ONE-PERIOD STABILITY ANALYSIS OF POLYGONAL SWEEPING PROCESSES WITH APPLICATION TO AN ELASTOPLASTIC MODEL

. We oﬀer a ﬁnite-time stability result for Moreau sweeping processes on the plane with periodically moving polyhedron. The result is used to establish the convergence of stress evolution of a simple network of elastoplastic springs to a unique cyclic response in just one cycle of the external displacement-controlled cyclic loading. The paper concludes with an example showing that smoothing the vertices of the polyhedron makes ﬁnite-time stability impossible.


Introduction
Aiming to design materials with better properties, there has been a great deal of work lately where a discrete structure comes from a certain microstructure formulated through a lattice of elastic springs [13] (metals), [15] (polymers), [6] (titanium alloys), [10] (biological materials).At the same time, recent findings show [4] that a pivotal role in the performance of heterogeneous materials under cyclic loading is played by micro-plasticity.Specifically, the regions of asymptotic concentration of plastic deformations are the likely candidates for fatigue crack initialization [3,4].Current methods of computing the asymptotic response (see e.g.[5,16]) run the numeric routine until the difference between the responses corresponding to two successive cycles of loading gets smaller than a prescribed tolerance, without any estimate as for how soon such a prescribed accuracy will be reached.Through a case study, this paper initiates the development of a theory where the distribution of plastic deformations in a network of elastoplastic springs with external loading can be evaluated in just two cycles of the loading.
We stick to the setting of ideal plasticity (the stress of each spring is constrained within so-called elastic limits beyond which plastic deformation begins) and address an equivalent problem of finding the asymptotic distribution of stresses s(t) = (s 1 (t), . . ., s m (t)) of a network of m elastoplastic springs.As a benchmark for the development of a general theory, this paper considers a network of elastoplastic springs given by Figure 1.We follow the Moreau sweeping process approach according to which the stress vector s(t) can be computed from Figure 1.Elastoplastic model with two displacement-controlled loadings.Throughout the paper we assume that the stiffnesses of all springs equal 1 and the elastic domains of all springs are [−1 , 1].Furthermore, we denote l(t) = (l 13 (t), l 24 (t)) .the m-dimensional solution x(t) of the differential inclusion where C ⊂ R m is a closed convex polyhedron, and c(t) ∈ R m is a T -periodic vector, that we compute in closed-form in the next section of the paper.Moreover, in Moreau's framework the moving set C + c(t) is always located in some subspace of a lower dimension d thus restricting the dynamics to the subspace.Specifically, in such case (1.1) can be reduced to where N V C+c(t) (y) is the normal cone in a d-dimensional subspace V of R m constructed as follows N V C+c(t) (y) = {ζ ∈ V : ζ, c − y 0, for any c ∈ C + c(t)} = N C+c(t) (y) ∩ V.
In the next section of the paper we construct V for the particular model of Figure 1, in which case we get d = dim V = 2.The goal of the paper is, therefore, to obtain conditions under which all the solutions to the planar sweeping process (1.3) with T -periodic t → c(t) converge to a unique asymptotic regime in exactly one time-interval [0, T ].When t → c(t) is Lipschitz-continuous (which is the case when the displacement-controlled loadings l 13 (t) and l 24 (t) of Figure 1 are Lipschitz-continuous), the existence of solutions to (1.3) and continuous dependence on the initial conditions is a well-known fact, see e.g.Kunze and Monteiro Marques [12].The asymptotic behavior of the sweeping process (1.3) was studied by Krejci [11] who proved that, when t → c(t) is T -periodic, any solution x(t) of (1.3) converges to a T -periodic solution to the process (1.3) as t → ∞.The uniqueness of such a T -periodic response was established in Gudoshnikov-Makarenkov [9] under the assumption that the set C is a simplex.The best result towards finite-time stability of solutions to (1.3) is obtained in Adly et al. [1], who dealt with a differential inclusion coming from frictional mechanics.According to [1] a solution x(t) of (1.3) is finite-time stable, if the vector x (t) lies strictly inside the normal cone N C+c(t) (x(t)) for a.a.t such that x(t) is located at a vertex of C + c(t).Denoting by ε > 0 the corresponding minimal margin between vector x (t) and the normal cone N C+c(t) (x(t)), the main result of the present paper relates the values of ε > 0 to the time T 0 required for the solution x(t) to reach its asymptotic limit.If T 0 (which depends on ε and x(0)) doesn't exceed T for any solution x(t) of (1.3), then any solution x(t) reaches the asymptotic limit in exactly one time-interval [0, T ], so that the complete information about the asymptotic dynamics of (1.3) can be gathered by computing x(t) on the time-interval [T, 2T ].
The paper is organized as follows.The next section of the paper is devoted to a formulation of the dynamics of the stress-vector s(t) = (s 1 (t), s 2 (t), s 3 (t)) in terms of a sweeping process of the form (1.3).Specifically, assuming that the stress of each spring is constrained within [−1, 1] and that the Hooke's constants of all springs is 1, we show that the closed convex polygon C of the associated sweeping process (1.3) has 6 vertexes.Section 4 establishes the main result of the paper (Thm.3.1) that relates properties of the function t → c(t) of (1.3) to the duration of finite-time convergence of solutions to the process (1.3) to the asymptotic limit.This relation is then used in Corollary 3.6 of the same section in order to give conditions for one-period stability of sweeping process (1.3), i.e. to give conditions which ensure that any solution x(t) of (1.3) merges with its asymptotic limit in a time not exceeding the period T of t → c(t).Corollary 3.6 is applied to the elastoplastic system of Figure 1 in Section 5, where, for each of the 6 vertexes of C, conditions for the mechanical parameters of the elastoplastic system of Figure 1 are obtained to ensure the one-period convergence to the asymptotic regime given by the vertex under consideration.In Section 6 we give a simple example with C being a circle, which shows that the polygonal shape of C is essential for finite-time stability.A conclusions section concludes the paper.

The sweeping process of the benchmark elastoplastic system
We consider a system of 3 elastoplastic springs connected as shown in Figure 1.We assume that the coordinate of each of the 4 nodes is described by a single scalar and that the distance between node 1 and node 3 is locked and controlled to be equal l 13 (t) (called displacement-controlled loading).Similarly, a displacement-controlled loading l 24 (t) locks the distance between nodes 2 and 4. If ξ = (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) are the coordinates of the nodes of the elastoplastic system of Figure 1, then introducing the total elongations x = (x 12 , x 23 , x 34 ) of the 3 springs of Figure 1 can be computed as Note, that solving The displacement-controlled loadings l 13 (t) and l 24 (t) lead to the following constraints (1, 1, 0)x = l 13 , (0, 1, 1)x = l 24 .
On the other hand, g ⊥ U implies that (1, −1, 1)g = 0, so that we can solve the above three equations for g obtaining g = 1 3 (2l 13 − l 24 , l 13 + l 24 , −l 13 + 2l 24 ) = 1 3 where When the stiffnesses of all springs equal 1 and the elastic domains are [−1, 1] (meaning that −1 and 1 are elastic limits for all springs), the equation for plastic elongations p = (p 1 , p 2 , p 3 ) of springs takes the form where s = (s 1 , s 2 , s 3 ) are stresses of the springs.According to Moreau [14] (see also ), if, at each time t, the elastoplastic system under consideration attains a static equilibrium, then the evolution (termed quasi-static evolution) of the variable can be described by the differential inclusion (1.3) with (see Fig. 2) from which we conclude that d = dim V = 2.

One-period stability result for planar sweeping processes
In this section we restrict ourselves to a 2-dimensional linear space V equipped with an inner product •, • and the consider sweeping process (1.3) with a closed convex polyhedron of a general form where I is a finite set of indices and {n i } i∈I are unit vectors.Without loss of generality, we will assume that 0 ∈ C is a vertex of C, i.e. there exist non-parallel unit vectors see Figure 3.In what follows we will assume that c : 3) is a Lipschitz-continuous function.This allows to get the existence and uniqueness of solutions to the process (1.3) by applying the standard existence and uniqueness theorem (see e.g.Kunze and Monteiro Marques [12]) successively on each of the intervals of continuous differentiability of c : [0, T ] → V .The main result of this paper is the following theorem.
Theorem 3.1.Let c : [0, T ] → V be a Lipschitz-continuous function such that the following conditions hold for all t ∈ [0, T ] where c (t) exists: see Figure 3. Consider a solution y to the sweeping process (1.3) with an initial condition y(0 It is always possible to find h satisfying (3.2).Indeed, one can use Figure 3 to show that a possible choice for h is e.g.h = n 1 + n −1 , see also the proofs below.The following two lemmas are required for the proof of Theorem 3.1.
.The notations of Lemma 3.3 are illustrated at Figure 4.
Proof.Observe, that N C (0) is a sector between vectors n −1 , n 1 of an angle less than π.We take the direction of a as the zero direction and parametrize two unit semicircles from a to −a by an angle ϕ ∈ Therefore, since cos is decreasing on [0, π], we have ϕ n min{ϕ −1 , ϕ In other words, We need to show that α = β.Since y (t * ) and c (t * ) exist we can use the definitions of derivative and limit, i.e. for each ε 1 > 0 there is δ 1 > 0 such that for all t satisfying 0 and for each ε 2 > 0 there is δ 2 > 0 such that for all t satisfying 0 and observe that for all t such that 0 Choose t > t * and conclude that which means the contradiction y(t) / ∈ C + c(t).If we assume α < β we proceed similarly and choose t < t * .Therefore α = β 0 which is the statement of the lemma.
Proof of Theorem 3.1.At first, notice, that 0 ∈ C and (A2) guarantee, that c(t) is the solution to (1.3) for i.e. the point c(t) is swept by two edges, corresponding to n 1 and n −1 .Condition (A3) means that n 1 = c (t) = n −1 hence by (A1) and (A2) we necessarily have By monotonicity of a normal cone this implies that for any other vertex v = 0 of C we have therefore a solution which will happen to be at any vertex of C other than 0 will leave the vertex immediately.
Choose any y 0 ∈ (C \ {0}) + c(0) and consider the corresponding solution y(t) of (1.3).For almost all t ∈ [0, T ] derivatives c (t) and y (t) exist and (1.3) holds as an inclusion.Therefore for almost all t ∈ [0, T ] either (a) y(t) is not swept, i.e. y (t) = 0, or (b) y(t) is swept by a single edge, i.e. y (t) = n i , c (t) n i for some i ∈ I and −c (t), n i > 0 (see lemma 3.5).
From this, taking in account (3.2), we get that Summing up the analysis in cases a and b, we conclude that we have y(T 0 ) − c(T 0 ), h = 0, i.e. y(T 0 ) − c(T 0 ) = 0, which concludes the proof.Theorem 3.1 finally allows us to formulate the following result about one-period stability of the solutions to the sweeping process (1.3).Corollary 3.6.Assume that c : [0, +∞) → V is a τ -periodic Lipschitz-continuous function.Assume that conditions (A1)-(A3) of Theorem 3.1 hold with some T ∈ (0, τ ] and h is chosen according to (3.2).Let T y0 be as given by (3.3).If T y0 ≤ T for all y 0 ∈ C + c(t), then the solution ŷ of (1.3) with the initial condition ŷ(T ) = c(T ) is a one-period globally stable τ -periodic solution.Specifically, any other solution y of (1.3) with y(0) ∈ C + c(0) merges with ŷ in time T (i.e.y(t) = ŷ(t), for all t ≥ T ). 4. Application of the one-period stability result to the sweeping process of the initial elastoplastic system In this section we apply Theorem 3.1 and Corollary 3.6 to the sweeping process of the elastoplastic system of Figure 1 derived in Section 2.
Let us denote the vertices of hexagon C by P 1 , . . ., P 6 .The coordinates of the vertices in R 3 are listed in the table below.One can verify that each of the vertices P 1 , . . ., P 6 simultaneously belongs to the edges of the cube C and to the plane V as defined in (2.6).In the same table we list the coordinates of P 1 , . . ., P 6 in the basis Λ given by (2.3).
Table 1 The coordinates of the vertices of the hexagon C of Figure 5 in bases R 3 and Λ. Basis Observing that one can also write down the coordinates in the basis Λ of the outward normals N 1 , . . ., N 6 as follows.
Table 2 The normals of the hexagon C of Figure 5 in the basis Λ.
and the actual vectors are As the vector h required by Theorem 3.1 we take the position-vectors of the vertices: see Figure 5. Observe from Figures 3 and 5 that the requirement (3.2) is always satisfied for h = H i , i ∈ 1, 6.We apply Theorem 3.1 to the 2-dimensional subspace V = (1, −1, 1) ⊥ of the space of elongations R 3 with the inner product •, • being the usual dot product from R 3 induced to the subspace.
We fix an arbitrary i ∈ 1, 6, translate the corresponding vertex P i to the origin, and redefine C and c(t) as In what follows, we apply Theorem 3.1 with n 1 and n −1 equal N i−1 and N i respectively (defining N 0 = N 6 ) on the subspace V .The reduction of the normal cone N C (0) to V takes the form that we use as N C (0) when applying Theorem 3.1.Finally, since the angles between the vectors N i and H i are the same for every vertex, the constant µ of (3.3) computes as To summarize, the following proposition follows directly from Theorem 3.1.
Proposition 4.1.Assume that, for some ε ∈ (0, 1) and for some i ∈ 1, 6, the Lipschitz-continuous function where Λ is given by (2.4).Consider the solution y of sweeping process (1.3) with the initial condition y(0) = y 0 ∈ C + c(0), where C and c(t) are as defined in (4.3), and V = (1, −1, 1) ⊥ .Let where H i is given by (4.2) with P i given by the R 3 -basis line of Table 1, and N i is given by (4.1) with N Λ i given by Table 2.If T y0 T, then the solution y(t) remains at the vertex P i − 1  3 Λl(t) of C + c(t) for all t T 0 .Likewise, the Corollary 3.6 leads to the following result.Corollary 4.2.Consider a sweeping process (1.3) with C and c(t) given by (4.3), and V = (1, −1, 1) ⊥ .Assume that, for some ε ∈ (0, 1), for some i ∈ 1, 6, and for some T ≤ τ , the τ -periodic Lipschitz-continuous function l : [0, ∞) → R 2 satisfies the condition (4.5) of Proposition 4.1.Let T y0 and P i be as defined in Proposition 4.1.If T y0 ≤ T for all y 0 ∈ C + c(0), then the solution ŷ of (1.3) with the initial condition ŷ(T ) = P i + c(T ) is a one-period globally stable τ -periodic solution.Specifically, any other solution y of (1.3) with y(0) ∈ C + c(0) merges with ŷ in time T (i.e.y(t) = ŷ(t), for all t ≥ T ).

Formulation of the results of the previous section in terms of the displacement-controlled loading explicitly
In this section we rewrite (4.5) in terms of l(t) only.Proceeding with (4.5) we could directly compute all N i = ΛN Λ i ∈ R 3 and plug them into (4.5)but it is more convenient to work using the coordinates of basis Λ in V .To do so we define an auxiliary inner product in R 2 , which takes its values on the pairs of coordinates of a vector as equal to the values of •, • on actual vectors: where the dot product • is from R 3 .Moreover, we can write We plug (5.1) into (4.5) and obtain the conditions for each i ∈ 1, 6 on the velocities of the displacementcontrolled loadings l (t): where H Λ i , N Λ i are coordinate vectors in the basis Λ of the respective vectors H i , N i taken from the tables above and ε > 0.
We can finally formulate Proposition 4.1 and Corollary 4.2 of the previous section referring to mechanical parameters of the elastoplastic system of Figure 1 only.
Proposition 5.1.Let s(t) = (s 1 (t), s 2 (t), s 3 (t)) be the stress response of the quasi-static evolution of the elastoplastic system of Figure 1 subject to a Lipschitz-continuous displacement-controlled loading l(t) = (l 13 (t), l 24 (t)) .Assume that, for some i ∈ 1, 6, the respective set of conditions of (5.3) holds on an interval [0, T ] with a fixed ε ∈ (0, 1).Define T 0 > 0 as where P i and H i are as defined in Proposition 4.1.Finally, assume that T 0 T. Then the stress-vector s(t) remains equal to P i for all t T 0 .To conclude the statement of Proposition 5.1 from Proposition 4.1, one uses formula (2.5) connecting s(t) and y(t) and observes that the setting of quasi-static evolution implies that the initial value of the stress satisfies (see Moreau [14] or Gudoshnikov-Makarenkov [9]) (where conv stands for a convex hull), which gives y(0) ∈ C + c(0) required for Proposition 4.1.
Corollary 5.2.Consider quasi-static evolution of the elastoplastic system of Figure 1 subject to a τ -periodic Lipschitz-continuous displacement-controlled loading l(t) = (l 13 (t), l 24 (t)) with τ > 4 √ 2 √ 3 .Assume that, for some 4 √ 2 τ √ 3 < ε < 1 and for some i ∈ 1, 6, the function l(t) satisfies the respective set of conditions of (5.3) on the interval [0, T ] with T = 4 √ 2 ε √ 3 .Then the stress evolution ŝ of Figure 1 defined by the initial condition ŝ(T ) = P i (where P i is as defined in Prop.5.1) is a one-period globally stable τ -periodic stress response.Specifically, any stress evolution s of elastoplastic system of Figure 1 takes the values s(t) = ŝ(t) for all t ≥ T .
To prove the corollary we refer to Figure 5 and estimate (5.4) as follows.
where in the expression after the first inequality we use H i = P i and assume P 0 = P 6 , P −1 = P 5 , P −2 = P 4 .We can interpret condition (5.3a) of convergence to P 1 = (1, 1, 0) ∈ R 3 as follows: if both controlled displacements are expanding, but l 13 is expanding faster then l 24 then (excluding a non-boundary case using ε) elastic elongations e 1 , e 2 and stresses s 1 , s 2 will reach their maximal value (so they will produce maximal stress) and spring 3 will have zero elastic elongation (i.e. it will be relaxed) by the time T 0 .
Analogously, for i = 3 we conclude from (5.3c) and P 3 = (−1, 0, 1) that if the controlled displacement l 13 is shrinking and l 24 is expanding, then by the time T 0 the variables e 1 , s 1 will be minimal (so it will produce minimal stress), e 3 , s 3 will be maximal, and spring 2 will be relaxed.
For i = 4 we conclude from (5.3d) and P 4 = (−1, −1, 0) that, in case of both controlled displacements shrinking and l 13 doing it faster, then by the time T 0 elastic elongations e 1 , e 2 and stresses s 1 , s 2 will be minimal but spring 3 will be relaxed.
These three cases are shown in Figure 6.The rest of the cases i = 2, 5, 6 are the same up to the spatial symmetry.

An example of asymptotic stability in a sweeping process
In this section we show that increasing the magnitude of a uni-directional loading is no longer capable to create finite-time stability when the boundary of the moving set is smooth.Indeed, let B r (0) ⊂ R 2 be the open ball of radius r > 0 centered at 0. For arbitrary r > 0 and T > 0, introduce the moving set C(t) as a T -periodic Let n * = n (u(t * ),t * ) .Using Heine's definition of limit and the definition of derivative we can show from (6.4) that there is τ > 0 small enough for n * , c(t * + τ ) − c(t * ) < 0. (6.5) From the other side, notice that u(t The contradiction with (6.5) proves our assumption wrong and finishes the proof of the lemma.
In what follows, we will approximate the solution u(t) on the interval t ∈ [t 0 , t 0 + ∆] using a so-called catching-up algorithm [14], [12].Specifically, for every n ∈ N we equally partition the interval by the points t 0 < t 1 < t 2 < . . .< t i < t i+1 < . . .< t n = t 0 + ∆, where |t i+1 − t i | = 1 n , and define an approximation u n (t) of the solution u(t) of (6.2) as According to the catching-up algorithm (see e.g.[12], Thm. 2), it holds that u n (t) → u(t) as n → ∞, pointwise on the interval [t 0 , t 0 + ∆].Let (v n (t), h n (t)) be the components of u n (t).Denoting ∆ i = ∆ n , we have Let α be the angle formed by the line segments [u n (t i ), O(t i )] and [O(t i ), (v n (t i ), 0)] and let β be the angle formed by the line segments [u n (t i ), O(t i+1 )] and [O(t i+1 ), (v n (t i ), 0)] (see Fig. 8).Then, by the definition of cotangent in the triangle formed by the vertices u n (t i ), H and O(t i+1 ), cot β = r cos α + L∆ i r sin α Therefore we can estimate the ratio hn(ti+1) hn(ti) = r sin β r sin α = sin 2 β sin 2 α as Collecting these inequalities through i ∈ 0, n − 1, we get .
Since e x = lim n→∞ 1 + x n n , we can pass to the limit as n → ∞ obtaining Recall that [t 0 , t 0 + ∆] ⊂ 0, T 4 .Arguments similar to above can be made for each of the two remaining intervals of monotonicity of the function x(t).Specifically, there are intervals t 0 , t 0 + ∆ ⊂ T 4 , 3T 4 , t 0 , t 0 + ∆ ⊂ Since there are only 3 intervals of monotonicity of x(t) on [0, T ], we can be sure that for t ∈ [0, T ] \ [t 0 , t 0 + ∆] ∪ t 0 , t 0 + ∆ ∪ t 0 , t 0 + ∆ , the solution u(t) is located in the interior of C(t) and it is not moving (u (t) = 0).Therefore, h(0) = h(t 0 ), h(t 0 + ∆) = h t 0 , h t 0 + ∆ = h t 0 , h t 0 + ∆ = h(T ), so that we can combine (6.6) and (6.7), and write the following estimate for the entire period [0, T ]: The same estimate can be obtained for each period [T (l − 1), T l], l ∈ N. It follows from the right-hand side of (6.8) that the distance from the solution to the x-axis decreases.The left-hand side implies that on the l-th period we have 0 < e −8l h(T l) h(0) .
Since h(0) = 0, we have that h(T l) = 0, l ∈ N, i.e. the solution u(t) never reaches the x-axis while approaching it arbitrary closely.In other words, the following proposition takes place.Proposition 6.2.Consider a planar sweeping process (6.2) with a moving set given by (6.1), where r > 0 and T > 0 are arbitrary given constants.Then, any solution u(t) = (v(t), h(t)) of (6.2) asymptotically approaches the solution u 0 (t) = (v 0 (t), 0), where Furthermore, if h(0) = 0, then the solution u(t) never reaches u 0 (t) in finite time.

Conclusions
The paper investigated finite-time stability of a network of 3 elastoplastic springs on 4 nodes subject to 2 cyclic displacement-controlled loadings.Though this looks simple, such a network led us to an interesting mathematical problem within the theory of sweeping processes.Specifically, in order to prove finite-time convergence of stresses of the springs of the network, a theory of finite-time stability for planar sweeping processes with periodically moving polyhedron has been developed.Our analysis clarified how different typical loading scenarii can lead to 6 different types of stress response corresponding to the 6 vertices of the associated polyhedron.
In order to emphasize the crucial role of the nonsmoothness of the boundary of the convex moving set for the finite-time stability, we showed that replacing the periodically moving polyhedron of the sweeping process by a circle is no longer a finite-time stable system.
Our approach is intrinsically two-dimensional.The idea of Adly et al. [1] (a Lyapunov function approach) seems to be promising for further generalizations of our work to higher dimensions.

Figure 2 .
Figure 2. Moving set of the sweeping process (1.1) shown in two projections.

Figure 3 .
Figure 3. Illustration of conditions of Theorem 3.1.

Figure 5 .
Figure 5.The moving set of the sweeping process from Section 2 and the elements which we use to apply Theorem 3.1.

Figure 6 .
Figure 6.Interpretations of the convergence of stress to P 1 (a), P 3 (b), P 4 (c).The top and bottom arrows surrounding l 13 , l 24 represent the directions of controlled displacements (double arrow represents the greater rate) and the arrows in between show the resulting state of the stresses.

3T 4 ,
T along which u(t) remains on the boundary of C(t).We may partition these intervals in the same manner and obtain the estimates analogous to(6