In der vorliegenden Arbeit wird die Aufklärung der Konformation von Peptiden mit speziell modifizierten Aminosäuren beschrieben. Die Methoden der theoretischen Chemie (Quantenchemie, Molekülmechanik, Moleküldynamik) bilden dabei die Grundlage der Konformationsanalysen. Durch systematische Anwendung dieser Methoden werden im ersten Teil der Arbeit die konformativen Eigenschaften verschiedener [beta]-Aminosäuren und ihrer Oligomere ([beta]-Peptide) untersucht. Aus diesen Ergebnissen werden anschließend Regeln für das Sekundärstrukturdesign von ß-Peptiden abgeleitet. Der zweite Teil beschäftigt sich mit der theoretischen Konformationsanalyse von [alpha]- Hydrazinosäuren und ihrer Oligomere (Hydrazinopeptide). Aus den gewonnenen Erkenntnissen über die Ausbildung charakteristischer Sekundärstrukturelemente in diesen Verbindungen wird ebenfalls ein Regelwerk für das Design von Sekundärstrukturen aufgestellt. / The present work describes the conformational characteristics of pepttides with specifically modified amino acid constituents. For this purpose, the methods of theoretical chemistry (quantum chemistry, molecular mechanics, molecular dynamics) are utilisied for the conformational analyses. The conformation of various [beta]-amino acids and their oligomers ([beta]-peptides) are inverstigated in the first part of this work applying these methods. Rules for the design of definite secondary structures in [beta]-peptides are then derived from the obtained results. In the second part, systematic theoretical conformational analyses on [alpha]-hydrazino acids and their oligomers (hydrazino peptides) are described. The results are then used to compile a set of rules for the formation of characteriasitc secondary structures in this class of compounds.

Quick and accurate predictions of equivalent properties for thin-walled composite structures are required in the preliminary design process. Existing literature provides analytical solutions to some structures but is limited to particular cases. No unified approach exists to tackle homogenization of thin-walled structures such as beams, plates, or three-dimensional structures using the thin-walled approximation. In this work, a unified approach is proposed to obtain equivalent properties for beams, plates, and three-dimensional structures for thin-walled composite structures using mechanics of structure genome. The adopted homogenization technique interprets the unit cell associated with the composite structures as an assembly of plates, and the overall strain energy density of the unit cell as a summation of the plate strain energies of these individual plates. The variational asymptotic method is then applied to drop all higher-order terms and the remaining energy is minimized with respect to the unknown fluctuating functions. This has been done by discretizing the two-dimensional unit cell into one-dimensional frame elements in a finite element description. This allows the handling of structures with different levels of complexities and internal geometry within a general framework. Comparisons have been made with other works to show the advantages which the proposed model offers over other methods.

<p>The Mechanics of Structural Genome (MSG) is a unified homogenization theory used to find equivalent constitutive models for beam, plate, and solid structures. It has been proven accurate for periodic structures. However, for certain applications such as non-prismatic wind turbine blades and helicopter flexbeams featuring ply drop-off, where there is no repeating structure and the periodic boundary condition cannot be used, MSG's accuracy is limited. In this work, we aim to extend MSG to find element stiffness matrices directly for aperiodic structures, instead of beam properties or three-dimensional (3D) solid material properties. Two finite elements based on MSG have been developed: Heterogeneous Beam Element (HBE) and Heterogeneous Solid Element (HSE).</p> <p><br></p> <p>For beam modeling, the beam-like structure is homogenized into a series of 3-node Heterogeneous Beam Elements (HBE) with 18×18 effective beam element stiffness matrices. These matrices are used as input for one-dimensional (1D) beam analysis using the Abaqus User Element subroutine (UEL). Using the macroscopic beam analysis results as input, we can also perform dehomogenization to predict the stresses and strains in the original structure. We use three examples (a prismatic composite beam, an isotropic homogeneous tapered beam, and a composite tapered beam) to demonstrate the capability of HBE and show its advantages over the MSG cross-sectional analysis approach. HBE can capture macroscopic behavior and detailed stresses due to non-prismatic geometry.</p> <p><br></p> <p>The Heterogeneous Solid Element (HSE) is developed based on MSG to model a heterogeneous body as an equivalent solid element using an effective element stiffness matrix. HSE modeling includes homogenization, macroscopic global analysis, and dehomogenization to recover local strains/stresses. HSE avoids the local periodicity assumption for traditional multiscale modeling techniques for composite structures that compute effective material properties instead. Abaqus composite solid element and MSG-based traditional multiscale modeling are used to validate the accuracy of HSE. All example results show that HSE is more accurate in predicting global structural behavior and local strains/stresses.</p> <p><br></p> <p>HBE and HSE provide a new concept for modeling aperiodic composite structures by modeling structures into equivalent beam or solid elements instead of beam properties of the reference line in 1D beam analysis or material properties of material points in solid structural analysis.</p>

Among the many active materials in use today, piezoelectric composite patches have enabled notable advances in emerging technologies such as disturbance sensing, control of flexible structures, and energy harvesting. The macro fiber composite (MFC), in particular, is well known for its outstanding performance. Multiscale models are typically required for smart-structure design with MFCs. This is due to the need for predicting the macroscopic response (such as tip deflection under a transverse load or applied voltage) while accounting for the fact that the MFC has microscale details. Current multiscale models of the MFC exclusively focus on predicting the macroscopic response with homogenized material properties. There are a limited number of homogenized properties available from physical experiments and various aspects of existing homogenization techniques for the MFC are shown here to be inadequate. Thus, new homogenized models of the MFC are proposed to improve smart-structure predictions and therefore improve device design. It is notable that current multiscale modeling efforts for MFCs are incomplete since, after homogenization, the local fields such as stresses and electric fields have not been recovered. Existing methods for obtaining local fields are not applicable since the electrodes of the MFC are embedded among passive layers. Therefore, another objective of this work was to find the local fields of the MFC without having the computational burden of fully modeling the microscopic features of the MFC over a macroscale area. This should enable smart-structure designs with improved reliability because failure studies of MFCs will be enabled. Large-scale 3D finite element (FE) models that included microscale features were constructed throughout this work to verify the multiscale methodologies. Note that after creating a free account on cdmhub.org, many files used to create the results in this work can be downloaded from https://cdmhub.org/projects/ernestocamarena.<br><br>First, the Mechanics of Structure Genome (MSG) was extended to provide a rigorous analytical homogenization method. The MFC was idealized to consist of a stack of homogeneous layers where some of the layers were homogenized with existing rules of mixtures. For the analytical model, the electrical behavior caused by the interdigitated electrodes (IDEs) was approximated with uniform poling and uniform electrodes. All other assumptions on the field variables were avoided; thus an exact solution for a stack of homogeneous layers was found with MSG. In doing so, it was proved that in any such multi-layered composite, the in-plane strains and the transverse stresses are equal in each layer and the in-plane electric fields and transverse electric displacement are constant between the electrodes. Using this knowledge, a hybrid rule of mixtures was developed to homogenize the entire MFC layup so as to obtain the complete set of effective device properties. Since various assumptions were avoided and since the property set is now complete, it is expected that greater energy equivalence between reality and the homogenized model has been made possible. The derivation clarified what the electrical behavior of a homogenized solid with internal electrodes should be—an issue that has not been well understood. The behavior was verified by large-scale FE models of an isolated MFC patch.<br> <br>Increased geometrical fidelity for homogenization was achieved with an FE-based RVE analysis that accounted for finite-thickness effects. The presented theory also rectifies numerous issues in the literature with the use of the periodic boundary conditions. The procedure was first developed without regard to the internal electrodes (ie a homogenization of the active layer). At this level, the boundary conditions were shown to satisfy a piezoelectric macrohomogeneity condition. The methodology was then applied to the full MFC layup, and modifications were implemented so that both types of MFC electrodes would be accounted for. The IDE case considered nonuniform poling and electric fields, but fully poled material was assumed. The inherent challenges associated with these nonuniformities are explored, and a solution is proposed. Based on the homogenization boundary conditions, a dehomogenization procedure was proposed that enables the recovery of local fields. The RVE analysis results for the effective properties revealed that the homogenization procedure yields an unsymmetric constitutive relation; which suggests that the MFC cannot be homogenized as rigorously as expected. Nonetheless, the obtained properties were verified to yield favorable results when compared to a large-scale 3D FE model.<br> <br>As a final test of the obtained effective properties, large-scale 3D FE models of MFCs acting in a static unimorph configuration were considered. The most critical case to test was the smallest MFC available. Since none of the homogenized models account for the passive MFC regions that surround the piezoelectric fiber array, some of the test models were constructed with and without the passive regions. Studying the deflection of the host substrate revealed that ignoring the passive area in smaller MFCs can overpredict the response by up to 20%. Satisfactory agreement between the homogenized models and a direct numerical simulation were obtained with a larger MFC (about a 5% difference for the tip deflection). Furthermore, the uniform polarization assumption (in the analytical model) for the IDE case was found to be inadequate. Lastly, the recovery of the local fields was found to need improvement.<br><br><br>

First, the theory of MSG is extended to aperiodic heterogeneous solid structures. Integral constraints are introduced to decompose the displacements and strains of the heterogeneous material into a fluctuating part and a macroscopic part, of which the macroscopic part represents the responses of the homogenized material. One advantage of this theory is that boundary conditions are not required. Consequently, it is capable of handling micro-structures of arbitrary shapes. In addition, periodic constraints can be incorporated into this theory as needed to model periodic or partially periodic materials such as textile composites. In this study, the newly developed method is employed to investigate the finite thickness effect of textile composites.<div><br></div><div>Second, MSG is enabled to deal with Timoshenko beam-like structures with spanwise heterogeneity, which provide higher accuracy than the previous available Euler–Bernoulli beam model. Its reduced form, the MSG beam cross sectional analysis, is found to be able to analyze generalized free-edge problems with arbitrary layups and subjected to general loads. In this method, the only assumption applied is that the laminate is long enough so that the Saint-Venant principle can be adopted. There is no limitation on the cross section of the laminate since no ad hoc assumption is involved with the microstructure geometry. This method solve the free-edge problem from a multiscale simulation point of view.<br></div><div><br></div>

Le nouvel élément de poutre est une évolution d'un élément de Timoshenko poutre avec un nœud supplémentaire situé à mi-longueur. Ce nœud supplémentaire permet l'introduction de trois composantes supplémentaires de contrainte afin que la loi constitutionnelle 3D complète puisse être utilisée directement. L'élément proposé a été introduit dans un code d'éléments finis dans Matlab et une série d'exemples de linéaires/petites contraintes ont été réalisées et les résultats sont systématiquement comparés avec les valeurs correspondantes des simulations ABAQUS/Standard 3D. Ensuite, la deuxième étape consiste à introduire le comportement orthotrope et à effectuer la validation de déplacements larges / petites contraintes basés sur la formulation Lagrangienne mise à jour. Une série d'analyses numériques est réalisée qui montre que l'élément 3D amélioré fournit une excellente performance numérique. En effet, l'objectif final est d'utiliser les nouveaux éléments de poutre 3D pour modéliser des fils dans une préforme composite textile. A cet effet, la troisième étape consiste à introduire un comportement de contact et à effectuer la validation pour un nouveau contact entre 3D poutres à section rectangulaire. La formulation de contact est dérivée sur la base de formulation de pénalité et de formulation Lagrangian mise à jour utilisant des fonctions de forme physique avec l'effet de cisaillement inclus. Un algorithme de recherche de contact efficace, qui est nécessaire pour déterminer un ensemble actif pour le traitement de contribution de contact, est élaboré. Et une linéarisation constante de la contribution de contact est dérivée et exprimée sous forme de matrice appropriée, qui est facile à utiliser dans l'approximation FEM. Enfin, on présente quelques exemples numériques qui ne sont que des analyses qualitatives du contact et de la vérification de l'exactitude et de l'efficacité de l'élément de 3D poutre proposé. / The new beam element is an evolution of a two nodes Timoshenko beam element with an extra node located at mid-length. That extra node allows the introduction of three extra strain components so that full 3D stress/strain constitutive relations can be used directly. The second step is to introduce the orthotropic behavior and carry out validation for large displacements/small strains based on Updated Lagrangian Formulation. A series of numerical analyses are carried out which shows that the enhanced 3D element provides an excellent numerical performance. Indeed, the final goal is to use the new 3D beam elements to model yarns in a textile composite preform. For this purpose, the third step is introducing contact behavior and carrying out validation for new 3D beam to beam contact with rectangular cross section. The contact formulation is derived on the basis of Penalty Formulation and Updated Lagrangian formulation using physical shape functions with shear effect included. An effective contact search algorithm is elaborated. And a consistent linearization of contact contribution is derived and expressed in suitable matrix form, which is easy to use in FEM approximation. Finally, some numerical examples are presented which are only qualitative analysis of contact and checking the correctness and the effectiveness of the proposed 3D beam element.

Les stratifiés composites sont de plus en plus utilisés dans les pièces de structures des aéronefs ce qui fait émerger de nouvelles problématiques comme celle des Impacts de Faible Energie (IFE). En effet, bien qu’ils possèdent des propriétés rapportées à leur masse très intéressantes ces matériaux peuvent être vulnérables aux petits chocs. Or, compte tenu des nombreux paramètres influents lors d’un tel impact (énergie, vitesse, stratification...), les essais actuellement majoritairement privilégiés à l’échelle industrielle sont long et coûteux. Ainsi, l’apport de la simulation numérique pourrait être d’une grande aide pour les constructeurs. La pratique du « virtual testing », en particulier, permettrait d’aller dans cette direction ce qui aurait pour effet de rationaliser les campagnes d’essais et les coûts financier qui en découlent. Cependant, elle peine à être mise en place ici car le temps CPU nécessaire pour la simulation fine des ndommagements induits par les IFE est trop important avec les méthodes actuelles. Partant de ce constat, ce travail a consisté à tirer avantageusement partie de la localisation spatiale et temporelle des délaminages, fissurations matricielles et ruptures de fibres qui peuvent apparaître pendant l’impact pour diminuer le coût de calcul. Ainsi une méthode multiéchelle en espace et en temps a été mise en place. Elle consiste à découper la structure impactée en deux zones. L’une est située autour du point d’impact, elle contient l’ensemble des non-régularités du problème (contact, loi adoucissante, modèle de zone cohésive). Elle est traitée avec le code de dynamique explicite Europlexus. L’autre correspond à la partie complémentaire. Le problème mécanique y est beaucoup plus régulier et il est traité avec le code de dynamique implicite Zset/Zébulon. Un couplage peu intrusif basé sur la méthode GC est donc réalisé entre ces deux codes. Il permet d’utiliser une modélisation adaptée dans chacune des deux régions ce qui permet en particulier d’utiliser des pas de temps différents. Un rapport supérieur à 1000 peut ainsi être obtenu entre celui du code explicite fixé par la condition de stabilité et celui utilisé dans la partie complémentaire. Un gain de temps CPU significatif confirmé par la simulation d’un impact réalisé sur un panneau composite raidi est ainsi obtenu. Il est également montré que la répartition implicite/explicite peut évoluer au cours du calcul. Pour cela un mécanisme de bascule a été mis en place. Il permet ainsi de faire transiter la résolution d’une partie de la structure initialement traitée dans le code Zebulon dans Europlexus. Un gain de temps supplémentaire est alors obtenu grâce à cette méthode sur le même cas d’application. / The composite laminates are increasingly used in aircraft structural parts which lead to new issues such as the Low Energy Impacts (LEI). Indeed, although they have well mechanical properties relative to their mass, small shocks may be very harmfull for laminates. Controlling such situations is essential for manufacturers that why lot of testing campaigns are currently performed. Yet, they are time consuming and expensive considering the many influential parameters (energy, speed, layup...). Numerical simulations of this phenomenon by practicing the so called “virtual testing” process could be really helpfull to rationalize testing campaigns in order to save money. Yet, this practice remain currently hard to do at the industrial scale due to the excessive CPU time required for fine simulation of damages induced by the LEI. Based on this observation, this work has consisted in taking advantage of the spatial and temporal location of delamination, matrix cracking and fiber breakage that can occur during impact in order to reduce the computational cost. Thus, a space and time multiscale method has been put in place. The impacted structure is split into two areas. One is located around the impacted point, it contains all the non-regularities of the problem (contact, softening law, cohesive zone model). This domain is treated with the explicit dynamics code Europlexus. The other one corresponds to the complementary part. The mechanical problem is much more regular and it is treated with the implicit dynamics code Zset / Zebulon. A low intrusive coupling based on the GC method is carried out between these two codes. It allows to use an adapted model in both regions different time step are in particular used. A time step ratio upper to 1000 can be reach between the one of the explicit code set by the stability condition and the one used in the complementary part. As a results, significant CPU time is saved. This is confirmed by the simulation of a stiffened composite panel impacted. It is also shown that the implicit / explicit allocation can change over the calculation. To do that, a switch mechanism has been established. It thus makes it possible to transit the resolution of a portion of the structure initially solved in the code Zebulon to Europlexus. As a results, further gain is obtained.

La thèse présente une méthodologie pour construire des éléments finis de type « solide-coque » avec intégration réduite en vue des applications à la simulation de la mise en forme tel que l’emboutissage des tôles où ces éléments finis doivent présenter de bonnes aptitudes à modéliser la flexion mais également les situations de laminage de la tôle. A partir des éléments volumiques à 8 nœuds et 3 degrés de liberté par nœud (les 3 composantes du déplacement), un neuvième nœud est rajouté au centre de l’élément. Ce neuvième nœud n’est pourvu que d’un seul degré de liberté, le déplacement le long de la direction de l’épaisseur. Cette direction privilégiée a un nombre de points d’intégration supérieur ou égal à 3 mais l’intégration est réduite au centre de l’élément diminuant très sensiblement les temps CPU par rapport à une intégration complète. Un soin particulier a été pris pour contrôler tous les modes à énergie nulle dus à l’intégration réduite. Ce nœud supplémentaire permet une distribution linéaire de la déformation normale. Avec les lois de comportement complètement 3D ces nouveaux éléments solide-coque donnent des résultats similaires en flexion à ceux obtenus avec des éléments coques et état plan de contrainte. Le neuvième nœud joue le rôle d’un paramètre supplémentaire pour l’interpolation quadratique du déplacement dans la direction de l’épaisseur. Ce degré de liberté a une signification physique et un effort équivalent à une pression normale peut être prescrit. Dans les situations de pression normale et dans le cas du contact, la contrainte normale obtenue est physique ce qui n’est pas le cas de nombreux éléments solide-coque de la littérature. Le pincement ou le laminage des tôles est correctement modélisé. Pour valider ces éléments, un module d’emboutissage en U avec passage et laminage de la bande de tôle sur des rouleaux a été construit au laboratoire. La comparaison entre les efforts d’emboutissage calculés et mesurés est très bonne ainsi que la géométrie des bandes de tôle obtenue après retour élastique. / This thesis presents a methodology for developing under-integrated “solid-shell” finite elements for sheet forming simulations like deep drawing where these elements must offer a bending capability and sheet thinning conditions as well. Starting from 8-node elements endowing three degrees of freedom per node (three displacement components), a ninth node is added at the centre of the element. This extra node has just one degree of freedom: a displacement along the ‘thickness’ direction. Several integration points are distributed along this privileged direction (5 points, generally) but the in-plane reduced integration at the centre of the element decreases CPU costs compared to a full integration. A special care has been taken to control all zero-energy modes due to the reduced integration. This additional node allows a linear distribution of the normal strain. With fully-3D constitutive laws, these new solid-shell elements give similar bending results as those obtained with shell elements and a plane stress state hypothesis. This ninth node acts as an additional parameter for the quadratic interpolation of the displacement in the ‘thickness’ direction. The corresponding degree of freedom has a physical meaning and a force, equivalent to a normal pressure for instance, may be prescribed. In situations of a normal pressure and in the case of contact, the obtained normal stress is physically defined, which is not the case for many solid-shell elements found in the literature. The pinching (or the thinning) of sheets is properly modelled. To validate these elements, an apparatus for U-drawing tests with ironing or thinning on strip sheets has been built in the laboratory. The comparison between numerical and experimental punch force during sheet forming is pretty good as well as the geometry of blank after springback.

La compréhension du comportement de structures jusqu'à leur ruine est nécessaire pour concevoir au mieux ces structures. Selon le matériau et les sollicitations considérées, les mécanismes physiques à l'origine de la rupture changent. Nous nous intéresserons à des matériaux homogènes pour lesquels la ruine passe par le développement de fissures autour desquelles les non-linéarités de comportement n'ont pas un rôle dominant. Ces conditions sont réunies pour les matériaux fragiles pour lesquels la source principale de dissipation est la génération non réversible d'une surface libre, et pour certaines fissures de fatigue. Sur un cycle de chargement, il existe de nombreuses applications pour lesquelles les non-linéarités restent confinées. La théorie de la mécanique linéaire élastique de la rupture est alors un modèle pertinent pour approcher le comportement de la structure. Sous ces hypothèses, le front de la fissure introduit une singularité. L'étude asymptotique de cette singularité dans des situations plane et anti-plane permet de définir les séries de Williams. La singularité est alors d'ordre un demi et elle est quantifiée par les facteurs d'intensité des contraintes (FIC) pour chacun des trois modes de sollicitations. En 3D, la fissure peut avoir une géométrie complexe, et aucune expression générale de la singularité n'existe. Dans cette thèse, les séries de Williams en déplacements sont utilisées et régularisées le long du front au sens des éléments finis. À partir de cette définition 3D des séries asymptotiques en pointe de fissure, une méthode d'extraction directe des FIC (DEK-FEM) est étendue au cas 3D. Le domaine est décomposé en deux domaines, raccordés en moyenne sur l'interface. Au voisinage du front, les champs mécaniques sont approchés par une troncature des champs asymptotiques. La singularité est donc traitée avec des champs adaptés, et les degrés de liberté associés sont directement les coefficients asymptotiques. Parmi ces coefficients asymptotiques, on retrouve les FIC et les T-stresses. Pour des raisons d'efficacité numérique et pour pouvoir relier l'échelle de la structure à l'échelle de la fissure, cette méthode est intégrée dans un contexte multigrilles localisées X-FEM. Ainsi nous montrons que cette approche permet une bonne évaluation des évolutions des FIC et du T-stress. Cette méthode est développée en parallèle d'une stratégie de post-traitement expérimental (mesure de champs de déplacements par corrélation d'images) basée sur les mêmes séries asymptotiques. Les images tridimensionnels d'un essai de fatigue in situ sont obtenues par micro-tomographie à rayons X et reconstruction. La corrélation et la régularisation basées sur les séries asymptotiques permettent d'obtenir la géométrie de la fissure et les FIC pour pouvoir identifier des lois de propagation de fissures 3D en fatigue. L'efficacité de cette méthode en parallèle d'une simulation DEK-FEM est illustrée en 2D. / It is necessary to understand the behavior of structures up to their failure to enhance their design. The mechanisms and phenomena undergoing failure vary according to the considered material and boundary conditions. We consider homogeneous materials for which cracks propagate in a context where behavior nonlinearities are not dominants. These conditions are matched for brittle and quasi-brittle materials and for some fatigue cracks. For the former, the main source of dissipation is the crack propagation which can be seen as the generation of a new free-surface. For the later, there is many applications where, in one loading cycle, the nonlinearities remains confined around the crack tip. The linear elastic fracture mechanics theory is then a pertinent model to approximate the structure behavior. Under such hypotheses, a singularity appears in the crack tip vicinity. The Williams' series expansion is computed from the asymptotic study of plane and anti-plane states. The stress is singular at the crack tip and the order of this singularity is one out of two. The singularity amplitude is quantified by the stress intensity factors (SIF), one for each of the three loading modes. In 3D, the crack shape is potentially complex (front curvature and non-planar crack), and no general asymptotic series expansion exists. In this PhD thesis, the 2D Williams' series in displacements are used and regularized with a finite element evolution along the front. From this 3D definition of the asymptotic fields in the crack tip vicinity, a numerical method for direct estimation of the SIF (DEK-FEM) is extended to 3D. This method is based on domain decomposition, the two domains are bounded in a weak sense on their interface. In the crack tip vicinity, the mechanical fields are approximated by a truncation of the asymptotic series expansion. Therefore, appropriate fields are used to deal with the singularity, and the associated degrees of freedom are directly the asymptotic coefficients. Among these coefficients are the SIF and the T-stresses. To bridge the scales between the structure and the crack front singularity and to increase the numerical efficiency, this method is embedded in a localized X-FEM multigrids approach. The proposed method is shown to provide an accurate evaluation of the SIF and T-stresses evolution. This approach has been developed in combination of an experimental post-processing method (full field displacement measurement through image correlation) based on the same asymptotic series expansion. The 3D images can be obtained for in situ fatigue experiments by X-ray microtomography and reconstruction. The crack geometry and the SIF are then provided by image correlation and regularization based on Williams series expansion. These data can be used for identifying a 3D fatigue crack growth law. The efficiency of the method is illustrated in 2D.

Mechanics of structure genome (MSG) is a unified homogenization theory that provides constitutive modeling of three-dimensional (3D) continua, beams and plates. In present work, the author extends the MSG to study the buckling of structures such as stiffened and sandwich panels. Such structures are usually slender or flat and easily buckle under compressive loads or bending moments which may result in catastrophic failure.<div><br><div>Buckling studies of stiffened and sandwich panels are found to be scattered. Most of the existed theories employ unnecessary assumptions or only apply to certain types of structures. There are few unified approaches that are capable of studying the buckling of different kinds of structures altogether. The main improvements of current approach compared with other methods in the literature are avoiding unnecessary assumptions, the capability of predicting all possible buckling modes including the global and local buckling modes, and the potential in studying the buckling of various types of structures.<br></div><div><br></div><div>For global buckling that features small local rotations, MSG mathematically decouples the 3D geometrical nonlinear problem into a linear constitutive modeling using structure genome (SG) and a geometrical nonlinear problem defined in a macroscopic structure. As a result, the original structures are simplified as macroscopic structures such as beams, plates or continua with effective properties, and the global buckling modes are predicted on macroscopic structures. For local buckling that features finite local rotations, Green strain is introduced into the MSG theory to achieve geometrically nonlinear constitutive modeling. Newton’s method is used to solve the nonlinear equilibrium equations for fluctuating functions. To find the bifurcated fluctuating functions, the fluctuating functions are then perturbed under the Bloch-periodic boundary conditions. The bifurcation is found when the tangent stiffness associated with the perturbed fluctuating functions becomes singular. Moreover, the arc-length method is introduced to solve the nonlinear equilibrium equations for post-local-buckling predictions because of its robustness. The imperfection is included in the form of geometrical imperfection by superimposing the scaled buckling modes in linear perturbation analysis on mesh.<br></div><div><br></div><div>Extensive validation case studies are carried out to assess the accuracy of the MSG theory in global buckling analysis and post-global-buckling analysis, and assess the accuracy of the extended MSG theory in local buckling and post-local-buckling analysis. Results using MSG theory and extended MSG theory in buckling analysis are compared with direct numerical solutions such as 3D FEA results and results in literature. Parametric studies are performed to reveal the relative influence of selective geometric parameters on buckling behaviors. The extended MSG theory is also compared with representative volume element (RVE) analysis with Bloch-periodic boundary conditions using commercial finite element packages such as Abaqus to assess the efficiency and accuracy of the present approach.<br></div></div>