minimum \(x\) coordinates. The differential form of Equation (1) will be as follows:  
\[\frac{\partial\sigma_{x}}{\partial x}+\frac{\partial\sigma_{yz}}{\partial y}+ \frac{\partial\sigma_{zx}}{\partial z}+X=0 \tag{2}\]  
Figure 4: Distribution of stresses acting on the tested sample with an elemental volume [25]. The numerical solution of the obtained set of basic equations was made using ThermoSon software:  
\[(\lambda+2G)\frac{\partial^{2}U}{\partial x^{2}}+G\left(\frac{\partial^{2}U}{ \partial y^{2}}+\frac{\partial^{2}U}{\partial z^{2}}\right)+(\lambda+G)\frac{ \partial^{2}V}{\partial x\partial y}+(\lambda+G)\frac{\partial^{2}W}{\partial x \partial z}=0 \tag{3}\]  
where \(\lambda\) is the Lame constant, \(G\) is the shear modulus, and \(U\), \(V\), and \(W\) are the displacement projections on the \(x,y,z\) coordinates. The right element of the Lame Equation (3) was replaced with the corresponding components of the inertia forces \(F_{in}\). This is because we are considering elastic waves in a solid. \(F_{in}\) in a chosen elementary volume. For instance, in the \(x\) direction:  
\[F_{in,x}=\rho\frac{\partial^{2}U}{\partial\tau^{2}} \tag{4}\]  
where \(\rho\) is the material density. The following model boundary conditions were adopted:  
1. normal and tangential stresses on the surface of the upper and lower plate do not occur
2. zero vertical displacements on the lower surface and forced harmonic vibrations at the point of stimulation. The internal fracture surface is characterized by nonlinear boundary conditions. At the crack compression stage, no tangential stresses are assumed, and at the crack deformation stage, the condition of zero normal stresses is added. Assuming that the appropriate elements of Equation (3) are equal to zero, the above conditions will be achieved.