Engineering Question
MEE 324 Structural Mechanics Lab Lab One: Tension Test (Elasticity & Plasticity) Instructors: Dr. Masoud Yekani Fard and Dr. Dallas Kingsbury 1. Objectives The objectives of this lab experiment are as follows: a. Carry out the tensile test on plain carbon steel and aluminum specimens to measure Young’s Modulus (E), yield strength (SY), ultimate tensile strength (SUTS), % elongation, percentage of reduction of area, fracture strength (SF), and power-law plasticity coefficients. b. Plot the tension stress-strain relationship them. Use linear regression and zero shift to discard non-meaningful data and verify the validity of your results using literature values. c. Observe necking and other failure analyses on and near the fracture surface of ductile samples. d. Prepare a lab report to (i) describe the experiment and the results; (ii) compare the results with the expectations from theory (MAE 213 and MEE 322); (iii) discuss the possible reasons for the differences between theory and the experimental observations. 2. Description A loading machine will be used (Fig. 1a) to apply axial tensile load “P” at the ends of a bone-shaped specimen. A load cell and a strain gauge (or an extensometer (Fig. 1b)) will measure load and displacement (or strain) data points for the entire test. The collected data must be analyzed to determine all the quantities of interest related to the objectives listed in Section 1. The experiment will be done in displacement control with a 1 mm/min speed. The load will be increased until the complete fracture of the specimen. The lab manager (or TAs) will show how to set the parameters before the test. Fig. 1 (a) Loading frame model Instron 4411, (b) Extensometer 3. Procedure The students will measure the diameter of “dog-bone” specimens (Fig. 2) at three different places along the gauge length of the specimens and calculate the average value. The students will measure the entire length of the specimens (L1) and the length of grip areas with the larger diameter (L2). These measurements must be done before the sample is mounted on the machine. The length of the gauge length area is calculated as L0 = L1 – L2. The lab manager (or the TAs) mount one of the specimens. You will be asked to mount another specimen or help in the process of doing so for the remaining samples. Please make sure you pay particular attention to the safety precautions required during sample mounting and dismounting. In addition, care should be exercised to keep the sample from deforming during mounting. The extensometer is a sensitive and delicate device that measures the elongation of the sample, which allows the calculation of the strain. The extensometer will be installed on the sample using rubber bands by the lab supervisor after the sample is mounted on the machine. Monitor the reading from the extensometer on the load frame control panel or the computer. The lab supervisor may have to manipulate the rubber bands if the reading deviates from zero until the reading is acceptable. A high deviation from zero means that the extensometer’s gauge length is different from its calibration point; it affects the strain values stored during the test. After the test is done, the students should dismount the broken sample from the machine, observe the sample and pay close attention to the fracture surfaces at both ends of the specimen. You should discuss the fracture surface in the lab report. The ductile samples should develop a localized region where the diameter decreases until a fracture occurs. This region is called the “neck.” Measure the diameter at the neck and far from it. In addition, put the two pieces together the best way you can and measure the total length of the specimens. Fig. 2 (a) A dog-bone sample, (b) Schematic diagram of a tension test 4. Theory 4.1 Engineering Stress and Engineering Strain Engineering stress and engineering strain are defined using the original cross-sectional area and length. The material properties are determined from the engineering stress-strain diagram. In the elastic regime, engineering stress and engineering strain can be defined as: 𝜎𝜎𝑒𝑒𝑒𝑒𝑒𝑒,𝑖𝑖 = 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒,𝑖𝑖 = 𝑃𝑃𝑖𝑖 (1) ∆𝑙𝑙𝑖𝑖 (2) 𝐴𝐴0 𝐿𝐿0 where σeng,i and εeng,i are the engineering stress and strain at stage “i”, Pi and ∆li are the load and elongation at stage “i”, and A0 and L0 are the initial cross section area and the initial length. The gauge length of the extensometer (0.5” = 12.7 mm) is used for L0. The students should plot the engineering stress-engineering strain curve to find the slope of the curve, i.e., Young’s modulus. The stress and strain tensors for the elastic behavior if the load is assumed in the Z direction is as follows: 0 𝜎𝜎 = �0 0 0 0 0 −𝜗𝜗∆𝐿𝐿𝑖𝑖 ⎡ 𝐿𝐿0 ⎢ 𝜀𝜀 = ⎢ 0 ⎢ ⎣ 0 0 0� (3) 𝑃𝑃𝑖𝑖 𝐴𝐴0 0 −𝜗𝜗∆𝐿𝐿𝑖𝑖 𝐿𝐿0 0 0⎤ ⎥ 0⎥ ∆𝐿𝐿𝑖𝑖 ⎥ 𝐿𝐿0 ⎦ (4) where υ is the Poisson’s ratio. The normal strain in Z direction is as follows. (5) 𝜎𝜎𝑧𝑧𝑧𝑧 = 𝐸𝐸. 𝜀𝜀𝑧𝑧𝑧𝑧 The students should use software to fit a straight line to the elastic part of the engineering stress and strain data. The slope of the line will be the value of E. Ensure that only data corresponding to the linear relationship is included in the analysis. Perform a zero shift if needed. There could be some data at the beginning of the test that you may have to discard before doing the linear fit. The value of the engineering stress at which for a slight increase in load (stress) there is a jump in deformation (strain) is called Yield Strength (SYield). One usual practice is to measure the stress at 0.2% plastic strain called the 0.2% offset yield strength. The value of the stress at the intersection of the line with a slope equal to E, starting at a strain equal to 0.002 (0.2%), with the engineering stress-engineering strain curve, is then taken as the yield strength (Fig. 3). Use the 0.2% offset yield strength technique in your lab report and report the yield strength. The ultimate tensile strength (σUTS) is the maximum load per unit of the original cross-section area that the sample can withstand. It is the maximum value of stress in the engineering stress-engineering strain curve. The Fracture strength (σf) is an estimate of the actual fracture stress. The presence of the neck changes the geometry so much that the state of stress at and around the neck is no longer uniaxial. The percentage of elongation (100 × εf) is the engineering strain at the fracture point. The Percentage of area reduction (%R.A.) measures ductility, which is the capacity of a material to deform plastically before breaking. %𝑅𝑅. 𝐴𝐴. = 100 × 𝐴𝐴0 − 𝐴𝐴𝑓𝑓 𝐴𝐴0 (6) where Af is the area at the fracture. The percentage of elongation and the percentage of area reduction are ductility measures; both properties should be reported to have a complete characterization of the material behavior. Fig. 3 Typical engineering stress-engineering strain curve True stress and true strain are calculated based on the actual dimensions with deformation. Note that within the elastic regime σtrue ≈ σeng and εtrue ≈ εeng. A more significant numerical difference appears once yielding is reached, and plastic deformation begins. True stress-true strain curve considers the significant changes in the geometry of the samples as the plastic strain increases. 𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑖𝑖 = 𝑃𝑃𝑖𝑖 𝐴𝐴𝑖𝑖 (7) 𝐿𝐿 𝜀𝜀𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑖𝑖 = ln � 𝑖𝑖 � = ln � 𝐿𝐿0 𝐿𝐿0 + ∆𝐿𝐿𝑖𝑖 𝐿𝐿0 � = ln�1 + 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒,𝑖𝑖 � (8) where σtrue,i and εtrue,i are the true stress and strain at stage “i”, Ai and Li are the cross section area and the length at stage “i”. The instantaneous cross-section area of the specimen can be calculated taking advantage of the fact that plastic deformation in metals is incompressible, i.e., volume is constant. For a cylindrical specimen, this implies that: 𝐴𝐴0 𝐿𝐿0 = 𝐴𝐴𝑖𝑖 𝐿𝐿𝑖𝑖 → 𝐴𝐴𝑖𝑖 = 𝐴𝐴0 𝐿𝐿0 𝐿𝐿𝑖𝑖 = 𝐴𝐴0 𝐿𝐿𝑖𝑖 𝐿𝐿0 = 𝐴𝐴0 �1+𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒,𝑖𝑖 � (9) The assumption of constant volume is valid for homogeneous plastic deformation in the gauge length of the specimen. Therefore, once the neck starts (localized deformation), the assumption used for the equations shown above is no longer valid, and the instantaneous cross-section area cannot be obtained. Hence, the true stress-true strain curve can be obtained until the ultimate tensile strength, which is the point where necking starts. The coefficients of the power-law equation relating the true stress and the true strain for the plastic regime are as follows: 𝑛𝑛 𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑝𝑝 = 𝜎𝜎0 𝜀𝜀𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑝𝑝 + 𝜎𝜎𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 (10) where σtrue,p and εtrue,p are the true stress and true strain within the plastic regime. σ0 and n are the stress coefficient and hardening coefficient and are obtained by plotting the natural log of the true stress vs. the natural log of the true strain. This should result in a straight line as follows: ln�𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑝𝑝 − 𝜎𝜎𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 � = ln(𝜎𝜎0 ) + 𝑛𝑛 ln 𝜀𝜀𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡,𝑝𝑝 (11) If the strain and the stress follow the power-law hardening, then a plot of the natural log of true stress vs. the natural log of true strain must be a straight line. The stress and hardening coefficients are obtained by fitting a straight line to the data between yielding and UTS. Students ensure that all the points are well within the plastic regime before plotting the appropriate curves. Including data from the elastic regime will result in erroneous values for stress and hardening coefficients. In the case of steel, the range of the yielding phenomenon (“wiggly” part of the curve) just after the elastic regime does not obey the power-law hardening; therefore, use the data between the end of yielding and UTS. Students must use the computer to fit a straight line to the data. Fittings done by hand are too inaccurate to be of any use. 5. Report guidelines The report should be no longer than seven pages, including the title and the references. Title page [5 points]: Must contain lab number and title; full name; date of the experiment; and due date. Abstract [5 points]: The abstract should not be more than 250 words, and it must contain a brief description of what was done, the objectives of the experiment, and the main results regarding the properties measured and the differences among the materials tested. Data analysis and discussions [For the 2 ductile materials (steel and 2024 aluminum) [35×2 = 70 points]]: a. Plot the engineering stress-engineering strain diagram; [8 pts] b. Perform linear regression on the diagram to estimate the Young’s modulus E; [3 pts] c. Determine the yielding strength (σyield), ultimate tensile strength (σUTS), and fracture strength (σf) from the diagram; [6 pts] d. Calculate the percentage of elongation and the percentage of area reduction; [4 pts] e. Plot the true stress-true strain diagram; [6 pts] f. Obtain a log-log true stress-true strain diagram for the data points between yielding and UTS; [4 pts] g. Perform linear regression on the log-log diagram to estimate stress coefficient and hardening coefficient; [4 pts] Literature Values [10 points]: Find “literature” values of the measured material properties (E, σYield , σUTS) in the literature and discuss the difference and similarities. Conclusion [8 points]: This section should summarize the principal ideas extracted from the results and discussion. References [2 points]: The report should include a list of all the books and journal papers used in writing the lab report. Note: Make sure to sign the attendance sheet.
