We present and compare the joint and dual variants of the extended Kalman filter for coupled state and parameter identification problems. With reference to nonlinear dynamics of layered composites, we assume that the elastic properties of the laminae are known, whereas the softening constitutive law of the interlaminar phases adopted to simulate delamination needs to be calibrated. Purpose of this study is the identification of the interlaminar model and of the debonding surface(s) on the basis of free-surface measurements only.
We show that, in the case of a dominant dilatational wave propagating in the through-the-thickness direction of the laminate, the free-surface displacement can be weakly sensitive to some constitutive parameters, and the relevant model calibration is not performed optimally. As far as delamination detection is concerned, the dual filter performs by far better than the joint filter, particularly in noisy environments.
Impacts of foreign objects on the surface of laminates can represent a major source of failure for layered composite structures. The final rupture mechanism can be linked to nonlinear irreversible phenomena inside each lamina as well as to the nucleation and propagation of cracks inside the resin-enriched interlaminar phases (see for a recent review). This latter phenomenon is commonly termed delamination.
In this work dynamic delamination is simulated assuming that: laminae behave elastically; interlaminar crack growth is governed by a softening interface model, whose mechanical properties are the peak strength and the fracture energy , , . Rate-effects, which should be taken into account when studying low-velocity impacts on polymer matrix composites , are here disregarded in order to simplify the analysis. Furthermore, intralaminar damage processes are not accounted for.
Because of the vanishing thickness of the resin-enriched phases, as compared to the thickness of the whole laminate, direct interlaminar testing can not be performed. Various experimental procedures have been proposed in the literature to determine strength and toughness of composite specimens; in these tests the local interphase properties can be shadowed by structural effects, leading to an incorrect interface model calibration (see, e.g., ). To avoid such a drawback, inverse analysis can be resorted to identify the constitutive parameters from loads and/or displacements recorded during standard laboratory tests. Following a methodology already pursued by the authors for composites loaded in the static , ,  and in the dynamic , ,  regimes, the extended Kalman filter (EKF) ,  is here used to calibrate a nonlinear, rate-independent interface constitutive law on the basis of results from impact tests on laminates.
The Kalman filter was originally proposed to estimate the state of dynamic systems on the basis of historical data (see, e.g., , ); nowadays, it is a well known procedure applied in many real-time control systems. Its use for parameter identification was first attempted in  (see also ) and subsequently applied in various fields, among which geotechnical, structural and mechanical engineering (, , ). The identification of model parameters is obtained by applying the Kalman filter to a dynamically evolving system that is described by a set of state variables, including the parameters to be identified. The filter thus furnishes an estimate of parameters recursively in time, treating the experimental information sequentially and improving the estimate according to a step-by-step procedure.
The algorithm for parameter identification reduces to the well known recursive least squares when the set of state variables simply coincides with the set of parameters to be identified and no additional features of the dynamic system are left to be determined , . The main advantage given by Kalman filtering is represented in this case by the non-deterministic, stochastic framework which allows the introduction of measurement and modeling errors. In the more general case of coupled state and parameter identification, the recursive estimate gives also some important information on system evolution. This approach appears to be well suited for dynamic structural systems.
It has been shown (see, e.g., ) that for nonlinear responses, a recursive procedure is usually more efficient than a batch approach; information on the nonlinear, irreversible processes can in fact be better captured selectively in time.
Anyway, the customary EKF does not perform optimally when structural nonlinearities are severe , . When softening is adopted to describe the interlaminar debonding process, the filter is prone to numerical instability, and quite poor model calibrations are obtained. To overcome instability effects, starting from recent progresses in the formulation of linearized Kalman filters, a dual extended Kalman filter (DEKF) suited for nonlinear structural dynamics is presented . We refer to the customary filter  as the joint extended Kalman filter (JEKF), for reasons that will become clear in Section 4, where the two algorithms are detailed.
To simplify the analyses, in this work we consider only delamination caused by the propagation of stress waves in the through-the-thickness direction, i.e. in the direction perpendicular to the stacking plane. The methodology can obviously be extended to full three-dimensional analyses; this work is at present left for future investigations.
The outline of the paper is as follows. In Section 2 we present the equations governing nonlinear composite dynamics. Section 3 describes the adopted interface constitutive modeling, highlighting the physical meaning of the model parameters to be identified. In Section 4 we present the JEKF and the DEKF. Section 5 reports results concerning pseudo-experimental impacts on two- and five-layer laminates, while in Section 6 actual testings on silicon carbide and glass fiber reinforced plastic composite are considered. Finally, some conclusions and a discussion on topics which deserve further studies are drawn in Section 7.
Nonlinear dynamics of layered composites
Let us consider a two-dimensional continuum Ω, which is divided by nΓ interfaces Γi into nΩ = nΓ + 1 disjoint parts Ωi (Fig. 1). The bulk material in Ω⧹Γ, with Γ = ∪iΓi, is assumed to obey a linear elastic constitutive law, while a nonlinear softening model along each Γi accounts for the possible, progressive failure of the interfaces.
After space discretization, the equation of motion of Ω in the small strain regime turns out to bewhere M and D are the mass and viscous damping
Interface constitutive model
To simulate interlaminar debonding in layered composites, the constitutive model for the resin-enriched interphases between plies has to account for strength degradation, up to complete failure, under tensile loadings.
Because of the small ratio between the thickness of a damaging interphase and the thickness of the whole laminate, the nonlinear response is lumped to a zero-thickness interface characterized by means of a traction vs displacement discontinuity relation. For mode I dynamic
Kalman filtering for nonlinear system dynamics
In this section we first use the typical formalism of the automatic control literature for nonlinear dynamic systems. Subsequently, we adapt it to the discretized equations governing composite dynamics, in order to highlight the main features of the EKF.
Let the nonlinear evolving system be described by the state-space model , :where z is the state vector, which collects all the nodal displacements used to characterize the current state; ϑ is
State and parameter identification in pseudo-experimental testing
In this section we consider a model problem for impact tests on laminates. With reference to Fig. 3, Fig. 14, we study only a portion of the composite (having vertical interlaminar surfaces Γi in the figures) and assume that a foreign object (impactor) is striking at t0 = 0 from the left the external surface of the specimen. We restrict the analyses to the region below the impact surface, where dilatational waves propagating in the through-the-thickness direction (the horizontal direction in the
State and parameter identification in actual experimental testing
To fully assess the performances of JEKF and DEKF, two actual impact testings are considered. In the first one a plane shock wave is generated inside silicon carbide plates to cause spalling, i.e. dynamic fracture of the specimens. In the second one an impact on a glass fiber reinforced composite panel is used to study the effects of delamination on the composite response.
In this paper we have compared two versions of the extended Kalman filter for coupled state and parameter identification problems, account taking of the nonlinear dynamics of the system in a noisy environment.
To identify model parameters, the JEKF is usually resorted. Stability and convergence analyses for this filter have been presented, but most of them are limited to linear systems. For nonlinear systems, a comprehensive mathematical analysis of the algorithm is at present missing; anyway,