应用数学和力学

1981 Vol. 2, No. 6

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Flow and Fracture of Rocks under General Triaxial Compression

Mogi

1981, 2(6): 585-598.

Abstract(1885) PDF(546)

Abstract:
Recent laboratory studies of the flow and fracture of rocks under general triaxial compression are reviewed. New developments in laboratory techniques have made it possible to measure three principal stresses and strains under general triaxial stress states, in which all three principal stresses are different.Strength and ductility of isotropic rocks are markedly affected not only by the least compression σ₃, but also by the intermediate compression σ₂, although these two effects are rather additional in strength, but opposite in ductility. The experimental results show that dilatancy is highly aniso-tropic under the general triaxial stress states.Deformational properties of anisotropic rocks have been also measured under the general triaxial compression. In this case, the effect of the intermediate compression markedly depends on the orientations of the weak planes.

The Symmetrical Deformation of Circular Membrane under the Action of Uniformly Distributed Loads in Its Central Portion

Chien Wei-zang, Wang Zhi-zhong, Xu Yin-ge, Chen Shan-lin

1981, 2(6): 599-612.

Abstract(1936) PDF(605)

Abstract:
This paper presents the solution of symmetrical deformation of circular membrane under the action of uniformly distributed loads in its central portion. Its limiting case is the solution of circular membrane under concentrated load at the center. This solution is the third solution of the circular membrane problems after the well-known Hencky's solution.

Representations of Linear, Isotropic Tensor Functions of an Asymmetric Argument

Guo Zhong-heng

1981, 2(6): 613-620.

Abstract(1705) PDF(575)

Abstract:
This paper offers a mathematical demonstration of the representation theorems for linear, isotropic scalar-and tensor-valued tensor functions of an asymmetric argument.

Analysis of Shallow Spherical Shell with Circular Base under Eccentrically Applied Concentrated Loads

Pao Shih-hua, Long Yu-qiu

1981, 2(6): 621-640.

Abstract(1497) PDF(500)

Abstract:
In this paper, problems of a shallow spherical shell with circular base under eccentrically applied concentrated loads are discussed. The solutions for six cases of eccentrically applied concentrated loads are given. They are:① Normal concentrated load,② Meridional tangential concentrated load,③ Circumferential tangential concentrated load,④ Concentrated moment in the tangential plane,⑤ Concentrated moment in the meridional normal plane,⑥ Concentrated moment in the circumferential normal plane.From the solutions of concentrated loads, the solutions of distributed line loads in the form of cos(nθ) along the circle are obtained.

Analysis of Ellipsoid Compressed by Two Axial Concentrated Forces at Two Ends

Yun Tian-quan, Shao Uong-chang, Qiu Chong-guang

1981, 2(6): 641-650.

Abstract(1804) PDF(712)

Abstract:
Integral equation method and photoetastic experiment are used for the stress analysis of an axial compressive ellipsoid. Let the concentrated forces and the centers of compression, with symmetrical unknown intensive functions x₁(c)=x2(-c) and x2(c)=x2(-c) respectively, be distributed-symmetrically to z=0 plane along the axis z(=-c) in [a,∞) and [-a,-∞) of the elastic space, in addition to a pair of equal and opposite axial forces acting on z=a and z=-a. We can reduce the problem of an axial compressive ellipsoid to two coupled Fredholm integral equations of the first kind. Furthermore, numerical calculation is then made. Two photo-elastic models of ellipsoid were analysed by "Freezing and Cutting" method, and the results, in which σ₂ is quite nearly to those obtained by integral equation method, had been used in the analysis of the data of compressive rock specimens.

The Moiré Geometry of Plane Finite Strain and Rotation

Chen Zhi-da

1981, 2(6): 651-657.

Abstract(1486) PDF(438)

Abstract:
We justified in this paper that the foundation of mathematical theory^[12] of finite deformation by the method of co-moving coordinate is identical to Moiré method in experimental mechanics. Hence, the important practical value of this theory is further ascertained.

An Approach to Determine the Stable Interval of the Constant Term of a Characteristic Equation

Wong Chia-ho

1981, 2(6): 659-670.

Abstract(1871) PDF(644)

Abstract:
In this paper, an approach is introduced to determine the stable interval of the constant term of a characteristic equation by using the theory of extended graphical representation of polynomials. Because the constant a_n itself is not taken into account and because this method is to get a stable interval of a_n, not merely to make a stability test for a set of known coefficients, this method of stability criteria has some advantages over the others. The interval of a_n can be obtained from the calculation of some algebraical expressions when n≤10 where n is the degree of the characteristic equation. It is very convenient to calculate for the cases n=5 and n=6. When n≥11, this interval can be found only by the method of numerical solutions.

The Compressive Properties of Long Bones

Sun Jia-ju, Geng Jie, He Fu-zhao

1981, 2(6): 671-678.

Abstract(1942) PDF(541)

Abstract:
In this study, compressive tests of the bones along the axial direction have been carried out on some wet specimens of the right femur and humerus, from which there have been obtained the elastic modulus of femur E=9.98×10⁹N/mM² and that of humerus E=11.37×10⁹N/m². Also comparisons and discussions have been made with reference to the available data reported abroad and at home.As indicated in this paper, bone tissues obviously possess viscoelastic properties. Their hysteresis loops are shown in Fig. 3 far and (b) and some mechanical phenomena observed during the test are illustrated elsewhere.

Finite Element Study of Dynamic Characteristics of Main Components of Machine Tools

He Qiong, Cheng Zheng-ding, Xiong Huan-guo

1981, 2(6): 679-688.

Abstract(1625) PDF(688)

Abstract:
Dynamic characteristics have been studied for main components of machine tools by finite element method. The multi-element static and dynamic structural analysis program makes possible the analysis of main components in various computing models.The eigen pairs are solved by the transfer sub-space iterative method with the advantages of high efficiency and precision. The program can be executed on the China-made computer TQ-16 and arbitrary sets of eigen pairs for the treatment of dynamic characteristics of complex structures of various kinds are thus obtained.

Some Theorems for Determining the Definiteness and Changeability of Sign of Functions and the Application to the Conditional Stability of the Motion of the Axis of a Sleeping Top

Ge Zheng-ming

1981, 2(6): 689-696.

Abstract(1750) PDF(432)

Abstract:
This paper gives some theorems for determining the definiteness and changeability of sign of functions as well as the application to the axis of a sleeping top. The necessary and sufficient condition of the conditional stability of the motion is obtained, which coincides with that of the stability of nutation angle and also with that of the stability of total variables in the equations of motion.

Energy Theorems for Solving Equations of Deflections

Fu Bao-lian

1981, 2(6): 697-707.

Abstract(1695) PDF(517)

Abstract:
In this paper the unit-dummy-load method is generalized on the basis of Castigliano's Theorem. On these grounds the general eguations of deflection surfaces of the structures, such as a kind of beams,plates and shells, are directly derived by the force method.We derived the eguations of the deflection surfaces of the rectangular thin plates and thick plates considering the effect of transverse shearing deformations with the inhomogeneous displacement boundary conditions. At the same time we give the equations of deflection axes of the corresponding straight beams.The applications of the reciprocal theorem are also generalized.Three simple calculated examples are given.

1981, 2(6): 709-713.

Abstract(1266) PDF(379)

Abstract: