Elastic Equilibrium Research Articles

Problems on elastic equilibrium of a wedge with cracks on the axis of symmetry

We use the Wiener-Hopf method to obtain exact solutions of plane deformation problems for an elastic wedge whose lateral sides are stress free and which has rectilinear cracks on its axis of symmetry. In problem 1, a finite crack issues from the wedge apex edge; in problem 2, a half-infinite crack originates at a certain distance from the wedge apex edge; and in problem 3, the wedge contains an internal finite crack.

Analysis of Carbon Nanotubes on the Mechanical Properties at Atomic Scale

This paper aims at developing a mathematic model to characterize the mechanical properties of single-walled carbon nanotubes (SWCNTs). The carbon-carbon (C–C) bonds between two adjacent atoms are modeled as Euler beams. According to the relationship of Tersoff-Brenner force theory and potential energy acting on C–C bonds, material constants of beam element are determined at the atomic scale. Based on the elastic deformation energy and mechanical equilibrium of a unit in graphite sheet, simply form ED equations of calculating Young's modulus of armchair and zigzag graphite sheets are derived. Following with the geometrical relationship of SWCNTs in cylindrical coordinates and the structure mechanics approach, Young's modulus and Poisson's ratio of armchair and zigzag SWCNTs are also investigated. The results show that the approach to research mechanical properties of SWCNTs is a concise and valid method. We consider that it will be useful technique to progress on this type of investigation.

A mixed problem of plate bending for doubly connected domains with partially unknown boundaries in the presence of cyclic symmetry

This paper addresses the problem of plate bending for a doubly connected body with outer and inner boundaries in the form of regular polygons with a common center and parallel sides. The neighborhoods of the vertices of the inner boundary are equal full-strength smooth arcs symmetric about the rays coming from the vertices to the center, but have unknown positions. Rigid bars are attached to the linear parts of the boundary. The plate bends by the moments applied to the middle point bars. The unknown arcs are free from external stresses. The same problem of plate bending is considered for a regular hexagon weakened by a full-strength hole. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body), the plate deflection and unknown parts of its boundary are determined under the condition that the tangential normal moment on that plate takes a constant value. Numerical analyses are also performed and the corresponding graphs are constructed.

Quasistatic bifurcation states of super-deep vertical drill strings

The problem of buckling of a long spinning rod containing an internal flow of a homogeneous liquid and undergoing compression and tension is considered in the paper. The authors derive the resulting equations to model stability of deep drilling strings according to the integrated calculation scheme, involving the string body and its lower section. The problem-solving procedure is proposed. Typical examples are discussed. The critical values of parameters for the elastic equilibrium of the system are found. The stability loss forms are determined.

Erratum to: Determination of stresses from their stress trajectories in plane elastic systems: the five constant theorem

Let K be a cylindrical region of finite height, having a generator parallel to the z-axis and a bounded, simply connected cross section S whose boundary is a sufficiently smooth Jordan curve. Moreover, let the material in K be elastic and isotropic, and in a state of plane stress or plane strain of elastic equilibrium, under the action of a conservative, z-independent body-force field and appropriate boundary tractions. We prove that at each z the principal stress-trajectory net corresponding to the stresses τxx, τxy, τyy is a 2–5 real-parameter family, as conjectured by Nemenyi.

A novel algorithm for area traffic capacity control with elastic travel demands

A non-linear area traffic control system with limited capacity is considered in this paper. Optimal signal settings and link capacity expansions can be determined while trip distribution and network flow are in equilibrium. This problem can be formulated as a non-linear mathematical program with equilibrium constraints. For the objective function a non-linear constrained optimization program for signal settings and link capacity expansion is determined. For the constraint set the elastic user equilibrium traffic assignment obeying Wardrop’s first principle can be formulated as a variational inequality. Since the constrained optimization problem is non-convex, only local optima can be obtained. In this paper, a novel algorithm using a non-smooth trust region approach is proposed. Numerical tests are performed using a real data city network and various example test networks in which the effectiveness and robustness of the proposed method are confirmed as compared to other well-known solution methods.

Relationships between soil fabric and suction cycles in compacted swelling soils

This study presents the influence of suction variation on the fabric of two clayey soils, using the mercury intrusion porosimetry (MIP) technique on dense and loose remoulded samples. The experimental study focussed on the evolution of the compacted soil fabric, including the macropores, mesopores, and micropores under various hydromechanical conditions. The variation of void ratio of both samples was initially studied during a single wetting and drying cycle. The experimental results confirm the existence of a suction limit between the meso- and macropores ( s m/M ) as well as the soil “shrinkage limit” suction ( s SL ). The suction increase between ( s m/M ) and ( s SL ) completely eliminated the macropores and produced mesostructural rearrangement. The higher suctions modified the meso- and micropores. The measured soil water retention curves (SWRC) of these two soils compared with the MIP calculations showed a more reasonable agreement for the dense soil with a fewer macropores. Moreover, several wetting and drying cycles were applied to the remoulded samples. The suction cycles produced a cumulative swelling strain for the dense soil while a shrinkage accumulation was observed for the loose samples. The wetting/drying cycles induced an equilibrium stage in which the samples behaved in an elastic way. The analysis of the soil fabric at this elastic equilibrium stage showed that the suction cycles created significant macrostructural modifications and reorganization of the mesopores towards smaller sizes.

Elastic strain relaxation in GaN/AlN nanowire superlattice

The molecular-beam epitaxy growth of AlN/GaN nanowire superlattices has been studied by using a combination of in situ x-ray diffraction experiments, high-resolution electron-microscopy analysis and theoretical calculations performed in a valence force field approach. It is found that the nanowire superlattices are in elastic equilibrium, in contrast with the two-dimensional case but in line with the predicted increase in the critical thickness in the nanowire geometry.

Determining the elastic equilibrium of a cylindrical shell by Vekua’s theory based on the classical elasticity theory and the theory of binary mixtures

Using the approach based on separation of variables, an analytic solution of the class of boundary value problems of the shallow cylindrical shell theory is constructed by Vekua’s method. The cylindrical shell is assumed to be rectangular in the plan. Conditions of a free support or sliding fixation are given on the sides of the rectangle; the load on the shell being arbitrary. The solution of boundary value problems is constructed using both a classical elastic medium and the theory of binary mixtures. Analysis of the constructed solutions is carried out.

Integrated reliability of travel time and capacity of urban road network under ice and snowfall conditions

In order to evaluate and integrate travel time reliability and capacity reliability of a road network subjected to ice and snowfall conditions, the conceptions of travel time reliability and capacity reliability were defined under special conditions. The link travel time model (ice and snowfall based-bureau public road, ISB-BPR) and the path choice decision model (elastic demand user equilibrium, EDUE) were proposed. The integrated reliability was defined and the model was set up. Monte Carlo simulation was used to calculate the model and a numerical example was provided to demonstrate the application of the model and efficiency of the solution algorithm. The results show that the intensity of ice and snowfall, the traffic demand and supply, and the requirements for level of service (LOS) have great influence on the reliability of a road network. For example, the reliability drops from 65% to 5% when the traffic demand increases by 30%. The comprehensive performance index may be used for network planning, design and maintenance.

Brownian Dynamics Simulation of FCS Measurements on Single Fluorophore-Labeled Superhelical DNA

We investigated the dynamics of a single-fluorophore-labeled pUC18 plasmid through a Brownian dynamics algorithm, followed up by a simulation of the Fluorescence Correlation Spectroscopy (FCS) process. Recent experimental FCS measurements indicated a sensitivity of the monomer mean square displacements in DNA circles towards superhelicity. Simulations with homogeneous DNA elasticity and local straight equilibrium are not sufficient to reproduce this observed behavior. But inserting permanently bent sequences into the DNA, which favor end loop formation, caused a dependence of the calculated FCS correlation curves on superhelical density. Furthermore, our simulations allow us to take into account the orientation of the fluorophore in polarized excitation, which might explain the observed appearance of a Rouse-like regime at intermediate time scales.

Elastic Bottleneck Equilibrium Model for Rail Transit Passengers at Rush Hours

As the traditional bottleneck models are failed to be fully utilized in the rail transit equilibrium at rush hours, the paper proposes the elastic bottleneck concepts which can reflect the trip character in each station, and develops a bottleneck equilibrium traffic flow model. First, with extending the definition of congestion risk cost, a more reasonable travel cost model for different stations at different time is developed at rush hours. Then, the peak hours computing equation is formulated to investigate the characteristics of rush hours. Considering the elastic features of different stations, the elastic bottleneck equilibrium model at the rush hours is developed. Finally, after model solution, the similarity between the model result and land value of space diminishing is analyzed comparatively, which is used to demonstrate the rationality of the proposed elastic bottlenecks model.

Analytical solutions for geometrically nonlinear trusses

This paper presents an analytical method for analyzing trusses with severe geometrically nonlinear behavior. The main objective is to find analytical solutions for trusses with different axial forces in the bars. The methodology is based on truss kinematics, elastic constitutive laws and equilibrium of nodal forces. The proposed formulation can be applied to hyper elastic materials, such as rubber and elastic foams. A Von Mises truss with two bars made by different materials is analyzed to show the accuracy of this methodology.

A stabilized forward elasticity finite element formulation yields a stable and convergent inverse elasticity solution.

“Elastography” refers to the procedure of using ultrasound to image tissue deformation. Images showing tissue deformation are sensitive to distributions of tissue mechanical properties. The mechanical properties themselves may be quantified by solving an inverse elasticity problem. An iterative approach to solving the inverse problem can be formulated by repeated solutions of the forward problem. That is, the shear modulus distribution sought is that which predicts a displacement field most consistent with the measured displacement field. Here we show that given plane displacement measurements for plane stress elasticity, the continuous elastic equilibrium equation uniquely determines the modulus distribution. On the other hand, we shall also demonstrate that the discrete elasticity equations from standard FEM discretization does not, even at the limit of infinite mesh refinement. We diagnose the problem as an underenforcement of the elasticity equations. With this knowledge, we have been able to design new forward elasticity FEM formulations that provide provably convergent forward and inverse elasticity solutions.

The numerical solution of boundary-value problems for an elastic body with an elliptic hole and linear cracks

Using the boundary-element method which is a combination of a fictitious load and a displacement discontinuity, numerical solutions are obtained for two-dimensional (plane deformation) boundary-value problems for the elastic equilibrium of infinite and finite homogeneous isotropic bodies having elliptic holes with cracks and cuts of finite length. Using the method of separation of variables, the boundary-value problem is solved in the case of an infinite domain containing an elliptic hole with a linear cut on whose contour the symmetry conditions are fulfilled.

Parameter identification method for determination of elastic modulus of rock based on adjoint equation and blasting wave measurements

AbstractThe purpose of this paper is to present a parameter identification method to determine the force of a blast and the elastic modulus of the ground using the measurements of a dynamic elastic wave, the adjoint equation method of optimal control theory, and the finite element method. Before the excavation of rocky ground, it is important to estimate the ground properties. In this paper, the elastic modulus is determined as the performance function is minimized using a technique based on the first‐order adjoint method. The performance function is a square sum of the discrepancies between the computed and the observed values of the velocities. After the determination of the magnitude of the blasting force, we can determine the elastic modulus of the rock. As the basic equation to calculate the velocities of dynamic elastic body, elastic equilibrium equations with linear viscosity are employed. The adjoint equation method has been utilized in order to calculate the gradient of the performance function with respect to the parameters. The gradient of the performance function is calculated using the first‐order adjoint equation. The weighted gradient method is applied for minimization. In order to solve the state equations in space and time, the finite element method and the Newmark \documentclass{article}\usepackage{amssymb}\footskip=0pc\pagestyle{empty}\begin{document}$\frac{1}{4}$\end{document} method are used. In this paper, we tested the practical application of our proposed method for determination of the elastic modulus of rock at the Ikawa tunnel located in the Tokushima prefecture, Japan. Copyright © 2009 John Wiley & Sons, Ltd.

Solution of some two-dimensional problems of elasticity

Using the method of boundary elements, we obtain numerical solutions of two-dimensional (plane deformation) boundary-value problems on the elastic equilibrium of infinite and finite homogeneous isotropic bodies having elliptic holes with cracks and cuts of finite length.

A computational framework of three-dimensional configurational-force-driven brittle crack propagation

We consider a variational formulation of quasi-static brittle fracture and develop a new finite-element-based computational framework for propagation of cracks in three-dimensional bodies. We outline a consistent thermodynamical framework for crack propagation in elastic solids and show that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality in the sense of Coleman’s method. Consequently, the crack propagation direction associated with the classical Griffith criterion is identified by the material configurational force which maximizes the local dissipation at the crack front. The variational formulation is realized numerically by a standard spatial discretization with finite elements which yields a discrete formulation of the global dissipation in terms configurational nodal forces. Therefore, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. In a consistent way with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface facets of the mesh. The crucial step for the success of this procedure is its embedding into an r-adaptive crack-facet reorientation procedure based on configurational-force-based indicators in conjunction with crack front constraints. We propose a staggered solution procedure that results in a sequence of positive definite discrete subproblems with successively decreasing overall stiffness, providing a robust algorithmic setting in the postcritical range. The predictive capabilitiy of the proposed formulation is demonstrated by means of representative numerical simulations.

Dynamics of a fluorophore attached to superhelical DNA: FCS experiments simulated by Brownian dynamics

We investigated the dynamics of a single-fluorophore-labeled pUC18 plasmid through a Brownian dynamics algorithm, followed by a simulation of the fluorescence correlation spectroscopy (FCS) process. Recent experimental FCS measurements indicated a sensitivity of the monomer mean square displacements in DNA circles towards superhelicity. Simulations with homogeneous DNA elasticity and local straight equilibrium are not sufficient to reproduce this observed behavior. But inserting permanently bent sequences into the DNA, which favor end loop formation, caused a dependence of the calculated FCS correlation curves on superhelical density. Furthermore, our simulations allow us to take into account the orientation of the fluorophore in polarized excitation, which might explain the observed appearance of a Rouse-like regime at intermediate time scales.

Phase-Field Simulation of the Effect of Elastic Inhomogeneity on Microstructure Evolution in Ni-Based Superalloys

Phase-field simulation is performed to examine the effect of elastic inhomogeneity between theand � 0 phases on microstructure evolution of the � 0 phase in Ni-based superalloys. For solving the elastic equilibrium equation numerically in the elastically inhomogeneous system, an iterative approach is used. In this simulation, both elastic anisotropy and shear modulus are varied independently based on the elastic constants of a practical Ni-based superalloy. Four different types of anti-phase domains in the � 0 phase are also considered in this simulation. The variation of elastic anisotropy affected significantly the manner of both morphology of the � 0 phase and distribution function of the � 0 particle size, whereas the variation of shear modulus did not affect them. (doi:10.2320/matertrans.MBW200826)