scholarly journals A New Method of Calculating the State of Stress in Granular Materials under Plane Strain Conditions

2017 ◽  
Vol 3 (4) ◽  
pp. 89-106
Author(s):  
Sergei E Alexandrov ◽  
Elena A Lyamina
Keyword(s):  
Equilibrium Equation ◽  
Yield Criterion ◽  
Yield Condition ◽  
Equation System ◽  
Plasticity Theory ◽  
Average Stress ◽  
Practical Significance ◽  
System Of Equations ◽  
Equilibrium Equations ◽  
Coordinate Method

The system of equations comprising the Mohr-Coulomb yield condition and the stress equilibrium equations may be studied independently of the flow law. This system of equations is hyperbolic. Accordingly, to solve the aforementioned system of equations, it is reasonable to apply the method of characteristics. In the special case of plasticity theory for materials whose yield criterion does not depend on the average stress, two methods are used to construct an orthogonal net of characteristics and to determine the stress field: the R-S method and Mikhlin’s coordinate method. In the case of the Mohr-Coulomb yield condition, the angle between the characteristic directions depends on the internal friction angle. Therefore, the above-mentioned methods should be generalised in accordance with this property of characteristics. Purpose. In the case of Plasticity theory for materials whose yield strength does not depend on the average stress, to calculate the stress filed, Mikhlin’s coordinate method is widely used. The purpose of this study is to generalise this method for the equation system consisting of the Mohr-Coulomb yield criterion and the pressure equilibrium equations. Methods. The geometrical properties of the characteristics of the equations’ system consisting of the Mohr-Coulomb yield condition and the equilibrium equations are used to introduce the generalised Mikhlin coordinates. Results. It’s been pointed out that solving equation system consisting of the MohrCoulomb yield condition and equilibrium equation comes to solving equation of telegraphy and to subsequent integration. Practical Significance. The developed method of system of equations’ solution, consisting of the Mohr-Coulomb yield condition and equilibrium equation enables obtaining high precision solutions at insignificant computer time expenditures.

scholarly journals A Method of Calculating Principal Stress Trajectories in Powder and Porous Materials Obeying a Piece-wise Linear Yield Criterion

2018 ◽  
Vol 220 ◽  
pp. 01002
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Prashant Date
Keyword(s):  
Principal Stress ◽  
Yield Criterion ◽  
System Of Equations ◽  
Equilibrium Equations ◽  
Cartesian Coordinates ◽  
Stress Equilibrium ◽  
Scale Factors ◽  
Stress Directions ◽  
Characteristic Coordinates ◽  
Principal Lines

The present paper deals with the system of equations comprising the pyramid yield criterion together with the stress equilibrium equations under plane strain conditions. The stress equilibrium equations are written relative to a coordinate system in which the coordinate curves coincide with the trajectories of the principal stress directions. The general solution of the system is found giving a relation connecting the two scale factors for the coordinate curves. This relation is used for developing a method for finding the mapping between the principal lines and Cartesian coordinates with the use of a solution of a hyperbolic system of equations. In particular, the mapping between the principal lines and Cartesian coordinates is given in parametric form with the characteristic coordinates as parameters.

scholarly journals Study on the theory of three-dimensional limit analysis of soil and practical calculation

2018 ◽  
Vol 175 ◽  
pp. 03013
Author(s):  
Bin Lia ◽  
Jianbao Fu
Keyword(s):  
Stress Analysis ◽  
Equilibrium Equation ◽  
Stress Component ◽  
Three Dimensional ◽  
Yield Condition ◽  
Computation Method ◽  
Practical Calculation ◽  
Equilibrium Equations ◽  
Fundamental Equations ◽  
The Relationship

Based on the stress analysis of a point in space, the stress analysis of a point on a surface is performed, and then the relationship among the normal stress, the shear stress, and the stress component of a point on a surface and the first order partial derivative of the surface equation is deduced. Equilibrium equations of soil columns between the sliding surface and the top surface of the slope are established, which include differential equilibrium equation of force, equilibrium equation of force, and equilibrium equation of moment. These equilibrium equations and Coulomb yield condition can form the fundamental equations of three-dimensional slope stability analysis. Applying the supposition similar to that applied in the simplified Bishop method, a kind of three-dimensional slope analysis method can be obtained. An example is presented to show that the computation method is reasonable and applicable.

scholarly journals Dynamic Equilibrium Equations in Unified Mechanics Theory

2021 ◽  
Vol 2 (1) ◽  
pp. 63-80
Author(s):  
Noushad Bin Jamal Bin Jamal M ◽  
Hsiao Wei Lee ◽  
Chebolu Lakshmana Rao ◽  
Cemal Basaran
Keyword(s):  
Energy Loss ◽  
Entropy Generation ◽  
Equilibrium Equation ◽  
Dynamic Equilibrium ◽  
Free Vibration Analysis ◽  
Newtonian Mechanics ◽  
Equilibrium Equations ◽  
Time Coordinate ◽  
Proposed Model ◽  
Laws Of Thermodynamics

Traditionally dynamic analysis is done using Newton’s universal laws of the equation of motion. According to the laws of Newtonian mechanics, the x, y, z, space-time coordinate system does not include a term for energy loss, an empirical damping term “C” is used in the dynamic equilibrium equation. Energy loss in any system is governed by the laws of thermodynamics. Unified Mechanics Theory (UMT) unifies the universal laws of motion of Newton and the laws of thermodynamics at ab-initio level. As a result, the energy loss [entropy generation] is automatically included in the laws of the Unified Mechanics Theory (UMT). Using unified mechanics theory, the dynamic equilibrium equation is derived and presented. One-dimensional free vibration analysis with frictional dissipation is used to compare the results of the proposed model with that of a Newtonian mechanics equation. For the proposed entropy generation equation in the system, the trend of predictions is comparable with the reported experimental results and Newtonian mechanics-based predictions.

scholarly journals A method of finding stress solutions for a general plastic material under plane strain and plane stress conditions

2020 ◽  
Vol 37 ◽  
pp. 100-107
Author(s):  
Sergei Alexandrov ◽  
Yeau-Ren Jeng
Keyword(s):  
Coordinate System ◽  
Plane Strain ◽  
Principal Stress ◽  
Plane Stress ◽  
Plastic Material ◽  
Yield Criterion ◽  
Boundary Value ◽  
Equilibrium Equations ◽  
Geometric Problem ◽  
Principal Lines

Abstract A general plastic material under plane strain and plane stress is classified by a yield criterion that depends on both the first and second invariants of the stress tensor. The yield criterion together with the stress equilibrium equations forms a statically determinate system. This system is investigated in the principal lines coordinate system (i.e. the coordinate curves of this coordinate system coincide with trajectories of the principal stress directions). It is shown that the scale factors of the principal lines coordinate system satisfy a simple equation. Using this equation, a method for constructing the principal stress trajectories is developed. Therefore, the boundary value problem of plasticity theory reduces to a purely geometric problem. It is believed that the method developed is useful for solving a wide class of boundary value problems in plasticity.

The Influence of Rotatory Inertia and Transverse Shear on the Dynamic Plastic Behavior of Beams

1979 ◽  
Vol 46 (2) ◽  
pp. 303-310 ◽  
Author(s):  
Norman Jones ◽  
J. Gomes de Oliveira
Keyword(s):  
Transverse Shear ◽  
Shear Force ◽  
Yield Criterion ◽  
Yield Condition ◽  
Bending Moment ◽  
Plastic Behavior ◽  
Plastic Response ◽  
Rotatory Inertia ◽  
Perfectly Plastic ◽  
Theoretical Procedure

The theoretical procedure presented herein examines the influence of retaining the transverse shear force in the yield criterion and rotatory inertia on the dynamic plastic response of beams. Exact theoretical rigid perfectly plastic solutions are presented for a long beam impacted by a mass and a simply supported beam loaded impulsively. It transpires that rotatory inertia might play a small, but not negligible, role on the response of these beams. The results in the various figures indicate that the greatest departure from an analysis which neglects rotatory inertia but retains the influence of the bending moment and transverse shear force in the yield condition is approximately 11 percent for the particular range of parameters considered.

scholarly journals On The Use of the Equilibrium Equations and Flow Law in Relating the Surface and Bed Topography of Glaciers and Ice Sheets

1968 ◽  
Vol 7 (50) ◽  
pp. 199-204 ◽  
Author(s):  
I. F. Collins
Keyword(s):  
Surface Topography ◽  
Equilibrium Equation ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Equilibrium Equations ◽  
Bed Topography ◽  
Flow Law ◽  
Mathematical Justification ◽  
Longitudinal Strains

In a recent paper Robin has developed a method of calculating the relation between bed and surface topography of an ice sheet. He found that by including the effect of longitudinal strains in the equilibrium equation the correlation between theory and observation could be much improved. This paper is concerned with the mathematical justification of the assumption made by Robin.

Overstraining of flush plain cross-bored cylinders

2004 ◽  
Vol 218 (2) ◽  
pp. 143-153 ◽  
Author(s):  
J M Kihiu ◽  
G O Rading ◽  
S M Mutuli
Keyword(s):  
Finite Element Method ◽  
Yield Criterion ◽  
Three Dimensional ◽  
Yield Condition ◽  
Solution Technique ◽  
Stress Distributions ◽  
Dimensional Finite Element ◽  
Von Mises ◽  
Three Dimensional Finite Element ◽  
Incremental Theory Of Plasticity

A three-dimensional finite element method computer program was developed to establish the elastic-plastic, residual and service stress distributions in thick-walled cylinders with flush and non-protruding plain cross bores under internal pressure. The displacement formulation and eight-noded brick isoparametric elements were used. The incremental theory of plasticity with a 5 per cent yield condition (an element is assumed to have yielded when the effective stress is within 5 per cent of the material yield stress) and von Mises yield criterion were assumed. The frontal solution technique was used. The incipient yield pressure and the pressure resulting in a 0.3 per cent overstrain ratio were established for various cylinder thickness ratios and cross bore-main bore radius ratios. For a thickness ratio of 2.25 and a cross bore-main bore radius ratio of 0.1, the stresses were determined for varying overstrain and an optimum overstrain ratio of 37 per cent was established. To find the accuracy of the results, the more stringent yield condition of 0.5 per cent was also considered. The benefits of autofrettage were presented and alternative autofrettage and yield condition procedures proposed.

Analytical Solution of a Borehole Problem Using Strain Gradient Plasticity

2002 ◽  
Vol 124 (3) ◽  
pp. 365-370 ◽  
Author(s):  
X.-L. Gao
Keyword(s):  
Analytical Solution ◽  
Strain Gradient ◽  
Plastic Region ◽  
Yield Condition ◽  
Plasticity Theory ◽  
Strain Gradient Plasticity ◽  
Spatial Gradient ◽  
Gradient Plasticity ◽  
Present Solution ◽  
Classical Plasticity

An analytical solution is presented for the borehole problem of an elasto-plastic plane strain body containing a traction-free circular hole and subjected to uniform far field stress. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. This gradient plasticity theory introduces a higher-order spatial gradient of the effective plastic strain into the yield condition to account for the nonlocal interactions among material points, while leaving other relations in classical plasticity unaltered. The solution gives explicit expressions for the stress, strain, and displacement components. The hole radius enters these expressions not only in nondimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect in a quantitative manner. The classical plasticity-based solution of the borehole problem is obtained as a special case of the present solution. Numerical results for the plastic region radius and the stress concentration factor are provided to illustrate the application and significance of the newly derived solution.

An Analytical Solution to Metal Plane Problem in Terms of Orthotropic Anisotropy and Strain Hardening

2010 ◽  
Vol 97-101 ◽  
pp. 348-356
Author(s):  
Yao He Liu ◽  
Guo Feng Yi ◽  
Jian Ming Xiong
Keyword(s):  
Stress State ◽  
Analytical Solution ◽  
Plane Stress ◽  
Yield Criterion ◽  
Research Result ◽  
Yield Condition ◽  
Plane Stress State ◽  
Yield Function ◽  
Anisotropic Parameter ◽  
Symmetric Plane

In this paper, the yield condition of Hill’s orthotropic yield criterion under axial symmetric plane stress state was discussed. The yield function of orthotropic material was proposed and the analytical solution to meet the condition of equations of equilibrium and compatibility under axial symmetric plane stress state is obtained, in which the conditions of power hardening materials was considered. The research result indicates that hardening coefficient and anisotropic parameter have substantial influence over stress and strain. However, in the presence of the coefficient R90=H/F，the influence appears to be quite weak.

A Review of Tubular Conical Transition Strength Design Equations

2020 ◽  
Author(s):  
Albert Ku ◽  
Jieyan Chen ◽  
Bernard Cyprian
Keyword(s):  
Thin Shell ◽  
Shell Theory ◽  
Yield Criterion ◽  
Plasticity Theory ◽  
Transition Strength ◽  
Ultimate Capacity ◽  
Design Standards ◽  
Fem Simulations ◽  
Column Type ◽  
Shell Equations

Abstract This paper consists of two parts. Part one presents a thin-shell analytical solution for calculating the conical transition junction loads. Design equations as contained in the current offshore standards are based on Boardman’s 1940s papers with beam-column type of solutions. Recently, Lotsberg presented a solution based on shell theory, in which both the tubular and the cone were treated with cylindrical shell equations. The new solution as presented in this paper is based on both cylindrical and conical shell theories. Accuracies of these various derivations will be compared and checked against FEM simulations. Part 2 of this paper is concerned with the ultimate capacity equations of conical transitions. This is motivated by the authors’ desire to unify the apparent differences among the API 2A, ISO 19902 and NORSOK design standards. It will be shown that the NORSOK provisions are equivalent to the Tresca yield criterion as derived from shell plasticity theory. API 2A provisions are demonstrated to piecewise-linearly approximate this Tresca yield surface with reasonable consistency. The 2007 edition of ISO 19902 will be shown to be too conservative when compared to these other two design standards.

Sign in / Sign up

Export Citation Format

Share Document