On the Constants of the Nonlinear Murnaghan’s Hyperelastic Material Model

Nonlinear Guided Wave Tomography for Detection and Evaluation of Early-Life Material Degradation in Plates

“…In order to formulate the field equations of nonlinear guided waves, the Lagrangian description is used, as shown in Figure 1 . An elastic potential function, W , of Murnaghan’s model can be expanded as [ 22 , 28 ]: where denotes the trace of the bracketed tensor, and are Lamé constants, are Murnaghan’s TOECs, which are directly related to Landau–Lifshitz’s TOECs [ 22 , 25 ], and are the invariants of the Lagrangian strain tensor which, related to the displacement vector . The second Piola–Kirchhoff stress tensor, T PK 2 , can be obtained from W : where I is the identity tensor.…”

“…The theory of elasticity shows that with linear kinematic equations, the physical nonlinearity is formally associated with the nonlinear relationship between the stress tensor and strain tensors [ 23 ]. Therefore, Murnaghan’s model acts as a weakly nonlinear model, as opposed to the linear (Hookean) model [ 24 , 25 ], to define the various degrees of degradation. On the other hand, it was demonstrated that the third-order elastic constants (TOECs) in Murnaghan’s model are sensitive to fatigue damage [ 26 ].…”