Publications

2017

• Modeling and Identification of the constitutive behavior of magneto-rheological elastomers
  
  • Voropaieff Jean-Pierre
  • Bodelot Laurence
  • Danas Kostas
  • Triantafyllidis Nicolas
  , 2017. In this paper, we study a class of active materials named magneto-rheological elastomers (MREs) with a main focus on their coupled magneto-mechanical response. Based on the theoretical framework proposed by Triantafyllidis [1], we develop the coupled magneto-mechanical constitutive laws in order to identify the corresponding model’s material parameters. The experimental data, obtained with a novel experimental setup allowing tensile tests up to large strains and under high magnetic fields, provide the constitutive parameters needed as input for subsequent numerical simulations.
• Buckling response of architectured columns: controlling instability across the scales through a rational design.
  
  • Tarantino Gabriella
  • Caruel Matthieu
  • Danas Kostas
  , 2017. In the present study, we combine numerical simulations with compressive testing of 3D printed polymer structures to investigate the buck ling response of a slender column, whose architecture employs unit-cell lattices which in turn consist of a sequence of columns uniformly spaced. The lower-scale columns are both of comparable size with the macro-geometry and a priori susceptible to buckling. Within the accuracy of data, the experimental trends are consistent with the numerical simulations and show that both the buckling and the post-buckling response at the macroscopic (i.e. continuum) level are dependent on the lower-scale microstructure.
• Instability of MRE film – substrate block under magneto-mechanical loadings
  
  • Psarra Erato
  • Danas Kostas
  • Bodelot Laurence
  , 2017. The present study pertains to the bifurcation modes of a MRE film bonded on a non–linear elastic and highly compliant substrate. The motivation derives from the increasing development of the magneto–sensitive elastomers and the magnetically trigered instabilities, possibly applicable in actively controlled haptic systems. The present study introduces a fully featured magneto–mechanical experiment that yields different film surface patterns, being activated by low magnetic fields and proper pre–straining of the bilayer system.
• Numerical and experimental study on effective elastic properties of porous materials: controlled porosity numerically generated and 3D printed
  
  • Zerhouni Othmane
  • Tarantino Gabriella
  • Danas Kostas
  , 2017. This work aims to introduce a methodology for the determination of the effective properties of random heterogeneous materials. Specifically, focus of this study are the effective properties of two phase porous material containing spherical voids which are investigated both experimentally and numerically. To this end, three-dimensional porous structures, with non-overlapping porosity, are first generated by means of a random sequential algorithm. Numerical simulations are conducted on a representative volume element, whose size is first determined. Specimens are fabricated by 3D-printing of the computer-generated microstructures and tested under uniaxial loading. Good agreement of Young modulus data is found, in the case of spherical monodisperse porosity.
• Etudes des modèles d'endommagement à gradient en grandes déformations
  
  • Crabbé Blandine
  • Marigo Jean-Jacques
  • Chamberland Eric
  • Guilié Joachim
  , 2017. Nous étudions les modèles d’endommagement à gradient dans le cadre des grandes dé- formations. Pour cela, nous établissons les solutions analytiques homogènes et localisées dans le cas unidimensionnel en choisissant un potentiel hyperélastique adapté. Les résultats sont ensuite confrontés à des simulations numériques effectuées via l’implémentation de ces modèles dans le logiciel éléments finis FEniCS.
• Strain effects in semiconductors
  
  • Guin Laurent
  • Jabbour Michel E
  • Triantafyllidis Nicolas
  , 2017. While it has been known, for long, that mechanical strains affect the electronic behavior of semiconductors, both the theoretical aspects of the coupling between mechanics and electronics and their practical consequences on the response of semiconductor devices remain to be investigated. In this work, we first develop a general fully-coupled mechanical, electrical and electronic continuum model of the finitely deformable semiconductor and subsequently use it to compute the strain-induced changes in the current-voltage characteristic of a p - n junction subjected to bending.
• Méthode de contrôle pour le calcul de la solution périodique en espace-temps d'un problème d'évolution
  
  • Le Tallec Patrick
  • Khristenko Ustim
  , 2017. Ce travail présente une méthode de contrôle pour calculer la solution périodique établie d’un problème d’évolution linéaire avec des conditions de périodicité dans l’espace-temps. L’idée consiste à modifier le problème à valeur initiale en introduisant un terme de «feedback control», basé sur l’erreur de périodicité et accélérant la convergence de la solution vers sa limite asymptotique. Pour justifier les résultats théoriques, la méthode est appliquée à un problème modèle 3D de roulage stationnaire d’un pneu viscoélastique avec sculpture périodique.
• Corrélation des propriétés mécaniques et micro-structurelles de la peau de souris à partir d'une sollicitation bi-axiale
  
  • Affagard Jean-Sébastien
  • Ducourthial Guillaume
  • Bonod-Bidaud Christelle
  • Ruggiero Florence
  • Schanne-Klein Marie-Claire
  • Allain Jean-Marc
  , 2017. Le derme, composant principal de la peau, est composée en majorité de collagène et présente une micro-structure très hiérarchisée qui influe sur son comportement mécanique aux différentes échelles. La prédiction de son comportement nécessite de caractériser l’influence de la micro-structure sur les propriétés mécaniques. Aussi, un test de traction bi-axiale couplé indépendamment à une mesure macroscopique et à une mesure microscopique a été développé. Ce travail a finalement permis d’identifier les paramètres d’une loi hyper-élastique anisotrope et de tester l’hypothèse de transformation affine sous-jacent au comportement adopté.
• Extension des schémas CSTS (Conservative Space- and Time-Staggered) à la géométrie axisymétrique et à l’ALE
  
  • Marboeuf Alexis
  • Claisse Alexandra
  • Le Tallec Patrick
  • Llor Antoine
  , 2017.
• Analyse isogéométrique : méthodes multipatchs et quadratures gaussiennes optimisées
  
  • Adam Nicolas
  • Zarroug Malek
  • Le Tallec Patrick
  , 2017. Le concept d’analyse isogéométrique (IGA) permet de lier plus directement la conception assistée par ordinateur à l’étape d’analyse par l’utilisation de fonctions B-splines communes, plus riches que les fonctions de Lagrange utilisées en éléments finis. La régularité élevée de ces fonctions ne permet pas de se soustraire aux problèmes de verrouillages si une quadrature Gaussienne complète est envisagée. Ce résumé propose des règles d’intégrations optimisées et validées dans le cas d’un domaine unique. Une extension est proposée dans le cadre de patchs multiples pour différents types de discrétisation spatiale.
• Complex structures from patterned cell sheets
  
  • Misra M.
  • Audoly B.
  • Shvartsman S. Y.
  Philosophical Transactions of the Royal Society B: Biological Sciences, Royal Society, The, 2017, 372 (1720), pp.20150515. The formation of three-dimensional structures from patterned epithelial sheets plays a key role in tissue morphogenesis. An important class of morphogenetic mechanisms relies on the spatio-temporal control of apical cell contractility, which can result in the localized bending of cell sheets and in-plane cell rearrangements. We have recently proposed a modified vertex model that can be used to systematically explore the connection between the two-dimensional patterns of cell properties and the emerging three-dimensional structures. Here we review the proposed modelling framework and illustrate it through the computational analysis of the vertex model that captures the salient features of the formation of the dorsal appendages during Drosophila oogenesis. This article is part of the themed issue ‘Systems morphodynamics: understanding the development of tissue hardware’. (10.1098/rstb.2015.0515)
  DOI : 10.1098/rstb.2015.0515
• Diffraction formulation for the sound radiation of a loudspeaker on a rigid convex edged shape enclosure
  
  • Roggerone Vincent
  • Corteel Étienne
  • Boutillon Xavier
  , 2017, 141 (5), pp.4034 - 4034. We present a method for calculating the sound radiation of a loudspeaker on a box-shape enclosure. This method is based on the iterative Asheim-Svensson formulation [Asheim et al. J. Acoust. Soc. Am., 2013]. In this geometrical acoustical model, the radiated sound is given by the addition of the direct sound of a baffled piston and the sound scattered by the edges of the loudspeaker enclosure. The latter are expressed in the form of multiple-order edge diffraction components. This method becomes particularly attractive at high frequencies since the maximum order required to get a correct estimation decreases with frequency. Altogether, the method becomes less costly than the Boundary Element Method (BEM) beyond a given frequency. The convergence of the method can be estimated by considering the distance traveled by the creeping wave on the enclosure instead of considering the successive orders of diffraction by each edge. An iterative formulation follows which is well-suited to slender objects such as a sound bar. Comparisons with measurements and results of a BEM calculation will be presented. [Work supported by the ANR-13-CORD-0008 EDISON 3D grant from the French National Agency of Research.] (10.1121/1.4989304)
  DOI : 10.1121/1.4989304
• Experimental Validation of the Multiphase Extended Leblond's Model
  
  • Weisz-Patrault Daniel
  , 2017. Transformation induced plasticity is a crucial contribution of the simulation of several forming processes involving phase transitions under mechanical loads, resulting in large irreversible strain even though the applied stress is under the yield stress. One of the most elegant and widely used models is based on analytic homogenization procedures and has been proposed by Leblond et al. [1-4]. Very recently, a simple extension of the Leblond's model has been developed by Weisz-Patrault [8]. Several product phases are taken into account and several assumptions are relaxed in order to extend the applicability of the model. The present contribution compares experimental tests with numerical computations, in order to discuss the validity of the developed theory. Thus, experimental results extracted from the existing literature are analyzed. Results show a good agreement between measurements and theoretical computations.
• Buckling of an elastic ridge: competition between wrinkles and creases
  
  • Lestringant Claire
  • Maurini Corrado
  • Lazarus Arnaud
  • Audoly Basile
  Physical Review Letters, American Physical Society, 2017, 118, pp.165501. We investigate the elastic buckling of a triangular prism made of a soft elastomer. A face of the prism is bonded to a stiff slab that imposes an average axial compression. We observe two possible buckling modes which are localized along the free ridge. For ridge angles $\phi$ below a critical value $\phi^\star\approx 90^\circ$ experiments reveal an extended sinusoidal mode, while for $\phi$ above $\phi^\star$ we observe a series of creases progressively invading the lateral faces starting from the ridge. A numerical linear stability analysis is set up using the finite-element method and correctly predicts the sinusoidal mode for $\phi \leq \phi^\star$, as well as the associated critical strain $\epsilon_{\mathrm{c}}(\phi)$. The experimental transition at $\phi^\star$ is found to occur when this critical strain $\epsilon_{\mathrm{c}}(\phi)$ attains the value $\epsilon_{\mathrm{c}}(\phi^\star) = 0.44$ corresponding to the threshold of the sub-critical surface creasing instability. Previous analyses have focused on elastic crease patterns appearing on planar surfaces, where the role of scale-invariance has been emphasized; our analysis of the elastic ridge provides a different perspective, and reveals that scale-invariance is not a sufficient condition for localization. (10.1103/PhysRevLett.118.165501)
  DOI : 10.1103/PhysRevLett.118.165501
• Spatial spectra of the eigenmodes of ribbed plates projected on dispersion branches
  
  • Lefebvre Gautier
  • Boutillon Xavier
  • Filoche Marcel
  , 2017. A vast literature has been devoted to the transverse vibration and the sound radiation of ribbed plates over the last decades. The present study has been motivated by the analysis of the dynamical behaviour of piano soundboards. As a rough approximation, a piano soundboard can be considered as an orthotropic ribbed plate. Our purpose is to establish condensed descriptions for their dynamics. For low frequencies, regularly ribbed plates can be considered as homogeneous plates. It is usually considered that homogenization is valid only up to a frequency corresponding roughly to the confinement of one half wave-length between the (periodically spaced) ribs. Beyond that frequency, depending on the relative characteristic mobility of the ribs and that of the base plate, the ribs may constrain transverse waves to be guided between them. We focus here on the spatial spectrum of the normal modes of the ribbed plate (2D Fourier transforms of the modal shapes). It appears that most of the peaks of each spectrum can be seen as belonging to one of a few dispersion branches in an appropriate (w; k)-plane. Interestingly, different peaks of a spectrum (of one given mode) usually "belong" to different dispersion branches. When valid, this description may prove an interesting intermediate step to derive approximations for the sound radiation of such plates.
• An asymptotic plate model for magneto-electro-thermo-elastic sensors and actuators
  
  • Bonaldi Francesco
  • Geymonat G.
  • Krasucki F.
  • Serpilli M.
  Mathematics and Mechanics of Solids, SAGE Publications, 2017, 22 (4), pp.798-822. We present an asymptotic two-dimensional plate model for linear magneto-electro-thermo-elastic sensors and actuators, under the hypotheses of anisotropy and homogeneity. Four different boundary conditions pertaining to electromagnetic quantities are considered, leading to four different models: the sensor-actuator model, the actuator-sensor model, the actuator model and the sensor model. We validate the obtained two-dimensional models by proving weak convergence results. Each of the four plate problems turns out to be decoupled into a flexural problem, involving the transversal displacement of the plate, and a certain partially or totally coupled membrane problem. (10.1177/1081286515612885)
  DOI : 10.1177/1081286515612885
• Second Order Homogenization of Subwavelength Stratified Media Including Finite Size Effect
  
  • Marigo Jean-Jacques
  • Maurel Agnes
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2017, 77 (2), pp.721 - 743. (10.1137/16M1070542)
  DOI : 10.1137/16M1070542
• Non-invasive Model-Based Assessment of Passive Left-Ventricular Myocardial Stiffness in Healthy Subjects and in Patients with Non-ischemic Dilated Cardiomyopathy
  
  • Hadjicharalambous Myrianthi
  • Asner Liya
  • Chabiniok Radomir
  • Sammut Eva
  • Wong James
  • Peressutti Devis
  • Kerfoot Eric
  • King Andrew D.
  • Lee Jack
  • Razavi Reza
  • Smith Nicolas
  • Carr-White Gerald
  • Nordsletten David
  Annals of Biomedical Engineering, Springer Verlag, 2017, 45 (3), pp.605-618. Patient-specific modelling has emerged as a tool for studying heart function, demonstrating the potential to provide non-invasive estimates of tissue passive stiffness. However, reliable use of model-derived stiffness requires sufficient model accuracy and unique estimation of model parameters. In this paper we present personalised models of cardiac mechanics, focusing on improving model accuracy, while ensuring unique parametrisation. The influence of principal model uncertainties on accuracy and parameter identifiability was systematically assessed in a group of patients with dilated cardiomyopathy (n 1⁄4 3) and healthy volunteers (n 1⁄4 5). For all cases, we examined three circumferentially symmetric fibre distributions and two epicardial boundary conditions. Our results demonstrated the ability of data-derived boundary conditions to improve model accuracy and highlighted the influence of the assumed fibre distribution on both model fidelity and stiffness estimates. The model personalisation pipeline—based strictly on non-invasive data—produced unique parameter estimates and satisfactory model errors for all cases, supporting the selected model assumptions. The thorough analysis performed enabled the comparison of passive parameters between volunteers and dilated cardiomyopathy patients, illustrating elevated stiffness in diseased hearts.
• Space/Time convergence analysis of a class of conservative schemes for linear wave equations
  
  • Chabassier Juliette
  • Imperiale Sebastien
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2017, 355 (3), pp.282-289. This paper concerns the space/time convergence analysis of conservative two-steps time discretizations for linear wave equations. Explicit and implicit, second and fourth order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. The convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value. (10.1016/j.crma.2016.12.009)
  DOI : 10.1016/j.crma.2016.12.009
• Investigation on fatigue behaviors of NiTi polycrystalline strips under stress-controlled tension via in-situ macro-band observation
  
  • Zheng Lin
  • He Y. J.
  • Moumni Ziad
  International Journal of Plasticity, Elsevier, 2017, 90, pp.116 - 145. (10.1016/j.ijplas.2016.12.008)
  DOI : 10.1016/j.ijplas.2016.12.008
• Fatigue crack growth in two TWIP steels with different stacking fault energies
  
  • H.K. Yang
  • Doquet Véronique
  • Z.F. Zhang
  International Journal of Fatigue, Elsevier, 2017. Fatigue crack growth tests with R=0 and 0.4 were carried out on Fe-22Mn-0.6C and Fe-22Mn-0.6C-3Al (wt. %) Twinning-Induced Plasticity (TWIP) steels with stacking fault energies around 21.5 and 37 mJ/m2, respectively. The former exhibits more crack closure effects, partly due to stronger asperity-induced closure. Strain-controlled push-pull tests followed by scanning electron microscope observations show that both steels are prone to mechanical twinning under low-cycle fatigue, associated with an increasing kinematic hardening. Twinning is however more profuse in Fe-22Mn-0.6C steel. Elastic-plastic finite elements simulations of crack growth, using specific constitutive equations able to capture the increasing kinematic hardening suggest that plasticity-induced crack closure is lower in Fe-22Mn-0.6C steel. Even after closure corrections, the Al-free steel, exhibits a lower resistance to fatigue crack growth, which is attributed to a pronounced strain localisation at the crack tip, and maybe also to environment effects. (10.1016/j.ijfatigue.2017.01.034)
  DOI : 10.1016/j.ijfatigue.2017.01.034
• Beat-by-beat assessment of cardiac afterload using descending aortic velocity–pressure loop during general anesthesia: a pilot study
  
  • Vallée Fabrice
  • Le Gall Arthur
  • Joachim Jona
  • Passouant Olivier
  • Mateo Joaquim
  • Mari Arnaud
  • Millasseau Sandrine
  • Mebazaa Alexandre
  • Gayat Etienne
  Journal of Clinical Monitoring and Computing, Springer Verlag, 2017, 113 (5), pp.727 - 735. Introduction: Continuous cardiac afterload evaluation could represent a useful tool during general anesthesia (GA) to titrate vasopressor effect. Using beat to beat descending aortic pressure(P)/flow velocity(U) loop obtained from esophageal Doppler and femoral pressure signals might allow to track afterload changes. Methods: We defined 3 angles characterizing the PU loop (alpha, beta and Global After-Load Angle(GALA)). Augmentation index (AIx) and total arterial compliance (Ctot) were measured via radial tonometry. Peripheral Vascular Resistances (PVR) were also calculated. Twenty patients were recruited and classified into low and high cardiovascular (CV) risk group. Vasopressors were administered, when baseline mean arterial pressure (MAP) fell by 20%. Results: We studied 118 pairs of pre/post bolus measurements. At baseline, patients in the lower CV risk group had higher cardiac output (6.1±1.7 vs 4.2±0.6 L/min; p= 0.005), higher Ctot (2.7±1.0 vs 2.0±0.4ml/mmHg, p = 0.033), lower Aix and PVR (13±10 vs 32±11 % and 1011±318 vs 1390±327 dyn.s.cm-5; p<0.001 and p=0.016, respectively) and lower GALA (41±15 vs ± 6°; p <0.001). GALA was the only PU Loop parameterassociated with Ctot, Aix and PVR. After vasopressors, MAP increase was associated with a decrease in Ctot, an increase in Aix and PVR and an increase in alpha, beta and GALA (p<0.001 for all). Changes in GALA and Ctot after vasopressors were strongly associated (p=0.004). Conclusions: PU Loop assessment from routine invasive hemodynamic optimization management during GA and especially GALA parameter could monitor cardiac afterload continuously in anesthetized patients, and may help clinicians to titrate vasopressor therapy. (10.1007/s10877-017-9982-5)
  DOI : 10.1007/s10877-017-9982-5
• Inverse problems and data assimilation methods applied to protein polymerisation
  
  • Armiento Aurora
  , 2017. The aim of this PhD thesis is to set up a mathematical strategy to investigate the physical process of protein aggregation. The study of this largely unknown process is particularly important since it has been identified as a key feature of a wide class of incurable diseases, called amyloid diseases. Prion diseases belong to this class and are caused by the aggregation of a misfolded configuration of the prion protein. Our work contributes to the research on prion diseases, by focusing on two kinds of aggregates : oligomers and fibrils. Oligomers, which are suspected of being the most toxic aggregates, are studied in the first part of this thesis. We base our work on the analysis of two types of experimental data. On the one hand, we consider Static Light Scattering (SLS) data, which can be interpreted biologically as the measurement of the average oligomer size and mathematically as the second moment of aggregate concentration. On the other hand, we consider oligomer size distribution data collected at several instants by using Size Exclusion Chromatography (SEC). Our study leads to the important conclusion that at least two different types of oligomers are present. Moreover, we provide a description of the interaction between these oligomers by proposing, for the first time, a two-species model. Our model is composed of a set of ODEs with the kinetic rates as parameters. The qualitative description provided by this model has been coupled to the information contained in the noisy experimental SLS data in a data assimilation framework. By means of the extended Kalman filter method, we solve a non-linear inverse problem, thereby estimating the kinetic coefficients associated to the experimental data. To validate this model we have compared our estimation to the experimental SEC data, observing a very good agreement between the two. Our oligomer species characterisation may lead to new strategies to design a first targeted treatment for prion diseases. The methodology applied to the study of oligomers can be seen as a first step in the analysis of fibrils. Due to the physical properties of these aggregates, fewer and less precise experiments can be performed and so a mathematical approach can provide a valuable contribution to their study. Our contribution is to propose a general strategy to estimate the initial condition of a fibril system. Inspired by the Lifshitz-Slyozov theory, we describe this system by a transport equation coupled with an integral equation. The estimation is performed making use of some empirical observations on the system. We consider the general case of observing a moment of order n. It is indeed possible to measure the first moment by Thioflavine T fluorescence or the second moment by SLS. We provide a theoretical and numerical solution of the initial condition estimation problem in the linear case of a depolymerising system. In particular, for constant depolymerisation rates, we propose a kernel regularisation strategy, that provides a first characterisation of the estimation. In the variable depolymerisation rates, we outline the variational data assimilation method 4d-Var. This method is more general and can be easily adapted to treat different problems. This inverse problem is particularly interesting since it can also be applied in other fields such as the cell cycle or dust formation.
• A novel microstructural interpretation for the biomechanics of mouse skin derived from multiscale characterization
  
  • Lynch Barbara
  • Bancelin Stéphane
  • Bonod-Bidaud Christelle
  • Gueusquin Jean-Baptiste
  • Ruggiero Florence
  • Schanne-Klein Marie-Claire
  • Allain Jean-Marc
  Acta Biomaterialia, Elsevier, 2017, 50, pp.302-311. Skin is a complex, multi-layered organ, with important functions in the protection of the body. The der-mis provides structural support to the epidermal barrier, and thus has attracted a large number of mechanical studies. As the dermis is made of a mixture of stiff fibres embedded in a soft non-fibrillar matrix, it is classically considered that its mechanical response is based on an initial alignment of the fibres, followed by the stretching of the aligned fibres. Using a recently developed setup combining mul-tiphoton microscopy with mechanical assay, we imaged the fibres network evolution during dermis stretching. These observations, combined with a wide set of mechanical tests, allowed us to challenge the classical microstructural interpretation of the mechanical properties of the dermis: we observed a continuous alignment of the collagen fibres along the stretching. All our results can be explained if each fibre contributes by a given stress to the global response. This plastic response is likely due to inner sliding inside each fibre. The non-linear mechanical response is due to structural effects of the fibres network in interaction with the surrounding non-linear matrix. This multiscale interpretation explains our results on genetically-modified mice with a simple alteration of the dermis microstructure. Statement of Significance: Soft tissues, as skin, tendon or aorta, are made of extra-cellular matrix, with very few cells embedded inside. The matrix is a mixture of water and biomolecules, which include the collagen fibre network. The role of the collagen is fundamental since the network is supposed to control the tissue mechanical properties and remodeling: the cells attach to the collagen fibres and feel the network deformations. This paper challenges the classical link between fibres organization and mechanical properties. To do so, it uses multiscale observations combined to a large set of mechanical loading. It thus appears that the behaviour at low stretches is mostly controlled by the network structural response, while, at large stretches, the fibre inner-sliding dominate. (10.1016/j.actbio.2016.12.051)
  DOI : 10.1016/j.actbio.2016.12.051
• Quasi-static Evolution, Variational Principles and Implicit Scheme in Gradient Plasticity
  
  • Nguyen Quoc-Son
  , 2017, 8, pp.143-152. This paper is devoted to the theory of gradient plasticity. Our attention is focussed on the description of the constitutive equations, on the formulation of the governing equations in terms of the energy potential and the dissipation potential of the solid. The evolution equation is discussed for quasi-static responses. A time-discretization by the implicit scheme of the evolution equation leads to the study of the incremental problem which is different from the rate problem. The incremental problem and associated incremental variational principles are discussed in relation with some existing results of the literature. (10.1007/978-3-319-48884-4_7)
  DOI : 10.1007/978-3-319-48884-4_7