Asymptotic formulae for the risk of failure related to an elasto-plastic problem with noise

Abstract

The risk of failure of mechanical structures under random forcing is an important concern in earthquake engineering. For a class of simple structures that can be modeled by an elasto-plastic oscillator, the risk of failure can be expressed in terms of the probability that, on a certain interval of time, the plastic deformation goes beyond a threshold related to a failure zone. In this note, asymptotic formulae for the risk of failure of an elasto-perfectly-plastic oscillator excited by a white noise are proposed. Our approach exploits the long cycle (repeating pattern) property of the aforementioned oscillator as introduced in A. Bensoussan, L. Mertz, S.C.P. Yam, Long cycle behaviour of the plastic deformation of an elasto-perfectly-plastic oscillator with noise, C. R. Acad. Sci. Paris Ser. I, 2012. We show that asymptotically the plastic deformation behaves like a Wiener process for which analytical formulae are available. Our result is a consequence of the Anscombe–Donsker Invariance Principle. Numerical experiments and comments are carried out.

Keywords

Risk analysis elasto-plastic oscillator white noise stochastic variational inequalities Donsker Invariance Principle

1. Introduction and scientific objectives

1.1. Background on a stochastic variational inequality modeling an elasto-plastic oscillator

Mechanical systems may be affected by vibrations. For instance, components (such as pipes) in a power plant may suffer from seismic vibrations. In this context, mechanical systems will accumulate fatigue and then face a risk of failure. From the modeling point of view, keeping track of the impact of past vibrations requires a specific description of the mechanical system under consideration. The state of the system must be described by a randomly forced dynamical system with memory. Random forcing here expresses the stochastic nature of the vibrations that apply to the mechanical structures.

In previous works [9,10], a basis prototype for modeling systems subject to vibrations has been investigated. It is referred as the elastic perfectly plastic oscillator (EPPO). In [7], it has been shown that the dynamics of such a nonlinear oscillator could be described in a relevant way by means of a stochastic variational inequality (SVI), as shown in ( SVI ) below. It turned out that this new mathematical framework is very interesting to derive new properties of a white noise driven EPPO such as (a) a new ergodic theorem [3], (b) a numerical PDE approach for its corresponding invariant measure [4] and (c) an application to engineering science [11]. Also, recently, nonlocal PDEs related to a colored noise driven EPPO have been proposed [6]. Here we are concerned with a white noise problem. The dynamics focuses on two quantities: the speed of deformation of the structure, when subject to the vibrations, and the stress supported by the structure. In simplest cases, the dynamics of the total deformation $x (t)$ reads $\begin{matrix} (1) & \ddot{x} (t) + c_{0} \dot{x} (t) + F (t) = \dot{w} (t) \end{matrix}$ where $w (t)$ is a Wiener process, $\dot{w} (t)$ describes the random forcing and $F (t)$ is some restoring force of the structure.

The so-called elasto-perfectly-plastic model corresponds to the case when the force $F (t)$ enters an linear regime in the elastic phase and a constant one in the plastic phase. The permanent (plastic) deformation at time t then writes $Δ (t) ≜ \int_{0}^{t} 1_{{| F (s) | = k Y}} y (s) d s$ , where $k Y$ describes the plastic phase through some stiffness coefficient k and some elasto-plastic bound Y. The variable ${z (t) ≜ x (t) - Δ (t), t ⩾ 0}$ describes the elastic deformation (i.e. the deformation without plastic deformation): when $| z (t) | < Y$ , the structure is in elastic regime; when $| z (t) | = Y$ , it is in plastic regime; the restoring force $F (t)$ always reads as $k z (t)$ . Using the notation $y (t) ≜ \dot{x} (t)$ , the pair ${(y (t), z (t)), t ⩾ 0}$ satisfies $\begin{matrix} (SVI) & \dot{y} (t) + c_{0} y (t) + k z (t) = \dot{w} (t), (\dot{z} (t) - y (t)) (ϕ - z (t)) ⩾ 0, \forall | ϕ | ⩽ Y . \end{matrix}$ Processes of this type belong to the class of degenerate reflected diffusions, we refer to [2] for a purely mathematical presentation of SVIs. Also, let us mention that one way to construct a solution to SVIs is to proceed by penalization, see [14] for the specific case of ( SVI ).

1.2. Derivation of the infinitesimal generator

Using Itô’s lemma for reflected diffusions, we can derive the infinitesimal generator of $(y (t), z (t))$ . Assume $(y (0), z (0)) = (y, z)$ . For all function φ regular enough ( $C^{2}$ w.r.t. to y and $C^{1}$ w.r.t. to z) defined on $R \times [- Y, Y]$ , we have $\begin{matrix} (2) & \{\begin{matrix} φ (y (t), z (t)) - φ (y, z) - \int_{0}^{t} L φ (y (s), z (s)) d s - \int_{0}^{t} R φ (y (s), z (s)) d Δ (s) \\ = \int_{0}^{t} \frac{\partial φ}{\partial y} (y (s), z (s)) d w (s) \end{matrix} \end{matrix}$ where the differential operators L (diffusion) and R (reflection) are defined by $\begin{matrix} L φ (y, z) ≜ \frac{1}{2} \frac{\partial^{2} φ}{\partial y^{2}} - (c_{0} y + k z) \frac{\partial φ}{\partial y} + y \frac{\partial φ}{\partial z} and R φ (y, z) ≜ - 1_{{| z | = Y}} (y, z) \frac{\partial φ}{\partial z} (y, z) . \end{matrix}$ Since $Δ (t) = \int_{0}^{t} 1_{{sign (y (s)) z (s) = Y}} y (s) d s$ , Equation (2) becomes: $\begin{matrix} \{\begin{matrix} φ (y (t), z (t)) - φ (y, z) - \int_{0}^{t} {L φ (y (s), z (s)) - 1_{{sign (y (s)) z (s) = Y}} y (s) \frac{\partial φ}{\partial z} (y (s), z (s))} d s \\ = \int_{0}^{t} \frac{\partial φ}{\partial y} (y (s), z (s)) d w (s) . \end{matrix} \end{matrix}$ Thus the generator is obtained as follows: $\begin{matrix} lim_{t \to 0} \frac{1}{t} (E φ (y (t), z (t)) - φ (y, z)) = \{\begin{matrix} \frac{1}{2} \frac{\partial^{2} φ}{\partial y^{2}} - (c_{0} y + k z) \frac{\partial φ}{\partial y} + y \frac{\partial φ}{\partial z}, & if | z | < Y, \\ \frac{1}{2} \frac{\partial^{2} φ}{\partial y^{2}} - (c_{0} y + k Y) \frac{\partial φ}{\partial y} + min (0, y) \frac{\partial φ}{\partial z}, & if z = Y, \\ \frac{1}{2} \frac{\partial^{2} φ}{\partial y^{2}} - (c_{0} y - k Y) \frac{\partial φ}{\partial y} + max (0, y) \frac{\partial φ}{\partial z}, & if z = - Y . \end{matrix} \end{matrix}$

1.3. Background on the long time behavior and the short cycles of the pair velocity-elastic deformation

The following theorem concerns the long time behaviour of $(y (t), z (t))$ .

Theorem 1 ([7]).

The pair velocity-elastic deformation $(y (t), z (t))$ is an ergodic Markov process. Therefore, there exists a unique invariant measure ν. Moreover, ν admits a probability density function composed of three positive $L^{1}$ functions $m (y, z), m (y, Y)$ and $m (y, - Y)$ . Using the notation $ν (f)$ for the application of the invariant measure on any bounded function f, it reads as follows $\begin{matrix} (ν) & ν (f) = \int_{D} f (y, z) m (y, z) d y d z + \int_{D^{+}} f (y, Y) m (y, Y) d y + \int_{D^{-}} f (y, - Y) m (y, - Y) d y \end{matrix}$ where $\begin{matrix} D ≜ R \times (- Y, Y), D^{+} ≜ (0, \infty) \times {Y}, D^{-} ≜ (- \infty, 0) \times {- Y} . \end{matrix}$

The following theorem provides an equivalent characterization of ν by duality.

Theorem 2 ([8]).

For all $λ > 0$ and for all bounded and measurable function $f (y, z)$ , there exists a unique continuous and bounded solution $u_{λ} (y, z)$ on $R \times [- Y, Y]$ to the following equation: $\begin{matrix} (P_{λ}) & \{\begin{matrix} λ u_{λ} + A u_{λ} = f (y, z) & in D, \\ λ u_{λ} + B_{+} u_{λ} = f (y, Y) & in D^{+}, \\ λ u_{λ} + B_{-} u_{λ} = f (y, - Y) & in D^{-} \end{matrix} \end{matrix}$ with the nonlocal boundary condition $\begin{matrix} u_{λ} (y, Y) and u_{λ} (y, - Y) are continuous \end{matrix}$ such that $\begin{matrix} ‖ u_{λ} ‖_{\infty} ⩽ \frac{‖ f ‖_{\infty}}{λ} and u_{λ} is continuous \end{matrix}$ and $\begin{matrix} lim_{λ \to 0} λ u_{λ} = ν (f) . \end{matrix}$ Here, we use the following notations $\begin{matrix} A φ ≜ - \frac{1}{2} \frac{\partial^{2} φ}{\partial y^{2}} + (c_{0} y + k z) \frac{\partial φ}{\partial y} - y \frac{\partial φ}{\partial z}, B_{\pm} φ ≜ - \frac{1}{2} \frac{\partial^{2} φ}{\partial y^{2}} + (c_{0} y \pm k Y) \frac{\partial φ}{\partial y} . \end{matrix}$

To be complete, we also recall the following the short cycle property of [3] which provides another equivalent characterization of ν.

Definition 3.
A short cycle associated to the solution $(y (t), z (t))$ of ( SVI ) is a piece of the trajectory starting in the elastic domain D and ending in one of the two transition points from the plastic state to the elastic state ${(0, - Y), (0, Y)}$ .

The interpretation of the short cycles, in terms of partial differential equations, can be given by the following problem: $\begin{matrix} (P_{v}) & A v = f in D, B_{+} v = f in D^{+}, B_{-} v = f in D^{-} \end{matrix}$ with the local boundary condition $v (0^{+}, Y) = 0$ and $v (0^{-}, - Y) = 0$ . The notation $v (y, z; f)$ is used and it is also called a short cycle. Here f is a bounded and measurable function.

From the analysis of ( P v ), a new formulation of the ergodic measure is obtained; it involves the values of discontinuities of the function $v (y, z; f)$ at the two phase transition points ${(0, - Y), (0, Y)}$ . It is expressed as follows: $\begin{matrix} ν (f) = \frac{v (0^{-}, Y; f) + v (0^{+}, - Y; f)}{2 v (0^{-}, Y; 1)} . \end{matrix}$ Besides, as $λ \to 0$ , an asymptotic expansion of the solution $u_{λ} (y, z; f)$ of ( P λ ) is obtained where the number $ν (f)$ appears as a term of first order $\begin{matrix} u_{λ} (y, z; f) = u (y, z; f) + \frac{ν (f)}{λ} + o (\frac{1}{λ}) \end{matrix}$ and where the function $u (y, z; f)$ satisfies the following Poisson equation: $\begin{matrix} (P_{u}) & A u = f - ν (f) in D, B_{+} u = f - ν (f) in D^{+}, B_{-} u = f - ν (f) in D^{-} \end{matrix}$ with the nonlocal boundary condition given by the fact that $y \mapsto u (y, \pm Y; f)$ are continuous.
1.4. Background on the long cycle behavior of the plastic deformation

Many works in engineering science, partly experimental, partly empirical on numerical simulations, have shown that, for large time, the variance of the total deformation grows linearly with time (see [9] and references therein). In [5], this property has been proved and the growth rate has been rigorously characterized. Relying on ( SVI ), the authors have proposed a novel and simple formulation of the evolution of the system in terms of stopping times in order to identify a repeating pattern in the trajectory: long cycles.

Definition 4 ([5]).

A long cycle of the solution $(y (t), z (t))$ of ( SVI ) is a path, enclosed by the stopping times $τ_{0}$ and $τ_{1}$ defined below, starting and ending in one of the two points ${(0, - Y), (0, Y)}$ which has touched the other point at least one time. Similarly, a half long cycle is a path enclosed by the stopping times $τ_{0}$ and $s_{0}$ , or by $s_{0}$ and $τ_{1}$ . In a recursive way, a sequence of stopping times $τ_{n}$ can be defined where $τ_{n}$ is the time when ends the nth long cycle.

Then, with the notation $\begin{matrix} τ_{0} ≜ inf {t > 0, y (t) = 0 and | z (t) | = Y}, \end{matrix}$ $δ ≜ sign (z (τ_{0}))$ which labels the first boundary reached by the process $(y (t), z (t))$ , and $\begin{matrix} \{\begin{matrix} s_{0} ≜ inf {t > τ_{0}, y (t) = 0 and z (t) = - δ Y}, \\ τ_{1} ≜ inf {t > s_{0}, y (t) = 0 and z (t) = δ Y}, \end{matrix} \end{matrix}$ they have obtained a probabilistic formula for the coefficient of the growth rate of the variance of $Δ (t)$ as follows: $\begin{matrix} (3) & lim_{t \to \infty} \frac{σ^{2} (Δ (t))}{t} = μ γ^{2} \end{matrix}$ where $\begin{matrix} (4) & γ^{2} ≜ E {(Δ (τ_{1}) - Δ (τ_{0}))}^{2} and μ ≜ \frac{1}{E (τ_{1} - τ_{0})} . \end{matrix}$ Recently (see [1]) an analytic framework for the growth rate of $σ^{2} (Δ (t))$ has been proposed by using a Fourier transform approach. The idea is simple. The Fourier transform of the plastic deformation over a long cycle, namely $\begin{matrix} Φ (ξ) ≜ E (exp (i ξ (Δ (τ_{1}) - Δ (τ_{0})))), \end{matrix}$ can be expressed as $E (exp (i ξ (Δ (τ_{1}) - Δ (s_{0}) + Δ (s_{0}) - Δ (τ_{0}))))$ and thus, using the fact that $Δ (s_{0}) - Δ (τ_{0})$ and $- (Δ (τ_{1}) - Δ (s_{0}))$ are independent and identically distributed, it yields $\begin{matrix} Φ (ξ) = φ (ξ) \bar{φ} (ξ) = {| φ (ξ) |}^{2} where φ (ξ) ≜ E (exp (i ξ (Δ (s_{0}) - Δ (τ_{0})))) . \end{matrix}$ Here $φ (ξ)$ is the Fourier transform of the plastic deformation over a half long cycle and can be interpreted as $v_{ξ} (0, Y)$ where $v_{ξ}$ solves the Problem ( P v ξ ), ξ is a parameter. Then, on the one hand, using the symmetry $Φ (ξ) = Φ (- ξ)$ for all ξ together with the following expansions: $\begin{matrix} \{\begin{matrix} Φ (ξ) = Φ (0) + Φ^{'} (0) ξ + Φ^{″} (0) \frac{ξ^{2}}{2} + o (ξ^{2}) \\ Φ (- ξ) = Φ (0) - Φ^{'} (0) ξ + Φ^{″} (0) \frac{ξ^{2}}{2} + o (ξ^{2}) \end{matrix} \end{matrix}$ which imply $\begin{matrix} Φ^{″} (0) = 2 (\frac{Φ (ξ) - Φ (0)}{ξ^{2}}) + o (1), \end{matrix}$ we deduce that $\begin{matrix} - Φ^{″} (0) = lim_{ξ \to 0} 2 (\frac{1 - | v_{ξ} (0, Y) |^{2}}{ξ^{2}}) . \end{matrix}$ Also, on the other hand, $E {(Δ (τ_{1}) - Δ (τ_{0}))}^{2}$ is obtained using the derivatives, with respect to ξ, of Φ by using the relationship $\begin{matrix} - Φ^{″} (0) = E {(Δ (τ_{1}) - Δ (τ_{0}))}^{2} . \end{matrix}$ In addition, the mean duration of a long cycle (which is twice the duration of a half long cycle) can be interpreted as $2 \bar{v} (0, Y)$ where $\bar{v}$ solves the Problem ( P v ¯ ). In this context, Formula (3) becomes $\begin{matrix} (5) & lim_{t \to \infty} \frac{σ^{2} (Δ (t))}{t} = lim_{ξ \to 0} \frac{1 - | v_{ξ} (0, Y) |^{2}}{ξ^{2} \bar{v} (0, Y)} \end{matrix}$ where $v_{ξ}$ satisfies $\begin{matrix} (P_{v_{ξ}}) & A v_{ξ} = 0 in D, B_{+} v_{ξ} = i ξ y v_{ξ} in D^{+}, B_{-} v_{ξ} = i ξ y v_{ξ} in D^{-} \end{matrix}$ with the boundary condition: $v_{ξ} (0^{-}, - Y) = 1$ and $y \mapsto v_{ξ} (y, Y)$ is continuous, and $\bar{v}$ satisfies $\begin{matrix} (P_{\bar{v}}) & A \bar{v} = 1 in D, B_{+} \bar{v} = 1 in D^{+}, B_{-} \bar{v} = 1 in D^{-} \end{matrix}$ with the boundary condition: $\bar{v} (0^{-}, - Y) = 0$ and $y \mapsto \bar{v} (y, Y)$ is continuous.

Remark 5.
The important remark is that Δ is a process with continuous trajectory and admits an asymptotic variance $γ^{2} μ t$ . Thus, with respect to an appropriate scaling in time and space, it is reasonable to expect that Δ behaves like a Brownian motion. This is interesting since analytical formulae can be employed for the risk analysis of failure of the structure modeled by ( SVI ). This remark was already formulated in [9] as a heuristic argument. In this note, we provide a mathematical explanation.

1.5. Goal of the paper and main ideas

In terms of dynamics of this oscillator, the risk of failure at a given risk level $b > 0$ and on a given time interval $(0, T)$ is described as the probability that the maximum of $| Δ (t) |$ goes beyond the level b before the time T, namely $\begin{matrix} (6) & W (b, T) ≜ P (max_{t \in (0, T)} | Δ (t) | ⩾ b) . \end{matrix}$ Such a function ${W (b, T), b ⩾ 0, T ⩾ 0}$ is particularly relevant for applications since it allows engineers to predict the fragility of a mechanical structure. However, on the one hand, there is no hope of having an explicit formula and, on the other hand, numerical solutions of $W (b, T)$ through Monte Carlo simulations are too slow.

In consequence, the purpose of our work is to provide an asymptotic explicit formula for $W (β, τ)$ , when β and τ are large enough in the sense $β \propto \sqrt{n}$ and $τ \propto n$ , as shown in Formula (9). The underlying mathematical machinery related to this formula relies on the convergence in law of the following rescaled plastic deformation process: $\begin{matrix} (7) & Δ^{n} (t) ≜ \frac{Δ (n t)}{γ \sqrt{μ} \sqrt{n}} \end{matrix}$ toward a standard Wiener process Z. In this context, as n goes to ∞, $W (\sqrt{n} b, n T)$ converges to $W_{⋆} (\frac{b}{γ \sqrt{μ}}, T) ≜ P ({sup}_{t \in (0, T)} | Z_{t} | ⩾ \frac{b}{γ \sqrt{μ}})$ which is known (see e.g. [12], p. 34) as $\begin{matrix} (8) & W_{⋆} (\frac{b}{γ \sqrt{μ}}, T) = 1 - \frac{4}{π} \sum_{k = 0}^{+ \infty} \frac{{(- 1)}^{k}}{2 k + 1} exp (- \frac{{(2 k + 1)}^{2} π^{2} T γ^{2} μ}{8 b^{2}}) . \end{matrix}$ The important consequence, as shown in Theorem 8-(ii), is the connection between W and $W_{⋆}$ in the following sense: $\begin{matrix} (9) & lim_{n \to \infty} W (\sqrt{n} b, n T) = 1 - \frac{4}{π} \sum_{k = 0}^{+ \infty} \frac{{(- 1)}^{k}}{2 k + 1} exp (- \frac{{(2 k + 1)}^{2} π^{2} T γ^{2} μ}{8 b^{2}}) . \end{matrix}$ From an engineering perspective, we believe that this approximation can improve analysis methods for the risk of failure of elasto-plastic oscillators. Indeed, for any $b > 0, T > 0$ and $ϵ > 0$ , there exists $N_{ϵ}$ (large enough, see Table 1 for an example) such that $\begin{matrix} (10) & W (\sqrt{n} b, n T) \in (W_{⋆} (\frac{b}{γ \sqrt{μ}}, T) - ϵ, W_{⋆} (\frac{b}{γ \sqrt{μ}}, T) + ϵ) provided that n ⩾ N_{ϵ} . \end{matrix}$

To the best of our knowledge, a theoretical value of $N_{ϵ}$ for a given ϵ is not known. This type of concerns seems to be an open problem even in simple cases related to the Donsker Principle. However, heuristically, it seems reasonable to believe that the rate of convergence of $W (\sqrt{n} b, n T)$ to $W_{⋆} (\frac{b}{γ \sqrt{μ}}, T)$ is order $1 / \sqrt{n}$ , as in the Central Limit Theorem (see Berry–Esseen Theorem, for instance Theorem 1 of Section XVI.5 of [12], p. 542), see Table 2 for an empirical study.

As shown in the proof of Theorem 8, the limit (9) is a consequence of the Anscombe–Donsker Invariance Principle (ADIP) (see e.g. Theorem 2.1 p. 158 of Chapter 5 in [13]). For the sake of completeness, it is recalled below.

Theorem 6 (Anscombe–Donsker Invariance Principle, ADIP).

Let ${x_{j}}_{j = 1}^{\infty}$ be a sequence of independent identically distributed (i.i.d.) random variables with zero mean and finite variance $γ^{2}$ , ${(K (t))}_{t ⩾ 0}$ be a nondecreasing, right-continuous family of positive, integer valued random variables satisfying $\begin{matrix} \frac{K (t)}{t} \overset{a . s .}{\to} μ as t \to \infty, 0 < μ < \infty \end{matrix}$ and, for $n ⩾ 1$ , consider $X^{n}$ the process defined by $\begin{matrix} X_{t}^{n} ≜ \frac{1}{γ \sqrt{n μ}} \sum_{j = 1}^{K (n t)} x_{j}, t ⩾ 0 . \end{matrix}$ Then $\begin{matrix} (11) & X^{n} \overset{J_{1}}{\to} W as n \to \infty \end{matrix}$ where $W$ is the Wiener measure on the space of continuous functions on $[0, \infty)$ , namely $C [0, \infty)$ .

Here, we recall the definition of the $J_{1}$ convergence for (11).

Definition 7 ( $J_{1}$ convergence).

Let $D$ denote the space of functions from $[0, \infty)$ into $R$ that are right continuous on $[0, \infty)$ and have finite left limits on $(0, \infty)$ . Let Γ denote the set of functions $α : [0, \infty) \to [0, \infty)$ that are strictly increasing and continuous with $α (0) = 0$ and ${lim}_{t \to \infty} α (t) = + \infty$ . A sequence ${(g_{n})}_{n}$ in $D$ converges to $g \in D$ in the $J_{1}$ -topology, if and only if for each $T > 0$ there is a sequence ${α_{n}}_{n}$ (possibly depending on T) in Γ such that $\begin{matrix} sup_{0 ⩽ t ⩽ T} | α_{n} (t) - t | \to 0 as n \to \infty and sup_{0 ⩽ t ⩽ T} | g_{n} (α_{n} (t)) - g (t) | \to 0 as n \to \infty . \end{matrix}$

1.6. Organization of the note

In Section 2, we present our main result together with a short proof. Then, numerical experiments in support of our formula (9) are provided in Section 3. In Section 4, further comments are also given.

2. Main result

In this section, we present our main result together with its proof. It provides bounds for the risk of failure of an EPPO excited by a white noise.

Theorem 8.
(i) Let $P_{n}$ be the measure induced by $Δ^{n}$ on $(D, B (D))$ . Then, ${P_{n}}_{n = 1}^{\infty}$ converges weakly to a measure $P_{⋆}$ under which the coordinate mapping process $W (t) = ω (t)$ on $C [0, \infty)$ is a standard, one-dimensional Wiener process. (ii) As a consequence, for any $b > 0$ and $T > 0$ , $\begin{matrix} (12) & lim_{n \to \infty} W (\sqrt{n} b, n T) = W_{⋆} (\frac{b}{γ \sqrt{μ}}, T) . \end{matrix}$
Proof.
As mentioned in the introduction, the main ingredient of the proof is ADIP combined with the notion of long cycles. In the framework of long cycles, for any time $t ⩾ 0$ , the plastic deformation breaks down as follows $\begin{array}{l} Δ (t) & = \int_{0}^{t} y (s) 1_{{sign (y (s)) z (s) = Y}} d s \\ = \sum_{k = 1}^{N (t)} \underset{δ_{k} ≜}{\underset{︸}{\int_{τ_{k - 1}}^{τ_{k}} y (s) 1_{{sign (y (s)) z (s) = Y}} d s}} + \underset{ϵ (t) ≜}{\underset{︸}{\int_{τ_{N (t)}}^{t} y (s) 1_{{sign (y (s)) z (s) = Y}} d s}} \\ ≜ \hat{Δ} (t) + ϵ (t) \end{array}$ where $N (t)$ is the number of long cycles up to the time t and satisfies $τ_{N (t)} ⩽ t < τ_{N (t) + 1}$ . Here $δ_{k}$ are i.i.d. where $E δ_{1} = 0, E δ_{1}^{2} = γ^{2}$ and ${lim}_{τ \to \infty} \frac{N (t)}{t} = μ$ .

On the one hand, ADIP can be applied to a rescaled version of $\hat{Δ}$ . Indeed, using the notation $\begin{matrix} {\hat{Δ}}^{n} (t) ≜ \frac{\sum_{k = 1}^{N (n t)} δ_{k}}{γ \sqrt{μ} \sqrt{n}}, \end{matrix}$ where we recall that $γ^{2} μ$ is the asymptotic growth rate of the variance of the plastic deformation, ADIP yields that ${\hat{Δ}}^{n}$ converges in law to a Wiener process as n goes to infinity. Here, it is worth mentioning that ADIP does not need to assume that $N (t)$ is independent of the random variables $δ_{k}$ . On the other hand, regarding the rescaled version of the remaining term $\begin{matrix} ϵ^{n} (t) ≜ \frac{ϵ (n t)}{γ \sqrt{μ} \sqrt{n}}, \end{matrix}$ $ϵ^{n}$ converges in law to 0 as n goes to infinity. Now, we are in position to identify the Wiener process as the limit in law for the rescaled plastic deformation $\begin{matrix} \frac{Δ (n t)}{γ \sqrt{μ} \sqrt{n}} = {\hat{Δ}}^{n} (t) + ϵ^{n} (t) . \end{matrix}$ Thus, with Z a standard Wiener process and $β ⩾ 0$ , $\begin{matrix} lim_{n \to + \infty} P (max_{0 ⩽ t ⩽ T} | \frac{Δ (n t)}{γ \sqrt{μ} \sqrt{n}} | ⩾ β) = P (max_{0 ⩽ t ⩽ T} | Z (t) | ⩾ β) . \end{matrix}$ Taking $β ≜ \frac{b}{γ \sqrt{μ}}$ , that gives the result $\begin{matrix} lim_{n \to + \infty} P (max_{0 ⩽ t ⩽ n T} | Δ (t) | ⩾ \sqrt{n} b) = P (max_{0 ⩽ t ⩽ T} | Z (t) | ⩾ \frac{b}{γ \sqrt{μ}}) . \end{matrix}$ □

3. Numerical experiments in support of Theorem 8

In this section, we provide numerical results of formula (12). A C code has been written to simulate $(y (t), z (t))$ . See [4] for the direct algorithmic scheme. For any $T > 0, b > 0, n ⩾ 1$ , to compute $W (\sqrt{n} b, n T)$ , we generate $MC$ numerical solutions and we approximate $W (\sqrt{n} b, n T)$ by an empirical estimation: $\begin{matrix} W_{MC} (\sqrt{n} b, n T) ≜ \frac{1}{MC} \sum_{i = 1}^{MC} 1_{{{max}_{t \in (0, T)} | Δ_{i} (t) | ⩾ b}}, \end{matrix}$ where $Δ_{i}$ is the plastic deformation of the ith sample, $i \in {1, \dots, MC}$ .

By way of example, we make the specific choice of $b_{α}$ such that $W_{⋆} (\frac{b_{α}}{γ \sqrt{μ}}, T) = α \in [0, 1]$ . Using the explicit formula (8) and Newton algorithm, we determine numerically an approximate value ${\tilde{b}}_{α}$ of $b_{α}$ . For the sake of clarity, we set $α = 0.5$ in the presentation. In our numerical experiment below, we check the convergence of $W_{MC} (\sqrt{n} b_{0.5}, n T)$ towards $α = 0.5$ and then we give an empirical value $N_{ϵ}$ (see Table below for different values of Y) for which $\begin{matrix} W_{MC} (\sqrt{n} b_{0.5}, n T) \in (0.5 - ϵ, 0.5 + ϵ), when n ⩾ N_{ϵ} . \end{matrix}$ In terms of engineering applications, the threshold coefficients and maturity of interest corresponding to the risk of failure $α = 0.5$ are not $b_{α}$ and T but any $(β, τ) \in {(\sqrt{n} b, n T), n ⩾ N_{ϵ}}$ . More precisely, if $(β, τ) \in {(\sqrt{N_{ϵ}} b_{0.5}, N_{ϵ} T), \dots, (\sqrt{N_{ϵ} + k} b_{0.5}, (N_{ϵ} + k) T), \dots}$ , the probability that the maximum of the absolute deformation goes above β before the time τ is around 0.5. Table 1 also shows numerical evidence that the average μ and $b_{α}$ decrease with Y, while $γ^{2}$ increases.

Monte Carlo error. To approximate (6) with $b = \sqrt{N_{ϵ}} {\tilde{b}}_{0.5}$ and $τ = N_{ϵ} T$ in Table 1, we generate $N_{MC}$ trajectories whose associated plastic deformations are denoted by ${(Δ_{i})}_{i = 1, \dots, N_{MC}}$ . Then we consider: $\begin{matrix} W_{MC} (b, τ) = \frac{1}{N_{MC}} \sum_{i = 1}^{N_{MC}} 1_{{{max}_{t \in (0, τ)} | Δ_{i} (t) | ⩾ b}} . \end{matrix}$ By the Central Limit Theorem, we have $\begin{matrix} W (b, τ) \in [W_{MC} (b, τ) - \frac{1.96 s_{W}}{\sqrt{N_{MC}}}, W_{MC} (b, τ) + \frac{1.96 s_{W}}{\sqrt{N_{MC}}}] \end{matrix}$ with $95 %$ confidence, where $s_{W} ≜ \sqrt{W_{MC} (b, τ) - {(W_{MC} (b, τ))}^{2}}$ . In Table 1, we used $N_{MC} = 10^{5}$ Monte Carlo samples. For instance, for $Y = 0.01$ , we obtain that with $95 %$ confidence: $\begin{matrix} W (b, τ) \in [0.474798 - 0.0031, 0.474798 + 0.0031] = [0.471698, 0.477898] . \end{matrix}$

Rate of convergence. To the best of our knowledge, the rate of convergence of $W (\sqrt{n} b, n T)$ towards $W_{⋆} (\frac{b}{γ \sqrt{μ}}, T)$ as $n \to \infty$ is not known. Numerical results tend to show that this rate is order $1 / \sqrt{n}$ , as shown in Fig. 1 and Table 2. With the same parameters as in Table 1, we have used linear least-square regression to fit a straight line $log (n) \mapsto a + c log (n)$ with the data curve $log (n) \mapsto log d (n)$ where $d (n) ≜ | W_{⋆} (\frac{{\tilde{b}}_{0.5}}{\sqrt{μ} γ}, T) - W_{MC} (\sqrt{n} {\tilde{b}}_{0.5}, n T) |$ is the distance between $W_{⋆}$ and the approximation of W obtained by Monte Carlo simulation. In other words, $d (n)$ is order $n^{c}$ , with $c \in [- 0.58, - 0.43]$ . Note that the asymptotic standard error (A.S.E.) on c is smaller than $0.5 %$ for every $Y \in {0.01, \dots, 0.1}$ .

Table 1
Empirical estimation of $N_{ϵ}$ by a probabilistic simulation of $W_{MC}$ with $N_{MC} = 10^{5}$ trajectories. Here $W_{⋆} ({\tilde{b}}_{0.5} / (\sqrt{μ} γ), T) = 0.499763$ for $Y \in {0.01, \dots, 0.1}$ . The coefficients μ and $γ^{2}$ are calculated by a Monte-Carlo approximation based on $10^{6}$ trajectories

Y $c_{0} = 1, k = 1, T = 1, ϵ = 0.025$

$W_{MC} (\sqrt{N_{ϵ}} {\tilde{b}}_{0.5}, N_{ϵ} T)$ $N_{ϵ}$ μ $γ^{2}$ ${\tilde{b}}_{0.5}$

0.01 0.474798 1188 0.390538 2.44725 1.12355

0.02 0.474968 1394 0.303235 3.07620 1.10999

0.03 0.474853 1392 0.261154 3.47465 1.09477

0.04 0.474907 1123 0.234129 3.77331 1.08021

0.05 0.474857 1103 0.215209 3.99652 1.06584

0.06 0.474851 1024 0.200707 4.17397 1.0519

0.07 0.474813 1359 0.188976 4.31854 1.03823

0.08 0.475012 1024 0.179479 4.42808 1.02455

0.09 0.474831 1446 0.171155 4.53941 1.01301

0.1 0.474907 1243 0.164293 4.57780 0.996686

Y	$c_{0} = 1, k = 1, T = 1, ϵ = 0.025$
0.01	0.474798	1188	0.390538	2.44725	1.12355
0.02	0.474968	1394	0.303235	3.07620	1.10999
0.03	0.474853	1392	0.261154	3.47465	1.09477
0.04	0.474907	1123	0.234129	3.77331	1.08021
0.05	0.474857	1103	0.215209	3.99652	1.06584
0.06	0.474851	1024	0.200707	4.17397	1.0519
0.07	0.474813	1359	0.188976	4.31854	1.03823
0.08	0.475012	1024	0.179479	4.42808	1.02455
0.09	0.474831	1446	0.171155	4.53941	1.01301
0.1	0.474907	1243	0.164293	4.57780	0.996686

Table 2

Values to fit a straight line $log (n) \mapsto a + c log (n)$ with the data $log (n) \mapsto log d (n)$ , where $d (n) ≜ | W_{⋆} (\frac{{\tilde{b}}_{0.5}}{\sqrt{μ} γ}, T) - W_{MC} (\sqrt{n} {\tilde{b}}_{0.5}, n T) |$ . The fit is obtained using linear least-square regression. The table includes asymptotic standard error (A.S.E.) for both a and c and reduced chi-squared

Y	$c_{0} = 1, k = 1, T = 1, ϵ = 0.05$

	a	c	A.S.E. on a	A.S.E. on c	Reduced $χ^{2}$
0.01	0.176908	−0.558621	±0.00639 (3.612%)	±0.0007726 (0.1383%)	0.00593928
0.02	0.0480492	−0.523418	±0.01075 (22.38%)	±0.0013 (0.2484%)	0.0168202
0.03	−0.375508	−0.460721	±0.00352 (0.9373%)	±0.0004255 (0.09236%)	0.00180176
0.04	−0.438347	−0.467042	±0.00461 (1.052%)	±0.0005574 (0.1193%)	0.0030914
0.05	−0.31323	−0.475275	±0.004668 (1.49%)	±0.0005644 (0.1188%)	0.00316958
0.06	−0.618525	−0.438052	±0.005041 (0.815%)	±0.0006095 (0.1391%)	0.0036957
0.07	0.409812	−0.579644	±0.01063 (2.595%)	±0.001285 (0.2218%)	0.0164417
0.08	0.0639668	−0.542532	±0.005863 (9.165%)	±0.0007088 (0.1306%)	0.00499889
0.09	−0.326126	−0.45783	±0.005836 (1.789%)	±0.0007055 (0.1541%)	0.0049529
0.1	−0.576013	−0.436479	±0.006747 (1.171%)	±0.0008157 (0.1869%)	0.00661985

Fig. 1.

For $Y = 0.01$ , distance $log (n) \mapsto log d (n)$ (purple) and its fit with a straight line $log (n) \mapsto f (n)$ (green, dashed line).

4. Conclusion and discussion

In this work, we have proposed asymptotic formulae for the risk of failure of an elasto-plastic oscillator excited by a white noise. The main result has been obtained by a combination of the novel notion of long cycle of an EPPO with white noise and the so-called Donsker Principle. We believe that these formulae can be useful for the risk analysis of fatigue for a certain type of mechanical structures under random forcing. Let us emphasize that our approach may not be used to estimate the risk of failure $W (b, T)$ for any time T or any fragility threshold b. Our approach is useful rather to solve the inverse problem of finding a family of time $T_{n}$ and a family of fragility threshold $b_{n}$ such that the corresponding risk of failure w.r.t. $(b_{n}, T_{n})$ is given by a certain $α \in [0, 1]$ .

4.1. Is the long cycle approach generalizable to different models?

4.1.1. EPPO excited by a Lévy noise

We believe that the long cycle approach is generalizable for an EPPO excited by a Lévy noise. Indeed, Lévy processes have independent and stationary increments. That would help finding an i.i.d. repeating pattern. The difficulty here is to provide a careful definition of the corresponding stopping times since the trajectory in $y (t)$ is not continuous (but $Δ (t)$ would still have a continuous trajectory). In our next work, we will investigate the extension of the present work to the case of an EPPO excited by a Lévy noise.

4.1.2. EPPO excited by a colored noise

In contrast with the Lévy noise case, we believe that the long cycle approach is not generalizable for an EPPO excited by a colored noise $ξ (t)$ , for instance $\begin{matrix} d ξ (t) = - γ ξ (t) d t + σ d w (t), γ, σ > 0 . \end{matrix}$ Indeed, in that case, the noise would be a part of the state variable and thus an additional dimension has to be taken into account. It is not clear where would be ξ at the beginning of each cycle and thus it is not clear how to identify a repeating pattern. However, numerical experiments show that the growth rate of the variance of the plastic deformation converges to a constant. In our next work, we will investigate an alternative method to the long cycle behavior in order to find this coefficient.

4.2. A possible alternative to the long cycle behavior

Let us conclude this work by a conjecture which seems reasonable as a general approach. We formulate it in the context of the EPPO excited by a white noise and numerical experiments in support of the conjecture are carried out, see Table 3.

Conjecture 9 (An alternative analytic approach).

We believe that $\begin{matrix} lim_{t \to + \infty} \frac{σ^{2} (Δ (t))}{t} = ν ({| \frac{\partial u}{\partial y} |}^{2}) \end{matrix}$ where ν and u have been defined in ( ν ) and ( P u ).

Table 3

Comparison of the two deterministic approaches solving PDEs and a probabilistic simulation for the last column. Using basic techniques and softwares

Y	$c_{0} = 1, k = 1$

	$\frac{1 - \| v_{ξ} (0, Y) \|^{2}}{ξ^{2} v (0, Y)}$	$ν (\| \frac{\partial u}{\partial y} \|^{2})$	$\frac{E {(Δ (t_{1}) - Δ (t_{0}))}^{2}}{E (t_{1} - t_{0})}$
0.1	0.712	0.712	0.781
0.2	0.501	0.523	0.576
0.3	0.383	0.386	0.440
0.4	0.279	0.278	0.325
0.5	0.214	0.206	0.241
0.6	0.151	0.146	0.174
0.7	0.110	0.112	0.127
0.8	0.076	0.076	0.089
0.9	0.055	0.054	0.063
1.0	0.039	0.039	0.045

Footnotes

Acknowledgements

The authors would like to thank George Papanicolaou for his suggestions helping to improve the manuscript. LM would like to thank François Delarue for his question (during the seminar of January 16 2014 at University of Nice Sophia Antipolis) regarding a functional central limit theorem for the plastic deformation of a white noise driven EPPO. We hope that this work answer his question. The authors would like to thank Alain Bensoussan, Stéphane Menozzi and Eric Vanden-Eijnden for stimulating discussions. LM expresses his sincere gratitude to the Courant Institute for being supported as Courant Instructor in 2014 and 2015, when this work was done. LM is also supported by a faculty discretionary fund from NYU Shanghai and the National Natural Science Foundation of China, Research Fund for International Young Scientists under the project #1161101053 entitled “Computational methods for non-smooth dynamical systems excited by random forces” and the Young Scientist Program under the project #11601335 entitled “Stochastic Control Method in Probabilistic Engineering Mechanics”.

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Y	$c_{0} = 1, k = 1, T = 1, ϵ = 0.025$

	$W_{MC} (\sqrt{N_{ϵ}} {\tilde{b}}_{0.5}, N_{ϵ} T)$	$N_{ϵ}$	μ	$γ^{2}$	${\tilde{b}}_{0.5}$
0.01	0.474798	1188	0.390538	2.44725	1.12355
0.02	0.474968	1394	0.303235	3.07620	1.10999
0.03	0.474853	1392	0.261154	3.47465	1.09477
0.04	0.474907	1123	0.234129	3.77331	1.08021
0.05	0.474857	1103	0.215209	3.99652	1.06584
0.06	0.474851	1024	0.200707	4.17397	1.0519
0.07	0.474813	1359	0.188976	4.31854	1.03823
0.08	0.475012	1024	0.179479	4.42808	1.02455
0.09	0.474831	1446	0.171155	4.53941	1.01301
0.1	0.474907	1243	0.164293	4.57780	0.996686

Asymptotic formulae for the risk of failure related to an elasto-plastic problem with noise

Abstract

Keywords

1. Introduction and scientific objectives

1.1. Background on a stochastic variational inequality modeling an elasto-plastic oscillator

1.2. Derivation of the infinitesimal generator

1.3. Background on the long time behavior and the short cycles of the pair velocity-elastic deformation

Theorem 1 ([7]).

Theorem 2 ([8]).

Definition 4 ([5]).

Theorem 6 (Anscombe–Donsker Invariance Principle, ADIP).

Definition 7 ( J 1 convergence).

1.6. Organization of the note

2. Main result

4.1. Is the long cycle approach generalizable to different models?

4.1.1. EPPO excited by a Lévy noise

4.1.2. EPPO excited by a colored noise

4.2. A possible alternative to the long cycle behavior

Conjecture 9 (An alternative analytic approach).

Footnotes

Acknowledgements

References

Definition 7 ( $J_{1}$ convergence).