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Topological surgery forms different shapes by gluing the sides with the same notation and direction.
Source publication
Boundary conditions are critical to the partial differential equations (PDEs) as they constrain the PDEs ensuring a unique and well defined solution. Based on combinatorial and surgery theory of manifolds, we develop multi-element boundary conditions as the generalization of the traditional boundary conditions in classical mechanics: Dirichlet boun...
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... research of surgery theory from the perspective of pure mathematics is abundant [6,7,8,9,10,11]. In 2D/3D, the examples of surgery theory are spheres, Möbius strip, Klein bottle and torus, being created from the fundamental polygons with different gluing strategies, as shown in Fig.1. ...
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... i-element boundary (domain) condition defines a tuple of i boundaries (domains), which can be understood as i boundaries (domains) are glued together as if there is no geometric distance between them. The gluing process can be applied on specific geometric boundaries or on virtual abstract spaces, where the former is the process as shown in Fig.1 and the latter transmits the interaction directly. ...
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... interaction happens between any two boundaries in i-element boundary. One illustration of the interaction is shown in Fig.10. The interaction between x 1 and x 2 follows the path p 1 in blue colour in traditional scenarios. ...
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... not only constrain the displacement in mechanics between two points (or between two regions), but also constrain the derivative of the displacement or its higher order derivatives. An example for a constraint of specific fields on two non-adjacent boundaries can be found in Fig.11. In Fig.11(b), the two boundaries by thick lines are glued with a virtual domain in green colour and we assume the virtual domain can transmit the interaction at a speed much larger than the sound speed in the grey domain. ...
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... example for a constraint of specific fields on two non-adjacent boundaries can be found in Fig.11. In Fig.11(b), the two boundaries by thick lines are glued with a virtual domain in green colour and we assume the virtual domain can transmit the interaction at a speed much larger than the sound speed in the grey domain. ...
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... the computational efficiency, we do not need to model the virtual domain directly but to use constraint equations to transmit the interaction. In Fig.11(c), the interaction can be enforced by smart devices, which detect the state of another device via the electric signal and apply influence of acceleration and velocity on material points they are attached to. ...
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... the other hand, r ij is the spatial vector formed in S j . One example illustrating the concepts of support and dualsupport is shown in Fig.12(b). A similar concept to dual-support is the dual-horizon in dual-horizon peridynamics [32,33]. ...
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... geometry and boundary conditions of the 2D model is depicted in Fig.13. The bottom nodes are fixed in all directions and the Neumann boundary condition is the upper nodes applied with a pressure load of 4.5 × 10 2 MPa/m. ...
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... contour of the displacement field U x and U y by NOM are compared with the finite element result by Abaqus, as shown in Fig.14. More detailed comparison on the displacement of the nodes on the Neumann boundary condition is depicted in Fig.15. ...
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... contour of the displacement field U x and U y by NOM are compared with the finite element result by Abaqus, as shown in Fig.14. More detailed comparison on the displacement of the nodes on the Neumann boundary condition is depicted in Fig.15. We can say the accuracy of NOM is very close to the finite element method. ...
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... quintic spline kernel function. The displacement of nodes on Neumann boundary is depicted in Fig.16, which shows that the displacement is quite close to each other. ...
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... consider a square with a square hole as shown in Fig.18. The boundaries are denoted ...
Citations
... In fact, certain constraints also exist on other order parameters such as chemical compositions in electrochemical reactions 21 and the quaternion representation of crystallographic orientations 22 . In addition to the constraints that need to be satisfied everywhere, there exist constraints on system boundaries, i.e., boundary conditions of partial differential equations (PDE) 23 . Conventionally, the Lagrange multiplier method is employed to obtain the minimum of a function/ functional under constraints, which represents the equilibrium state of the system. ...
Modeling of ductile fracture in polycrystalline structures is challenging, since it requires integrated modeling of cracks, crystal plasticity, and grains. Here we extend the typical phase-field framework to the situations with constraints on the order parameters, and formulate two types of phase-field models on ductile fracture. The Type-I model incorporates three sets of order parameters, which describe the distributions of cracks, plastic strain, and grains, respectively. Crystal plasticity is employed within grain interiors accommodated by J 2 plasticity at grain boundaries. The applications of the Type-I model to single crystals and bicrystals demonstrate the influences of grain orientations and grain boundaries on crack growth. In the Type-II model, J 2 plasticity is assumed for the whole system and grain structures are neglected. Taking advantage of the efficiency of the fast Fourier transform, our Type-II model is employed to study low cycle fatigue. Crack closure and striation-like patterning of plastic strain are observed in the simulations. Crack growth rate is analyzed as a function of the J -integral, and the simulated fatigue life as a function of plastic strain agrees with the Coffin–Manson relation without a priori assumption.
In recent years, mechanical characteristics including vibration, bending, buckling and postbuckling, stability and instability, etc. of small-scaled structures (such as micro/nanowires, micro and nano plates, micro and nano tubes, micro and nano beams, carbon micro/nano-tubes, micro and nano shells, micro and nano probes, etc.) have been broadly investigated, because of recent advances in nanotechnologies and due to their astonishing characteristics. Also, these small-scaled structures are extensively applied in several areas. However, comprehending the mechanical characteristics of these small-scaled structures is of great prominence and yet a daunting task. To understand the mechanical behaviors and capture the effects (including small-scale effect) of these different advanced small-scaled structures (at micro and nano levels), it is very essential to implement them into classical/nonclassical continuum elasticity based on a suitable classical/nonclassical elasticity formulation or theory. Thus, this work searches the literature and hereby presents an ample literature review study with a special focus primarily on the novel progress of vibration analyses of microplates and nanoplates using nonlocal/nonclassical continuum theories of elasticity. Scopus and web of science databases were used as the primary scholarly-databases to acquire the documents reviewed in the current research. Furthermore, several perspectives of classifications are considered while reviewing and grouping these acquired documents. The main purpose was to illustrate existing research trends in the vibrational analysis of small-scaled plate-based structures utilizing nonlocal plate-based theories and nonclassical continuum elasticity theories, and to provide a foundation and guidance for future research on the mechanics of small-scaled plate-based structures.
