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The calculations and figures for determining the elastic modulus, yield point, and ultimate strength of 1018 steel in uniaxial tension. sample calculations for converting force to stress and chart displacement to strain, as well as figures showing the stress-strain relationship and the elastic modulus prediction.
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Example Long Laboratory Report
I. R. Student
Lab Partners: I. R. Confused I. Dont Care
Properties of Materials Laboratory #
Date of Experiment: Jan. 15, 1888 Submission Date: Feb. 30, 2010 Submitted To: C. M. Fail
LIST OF FIGURES ................................................................................................................................................. i LIST OF TABLES ................................................................................................................................................... i LIST OF SYMBOLS ............................................................................................................................................... ii ABSTRACT ............................................................................................................................................................. 1 INTRODUCTION .................................................................................................................................................... 1 THEORY .................................................................................................................................................................. 1 EXPERIMENTAL PROCEDURE .......................................................................................................................... 2 RESULTS AND DISCUSSION............................................................................................................................... 3 CONCLUSIONS ...................................................................................................................................................... 5 REFERENCES ......................................................................................................................................................... 7 APPENDICES:
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Figure 1. "Dogbone" specimen geometry used for tensile test ................................................................................ 2 Figure 2. Force as a function of chart displacement for 1018 steel tested in tension ................................................................................................................ 3 Figure 3. Stress-strain plot for 1018 steel in tension ............................................................................................... 3 Figure 4. Low strain region of the stress-strain plot of 1018 steel showing two linear regions and predicted regression line ............................................................................................................................ 4 Figure 5. Determination of yield point by the 0.2% offset method ......................................................................... 5 Figure 6. Determination of the ultimate strength of 1018 steel in tension .................................................................................................................................... 6
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Table 1. Summary of elastic modulus, yield point, and ultimate strength of 1018 steel tested in uniaxial tension ........................................................................ 7 Table 2. Sample calculations - results of linear regression analysis ....................................................................... 7
The elastic modulus, yield point, and ultimate strength of 1018 steel were determined in uniaxial tension. The "dogbone" specimen geometry was used with the region of minimum cross section having the dimensions: thickness = 3.18 ± 0.05 mm, width = 6.35 ± 0.05 mm, and gage section = 38.1 ± 0.05 mm. The elastic modulus of the specimen was determined to be 196.7 GPa with a standard deviation of 2.0 GPa. The lower limit on the yield point was determined to be 357.1 MPa with a standard deviation of 6.3 MPa. The upper limit on the yield point was determined to be385.7 MPa with a standard deviation of 6.8 MPa. The ultimate strength was determined to be 487.7 MPa with a standard deviation of 8.6 MPa.
INTRODUCTION Mechanical properties are of interest to engineers utilizing materials in any application where forces are applied, dimensions are critical, or failure is undesirable. Three fundamental mechanical properties of metals are the elastic
determine the elastic modulus, yield point, and ultimate strength of 1018 steel.
THEORY When forces are applied to materials, they deform in reaction to those forces. The magnitude of the deformation for a constant force depends on the geometry of the materials. Likewise, the magnitude of the force required to cause a given deformation, depends on the geometry of the material. For these reasons, engineers define stress and strain. Stress (engineering definition) is given by:
Defined in this manner, the stress can be thought of as a normalized force. Strain (engineering definition) is given by:
The strain can be thought of as a normalized deformation. While the relationship between the force and deformation depends on the geometry of the material, the relationship between the stress and strain is geometry independent. The relationship between stress and strain is given by a simplified form of Hooke's Law [1]:
Since E is independent of geometry, it is often thought of as a material constant. However, E is known to depend on both the chemistry, structure, and temperature of a material. Change in any of these characteristics must be known before using a "handbook value" for the elastic modulus. Hooke's Law (Equation 3) predicts a linear relationship between the strain and the stress and describes the elastic response of a material. In materials where Hook's Law describes the stress-strain relationship, the elastic response is the dominant deformation mechanism. However, many materials exhibit nonlinear behavior at higher levels of stress. This nonlinear behavior occurs when plasticity becomes the dominant deformation mechanism. Metals are known to exhibit both elastic and plastic response regions [2]. The transition from an elastic response to a plastic response occurs at a
(bending), the yield point is an important characteristic to know. In practice, the yield point is the stress where the stress-strain behavior transforms from a linear relationship to a non-linear relationship. The most commonly used method to experimentally determine the yield point is the 0.2% offset method [3]. In this method, a line is drawn from
elastic modulus. The yield point is then determined as the intersection of this line with the experimental data. In materials that exhibit a large plastic response, the deformation tends to localize. Continued deformation occurs only in this local region, and is known as necking [4]. Necking begins at a critical point known as the ultimate stress
While many experimental tests exist to determine the mechanical properties, the simplest is the tension test. A convenient sample geometry for the tensile test is the "dogbone" geometry (Figure 1). In this test geometry, one end of
i
i o (^) i
ε
NOTE: In this paper the author is reporting values as (mean value) +/- (standard deviation ) of the data. For ME Lab I, you will usually report your values as (mean value) +/- (maximum probable error associated with mean value).
Figure 1. "Dogbone" specimen geometry used for tensile test.
the test specimen is held fixed while the other end is pulled in uniaxial tension collinear with the long axis of the sample. The forces throughout the sample and test machine are constant, but the stress varies with cross sectional area. The stress reaches critical values first in the region of the sample of minimum cross section, and the minimum cross section is in the sample. Therefore, the properties of the material are determined in this region.
A tensile test sample was machined from 1018 steel stock (106.1 mm X 19.05 mm X 3.18 mm) to the geometry shown in Figure 1. The region of minimum cross section had dimensions 6.35 mm in width, 3.18 mm in thickness, and 38.1 mm in length. The error in these dimensions was ± 0.05 mm. This sample was clamped into an Instron Universal Test Machine (Model 1125). The Instron test machine is a displacement controlled machine. One end of the sample was held at a fixed position with the other end was displaced at a constant rate. A load cell (Instron model 2511-319) was used to determine the force required to maintain a constant displacement rate. The accuracy of the load cell was ± 1 N. Data was collected on a strip chart (Instron model A1030) that monitored the force as a function of chart displacement. The stress in the sample at any force level can be determined from Equation 1. The calculation of strain requires the
from:
The rate of sample displacement was 1 ± 0.01 mm/min. The rate of chart displacement was 10 ± 0.1 mm/min. The elastic modulus, yield point, and ultimate stress were determined from the stress-strain plot.
RESULTS AND DISCUSSION The force-chart displacement graph for the 1018 steel examined is shown in Figure 2. The data shown in Figure 2 were converted to a corresponding stress-strain graph, Figure 3. Figure 3 clearly indicates two regions of linear behavior in the low strain region of the stress-strain graph. This behavior suggests that the sample was very compliant at low stress levels, and very stiff at high stress levels. Unfortunately, there is no structural or chemical reason why steel should exhibit an increasing modulus with increasing stress. Therefore, a more probable explanation is realignment and rotation of the test fixture in the low stress (low force) region. Remember, that the text fixture and the sample are under the same applied force. Under these experimental conditions, the most compliant member will dominate the stress-strain behavior. While the fixture appears very compliant during realignment and rotation, the fixture appears very stiff due to
sample
chart displacment chart
δ
δ (4)
The regression line shown in Figure 4 predicts an intercept value of -103.2 MPa. This non-zero intercept suggests that the sample must be compressed in order to obtain zero strain. It should be noted, that this non-zero intercept resulted from the shift in strain values due to realignment and rotation of the fixture, and not due to permanent deformation or violation of Hooke's Law. The region of fixture realignment and rotation creates some difficulties in determining the yield point by the 0.2% offset method. In the initial linear region, the majority of the deformation occurs in the fixture and not in the sample. At the same time, some small amount of deformation must occur in the sample since the force in the sample is greater than zero. The linear region used to estimate the elastic modulus of the sample, Figure 4, must be extrapolated to zero stress to determine the onset of deformation in the sample. A general regression line can be predicted:
The strain at zero stress (ε σ (^) =0 ) can be predicted from:
In the case of the relationship shown in Figure 4 (E=196.7 GPa and φ 3=-103.2 MPa), the strain at zero stress was
calculated to be 5.2 x 10-4. However, since both E and φ were predicted, and contain error associated with that prediction, there must also be error in calculated strain at zero stress. That error can be approximated according to:
where the standard deviation of the predicted intercept was 3.3 MPa, and the standard deviation of the predicted elastic modulus was 2 GPa. Solving Equation 7 yields:
ε σ
φ
=
2 2
Figure 4. Low strain region of the stress-strain plot of 1018 steel showing two linear regions and predicted regression line.
Since the errors terms input into Equation were standard deviations, the resulting ∆εσ=0 calculated (Equation 8) is also a standard deviation. It is appropriate to use the strain at zero stress (εσ=0 in Equation 6) minus some error term as the minimum limit for the starting point of the 0.2% offset line. Likewise, it is appropriate to use the strain at zero stress plus some error term as the maximum limit for the starting point of the 0.2% offset line. Two error terms can be used. The first is the standard deviation (∆εσ=0). However since the standard deviation is less than the corresponding distribution width, and alternate approach is to use three times the standard deviation as the appropriate error term [4]. While both approaches are common, the second approach produces the largest range of starting points for the 0.2% offset line, and would be considered the worst case scenario. This worst case scenario will be utilized in this instance. Therefore, the starting point for the line of 0.2% offset must be between 2.01 x 10-3^ and 3 x 10-3^ strain. The range of 0.2% offset lines is shown in Figure 5. The yield point is between 357.1 MPa and 385.7 MPa based upon the intersection of these lines with the experimental data. Addition error in these stress resulted form errors in determining the force and errors in the dimensions. The force was determined to the nearest 25 N. The error in stress can be approximated from:
The error in the force was 12.5 N; the error in both the width and thickness was 0.05 mm. It should be noted that Equation 9 predicts an error in stress that depends on the force. Forces of 7211 N and 7788 N correspond to stresses of 357.1 MPa and 385.7 MPa respectively. The error in stress for a stress of 357.1 MPa was calculated to be 6.3 MPa. The error in stress for a stress of 385.7 MPa was calculated to be 6.8 MPa. The ultimate stress (maximum in the stress-strain graph) can be determined graphically as shown in Figure 6. The ultimate stress was determined to be 487.7 MPa. The error in the ultimate stress as calculated from Equation 9 at a force level of 9848 N was ± 8.6 MPa.
∆ =1.7x 10 −^4
2 2
2
2
2
Figure 5. Determination of yield point by the 0.2% offset method.
TABLE 2. Sample Calculations - Results of linear regression analysis from Excel.
Stress range = 50 to 300 MPa Strain range = 0.0007 to 0.
Regression Output: Constant -103. Std Err of Y Est 3. R Squared 0. No. of Observations 15 Degrees of Freedom 13
X Coefficient(s) 196714. Std Err of Coef. 1990.
Displacement Force
Strain Stress (MPa)
