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.

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.

Analytical Solutions of Crack Tip Plasticity Zone Shape for a Semi-Infinite Crack

2003 ◽  
Author(s):  
Peihua Jing ◽  
Tariq Khraishi ◽  
Larissa Gorbatikh
Keyword(s):  
Plane Strain ◽  
Plane Stress ◽  
Analytical Solutions ◽  
Yield Criterion ◽  
Plasticity Zone ◽  
Linear Elastic ◽  
Perfectly Plastic ◽  
Classical Plasticity ◽  
Elastic Perfectly Plastic ◽  
Von Mises

In this work, closed-form analytical solutions for the plasticity zone shape at the lip of a semi-infinite crack are developed. The material is assumed isotropic with a linear elastic-perfectly plastic constitution. The solutions have been developed for the cases of plane stress and plane strain. The three crack modes, mode I, II and III have been considered. Finally, prediction of the plasticity zone extent has been performed for both the Von Mises and Tresca yield criterion. Significant differences have been found between the plane stress and plane strain conditions, as well as between the three crack modes’ solutions. Also, significant differences have been found when compared to classical plasticity zone calculations using the Irwin approach.

scholarly journals Distribution of Zones of Elastic and Plastic Deformation Appearing in a Layer under Compression by Two Rigid Parallel Plates

2001 ◽  
Vol 14 (4) ◽  
pp. 492-496
Author(s):  
Olga V. Gomonova ◽  
Keyword(s):  
Plastic Deformation ◽  
Conservation Laws ◽  
Plane Strain ◽  
Strain State ◽  
Plastic Material ◽  
Yield Criterion ◽  
Parallel Plates ◽  
Plane Strain State ◽  
Elastic And Plastic Deformation

A problem of distribution of zones of elastic and plastic deformation appearing in a layer of elasto-plastic material under compression by two rigid parallel plates, for the case of plane strain state with Tresca – Saint-Venant yield criterion is solved. The technique based on application of conservation laws is used to solve the problem

Effect of State-of-Stress and Yield Criterion on the Bauschinger Effect

1968 ◽  
Vol 90 (3) ◽  
pp. 403-408 ◽  
Author(s):  
S. T. Rolfe ◽  
R. P. Haak ◽  
J. H. Gross
Keyword(s):  
Yield Strength ◽  
Plane Strain ◽  
Plane Stress ◽  
Bauschinger Effect ◽  
Yield Criterion ◽  
Cold Deformation ◽  
Stress Relief ◽  
Cold Forming ◽  
State Of Stress ◽  
Direction Opposite

During fabrication, the cold forming of structural components may reduce the yield strength of a component if it is loaded in a direction opposite to that of the cold forming. This reduction in yield strength, referred to as the Bauschinger effect, is influenced by the state-of-stress under which the cold forming is performed, by the criterion used to determine the yield strength, and by the use of post-forming stress relief. To establish the importance and magnitude of these effects, specimens from 2 1/2-in-thick plates of HY-80 steel, cold-formed by plane strain bending, were tested along with specimens that were cold-formed by plane-stress axial straining. For material tested in a direction opposite to that of cold forming, the Bauschinger effect was observed both in tension and compression, whereas for material tested at 90 deg to the direction of cold forming in plane strain, both the tensile and compressive yield strengths increased and no Bauschinger effect was observed. Because of the difference in restraint, the Bauschinger effect was greater for plane-stress axial deformation than for plane-strain bending deformation. The Bauschinger effect was greater when the yield strength was determined at small offsets and was essentially eliminated at an offset greater than 0.5 percent. In addition, the Bauschinger effect was greatest for small amounts of cold deformation and was progressively decreased by strain hardening at large amounts of cold deformation. The reduction in secant modulus and in yield strength (Bauschinger effect) in cold-formed material was essentially eliminated by stress-relief treatment at 1025 deg. F. The results indicate the importance of knowing the cold-forming state-of-stress, the criterion used in determining yield strength, and the effects of stress relief when assessing the effects of cold deformation on mechanical properties.

Improved Compliance Solutions for C(T), SE(B) and Clamped SE(T) Specimens Including Side-Grooves, Varying Thicknesses and 3-D Effects

2014 ◽  
Author(s):  
Gustavo Henrique B. Donato ◽  
Felipe Cavalheiro Moreira
Keyword(s):  
Crack Growth ◽  
Plane Strain ◽  
Plane Stress ◽  
Potential Drop ◽  
Crack Size ◽  
Growth Conditions ◽  
Size Estimation ◽  
Unloading Compliance ◽  
Integrity Assessments ◽  
Elastic Unloading

Fracture toughness and Fatigue Crack Growth (FCG) experimental data represent the basis for accurate designs and integrity assessments of components containing crack-like defects. Considering ductile and high toughness structural materials, crack growing curves (e.g. J-R curves) and FCG data (in terms of da/dN vs. ΔK or ΔJ) assumed paramount relevance since characterize, respectively, ductile fracture and cyclic crack growth conditions. In common, these two types of mechanical properties severely depend on real-time and precise crack size estimations during laboratory testing. Optical, electric potential drop or (most commonly) elastic unloading compliance (C) techniques can be employed. In the latter method, crack size estimation derives from C using a dimensionless parameter (μ) which incorporates specimen’s thickness (B), elasticity (E) and compliance itself. Plane stress and plane strain solutions for μ are available in several standards regarding C(T), SE(B) and M(T) specimens, among others. Current challenges include: i) real specimens are in neither plane stress nor plane strain - modulus vary between E (plane stress) and E/(1-ν2) (plane strain), revealing effects of thickness and 3-D configurations; ii) furthermore, side-grooves affect specimen’s stiffness, leading to an “effective thickness”. Previous results from current authors revealed deviations larger than 10% in crack size estimations following existing practices, especially for shallow cracks and side-grooved samples. In addition, compliance solutions for the emerging clamped SE(T) specimens are not yet standardized. As a step in this direction, this work investigates 3-D, thickness and side-groove effects on compliance solutions applicable to C(T), SE(B) and clamped SE(T) specimens. Refined 3-D elastic FE-models provide Load-CMOD evolutions. The analysis matrix includes crack depths between a/W=0.1 and a/W=0.7 and varying thicknesses (W/B = 4, W/B = 2 and W/B = 1). Side-grooves of 5%, 10% and 20% are also considered. The results include compliance solutions incorporating all aforementioned effects to provide accurate crack size estimation during laboratory fracture and FCG testing. All proposals revealed reduced deviations if compared to existing solutions.

scholarly journals Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 145
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Yeong-Maw Hwang
Keyword(s):  
Plane Strain ◽  
Yield Criterion ◽  
Large Strains ◽  
Rigid Plastic ◽  
Plastic Properties ◽  
Elastic Plastic ◽  
Starting Point ◽  
The Difference ◽  
Inside Radius ◽  
Elastic Unloading

The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution’s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill’s quadratic yield criterion is adopted. Both elastic/plastic and rigid/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic/plastic and rigid/plastic solutions is negligible, except at the very beginning of the process. However, the rigid/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses.

scholarly journals A Study of Yielding and Plasticity of Rapid Prototyped ABS

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1495
Author(s):  
Dan-Andrei Șerban ◽  
Cosmin Marșavina ◽  
Alexandru Viorel Coșa ◽  
George Belgiu ◽  
Radu Negru
Keyword(s):  
Experimental Data ◽  
Stress State ◽  
Plastic Flow ◽  
Plane Stress ◽  
Yield Criterion ◽  
Numerical Analyses ◽  
Flow Potential ◽  
Stress States ◽  
State Configuration

In this article, the yielding and plastic flow of a rapid-prototyped ABS compound was investigated for various plane stress states. The experimental procedures consisted of multiaxial tests performed on an Arcan device on specimens manufactured through photopolymerization. Numerical analyses were employed in order to determine the yield points for each stress state configuration. The results were used for the calibration of the Hosford yield criterion and flow potential. Numerical analyses performed on identical specimen models and test configurations yielded results that are in accordance with the experimental data.

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