13. Damage Detection in an Energy Flow Model Including Parameter Uncertainty

Structural energy dissipation pattern is modified by the presence of discontinuities like a crack. Crack nucleation and growth reduces the structural stiffness which makes this effect useful as a damage indicator. Computational models have become the main tool for understanding the behavior of complex structures when experimental evaluation can be difficult to perform. However, many of this classical numerical analysis assumes a deterministic model and almost nothing is understood about the effect of uncertainty in the parameters, external forces and boundary conditions. This work presents a study about the energy flow patterns caused by localized damage in structures like rod, including uncertainties in a geometric parameter. The problem is solved in two steps. First, the structure is modeled by the Spectral Element Method (SEM). The mean and variance of displacement responses are obtained by using the Polynomial Chaos (PC) expansion. In PC the stochastic solutions are expanded as orthogonal polynomials of the input random parameters. Second, by using the displacements obtained in the step before, the mean and variance of energies are calculated by applying the expectation into the equations of energy density and energy flow. However, this approach produces unusual equations for expected values and covariances. Like, the expected value for a product of three random correlated variables, whose solution includes the covariance between one variable and a product of two others variables. A formulation is developed and proposed to solve this problem. Monte Carlo Simulation (MCS) is used to validate the results obtained by these solutions. Numerical examples are analyzed for some different cases, which present good approximation as compared with MCS results.