A Numerical Study on CMOD Compliance for Single-Edge Notched Bending Specimens

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Enyang Wang ◽  
Wenxing Zhou ◽  
Guowu Shen ◽  
Daming Duan
Keyword(s):  
Crack Length ◽  
Plane Strain ◽  
Plane Stress ◽  
Stress Condition ◽  
Three Dimensional ◽  
Plane Strain Condition ◽  
Strain Condition ◽  
Two Dimensional ◽  
Single Edge ◽  
Plane Stress Condition

Several well-known equations for estimating the crack length in the single-edge notched bending (SE(B)) specimens from the normalized crack mouth opening displacement (CMOD) compliance are evaluated based on two-dimensional (2D) and three-dimensional (3D) finite element analyses (FEAs). Two-dimensional FEAs are first carried out to verify the reported accuracy and applicable ranges for each equation based on the plane strain models with six different crack lengths. Three-dimensional FEAs are then carried out to estimate the errors of prediction of equations that evaluate the crack length from the plane stress- and plane strain-based CMOD compliances. Both plane-sided and side-grooved models are included in 3D FEAs and have seven different thickness-to-width ratios. The error of prediction of a given equation is largely impacted by the thickness-to-width ratio, the crack length, the presence of side grooves, and the use of the plane stress- or plane strain-normalized CMOD compliance. Based on the errors of prediction, the relevance of the actual state of stress in the ligament of the SE(B) specimens to the plane strain condition or the plane stress condition is inferred. Knowledge of the relevance of the plane stress condition or the plane strain condition can be used to select the corresponding CMOD compliance in crack length-CMOD equations, and, therefore, the corresponding predictive accuracy can be improved.

Effectiveness of Non-Linear Crack Mechanics under Plane Strain Conditions

2007 ◽  
Vol 348-349 ◽  
pp. 497-500
Author(s):  
T. Teranishi ◽  
Hironobu Nisitani
Keyword(s):  
Plane Strain ◽  
Plane Stress ◽  
Large Scale ◽  
Stress Condition ◽  
Plane Strain Condition ◽  
Strain Condition ◽  
Plane Stress Condition ◽  
Crack Mechanics ◽  
Non Linear ◽  
Scale Yielding

The non-linear crack mechanics (NLCM) is a concept assuring the occurrence of the same phenomena in two cracked bodies under large scale yielding. It has been recognized that NLCM is effective in the cases of plane stress conditions. In this study, it was made clear that NLCM is effective not only in the case of plane stress condition but also in the case of plane strain condition.

scholarly journals Analytical Solution of Mixed Boundary Value Problems Using the Displacement Potential Approach for the Case of Plane Stress and Plane Strain Conditions

2017 ◽  
Vol 22 (2) ◽  
pp. 269-291
Author(s):  
S.K. Deb Nath
Keyword(s):  
Plane Strain ◽  
Plane Stress ◽  
Mixed Boundary ◽  
Stress Component ◽  
Plane Strain Condition ◽  
Strain Condition ◽  
Plane Stress Condition ◽  
Potential Approach ◽  
Displacement Potential ◽  
Fixed End

AbstractTwo elastic plate problems made of duralumin are solved analytically using the displacement potential approach for the case of plane strain and plane stress conditions. Firstly, a one end fixed plate is considered in which the rest of the edges are stiffened and a uniform load is applied to the opposite end of the fixed end. Secondly, a plate is considered in which all of the edges are stiffened and a uniform tension is applied at its both ends. Solutions to both of the problems are presented for the case of plane stress and plane strain conditions. The effects of plane stress and plane strain conditions on the solutions are explained. In the case of stiffening of the edges of the plate, the shape of the plate does not change abruptly, which is clearly observed in both of the cases. For the plane strain condition, the plates become stiffer in the loading direction as compared to the plane stress condition. For the plane strain condition, there is a significant variation of the normal stress component, σzzat different sections of the plate. The graphical results, clearly identify the critical regions of the plate for the case of the plane stress and plane strain condition.

A Numerical Study on CMOD Compliance for Single-Edge Bending Specimens

2012 ◽  
Author(s):  
Enyang Wang ◽  
Wenxing Zhou ◽  
Guowu Shen ◽  
Daming Duan
Keyword(s):  
Crack Length ◽  
Plane Strain ◽  
Plane Stress ◽  
Numerical Study ◽  
Three Dimensional ◽  
Actual State ◽  
Width Ratio ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element ◽  
Different Thickness

Several well-known equations for estimating the crack length in the SE(B) specimens from the normalized CMOD compliance are evaluated based on two- and three-dimensional finite element analyses. Two-dimensional analyses are carried out first to verify the reported accuracy and applicable ranges for each equation based on the plane strain models with six different crack lengths. Three-dimensional analyses are then carried out to estimate the errors of prediction of the equations that evaluate the crack length from the plane stress- and plane strain-based CMOD compliances. Both plain-sided and side-grooved models are included in the three-dimensional analyses and have seven different thickness-to-width ratios. The error of prediction of a given equation is largely impacted by the thickness-to-width ratio, crack length, presence of side grooves, and use of the plane stress- and plane strain-normalized CMOD compliance. Based on the errors of prediction, the relevance of the plane strain and plane stress conditions to the actual state of stress in the ligament of the SE(B) specimens is inferred. Knowledge of the relevance of the plane stress and plane strain conditions can be used to select either plane stress- or plane strain-based CMOD compliance in the crack length-CMOD equations and improve the accuracy of the prediction.

Three-dimensional thermo-mechanical analysis of continuous casting and comparison with two-dimensional models

2018 ◽  
Vol 53 (6) ◽  
pp. 421-434
Author(s):  
Reza Vaghefi ◽  
MR Hematiyan ◽  
Ali Nayebi
Keyword(s):  
Continuous Casting ◽  
Phase Change ◽  
Galerkin Method ◽  
Plane Strain ◽  
Plane Stress ◽  
Three Dimensional ◽  
Casting Process ◽  
Mechanical Analysis ◽  
Two Dimensional ◽  
Generalized Plane Strain

In this study, a three-dimensional thermo-elasto-plastic model is developed for simulating a continuous casting process. The obtained results are compared with those from different two-dimensional analyses, which are based on plane stress, plane strain, and generalized plane strain assumptions. All analyses are carried out using the meshless local Petrov–Galerkin method. The effective heat capacity method is employed to simulate the phase change process. The von Mises yield criterion and elastic–perfectly-plastic model are used to simulate the stress state during the casting process; while, material parameters are assumed to be temperature-dependent. Based on the three-dimensional and two-dimensional models, numerical results are provided to determine the stress, displacement, and temperature fields induced in the cast material. It is observed that the present meshless local Petrov–Galerkin method is accurate in three-dimensional thermo-mechanical analysis of highly nonlinear phase change problems. Reasonable agreements are observed between the results obtained from the three-dimensional analysis with those retrieved by the generalized plane strain assumption. However, it is observed that the results obtained under plane stress/strain conditions have some significant differences with the results obtained from three-dimensional modeling of continuous casting.

Drained solution for cylindrical cavity expansion in modified Cam clay soil under constant vertical stress

2020 ◽  
pp. 1-14
Author(s):  
D. Su
Keyword(s):  
Plane Strain ◽  
Cylindrical Cavity ◽  
Stress Condition ◽  
Vertical Stress ◽  
Plane Strain Condition ◽  
Cavity Expansion ◽  
Strain Condition ◽  
Cylindrical Cavity Expansion ◽  
Modified Cam Clay ◽  
Cam Clay

Cavity expansion is a fundamental theoretical problem in geomechanics. For the cylindrical cavity expansion problem, derivations of solutions are usually based on the assumption that the soil is subject to the plane strain condition. However, this is untrue for cavity expansion in pressuremeter tests. This study first derived the equilibrium equation for cylindrical cavity expansion under the constant vertical stress condition. Then, the equilibrium equation was solved for modified Cam clay soils with different overconsolidation ratios (OCRs). The solutions were compared with the responses in the same soils under the plane strain condition. It was found that the ratio of the limiting cavity pressure in the latter to that in the former ranged from 1.31 to 2.76 and increased with an increase in the OCR. Under the constant vertical stress condition, significant heaving occurred in the vicinity of the cavity, and volumetric strain evolved from contraction to dilation as the OCR increased. Significant differences were noted in the stress paths of the two different loading conditions. These results indicate that the assumption of the plane strain condition will lead to overestimation of the limiting cavity pressure and inaccurate prediction of the stress path in the pressuremeter test, especially for heavily overconsolidated soils.

Dimensionless Compliance With Effective Modulus in Crack Length Evaluation for B × B Single Edge Bend Specimens

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Guowu Shen ◽  
William R. Tyson ◽  
James A. Gianetto ◽  
Jie Liang
Keyword(s):  
Fracture Toughness ◽  
Crack Length ◽  
Plane Strain ◽  
Plane Stress ◽  
Crack Size ◽  
Effective Modulus ◽  
Single Edge ◽  
Opening Displacement ◽  
Size Evaluation ◽  
Elastic Unloading

In ASTM standard E1820, the crack size, a, may be evaluated during J-integral or crack-tip opening displacement (CTOD) resistance testing using the measured crack-mouth opening displacement (CMOD) elastic unloading compliance C (UC). The equation given to relate a to dimensionless compliance BCE (the product of thickness B, the compliance C and the modulus of elasticity E) in E1820 incorporates Young's modulus E rather than the plane-strain modulus E/(1 − υ2) where υ is Poisson's ratio. However, the three-dimensional (3-D) single edge bend (SE(B)) specimens used in fracture toughness tests are in neither plane-stress nor plane-strain condition, especially for B×B SE(B) specimens which are popular in characterizing fracture toughness of pipes with surface notches. In the present study, 3-D finite element analysis (FEA) was used to evaluate the CMOD compliance of plain- and side-grooved B×B SE(B) specimens with shallow and deep cracks. Crack sizes evaluated using plane-stress and plane-strain assumptions with the CMOD compliance calculated from FEA for the 3-D specimen were compared with the actual crack size of the specimens used in FEA. It was found that the errors using plane-strain or plane-stress assumptions can be as high as 5–10%, respectively, especially for shallow-cracked specimens. In the present study, an effective modulus with value between plane-stress and plane-strain is proposed and evaluated by FEA for the 3-D B×B SE(B) specimens for use in estimating the dimensionless compliance for crack size evaluation of B×B SE(B) specimens. It is shown that the errors in crack size evaluation can be reduced to 1% and 2% for plain-sided and side-grooved specimens, respectively, using this effective modulus. The effect of material removal to accommodate integral knife edges on the CMOD compliance was studied and taken into account in the crack length evaluations in the present study. Elastic unloading tests were conducted to measure the compliance of SE(B) specimens with two widths W and notch depths a/W from 0.1 to 0.5. Notch depths of the specimens evaluated by using the measured compliance and assumptions of plane stress, plane strain, and effective moduli were compared with the notch depths of the specimens used in the tests. It was found that best agreement of notch depth was achieved using the effective modulus.

On Systematic Errors of Two-Dimensional Finite Element Modeling of Right Circular Planar Flexure Hinges

2004 ◽  
Vol 127 (4) ◽  
pp. 782-787 ◽  
Author(s):  
B. Zettl ◽  
W. Szyszkowski ◽  
W. J. Zhang
Keyword(s):  
Finite Element ◽  
Plane Strain ◽  
Plane Stress ◽  
Compliant Mechanisms ◽  
Three Dimensional ◽  
Systematic Errors ◽  
Plane Stress State ◽  
3D Analysis ◽  
Two Dimensional ◽  
Flexure Hinges

This paper discusses the finite element method (FEM) based modeling of the behavior of typical right circular flexure hinges used in planar compliant mechanisms. Such hinges have traditionally been approximated either by simple beams in the analytical approach or very often by two-dimensional (2D) plane stress elements when using the FEM. The three-dimensional (3D) analysis presented examines these approximations, focusing on systematic errors due to 2D modeling. It is shown that the 2D models provide only the lower (assuming the plane stress state) or the upper (assuming the plane strain state) limits of the hinge’s stiffness. The error of modeling a particular hinge by 2D elements (with either the plane stress or the plane strain assumptions) depends mainly on its depth-to-height ratio and may reach up to about 12%. However, this error becomes negligible for hinges with sufficiently high or sufficiently low depth-to-height ratios, in which either the plane strain or stress states dominate respectively. It is also shown that the computationally intensive 3D elements can be replaced, without sacrificing accuracy, by numerically efficient 2D elements if the material properties are appropriately manipulated.

The Influence of In-Plane Constraint on Void Behavior in Front of a Crack in Plane Strain

2014 ◽  
Vol 224 ◽  
pp. 139-144
Author(s):  
Jaroslaw Galkiewicz
Keyword(s):  
Boundary Layer ◽  
Plane Strain ◽  
Plastic Material ◽  
Three Dimensional ◽  
Plane Strain Condition ◽  
Strain Condition ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Elastic Plastic ◽  
Elastic Plastic Material

This paper investigates voids’ behavior in front of a crack in elastic-plastic material under plane strain condition. Using the modified boundary layer approach for selected values of Q-stress it evaluates the deformations of a material cell. The deformations are recomputed with an exact three-dimensional model.

scholarly journals A displacement-based approach to geometric instabilities of a film on a substrate

2019 ◽  
Vol 24 (9) ◽  
pp. 2999-3023 ◽  
Author(s):  
Ali Javili ◽  
A. Derya Bakiler
Keyword(s):  
Plane Strain ◽  
Plane Stress ◽  
Three Dimensional ◽  
Current Approach ◽  
Original Form ◽  
Differential Growth ◽  
Plane Stress Condition ◽  
Subtle Difference ◽  
Compressive Stresses ◽  
Displacement Based

When a thin film adhered to a compliant substrate is growing, it will eventually buckle in order to release the compressive stresses accumulated within the film due to growth. Such geometric instabilities caused by compressive stresses prevail among all living systems in nature and their outcomes range from highly beneficial to destructive. Therefore, understanding compression induced instabilities is of crucial importance. Note that the origin of the “compression” need not necessarily be differential growth, as it may be due to pre-stretch or thermal expansion. A commonly accepted solution strategy for instabilities in bilayer structures dates back to the seminal work of Allen and employs the Airy stress functions. Owing to its reliance on a stress-based approach, the Allen solution is limited to linear two-dimensional problems and its success depends entirely on choosing an appropriate Airy function. The main objective of this contribution is to circumvent these limitations via a displacement-based approach formally suitable for three-dimensional problems, anisotropic materials, and even applicable to finite deformations. Furthermore, the Allen solution in its original form is valid for the plane-stress condition but often it is mistakenly compared with the numerical simulations corresponding to the plane-strain condition. We analyze the subtle difference between the solutions associated with the plane-strain and plane-stress conditions. Next, the analytical solution is compared against the computational results using the finite element method via eigenvalue analysis. Finally, it is briefly explained how the current approach can be utilized beyond the classical bilayer systems.

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