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Liquid crystal elastomers are cross-linked polymer networks covalently bonded with liquid crystal mesogens. In the nematic phase, due to strong coupling between mechanical strain and orientational order, these materials display strain-induced instabilities associated with formation and evolution of orientational domains. In building a simulation model of these materials, we consider the limit in which the orientational order equilibrates rapidly compared to the strain, so that the local order tensor remains in continuously evolving quasi- static equilibrium as the strain relaxes. Our method allows us to study the onset of stripe formation in a monodomain film stretched along an axis perpendicular to the nematic director, the transition from polydomain to monodomain states, and the interaction of nematic liquid crystal elastomers with external stimuli such as an electric field. We intend through this model to further our understanding of the basic physics governing the dynamic mechanical response of nematic elastomers and also provide a useful computational tool for design and testing of potential engineering device applications.
We investigate two soft matter systems that display novel behaviors when driven out of equilibrium by internal stresses, fueled by energy derived from the environment. First, we model shape-morphing dynamics of liquid crystal elastomers. We investigate photoactuation in thin polymer films that exhibit continuous, directional, macroscopic mechanical waves under constant light illumination. These polymer materials deform mechanically in response to any stimulus that modifies the strength of their nematic order. Doping the polymer with an azobenzene derivative enables the material to actuate in response to light. The trajectory of mechanical response can be controlled by patterning the orientation of the nematic director field during cross-linking, a process known as "blueprinting", which defines the local axis of induced contraction. We model the mechanics of photo-actuation via a Hamiltonian-based nonlinear finite element elastodynamics simulation. We find that the underlying mechanism enabling continuous wave generation is a feedback loop driven by self-shadowing, along with coupling between the nematic order and illumination. The model further demonstrates the mechanism by which wave direction and propagation speed depend on the blueprinted director pattern. These results explain experimental observations by our collaborators, who exploited this mechanical wave generation effect to produce robotic devices that undergo autonomous light-powered locomotion. Second, we carry out computer simulation studies of pattern formation in an active nematic fluid composed of flexible filaments, modeled as a thin layer of bead-spring polymer chains with active driving forces and thermal noise. We consider filaments that self-propel, representing e.g. gliding motility of filamentous bacteria on a smooth or bumpy surface. We investigate phase behavior and diffusive transport as a function of filament density and bending modulus, and demonstrate that Gaussian curvature of the substrate controls filament density. Next, we consider an active nematic fluid with extensile interactions, representing e.g. microtubules that slide against their neighbors driven by kinesin molecular motors. We analyze emergent non-equilibrium dynamics and microstructural evolution as a function of filament bending modulus and the magnitude of active forces. Analysis of the resulting flow includes tracking nucleation, motion, and annihilation of topological defects. Results from this filament-based model are compared to related experiments and continuum models in the literature. The projects both examine soft matter systems exhibiting emergent phenomena when driven from equilibrium via couplings to their environment. In building simulation codes for both projects, we utilize highly parallelized GPU (Graphics Processing Unit) implementations that decrease the runtime and increase the achievable length scale and duration of simulation studies. Computational models are important tools in understanding and ex- plaining the fundamental mechanisms of complex systems, and have potential for future use in design and optimization of smart material applications.
Liquid crystal elastomers (LCEs) are a class of polymer networks which involve the incorporation of liquid crystal (LC) molecules into their polymer backbone or side chain. This results in anisotropy in their mechanical, optical, and electromagnetic properties similar to those exhibited by traditional LC materials. Their mechanical properties are highly coupled to the internal state of LC order, which can result in large mechanical deformations as LC order changes. This can occur in response to a variety of external stimuli such as changes in temperature, exposure to light, and application of external fields. The interplay between LC order and mechanical properties makes LCEs a highly promising class of functional materials and subsequently, they have been the subject of much research over the past several decades. However, developing an application of LCEs remains difficult in that their mechanical response is both complex and coupled to the state of liquid crystal order prior to cross-linking. Their physics are sufficiently complicated that in most cases, the use of pen-and-paper analysis is precluded. Additionally, the LCE fabrication process is complex and expensive, making trial-and-error experimental design methods unsuitable. This motivates the development and use of simulation-based methods to augment traditional experimental design methods. The two main contributors to the complexity of the design of LCE applications are the choice and imposition of liquid crystal order, or "texture", prior to cross-linking. In this work, simulation-based methods are developed and partially validated for use in applications-focused design of temperature-responsive nematic LCEs. These methods enable the simulation of LCEs of macroscopic size and of non-trivial geometry through the use of continuum mechanics and suitable numerical methods (the finite element method). LC texture is an input parameter in the presented method, allowing many choices of texture to be explored at low cost given that the textures are physically accessible. In addition to methods development and validation results, proof-of-concept simulation-based design studies were performed for two types of LCE-based actuators that are of current interest in the field: grippers and hinge mechanisms. Finally, preliminary results are presented resulting from the integration of nematic texture dynamics simulation (pre-cross-linking) and LCE mechanical simulations (post-cross-linking) which address the two main sources of complexity in the design process of LCE functional materials.
This text is a primer for liquid crystals, polymers, rubber and elasticity. It is directed at physicists, chemists, material scientists, engineers and applied mathematicians at the graduate student level and beyond.
This Encyclopedia covers the entire science of continuum mechanics including the mechanics of materials and fluids. The encyclopedia comprises mathematical definitions for continuum mechanical modeling, fundamental physical concepts, mechanical modeling methodology, numerical approaches and many fundamental applications. The modelling and analytical techniques are powerful tools in mechanical civil and areospsace engineering, plus in related fields of plasticity, viscoelasticity and rheology. Tensor-based and reference-frame-independent, continuum mechanics has recently found applications in geophysics and materials.
The derivation and understanding of Partial Differential Equations relies heavily on the fundamental knowledge of the first years of scientific education, i.e., higher mathematics, physics, materials science, applied mechanics, design, and programming skills. Thus, it is a challenging topic for prospective engineers and scientists. This volume provides a compact overview on the classical Partial Differential Equations of structural members in mechanics. It offers a formal way to uniformly describe these equations. All derivations follow a common approach: the three fundamental equations of continuum mechanics, i.e., the kinematics equation, the constitutive equation, and the equilibrium equation, are combined to construct the partial differential equations.
A multidisciplinary perspective on the dynamic processes occurring in Earth's mantle The convective motion of material in Earth's mantle, powered by heat from the deep interior of our planet, drives plate tectonics at the surface, generating earthquakes and volcanic activity. It shapes our familiar surface landscapes, and also stabilizes the oceans and atmosphere on geologic timescales. Mantle Convection and Surface Expressions brings together perspectives from observational geophysics, numerical modelling, geochemistry, and mineral physics to build a holistic picture of the deep Earth. It explores the dynamic processes occurring in the mantle as well as the associated heat and material cycles. Volume highlights include: Perspectives from different scientific disciplines with an emphasis on exploring synergies Current state of the mantle, its physical properties, compositional structure, and dynamic evolution Transport of heat and material through the mantle as constrained by geophysical observations, geochemical data and geodynamic model predictions Surface expressions of mantle dynamics and its control on planetary evolution and habitability The American Geophysical Union promotes discovery in Earth and space science for the benefit of humanity. Its publications disseminate scientific knowledge and provide resources for researchers, students, and professionals.
Comprehensive introduction to nonlinear elasticity for graduates and researchers, covering new developments in the field.
This monograph is centered on mathematical modeling, innovative numerical algorithms and adaptive concepts to deal with fracture phenomena in multiphysics. State-of-the-art phase-field fracture models are complemented with prototype explanations and rigorous numerical analysis. These developments are embedded into a carefully designed balance between scientific computing aspects and numerical modeling of nonstationary coupled variational inequality systems. Therein, a focus is on nonlinear solvers, goal-oriented error estimation, predictor-corrector adaptivity, and interface conditions. Engineering applications show the potential for tackling practical problems within the fields of solid mechanics, porous media, and fluidstructure interaction.
This book gathers outstanding papers presented at the European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2019). The conference was organized by Delft University of Technology and was held in Egmond aan Zee, the Netherlands, from September 30 to October 4, 2019. Leading experts in the field presented the latest results and ideas regarding the design, implementation and analysis of numerical algorithms, as well as their applications to relevant societal problems. ENUMATH is a series of conferences held every two years to provide a forum for discussing basic aspects and new trends in numerical mathematics and scientific and industrial applications, all examined at the highest level of international expertise. The first ENUMATH was held in Paris in 1995, with successive installments at various sites across Europe, including Heidelberg (1997), Jyvaskyla (1999), lschia Porto (2001), Prague (2003), Santiago de Compostela (2005), Graz (2007), Uppsala (2009), Leicester (2011), Lausanne (2013), Ankara (2015) and Bergen (2017).