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

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 ( ν ) ν ( f ) = ∫ D f ( y , z ) m ( y , z ) d y d z + ∫ D + f ( y , Y ) m ( y , Y ) d y + ∫ D − f ( y , − Y ) m ( y , − Y ) d y where D ≜ R × ( − Y , Y ) , D + ≜ ( 0 , ∞ ) × { Y } , D − ≜ ( − ∞ , 0 ) × { − Y } .

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 × [ − Y , Y ] to the following equation: ( P λ ) 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 − with the nonlocal boundary condition u λ ( y , Y ) and u λ ( y , − Y ) are continuous such that ‖ u λ ‖ ∞ ⩽ ‖ f ‖ ∞ λ and u λ is continuous and lim λ → 0 λ u λ = ν ( f ) . Here, we use the following notations A φ ≜ − 1 2 ∂ 2 φ ∂ y 2 + ( c 0 y + k z ) ∂ φ ∂ y − y ∂ φ ∂ z , B ± φ ≜ − 1 2 ∂ 2 φ ∂ y 2 + ( c 0 y ± k Y ) ∂ φ ∂ y .

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: ( P v ) A v = f in D , B + v = f in D + , B − v = f in D − 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: ν ( f ) = v ( 0 − , Y ; f ) + v ( 0 + , − Y ; f ) 2 v ( 0 − , Y ; 1 ) . Besides, as λ → 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 u λ ( y , z ; f ) = u ( y , z ; f ) + ν ( f ) λ + o ( 1 λ ) and where the function u ( y , z ; f ) satisfies the following Poisson equation: ( P u ) A u = f − ν ( f ) in D , B + u = f − ν ( f ) in D + , B − u = f − ν ( f ) in D − with the nonlocal boundary condition given by the fact that y ↦ u ( y , ± 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.

Then, with the notation τ 0 ≜ inf { t > 0 , y ( t ) = 0 and | z ( t ) | = Y } , δ ≜ sign ( z ( τ 0 ) ) which labels the first boundary reached by the process ( y ( t ) , z ( t ) ) , and 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 } , they have obtained a probabilistic formula for the coefficient of the growth rate of the variance of Δ ( t ) as follows: (3) lim t → ∞ σ 2 ( Δ ( t ) ) t = μ γ 2 where (4) γ 2 ≜ E ( Δ ( τ 1 ) − Δ ( τ 0 ) ) 2 and μ ≜ 1 E ( τ 1 − τ 0 ) . 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 Φ ( ξ ) ≜ E ( exp ( i ξ ( Δ ( τ 1 ) − Δ ( τ 0 ) ) ) ) , 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 Φ ( ξ ) = φ ( ξ ) φ ¯ ( ξ ) = | φ ( ξ ) | 2 where φ ( ξ ) ≜ E ( exp ( i ξ ( Δ ( s 0 ) − Δ ( τ 0 ) ) ) ) . 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: Φ ( ξ ) = Φ ( 0 ) + Φ ′ ( 0 ) ξ + Φ ″ ( 0 ) ξ 2 2 + o ( ξ 2 ) Φ ( − ξ ) = Φ ( 0 ) − Φ ′ ( 0 ) ξ + Φ ″ ( 0 ) ξ 2 2 + o ( ξ 2 ) which imply Φ ″ ( 0 ) = 2 ( Φ ( ξ ) − Φ ( 0 ) ξ 2 ) + o ( 1 ) , we deduce that − Φ ″ ( 0 ) = lim ξ → 0 2 ( 1 − | v ξ ( 0 , Y ) | 2 ξ 2 ) . 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 − Φ ″ ( 0 ) = E ( Δ ( τ 1 ) − Δ ( τ 0 ) ) 2 . In addition, the mean duration of a long cycle (which is twice the duration of a half long cycle) can be interpreted as 2 v ¯ ( 0 , Y ) where v ¯ solves the Problem ( P v ¯ ). In this context, Formula (3) becomes (5) lim t → ∞ σ 2 ( Δ ( t ) ) t = lim ξ → 0 1 − | v ξ ( 0 , Y ) | 2 ξ 2 v ¯ ( 0 , Y ) where v ξ satisfies ( 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 − with the boundary condition: v ξ ( 0 − , − Y ) = 1 and y ↦ v ξ ( y , Y ) is continuous, and v ¯ satisfies ( P v ¯ ) A v ¯ = 1 in D , B + v ¯ = 1 in D + , B − v ¯ = 1 in D − with the boundary condition: v ¯ ( 0 − , − Y ) = 0 and y ↦ 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 (6) W ( b , T ) ≜ P ( max t ∈ ( 0 , T ) | Δ ( t ) | ⩾ b ) . 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 ∝ n and τ ∝ 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: (7) Δ n ( t ) ≜ Δ ( n t ) γ μ n toward a standard Wiener process Z. In this context, as n goes to ∞, W ( n b , n T ) converges to W ⋆ ( b γ μ , T ) ≜ P ( sup t ∈ ( 0 , T ) | Z t | ⩾ b γ μ ) which is known (see e.g. [12], p. 34) as (8) W ⋆ ( b γ μ , T ) = 1 − 4 π ∑ k = 0 + ∞ ( − 1 ) k 2 k + 1 exp ( − ( 2 k + 1 ) 2 π 2 T γ 2 μ 8 b 2 ) . The important consequence, as shown in Theorem 8-(ii), is the connection between W and W ⋆ in the following sense: (9) lim n → ∞ W ( n b , n T ) = 1 − 4 π ∑ k = 0 + ∞ ( − 1 ) k 2 k + 1 exp ( − ( 2 k + 1 ) 2 π 2 T γ 2 μ 8 b 2 ) . 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 (10) W ( n b , n T ) ∈ ( W ⋆ ( b γ μ , T ) − ϵ , W ⋆ ( b γ μ , T ) + ϵ ) provided that n ⩾ N ϵ .

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 ( n b , n T ) to W ⋆ ( b γ μ , T ) is order 1 / 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.

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 , ∞ ) into R that are right continuous on [ 0 , ∞ ) and have finite left limits on ( 0 , ∞ ) . Let Γ denote the set of functions α : [ 0 , ∞ ) → [ 0 , ∞ ) that are strictly increasing and continuous with α ( 0 ) = 0 and lim t → ∞ α ( t ) = + ∞ . A sequence ( g n ) n in D converges to g ∈ 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 sup 0 ⩽ t ⩽ T | α n ( t ) − t | → 0 as n → ∞ and sup 0 ⩽ t ⩽ T | g n ( α n ( t ) ) − g ( t ) | → 0 as n → ∞ .

Conjecture 9 (An alternative analytic approach).

We believe that lim t → + ∞ σ 2 ( Δ ( t ) ) t = ν ( | ∂ u ∂ y | 2 ) where ν and u have been defined in ( ν ) and ( P u ).