In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is...In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.展开更多
Based on the observation of importance sampling and second order information about the failure surface of a structure, an importance sampling region is defined in V-space which is obtained by rotating a U-space at the...Based on the observation of importance sampling and second order information about the failure surface of a structure, an importance sampling region is defined in V-space which is obtained by rotating a U-space at the point of maximum likelihood. The sampling region is a hyper-ellipsoid that consists of the sampling ellipse on each plane of main curvature in V-space. Thus, the sampling probability density function can be constructed by the sampling region center and ellipsoid axes. Several examples have shown the efficiency and generality of this method.展开更多
Flexural toppling occurs when a series of layered rock masses bend towards their free face.It is important to evaluate the maximum bending degree and the requirement of supports of flexural toppling rock mass to preve...Flexural toppling occurs when a series of layered rock masses bend towards their free face.It is important to evaluate the maximum bending degree and the requirement of supports of flexural toppling rock mass to prevent rock mass cracking and even failure leading to a landslide.Based on the rock tensile strain-softening model,this study proposes a method for calculating the maximum curvature(C_(ppmax))of flexural toppling rock masses.By applying this method to calculate Cppmax of 9 types of rock masses with different hardness and rock layer thickness,some conclusions are drawn:(1)the internal key factors affecting C_(ppmax)are E^(⋆)(E^(⋆)=E_(ss)/E_(0),where E_(0)and E_(ss)are the mean deformation moduli of the rock before and after reaching its peak tensile strength,respectively),the strainεt corresponding to the tensile strength of rock,and the thickness(h)of rock layers;(2)hard rock layers are more likely to develop into block toppling than soft rock layers;and(3)thin rock layers are more likely to remain in flexural toppling state than thick rock layers.In addition,it is found that C_(ppmax)for flexural toppling rock masses composed of bedded rocks such as gneiss is related to the tensile direction.展开更多
基金This work was partially supported by the National Natural Science Foundation of China (Grant No. 10631020)Basic Research Grant of Tsinghua University (Grant No. JCJC2005071).
文摘In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.
文摘Based on the observation of importance sampling and second order information about the failure surface of a structure, an importance sampling region is defined in V-space which is obtained by rotating a U-space at the point of maximum likelihood. The sampling region is a hyper-ellipsoid that consists of the sampling ellipse on each plane of main curvature in V-space. Thus, the sampling probability density function can be constructed by the sampling region center and ellipsoid axes. Several examples have shown the efficiency and generality of this method.
基金funded by the National Natural Science Foundation of China(No.41972264)Zhejiang Provincial Natural Science Foundation of China(No.LR22E080002)the Key R&D Project of Zhejiang Province(No.2021C03159).
文摘Flexural toppling occurs when a series of layered rock masses bend towards their free face.It is important to evaluate the maximum bending degree and the requirement of supports of flexural toppling rock mass to prevent rock mass cracking and even failure leading to a landslide.Based on the rock tensile strain-softening model,this study proposes a method for calculating the maximum curvature(C_(ppmax))of flexural toppling rock masses.By applying this method to calculate Cppmax of 9 types of rock masses with different hardness and rock layer thickness,some conclusions are drawn:(1)the internal key factors affecting C_(ppmax)are E^(⋆)(E^(⋆)=E_(ss)/E_(0),where E_(0)and E_(ss)are the mean deformation moduli of the rock before and after reaching its peak tensile strength,respectively),the strainεt corresponding to the tensile strength of rock,and the thickness(h)of rock layers;(2)hard rock layers are more likely to develop into block toppling than soft rock layers;and(3)thin rock layers are more likely to remain in flexural toppling state than thick rock layers.In addition,it is found that C_(ppmax)for flexural toppling rock masses composed of bedded rocks such as gneiss is related to the tensile direction.
