Inverse Calculation of Uniaxial Stress-Strain Curves From Bending Test Data

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
G. S. Schajer ◽  
Y. An
Keyword(s):  
A Priori ◽  
Strain Curve ◽  
Uniaxial Stress ◽  
Inverse Solution ◽  
Surface Strain ◽  
Bending Test ◽  
Stress Strain ◽  
Strain Data ◽  
Tension And Compression ◽  
Priori Information

Uniaxial tension and compression stress-strain curves are simultaneously evaluated from load and surface strain data measured during a bending test. The required calculations for the uniaxial results are expressed as integral equations and solved in that form using inverse methods. This approach is taken to reduce the extreme numerical sensitivity of calculations based on equations expressed in differential form. The inverse solution method presented addresses the numerical sensitivity issue by using Tikhonov regularization. The use of a priori information is explored as a means of further stabilizing the stress-strain curve evaluation. The characteristics of the inverse solution are investigated using experimental data from bending and uniaxial tests.

Uniaxial Deformation of Rubber Cylinders

1995 ◽  
Vol 68 (5) ◽  
pp. 739-745 ◽  
Author(s):  
P. H. Mott ◽  
C. M. Roland
Keyword(s):  
Strain Curve ◽  
Uniaxial Deformation ◽  
Deflection Curve ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Optical Birefringence ◽  
Parabolic Profile ◽  
Strain Data ◽  
Tension And Compression ◽  
Actual Stress

Abstract Stress, strain and optical birefringence measurements were made on elastomeric cylinders deformed in tension and compression. The birefringence data enables the actual stress to be determined even when the deformation is not homogeneous. In the absence of lubricant, uniaxially loaded rubber cylinders deviate from homogeneous deformation due to bonding of the cylinder ends. The existing analysis to correct the force-deflection curve for the effect of this sticking, strictly valid for infinitesimal strains, is premised on the idea that the deformed cylinder has a parabolic profile. Experimentally, however, it was found that slender rubber cylinders assume a much flatter profile, while maintaining constant volume, when deformed. Nevertheless, the accuracy of the stress-strain curve when the standard correction is applied turns out to be quite good, partially a result of cancellation of two, relatively small, errors. This accuracy was assessed by comparison of force-deflection data from bonded cylinders both to stress-strain data obtained on lubricated cylinders and to the stresses deduced from the measured birefringence.

Tension-compression stress-strain curves from bending tests

1971 ◽  
Vol 6 (4) ◽  
pp. 286-292 ◽  
Author(s):  
P W J Oldroyd
Keyword(s):  
Strain Curve ◽  
Bending Moment ◽  
Uniaxial Stress ◽  
Bending Test ◽  
Compression Stress ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Bending Tests ◽  
Original Length ◽  
Annealed Copper

A formula—Nadai's bending formula—is derived which enables the tension (or compression) stress-strain curve for a material to be obtained from the curve relating bending moment to curvature for a beam of solid rectangular section. The method is extended to give a formula which covers deformations in which reversals of plastic strain occur. The results obtained from a unidirectional bending test made on annealed copper are compared with those obtained from a tensile test made on the same material and the accuracy of the stress-strain values obtained from the bending test is discussed. The results obtained from a reversed bending test are also compared with those obtained from a tension-compression test in which a specimen was first stretched and then compressed to its original length. The limitations imposed by this method of obtaining the stress-strain curve for a material are examined and the advantages its presents in the study of the behaviour of materials under uniaxial stress are outlined.

scholarly journals Uniaxial Extension and Compression in Stress-Strain Relations of Rubber

1978 ◽  
Vol 51 (4) ◽  
pp. 840-851 ◽  
Author(s):  
Lawrence A. Wood
Keyword(s):  
Experimental Data ◽  
Tensile Deformation ◽  
Uniaxial Extension ◽  
Strain Curve ◽  
Uniaxial Stress ◽  
Stress Strain ◽  
General Applicability ◽  
Empirical Constant ◽  
Strain Data ◽  
Theoretical Predictions

Abstract A comprehensive literature survey shows the general applicability of the generalized normalized Martin, Roth and Stiehler equation to uniaxial stress-strain data in extension and compression on rubber vulcanizates. The equation can be expressed as F/M=(L−1−L−2) expA (L−L−1) where F is the stress on the undeformed section and L the ratio of stressed to unstressed length. The equation contains two constants—M, Young's Modulus, the slope of the stress-strain curve at L=1, and A an empirical constant. The conformity of stress-strain data to the equation can readily be determined by a plot of logF/(L−1−L−2) against (L−L−1). In almost every case a straight line is obtained, from the slope and intercept of which both the constants can be determined. The range of validity of the equation usually begins near L=0.5 (in the compression region) and continuing through the region of low deformations often extends to the region of rupture in extension. If uniaxial compression data are available the modulus can thus be obtained by interpolation through the region of low deformations, where experimental data are often somewhat unreliable. The value of the modulus M varies with the nature of the rubber, the extent of vulcanization, and the time and temperature of creep or stress relaxation. The value of the constant A is near 0.4 for pure-gum vulcanizates, increasing to values near 1.0 with increasing filler content, and showing an abrupt increase when crystallization occurs. Direct experimental observations where the deformation of a single specimen is varied continuously from compressive to tensile deformation, are cited to show that M, defined as the limit of the ratio of stress to strain, is independent of the direction of approach to the limit at L=0.5. The normalized Mooney-Rivlin plots show F/[2M (L−L−2)] against L−1. These graphs have only limited regions of linearity corresponding to constant values of the coefficients C1 and C2. Since these regions do not include the undeformed state the Mooney-Rivlin equation cannot be used at low elongations or in compression. The values of C1 and C2 show very wide fluctuations for the Mooney-Rivlin plots of experimental data, which are themselves usually well represented by the Martin, Roth, and Stiehler equation with different values of the constant A. In view of all these considerations the conclusion of the present study confirms that of Treloar in his recent publications in failing to find much utility in making Mooney-Rivlin plots. The failure to represent the experimental data at low elongations and the inability to correlate the constants with theoretical predictions based on strain energy or statistical theory considerations are the most serious objections.

Hole-Drilling Residual Stress Profiling With Automated Smoothing

2007 ◽  
Vol 129 (3) ◽  
pp. 440-445 ◽  
Author(s):  
Gary S. Schajer
Keyword(s):  
Residual Stress ◽  
A Priori ◽  
Hole Drilling ◽  
Simple Method ◽  
Strain Measurements ◽  
Stress Solution ◽  
Strain Data ◽  
Data Content ◽  
Small Increments ◽  
Priori Information

An effective procedure is presented that allows stable hole-drilling residual stress calculations using strain data from measurements taken at many small increments of hole depth. This use of many strain measurements is desirable because it improves the data content of the calculation, and the statistical reliability of the residual stress results. The use of Tikhonov regularization to reduce the noise sensitivity that is characteristic of a fine-increment calculation is described. This mathematical procedure is combined with the Morozov criterion to identify the optimal amount of regularization that balances the competing tendencies of noise reduction and stress solution distortion. A simple method is described to estimate the standard error in the strain measurements so that the optimal regularization can be chosen automatically. The possible use of a priori information about the trend of the expected solution is also discussed as a further means of improving the stress solution. The application of the described method is demonstrated with some experimental measurements, and realistic results are obtained.

scholarly journals The Neuroelectromagnetic Inverse Problem and the Zero Dipole Localization Error

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Rolando Grave de Peralta ◽  
Olaf Hauk ◽  
Sara L. Gonzalez
Keyword(s):  
Inverse Problem ◽  
Unique Solution ◽  
A Priori ◽  
Inverse Solution ◽  
Localization Error ◽  
A Priori Information ◽  
Dipole Localization ◽  
Inverse Solutions ◽  
Priori Information ◽  
Additional Constraints

A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP) is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA) attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.

Elastomer Behavior IV. Loop Structure of Elastomer Networks

1966 ◽  
Vol 39 (5) ◽  
pp. 1489-1495
Author(s):  
L. C. Case ◽  
R. V. Wargin
Keyword(s):  
Experimental Data ◽  
Crosslink Density ◽  
Strain Curve ◽  
Uniaxial Stress ◽  
Polymer Chains ◽  
Theoretical Treatment ◽  
Loop Structure ◽  
Stress Strain ◽  
Strain Behavior ◽  
Selection Of

Abstract A new theoretical treatment strongly indicates that an elastomer network actually consists of a system of fused, closed, interpenetrating loops of polymer chains. This interpenetrating loop structure restricts the movement of the chains and thereby affects the stress-strain behavior of the elastomer. Methods have been developed to enable the calculation of the number of effective crosslinks caused by loop interpenetrations (virtual crosslinks). The uniaxial stress-strain behavior of an elastomer predicted using our methods can be fitted almost perfectly to published experimental data by proper selection of chain parameters. Previous theoretical treatments gave only a qualitative fit to the experimental data for the stress-strain behavior of elastomers and were not capable of predicting the correct shape of the experimental stress-strain curve. The present treatment gives a nearly perfect fit for both stress as a function of strain at constant crosslink density, and stress as a function of crosslink density at constant strain, and thus represents a vast improvement.

Compression Stress Strain of Rubber

1933 ◽  
Vol 6 (1) ◽  
pp. 126-150 ◽  
Author(s):  
J. R. Sheppard ◽  
W. J. Clapson
Keyword(s):  
Hollow Sphere ◽  
Compressive Force ◽  
Strain Curve ◽  
The Other ◽  
One Dimension ◽  
Compression Stress ◽  
Stress Strain ◽  
Strain Data ◽  
Tensile Forces ◽  
Strain Axis

Abstract 1. A relation of simple form between compressive force and equivalent two-way tensile forces is developed. 2. Based on this relation, a new method for determining the compression stress strain of rubber is outlined, which avoids difficulties and errors inherent in direct compression. It consists in applying tensile forces simultaneously in two directions, and, from these and the strained dimensions, in computing the compressive force that would have produced the same deformation. 3. The mode of applying the two-way tensiles is to inflate a hollow sphere of rubber; the experimental data required to determine the compression stress strain are pressure of gas in, and dimensions of, the inflating hollow sphere. 4. The method has been applied to cold-cured pure-gum rubber in the form of toy balloons which, in its ordinary elongation stress strain, shows a breaking elongation of about 650 to 700 per cent and a tensile of 30 to 40 kg. per square centimeter. While the numerical values obtained on this stock have no special significance, as they will vary from stock to stock, the following are examples: breaking compression, about 97.3 per cent; breaking compressive force, 6000 to 9000 kg. per square centimeter (on original cross section); hysteresis, 29 to 35 per cent of work of compression to near rupture. 5. As a common measuring stick by which to gage degree of strain in deformations of different types—e. g., increasing one dimension (and diminishing the other two) as against diminishing one dimension (and increasing the other two)—energy seems the best. Energy at break for ordinary elongation stress strain was 50 to 70 kg. cm. per cubic centimeter, and for compression stress strain was 89 to 103 kg. cm. per cubic centimeter. 6. The compression stress-strain data may, if desired, be expressed in terms of two-way tensiles vs. two-way elongations. Energy of compression may be computed either as twice the area subtended between such a curve and the strain axis, or as the area between the compression stress strain and the strain axis. 7. It is strongly indicated that the compression stress strain of rubber is continuous with the ordinary elongation stress strain when both are plotted in the same units, and that the complete stress strain should accordingly be considered as a single continuous curve having an elongation branch and a compression branch with the origin as dividing point. 8. The analytic features of the complete stress strain are described. 9. Granting the observed concavity of the upper part of the elongation stress strain, and the thesis of continuity between elongation and compression, a point of inflection is bound to exist theoretically. 10. Implications of the thesis of continuity are: (1) An equation for the stress-strain curve must fit the complete curve; it is not sufficient that it fit the elongation branch only. (2) It is impossible to compute the compression stress strain from the ordinary (one-way) elongation stress-strain data. The two sets of data are related empirically. 11. When compressive force and equivalent two-way tensiles are based on actual cross sections, stress conditions at a point are expressed and we have the simple rule: Pressure at a point is numerically equal to the transverse tensions which, substituted therefor, will maintain the same strain.

On the Estimation of Dynamic Stress-Strain Curve By Impact Bending Test

2004 ◽  
Vol 2004.1 (0) ◽  
pp. 195-196
Author(s):  
Akihiro HOJO ◽  
Akiyosi CHATANI ◽  
Hiroshi TACHIYA
Keyword(s):  
Dynamic Stress ◽  
Strain Curve ◽  
Bending Test ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Impact Bending

Strain Localisation during a Tensile Test of Metallic Plates: Experimental and Numerical Results

2009 ◽  
Vol 417-418 ◽  
pp. 569-572
Author(s):  
D.A. Cendón ◽  
Jose M. Atienza ◽  
Manuel Elices Calafat
Keyword(s):  
Tensile Test ◽  
Strain Curve ◽  
Maximum Load ◽  
Surface Strain ◽  
Displacement Curve ◽  
Strain Localisation ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Strain Fields ◽  
Developed Surface

The stress-strain curve of a material is usually obtained from the load-displacement curve measured in a tensile test, assuming no strain localisation up to maximum load. However, strain localisation and fracture phenomena are far from being completely understood. Failure and strain localisation on plane tensile specimens has been studied in this work. A deeply instrumented experimental benchmark on steel specimens has been developed. Surface strain fields have been recorded throughout the tests, using an optical extensometer. This allowed characterisation of the strain localisation and failure processes. Tests have been numerically modelled for a more detailed analysis. Preliminary results show a substantial influence of geometrical specimen defects on the strain localisation phenomena that may be critical on the stress-strain curves obtained and in the failure mechanisms.

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