Meridian Surfaces Research Articles

A modified HJC model for geological materials subjected to blasting loadings

In this paper, a modified HJC model is proposed to address the problem that the HJC model cannot simulate tensile damage cracks. The modified HJC model modifies the strength surface, strain rate effect and damage model of the original HJC model, respectively. The modified model is embedded in LS-DYNA software for single and multiple finite element validation. The effects of different parameters on the modified HJC model stress–strain curves are analyzed in the single finite element validation. In multiple finite element verification, uniaxial and triaxial compression experiments, Brazilian disc splitting experiments, SHPB dynamic compression experiments, and SHPB dynamic splitting experiments are performed. The results of single and multiple finite element numerical validation illustrate that the modified HJC model can well simulate the pre-peak and post-peak nonlinear mechanical behaviors of geological materials such as rocks and concrete. Compared with the HJC model, the damage pattern of the modified HJC model is closer to the real damage pattern. For the strain rate effect, the modified HJC model can reflect the real compressive strain rate effect and improve the sensitivity of tensile strength to strain rate. The results of the numerical simulation of the blasting funnel illustrate that: the modified HJC model can simulate explosive tensile cracking well and can explain the formation mechanism of blasting funnel due to the distinction between tensile and compressive meridian surfaces and the addition of tensile damage. The modified HJC model is more applicable to the numerical simulation of engineering blasting.

Surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curvature vector field parametrized by canonical parameters is determined uniquely up to a motion in Euclidean (or Minkowski) space by the three invariant functions satisfying a system of three partial differential equations. We find examples of surfaces with parallel normalized mean curvature vector field and solutions to the corresponding systems of PDEs in Euclidean or Minkowski space in the class of the meridian surfaces.

On biconservative surfaces in 4-dimensional Euclidean space

In this paper, we study non-CMC biconservative surfaces with flat normal bundle in the Euclidean 4-space E4. First, we determine locally all parallel normalized mean curvature vector (PNMCV) biconservative surfaces in E4, showing that they are certain rotational surfaces. Then, we consider meridian surfaces lying on a rotational hypersurface of E4 and construct a family of biconservative surfaces with flat normal bundle and non-parallel normalized mean curvature vector.

Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-Space with Neutral Metric

In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of minimal and quasi-minimal meridian surfaces. We also classify the meridian surfaces with non-zero constant mean curvature.

Meridian Surfaces of Elliptic or Hyperbolic Type with Pointwise $1$-type Gauss Map in Minkowski $4$-space

In the present paper we consider a special class of spacelike surfaces in the Minkowski $4$-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respect to their Gauss map. We find all meridian surfaces of elliptic or hyperbolic type with harmonic Gauss map and give the complete classification of meridian surfaces of elliptic or hyperbolic type with pointwise $1$-type Gauss map.

Meridian Surfaces of Parabolic Type in the Four-Dimensional Minkowski Space

We construct a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with lightlike axis and call these surfaces meridian surfaces of parabolic type. They are analogous to the meridian surfaces of elliptic or hyperbolic type. Using the invariants of these surfaces we give the complete classification of the meridian surfaces of parabolic type with constant Gauss curvature or constant mean curvature. We also classify the Chen meridian surfaces of parabolic type and the meridian surfaces of parabolic type with parallel normal bundle.

SPECIAL CLASSES OF MERIDIAN SURFACES IN THE FOUR-DIMENSIONAL EUCLIDEAN SPACE

Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants. In the present paper we give the complete classification of Chen meridian surfaces and meridian surfaces with parallel normal bundle.

SEMI-PARALLEL MERIDIAN SURFACES IN E^4

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We classified semi-parallel meridian surface in 4-dimensional Euclidean space $E^4$.

MERIDIAN SURFACES IN 𝔼4WITH POINTWISE 1-TYPE GAUSS MAP

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Further, we give necessary and sufficient conditions for a meridian surface to have pointwise 1-type Gauss map and find all meridian surfaces with pointwise 1-type Gauss map.

MARGINALLY TRAPPED MERIDIAN SURFACES OF PARABOLIC TYPE IN THE FOUR-DIMENSIONAL MINKOWSKI SPACE

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We introduce meridian surfaces of parabolic type as one-parameter systems of meridians of a rotational hypersurface with lightlike axis in Minkowski 4-space and find their basic invariants. We find all marginally trapped meridian surfaces of parabolic type and give a geometric construction of these surfaces.

An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our main theorem states that these seven invariants determine the surface up to a motion in Minkowski space. We introduce meridian surfaces as one-parameter systems of meridians of a rotational hypersurface in the four-dimensional Minkowski space. We find all marginally trapped meridian surfaces.