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Algorithms 2009, 2(2), 808827; https://doi.org/10.3390/a2020808
Article
Failure Assessment of Layered Composites Subject to Impact Loadings: a Finite Element, SigmaPoint Kalman Filter Approach
Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Received: 1 April 2009 / Accepted: 27 May 2009 / Published: 4 June 2009
Abstract
:We present a coupled finite element, Kalman filter approach to foresee impactinduced delamination of layered composites when mechanical properties are partially unknown. Since direct numerical simulations, which require all the constitutive parameters to be assigned, cannot be run in such cases, an inverse problem is formulated to allow for modeling as well as constitutive uncertainties. Upon space discretization through finite elements and time integration through the explicit $\alpha $method, the resulting nonlinear stochastic state model, wherein nonlinearities are due to delamination growth, is attacked with sigmapoint Kalman filtering. Comparison with experimental data available in the literature and concerning interlaminar failure of layered composites subject to lowvelocity impacts, shows that the proposed procedure leads to: an accurate description of the failure mode; converged estimates of interlaminar strength and toughness in good agreement with experimental data.
Keywords:
layered composites; impactinduced delamination; Kalman filter; sigmapoint transformation1. Introduction
Composite structures are often exposed to impact loadings during their life cycle [1]; this kind of loading conditions may become a serious issue, since composites can be very sensitive to imperfections beyond a critical threshold load. Depending on the impact energy, i.e. on the momentum of the impactor when it strikes the structural component, two different failure scenarios can be expected: (i) in case of socalled low velocity (and, therefore, low energy) impacts, stress waves propagate inside the composite leading to diffused damage processes that progressively decrease or even annihilate the residual strength and stiffness of the structure; (ii) in case of high velocity (and, therefore, high energy) impacts, local perforation of the laminate occurs, giving rise to very localized failure mechanisms. In the former case the response of a laminated plate to the impact is characterized by forced vibrations of the whole structure; in the latter case, the response of a plate and the damaging processes are very localized around the impact location, and deformation modes of the structure as a whole are only marginally excited.
Having as a target the development of a realtime health monitoring (and, hopefully, control) strategy for composite structures, in this work we focus on low velocity impact conditions. In such cases, the aforementioned diffused degradation processes consist in interlaminar debonding (i.e. delamination) and intralaminar damage, the latter being related to fibers failure, matrix cracking and fiber/matrix decoherence. Typically, a major role is played by delamination; therefore, throughout the whole paper we shall assume the laminae of the composite to behave elastically, while reduction of the structural loadcarrying capacity may occur only because of interlaminar damage.
To simplify modeling, the small ratio between the thickness of each interlaminar phase and the thickness of the whole laminate is exploited by numerically treating interlaminar phases as lumped interfaces. The relevant constitutive laws locally link the tractions acting upon the surfaces of the two adjacent layers to the displacement jump occurring across the interface, allowing for both opening and sliding (or shear) deformation modes. This lumped interface formulation has to be considered as a coarse grained description of the actual interphase behavior, as obtained via homogenization along the thickness direction (i.e. in the direction perpendicular to the stacking plane); the displacement jump thus accounts for both the elastic response of the interphase material and the possible discontinuity in the displacement field due to microcracking. To model microcracking processes and the relevant reduction of the interlaminar strength, nonlinear constitutive laws are adopted; beyond the attainment of peak tractions, delamination inception is modeled through a softening regime. Since interphase materials are usually quasibrittle, dissipative phenomena due to yielding can be disregarded. Moreover, viscous and ratedependent properties of the composite phases are not considered here; therefore, only rateindependent constitutive laws will be adopted.
Within this framework, laminated plates are treated as a stacking sequence of laminae and interfaces. This scheme, sometimes referred to as mesoscale modeling (see, e.g. [2]), was proved to furnish accurate outcomes at the length scale of the whole laminate (also in terms of delamination prediction), provided the interface constitutive laws are appropriately calibrated. Since they are typically formulated on a pure phenomenological basis and contain several parameters to be tuned, this calibration task turns out to be difficult, also because of the lower dimensionality of interfaces (twodimensional loci in a threedimensional setting) and because of shadowing effects due to the global laminate response, see e.g. [3].
To accurately predict the failure mode of layered composites subject to low velocity impacts and simultaneously calibrate interface constitutive laws, a coupled finite element (FE) Kalman filter approach is here proposed. Upon space discretization (through displacementbased finite elements) and time integration (through the explicit $\alpha $method, in order to damp spurious high frequency oscillations in the fully discretized solution), Kalman filtering is adopted to fully track the structural state in a stochastic frame [4,5,6], exploiting system evolution to continuously improve estimates. Because of the nonlinearities induced by interlaminar softening, the customarily adopted extended Kalman filter (EKF) is not able to provide robust state tracking and accurate estimations of interlaminar properties like e.g. strength and toughness [7]. We therefore attack the problem with the newly proposed sigmapoint Kalman filter (SPKF) [8,9,10], which avoids the occurrence of unstable evolution of the estimates (see also [11]). While the EKF linearizes the system evolution equations (through a Taylor series expansion of the relevant mapping, thought to be analytic everywhere, arrested at first order) and accordingly updates the statistics of the structural state, the SPKF samples the probability distribution of the state via an appropriately chosen set of sigmapoints, which are then evolved according to the actual nonlinear system dynamics to obtain the updated statistics of the state itself. Among alternative approaches to system identification and, specifically, to model calibration, it is worth quoting here [12].
In this work the capability of the proposed FESPKF approach is assessed through comparison with experimental data available in the literature and concerning failure of laminated plates subject to plane impacts [13,14]; in such tests, the composites are mainly stressed in the region surrounding the impact location by dilatational waves propagating in the throughthethickness direction and causing mode I delamination. Outcomes show that the failure mode is always correctly captured, and estimated values of interlaminar strength and toughness well agree with available data.
Plane impact tests do not allow a complete calibration of the interface constitutive law, since only model parameters relevant to mode I failure can be estimated. Anyway, forthcoming results show that under very adverse conditions, as due to fast completion of the whole debonding process, the SPKF proves to be robust and accurate; on the contrary, under similar conditions the EKF was shown in [7,15] to either diverge or provide biased estimates. An extensive assessment of the SPKF performance in the presence of diffused damage/delamination patterns is beyond the scope of this paper and will be presented in future works.
As far as notation is concerned, a matrix one will be adopted throughout, with uppercase and lowercase bold symbols respectively denoting matrices and vectors. A superscript ${}^{\text{T}}$ will stand for transpose, $\square $ will denote the norm of vector □, and a superposed dot will represent time rates.
2. Forward problem: layered composite dynamics
Let us consider a layered body Ω, possessing a smooth outer boundary with unit outward normal $\mathbf{m}$ (see Figure 1). Ω be crossed by ${n}_{\Gamma}$ nonintersecting surfaces, or interfaces ${\Gamma}_{j}$, $j=1,...,{n}_{\Gamma}$. Even though the local orientation ${\mathbf{n}}_{j}$ of these interfaces may vary in Ω, for ease of notation we shall assume it to be constant and equal to $\mathbf{n}$ along every ${\Gamma}_{j}$: interfaces are thus flat and parallel to each other.
The equilibrium of Ω at time t is governed by:
where: $\Gamma ={\cup}_{j=1}^{{n}_{\Gamma}}{\Gamma}_{j}$; ${\Gamma}_{\tau}$ is the portion of the outer boundary of Ω where tractions $\overline{\tau}$ are assigned. Treating each ${\Gamma}_{j}$ as a locus where the displacement field $\mathbf{u}$ may suffer jumps (alike a crack), ${\Gamma}_{j}^{+}$ and ${\Gamma}_{j}^{}$ are the two sides of ${\Gamma}_{j}$; these two sides can not be distinguished in the initial configuration (at $t=0$) but, as soon as delamination occurs, they experience a relative movement. In the above equations, according to Voigt notation: σ is the stress vector; ${\tau}_{j}$ is the traction vector acting along surface ${\Gamma}_{j}$; $\overline{\mathbf{b}}$ is the assigned bulk load in $\Omega \backslash \Gamma $; ϱ is the bulk mass density; $\ddot{\mathbf{u}}$ is the acceleration field in $\Omega \backslash \Gamma $; $\mathcal{C}$ is the differential compatibility operator; $\mathcal{M}$ and $\mathcal{N}$ are the matrices collecting the components of unit vectors $\mathbf{m}$ and $\mathbf{n}$, respectively.
$$\begin{array}{c}\hfill {\mathcal{C}}^{\text{T}}\sigma +\overline{\mathbf{b}}=\varrho \ddot{\mathbf{u}}\phantom{\rule{2.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\Omega \backslash \Gamma \end{array}$$
$$\mathcal{N}\sigma ={\tau}_{j}\phantom{\rule{2.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Gamma}_{j}^{+}\text{,}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\mathcal{N}\sigma ={\tau}_{j}\phantom{\rule{2.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Gamma}_{j}^{}\text{,}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}j=1,...,{n}_{\Gamma}$$
$$\begin{array}{c}\hfill \mathcal{M}\sigma =\overline{\tau}\phantom{\rule{2.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Gamma}_{\tau}\end{array}$$
Assuming linearized kinematics, compatibility reads:
$$\begin{array}{c}\hfill \epsilon =\mathcal{C}\mathbf{u}\phantom{\rule{2.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\Omega \backslash \Gamma \end{array}$$
$${\left[\mathbf{u}\right]}_{j}=\mathbf{u}{}_{{\Gamma}_{j}^{+}}\mathbf{u}{}_{{\Gamma}_{j}^{}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}j=1,...,{n}_{\Gamma}$$
Figure 2.
Piecewise linear (PWL) and linearexponential (LE) effective tractiondisplacement discontinuity relationships.
$$\begin{array}{c}\hfill \mathbf{u}=\overline{\mathbf{u}}\phantom{\rule{2.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Gamma}_{u}\end{array}$$
The body is assumed to be initially at rest, in an undeformed and unstressed state:
$${\mathbf{u}}_{0}=\mathbf{0},\phantom{\rule{2.em}{0ex}}{\dot{\mathbf{u}}}_{0}=\mathbf{0}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\Omega $$
In case of low velocity impacts, dissipation due to intralaminar damage is assumed to be negligible when compared to that due to delamination. Hence, laminae are thought to behave elastically according to:
where ${\mathbf{E}}_{\Omega}$ is the elasticity matrix of the bulk.
$$\begin{array}{c}\hfill \sigma ={\mathbf{E}}_{\Omega}\epsilon \phantom{\rule{2.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\Omega \backslash \Gamma \end{array}$$
As for interlaminar phases, to be treated as lumped surfaces along which debonding is the result of complex microscale damaging processes, a phenomenological coarsegrained modeling is adopted. Strength reduction preceding delamination is governed by softening constitutive laws [3,16,17,18,19,20,21], which locally link tractions ${\tau}_{j}$ (see Eq. (2) to the displacement jump ${\left[\mathbf{u}\right]}_{j}$ (see Eq. (5). Since both opening/closing (along $\mathbf{n}$) and sliding (in the ${\mathbf{s}}_{1}^{j}{\mathbf{s}}_{2}^{j}$ plane) discontinuities are expected to take place under general loading conditions, interface kinematics is effectively represented by the displacement discontinuity $\left[u\right]$ (in what follows, for ease of notation, we shall drop subscript j), defined as [22,23,24]:
where: ${\left[u\right]}_{n}={\left[\mathbf{u}\right]}^{\text{T}}\mathbf{n}$ and ${\left[u\right]}_{s}=\left\left[\mathbf{u}\right]{\left[u\right]}_{n}\mathbf{n}\right$ are the opening and sliding displacement discontinuities, respectively; κ is a coupling coefficient, which phenomenologically accounts for the local ongoing mixedmode damaging in the interlaminar phase. The effective traction τ, workconjugate to $\left[u\right]$, reads (see [22,23]):
where ${\tau}_{n}$ and ${\tau}_{s}$ are the normal and shear traction components.
$$\begin{array}{c}\hfill \left[u\right]=\sqrt{{\left[u\right]}_{n}^{2}+{\kappa}^{2}{\left[u\right]}_{s}^{2}}\end{array}$$
$$\begin{array}{c}\hfill \tau =\sqrt{{\tau}_{n}^{2}+{\displaystyle \frac{{\tau}_{s}^{2}}{{\kappa}^{2}}}}\end{array}$$
Since quasibrittle materials (like interphase resins) show a high ratio between compressive and tensile strengths, interlaminar damage in compression due to low velocity impacts can be disregarded. On the other hand, under tensile stress pulses the response of interphase materials is characterized by softening beyond the attainment of a peak traction. Two simple effective descriptions of this behavior are here envisaged: a piecewise linear (PWL) law
and a linearexponential (LE) law
In these cohesivelike envelopes: ${\left[u\right]}_{e}$ is the effective displacement discontinuity at the attainment of the peak traction ${\tau}_{M}=K{\left[u\right]}_{e}$ (elastic limit); K is the stiffness in the elastic regime; Q is the (negative) slope of the softening branch in the PWL case; ${\left[u\right]}_{U}=\left(1\frac{K}{Q}\right){\left[u\right]}_{e}$ is the effective displacement discontinuity beyond which the interaction of opened crack faces is annihilated in the PWL case; ς defines the slope of the softening branch in the LE case. The two effective models (11) and (12) are compared, for $\left[u\right]>0$, in Figure 2 at assigned effective fracture toughness, or work of separation $G=\underset{0}{\overset{\infty}{\int}}\tau \phantom{\rule{0.166667em}{0ex}}d\left[u\right]$. It is worth noting that, since the LE law assumes the interaction between crack faces to continue up to $\left[u\right]\to \infty $, delamination growth is modeled via a breakdown threshold; according to [25,26], we assume this interaction to suddenly vanish as soon as the current traction τ becomes smaller than a predefined fraction (say 510 %) of the peak value ${\tau}_{M}$.
$$\left\{\begin{array}{cc}\tau =K\left[u\right]\phantom{\rule{2.em}{0ex}}\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}\left[u\right]\le {\left[u\right]}_{e}\hfill \\ \tau ={\tau}_{M}+Q\left(\left[u\right]{\left[u\right]}_{e}\right)\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}{\left[u\right]}_{e}<\left[u\right]\le {\left[u\right]}_{U}\hfill \\ \tau =0\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}\left[u\right]>{\left[u\right]}_{U}\hfill \end{array}\right.$$
$$\left\{\begin{array}{cc}\tau =K\left[u\right]\phantom{\rule{2.em}{0ex}}\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}\left[u\right]\le {\left[u\right]}_{e}\hfill \\ \tau ={\tau}_{M}exp\left(\varsigma (\left[u\right]{\left[u\right]}_{e})\right)\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}\left[u\right]>{\left[u\right]}_{e}\hfill \end{array}\right.$$
To allow for irreversibility of the micromechanical damage processes, unloading from the tensile envelope follows a radial path to the origin of the $\tau \left[u\right]$ plane. In rate form, constitutive modeling for interface ${\Gamma}_{j}$ can thus be written:
where ${\mathbf{E}}_{\Gamma}$ is the interface tangent stiffness matrix. For additional details, readers are referred to [23,24].
$$\begin{array}{c}\hfill {\dot{\tau}}_{j}={\mathbf{E}}_{\Gamma}{\left[\dot{\mathbf{u}}\right]}_{j}\end{array}$$
Upon space discretization via displacementbased finite elements, the weak form of the equilibrium equations for the layered body reads:
where ${\mathbf{u}}^{h}$ is the vector of nodal displacements. In (14) the mass matrix $\mathbf{M}$, the bulk stiffness matrix ${\mathbf{K}}_{\Omega}$, the internal force vectors ${\mathbf{R}}^{j}$, $j=1,...,{n}_{\Gamma}$, and the external load vector $\mathbf{F}$ are respectively given by:
Here: Φ is the matrix of nodal shape functions; ${\mathbf{B}}_{\Omega}$ and ${\mathbf{B}}_{{\Gamma}_{j}}$ are, respectively, the compatibility matrices for the bulk $\Omega \backslash \Gamma $ and surface ${\Gamma}_{j}$ [7,27].
$$\mathbf{M}{\ddot{\mathbf{u}}}^{h}+{\mathbf{K}}_{\Omega}{\mathbf{u}}^{h}+\sum _{j=1}^{{n}_{\Gamma}}{\mathbf{R}}^{j}=\mathbf{F}$$
$$\begin{array}{cc}\hfill \mathbf{M}=& \underset{\Omega \backslash \Gamma}{\int}\varrho \phantom{\rule{0.166667em}{0ex}}{\Phi}^{\text{T}}\Phi \phantom{\rule{0.166667em}{0ex}}d\Omega \hfill \\ \hfill {\mathbf{K}}_{\Omega}=& \underset{\Omega \backslash \Gamma}{\int}{\mathbf{B}}_{\Omega}^{\text{T}}{\mathbf{E}}_{\Omega}{\mathbf{B}}_{\Omega}\phantom{\rule{0.166667em}{0ex}}d\Omega \hfill \\ \hfill {\mathbf{R}}^{j}=& \underset{{\Gamma}_{j}}{\int}{\mathbf{B}}_{{\Gamma}_{j}}^{\text{T}}{\tau}_{j}\phantom{\rule{0.166667em}{0ex}}d{\Gamma}_{j}\hfill \\ \hfill \mathbf{F}=& \underset{\Omega \backslash \Gamma}{\int}{\Phi}^{\text{T}}\overline{\mathbf{b}}\phantom{\rule{0.166667em}{0ex}}d\Omega +\underset{{\Gamma}_{\tau}}{\int}{\Phi}^{\text{T}}\overline{\tau}\phantom{\rule{0.166667em}{0ex}}d{\Gamma}_{\tau}\hfill \end{array}$$
In the spacediscretized formulation here above we have implicitly assumed that the displacement field ${\mathbf{u}}^{h}$ may be discontinuous only along interelement boundaries. Smarter formulations [23,28,29,30,31] allow also for intraelement discontinuities by exploiting the partition of unity property of standard nodal shape functions [32]; since delamination in layered continua occurs along the apriori known interlaminar surfaces, the description of the discontinuity topology is simple and the aforementioned feature of the smart (usually termed extended) finite element formulation is of no help.
We now need to adopt a time stepping scheme for (14), able to reduce to a minimum or even avoid spurious high frequency oscillations of the velocity field in the fullydiscretized solution. In [33] it was shown that local corrective flux terms can be adopted to accurately track discontinuities in the velocity field due to the propagation of stress wave fronts. Since this time stepping scheme is computationally expensive while our target is a lowcost scheme, a less expensive (and, indeed, less accurate) explicit $\alpha $method is adopted [34,35]. Having partitioned the time interval of interest according to $\left[{t}_{0}\phantom{\rule{1.em}{0ex}}{t}_{N}\right]={\cup}_{i=0}^{N1}\left[{t}_{i}\phantom{\rule{1.em}{0ex}}{t}_{i+1}\right]$, the solution of (14) at time ${t}_{i+1}$ in terms of nodal displacements, velocities and accelerations is obtained according to:
 prediction:$$\begin{array}{cc}\hfill {\tilde{\mathbf{u}}}_{i+1}^{h}=& {\mathbf{u}}_{i}^{h}+\Delta t\phantom{\rule{0.166667em}{0ex}}{\dot{\mathbf{u}}}_{i}^{h}+\Delta {t}^{2}({\displaystyle \frac{1}{2}}\beta ){\ddot{\mathbf{u}}}_{i}^{h}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\tilde{\dot{\mathbf{u}}}}_{i+1}^{h}=& {\dot{\mathbf{u}}}_{i}^{h}+\Delta t(1\gamma ){\ddot{\mathbf{u}}}_{i}^{h}\hfill \end{array}$$
 integration:$${\ddot{\mathbf{u}}}_{i+1}^{h}={\mathbf{M}}^{1}\left({\mathbf{F}}_{i+1+\alpha}(1+\alpha )\left(\mathbf{K}{\tilde{\mathbf{u}}}_{i+1}^{h}+\sum _{j}{\tilde{\mathbf{R}}}_{i+1}^{j}\right)+\alpha \left(\mathbf{K}{\mathbf{u}}_{i}+\sum _{j}{\mathbf{R}}_{i}^{j}\right)\right)$$
 correction:$$\begin{array}{cc}\hfill {\mathbf{u}}_{i+1}^{h}=& {\tilde{\mathbf{u}}}_{i+1}^{h}+\Delta {t}^{2}\beta \phantom{\rule{0.166667em}{0ex}}{\ddot{\mathbf{u}}}_{i+1}^{h}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\dot{\mathbf{u}}}_{i+1}^{h}=& {\tilde{\dot{\mathbf{u}}}}_{i+1}^{h}+\Delta t\gamma \phantom{\rule{0.166667em}{0ex}}{\ddot{\mathbf{u}}}_{i+1}^{h}\hfill \end{array}$$
The time step size $\Delta t$ has been set so as to fulfill the Courant condition in the bulk. In case of growing delamination, interface elements in the process zone (where damage is locally reducing the interlaminar strength and stiffness) need to be sized so as to accurately resolve the traction field; this requirement is satisfied if the characteristic size of interface elements is a fraction (say $\frac{1}{3}\frac{1}{5}$) of the process zone length, estimated e.g. through Irwin’s law [37]. Since the surrounding FEs in the bulk are associated to interface elements via matching grids, it turns out that interlaminar strength and toughness, which affect the process zone length, actually affect the time step size $\Delta t$ too; for details, readers are referred to [38,39].
3. Inverse problem: constrained sigmapoint Kalman filter
In our study the inverse problem has to provide realtime forecasts of the delamination processes, along with estimates of uncertain model parameters. Allowing for the above described explicit time integration scheme, a state vector $\mathbf{x}$ gathering all the information on the evolving system is obtained by joining the nodal displacement, velocity and acceleration vectors to a vector ϑ gathering model parameters in need of calibration:
Since laminate failure due to delamination is represented by displacement discontinuities arising along the interlaminar surfaces, vector $\mathbf{x}$ in (21) also provides the requested details of the failure mode. State vectors shaped in this way, with model parameters explicitly accounted for, are typically adopted in extended Kalman filtering [5,40]. While the current structural state (here ${\mathbf{u}}^{h}$, ${\dot{\mathbf{u}}}^{h}$ and ${\ddot{\mathbf{u}}}^{h}$) is always at least partially observed, model parameters to be calibrated can not be directly measured; by joining structural and parameter vectors in $\mathbf{x}$, state tracking can consistently enhance model calibration.
$$\begin{array}{c}\hfill \mathbf{x}=\left\{\begin{array}{c}{\mathbf{u}}^{h}\\ {\dot{\mathbf{u}}}^{h}\\ {\ddot{\mathbf{u}}}^{h}\\ \vartheta \end{array}\right\}\end{array}$$
Allowing for modeling and measurement errors in a stochastic framework, the discretized statespace model of the system within the time interval $\left[{t}_{i}\phantom{\rule{1.em}{0ex}}{t}_{i+1}\right]$ reads:
where: ${\mathbf{y}}_{i}$ is the observation vector, which collects measurables; ${\mathbf{v}}_{i}$ and ${\mathbf{w}}_{i}$ are the process and measurement noises. These noises are assumed to be additive, uncorrelated white and Gaussian processes, with zero mean and timeinvariant covariance matrices $\mathbf{V}$ and $\mathbf{W}$ [6,40]. Because of interlaminar strength degradation, mapping ${\mathbf{f}}_{i}$ turns out to be highly nonlinear; on the other hand, since either surface displacements or velocities are measured during testing, the observation equation (22${}_{2}$) shows up as a linear relation between ${\mathbf{y}}_{i}$ and ${\mathbf{x}}_{i}$ (see Eq. (21).
$$\left\{\begin{array}{cc}{\mathbf{x}}_{i+1}\hfill & ={\mathbf{f}}_{i}\left({\mathbf{x}}_{i}\right)+{\mathbf{v}}_{i}\hfill \\ {\mathbf{y}}_{i}\hfill & =\mathbf{H}{\mathbf{x}}_{i}+{\mathbf{w}}_{i}\hfill \end{array}\right.$$

The mentioned nonlinearities of mapping ${\mathbf{f}}_{i}$ cause estimates obtained through a standard EKF to be affected by biases [41,42]; to get accurate results, the recently proposed SPKF [8,9,43,44] is instead here adopted. While the EKF linearizes the mapping ${\mathbf{f}}_{i}$, the SPKF deterministically samples the probability distribution of $\mathbf{x}$ at time ${t}_{i}$ (i.e. at the beginning of the time step) through a set of properly chosen sigmapoints ${\widehat{\chi}}_{i,\mathsf{j}}$, $\mathsf{j}=0,...,{N}_{\chi}$. These sigmapoints are allowed to evolve according to the nonlinear mapping ${\mathbf{f}}_{i}$; the statistics of $\mathbf{x}$ at time ${t}_{i+1}$ (i.e. at the end of the time step) are then obtained through a weighted averaging procedure performed on the evolved sigmapoints [8]. The filtering procedure is detailed in Table 1, where $\mathbb{E}[\square ]$ represents the expected value of □. In this scheme, the number ${N}_{\chi}$ of sigmapoints, terms $\Delta {\chi}_{i,\mathsf{j}}$, and weights ${\omega}_{\mathsf{j}}$ and ${\omega}_{\mathsf{j}}^{\u2605}$ need to be set so as to achieve maximum accuracy; such issue is treated in Appendix A, where fundamental results are summarized.
In Kalman filtering it is usually assumed that all the state vector components are Gaussian variables; hence, the relevant probability distributions cannot be compliant with physical side constraints (like, e.g. positiveness of interlaminar strength and toughness). In this work we do not aim to modify this filtering frame with the adoption of alternative probability distributions; instead, we define terms $\Delta {\chi}_{i,\mathsf{j}}$ so that all the sigmapoints be compliant with the aforementioned side constraints (see Appendix A). Through this procedure, which was proposed in [27], we can assure that all the sigmapoints do not follow unphysical paths in the state vector space. It is worth noting that this procedure plays a role only when the sigmapoints are well scattered around the mean ${\widehat{\mathbf{x}}}_{i}$ (see the prediction phase in Table 1), i.e. mainly at the beginning of filtering.
4. Failure assessment of laminates exposed to impacts
Figure 3.
Impact on a 7layer composite [13]: spacetime diagram (the vertical dashed line here represents the possibly debonding surface of a brittle, homogeneous material).
To assess the capability of the proposed FESPKF approach to provide accurate failure forecasts for layered composites subject to low velocity impacts and to simultaneously calibrate the interlaminar constitutive laws, two experiments reported in the literature have been considered: an impact test on a 7layer composite plate (experiment FY06001 in [13]), and an impact test on a 11layer composite plate backed by another 11layer composite plate (experiment 1 in [14]). In both cases, plane impacts caused the propagation of compressive waves behind the impact location; upon reflection at the rear surface of the plates, a spalllike failure governed by tensile tractions perpendicular to the interlaminar surfaces took place. Because of the experimental setup, shear tractions do not play any role in the failure events; in what follows, for ease of notation we therefore avoid using subscript n to denote (mode I) opening/closing displacement discontinuities along interfaces.
Figure 4.
Impact on a 7layer composite [13]: (a) comparison between experimental and tracked free surface velocity records; (bc) estimated values of interlaminar openings ${\left[u\right]}_{j}$, $j=1,...,6$ (b: PWL law; c: LE law).
Figure 5.
Impact on a 7layer composite [13]: evolution in time of estimated interlaminar (a) strength ${\tau}_{M}$ and (b) toughness G.
As far as space discretization is concerned, owing to the major role played by dilatational waves the full threedimensional problem can be approached as a onedimensional one featuring constrained deformation in the plane perpendicular to the wave propagation direction. Accordingly, onedimensional meshes of constant strain elements have been adopted for the bulk; the characteristic size (length) of FEs in the throughthethickness direction was determined by a tradeoff between accuracy requirements and computational economy (leading to around 10 elements to discretize each lamina).
In the first experiment, each lamina was 1.37 mm in thickness, and was made of a balanced 5harness satin weave Eglass and LY564 epoxy. The overall wave speed in the throughthethickness direction was measured as 3.34 km/s, while the mass density was $\varrho =1885$ kg/m${}^{3}$; additional details and a thorough discussion of the results from an experimental perspective can be found in [13]. The specimen was stricken by a 12.5 mm thick aluminum impactor, flying at velocity $v=71$ m/s, which led to dynamic delamination. The free surface velocity profile ${\dot{u}}_{r}$ was measured during the test with a velocity interferometer for any reflector (VISAR).
Figure 6.
Impact on a 7layer composite [13]: evolution in time of estimated interlaminar strength ${\tau}_{M}$ and toughness G, and of the relevant covariances (a: PWL law; b: LE law). In the plots the squares denote the common initial guess ${\widehat{\vartheta}}_{0}$, whereas the circles denote the final estimates at time ${t}_{N}=10$μs.
The spacetime diagram of this test is shown in Figure 3: in this diagram, wave reflection and dispersion due to the interlaminar phases [45] have been neglected. Moreover, since failure is actually caused by delamination, the possibly debonding surface highlighted in the diagram is here to be considered as a reference one: even in the presence of wave dispersion, the actual failure of the laminate is expected to occur close to this reference surface.
Figure 4 shows a comparison between the estimated and the experimentally observed free surface velocity ${\dot{u}}_{r}$, and the time evolution of the tracked interlaminar openings ${\left[u\right]}_{j}$, $j=1,...,6$; filter results are shown for both linear and exponential softening models. Before $t\approx 7$μs information on the mechanical properties of the failing interlaminar surface are not filtered, since they are not yet available through measurables in $\mathbf{y}$ (see Figure 3), and some discrepancies between experimental and numerical graphs of ${\dot{u}}_{r}$ are shown (see Figure 4(a)). Delamination is foreseen to occur along the third interlaminar surface away from the impact location; in fact, only ${\left[u\right]}_{3}$ is diverging in Figure 4(b) and Figure 4(c), while all the other interface openings remain bounded. This latter outcome, which is independent of the shape of the softening envelope, well agrees with the statespace diagram of Figure 3 once wave dispersion caused by interlaminar phases and other inner composite inhomogeneities is allowed for.
Previous results are accompanied by model calibration; in Figure 5 the time evolution of estimated interlaminar strength ${\tau}_{M}$ and toughness G are reported at varying initialization of ϑ within the domain:
These estimates are not modified by the filter (apart from small fluctuations due to the stochastic environment) till the socalled pull back signal is processed at $t\approx 7$μs. Since ${\left[u\right]}_{3}$ is already diverging at that time (see Figure 4), the SPKF cannot obtain much information on the interlaminar properties; hence, depending on the initialization in ${\widehat{\mathbf{x}}}_{0}$, it sometimes happens that parameter estimates are eventually affected by biases. Anyway, it is worth noting that, under the same conditions, the standard EKF is not able to provide results featuring the same degree of accuracy [7]. In fact, the average converged estimates of ${\tau}_{M}$ are in good agreement with the tensile strength of 119.5 MPa reported in [13], while final estimates of G well represent the behavior of this kind of composites. On the basis of the similar accuracy in the calibration of the softening envelopes here considered, it can be said that the shape of the softening regime does not play a major role in this kind of laminate failure.
$${\mathcal{C}}_{\vartheta}=\left\{\begin{array}{c}50\le {\tau}_{M,0}\le 200\phantom{\rule{4.pt}{0ex}}\text{(MPa)}\hfill \\ 0.5\le {G}_{0}\le 2.0\phantom{\rule{4.pt}{0ex}}\text{(N/mm)}\hfill \end{array}\right.$$
To provide further insights into the performance of the FESPKF approach, Figure 6 shows an example of the time evolution of estimated model parameters along with the relevant statistical dispersion (here represented through error bars) around the expected values. Because of the aforementioned fast opening of the only decohering interlaminar surface, entries in the covariance matrix $\mathbf{P}$ are not enhanced much by the SPKF.
Figure 7.
Impact on a 11+11layer composite [14]: spacetime diagram (the vertical dashed line here represents the possibly debonding surface of a brittle, homogeneous material).
In the second experiment a glass fiber reinforced plastic (GRP) specimen, $7.02$ mm thick, was backed by a 6.91 mm thick GRP plate. The overall wave speed in the throughthethickness direction of the plates amounted to 3.19 km/s, and the mass density was $\varrho =1867$ kg/m${}^{3}$[14]. The specimen was stricken by a 5layer GRP flyer, $2.96$ mm in thickness and flying at velocity $v=85$ m/s, which caused dynamic delamination. During the test the free surface velocity profile was again measured via a VISAR. The reference spacetime diagram is depicted in Figure 7; it is here shown that, because of the test setup, release waves interact causing delamination inside the back plate.
Results of the FESPKF approach are reported in Figure 8 in terms of comparison between experimental and tracked free surface velocity ${\dot{u}}_{r}$, and in terms of estimated displacement jumps along all the interlaminar surfaces of the back plate (sequence starting at the specimen/back plate contact surface).
Figure 8.
Impact on a 11+11layer composite [14]: (a) comparison between experimental and tracked free surface velocity records; (b) estimated values of interlaminar openings ${\left[u\right]}_{j}$, $j=1,...,10$.
Tracking capabilities turn out again to be independent of the interface law: in Figure 8 discrepancies due to the shape of the softening regime cannot be detected. Such results have been obtained for any initialization values of ${\tau}_{M}$ and G inside the domain:
Delamination is foreseen to take place along the 7th interlaminar surface, in good agreement with the spacetime diagram of Figure 7 and with the results of [7,14]. As for model calibration, outcomes are qualitatively similar to those reported for the previous test in terms of accuracy and convergence of the estimates.
$${\mathcal{C}}_{\vartheta}=\left\{\begin{array}{c}25\le {\tau}_{M,0}\le 100\phantom{\rule{4.pt}{0ex}}\text{(MPa)}\hfill \\ 0.1\le {G}_{0}\le 0.6\phantom{\rule{4.pt}{0ex}}\text{(N/mm)}\hfill \end{array}\right.$$
5. Concluding remarks
In this paper we have proposed a finite element, sigmapoint Kalman filter approach to the failure assessment of layered composites subject to low velocity impacts. Since under such loading conditions the damage and energy dissipation are mainly linked to the propagation of interlaminar cracks (delamination), softening interface constitutive laws have been adopted to locally link the tractions transmitted between adjacent laminae to the opening/sliding displacement discontinuities occurring across the debonding surfaces.
To track in real time the whole state of the laminate and predict failure, a Kalman filter has been adopted. The filter has also allowed to calibrate the interface constitutive laws, whose parameters can be difficult to estimate due to the lower dimensionality of the interlaminar surfaces. To properly deal with the nonlinearities induced by strength degradation phenomena, the sigmapoint version of the Kalman filter has been implemented: this filter was already proved to be robust in highly nonlinear regimes, and to furnish more accurate outcomes than the customarily used extended Kalman filter (see [27,35]).
Through analysis of failures induced by plane waves propagating in the throughthethickness direction of stricken laminates, it has been shown that debonding events are accurately tracked by the offered procedure. Furthermore, interlaminar strength and toughness, which have been assumed unknown in the simulations, turned out to be accurately estimated.
In future works, to track and monitor diffused damage/delamination events, even featuring more complicated patterns than those here investigated, order reduction techniques will be coupled to the present methodology; this coupling will allow to get closer to the design of health monitoring systems for complex heterogeneous structures, able to detect in real time possible strength degradation phenomena induced by loading and environment.
A. Sigmapoint transformation
In this Appendix we summarize the fundamentals of sigmapoint transformation, with focus on its performance in representing the probability distribution of a random variable vector undergoing a nonlinear transformation [8,9].
Let $\mathbf{x}$ be a Gaussian random vector, with mean ${\widehat{\mathbf{x}}}_{i}$ and covariance ${\mathbf{P}}_{i}$ at time ${t}_{i}$. If $\mathbf{x}$ undergoes an analytic, nonlinear transformation ${\mathbf{f}}_{i}$ within the time interval $\left[{t}_{i}\phantom{\rule{1.em}{0ex}}{t}_{i+1}\right]$, its expected value at time ${t}_{i+1}$ reads:
where:
${N}_{x}$ being the dimension of $\mathbf{x}$. The relevant error covariance matrix is:
$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{i+1}^{}=\mathbb{E}\left[{\mathbf{x}}_{i+1}^{}\right]={\mathbf{f}}_{i}\left({\widehat{\mathbf{x}}}_{i}\right)+\sum _{n=1}^{\infty}{\displaystyle \frac{1}{n!}}\mathbb{E}\left[{\mathbf{D}}_{\Delta \mathbf{x}}^{n,i}\right]\end{array}$$
$$\begin{array}{c}\hfill {\mathbf{D}}_{\Delta \mathbf{x}}^{n,i}\equiv {\left(\sum _{\ell =1}^{{N}_{x}}{\displaystyle \frac{\partial {\mathbf{f}}_{i}}{\partial {x}_{\ell}}}{}_{\mathbf{x}={\widehat{\mathbf{x}}}_{i}}({x}_{\ell}{\widehat{x}}_{i,\ell})\right)}^{n}\end{array}$$
$$\begin{array}{cc}\hfill {\mathbf{P}}_{i+1}^{}=& \mathbb{E}\left[\left({\mathbf{x}}_{i+1}{\widehat{\mathbf{x}}}_{i+1}^{}\right){\left({\mathbf{x}}_{i+1}{\widehat{\mathbf{x}}}_{i+1}^{}\right)}^{\text{T}}\right]\hfill \\ \hfill =& \sum _{n=1}^{\infty}\sum _{m=1}^{\infty}{\displaystyle \frac{1}{n!}}{\displaystyle \frac{1}{m!}}\left(\mathbb{E}\left[{\mathbf{D}}_{\Delta \mathbf{x}}^{n,i}\phantom{\rule{0.166667em}{0ex}}{\mathbf{D}}_{\Delta \mathbf{x}}^{m,i\text{T}}\right]\mathbb{E}\left[{\mathbf{D}}_{\Delta \mathbf{x}}^{n,i}\right]\mathbb{E}\left[{\mathbf{D}}_{\Delta \mathbf{x}}^{m,i\text{T}}\right]\right)\hfill \end{array}$$
Through the sigmapoint transformation, the probability distribution of $\mathbf{x}$ at time ${t}_{i}$ is sampled via a set of sigmapoints ${\widehat{\chi}}_{i,\mathsf{j}}$ scattered around ${\widehat{\mathbf{x}}}_{i}$ according to (see Table 1):
where: ψ is a scaling factor to be determined; $\sqrt{{\mathbf{P}}_{i}}$ is the square root of the (positive definite) covariance matrix ${\mathbf{P}}_{i}$; ${\mathbf{1}}_{\mathsf{k}}$ is a unit vector pointing towards the $\mathsf{k}$th component of the state vector in the relevant space, whose role in (28) is to sample the $\mathsf{k}$th column of matrix $\sqrt{{\mathbf{P}}_{i}}$; ${N}_{\chi}=2{N}_{x}$.
$$\left\{\begin{array}{cc}\Delta {\chi}_{i,0}\hfill & =\mathbf{0}\hfill \\ \Delta {\chi}_{i,\mathsf{k}}\hfill & =+\psi \sqrt{{\mathbf{P}}_{i}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{1}}_{\mathsf{k}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\mathsf{k}=1,...,\frac{{N}_{\chi}}{2}\hfill \\ \Delta {\chi}_{i,\frac{{N}_{\chi}}{2}+\mathsf{k}}\hfill & =\psi \sqrt{{\mathbf{P}}_{i}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{1}}_{\mathsf{k}}\hfill \end{array}\right.$$
Information conveyed by the evolved sigmapoints are merged at time ${t}_{i+1}$ via a weighted averaging scheme, according to:
and:
where:
$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_{i+1,\text{SPT}}^{}=& \sum _{\mathsf{j}=0}^{{N}_{\chi}}{\omega}_{\mathsf{j}}\phantom{\rule{0.166667em}{0ex}}{\widehat{\chi}}_{i+1,\mathsf{j}}^{}\hfill \\ \hfill =& \left(\sum _{\mathsf{j}=0}^{{N}_{\chi}}{\omega}_{\mathsf{j}}\right){\mathbf{f}}_{i}\left({\widehat{\mathbf{x}}}_{i}\right)+\sum _{n=1}^{\infty}{\displaystyle \frac{1}{n!}}\left(\sum _{\mathsf{j}=0}^{{N}_{\chi}}{\omega}_{\mathsf{j}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{D}}_{\Delta {\chi}_{i,\mathsf{j}}}^{n,i}\right)\hfill \end{array}$$
$$\begin{array}{cc}\hfill {\mathbf{P}}_{i+1,\text{SPT}}^{}=& \sum _{\mathsf{j}=0}^{{N}_{\chi}}{\omega}_{\mathsf{j}}^{\u2605}\left({\widehat{\chi}}_{i+1,\mathsf{j}}^{}{\widehat{\mathbf{x}}}_{i+1,\text{SPT}}^{}\right){\left({\widehat{\chi}}_{i+1,\mathsf{j}}^{}{\widehat{\mathbf{x}}}_{i+1,\text{SPT}}^{}\right)}^{\text{T}}\hfill \\ \hfill =& \sum _{\mathsf{j}=0}^{{N}_{\chi}}{\omega}_{\mathsf{j}}^{\u2605}\sum _{n=1}^{\infty}\sum _{m=1}^{\infty}{\displaystyle \frac{1}{n!}}{\displaystyle \frac{1}{m!}}\left({\mathbf{D}}_{\Delta {\chi}_{i,\mathsf{j}}}^{n,i}\sum _{\mathsf{r}=0}^{{N}_{\chi}}{\omega}_{\mathsf{r}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{D}}_{\Delta {\chi}_{i,\mathsf{r}}}^{n,i}\right){\left({\mathbf{D}}_{\Delta {\chi}_{i,\mathsf{j}}}^{m,i}\sum _{\mathsf{s}=0}^{{N}_{\chi}}{\omega}_{\mathsf{s}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{D}}_{\Delta {\chi}_{i,\mathsf{s}}}^{m,i}\right)}^{\text{T}}\hfill \end{array}$$
$$\begin{array}{c}\hfill {\mathbf{D}}_{\Delta {\chi}_{i,\mathsf{j}}}^{n,i}\equiv {\left(\sum _{\ell =1}^{{N}_{x}}{\displaystyle \frac{\partial {\mathbf{f}}_{i}}{\partial {x}_{\ell}}}{}_{\mathbf{x}={\widehat{\mathbf{x}}}_{i}}\Delta {\chi}_{i,\mathsf{j}\ell}\right)}^{n}\end{array}$$
Weights ${\omega}_{\mathsf{j}}$ in (29) can be determined by matching the series expansions (25) and (29) up to third order; by assuming ${\omega}_{\mathsf{j}}=\omega $ for $\mathsf{j}=1,...,{N}_{\chi}$ this requirement results in:
To also set the value of the scaling factor ψ, a further condition needs to be established. Usually, it is suggested to match the diagonal entries of the fourth order terms in (25) and (29): this leads to $\psi =\sqrt{3}$. In [27], partially exploiting the features of the socalled scaled sigmapoint transformation [46], we proposed to adaptively set ψ so as to always fulfill the physical constraints:
where ${\vartheta}^{m}$ and ${\vartheta}^{M}$ are, respectively, the minimum and maximum (if applicable) allowed values of model parameters. These side constraints need to be fulfilled by all the sigmapoints ${\widehat{\chi}}_{i,\mathsf{j}}$: for $\mathsf{j}=0$, conditions (33) are automatically satisfied; for $\mathsf{j}=1,...,{N}_{\chi}$, conditions (33) are satisfied if:
where: ${\mathbf{a}}_{\mathsf{k}}={\mathbf{B}}_{\vartheta}\sqrt{{\mathbf{P}}_{i}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{1}}_{\mathsf{k}}$; ${\mathbf{B}}_{\vartheta}$ is a Boolean matrix, defined as $\vartheta ={\mathbf{B}}_{\vartheta}\mathbf{x}$; ${N}_{\vartheta}$ is the dimension of vector ϑ.
$$\left\{\begin{array}{c}{\omega}_{0}+{N}_{\chi}\omega =1\hfill \\ 2{\psi}^{2}\omega =1\hfill \end{array}\right.$$
$${\vartheta}^{m}\le \vartheta \le {\vartheta}^{M}$$
$$\psi \le min\left\{{\displaystyle \frac{{\widehat{\vartheta}}_{\ell}{\vartheta}_{\ell}^{m}}{{a}_{\mathsf{k}}^{\ell}}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\displaystyle \frac{{\vartheta}_{\ell}^{M}{\widehat{\vartheta}}_{\ell}}{{a}_{\mathsf{k}}^{\ell}}}\right\},\phantom{\rule{2.em}{0ex}}\mathsf{k}=1,...,\frac{{N}_{\chi}}{2};\ell =1,...,{N}_{\vartheta}$$
Weights ${\omega}_{\mathsf{j}}^{\u2605}$ in (30) are set by first letting ${\omega}_{\mathsf{j}}^{\u2605}={\omega}_{\mathsf{j}}=\omega $ for $\mathsf{j}=1,...,{N}_{\chi}$. Value of ${\omega}_{\mathsf{0}}^{\u2606}$ is then obtained by matching the fourth order terms involving ${\mathbf{D}}_{\diamond}^{2,i}{\mathbf{D}}_{\diamond}^{2,i\text{T}}$ in (27) and (30), thus getting ${\omega}_{\mathsf{0}}^{\u2605}=4\frac{{N}_{\chi}}{{\psi}^{2}}{\psi}^{2}$ [27].
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