A CLASS OF MIXED-TYPE FUNDAMENTAL PROBLEMS ON ELASTIC HALF-PLANE

The problems of equilibrium of an elastic half-plane with loads on some intervals of the boundary and zero displacements on its other parts are discussed. The solutions of such problems are expressed in terms of integrals by reducing them to Riemann boundary value problems. The analytic expressions of the solutions are obtained in cases of uniform loads. In particular, the solution is written in detail for the important case when uniform pressure is given on a single interval or two equal intervals.

Fundamental solution method for inverse source problem of plate equation

The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method （FSM） and the Tikhonov regularization technique axe used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.

Fundamental Solution for Welding Problem by Two Dissimilar Isotropic Semi－Planes

A fundamental solution was obtained for an infinite plane bonded by two dissimilar isotropic semi-planes by employing plane elastic complex variable method and theory of boundary value problems for analytic functions.Fundamental solution was prepared for solving these types of problems with boundary element method.

Mysteries of the geometrization of gravitation

As we now know, there are at least two major difficulties with general rel- ativity （GR）. The first one is related to its incompatibility with quantum mechanics, in the absence of a consistent, widely accepted theory that combines the two theo- ries. The second problem is related to the requirement of the dark sectors-inflaton, dark matter and dark energy by the energy-stress tensor, which are needed to explain a variety of astronomical and cosmological observations. Research has indicated that the dark sectors themselves do not have any non-gravitational or laboratory evidence. Moreover, the dark energy poses, in addition, a serious confrontation between funda- mental physics and cosmology. Guided by theoretical and observational evidences, we are led to an idea that the source of gravitation and its manifestation in GR should be modified. The result is in striking agreement with not only the theory, but also the ob- servations, without requiring the dark sectors of the standard approach. Additionally, it provides natural explanations to some unexplained puzzles.

Dark Future for Dark Matter

The prevailing cosmological constant and cold dark matter (ΛCDM) cosmic concordance model accounts for the radial expansion of the universe after the Big Bang. The model appears to be authoritative because it is based on the Einstein gravitational field equation. However, a thorough scrutiny of the underlying theory calls into question the suitability of the field equation, which states that the Einstein tensor Gμv is a constant multiple of the stress-energy tensor Tμv when they both are evaluated at the same 4D space-time point: Gμv = 8πkTμv, where k is the gravitational constant. Notwithstanding its venerable provenance, this equation is incorrect unless the cosmic pressure is p = 0;but then all that remains of the Einstein equation is the Poisson equation which models the Newtonian gravity field. This shortcoming is not resolved by adding the cosmological constant term to the field equation, Gμv +Λ gμv =8πkTμv, as in the ΛCDM model, because then p = Λ, so the pressure is a universal constant, not a variable. Numerous studies support the concept of a linearly expanding universe in which gravitational forces and accelerations are negligible because the baryonic mass density of the universe is far below its critical density. We show that such a coasting universe model agrees with SNe Ia luminosity vs. redshift distances just as well or even better than the ΛCDM model, and that it does so without having to invoke dark matter or dark energy. Occam’s razor favors a coasting universe over the ΛCDM model.

Mysteries of R^ik=0： A novel paradigm in Einstein＇s theory of gravitation

Despite a century-long effort, a proper energy-stress tensor of the gravitational field, could not have been discovered. Furthermore, it has been discovered recently that the standard formulation of the energy-stress tensor of matter, suffers from various inconsistencies and paradoxes, concluding that the tensor is not consistent with the geometric formulation of gravitation [Astrophys. Space Sci., 2009, 321： 151; Astrophys. Space Sei., 2012, 340： 373]. This perhaps hints that a consistent theory of gravitation should not have any bearing on the energy-stress tensor. It is shown here that the so-called ＂vacuum＂ field equations Rik = 0 do not represent an empty spacetime, and the energy, momenta and angular momenta of the gravitational and the matter fields are revealed through the geometry, without including any formulation thereof in the field equations. Though, this novel discovery appears baffling and orthogonal to the usual understanding, is consistent with the observations at all scales, without requiring the Moreover, the resulting theory circumvents the besides explaining some unexplained puzzles. hypothetical dark matter, dark energy or inflation long-standing problems of the standard cosmology