A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements

Geometric partial differential equations: Surface and bulk processes

The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems

A theoretical and computational study of the mechanics of biomembranes at multiple scales

Lipid membranes are thin objects that form the main separation structure in cells. They have remarkable mechanical properties; while behaving as a solid shell against bending, they exhibit in-plane fluidity. These two aspects of their mechanics are not only interesting from a physical viewpoint, but also fundamental for their biological function. Indeed, the equilibrium shapes of different organelles in the cell rely on the bending elasticity of lipid membranes. On the other hand, the in-plane fluidity of the membrane is essential in functions such as cell motility, mechano-adaptation, or for the lateral diffusion of proteins and other membrane inclusions. The bending rigidity of membranes can be motivated from microscopic models that account for the stress distribution across the membrane thickness. In particular, the microscopic stress across the membrane is routinely computed from molecular dynamics simulations to investigate how different microscopic features, such as the addition of anesthetics or cholesterol, affect their effective mechanical response. The microscopic stress bridges the gap between the statistical mechanics of a set of point particles, the atoms in a molecular dynamics simulation, and continuum mechanics models. However, we lack an unambiguous definition of the microscopic stress, and different definitions of the microscopic stress suggest different connections between molecular and continuum models. In the first Part of this Thesis, we show that many of the existing definitions of the microscopic stress do not satisfy the most basic balance laws of continuum mechanics, and thus are not physically meaningful. This striking issue has motivated us to propose a new definition of the microscopic stress that complies with these fundamental balance laws. Furthermore, we provide a freely available implementation of our stress definition that can be computed from molecular dynamics simulations (mdstress.org). Our definition of the stress along with our implementation provides a foundation for a meaningful analysis of molecular dynamics simulations from a continuum viewpoint. In addition to lipid membranes, we show the application of our methodology to other important systems, such as defective crystals or fibrous proteins. In the second part of the Thesis, we focus on the continuum modeling of lipid membranes. Because these membranes are continuously brought out-of-equilibrium by biological activity, it is important to go beyond curvature elasticity and describe the internal mechanisms associated with bilayer fluidity. We develop a three-dimensional and non-linear theory and a simulation methodology for the mechanics of lipid membranes, which have been lacking in the field. We base our approach on a general framework for the mechanics of dissipative systems, Onsager's variational principle, and on a careful formulation of the kinematics and balance principles for fluid surfaces. For the simulation of our models, we follow a finite element approach that, however, requires of unconventional dicretization methods due to the non-linear coupling between shape changes and tangent flows on fluid surfaces. Our formulation provides the basis for further investigations of the out-of-equilibrium chemo-mechanics of lipid membranes and other fluid surfaces, such as the cell cortex.Las membranas lipídicas son estructuras delgadas que forman la separación fundamental de las células. Tienen propiedades físicas notables: mientras que se comportan como láminas delgadas sólidas frente a curvatura, presentan fluidez interfacial. Estos dos aspectos de su mecánica son interesantes desde un punto de vista físico e ingenieril, pero además son fundamentales para su función biológica. Las formas de equilibrio de diferentes organelos celulares dependen de la elasticidad frente a curvatura de la membrana lipídica. Por otro lado, la fluidez interfacial es esencial en funciones como la movilidad celular, la adaptación mecánica a deformaciones, o para la difusión lateral de proteínas. La elasticidad frente a curvatura de las membranas lipídicas puede motivarse a través de modelos microscópicos que tienen en cuenta la distribución de esfuerzos a lo largo del espesor de la membrana. En particular, el tensor de esfuerzos microscópico se calcula habitualmente en simulaciones de dinámica molecular a lo largo del espesor de la membrana para investigar cómo diferentes características microscópicas, como la adición de anestésicos o colesterol, afecta la respuesta mecánica efectiva. El tensor de esfuerzos microscópico tiende un puente entre la mecánica estadística de un conjunto de partículas puntuales, los átomos de una simulación de dinámica molecular, y modelos de mecánica de medios continuos. Sin embargo, no disponemos de una definición única del tensor de esfuerzos microscópico, y diferentes definiciones dan lugar a diferentes interpretaciones de la conexión entre modelos moleculares y continuos. En la primera parte de la tesis, mostramos que muchas de las definiciones del tensor de esfuerzos microscópico no satisfacen las leyes más básicas de la mecánica de medios continuos, y por tanto no son físicamente relevantes. Este problema nos ha motivado a proponer una nueva definición del tensor de esfuerzos microscópicos que cumpla las leyes fundamentales de la mecánica de medios continuos por construcción. Además, hemos desarrollado (y puesto a disposición del público libremente) una implementación numérica de nuestra definición del tensor de esfuerzos microscópico que puede calcularse mediante simulaciones de dinámica molecular (mdstress.org). Nuestra definición del tensor de esfuerzos, así como nuestra implementación del mismo, proporcionan una base sólida para el análisis de simulaciones de dinámica molecular desde un punto de vista continuo. Además de membranas lipídicas, mostramos la aplicación de nuestro método en otros sistemas relevantes, como cristales con defectos o proteínas fibrosas. En la segunda parte de esta tesis nos hemos focalizado en el modelado continuo de membranas lipídicas. Ya que estas membranas están constantemente sufriendo actividad biológica que las lleva fuera de equilibrio, es importante tener en cuenta no sólo la elasticidad de curvatura, sino también los grados de libertad internos asociados a la fluidez de la membrana. Para ello, desarrollamos un nuevo marco teórico y computacional general, tridimensional y no-lineal, para la mecánica de membranas lipídicas. Nuestro enfoque se basa en un marco general para la mecánica de sistemas disipativos, el principio variacional de Onsager, y en una formulación cuidadosa de la cinemática y las ecuaciones de balance para superficies fluídas. Para la simulación de nuestros modelos, seguimos una aproximación basada en elementos finitos que, sin embargo, requiere de métodos no convencionales debido al acoplamiento no-lineal entre cambios de forma y los campos de velocidad tangentes en superficies fluídas. Nuestra formulación proporciona la base para futuras investigaciones de la quimiomecánica fuera de equilibrio de membranas lipídicas y otras superficies fluídas, como el cortex celula

Modeling and Programming Shape-Morphing Structured Media

Shape-morphing and self-propelled locomotion are examples of mechanical behaviors that can be "programmed" in structured media by designing geometric features at micro- and mesostructural length scales. This programmability is possible because the small-scale geometry often imposes local kinematic modes that are strongly favored over other deformations. In turn, global behaviors are influenced by local kinematic preferences over the extent of the structured medium and by the kinematic compatibility (or incompatibility) between neighboring regions of the domain. This considerably expands the design space for effective mechanical properties, since objects made of the same bulk material but with different internal geometry will generally display very different behaviors. This motivates pursuing a mechanistic understanding of the connection between small-scale geometry and global kinematic behaviors. This thesis addresses challenges pertaining to the modeling and design of structured media that undergo large deformations. The first part of the thesis focuses on the relation between micro- or mesoscale patterning and energetically favored modes of deformation. This is first discussed within the context of twisted bulk metallic glass ribbons whose edges display periodic undulations. The undulations cause twist concentrations in the narrower regions of the structural element, delaying the onset of material failure and permitting the design of structures whose deployment and compaction emerge from the ribbons' chirality. Following this discussion of a periodic system, we study sheets with non-uniform cut patterns that buckle out-of-plane. Motivated by computational challenges associated with the presence of geometric features at disparate length scales, we construct an effective continuum model for these non-periodic systems, allowing us to simulate their post-buckling behavior efficiently and with good accuracy. The second part of the thesis discusses ways to leverage the connection between micro/mesoscale geometry and energetically favorable local kinematics to create "programmable matter" that undergo prescribed shape changes or self-propelled locomotion when exposed to an environmental stimulus. We first demonstrate the capabilities of an inverse design method that automates the design of structured plates that morph into target 3D geometries over time-dependent actuation paths. Finally, we present devices made of 3D-printed liquid crystal elastomer (LCE) hinges that change shape and self-propel when heated.</p