Mandar Chandorkar

Cyril Furtlehner

Enrico Camporeale

Michèle Sebag

DYNAMIC TIME LAG REGRESSION: PREDICTING WHAT & WHEN

This paper tackles a new regression problem, called Dynamic Time-Lag Regression (DTLR), where a cause signal drives an effect signal with an unknown time delay. The motivating application, pertaining to space weather modelling, aims to predict the near-Earth solar wind speed based on estimates of the Sun's coronal magnetic field. DTLR differs from mainstream regression and from sequence-to-sequence learning in two respects: firstly, no ground truth (e.g., pairs of associated subsequences) is available; secondly, the cause signal contains much information irrelevant to the effect signal (the solar magnetic field governs the solar wind propagation in the heliosphere, of which the Earth's magnetosphere is but a minuscule region). A Bayesian approach is presented to tackle the specifics of the DTLR problem, with theoretical justifications based on linear stability analysis. A proof of concept on synthetic problems is presented. Finally, the empirical results on the solar wind modelling task improve on the state of the art in solar wind forecasting.

INTRODUCTION

A significant body of work in machine learning concerns the modeling of spatio-temporal phenomena [START_REF] Shi | Machine learning for spatiotemporal sequence forecasting: A survey[END_REF][START_REF] Sundar Rangapuram | Deep state space models for time series forecasting[END_REF], including the causal analysis of time series [START_REF] Peters | Elements of Causal Inference -Foundations and Learning Algorithms[END_REF], with applications ranging from markets [START_REF] Pennacchioli | The retail market as a complex system[END_REF] to bioinformatics [START_REF] Brouard | Fast metabolite identification with input output kernel regression[END_REF] to climate [START_REF] Nooteboom | Using network theory and machine learning to predict el niño[END_REF]. This paper focuses on the problem of modeling the temporal dependency between two spatio-temporal phenomena, where the latter one is caused by the former one [START_REF] Granger | Investigating causal relations by econometric models and cross-spectral methods[END_REF][START_REF] Runge | Causal network reconstruction from time series: From theoretical assumptions to practical estimation[END_REF] with a non-stationary time delay.

The motivating application domain is that of space weather. The sun, a perennial source of charged energetic particles, is at the origin of geomagnetic phenomena within the sun-earth system. Specifically, the sun ejects charged particles into the surrounding space in all directions and some of these particle clouds, a.k.a. solar wind, reach the Earth's vicinity. High speed solar wind is a major threat for the modern world, causing severe damages to e.g., satellites, telecommunication infrastructures, under sea pipelines, among others. 1A key prediction task thus is to forecast the speed of the solar wind in the vicinity of the Earth [START_REF] Munteanu | Propagation delay of solar wind discontinuities: Comparing different methods and evaluating the effect of wavelet denoising[END_REF][START_REF] Haaland | Solar wind propagation delay: Comment on minimum variance analysis-based propagation of the solar wind observations: Application to real-time global magnetohydrodynamic simulations by A. Pulkkinen and L. Raststatter[END_REF][START_REF] Reiss | Forecasting the ambient solar wind with numerical models. i. on the implementation of an operational framework[END_REF], sufficiently early to emit an alarm and be able to prevent the damage to the best possible extent. Formally the goal is to model the dependency between heliospheric observations (available at light speed), referred to as cause series, and the solar wind speed series recorded at the Lagrangian point L 1 (a point on the Sun-Earth line 1.5 million kilometers away from the Earth), referred to as effect series. The key difficulty is that the time lag between an input and its effect, the solar wind recorded at L 1 , varies from circa 2 to 5 days depending on, among many factors, the initial direction of emitted particles and their energy. Would the lag be constant, the solar wind prediction problem would boil down to a mainstream regression problem. The challenge here is to predict, from the solar image x(t) at time t the value y(t + ∆t) of the solar wind speed reaching the earth at time t + ∆t where both the value y(t + ∆t) and the time lag ∆t depend on x(t).

Related work. Indeed, the modeling of dependencies among time series has been intensively tackled (see e.g., [START_REF] Zhou | Non-parametric determination of real-time lag structure between two time series: The optimal thermal causal path method with applications to economic data[END_REF]; [START_REF] Runge | Causal network reconstruction from time series: From theoretical assumptions to practical estimation[END_REF]). When considering varying time lag, many approaches rely on dynamic time warping (DTW) [START_REF] Sakoe | Dynamic programming algorithm optimization for spoken word recognition[END_REF]. For instance, DTW is used in [START_REF] Gaskell | Signal diffusion mapping: Optimal forecasting with time-varying lags[END_REF], taking a Bayesian approach to achieve the temporal alignment of both series under some restricting assumptions (considering slowly varying time lags and linear relationships between the cause and effect time series). More generally, the use of DTW in time series analysis relies on simplifying assumptions on the cause and effect series (same dimensionality and structure) and builds upon available cost matrices for the temporal alignment. Also related is sequence-to-sequence learning (Sutskever et al., 2014), primarily aimed to machine translation. While Seq2Seq modelling relaxes some former assumptions (such as the fixed or comparable sizes of the source and target series), it still relies on the known segmentation of the source series into disjoint units (the sentences), each one being mapped into a large fixed-size vector using an LSTM; and this vector is exploited by another LSTM to extract the output sequence. Attention-based mechanisms [START_REF] Graves | Generating sequences with recurrent neural networks[END_REF]; [START_REF] Bahdanau | Neural machine translation by jointly learning to align and translate[END_REF] alleviate the need to encode the full source sentence into a fixed-size vector, by learning the alignment and allowing the model to search for the parts of the source sentence relevant to predict a target part. More advanced attention mechanisms [START_REF] Kim | Structured attention networks[END_REF]Vaswani et al., 2017) refine the way the source information is leveraged to produce a target part. But to our best knowledge, the end-to-end learning of the sequence-to-sequence modelling relies on the segmentation of the source and target series, and the definition of associated pairs of segments (e.g. the sentences).

Our claim is that the regression problem of predicting both what the effect is and when the effect is observed, called Dynamic TimeLag Regression (DTLR), constitutes a new ML problem: With respect to the modeling of dependencies among time series, it involves stochastic dependencies of arbitrary complexity; the relationship between the cause and the effect series can be non-linear (the what model). Furthermore, the time lag phenomenon (the when model) can be non smooth (as opposed to e.g. [START_REF] Zhou | Non-parametric determination of real-time lag structure between two time series: The optimal thermal causal path method with applications to economic data[END_REF]). With respect to sequence-to-sequence translation, a main difference is that the end-to-end training of the model cannot rely on pairs of associated units (the sentences), adversely affecting the alignment learning. Lastly, and most importantly, in the considered DTLR problem, even if the cause series has high information content, only a small portion of it is relevant to the prediction of the effect series. On one hand, the cause series might be high dimensional (images) whereas the effect series is scalar; on the other hand, the cause series governs the solar wind speed in the whole heliosphere and not just in near-Earth space. In addition to avoiding typically one or two orders of magnitude expansion of an already large input signal dimension, inserting the time-lag inference explicitly in the model can also potentially improve its interpretability.

Organization of the paper. The Bayesian approach proposed to tackle the specifics of the DTLR regression problem is described in section 2; the associated learning equations are discussed, followed by a stability analysis and a proof of consistency (section 3). The algorithm is detailed in section 4. The experimental setting used to validate the approach is presented in section 5; the empirical validation on toy problems and on the real-world problem are discussed in section 6 2 PROBABILISTIC DYNAMICALLY DELAYED REGRESSION

POSITION OF THE PROBLEM

Given two time series, the cause series x(t) (x(t) ∈ X ⊂ R D ) and the observed effect series y(t), the sought model consists of a mapping f (.) which maps each input pattern x(t) to an output y(φ(t)), and a mapping g(.) which determines the time delay φ(t) -t between the input and output patterns:

y(φ(t)) = f [x(t)] (1) φ(t) = t + g[x(t)]
(2) with f : X → R, and g : X → R + , where t ∈ R + represents the continuous temporal domain. The input signal x(t) is possibly high dimensional and contains the hidden cause to the effect y(t) ∈ R; y(t) is assumed to be scalar in the remainder of the paper. Function g : X → R + represents the time delay between inputs and outputs. Vectors are written using bold fonts.

As said, Eqs 1-2 define a regression problem that differs from standard regression in two ways: Firstly, the time lag g[x(t)] is non-stationary as it depends on x(t). Secondly, g[x(t)] is unknown, i.e. it is not recorded explicitly in the training data.

Assumption. For the sake of the model identifiability and computational stability, the time warping function φ(t) = t + g[x(t)] is assumed to be sufficiently regular w.r.t. t. Formally, φ(.) is assumed to be continuous2 .

PROBABILISTIC DYNAMIC TIME-LAG REGRESSION

For practical reasons, cause and effect series are sampled at constant rate, and thereafter noted x t and y t with t in N . Accordingly, mapping g maps x t onto a finite set T of possible time lags, with T = {∆t min . . . , ∆t max } an integer interval defined from domain knowledge. The unavoidable error due to the discretization of the continuous time lag is mitigated along a probabilistic model, associating to each cause x, a set of predictors ŷ(x) = {ŷ i (x), i ∈ T } and a probability distribution p(x) on T estimating the probability of delay of the effects of x. Overall, the DTLR solution is sought as a probability distribution conditioned on cause x, mixture of Gaussians3 centered on the predictors ŷi (x), where the mixture weights are defined from p(x). More formally, letting y t denote the vector of random variables {y t+i , i ∈ T }:

P y t |x t = x = {τi∈{0,1},i∈T } p τ 1 , . . . , τ |T | |x)N ŷ(x), Σ(τ ) (3)
with Σ = Diag(σ i (τ ) 2 ) the diagonal matrix of variance parameters attached to each time-lag i ∈ T . Two simplifications are made for the sake of the analysis. Firstly, the stochastic time lag is modelled as the vector (τ i ), i ∈ T of binary latent variables, where τ i indicates whether x t drives y t+i (τ i = 1) or not (τ i = 0). The assumption that every cause has a single effect is modelled by imposing:4

i∈T τ i = 1. (4)
From constraint (4), probability distribution p(x) thus is sought as the vector (p i (x)) for i in T , summing to 1, such that pi (x) stands for the probability of the effect of x t = x to occur with delay i.

The second simplifying assumption is that the variance σ 2 i (τ ) of predictor ŷi does not depend on x, by setting:

σ i (τ ) -2 = 1 + j α ij τ j σ -2 ,
with σ 2 a default variance and α ij ≥ 0 a matrix of non-negative real parameters. This particular formulation supports the tractable analysis of the posterior probability of τ i (in supplementary material). The fact that x can influence y i through predictor ŷi (x) even when τ i = 0 reflects an indirect influence due to the auto-correlation of the y series. This influence comes with a higher variance, enforced by making α ij a decreasing function of |i -j|. More generally, a large value of α ii compared to α ij for i = j corresponds to a small auto-correlation time of the effect series.

LEARNING CRITERION

The joint distribution is classically learned by maximizing the log likelihood of the data, which can here be expressed in closed form. Let us denote respectively the dataset and parameters as {(x, y)} data and θ = (ŷ, p, σ, α). From Eq. (3) the conditional probability q i (x, y) def = P (τ i = 1|x, y) reads:

q i (x, y) = 1 Z(x, y|θ) pi (x) exp - 1 2σ 2 j∈T α ji y j -ŷj (x) 2 + 1 2 j∈T log(1 + α ji ) (5)
with normalization constant

Z(x, y|θ) = i∈T pi (x) exp - 1 2σ 2 j∈T α ji y j -ŷj (x) 2 + 1 2 j∈T log(1 + α ji ) .
The log-likelihood then reads (intermediate calculations in supplementary material, appendix A):

L[{(x, y)} data |θ] = -|T | log(σ) -E data i∈T 1 2σ 2 y i -ŷi (x) 2 -log Z(x, y|θ) (6)
where E data denotes averaging over the dataset. For notational simplicity, the time index t is omitted in the following and the empirical averaging on the data is noted E data . The hyper-parameters σ and matrix α of the model are obtained by optimizing L:

σ 2 1 + α ij = E data y i -ŷi (x) 2 q j (x, y) E data q j (x, y) , (7)
In addition the optimal ŷ and p reads:

ŷi (x) = E data y i 1 + j∈T α ij q j (x, y) x E data 1 + j∈T α ij q j (x, y) x (8) pi (x) = E data q i (x, y) x , (9)
where the above conditional empirical averaging operates as an averaging over samples close to x.

These are self-consistent equations, since q i (x, y) depends on the parameters σ 2 and α ij , ŷ and p.

The proposed algorithm detailed in section 4 implements the saddle point method defined from Eqs (7,5,8,9): alternatively, hyper-parameters σ and α ij are updated from Eq. ( 7) based on the current ŷi and pi ; and predictors ŷi and mixture weights pi are updated according to Eqs ( 8) and ( 9) respectively.

THEORETICAL ANALYSIS

The proposed DTLR approach is shown to be consistent and analyzed in the simple case where α is a diagonal matrix (α ij = αδ ij ).

LOSS FUNCTION AND RELATED OPTIMAL PREDICTOR

Let us assume that the hyper-parameters of the model have been identified together with predictors ŷi (x) and weights pi (x). These are leveraged to achieve the prediction of the effect series. For any given input x, the sought eventual predictor is expressed as (ŷ(x), Î(x)) where Î(x) is the predicted time lag and ŷ(x) the predicted value. The associated L 2 loss is:

L 2 (ŷ, Î) = E data y Î(x) -ŷ(x) 2 . ( 10
)
Then it comes:

Proposition 3.1. With same notations as in Eq. ( 3), with α ij = αδ ij , α > 0, the optimal composite predictor (y , I ) is given by y

(x) = ŷI (x) (x) with I (x) = arg max i pi (x) ,
Proof. In supplementary material, Appendix C.

LINEAR STABILITY ANALYSIS

The saddle point (Eqs 7, 5, 8, 9) admits among others a degenerate solution, corresponding to pi (x) = 1/|T |, α ij = 0 for all pairs (i, j), with σ 2 = σ 2 0 . Informally the model converges toward this degenerate trivial solution when there is not enough information to build specialized predictors ŷi .

Let us denote ∆y

2 i (x) = y i -ŷi (x)
2 the square error made by predictor ŷi for x, and

σ 2 0 = 1 |T | E data i∈T ∆y 2 i (x)
the average of MSE over the set of the predictors ŷi , i ∈ T .

Let us investigate the conditions under which the degenerate solution may appear, by computing the Hessian of the log-likelihood and its eigenvalues. Under the simplifying assumption α ij = αδ ij , the model involves 2|T | functional parameters ŷ and p and two hyper-parameters α and r = σ 2 /σ 2 0 . After the computation of the Hessian (in supplementary material, Appendix B) the system involves three key statistical quantities, two global ones:

C 1 [q] = 1 σ 2 0 E data i∈T q i (x, y)∆y 2 i (x) , (11)
C 2 [q] = 1 σ 4 0 E data i∈T q i (x, y) ∆y 2 i (x) - j∈T q j (x, y)∆y 2 j (x) 2 , ( 12
)
and a local |T |-vector of components

u i [x, q] = 1 σ 2 0 E data q i (x, y) ∆y 2 i (x) - j∈T q j (x, y)∆y 2 j (x) x .
Up to a constant, C 1 represents the covariance between the latent variables {τ i } and the normalized predictor errors. C 1 smaller than one indicates a positive correlation between the latent variables and small errors; the smaller the better. For the degenerate solution, i.e. q = q 0 uniform, C 1 [q 0 ] = 1 and C 2 [q 0 ] represents the default variability among the prediction errors. u i [x, q] informally measures the quality of predictor ŷi relatively to the other ones at x. More precisely, a negative value of u i [x, q] indicates that ŷi is doing better than average in the neighborhood of x.

At a saddle point the parameters are given by:

σ 2 σ 2 0 = |T | -C 1 [q] |T | -1 and α = |T | |T | -1 1 -C 1 [q] C 1 [q] .
The predictors ŷ are decoupled from the rest whenever they are centered, which we assume. So the analysis can focus on the other parameters.

If p is fixed a saddle point is stable iff

C 2 [q] < 2C 2 1 [q] + O 1 |T | .
In particular, the degenerate solution is unstable if

C 2 [q 0 ] > 2 1 - 1 |T | .
Note that for ∆y i (x) iid centered with variance σ 2 0 and relative kurtosis κ (conditionally to x) one has C 2 = (2 + κ)(1 -1/|T |). Therefore, whenever ∆y 2 i (x) fluctuates and the relative kurtosis is non-negative, the degenerate solution is unstable and will thus be avoided.

If p is allowed to evolve (after Eq. ( 9)) the degenerate trivial solution becomes unstable as soon as C 2 [q 0 ] is non-zero, due to the fact that the gradient points in the opposite direction to u(x) (with dp(x) ∝ -C 2 [q 0 ]u(x)), thus rewarding the predictors with lowest errors by increasing their weights.

The system is then driven toward other solutions, among which the localized solutions of the form:

pi (x) = δ i,I(x) ,
with an input dependent index I(x) ∈ T . As shown (in supplementary material, Appendix C) the maximum likelihood localized solution also minimizes the loss function (Eq. 10). The stability of such localized solutions and the existence of other (non-localized) solutions is left for further work.

THE DTLR ALGORITHM

The DTLR algorithm learns both regression models ŷ(x) and p(x) from series x t and y t , using alternate optimization of the model parameters and the model hyper-parameters α and σ 2 , after Eqs (7,5,8,9). The model search space is that of neural nets, parameterized by their weight vector θ. The inner optimization loop updates θ using mini-batch based stochastic gradient descent. At the end of each epoch, after all minibatches have been considered, the outer optimization loop computes hyper-parameters α and σ 2 on the whole data.

Initialization of α and σ it ←-0 ; while it < max do while epoch do θ ←-Optimize(L(θ, α, σ 2 )) ; end σ 2 ←-σ 2 0 |T |-C1[q] |T |-1 ; α ←-|T | |T |-1 1-C1[q]
C1[q] ; end Result: Model parameters θ = {ŷ, p}, hyper-parameters α, σ 2

Algorithm 1: DTLR algorithm

The algorithm code is available in supplementary material and will be made public after the reviewing period. The initialization of hyper-parameters α and σ is settled using preliminary experiments (same setting for all considered problems: α ∼ U (0.75, 2); σ 2 ∼ U (10 -5 , 5)).

The neural architecture implements predictors ŷ(x) and weights p(x) on the top of a same feature extractor from input x. In the experiments, the architecture of the feature extractor is a 2-hidden layer fully connected network. On the top of the feature extractor are the single layer ŷ and p models, each with |T | output neurons, with |T | the size of the chosen domain for the time lag.

EXPERIMENTAL SETTING

The goal of experimental validation is threefold. A first issue regards the accuracy of DTLR, measured from the mean absolute error (MAE), root mean square error (RMSE) and Pearson correlation of the learned DTLR model (y (x t ), I (x)). DTLR is compared to the natural baseline defined as the regressor with constant time lag, ŷ ∆(x t ), with ∆ being the average of all possible time lags in T . The predictions of DTLR and the baseline are compared with the ground truth value of the effect series. However the predicted time lag I (x t ) can only be assessed if the ground truth time-lag relationship is known. In order to do so, three synthetic problems of increasing difficulty are defined below.

Secondly, the stability and non-degeneracy of the learned model are assessed from the statistical quantities σ 0 and C 1 (section 3.2), compared to the degenerate solution pi (x) = 1/|T |. For C 1 < 1, the model accurately specializes the found predictors pi .

Lastly, and most importantly, DTLR is assessed on the solar wind prediction problem, and compared to the best state of the art in space weather.

Synthetic Problems. Four synthetic problems of increasing difficulty are generated using Stochastic Langevin Dynamics. In all problems, the cause signal x t ∈ R 10 and the effect signal y t are generated as follows (with η = 0.02, s 2 = 0.7):

x t+1 = (1 -η)x t + N (0, s 2 ) (13) v t = k||x t || 2 + c (14) y t+g(xt) = f (v t ), (15)
with time-lag mapping g(x t ) ranges in a time interval with width 20 (except for problem I where |T | = 15). The complexity of the synthetic problems is governed by the amplitude and time-lag functions f and g (more in appendix, Table 2): Problem

f (vt) g(xt)
Other

I vt 5 k=10,c=0 II vt 100/vm k=1,c=10 III √ v 2 t +2ad ( √ v 2 m +2ad-v)/a k=5,a=5,d=1000,c=100 IV vt g(xt)=exp(vt)/(1+exp(vt/20)) k=10,c=40
Solar Wind Speed Prediction. The challenge of predicting solar wind speed from heliospheric data is due to the non-stationary propagation time of the solar plasma through the interplanetary medium. For the sake of a fair comparison with the best state of the art Reiss et al. ( 2019), the same experimental setting is used. The cause series x t includes the solar magnetic (flux tube expansion, FTE) and the coronal magnetic field strength estimates produced by the current sheet source surface [START_REF] Zhao | Prediction of the interplanetary magnetic field strength[END_REF] model, exploiting the hourly magnetogram data recorded by the Global Oscillation Network Group from 2008 to 2016. The effect series, the hourly solar wind data is available from the OMNI data base from the Space Physics Data Facility5 . After domain knowledge, the time-lag ranges from 2 to 5 days, segmented in six-hour segments (thus |T | = 12). For the i-th segment, the "ground truth" solar wind y i is set to its median value over the 6 hours.

DTLR is validated using a nine fold cross-validation ( On the easy Problem I, DTLR predicts the correct time lag for 97.93% of the samples. The higher value of σ 0 in problems I and II compared to the other problems is explained from the higher variance in the effect series y(t).

On Problem II, DTLR accurately learns the inverse relationship between x t , g(x t ) and y t on average. The time lag is overestimated in the regions with low time lag (with high velocity), which is blamed on the low sample density in this region, due to the data generation process. Interestingly, Problems III and IV are better handled by DTLR, despite a more complex dynamic time lag relationship. In both latter cases however, the model tends to under-estimate the time lag in the high time lag region and conversely to over-estimate it in the low time lag region.

THE SOLAR WIND PROBLEM

DTLR finds an operational solar wind model (

DISCUSSION AND PERSPECTIVES

The contribution of the paper is twofold. A new ML setting, Dynamic Time Lag Regression has been defined, aimed at the modelling of varying time-lag dependency between time series. The introduction of this new setting is motivated by an important scientific and practical problem from the domain of space weather, an open problem for over two decades.

Secondly, a Bayesian formalization has been proposed to tackle the DTLR problem, relying on a saddle point optimization process. A closed form analysis of the training procedure stability under simplifying assumptions has been conducted, yielding a practical alternate optimization formulation, implemented in the DTLR algorithm. This algorithm has been successfully validated on synthetic and real-world problems, although some bias toward the mean has been detected in some cases.

On the methodological side, this work opens a short term perspective (handling the bias) and a longer term perspective, extending the proposed nested inference procedure and integrating the model selection step within the inference architecture. The challenge is to provide the algorithm with the means of assessing online the stability and/or the degeneracy of the learning trajectory.

Regarding the motivating solar wind prediction application, a next step consists of enriching the data sources and the description of the cause series x t , typically by directly using the solar images.

Another perspective is to consider other applications of the general DTLR setting, e.g. considering fine-grained modelling of diffusion phenomena.

APPENDIX A LOG LIKELIHOOD OF THE LATENT MODEL (3)

A.1 DIRECT COMPUTATION Due to the single effect constraint (4) the mixture model ( 3) can be expressed simply as

P (y|x) = i∈T pi (x) j∈T 1 + α ji 2πσ 2 e -1 2σ 2 (1+αji) yj -ŷj (x) 2 = i∈T pi (x) j∈T 1 + α ji 2πσ 2 e -1 2σ 2 αji yj -ŷj (x) 2 exp - 1 2σ 2 j∈T y j -ŷj (x) 2
Let θ def = (ŷ, p, σ, α) denote the parameters of the model and consider the probability that predictor ŷi is the good one conditionally to a pair of observation (x, y):

q i (x, y) = P (τ i = 1|x, y) = 1 Z(x, y|θ) pi (x) exp - 1 2σ 2 j∈T α ji y j -ŷj (x) 2 + 1 2 j∈T log(1 + α ji ) with Z(x, y|θ) = i∈T pi (x) exp - 1 2σ 2 j∈T α ji y j -ŷj (x) 2 + 1 2 j∈T log(1 + α ji ) .
This gives immediately

L[{(x, y)} data |θ] = -|T | log(σ) -E data i∈T 1 2σ 2 y i -ŷi (x) 2 -log Z(x, y|θ) A.2 LARGE DEVIATION ARGUMENT
Even though the log likelihood can be obtained by direct summation, for sake of generality we show how this can result from a large deviation principle. Assume that the number of learning samples tends to infinity, and so that in a small volume dv = dxdy around a given joint configuration (x, y), the number of data N x,y becomes large. Restricting the likelihood to this subset of the data yields the following:

L x,y = Nx,y m=1 {τ (m) } p(τ (m) |x) i∈T √ 2π σ i (τ (m) ) exp - 1 2 i∈T y i -ŷi (x) 2 σ i (τ (m) ) 2 .
Upon introducing the relative frequencies:

q i (x, y) = 1 N x,y Nx,y m=1 τ (m) i satisfying i∈T q i (x, y) = 1, the sum over the τ (m) i
is replaced by a sum over these new variables, with the summand obeying a large deviation principle L x,y q exp -N x,y F x,y q where the rate function reads

F x,y q = |T | log(σ) + i∈T y i -ŷi (x) 2 1 + j∈T α ij q j 2σ 2 - 1 2 q i j∈T log(1 + α ji ) + q i log q i pi .
Taking the saddle point for q i yield as a function of (x, y) expression ( 7). Inserting this into F and taking the average over the data set yields the log likelihood (5) with opposite sign:

L[{(x, y)} data |θ] = -E data F
x,y q(x, y) .

A.3 SADDLE POINT EQUATIONS

Now we turn to the self-consistent equations relating the parameters θ of the model at a saddle point of the log likelihood function. First, the optimization of the predictors ŷ yields:

∂L ∂ ŷi (x) = 1 σ 2 E data y i -ŷi (x) 1 + j∈T α ij q j (x, y) x .
Then the optimization of p gives:

∂L ∂ pi (x) = E data q i (x, y) pi (x) -λ(x) x , = 1 pi (x) E data q i (x, y) x -λ(x)
with λ(x) a Lagrange multiplier to insure that i pi (x) = 1 This gives

pi (x) = 1 λ(x) E data q i (x, y) x Hence i∈T pi (x) = 1 λ(x)
= 1 ∀x in order to fulfill the normalization constraint, yielding finally expression (9).

Finally the optimization of α reads:

∂L ∂α ij = 1 2(1 + α ij ) E data q j (x, y) - 1 2σ 2 E data y i -ŷi (x) 2 q j (x, y) .
APPENDIX B PROOF OF PROPOSITION 3.1

Given I(x) a candidate index function we associate the point-like measure

p i (x) = δ i,I(x) .
Written in terms of p the loss function reads

L 2 (ŷ, p) = E x,y i∈T p i (x) y i -ŷ(x) 2 .
Under (3) (with α ij = αδ ij ) the loss is equal to

L 2 (ŷ, p) = E x i∈T p i (x) ŷi (x) -ŷ(x) 2 -pi (x) ασ 2 1 + α + σ 2
The minimization w.r.t. ŷ yields

ŷ(x) = i∈T p i (x)ŷ i (x). ( 16
)
In turn, as a function of p i the loss being a convex combination, its minimization yields

p i (x) = δ i,I(x) , (17)
I(x) = arg min i∈T ŷi (x) -ŷ(x) 2 -pi (x) ασ 2 1 + α . ( 18
)
Combining these equations (16,17,18) we get

I(x) = arg max i∈T pi (x) ,
which concludes the proof.

L(r, β) = - |T | 2 log(r) - |T | 2r + 1 2 log(1 + rβ) + E data log(Z) -λ(x) i∈T pi (x)
with

Z = i pi (x) exp - β 2σ 2 0 ∆y 2 i (x) ,
and where λ(x) is a Lagrange multiplier which has been added to impose the normalization of p.

The gradient reads

∂L ∂r = 1 2r 2 |T |(1 -r) + βr 2 1 + βr , ∂L ∂β = r 2(1 + rβ) - 1 2 C 1 [q], ∂L ∂ ŷi (x) = 1 σ 2 E data y i -ŷi (x) 1 + αq i (x, y) x . ∂L ∂ pi (x) = E data q i (x, y)|x pi (x) -λ(x), with C 1 [q] = 1 σ 2 0 E data i∈T q i (x, y)∆y 2 i (x) ,
This leads to the following relation at the saddle point:

r = |T | -C 1 [q] |T | -1 , α = |T | |T | -1 1 -C 1 [q] C 1 [q] , ŷi (x) = E data y i 1 + αq i (x, y) x E data 1 + αq i (x, y) x pi (x) = E data q i (x, y)|x .
Let us now compute the Hessian. It is easy to see that the block corresponding to the predictors ŷ decouples from the rest as soon as these predictors are centered.

Denoting

C 2 [q] = 1 σ 4 0 E data i∈T q i (x, y) ∆y 2 i (x) - n j=1 q j (x, y)∆y 2 j (x) 2 , we have ∂ 2 L ∂r 2 = 1 2r 2 -|T | + 2 |T | |T | -1 C 1 [q] -1 -β 2 C 2 1 [q] ∂ 2 L ∂r∂β = 1 2r 2 C 2 1 [q] ∂ 2 L ∂β 2 = 1 4 C 2 [q] -2C 2 1 [q] ∂ 2 L ∂ pi (x)∂ pj (x) = - E data q i (x, y)q j (x, y)|x pi (x)p j (x) ∂ 2 L ∂r∂ pi (x) = 0 ∂ 2 L ∂β∂ pi (x) = - u i [x, q] 2p i (x)
,

where u i [x, q] def = 1 σ 2 0 E data q i (x, y) ∆y 2 i (x) - j∈T q j (x, y)∆y 2 j (x) |x .
There are two blocks in this Hessian, the one corresponding to r and β and the one corresponding to derivatives with respect to pi . The stability of the first one depends on the sign of C 2 [q] -2C 2 1 [q] for |T | large while the second block is always stable as being an average of the exterior product of the vector (q 1 (x, y)/p 1 (x), . . . , q |T | (x, y)/p |T | (x)) by itself. At the degenerate point α = 0, r = 1, pi = 1/|T | the Hessian simplifies as follows. Denote

dη = dre 1 + dβe 2 + dx i∈T dp i (x)e i+2 (x)
a given vector of perturbations, decomposed onto a set of unit tangent vectors, {e 1 and e 2 } being respectively associated to r and β, while e i (x) associated to pi (x) for all i ∈ T and x ∈ X . Denote

u = i∈T dxu i [x]e i (x) v(x) = i∈T e i (x) with C 2 = 1 |T |σ 4 0 E data i∈T ∆y 2 i (x) - 1 |T | j∈T ∆y 2 j (x) 2 . u i [x] = 1 σ 2 0 E data ∆y 2 i (x) -σ 2 0 |x .
With these notations the Hessian reads:

H = 1 2 -|T |e 1 e t 1 + e 1 e t 2 + e 2 e t 1 + C 2 2 -1 e 2 e t 2 -ue t 2 -e 2 u t -dxv(x)v t (x) .
In fact we are interested in the eigenvalues of H in the subspace of deformations which conserve the norm of p, i.e. orthogonal to v(x), thereby given by η = η 1 e 1 + η 2 e 2 + η 3 u. In this subspace the Hessian reads

H = 1 2       -|T | 1 0 1 C 2 2 -1 -M |T |C 2 0 -M |T |C 2 0      
, where M is the number of data points, resulting from the fact that

i∈T dxu i [x] 2 = M σ 4 0 E data i∈T ∆y 2 i (x) -σ 2 0 2 , = M C 2 ,
because E data (•|x) as a function of x is actually a point-wise function on the data. If |u| 2 > 0 or if |u| = 0 and 1 + |T |(C 2 /2 -1) > 0 there is at least one positive eigenvalue. Let Λ be such an eigenvalue. After eliminating dr and dβ from the eigenvalue equations in dη, the deformation along this mode verifies

dη ∝ Λe 1 + Λ(|T | + Λ)e 2 -M |T |(|T | + Λ)C 2 u,
which corresponds to increasing r and α while decreasing for each x the pi having the highest mean relative error

u i [x].
Concerning solutions for which pi

(x) = δ i Î(x)
is concentrated on some index Î(x), the analysis is more complex. In that case C 2 [p] = 0 and C 1 [p] > 0. The (r, β) sector has 2 negative eigenvalues, while the p block is (-) a covariance matrix, so it has as well negative eigenvalues. The coupling between these two blocks could however in principle generate in some cases some instabilities.

Still, the log likelihood of such solutions reads

L = - |T | 2 log(σ 2 ) + 1 2 log(1 + α) - 1 2σ 2 E data i∈T ∆y 2 i (x) - α 2σ 2 E data ∆y 2 I(x) (x)
so we get the following optimal solution

σ 2 = 1 |T | E data i∈T ∆y 2 i (x) , 1 1 + α = E data ∆y 2 I(x) (x) σ 2 , I(x) = arg min i∈T E data ∆y 2 i (x)|x .

APPENDIX D EXPERIMENTS: ADDITIONAL DETAILS

Here we provide some additional details and context to the experimental validation of the DTLR methodology described in section 5. Table 2 provides some information about the datasets used in the synthetic and solar wind prediction problems8 . Sections D.1.1 and D.1.2 give additional plots for evaluating the experimental results.

For the solar wind prediction task, the solar wind data was mapped into standardized Gaussian space using a quantile-quantile and inverse probit mapping. Nine fold cross-validation was performed using splits as specified in table 3. To compare the DTLR results with the state of the art solar wind forecasting, we used results from Reiss et al. (2019, Table 1). Since [START_REF] Reiss | Forecasting the ambient solar wind with numerical models. i. on the implementation of an operational framework[END_REF] compared the various forecasting methods on only one solar rotation (first row of table 3), comparing these results with DTLR can be considered as a preliminary examination. Nevertheless, the results presented in table 1a show encouraging signs for the competitiveness and usefulness of the DTLR method. D.1.2 SOLAR WIND PREDICTION q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 300 400 500 600 0 200 400 600 t (hours) km/s q q

Actual Prediction (a) Hourly forecasts for period 2008-11-20 07:00 to 2008-12-17 14:00 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 300 400 500 600 700 0 200 400 600 t (hours) km/s q q

Actual Prediction (b) Hourly forecasts for period 2016-11-16 17:00 to 2016-12-14 01:00

Figure 4 :

 4 Figure 2: Problem II, Results

Figure 5 :

 5 Figure 5: Solar Wind Prediction: reconstructed time series predictions

Table 1 :

 1 DTLR performance: accuracy (MAE and RMSE, the lower the better; Pearson, the higher the better) and stability σ 0 and C 1 (the lower the better). For each indicator, is reported the DTLR value (9-fold CV), the baseline value and the time-lag error.

	Problem	M.A.E	R.M.S.E	Pearson Corr.	σ 0	C 1
	I	8.82 / 21.79 / 0.021 12.35 / 28.79 / 0.26 0.98 / 0.87 / -	29.8	0.14
	II	10.15 / 27.40 / 0.4	13.70 / 35.11 / 0.67 0.95 / 0.73 / 0.70 26.83 0.16
	III	3.17 / 11.01 / 0.17	4.63 / 14.99 / 0.42	0.98 / 0.79 / 0.84 11.84 0.09
	IV	3.88 / 12.28 / 0.34	5.33 / 15.89 / 0.64	0.98 /0.79/ 0.81	12.18 0.13
	Solar Wind 56.35 / 66.45 / -	74.20 / 84.53 / -	0.6 / 0.41 / -	76.46 0.89
	6.1 SYNTHETIC PROBLEMS

Table 3 in appendix), where each fold is a continuous period corresponding to a solar rotation. 6 6 EMPIRICAL VALIDATION Table 1 summarizes the DTLR performance on the synthetic and solar wind problems (detailed results are provided in the appendix).

Table 1 )

 1 , though the significantly higher difficulty of the solar wind problem is witnessed by the C 1 value close to the degenerate value 1. The detailed comparison with the state of the artReiss et al. (2019) (Fig. 1, Left) shows that DTLR improves on the current best state of the art (on all variants including ensemble approaches, and noting that median models are notoriously hard to beat). (Fig.1, Right) shows the good correlation between the predicted solar wind 7 and the measured solar wind.

	Model	M.A.E R.M.S.E
	WS	74.09	85.27		600
	DCHB	83.83	103.43
	WSA	68.54	82.62
	Ensemble Median (WS) Ensemble Median (DCHB) 78.27 71.52	83.36 100.04	v ^ (km/s)	500
	Ensemble Median (WSA)	62.24	74.86
	Persistence (4 days)	130.48 161.99		400
	Persistence (27 days)	66.54	78.86
	Fixed Lag Baseline	67.33	80.39
	DTLR	60.19	72.64		300
	(a) Comparative assessment on the Solar			300	400	500 v (km/s)	600	700
	Wind problem compared to the state of the
	art Reiss et al. (2019, Table 1)				(b) Scatter Chart (9 fold CV)

Figure 1: DTLR on the solar wind problem. Left: comparative quantitative assessment w.r.t. the state of the art. Right: qualitative assessment of the prediction.

Table 2 :

 2 Synthetic and Real-World Problems

	Problem # train	# test	d	|T |
	I 10, 000 2, 000 10	15
	II 10, 000 2, 000 10	20
	III 10, 000 2, 000 10	20
	IV 10, 000 2, 000 10	20
	Solar Wind 77, 367 2, 205 374 12

Table 3 :

 3 Cross validation splits used to evaluate DTLR on the solar wind forecasting task

	Split Id Carrington Rotation Start	End
	1	2077	2008/11/20 07:00:04 2008/12/17 14:38:34
	2	2090	2009/11/09 20:33:43 2009/12/07 04:03:59
	3	2104	2010/11/26 17:32:44 2010/12/24 01:15:56
	4	2117	2011/11/16 07:04:41 2011/12/13 14:39:28
	5	2130	2012/11/04 20:39:43 2012/12/02 04:06:23
	6	2143	2013/10/25 10:17:52 2013/11/21 17:36:35
	7	2157	2014/11/11 07:09:56 2014/12/08 14:41:02
	8	2171	2015/11/28 04:09:27 2015/12/25 11:53:33
	9	2184	2016/11/16 17:41:04 2016/12/14 01:16:43

The adverse impact of space weather is estimated to cost

to 400 million USD per year, but can sporadically lead to much larger losses.

For some authors[START_REF] Zhou | Non-parametric determination of real-time lag structure between two time series: The optimal thermal causal path method with applications to economic data[END_REF],the monotonicity of φ(.) is additionally required and enforced using constraints:φ(t1) ≤ φ(t2), ∀t1 ≤ t2. This assumption is not retained as one may achieve a similar effect by using regularization based smoothness penalties.

Note that pre-processing can be used if needed to map non-Gaussian data onto Gaussian data.

Note however that the cause-effect correspondence might be many-to-one, with an effect depending on several causes.

https://omniweb.gsfc.nasa.gov

The Sun completes a rotation (or Carrington rotation) in approximately 27 days.

The predicted values, every 6 hours, are interpolated for comparison with the hourly measured solar wind.

In the solar wind problem, the training and test data sizes correspond to one cross-validation split

APPENDIX C STABILITY ANALYSIS

The analysis is restricted for simplicity to the case α ij = αδ ij . The log likelihood as a function of r = σ 2 /σ 2 0 and β = α/r after inserting the optimal q = q(x, y) reads in that case APPENDIX E NEURAL NETWORK ARCHITECTURE DETAILS


Georges Griso
email: griso@ljll.math.upmc.fr

Julia Orlik
email: orlik@itwm.fhg.de

Stephan Wackerle
email: wackerle@itwm.fhg.de

Asymptotic Behavior for Textiles in von-Kármán regime

Keywords: Homogenization, periodic unfolding method, dimension reduction, von-Kármán orthotropic plate, Energy minimization under pre-strain, buckling under homogenized pre-strain Mathematics Subject Classification (2010): 35B27, 35J86, 47H05, 74Q05, 74B05, 74K10, 74K20

The paper is dedicated to the investigation of simultaneous homogenization and dimension reduction of textile structures as elasticity problem with an energy in the von-Kármán-regime. An extension for deformations is presented allowing to use the decomposition of plate-displacements. The limit problem in terms of displacements is derived with the help of the unfolding operator and yields in the limit the von-Kármán plate with linear elastic cell-problems. It is shown, that for homogeneous isotropic beams in the structure, the resulting plate is orthotropic. As application of the obtained limit plate we study the buckling behavior of orthotropic textiles.

Introduction

In contrast to our first paper about homogenization of textiles [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF], where a geometrical linear elasticity is considered, we investigate here textiles with von-Kármán energy. The von-Kármán model is a nonlinear plate model, which is stated with respect to displacements and widely used by mathematicians and engineers, see [START_REF] Blanchard | Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant's strain tensor[END_REF]2,[START_REF] Ciarlet | A justification of the von Kármán equations[END_REF][START_REF] Ciarlet | Mathematical Elasticity: Theory of Plates[END_REF]14,[START_REF] Friesecke | A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence[END_REF][START_REF] Puntel | Wrinkling of a stretched thin sheet[END_REF]. To achieve the von-Kármán model in the limit we consider an elastic energy of order ||e(u ε )|| L 2 (Ω * ε ) ≤ Cε 5/2 , which is in consensus with [START_REF] Bourne | Energy Bounds for a Compressed Elastic Film on a Substrate[END_REF][START_REF] Ciarlet | A justification of the von Kármán equations[END_REF]14,[START_REF] Friesecke | A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence[END_REF]. A simultaneous homogenization and dimension reduction of a von-Kármán plate was already studied in [START_REF] Neukamm | Derivation of the homogenized von Kármán plate theory from 3d elasticity[END_REF], however in our paper we give the corrector results related on the periodic topology of the textile.

Since the von-Kármán plate arises as Γ-Limit from geometrical nonlinear problems, it is necessary to consider initially deformations. Hence, the homogenization of the textile for von-Kármán energy begins with the extension of a deformation into the holes of the structure. This extension is applied onto the deformations of the textile beam structure for glued beams. Due to the fact that the limit-plate is stated with respect to displacements we directly introduce the decomposition of the displacement associated to the extended deformation, see [START_REF] Blanchard | Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant's strain tensor[END_REF][START_REF] Griso | Asymptotic behavior of structures made of curved rods[END_REF][START_REF] Griso | Decomposition of displacements of thin structures[END_REF]. For the elementary displacements we establish the Korn-type estimates giving rise to the asymptotic behavior of the fields. To derive the homogenized model the unfolding and rescaling operator (see for instance [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF]11,[START_REF] Griso | Asymptotic Analysis for Periodic Perforated Shells[END_REF][START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF]) accounting for both homogenization and dimension reduction is used. For the von-Kármán-plate studied here it is necessary to investigate also the nonlinear term in the Green-St.Venant strain tensor. The additional term yield the von-Kármán nonlinearities in the limit. The derived asymptotic limits allow to prove with arguments of Γ-convergence to show that the limit energy in in fact of von-Kármán-type. It is proven, that the homogenized limit energy admits minima. The uniqueness, though, is not provable.

Although the initial and the homogenized problem are nonlinear, the cell-problems for the von-Kármán plate are linear and in fact the same as achieved for a linear elastic plate. Furthermore, the cell-problems yield for isotropic homogeneous beams an orthotropic plate, which is valid for the von-Kármán energy and linear elasticity.

The final part of the paper is devoted to homogenization of the pre-stress in yarns and modeling of the buckling of the von-Kármán plate under pre-strain, like in [START_REF] Lecumberry | Stability of Slender Bodies under Compression and Validity of the von-Kármán Theory[END_REF][START_REF] Cerda | Geometry and physics of wrinkling[END_REF][START_REF] Puntel | Wrinkling of a stretched thin sheet[END_REF][START_REF] Berdichevsky | Variational Principles of Continuum Mechanics I. Fundamentals[END_REF], but for an orthotropic plate.

Preliminary extension results for deformations and displacements

In this section, Ω and Ω are two bounded domains in R n containing the origin with Lipschitz boundaries and such that Ω ⊂ Ω . For every ε > 0, we denote Ω ε = ε Ω and Ω ε = ε Ω . In Lemma 2.1 we prove an extension result for deformations in H 1 (Ω ε ) n . The lemma below is based on the rigidity theorem obtained by G. Friesecke, R. James and S. Müller in [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity[END_REF]. Here, for the starshaped open sets with respect to a ball, we use a variant of this theorem which explicitly gives the dependence of the constants in the estimates in terms of only two parameters which depend on the geometry of the domain: its diameter and the radius of the ball (see [START_REF] Blanchard | Decomposition of deformations of thin rods. Application to nonlinear elasticity[END_REF]19]).

Lemma 2.1. For every deformation v in H 1 (Ω ε ) n there exists a deformation v in H 1 (Ω ε ) n satisfying

v |Ωε = v, dist v, SO(n) L 2 (Ω ε ) ≤ C dist(v, SO(n)) L 2 (Ωε) .
(2.1)

The constant does not depend on ε.

Proof. First, some classical recalls and then the proof.

• (i) Since Ω is a bounded domain with Lipschitz boundary, there exist N ∈ N * , R 1 and R 2 two strictly positive constants and a finite set {O 1 , . . . , O N } of open subsets of Ω, each of diameter less than R 1 and starshaped with respect to a ball of radius R 2 (B(A i , R 2 ),

A i ∈ O i ) such that Ω = N k=1 O k .
As a consequence, there exists r such that for every O i , i ∈ {1, . . . , N } there exists a chain from

O 1 to O i O l1 = O 1 , O l2 , . . . , O lp = O i , p ∈ {1, . . . , N }
such that, if p > 1 one has O lj ∩ O lj+1 , j ∈ {1, . . . , p -1}, contains a ball of radius r.

• (ii) Let O be an open set in R n included in the ball B(A; R 1 ) and starshaped with respect to the ball

B(A, R 2 ), R 1 > 0, R 2 > 0. Theorem II.1.1 in [3] claims that for every deformation v ∈ H 1 (O) n , there exist a matrix R ∈ SO(n) and a ∈ R n such that v -a-R x L 2 (O) ≤ CR 1 dist(∇v, SO(3)) L 2 (O) , ∇v -R L 2 (O) ≤ C dist(∇v, SO(n)) L 2 (O) .
The constant C depends only on R 1 R 2 and n.

Transform O by a dilation of ratio ε > 0 and center A, the above result gives: for every deformation

v ∈ H 1 (O ε ) n where O ε . = εO, there exist a matrix R ∈ SO(n) and a ∈ R n such that v-a-R x L 2 (Oε) ≤ Cε dist(∇v, SO(n)) L 2 (Oε) , ∇v-R L 2 (Oε) ≤ C dist(∇v, SO(n)) L 2 (Oε) .
The constant C does not depend on ε.

• (iii) Ω and Ω being two bounded domains in R n with Lipschitz boundaries and such that Ω ⊂ Ω . There exists a continuous linear extension operator

P from H 1 (Ω) n into H 1 (Ω ) n satisfying ∀v ∈ H 1 (Ω) n , P (v) |Ω = v, P (v) L 2 (Ω ) ≤ C v L 2 (Ω) , P(v) H 1 (Ω ) ≤ C v H 1 (Ω) .
If we transform Ω and Ω by the same dilation of ratio ε (and center O ∈ Ω), this extension operator induces an extension operator

P ε from H 1 (Ω ε ) 3 into H 1 (Ω ε ) 3 satisfying ∀v ∈ H 1 (Ω ε ) 3 , P ε (v) |Ωε = v, P ε (v) L 2 (Ω ε ) ≤ C v L 2 (Ωε) , P ε (v) L 2 (Ω ε ) + ε ∇P ε (v) L 2 (Ω ε ) ≤ C v L 2 (Ωε) + ε ∇v L 2 (Ωε) .
The constants do not depend on ε.

Now, consider a deformation v ∈ H 1 (Ω ε ) n . We apply (ii) with the open sets O i,ε = A i + ε(O i -A i ), there exist matrices R i ∈ SO(n) and vectors a i ∈ R n such that v -a i -R i x L 2 (εOi,ε) ≤ Cε dist(∇v, SO(n)) L 2 (εOi,ε) , ∇v -R i L 2 (εOi,ε) ≤ C dist(∇v, SO(n)) L 2 (εOi,ε) .
The constant C does not depend on ε.

Then, using the second part of (i), we compare R i to R 1 as well as a i to a 1 , i ∈ {1, . . . , N }. As a consequence, one obtains that

v -a 1 -R 1 x L 2 (Ωε) ≤ Cε dist(∇v, SO(3)) L 2 (Ωε) , ∇v -R 1 L 2 (Ωe) ≤ C dist(∇v, SO(n)) L 2 (Ωε) .
The constants do not depend on ε. Now, we define the extension of v. We set

v = P ε (v -a 1 -R 1 x) + a 1 + R 1 x a.e. in Ω ε .
We easily check (2.1).

3 The structure

Parameterization of the yarns

To see the parametrization of yarns and the structure we refer to [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF]. Nevertheless, we shortly repeat the most important definitions and results. The middle line of a beam is paramtrized by rescaled

function Φ ε = εΦ( z ε ) of Φ(z) =            -κ, if z ∈ [0, κ], κ 6 (z-κ) 2 (1-2κ) 2 -4 (z-κ) 3 (1-2κ) 3 -1 if z ∈ [κ, 1 -κ], κ if z ∈ [1 -κ, 1], Φ(2 -z) if z ∈ [1, 2]. (3.1)
Then the beams in the structure are defined by

P (1) r . = z ∈ R 3 | z 1 ∈ (0, L), (z 2 , z 3 ) ∈ (-κε, κε) 2 , P (2)
r . = z ∈ R 3 | z 2 ∈ (0, L), (z 1 , z 3 ) ∈ (-κε, κε) 2 .
for the reference beams in the two directions. Then the curved beams are defined by

P (1,q) ε . = x ∈ R 3 | x = ψ (1,q) ε (z), z ∈ P (1) r , P (2,p) ε . = x ∈ R 3 | x = ψ (2,p) ε (z), z ∈ P (2) r , 1 2κ 2κ 1 Figure 1: The domain Y * ⊂ Y = (0, 1) 2 ×(-2κ, 2κ
), a quarter of the periodicity cell of the full structure.

with the diffeomorphisms

ψ (1,q) ε (z) . = M (1,q) ε (z 1 ) + z 2 e 2 + z 3 n (1,q) ε (z 1 ), ψ (2,p) ε (z)
. = M (2,p) ε (z 2 ) + z 1 e 1 + z 3 n (2,p) ε (z 2 ),
and the corresponding middle lines

M (1,q) ε (z 1 ) . = z 1 e 1 + qεe 2 + (-1) q+1 Φ ε (z 1 )e 3 , M (2,p) ε (z 2 ) . = pεe 1 + z 2 e 2 + (-1) p Φ ε (z 2 )e 3 .

Parameterization of the whole structure

Denote Ω * ε the whole structure (see [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF] for details)

Ω * ε . = Ω ε ∩ 2Nε p=0 P (1,q) ε ∪ 2Nε q=0 P (2,p) ε , Ω ε . = ω × (-2κε, 2κε), ω = (0, L) 2 . (3.2)

An extension result

The presented extension heavily depends on the fact that the beams are glued. For a more general contact condition as in [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF] it is necessary to treat the two directions separately and obtain two deformations, which give the same limit for g ε ∼ ε 4 . Nevertheless, the more general case would exceed the bounds of this paper.

Proposition 3.1. For every deformation v in H 1 (Ω * ε ) 3 there exists a deformation v in H 1 (Ω ε ) 3 sat- isfying v |Ω * ε = v, dist v, SO(3) L 2 (Ωε) ≤ C dist(v, SO(3)) L 2 (Ω * ε ) . (3.
3)

The constant does not depend on ε.

Proof. Now that the general extension for Lipschitz domains in nonlinear elasticity is recalled in the above lemma, we specify the extension procedure for the the domain Ω * ε . First, we divide the domain Ω * ε into portions included in domains isometric to the parallelotope (0, ε + 2κε) × (0, 2κε) × (0, 4κε)1 as depicted in Figure 2. These portions include a curved beam and parts of the beams in the perpendicular direction with which the beam is in contact. Besides, after a rotation and/or a reflection, all the portions are of the same form and itself Lipschitz-domains. Furthermore, note that these portions intersect each other and every contact cylinder C pq ×(-2κε, 2κε) with C pq = (pε -κε, pε + κε) × (qε -κε, qε + κε) is used in four of such domains.

Since every portion is a Lipschitz domain the extension procedure given in Lemma 2.1 is applicable for every v ∈ H 1 (Ω * ε ) and yields an extension to the parallelotope (e.g. (pε -κε, (p + 1)ε + κε) × (qεκε, qε + κε) × (-2κε, 2κε)). As second step, we define new domains included in (pε -κε, (p + 1)ε + κε) × (qε -κε, (q + 1)ε + κε) × (-2κε, 2κε) by collecting four of the above portions as depicted in Figure 3. Note that the contact cylinders in every corner of the new domain is used by two portions. Obviously this domain is again a Lipschitz domain and hence we extend all the fields into the holes using again Lemma 2.1.

To obtain the full extension we reassemble the structure. To do this, note that the domains (pεκε, (p + 1)ε + κε) × (qε -κε, (q + 1)ε + κε) × (-2κε, 2κε) have an overlap. This overlap includes every beam twice and the contact cylinders again fourfold, i.e. the overlap consists exactly of the domains before the last extension. Together with the interportions from the step before we obtain that the contact cylinders are the most used domains for the extension, namely eight times. This influences the estimate and finally give the final extension v which satisfies

dist v, SO(3) L 2 (Ωε) ≤ C dist(v, SO(3)) L 2 (Ω * ε )
where the constant does not depend on ε. By construction, we have v |Ω * ε = v. Henceforth, we use the extended deformation v ∈ H 1 (Ω ε ), which is a deformation of a periodic plate without holes. This allows to use the results in the papers [START_REF] Blanchard | Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant's strain tensor[END_REF] and general results of [12], [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF], [2]. The structure is clamped on its lateral boundary. Moreover, in contrast to [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF] here we assume a glued contact, which corresponds to the case g ε ≡ 0 in [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF]. This allows to obtain one deformation field for the whole structure Ω * ε instead of one for each beam as in [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF].

Denote γ = ∂ω ∩ {x 2 = 0} = (0, L) × {0}, Γ ε = γ × (-2κε, 2κ ε ).
The set of the admissible deformations are 4

V ε . = v ∈ H 1 (Ω * ε ) 3 | such that v = I d a.e. on ∂Ω * ε ∩ Γ ε , D ε . = v ∈ H 1 (Ω ε ) 3 | such that v = I d a.e.
The non-linear elasticity problem

Set Y := (0, 2) 2 , Y := (0, 2) 2 × (-2κ, 2κ).
Let Y * ⊂ Y be the reference cell of the beam structure. The cell Y * is deduced from Y * (see Figure 1) after two symmetries with respect to the planes y 1 = 1 and y 2 = 1. Denote W the local elastic energy density, then the total elastic energy is

J ε (v) = Ω * ε W ε •, ∇v dx - Ω * ε f ε • (v -I d ) dx, ∀v ∈ V ε , (4.1)
where I d is the identity map. The local density energy W : Y * × S 3 -→ R + ∪ {+∞} is given by

W ε (•, F ) =    Q • ε , 1 2 (F T F -I 3 ) if det(F ) > 0, + ∞ if det(F ) ≤ 0,
where S 3 is the space of 3 × 3 symmetric matrices. The quadratic form Q is defined by Q(y, S) = a ijkl (y)S ij S kl for a.e. y ∈ Y * and for all S ∈ S 3 ,

where the a ijkl 's belong to L ∞ (Y * ) and are periodic with respect to e 1 and e 2 . Moreover, the tensor a is symmetric, i.e., a ijkl = a jikl = a klji . Also it is positive definite and satisfies

∃c 0 > 0, such that c 0 S ij S ij ≤ a ijkl (y)S ij S kl for a.e. y ∈ Y * and for all S ∈ S 3 . (4.2)
Note, that the energy density

W ε (x, ∇v(x)) =    Q x ε , E(v)(x) if det(∇v(x)) > 0, + ∞ if det(∇v(x)) ≤ 0, for a.e. x ∈ Ω * ε depends on the strain tensor E(v) = 1 2
(∇v) T ∇v -I 3 with I 3 the unit 3 × 3 matrix.

Remark 4.1. As a classical example of a local elastic energy satisfying the above assumptions, we mention the following St Venant-Kirchhoff 's law for which

W (F ) =    λ 8 tr(F T F -I 3 ) 2 + µ 4 tr (F T F -I 3 ) 2 if det(F ) > 0 + ∞ if det(F ) ≤ 0.
Now we are in the position to state the problem. Therefore, set

m ε = inf v∈Vε J ε (v) 2 .
5 Preliminary estimates 5.1 Recalls about the plate deformations

Denote x = (x 1 , x 2 ) ∈ R 2 and U ε . = u ∈ H 1 (Ω ε ) 3 | u = 0 a.e. on Γ ε .
The deformations and the terms of their decompositions are estimated in terms of dist(∇v, SO(3)) L2 (Ω * ε ) , this is why the lemma below plays a crucial role Lemma 5.1. Let v ∈ V ε be a deformation and v ∈ D ε the extended deformation given by Proposition 3.1 and Remark 3.2. The associated displacement u = v -I d belongs to U ε and satisfies

e(u) L 2 (Ωε) ≤ C 0 dist(∇v, SO(3)) L 2 (Ω * ε ) + C 1 ε 5/2 dist(∇v, SO(3)) 2 L 2 (Ω * ε ) (5.1)
The constants do not depend on ε and v (they depend on ω, Y * and κ).

Proof. In [4, Lemma 4.3] it is proved that there exists a constant which does not depend on ε and v such that

e(u) L 2 (Ωε) ≤ C dist(∇ v, SO(3)) L 2 (Ωε) 1 + 1 ε 5/2 dist(∇ v, SO(3)) L 2 (Ωε) .
Then, Proposition 3.1 gives a constant which does not depend on ε and v such that

dist v, SO(3) L 2 (Ωε) ≤ C dist(v, SO(3)) L 2 (Ω * ε )
. This ends the proof of the lemma.

Recalls about the plate displacements

Set H 1 γ (ω) . = φ ∈ H 1 (ω) | φ = 0 a.e. on γ , H 2 γ (ω) . = φ ∈ H 1 (ω) | φ = 0, ∇φ = 0 a.e. on γ .
Below we recall a definition from [11, Chapter 11] (see also [START_REF] Griso | Asymptotic Analysis for Periodic Perforated Shells[END_REF][START_REF] Griso | Asymptotic behavior of structures made of plates[END_REF])

Definition 5.2. Elementary displacement are elements u e of H 1 (Ω ε ) 3 satisfying for a.e. x = (x , x 3 ) ∈ Ω ε (where x ∈ ω) u e,1 (x) = U 1 (x ) + x 3 R 1 (x ), u e,2 (x) = U 2 (x ) + x 3 R 2 (x ), u e,3 (x) = U 3 (x )
.

Here U = (U 1 , U 2 , U 3 ) ∈ H 1 (ω) 3 and R = R 1 e 1 + R 2 e 2 ∈ H 1 (ω) 2 .
The following lemma is proved in [11, Theorem 11.4 and Proposition 11.6] Lemma 5.3. Let u be in U ε . The displacement u can be decomposed as the sum

u = u e + u (5.2)
of an elementary displacement u e and a residual displacement u, both belonging to U ε and satisfying

U ∈ H 1 γ (ω) 3 , R ∈ H 1 γ (ω) 2 , u L 2 (Ωε) + ε ∇u L 2 (Ωε) ≤ Cε e(u) L 2 (Ωε) . (5.3)
Moreover, one has

U α H 1 (ω) + ε U 3 H 1 (ω) + R H 1 (ω) ≤ C ε 1/2 e(u) L 2 (Ωε) , ∂ α U 3 + R α L 2 (ω) ≤ C ε 1/2 e(u) L 2 (Ωε) , u α L 2 (Ωε) + ε u 3 L 2 (Ωε) ≤ C e(u) L 2 (Ωε) , 2 α,β=1 ∂u α ∂x β L 2 (Ωε) + ∂u 3 ∂x 3 L 2 (Ωε) ≤ C e(u) L 2 (Ωε) , 2 α=1 ∂u α ∂x 3 L 2 (Ωε) + ∂u 3 ∂x α L 2 (Ωε) ≤ C ε e(u) L 2 (Ωε) .
(5.4)

The constants do not depend on ε.

Assumptions on the forces

The forces have to admit a certain scaling with respect to the ε-scaling of the domain. For the textile we require forces of the type

f ε,1 = ε 2 f 1 , f ε,2 = ε 2 f 2 , f ε,3 = ε 3 f 3 , a.e. in Ω * ε , (5.5)
with f ∈ L 2 (ω) 3 . In order to obtain at the limit a von-Kármán model, the applied forces must satisfy a condition

f L 2 (ω) ≤ C * . (5.6)
This constant depends on the reference cell Y * , the mid-surface ω of the structure and the local elastic energy W (see Lemma 5.4).

The scaling of the force gives rise to the order of the energy in the elasticity problem. We prove this, in lemma below. Lemma 5.4. Let v ∈ V ε be a deformation such that J ε (v) ≤ 0. Assume (5.5) on the forces. There exists a constant C * independent of ε and the applied forces such that, if

f L 2 (ω) < C * one has dist(∇v, SO(3)) L 2 (Ω * ε ) ≤ Cε 5/2 f L 2 (Ω) . The constant C does not depend on ε.
Proof. Using (4.2) gives rise to the estimation

c 0 dist(∇v, SO(3)) 2 L 2 (Ω * ε ) ≤ Ω * ε f ε • (v -I d ) dx .
(5.7)

Introduce u = v -I d ∈ U ε the associated displacement to the extended deformation (see Lemma 5.1). Then, with (5.5) and the estimates (5.4) 3 we obtain

Ω * ε f ε • (v -I d ) dx ≤ ε 5/2 f α L 2 (ω) u α L 2 (Ω * ε ) + ε 7/2 f 3 L 2 (ω) u 3 L 2 (Ω * ε ) ≤ ε 5/2 f α L 2 (ω) u α L 2 (Ωε) + ε 7/2 f 3 L 2 (ω) u 3 L 2 (Ωε) ≤ C 2 ε 5/2 f L 2 (ω) e(u) L 2 (Ωε) .
(5.8)

Eventually, the above inequality with (5.7) and Lemma 5.1 give

c 0 dist(∇v, SO(3)) 2 L 2 (Ω * ε ) ≤ C 2 C 0 ε 5/2 f L 2 (Ω) dist(∇v, SO(3)) L 2 (Ω * ε ) + C 2 C 1 f L 2 (Ω) dist(∇v, SO(3)) 2 L 2 (Ω * ε ) . If C 2 C 1 f L 2 (ω) < c 0 , then dist(∇v, SO(3)) L 2 (Ω * ε ) ≤ C 2 ε 5/2 c 0 -C 2 C 1 f L 2 (Ω) f L 2 (Ω) .
Now, if the deformation v ∈ V ε satisfies J ε (v) ≤ 0, one can give a lower bound of the infimum of the functional J ε . To do this, use the assumptions (4.2) on the problem and (5.5)-(5.6) on the forces together with the Lemmas 5.1-5.4 and inequality (5.8) lead to

c 0 (∇v) T ∇v -I 3 2 L 2 (Ω * ε ) ≤ Ω * ε W (∇v)dx ≤ Ω * ε f ε • (v -I d ) dx ≤ Cε 5 f 2 L 2 (ω) . (5.9)
As a consequence, there exists a constant c independent of ε such that

-cε 5 ≤ J ε (v) ≤ 0 (5.10) Recalling that m ε = inf v∈Vε J ε (v) yields -c ≤ m ε ε 5 ≤ 0. (5.11)
Our aim is to give the asymptotic behavior of the rescaled sequence m ε ε 5 ε and to characterize its limit as the minimum of a functional.

Asymptotic behavior

In this section, we consider a sequence {v ε } ε of deformations satisfying

dist(∇v ε , SO(3)) L 2 (Ω * ε ) ≤ Cε 5/2 . (6.1)
Below we are interested by the asymptotic behavior of the sequence of displacements

{u ε } ε = {v ε -I d } ε .
Here, for every ε, u ε is the associated displacement to the extended deformation v ε . From Lemma 5.1, one has e(u ε ) L 2 (Ωε) ≤ Cε 5/2 . (6.2)

Below, we recall some estimates of Lemma 5.3 under this assumption. One has

u ε L 2 (Ωε) + ε ∇u ε L 2 (Ωε) ≤ Cε 7/2 , U ε,3 H 1 (ω) + R ε H 1 (ω) ≤ Cε, U ε,α H 1 (ω) ≤ Cε 2 , ∂ α U ε,3 + R ε,α L 2 (ω) ≤ Cε 2 . (6.
3)

The constants do not depend on ε.

Lemma 6.1 (See [4, Section 7]). Under the assumptions of Lemma 5.3, there exist a subsequence of {ε}, still denoted {ε},

U 3 ∈ H 2 γ (ω) and U α , R α ∈ H 1 γ (ω) (α ∈ {1, 2}) such that 1 ε U ε,3 → U 3 strongly in H 1 γ (ω), 1 ε R ε,α R α weakly in H 1 γ (ω), and strongly in L 4 (ω) 1 ε 2 U ε,α U α weakly in H 1 γ (ω), 1 ε 2 ∂ α U ε,3 + R ε,α Z α weakly in L 2 (ω). (6.4)
Moreover, one has

∂ α U 3 + R α = 0. (6.5)
Proof. The convergences and equalities are easy consequences of the above estimates (6.3). To see the strong convergence of (6.4) 1 note that (5.4) 1,2 and the strong convergence of

1 ε R ε,α in L 2 (Ω) imply 1 ε ∂ α U ε,3 = 1 ε (∂ α U ε,3 + R ε,α ) - 1 ε R ε,α → 0 -R α = ∂ α U 3 , strongly in L 2 (ω).
the last equality comes from (6.5).

The unfolding and the unfolding and rescaling operators

For the asymptotic behavior we introduce two operators: T ε for the homogenization in ω and T ε for the homogenization and dimension reduction in Ω * ε . Both operators can be found in [11] thus we recall here only the important properties. Definition 6.2. For every measurable function φ ∈ L 1 (ω) we recall the definition of the unfolded function

T ε (φ) ∈ L 1 (ω × Y ) T ε (φ)(x , y ) = φ ε x ε + εy for a.e. (x , y ) ∈ ω × Y .
For every measurable function ψ ∈ L 1 (Ω * ε ) the unfolding and the rescaling operator T ε is defined by

T ε (ψ)(x , y) = ψ ε x ε + εy for a.e. (x , y) ∈ ω × Y * . Lemma 6.3. There exist a subsequence of {ε}, still denoted {ε}, U α , R α ∈ L 2 (ω; H 1 per (Y )) and u ∈ L 2 (ω; H 1 per (Y )) such that 1 ε T ε (∇U ε,3 ) -→ ∇U 3 strongly in L 2 (ω × Y ) 2 1 ε T ε (R ε ) -→ R strongly in L 2 (ω × Y ) 2 1 ε 2 T ε (∇R ε,α ) ∇R α + ∇ y R α weakly in L 2 (ω × Y ) 2 1 ε 2 T ε (∇U ε,α ) ∇U α + ∇ y U α weakly in L 2 (ω × Y ) 2 1 ε 2 T ε (∂ α U ε,3 + R ε,α Z α + ∇ yα u + R weakly in L 2 (ω × Y ). (6.6)
Moreover, there exists u ∈ L 2 (ω;

H 1 per (Y * )) 3 such that 1 ε 2 T ε (u ε ) u weakly in L 2 (ω; H 1 (Y * )), 1 ε 2 T ε (∇u ε ) ∇ y u weakly in L 2 (ω × Y * ) 9 .
(6.7)

Furthermore, one has

1 ε 2 T ε ∇u ε (∇u ε ) T -→   ∂ 1 U 3 ∂ 1 U 3 ∂ 1 U 3 ∂ 2 U 3 0 ∂ 1 U 3 ∂ 2 U 3 ∂ 2 U 3 ∂ 2 U 3 0 0 0 ∇U 3 • ∇U 3   strongly in L 1 (ω × Y * ) 9 . (6.8)
The above convergence is weak in

L 2 (ω × Y * ) 9 .
Proof. The first convergence (6.6) 1 is a consequence of (6.4) and the classical results of the PUM. Convergences (6.6) 2,3,4 come from the convergences in Lemma 6.1 and again of the classical results of the PUM (see [11]). The last convergence (6.6) 5 is a consequence of [11, Lemma 11.11], together with the convergences (6.4) 4 and (6.6) 3 .

∇u ε =   ∂ 1 U ε,1 + x 3 ∂ 1 R ε,1 ∂ 2 U ε,1 + x 3 ∂ 2 R ε,1 R ε,1 ∂ 1 U ε,2 + x 3 ∂ 1 R ε,2 ∂ 2 U ε,2 + x 3 ∂ 2 R ε,2 R ε,2 ∂ 1 U ε,3 ∂ 2 U ε,3 0   + ∇u ε
Then since (6.6) 1,2 are strong convergences and the other fields converge to zero due to (6.6) 3,2 and (6.8) we obtain

1 ε T ε ∇u ε -→   0 0 R 1 0 0 R 2 ∂ 1 U 3 ∂ 2 U 3 0   strongly in L 2 (ω × Y * ) 9 .
(6.9)

Hence, using (6.5) this yields

1 ε 2 T ε ∇u ε (∇u ε ) T -→   0 0 R 1 0 0 R 2 -R 1 -R 2 0     0 0 -R 1 0 0 -R 2 R 1 R 2 0   =   R 1 R 1 R 1 R 2 0 R 1 R 2 R 2 R 2 0 0 0 R 2 1 + R 2 2   strongly in L 1 (ω × Y * ) 9 .
Now note that the sequence is actually bounded in L 2 (Ω × Y * ) by

∇u ε (∇u ε ) T L 2 (Ωε) ≤ ∇u ε (∇u ε ) T + 2e(u ε ) L 2 (Ωε) + 2 e(u ε ) L 2 (Ωε) = ∇v ε (∇v ε ) T -I 3 L 2 (Ωε) + 2 e(u ε ) L 2 (Ωε) ≤ Cε 5/2
Hence, the sequence 1 9 . This ensures that (6.8) is also

ε 2 T ε ∇u ε (∇u ε ) T ε is bounded in L 2 (Ω × Y * )
weakly converging in L 2 (Ω × Y * ).
Eventually, we find with (6.9)

T ε (∇v ε ) → I 3 strongly in L 2 (ω × Y * ) 9 . (6.10)
Additionally, the displacements converge as follows

1 ε 2 T ε (u ε,α ) → U α -y 3 ∂ α U 3 strongly in L 2 (Ω × Y), 1 ε 1 T ε (u ε,3 ) → U 3 strongly in L 2 (Ω × Y).
(6.11)

The above convergences show that the limit displacement is of Kirchhoff-Love type. Then, we have Lemma 6.4. For a subsequence we have

1 2ε 2 T ε (∇v ε ) T ∇v ε -I 3 E(U) + e y ( u) weakly in L 2 (ω × Y * ) 9 , (6.12)
where the symmetric matrix E(U) is defined by The first term on the right hand side is already covered in (6.8). Hence, consider now 1 ε 2 T ε (e(u)). However, this is already done in [11] and yields

E(U) =    -y 3 ∂ 2 U3 ∂x 2 1 + Z 11 -y 3 ∂ 2 U ∂x1∂x2 + Z 12 0 * -y 3 ∂ 2 U3 ∂x 2 2 + Z 22 0 0 0 0    where u(x , y) = u(x , y) + y 3 2 (Z 1 (x ) • e 3 )e 1 + y 3 2 (Z 2 (x ) • e 3 )e 2 + y 3 R(x , y ) ∧ e 3 + u(x , y ) + y 3 |∇U 3 (x )| 2 e 3
1 ε 2 T ε (e(u))    e 11 (U) -y 3 ∂ 2 U3 ∂x 2 1 e 12 (U) -y 3 ∂ 2 U3 ∂x1∂2 0 * e 22 (U) -y 3 ∂ 2 U3 ∂x 2 2 0 0 0 0    + e y ( u), weakly in L 2 (Ω × Y * ), (6.14)
where we define

u(x , y) = u(x , y) + y 3 2 (Z 1 (x ) • e 3 )e 1 + y 3 2 (Z 2 (x ) • e 3 )e 2 + y 3 R(x , y ) ∧ e 3 + u(x , y ) + y 3 |∇U 3 (x )| 2 e 3
for a.e. (x , y) ∈ ω × Y * . Upon rewriting the result this yields the claim.

Note, that the antisymmetric part is responsible for the nonlinearity of the problem. Finally we prove that in the limit problems and in the case of glued yarns, one can replace the e αβ (U)'s with the Z αβ (U)'s.

The limit problem

The limits of the previous section allow to investigate the limit of the elastic problem. Therefore recall the energy of the elasticity problem in the limit

J (U, u) = ω Y * W y, E(U) + e y ( u) dydx -|Y * | ω f • U dx .
(6.15)

Define the limit space

U := U = U 1 , U 2 , U 3 ∈ H 1 (ω) 2 × H 2 (ω) | U = 0, ∂ α U 3 = 0 a.e. on γ
Furthermore, set

J(U, u) = ω Y * W y, E(U) + e y ( u) dydx , (6.16)
the part of the energy without the external force. Thus we can write

J (U, u) = J(U, u) -|Y * | ω f • U dx .
First we prove that the functional J admits a minimum on U × L 2 (ω;

H 1 per (Y * )) 3 . For every (ξ, ζ, w) ∈ S . = R 3 × R 3 × H 1 per,0 (Y * ) 3 denote E the symmetric matrix E(ξ, ζ, w) =   ξ 1 -y 3 ζ 1 + e 11,y ( w)
E ij (ξ, ζ, w) 2 L 2 (Y * ) .
Then, this expression actually defines a norm on S equivalent to the product-norm.

Proof. To show that the semi-norm is actually a norm it is necessary to show the positive definiteness, i.e., (ξ, ζ, w) S = 0 implies (ξ, ζ, w) = 0. Let (ξ, ζ, w) ∈ S satisfy (ξ, ζ, w) S = 0 and define the map

τ (y) =   y 1 (ξ 1 -y 3 ζ 1 ) + y 2 (ξ 3 -y 3 ζ 3 ) y 1 (ξ 3 -y 3 ζ 3 ) + y 2 (ξ 2 -y 3 ζ 2 ) - y 2 1 2 ζ 1 - y 2 2 2 ζ 2 -y 1 y 2 ζ 3   .
Then rewrite E(ξ, ζ, w) = e y (τ + w). (6.17 Finally, by a contradiction argument it is easy to prove that there exists a constant C such that

ξ 2 + ζ 2 + w H 1 per,0 (Y * ) ≤ C (ξ, ζ, w) S (6.18)
holds for all (ξ, ζ, w) ∈ S.

Lemma 6.6. The functional J admits a minimum on U × L 2 (ω; H 1 per,0 (Y * )) 3 .

Proof. First, from (4.2) and Lemma 6.5, there exists a constant C > 0 such that

C 2 α,β=1
e αβ (W) + 1 2

∂W 3 ∂x α ∂W 3 ∂x β 2 L 2 (ω) + ∂ 2 W 3 ∂x α ∂x β 2 L 2 (ω) + w 2 L 2 (ω;H 1 (Y * )) ≤ J(W, w), for all (W, w) ∈ U × L 2 (ω; H 1 per,0 (Y * )) 3 . (6.19) Set m = inf (U , u)∈U×L 2 (ω;H 1 per,0 (Y * )) 3 J (U, u)
where m ∈ [-∞, 0].

Step 1. We show that m ∈ (-∞, 0].

To show that m is actually finite we show that the sequence is bounded and thus admits a weak convergent subsequence and then use the weak sequential continuity of J . The boundedness of U i are show with the help of the functional J. Now, consider first U 3 , which using (6.19) together with the boundary conditions satisfies

u 2 L 2 (ω;H 1 (Y * )) ≤ J(U, u), U 3 2 H 2 (ω) ≤ C 2 α,β=1 ∂ 2 U 3 ∂x α ∂x β 2 L 2 (ω) ≤ C 2 J(U, u). (6.20)
Similarly, the estimate for U α is obtained. For this keep in mind that in the energy only Z αβ arise and we arrive at

2 α,β=1 e αβ (U) 2 L 2 (ω) ≤ cJ(U, u) + ∇U 3 4 L 4 (ω) ≤ cJ(U, u) + U 3 4 H 2 (ω) ≤ cJ(U, u) + [C 1 C 2 J(U, u)] 2 .
Note that we used here the embedding H 2 (ω) → W 1,4 (ω). The 2D-Korn inequality then yields

U 1 2 H 1 (Ω) + U 2 2 H 1 (Ω) ≤ cJ(U, u) + [C 1 C 2 J(U, u)] 2 . ( 6.21)
With (6.20) and (6.21) we have for the sequence that

J(U, u) ≤ f 3 L 2 (ω) U 3 L 2 (ω) + f 1 2 L 2 (ω) + f 2 2 L 2 (ω) U 1 L 2 (ω) + U 2 L 2 (ω) ≤ f 3 L 2 (ω) J(U, u) + f 1 2 L 2 (ω) + f 2 2 L 2 (ω) c J(U, u) + C 1 C 2 J(U, u) (6.22)
Thus we also have

J(U, u) ≤ c f L 2 (ω) J(U, u) + C 1 C 2 f L 2 (ω) J(U, u) (6.23)
which shows that J(U, u) is bounded if and only if C 1 C 2 f L 2 (ω) ≤ 1, which is the same constraint as before in Lemma 5.4. Then, we have

∀(U, u) ∈ U × L 2 (ω; H 1 per,0 (Y * )) 3 , J (U, u) ≤ 0 =⇒ U 1 H 1 (ω) + U 2 H 1 (ω) + U 3 H 2 (ω) + u L 2 (ω;H 1 (Y * )) ≤ C.
Then we easily show that m ∈ (-∞, 0].

Step 2. We show that m is a minimum.

Consider a minimizing sequence

{(U n , u n )} n ⊂ U×L 2 (ω; H 1 per,0 (Y * )) 3 satisfying J (U n , u n ) ≤ J (0, 0) = 0 and m = inf (U , u)∈U J (U, u) = lim n→+∞ J (U n , u n ).
From step 1, one has

U n 1 H 1 (ω) + U n 2 H 1 (ω) + U n 3 H 2 (ω) + u n L 2 (ω;H 1 (Y * )) ≤ C
where the constant does not depend on n.

Hence, there exists a subsequence of

{(U n , u n )} n , still denoted {(U n , u n )} n , such that (U n , u n ) (U , u ) weakly in U × L 2 (ω; H 1 per,0 (Y * )) 3 .
Furthermore, by the lower semi-continuity of J it is clear that

J (U , u ) = lim inf n→+∞ J (U n , u n ) ≤ lim n→+∞ J (U n , u n ) ≤ m (6.24)
However, since m = inf (U , u)∈U J (U, u) we conclude that for every (U, u)

∈ U × L 2 (ω; H 1 per,0 (Y * )) 3 it holds J (U , u ) ≤ m ≤ J (U, u).
This proves that the infimum is in fact a minimum. Theorem 6.7. Under the assumptions on the forces (5.5)-(5.6) we have

m = lim ε→0 m ε ε 5 = min (U , u)∈U×L 2 (ω;H 1 per (Y * )) 3 J (U, u). (6.25)
Proof. To show this result, we use a kind of Γ-convergence technique.

Step 1. In this step we show that min

(U , u)∈U×L 2 (ω;H 1 per (Y * )) 3 J (U, u) ≤ lim inf ε→0 m ε ε 5 .
To show this, let {v ε } ε , v ε ∈ V ε , be a minimizing sequence of deformations. It satisfies

lim ε→0 J ε (v ε ) ε 5 = lim inf ε→0 m ε ε 5 .
Without lost of generality, we can assume that the sequence satisfies J ε (v ε ) ≤ 0 and hence the estimates of the previous sections yield

dist(∇v ε , SO(3)) 2 L 2 (Ω * ε ) ≤ Cε 5 and (∇v ε ) T ∇v ε -I 3 2 L 2 (Ω * ε ) ≤ Cε 5 . (6.26)
Therefore, we are allowed to use the decomposition defined in 5.2 and yields the estimates (6.3) and convergences as in Lemma 6.4 and 6.3. Then the assumptions on the force lead to

lim ε→0 1 ε 5 ω×Y * T ε (f ε • (v ε -I d ))dx dy = lim ε→0 1 ε 5 ω×Y * T ε (f ε • u ε )dx dy = |Y * | ω f • U dx ,
converging as a product of a weak and a strong convergence. As consequence, we have with the weak convergence of the strain tensor 6.12 together with the weak lower semi-continuity of J that lim inf

ε→0 J ε (v ε ) ε 5 ≥ J(U, u) -|Y * | ω f • U dx (6.27)
Step 2. We show that for every (U , u

) ∈ U × L 2 (ω; H 1 per (Y * )) 3 , one has lim sup ε→0 m ε ε 5 ≤ J (U , u ). (6.28)
To do that, let (U , u ) be in U × L 2 (ω; H 1 per (Y * )) 3 . We will build a sequence {v ε } ε of admissible deformations such that lim sup 3 , where we additionally assume that u n|x2=0 = 0, such that

ε→0 m ε ε 5 ≤ lim n→+∞ lim ε→0 J ε (V (n) ε ) ε 5 = J (U , u ). Consider a sequence {U (n) } n in V ∩ C 1 (ω) 2 × C 2 (ω) and { u (n) } n in L 2 (Ω; H 1 per (Y * )) 3 ∩ C 1 (ω × Y * )
U (n) α → U α strongly in H 1 (ω) U (n) 3 → U 3 strongly in H 2 (ω) u (n) → u strongly in L 2 (ω; H 1 per (Y * )).
(6.29)

Now, we show that there exists a sequence {v ε } ε such that

lim sup ε→0 m ε ε 5 ≤ J (U (n) , u (n) ).
We define the sequence of deformations

V (n) ε,1 (x) = x 1 + ε 2 U (n) 1 (x 1 , x 2 ) - x 3 ε ∂ 1 U (n) 3 (x 1 , x 2 ) + ε u (n) 1 (x 1 , x 2 , x 3 ε ) V (n) ε,2 (x) = x 2 + ε 2 U (n) 2 (x 1 , x 2 ) - x 3 ε ∂ 2 U (n) 3 (x 1 , x 2 ) + ε u (n) 2 (x 1 , x 2 , x 3 ε ) V (n) ε,1 (x) = x 3 + ε U (n) 3 (x 1 , x 2 ) + ε 2 u (n) 3 (x 1 , x 2 , x 3 ε )
and by construction we have V ∈ V ε . Obviously, the deformation can be further restricted to the original structure Ω * ε . Now we are interested in the convergences of the deformations {V (n) ε } ε . Note that they satisfy

∇V (n) ε -I 3 L ∞ (Ωε) ≤ C(n)ε,
which is why we can assume that det(∇V (n) ε ) > 0 for all n ∈ N and all x ∈ Ω * ε (if ε is small enough). This leads us together with the right-hand-side to

m ε ≤ J ε (V (n) ε ). (6.30)
Since the convergence of the deformation components are known, we obtain

1 2ε 2 T ε ∇V (n) ε T ∇V (n) ε -I 3 -→ E(U (n) ) + e y u (n) strongly in L 2 (ω × Y * )
defined as in Lemma 6.4. This convergence gives rise to the convergence of the elastic energy

lim ε→0 1 ε 5 J ε (V (n) ε ) = lim ε→0 1 ε 5 ω×Y * T ε W y, ∇V (n) ε dx dy = lim ε→0 1 ε 5 ω×Y * T ε Q y, (∇V (n) ε ) T ∇V (n) ε -I 3 dx dy = ω×Y * Q y, E(U (n) ) + e y ( u (n) ) dx dy
and the right-hand-side

lim ε→0 1 ε 5 ω×Y * T ε f ε • V (n) ε -I d dx dy -→ |Y * | ω f ε • U (n) dx dy
Hence with (6.30) we obtain lim sup

ε→0 m ε ε 5 ≤ lim ε→0 1 ε 5 J ε (V (n) ε ) = J U (n) , u (n) .
Since this holds for every n ∈ N, then consider the limit for n to infinity. The strong convergences (6.29) yield lim sup

ε→0 m ε ε 5 ≤ lim n→+∞ J U (n) , u (n) = J U , u ,
which concludes the proof of (6.28).

Step 3. Hence, combining both steps we obtain for every (U , u

) ∈ U × L 2 (ω, H 1 per (Y * )) J (U, u) ≤ lim inf ε→0 m ε ε 5 ≤ lim sup ε→0 m ε ε 5 ≤ J (U , u ). (6.31) Thus, choosing (U , u ) = (U, u) gives J (U, u) = lim ε→0 m ε ε 5 and finally, one obtains lim ε→0 m ε ε 5 = J (U, u) = min (U , u )∈U×L 2 (ω,H 1 per (Y * ))
J (U , u ).

The cell problems

Recall the energy (6.15):

J (U, u) = 1 2 ω Y * a E(U) + e y ( u) : E(U) + e y ( u) dydx -|Y * | ω f • U dx . (6.32)
To obtain the cell problems consider the variational formulation for u associated to the functional J . For this we use the Euler-Lagrange equation (since it is a quadratic form in e( u) over a Hilbert-space) and we obtain:

Find u ∈ L 2 (Ω; H 1 (Y * )) 3 such that Ω×Y * a E(U) + e y ( u) : e y ( w) dy = 0, for all w ∈ L 2 (Ω; H 1 per,0 (Y * )) 3 . ( 6
       for all w ∈ H 1 per,0 (Y * ) 3 , (6.36)
where we denote

M 11 =   1 0 0 0 0 0 0 0 0   , M 22 =   0 0 0 0 1 0 0 0 0   , M 12 = M 21 =   0 1 0 1 0 0 0 0 0   . (6.37)
Then, set the homogenized coefficients

a hom αβα β = 1 |Y * | Y * a ijkl (y) M αβ ij + e y,ij ( χ m αβ ) M α β kl dy, b hom αβα β = 1 |Y * | Y * a ijkl (y) y 3 M αβ ij + e y,ij ( χ b αβ ) M α β kl dy, c hom αβα β = 1 |Y * | Y * a ijkl (y) y 3 M αβ ij + e y,ij ( χ b αβ ) y 3 M α β kl dy. (6.38)
Accordingly, the homogenized energy is defined by

J hom vK (U) = 1 |Y * | J (U, u) 1 2 ω a hom αβα β Z αβ Z α β + b hom αβα β Z αβ ∂ α β U 3 + c hom αβα β ∂ αβ U 3 ∂ α β U 3 dx - ω f • U dx . (6.39)
with

Z αβ = e αβ (U) + 1 2 ∂ α U 3 ∂ β U 3 .
m = lim ε→0 m ε ε 5 = min (U , u)∈U×L 2 (ω;H 1 per (Y * )) 3 J (U, u) = |Y * | min U ∈U
J hom vK (U).

Yarn made of isotropic and homogeneous material

Let us here assume that the yarns are made from an isotropic and homogeneous material whose Lamé's constants are λ, µ. The following Lemma shows that the homogenized textile is then orthotropic. Proof. Consider the following transformation:

φ ∈ H 1 per (Y * ) 3 -→ φ ∈ H 1 per (Y * ) 3 φ(y) = -φ 1 ( y)e 1 +
φ 2 ( y)e 2 + φ 3 ( y)e 3 where y = (2 -y 1 )e 1 + y 2 e 2 + y 3 e 3 , for a.e. y ∈ Y * . Using this transformation in problem (6.36) 2 gives

One has

             e y,ii ( φ)
χ b αα (2 -y 1 , y 2 , y 3 ) = χ b αα (y), χ b 12 (2 -y 1 , y 2 , y 3 ) = -χ b 12 (y)
b hom αβα β = 1 |Y * | Y * σ α β ( χ b αβ ) dy.
Hence b hom αα12 = 0, α ∈ {1, 2}. Now, from (6.43), we get (6.44) Equality (6.44) and the periodicity lead to equalities below of the traces

     χ b 12,1 (2 -y 1 , y 2 , y 3 ) = χ b 12,1 (y), χ b 12,2 (2 -y 1 , y 2 , y 3 ) = -χ b 12,2 (y), χ b 12,3 (2 -y 1 , y 2 , y 3 ) = -χ b 12,3 (y),      χ b αα,1 (2 -y 1 , y 2 , y 3 ) = -χ b αα,1 (y), χ b αα,2 (2 -y 1 , y 2 , y 3 ) = χ b αα,2 (y)
χ b 12,i (0, y 2 , y 3 ) = χ b 12,i (1, y 2 , y 3 ) = χ b 12,i (2, y 2 , y 3 ) = 0, i ∈ {2, 3}, χ b αα,1 (0, y 2 , y 3 ) = χ b αα,1 (1, y 2 , y 3 ) = χ b αα,1
(2, y 2 , y 3 ) = 0. Now using the symmetry with respect to the plane y 2 = 1, we obtain

χ b αα (y 1 , 2 -y 2 , y 3 ) = χ b αα (y), χ b 12 (y 1 , 2 -y 2 , y 3 ) = -χ b 12 (y),
for a.e. y ∈ Y * . (6.45)

Hence χ b 12,i (y 1 , 0, y 3 ) = χ b 12,i (y 1 , 1, y 3 ) = χ b 12,i (y 1 , 2, y 3 ) = 0, i ∈ {1, 3}, χ b αα,2 (y 1 , 0, y 3 ) = χ b αα,2 (y 1 , 1, y 3 ) = χ b αα,2 (y 1 , 2, y 3 ) = 0.
The results above allow to replace problem (6.36) 2 by the following ones: where Y * is the part of the cell included in (0, 1) 2 × (-2κ, 2κ) and (i ∈ {2, 3}, j ∈ {1, 3}) which ends the proof of (6.41). Similarly one obtains (6.42).

         Find χ b 12 ∈ G(Y * ) such that Y * σ y,ii ( χ b 12 ) e y,ij (
G(Y * ) = φ ∈ H 1 (Y * ) 3 | φ i (0, y 2 , y 3 ) = φ i (1, y 2 , y 3 ) = 0, φ j (y 1 , 0, y 3 ) = φ j (y 1 , 1, y 3 ) = 0 , H(Y * ) = φ ∈ H 1 (Y * ) 3 | φ 1 (0, y 2 , y 3 ) = φ 1 (1, y 2 , y 3 ) = 0, φ 2 (y 1 , 0, y 3 ) = φ 1 (y 1 , 1, y 3 ) = 0 .
As a consequence of the above lemma in the expressions of the energy (6.39) and (6.51) they remain three coefficients a hom (a hom 1111 , a hom 1122 , a hom 1212 ) and c hom (c hom 1111 , c hom 1122 , c hom 1212 ).

The linear problem

The analysis presented in this paper is stated especially for the von-Kármán limit. Although this is a nonlinear model the problem is stated with displacments and not deformations as usual in nonlinear elasticity. In fact the von-Kármán plate is the critical case for the choice of the geometric energy dist(∇v, SO(3)) L 2 (Ωε * ) ∼ Cε 5/2 in between linear and nonlinear plates, as it can be seen in [START_REF] Blanchard | Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant's strain tensor[END_REF]2,[START_REF] Friesecke | A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence[END_REF][START_REF] Ciarlet | A justification of the von Kármán equations[END_REF][START_REF] Bourne | Energy Bounds for a Compressed Elastic Film on a Substrate[END_REF][START_REF] Neukamm | Derivation of the homogenized von Kármán plate theory from 3d elasticity[END_REF]14].

To obtain the linear problem one simply considers the symmetric strain tensor e(u) instead of the Green-Lagrangian strain tensor e(u) + 1 2 ∇u(∇u) T = 1 2 (∇v(∇v) T -I 3 ). All results in this paper remain true but the Z αβ (U) are replaced by e αβ (U) in the limit.

The resulting linear limit energy

J lin (U, u) = ω Y * W y, E lin (U) + e y ( u) dydx -|Y * | ω f • U dx , (6.49)
with

E lin (U) =    e 11 (U) -y 3 ∂ 2 U3 ∂x 2 1 e 12 (U) -y 3 ∂ 2 U3 ∂x1∂2 0 * e 22 (U) -y 3 ∂ 2 U3 ∂x 2 2 0 0 0 0    (6.50)
found in (6.14).

Then, with the same steps as in section 6.3 the cell problems are given by (6.36) and yield the homogenized linear plate equation also found in [11,Thm. 11.21]. Theorem 6.10. Assume that the force satisfies f ε = ε 2+ν f 1 e 1 + ε 2+ν f 2 e 2 + ε 3+ν f 3 e 3 with f ∈ L 2 (ω) and ν > 0. Then, J lin is the unfolded limit energy. Furthermore, the cell problems are again given by (6.36) and yield the homogenized energy

J hom lin (U) = 1 2 ω a hom αβα β e αβ (U)e α β (U) + b hom αβα β e αβ (U)∂ α β U 3 + c hom αβα β ∂ αβ U 3 ∂ α β U 3 dx - ω f • U dx . (6.51)
The minimizer of this functional satisfies the variational problem Find U ∈ U such that for all V ∈ U:

∂ α β V 3 + e αβ (V)∂ α β U 3 ) + c hom αβα β ∂ αβ U 3 ∂ α β V 3 dx = ω f • V dx . (6.52)
Note that this is the same energy as for the problem presented in [11, Ch. 11] for the case θ = 1. The existence and uniqueness of a solution for this linear problem is for instance investigated in [11,[START_REF] Ciarlet | A justification of the two dimensional linear plate model[END_REF][START_REF] Oleinik | Mathematical Problems in Elasticity and Homogenization[END_REF][START_REF] Ciarlet | Mathematical Elasticity: Theory of Plates[END_REF]. Remark 6.11. The shown derivation of a homogenized von-Kármán plate is also valid for other microstructures for which the extension in section 2 holds true, e.g. shells whose mid-surfaces are developable surfaces. Remark 6.12. It is also possible to derive the von-Kármán plate on the level of deformations and the decomposition of deformation, see [START_REF] Blanchard | Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant's strain tensor[END_REF]2]. However, this needs a more involved analysis of the decomposed fields, since there exist more degrees of freedom. This different approach yields some insights into nonlinear elasticity and the connection between nonlinear decomposition and linear decompositions (see also [2,[START_REF] Griso | Decomposition of displacements of thin structures[END_REF]), yet the result is the same as presented here.

7 Comparison to [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF] In fact, the cell problems derived here are the same in [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF]. However, it is not obvious on the first sight because of a different point of view therein. The coincidence of both cases can be explained by the fact, that the homogenization presented here is also valid for the problem stated in [START_REF] Griso | Asymptotic Behavior for Textiles. hal-02279251[END_REF] with fixed junctions between the beams, i.e. the gap-function g ≡ 0, for linear elasticity. Besides replacing Z αβ by e αβ the result analogous to (6.39).

Stability of a plate with von-Kármán energy

Here, we give an example of the pre-strain in an orthotropic plate. Linearization of the Kármán plate was considered in [START_REF] Lecumberry | Stability of Slender Bodies under Compression and Validity of the von-Kármán Theory[END_REF], [START_REF] Puntel | Wrinkling of a stretched thin sheet[END_REF], [START_REF] Berdichevsky | Variational Principles of Continuum Mechanics I. Fundamentals[END_REF], under an assumption of scalar fields and that the in-plane elastic strain is equal to the given in-plane strain e * .

Homogenization of the in-plane pre-stress

In counterparts of 4.1, we can take a given pre-stress in yarns, σ * ij,ε (x) := ε 2 a ijkl ( x ε )e * kl,ε (x), due to a thermal, chemical or electric expansion, with e

* kl,ε ∈ L 2 (Ω * ε ), (k, l) ∈ {1, 2, 3} 2 . We assume that T ε (e * ε ) e * weakly in L 2 (ω × Y * ; R 3×3 ). ( 8.1)
In 4.1, we replace the term

Ω * ε f ε • (v -I d )
dx by the following:

Ω * ε ε 2 a e * ε : e(v -I d ) dx.
Then, as in Subsection 5.3, we prove that there exists a constant

C * such that, if e * ε L 2 (Ω * ε ) ≤ C * √ ε then for every v ∈ V ε such that J ε (v) ≤ 0 one has dist(∇v, SO(3)) L 2 (Ω * ε ) ≤ C * * ε 5/2
where the constants C * and C * * do not depend on ε. Proceeding as in Section 6, we obtain the limit functional Then, the effective pre-strain is

J (U, u) = 1 2 ω×Y * a E(U) + e y ( u) : E(U) + e y ( u) dydx - 1
a hom αβα β (e * αβ ) hom (x ) = 1 |Y * | Y * a(y)e * (x , y) : M α β + e y ( χ p )(x , y) dy. (8.3)
So, in te limit, the force functional will be replaced by

|Y * | ω a hom (e * ) hom (x ) : Z(x ) dx , (8.4)
with

Z = Z 11 Z 12 * Z 22 , (8.5)
Note that the macroscopic pre-stress will act in-plane, i.e. there will be no entry in the bending term from this pre-stress, because of the rotational symmetry of the periodicity cell. Note, that since the minimization of a functional is up to a given additive term, the pre-strain can be insert into the left-hand side (energetic part) of the functional and substructed from the elastic strain.

Uni-axial compression

Now, all displacements must satisfy the following inhomogeneous Dirichlet conditions:

U(0, x 2 ) =     e * L 2 0 0     , U(L, x 2 ) =     - e * L 2 0 0     ,
for a.e. x 2 ∈ (0, L). Now, our aim is to minimize over U N ew the functional We know that the infimum of this functional on U N ew is reached. Here, we want to get buckling. This means that the solutions of the above problem are not only in-plane displacements. Suppose that there is no buckling. This means that the solutions of the above minimization problem are in fact the solution of the following minimization problem: The above minimization problem admits a unique solution (an in-plane displacement U * lin ). We have 0 < J hom lin (U * lin + U) = C * (e * ) 2 ≤ J hom lin ( U) = (e * ) 2 a hom 1111 L 23 .

J hom vK (U + U) =
Now, if we find a displacement of type V 3 e 3 such that J hom vK (V 3 e 3 + U) < J hom lin (U * lin + U) = J hom vK (U * lin + U) this will show that there is buckling.

To do this, we seek a flexion independent of x 2 (taking into account the boundary conditions) A necessary condition to obtain a buckling is J hom vK (V 3 e 3 + U) < (e * ) 2 a hom 1111 L 2 . Hence To get a sufficient condition we need to know a lower bound of C * .

Remark 8.1. A weaker condition on e * , which recovers conditions as in [START_REF] Berdichevsky | Variational Principles of Continuum Mechanics I. Fundamentals[END_REF], is given by the bound from below on the energy. where c does not depend on ω.

Figure 2 :Figure 3 :

 23 Figure 2: First extension domain

on Γ ε . ( 3 . 4 )

 34 Remark 3.2. Every deformation belonging to V ε is extended in (0, L) × (-κε, 0) × (-2κε, 2κ ε ) by setting v = I d in this open set. Then, Proposition 3.1 gives an extension of v whose restriction to Ω ε belongs to D ε and satisfies (3.3).

  for a.e. (x , y) ∈ ω × Y * and Z αβ = e αβ (β) ∈ {1, 2} 2 . Proof. First, in the strain tensor ∇v(∇v) T -I 3 replace the deformation by its associated displacement u = v -I d . This yields ∇v(∇v) T -I 3 = ∇u(∇u) T + ∇u + (∇u) T = ∇u(∇u) T + 2e(u). (6.13)

ξ 3 -Lemma 6 . 5 .

 365 y 3 ζ 3 + e 12,y ( w) e 13,y ( w) * ξ 2 -y 3 ζ 2 + e 22,y ( w) e 23,y ( w) * * e 33,y ( w)  We equip the space S . = R 3 × R 3 × H 1 per,0 (Y * ) 3 with the semi-norm (ξ, ζ, w)

  )Hence, τ (y) + w(y) = a + b ∧ y is a rigid motion. Then the properties of w ∈ H 1 per,0 (Y * ) (periodicity in the directions e 1 , e 2 and vanishing mean) imply that a = b = ξ = ζ = 0 and thus also w = 0.

  .33) Upon this, we use the periodicity w.r.t. Y * to restrict the cell problems to Find u ∈ H 1 per,0 (Y * ) 3 such that Y * a E(U) + e y ( u) : e y ( w) dy = 0, for all w ∈ H 1 per,0 (Y * ) 3 . (6.34) Hence, the fields u depend linearly on E(U), one has u(x , y) = 2 α,β=1 Z αβ (x ) χ m αβ (y) + 2 α,β=1 ∂ αβ U 3 (x ) χ b αβ (y). (6.35) This leads directly to the typical cell problems Find χ m 11 , χ m 12 , χ m 22 , χ b 11 , χ b 12 , χ b 22 ∈ H 1 per,0 (Y * ) 3×6 such that Y * a(y)(M αβ + e y ( χ m αβ )) : e y ( w) dy = 0, Y * a(y)(-y 3 M αβ + e y ( χ b αβ )) : e y ( w) dy = 0,

Theorem 6 . 8 .

 68 Under the assumptions (5.5)-(5.6) the problem min U ∈U J hom vK (U) (6.40) admits solutions. Moreover, one has

Lemma 6 . 9 . 4 ( 6

 6946 Under the above assumption on the material, one has b hom αβα β = 0 ∀(α, β, α , β ) ∈ 1, 2 .41) and also a hom 1111 = a hom 2222 and a hom αα12 = 0, α ∈ 1, 2 , c hom 1111 = c hom 2222 and c hom αα12 = 0, α ∈ 1, 2 (6.42)

  (y) = e y,ii (φ)( y), i ∈ {1, 2, 3}, e y,12 ( φ)(y) = -e y,12 (φ)( y), e y,13 ( φ)(y) = -e y,13 (φ)( y), e y,23 ( φ)(y) = e y,23 (φ)( y), for a.e. y ∈ Y * .

  , for a.e. y ∈ Y * . (6.43) Since Y * y 3 dy = 0, one has

, χ b αα, 3

 3 (2 -y 1 , y 2 , y 3 ) = χ b αα,3 (y), for a.e. y ∈ Y * .

  w) dy = Y * y 3 σ y,12 ( w) dy, for all w ∈ G(Y * ) Find χ b αα ∈ H(Y * ) such that Y * σ y,ii ( χ b αα ) e y,ij ( w) dy = Y * y 3 σ y,αα ( w) dy, for all w ∈ H(Y * ) (6.46)

Now, consider the

  transformation φ ∈ H(Y * ) -→ φ ∈ H(Y * ), (resp. φ ∈ G(Y * ) -→ φ ∈ G(Y * )) φ(y) = φ 2 (y)e 1 + φ 1 (y)e 2 -φ 3 (y)e 3 where y = y 2 e 1 + y 1 e 2 -y 3 e 3 , for a.e. y ∈ Y * . One has e y,11 (φ)(y) = e y,22 (φ)(y), e y,22 (φ)(y) = e y,11 (φ)(y), e y,33 (φ)(y) = e y,33 (φ)(y), e y,13 (φ)(y) = -e y,23 (φ)(y), e y,12 (φ)(y) = e y,12 (φ)(y), e y,23 (φ)(y) = -e y,13 (φ)(y), for a.e. y ∈ Y . We use the above transformation in problems (6.46) that gives χ b 12 (y 2 , y 1 , -y 3 ) = -χ b 12 (y), χ b 11 (y 2 , y 1 , -y 3 ) = -χ b 22 (y), for a.e. y ∈ Y * . (6.47) These equalities lead to b hom 1212 = b hom 1122 = 0, b hom 1111 = -b hom 2222 . The last transformation φ ∈ H(Y * ) -→ φ ∈ H(Y * ), φ(y) = -φ 2 (y)e 1 + φ 1 (y)e 2 + φ 3 (y)e 3 where y = (1 -y 2 )e 1 + y 1 e 2 + y 3 e 3 , for a.e. y ∈ Y * . One has e y,11 (φ)(y) = e y,22 (φ)(y), e y,22 (φ)(y) = e y,11 (φ)(y), e y,33 (φ)(y) = e y,33 (φ)(y), e y,13 (φ)(y) = -e y,23 (φ)(y), e y,12 (φ)(y) = -e y,12 (φ)(y), e y,23 (φ)(y) = -e y,13 (φ)(y), for a.e. y ∈ Y * . We use the above transformation in problem (6.46) 2 that gives χ b 11 (1 -y 2 , y 1 , y 3 ) = χ b 22 (y), for a.e. y ∈ Y * . (6.48) This equality gives b hom 1111 = b hom 2222

∂

  2 ω×Y * a e * : E(U) + e y ( u) dx dy. Now, we search the field u with an additional corrector, responsible for the right-hand side, to homogenize the pre-stress, similar to[START_REF] Orlik | Two-scale homogenization in transmission problems of elasticity with interface jumps[END_REF]. Let χ p be in L 2 (ω; H 1 per,0 (Y * )) 3 the solution of Y * a e y ( χ p ) : e y ( w) dy = Y * a e * (x, •) : e y ( w) dy, for a.e. x ∈ ω and for all w ∈ H 1 per,0 (Y * ) 3 . αβ U 3 (x ) χ b αβ (y) + χ p (x , y), for a.e. (x , y) ∈ ω × Y * . (8.2)

  ) = e * L 2 -x 1 e 1 for a.e. x = (x 1 , x 2 ) ∈ ω. (8.7) This displacement satisfies the above Dirichlet conditions and one has e( U) = -e * 0 0 0 .DenoteU N ew = U = U 1 , U 2 , U 3 ∈ H 1 (ω) 2 × H 2 (ω) | U = 0, ∂ 1 U 3 = 0 a.e. on Γ D , Γ D = {0, L} × (0, L).

Z

  αβ = e αβ (U + U) + ∂ α U 3 ∂ β U 3 = Z αβ + Z * αβ where Z * αβ = e αβ ( U). Again, we want to solve the minimization problem    Find U * ∈ U N ew such that min U ∈U N ew J hom vK (U + U) = J hom vK (U * + U).

Find

  U * lin ∈ U N ew such that min U ∈U N ew J hom lin (U + U) = J hom lin (U * lin + U) where J hom lin (U+ U) = 1 2 ω a hom 1111 (e 11 (U+ U)) 2 +(e 22 (U+ U)) 2 +4a hom 1212 (e 12 (U+ U)) 2 +2a hom 1122 e 11 (U+ U)e 22 (U+ U) dx

V 3 (

 3 x ) = V 3 (x 1 ) for a.e. x = (x 1 , x 2 ) ∈ ω with V 3 ∈ H 2 0 (0, L) in order to get an admissible displacement (V 3 e 3 ∈ U N ew ). Hence, one hase 11 ( U) + (∂ 1 V 3 ) 2 = -e * + (V 3 (x 1 )) 2 , e 12 ( U) + ∂ 1 V 3 ∂ 2 U 3 = 0, e 22 ( V) + ∂ 2 V 3 ∂ 2 U 3 = 0, (∂ 11 V 3 ) 2 = (V 3 (x 1 )) 2 , (∂ 22 V 3 ) 2 = 0, (∂ 12 V 3 ) 2 = 0, ∂ 11 V 3 ∂ 22 V 3 = 0.That givesJ hom vK (V 3 e 3 + U) 1111 (e * ) 2 -2e * (V 3 (x 1 )) 2 + (V 3 (x 1 )) 4 + c hom 1111 (V 3 (x 1 )) 2 dx 1 .

3 (x 1 )) 2 dx 1 . 4 8L 3 + c hom 1111 2π 4 L 3 <

 3114343 Choose the functionV 3 (x 1 ) = sin 2 πx 1 L for all x 1 ∈ [0, L].A straight forward calculation leads to a hom 1111 3π 2e * a hom 1111

0 (V 3

 03 1111 -e * L 2 2π 2 a hom 1111 (V 3 (x 1 )) 2 dx 1 . (8.9)where we used the Poincaré inequalityL (x 1 )) 2 dx 1 ≤ L 2 (2π) 2 L 0 (V 3 (x 1 )) 2 dx 1 .(8.10)Then, we have(e * ) 2 a hom 1111 L 2 > J hom vK (V 3 e 3 + U) ≥ 1111 -e * L 2 2π 2 a 1111 (V 3 (x 1)) 2 dx 1 < 0. condition, note that the coercivity yields c e(U * lin -U) 2 ≤ J hom lin (U * lin -U). (8.13) Furthermore, note that e(U * lin -U) > 0 since the fields satisfy different boundary conditions. Indeed, we know that by the Korn-inequality, the trace-estimatione(U * lin -U) L 2 (ω) ≥ c U * lin -U H 1 (ω) ≥ c U * lin -U L 2 (Γ) ≥ cL 2 e * > 0 (8.14)

  1 2 ω a hom 1111 (Z 11 ) 2 + (Z 22 ) 2 + 4a hom 1212 (Z 12 ) 2 + 2a hom 1122 Z 11 Z 22 + c hom 1111 (∂ 11 U 3 ) 2 + (∂ 22 U 3 ) 2 + 4c hom 1212 (∂ 12 U 3 ) 2 + 2c hom 1122 ∂ 11 U 3 ∂ 22 U 3 dx , U ∈ U N ew

We reduce the parallelotopes that are in contact with the boundary of ω.

It is well known that the existence of a minimizer for Jε is still an open problem.

To get the exact value of the constant C * we have to solve the corresponding linear problem.