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Register a new userOn a consistent finite-strain plate theory based on 3-D energy principle
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This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional (3-D) potential energy under general loadings with a third-order error. Staring from the 3-D nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the 3-D field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a 2-D virtual work principle. An alternative approach based on a 2-D truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a 2-D energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Comparing with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of the loadings, applicability to finite-strain problems and no involvement of unphysical quantities.
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Electromagnetic (EM) scattering systems widely exist in EM engineering domain. For a certain objective scattering system, all of its working modes constitute a linear space, i.e. modal space. Characteristic mode theory (CMT) can effectively construct a basis of the space, i.e. characteristic modes (CMs), and the CMs only depend on the inherent physical properties of the objective system, such as the topological structure and the material parameter of the objective system. Thus, CMT is very valuable for analyzing and designing the inherent EM scattering characters of the objective system. This work finds out that integral equation (IE) is not the best framework for carrying CMT. This dissertation proposes a completely new framework for carrying CMT, i.e. work-energy principle (WEP) framework, and at the same time proposes a completely new method for constructing CMs, i.e. orthogonalizing driving power operator (DPO) method. In new WEP framework and based on new orthogonalizing DPO method, this work resolves 5 pairs of important unsolved problems existing in CMT domain.
In this paper, we generalize Huygens principle (HP), extinction theorem (ET), and Franz-Harrington formulation (FHF). In our previous works, the traditional HP, ET, and FHF in homogeneous isotropic environment are generalized to inhomogeneous anisotropic lossy environment; the traditional FHF of homogeneous isotropic material system is generalized to inhomogeneous anisotropic lossy material system and then to piecewise inhomogeneous anisotropic lossy material system; the traditional HP, ET, and FHF of simply connected material system are generalized to multiply connected system and then to non-connected system; the traditional FHF of external scattering field and internal total field are generalized to internal scattering field and internal incident field. In previous work, it is proved that the generalized HP (GHP) and generalized ET (GET) are equivalent to each other; the GHP, GET, and generalized FHF (GFHF) satisfy so-called topological additivity, i.e., the GHP/GET/GFHF of whole electromagnetic (EM) system equals to the superposition of the GHP/GET/GFHF corresponding to all sub-systems. In this paper, the above results obtained in our previous works, which focuses on the EM system constructed by material bodies, are further generalized to the metal-material combined EM system in inhomogeneous anisotropic lossy environment, and traditional surface equivalence principle is generalized to line-surface equivalence principle.
In this paper, we consider an imperfect finite beam lying on a nonlinear foundation, whose dimensionless stiffness is reduced from $1$ to $k$ as the beam deflection increases. Periodic equilibrium solutions are found analytically and are in good agreement with a numerical resolution, suggesting that localized buckling does not appear for a finite beam. The equilibrium paths may exhibit a limit point whose existence is related to the imperfection size and the stiffness parameter $k$ through an explicit condition. The limit point decreases with the imperfection size while it increases with the stiffness parameter. We show that the decay/growth rate is sensitive to the restoring force model. The analytical results on the limit load may be of particular interest for engineers in structural mechanics
In [AIP Advances 6, 121707 (2016)], a soil structured with concrete columns distributed within two specially designed seismic cloaks thanks to a combination of transformational elastodynamics and effective medium theory was shown to detour Rayleigh waves of frequencies lower than 10 Hz around a cylindrical region. The aforementioned studies motivate our exploration of interactions of surface elastic waves propagating in a thick plate (with soil parameters) structured with concrete pillars above it. Pillars are 40 m in height and the plate is 100 m in thickness, so that typical frequencies under study are below 1 Hz, a frequency range of particular interest in earthquake engineering. We demonstrate that three seismic cloaks allow for an unprecedented flow of elastodynamic energy. These designs are achieved by first computing ideal cloaks parameters deduced from a geometric transform in the Navier equations that leads to almost isotropic and symmetric elasticity (4th order) and density (2nd order) tensors. To do this we extend the theory of Non-Euclidean cloaking for light as proposed by the theoretical physicists Leonhardt and Tyc. In a second step, ideal heterogeneous nearly isotropic cloaks parameters are approximated by averaging elastic properties of sets of pillars placed at the nodes of a bipolar coordinate grid, which is an essential ingredient in our Non-Euclidean cloaking theory for elastodynamic waves. Cloaking effects are studied for a clamped obstacle (reduction of the disturbance of the wave wavefront and its amplitude behind a clamped obstacle). Protection is achieved through reduction of the wave amplitude within the center of the cloak.These results represent a first step towards designs of Non-Euclidean seismic cloaks for surface (Rayleigh and Love) waves propagating in semi-infinite elastic media structured with pillars.
Based on previous work for the static problem, in this paper we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional (2D) shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.
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