Subdivision Shell Elements with Anisotropic Growth

A thin shell finite element approach based on Loop's subdivision surfaces is proposed, capable of dealing with large deformations and anisotropic growth. To this end, the Kirchhoff-Love theory of thin shells is derived and extended to allow for arbitrary in-plane growth. The simplicity and computational efficiency of the subdivision thin shell elements is outstanding, which is demonstrated on a few standard loading benchmarks. With this powerful tool at hand, we demonstrate the broad range of possible applications by numerical solution of several growth scenarios, ranging from the uniform growth of a sphere, to boundary instabilities induced by large anisotropic growth. Finally, it is shown that the problem of a slowly and uniformly growing sheet confined in a fixed hollow sphere is equivalent to the inverse process where a sheet of fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless, quasi-static, elastic limit.Comment: 20 pages, 12 figures, 1 tabl

Compaction of Quasi One-Dimensional Elastoplastic Materials

Insight in the crumpling or compaction of one-dimensional objects is of great importance for understanding biopolymer packaging and designing innovative technological devices. By compacting various types of wires in rigid confinements and characterizing the morphology of the resulting crumpled structures, here we report how friction, plasticity, and torsion enhance disorder, leading to a transition from coiled to folded morphologies. In the latter case, where folding dominates the crumpling process, we find that reducing the relative wire thickness counter-intuitively causes the maximum packing density to decrease. The segment-size distribution gradually becomes more asymmetric during compaction, reflecting an increase of spatial correlations. We introduce a self-avoiding random walk model and verify that the cumulative injected wire length follows a universal dependence on segment size, allowing for the prediction of the efficiency of compaction as a function of material properties, container size, and injection force.Comment: 7 pages, 6 figure

Scaling of the buckling transition of ridges in thin sheets

When a thin elastic sheet crumples, the elastic energy condenses into a network of folding lines and point vertices. These folds and vertices have elastic energy densities much greater than the surrounding areas, and most of the work required to crumple the sheet is consumed in breaking the folding lines or ``ridges''. To understand crumpling it is then necessary to understand the strength of ridges. In this work, we consider the buckling of a single ridge under the action of inward forcing applied at its ends. We demonstrate a simple scaling relation for the response of the ridge to the force prior to buckling. We also show that the buckling instability depends only on the ratio of strain along the ridge to curvature across it. Numerically, we find for a wide range of boundary conditions that ridges buckle when our forcing has increased their elastic energy by 20% over their resting state value. We also observe a correlation between neighbor interactions and the location of initial buckling. Analytic arguments and numerical simulations are employed to prove these results. Implications for the strength of ridges as structural elements are discussed.Comment: 42 pages, latex, doctoral dissertation, to be submitted to Phys Rev

A model for the fragmentation kinetics of crumpled thin sheets

As a confined thin sheet crumples, it spontaneously segments into flat facets delimited by a network of ridges. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Experiments have shown that the total crease length accrues logarithmically when repeatedly compacting and unfolding a sheet of paper. Here, we offer insight to this unexpected result by exploring the correspondence between crumpling and fragmentation processes. We identify a physical model for the evolution of facet area and ridge length distributions of crumpled sheets, and propose a mechanism for re-fragmentation driven by geometric frustration. This mechanism establishes a feedback loop in which the facet size distribution informs the subsequent rate of fragmentation under repeated confinement, thereby producing a new size distribution. We then demonstrate the capacity of this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon.Comment: 11 pages, 7 figures (+ Supplemental Materials: 15 pages, 9 figures); introduced a simpler approximation to model, key results unchanged; added references, expanded supplementary information, corrected Fig. 2 and revised Figs. 4 and 7 for clearer presentation of result

Atomistic continuum simulations for nano-indentation and compression of multi-layer graphene

Graphene has attracted a great share of research interest due to its extraordinary electrical, thermal, mechanical, and physical properties. Such spectacular properties of graphene open a wide range potential of applications in electronics, energy storage, composites, and biomedical fields. The mechanical properties of graphene can have a huge impact on its performance in graphene-based devices and thus it is important to study them. But the difficulties in experimental characterization and computational limitations to simulate large graphene sample consisting of billions of atoms makes it a challenging task. Thus, accurate and efficient simulation tools to predict the complex deformation of large graphene samples are needed but are still elusive. The objective of this thesis is to utilize the atomistic-continuum foliation (AC) model developed by Ghosh and Arroyo (2013) and modified by Upendra Yadav, to reproduce the Nano-indentation experiments accurately. This atomistic - continuum foliation (AC) model enables one to directly reproduce the experimental results, something that was not possible before. Using this model we can study the effects of different variables like adhesion, frictional force, indenter radius and stress concentration which is not possible to obtain from the experiments. This thesis also includes the simulation results for different surface morphology like wrinkles and ripples generated on the multi-layer graphene samples under uniaxial and biaxial compression for different van der Waals potential, which paves a way to study the multifunctionality and control of crumpling and unfolding of graphene to enhance its performance in graphene-based devices

Rim curvature anomaly in thin conical sheets revisited

This paper revisits one of the puzzling behaviors in a developable cone (d-cone), the shape obtained by pushing a thin sheet into a circular container of radius $R$ by a distance $\eta$ [E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999)]. The mean curvature was reported to vanish at the rim where the d-cone is supported [T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006)]. We investigate the ratio of the two principal curvatures versus sheet thickness $h$ over a wider dynamic range than was used previously, holding $R$ and $\eta$ fixed. Instead of tending towards 1 as suggested by previous work, the ratio scales as $(h/R)^{1/3}$. Thus the mean curvature does not vanish for very thin sheets as previously claimed. Moreover, we find that the normalized rim profile of radial curvature in a d-cone is identical to that in a "c-cone" which is made by pushing a regular cone into a circular container. In both c-cones and d-cones, the ratio of the principal curvatures at the rim scales as $(R/h)^{5/2}F/(YR^{2})$, where $F$ is the pushing force and $Y$ is the Young's modulus. Scaling arguments and analytical solutions confirm the numerical results.Comment: 25 pages, 12 figures. Added references. Corrected typos. Results unchange

Discrete Differential Geometry of Thin Materials for Computational Mechanics

Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structure-respecting discrete-differential-geometric (DDG) approach often leads to new algorithms that more accurately track the physically behavior of the system with less computational effort. Thin objects, such as pieces of cloth, paper, sheet metal, freeform masonry, and steel-glass structures are particularly rich in geometric structure and so are well-suited for DDG. I show how understanding the geometry of time integration and contact leads to new algorithms, with strong correctness guarantees, for simulating thin elastic objects in contact; how the performance of these algorithms can be dramatically improved without harming the geometric structure, and thus the guarantees, of the original formulation; how the geometry of static equilibrium can be used to efficiently solve design problems related to masonry or glass buildings; and how discrete developable surfaces can be used to model thin sheets undergoing isometric deformation

Non-smooth developable geometry for interactively animating paper crumpling

International audienceWe present the first method to animate sheets of paper at interactive rates, while automatically generating a plausible set of sharp features when the sheet is crumpled. The key idea is to interleave standard physically-based simulation steps with procedural generation of a piecewise continuous developable surface. The resulting hybrid surface model captures new singular points dynamically appearing during the crumpling process, mimicking the effect of paper fiber fracture. Although the model evolves over time to take these irreversible damages into account, the mesh used for simulation is kept coarse throughout the animation, leading to efficient computations. Meanwhile, the geometric layer ensures that the surface stays almost isometric to its original 2D pattern. We validate our model through measurements and visual comparison with real paper manipulation, and show results on a variety of crumpled paper configurations

Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh

A method of combining 1-d and 2-d structural finite elements to capture the fundamental mechanical properties of engineering fabrics subject to finite strains is introduced. A mutually constrained pantographic beam and membrane mesh is presented and simple homogenisation theory is developed to relate the macro-scale properties of the mesh to the properties of the elements within the mesh. The theory shows that each of the macro-scale properties of the mesh can be independently controlled. An investigation into the performance of the technique is conducted using tensile, cantilever bending and uniaxial bias extension shear simulations. The simulations are first used to verify the accuracy of the homogenisation theory and then used to demonstrate the ability of the modelling approach in accurately predicting the shear force, shear kinematics and out-of-plane wrinkling behaviour of engineering fabrics