The nonlinear mechanics of continual damage and its application to problems of creep and fatigue

The results of investigations of the mechanics of continual damage of materials and structural elements are reviewed. The focus is on nonlinea models of damage accumulation for problems of creep and high-cyclic fatigue. Nonlinearity criteria are formulated, and methods of construction of nonlinear models are analyzed. Approaches to the solution of problems of prediction of long-term strength and lifetime under creep and interaction of creep and fatigue are considered. S. P. Timoshenko Institute of Mechanics, national Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 31–66, March, 2000.     

Computational model of boundary integral equation in solid mechanics

In the first part of the paper, the computational model of boundary integral equation in solid mechanics is presented while in the second part the model is used in the solution of two problems of solid mechanics.     

A novel twice-interpolation finite element method for solid mechanics problems

Formulation and numerical evaluation of a novel twice-interpolation finite element method （TFEM） is presented for solid mechanics problems. In this method, the trial function for Galerkin weak form is constructed through two stages of consecutive interpolation. The primary interpolation follows exactly the same procedure of standard FEM and is further reproduced according to both nodal values and averaged nodal gradients obtained from primary interpolation. The trial functions thus constructed have continuous nodal gradients and contain higher order polynomial without increasing total freedoms. Several benchmark examples and a real dam problem are used to examine the TFEM in terms of accuracy and convergence. Compared with standard FEM, TFEM can achieve significantly better accuracy and higher convergence rate, and the continuous nodal stress can be obtained without any smoothing operation. It is also found that TFEM is insensitive to the quality of the elemental mesh. In addition, the present TFEM can treat the incompressible material without any modification.     

A damage mechanics approach to stress softening and its application to rubber

We analyze stress softening phenomena within the framework of the ‘generalized standard material’ based on the notion of a ‘normal dissipative mechanism’. We prove that the monotonicity properties of the ‘yield function’ governing such mechanism lead to local and global uniqueness of the response. Applications to oscillators with a single degree of freedom, whose anharmonic spring exhibits stress softening, are also presented.     

Existence of boundary value problems for TFD equation in quantum mechanics

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一种SPH应力修正算法及自由表面流中的应用

提出了一种SPH应力修正算法,即模型中的拉应力和压应力分别采用不同的插值核函数和状态方程来处理,改善应力稳定性问题。介绍了一种改进的Quintic核函数,用于改善模型中压应力的稳定性。通过增加钟型核函数的光滑长度,改善模型中拉应力的稳定性。采用该应力修正算法模拟了无重力条件下方形液滴的震荡变形过程,对比分析了不同算法的模拟结果。此外,为进一步验证算法的适用性,模拟了溃坝算例。研究表明,改进的Quintic型核函数明显改善了粒子聚集现象,该SPH应力修正方法可以使液滴具有更均匀的粒子分布以及更光滑的自由表面,有效改善了SPH方法中的压应力不稳定作用以及自由表面流的模拟精度。     

A finite-element approach in stream-tube method for solving fluid and solid mechanics problems

This paper presents a theoretical formulation in which the stream-tube method (STM) is examined through a variational approach for solving solid strain and fluid flow problems with finite elements. The analysis considers a reference domain, used as computational domain, related to the physical domain by an unknown transformation function to be determined numerically. Mass conservation is automatically verified by STM. The variational approach leads to eliminate the pressure in fluid problems and avoids to set up a mixed displacement–pressure procedure in the case of incompressible solids. Examples are given for fluid flows, applications and comparisons are also provided in the bending problem in elasticity.     

On the derivation of well posed boundary value problems in structural mechanics

The paper starts with a discussion of a number of published boundary value problems in structural mechanics, whose intuitive formulation led to incorrect boundary or matching conditions. It is then shown on three examples, using variational methods, how to obtain well posed formulations for problems of this type. For uniformity of presentation, all examples deal with continuously supported beams. The first two examples exhibit boundary and matching points whose position is fixed along the beam axis. In the third example it is shown how to use the method of variational calculus for a variable end point to formulate structural problems with variable matching points. This is demonstrated on the problem of a beam which rests on, but is not attached to, a Pasternak base; thus where parts of the beam may lift off the base. Each example concludes with a comparison and discussion of related formulations published in the literature.     

Infinite spline boundary element method and its application to structure-foundation interaction

In view of the infinity behaviors of3-D Kelvin solution,we constructed an infinite spline boundary element which has fine precision in the analysis of the half space foundation subjected to uniform pressure on the circular domain.We also analysed a square plate resting on elastic half space foundation.The results indicate that this model not only fits for the coupled analysis of foundation and structures but also has the advantage of fewer degrees of freedom and fine precision.     

A transformation of elastic boundary value problems with application to anisotropic behavior

A general geometrical transformation of the coordinates and of the displacement field is proposed; it is used to convert any boundary value problem for a linear elastic body into another one with different geometry, elastic moduli and boundary conditions. With this method, new problems, especially for inhomogeneous anisotropic bodies, may be solved by use of solutions of simpler ones. After a derivation of sufficient conditions to be fulfilled by such a transformation, the case of a linear homogeneous transformation is investigated in more detail. It is shown that a number of situations exist for which the transformed problem has a known analytical solution which can be used to derive the solution of the original problem straightforwardly. Special attention is paid to Saint-Venant-type anisotropy and to the derivation of the Green function for an infinite or a semi-infinite body.     

有限体积法在二维固体力学问题中的应用

推导了矩形单元的二维固体力学有限体积法公式，与解析解及有限元解进行了比较，讨论了有限体积法在非线性问题中的应用     

DSPH 方法的有效性验证及应用

讨论了不连续的光滑粒子流体动力学(discontinuoussmoothedparticlehydrodynamics,DSPH)方 法对粒子均匀性的敏感度,同时通过对DSPH 计算公式的详细推导,分析了该方法在界面处提高计算精度的 本质原因。通过简单的对碰模型,将DSPH 方法应用于实际问题中,研究了使用DSPH 方法计算密度、应变 率和能量时对模拟结果的影响,将多维的DSPH 方法推广到实际的物理计算中。     

Improving the differentiability of generalized fourier series solutions of boundary value problems of mechanics by using boundary functions

We prove a theorem on conditions for the differentiation of generalized Fourier series. We show that Fourier series solutions of boundary value problems can in general be differentiated term by term only once. To improve the differentiability properties of such series, we suggest to use pth-order boundary functions. We suggest an algorithm for constructing boundary functions for classical domains. This approach is illustrated by a new solution, with improved differentiability properties, of the problem on the torsion of an elastic rod of rectangular cross-section.