-Ki Hong

Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply connected problem

The spurious eigenvalues of an annular domain have been verified for the singular and hypersingular boundary‐element methods (BEMs) and circumvented by using the Burton‐Miller approach. Do they also occur in other formulations: continuous formulations such as the singular and hypersingular boundary integral equations (BIEs), the null‐field BIEs and the fictitious BIEs, or such discrete formulations as the null‐field BEMs and the fictitious BEMs? For the ten formulations of the multiply connected problem the study of otherwise the same issues is continued in the present paper. By using the degenerate kernels and the Fourier series, it is demonstrated analytically for the six continuous formulations of BIEs that spurious eigensolutions depend on the geometry of the inner boundary but not on that of the outer boundary. This conclusion can be extended to the six discrete formulations of BEMs. To filter out the spurious eigenvalues, the CHIEF (combined Helmholtz integral equation formulation) method is used here instead of the Burton‐Miller approach. The optimum number and appropriate positions of the CHIEF points are also addressed. It is then shown that, in the null‐field and fictitious BEMs, the spurious and true eigenvalues can be detected and distinguished by using the singular‐value‐decomposition‐updating techniques in conjunction with the Fredholm alternative theorem. Illustrative examples show the validity of the proposed methodologies.

Internal symmetry in the constitutive model of perfect elastoplasticity

Abstract Internal symmetry in the constitutive model of perfect elastoplasticity is investigated here. Using homogeneous coordinates, we convert the non-linear model to a linear system X = AX . In this way the inherent symmetry in the constitutive model of perfect elastoplasticity (in the on phase) is brought out. The underlying structure is found to be the cone of Minkowski spacetime M n+1 on which the proper orthochronous Lorentz group SO 0 (n, 1) left acts. When the plasticity mechanism is shut off by the input path, the internal symmetry is switched to a translation group T(n) acting on the closed disc D n of Euclidean space E n . Based on the group properties a Cayley transformation is developed, which updates the stress points to be automatically on the yield surface at every time increment. These results (and their generalizations to more sophisticated models) are essential for computational plasticity. As an example, the results calculated using the group-preserving scheme and the exact constitutive solutions for a rectilinear path are compared.

Boundary element analysis and design in seepage problems using dual integral formulation

Abstract A dual integral formulation with a hypersingular integral is derived to solve the boundary value problem with singularity arising from a degenerate boundary. A seepage flow under a dam with sheet piles is analyzed to check the validity of the mathematical model. The closed-form integral formulae containing the four kernel functions in the dual integral equations are presented and clearly reveal the properties of the single- and double-layer potentials and their derivatives. The field and boundary quantities of the potential heads and normal fluxes can thus be expressed in terms of both boundary potentials and boundary normal fluxes through the dual boundary integral equations. To facilitate the computation of the seepage flow along and near the boundary, an expression for the flux tangential to the boundary is also derived. The numerical implementations are compared with analytical solutions and the results of a general purpose commercial finite element program. Finally, four design cases of sheet piles are examined, and the best choice among them is suggested.

Internal symmetry in bilinear elastoplasticity

Abstract Internal symmetry of a constitutive model of bilinear elastoplasticity (i.e. linear elasticity combined with linear kinematic hardening–softening plasticity) is investigated. First the model is analyzed and synthesized so that a two-phase two-stage linear representation of the constitutive model is obtained. The underlying structure of the representation is found to be Minkowski spacetime, in which the augmented active states admit of a Lorentz group of transformations in the on phase. The kinematic rule of the model renders the transformation group inhomogeneous, resulting in a larger group—the proper orthochronous Poincare group.

INTEGRAL REPRESENTATIONS AND REGULARIZATIONS FOR A DIVERGENT SERIES SOLUTION OF A BEAM SUBJECTED TO SUPPORT MOTIONS

SUMMARY Derived herein is the integral representation solution of a Rayleigh-damped Bernoulli-Euler beam subjected to multi-support motion, which is free from calculation of a quasi-static solution, and in which the modal participation factor for support motion is formulated as a boundary modal reaction, thus making efficient calculation feasible. Three analytical methods, including ( 1 ) the quasi-static decomposition method, (2) the integral representation with the Cesaro sum technique, and (3) the integral representation in conjunction with Stokes’ transformation, are presented. Two additional numerical methods of (4) the large mass FEM simulation technique and (5) large stiffness FEM simulation technique are easily incorporated into a commercial program to solve the problem. It is found that the results obtained by using these five methods are in good agreement, and that both the Cesiro sum and Stokes’ transformation regularization techniques can extract the finite part of the divergent series of the integral representation. In comparison with the Mindlin method and Cesaro sum technique, Stokes’ transformation is the best way because it is not only free of calculation of the quasi-static solution, but also because it can obtain the convergence rate as rapidly as the mode acceleration method can.

Application of Cesàro mean and the L-curve for the deconvolution problem

In this paper, the Cesaro mean technique is applied to regularize the divergent problem which occurs in ground motion deconvolution analysis in geotechnical engineering. To deal with this ill-posed problem, we use the corner of the L-curve as the compromise point to determine the optimal order of Cesaro mean so that the high frequency content can be suppressed instead of engineering judgement using the concept of a cutoff frequency. The fractional order of Cesaro mean has been derived and used to fulfill this purpose. From the examples shown, it is found that the wave form including maximum acceleration can be accurately predicted and that both the high frequency content and divergent results can be avoided by using the proposed regularization technique.

Prandtl-Reuss elastoplasticity: On-off switch and superposition formulae

Abstract Constitutive postulates for Prandtl-Reuss elastoplasticity are selected. Based on them, sufficient and necessary conditions for plastic irreversibility are found to be the yield condition and the straining condition. This is then analyzed and it is pointed out that Prandtl-Reuss elastoplasticity is nothing but a two-phase linear system with an on-off switch, which is operated in the pace of an intrinsic measure of plastic irreversibility. Then the temporally global concept of the switch-on time t on and the switch-off time t off and their determination and bound estimation is developed. Owing to the implicit linearity, superposition formulae for the stress response are discovered. As an example, the exact constitutive solutions for circular paths based on the superposition formulae are obtained and t on and t off are determined.

Anisotropic plasticity with application to sheet metals

Abstract Hill’s 1948 anisotropic theory of plasticity is extended to include the concept of isotropic–kinematic hardening. The “anomalous” effect can be accounted for by kinematic hardening. It is shown that the quadratic yield function can be used for sheet metals irrespective of its plastic strain ratio R. It is further shown that effects of thickness reduction due to further rolling may be accounted for by kinematic hardening.

Using comparison theorem to compare corotational stress rates in the model of perfect elastoplasticity

Abstract For the simple shear problem of a perfectly elastoplastic body, we convert the non-linear governing equations into a third order linear differential system, then into a second order linear differential system, and further into a Sturm–Liouville equation. Thus Sturms comparison theorem can be employed and extended to compare the simple shear responses based on different objective corotational stress rates. It is proved that the rates of Jaumann, Green–Naghdi, Sowerby–Chu, Xiao–Bruhns–Meyers, and Lee–Mallett–Wertheimer render non-oscillatory stress responses, with the Jaumann equation as a Sturm majorant for the other four equations. For an objective corotational stress rate with the general plane spin a sufficient non-oscillation criterion is found to be that the plane spin must not exceed the shear strain rate.

Lorentz group on Minkowski spacetime for construction of the two basic principles of plasticity

Abstract We show that a model of plasticity is a necessary consequence of the two basic principles: (1) causality in the truncated future cone of the Minkowski spacetime (or its generalization) of augmented states, and (2) controllability and non-generativity in a reachable, bounded space of states. To consider the symmetry switching between PSO o (n, 1) and SE(n) due to switching on and off the plasticity mechanism, the model is reconstructed as a dynamical system on a composite space, which results from a surgery on Minkowski spacetime.