Geol 542: Advanced Structural Geology
Fall 2013
Problem Set #6: Experimental Determination of Rock Strength 1. The attached figure shows a plot of the strain data collected on a cylindrical sample of Berea sandstone during a constant strain rate, uniaxial compression test. The actual data are shown in the attached Table. Note the first lateral strain is a negative number (negative stresses/strains are compressional/contractional). Use an Excel spreadsheet to tabulate the data and answer the following questions. In all graphs that you plot, use the absolute value of the axial stress and strain (i.e., plot axial stress and strain as positive numbers). a) Use the data to determine the Poisson's ratio, ν, of the sample at each load step. Print out and include your Excel spreadsheet table. (8) See table below. Poisson’s ratio is in red. (8) b) Make a plot of your calculated values of ν as a function of the load (i.e., axial stress). (5) (5) Geol 542: Advanced Structural Geology
Fall 2013
c) What is happening at the very start of the experiment (i.e., in the first load increment) to give such an unusual ν value? Hint: think about what happens to most rocks at the very start of a rock compression test. (3) In the first load increment, the sample exhibits a negative Poisson’s ratio, which is not expected in elastic materials. It essentially indicates that the sample contracts radially at the same time that it is contracting axially. This behavior likely reflects the closing of preexisting microcracks and other holes or spaces in the sample before elastic deformation commences. (3) d) Now make a plot of axial stress vs axial strain (i.e., you will be recreating one of the curves on the attached graph). Calculate the Young’s modulus E for each increment of loading (i.e., slope of line) and plot it on a separate graph as a function of loading. Include a table showing your calculated values of E (this can be another column added to the spreadsheet in part (a). To calculate E, use the formula (yj – yi) / (xj – xi), where (xi, yi) and (xj, yj) are consecutive points on the graph. (9) See chart in (a) for calculated values of E. (3) (3) Geol 542: Advanced Structural Geology
Fall 2013
(3) e) Based on your two charts in question (d), comment on whether this sandstone can be considered to behave in a linear elastic manner as it deforms (in your answer, refer to different stages of the experiment in terms of the load increments from 1-­‐16). Make sure you comment on the expected nature of Young’s modulus for a linear elastic material. In explaining your range of values of E, be sure to compare your stress vs strain curve with the expected shape of this curve for most rocks, how this translates into what is physically happening within the rock during different stages of the experiment, and why this affects your calculated values of E. (8) In a linear elastic material, stress is linearly related to strain on a stress versus strain graph. The slope of the line defines the Young’s modulus, which is a constant. In the sandstone experimental data, the stress versus strain line is not everywhere linear. In the first 3 loading increments (up to 4.1 MPa), the curve is slightly concave upward. It is then approximately linear up to loading increments 12-­‐14 (30.65 – 36.04 MPa). Beyond this point, the line is slightly convex upward. This stress/strain behavior is not unexpected in otherwise linearly elastic materials. Clearly, the sandstone does behave in a linear elastic manner during a portion of the experiment. Accordingly, the Young’s modulus is very nearly constant during this part of the deformation (~18.6 – 24.4 MPa). Note: Young’s modulus is calculated by considering the slope of the line (i.e., Ei = (yj – yi)/(xj – xi), where i and j are consecutive data points). The first value of E is the slope from the origin to the first data point.(8) f) Do the values of the Poisson’s ratio you calculated make sense based on what you know about this parameter and how it describes the behavior of elastic materials? (3) For elastic materials, ν should be in the range 0 – 0.5. Other than the initial negative value of ν, calculated values of ν range from 0.17 to 1.51. All values in loading increments 2-­‐9 (2.27 – 20.80 MPa axial loading) are <0.5, suggesting elastic behavior. Beyond loading increment 9, some of the deformation must be inelastic (e.g., formation of axial cracks within the sample) despite the seemingly linear elastic stress vs. strain behavior of the bulk sample up to loading increment 12. (3) Geol 542: Advanced Structural Geology
Fall 2013
g) What do we call the point along the stress vs strain curve where the behavior begins to be no longer elastic, and what is the value of the stress at this point? (2) This point (likely at loading increment 14) is called the yield point (with a corresponding yield stress of 36.04 MPa). (2) h) What do we call the maximum point on this particular stress vs strain graph (a property of this rock sample)? (2) Uniaxial compressive strength. (2) i) Why do the lateral strains suddenly increase rapidly as the axial strain reaches around -­‐2.3 millistrains? Explain what is physically happening in the rock at this point. Hint: think about how deformation progresses in a uniaxial experiment. (3) The lateral strains increase suddenly due to brittle failure of the specimen. In a uniaxial experiment, there is no confining stress so failure is likely to occur as axial splitting (i.e., the creation of a vertical tension fracture), which causes the two halves of the sample to separate (i.e., the lateral strain will suddenly increase). (3) j) Was this experiment conducted in a soft or stiff testing machine? Explain your reasoning. (2) Probably a soft testing machine because the experiment abruptly ends at the point of brittle failure. (2) [46] Geol 542: Advanced Structural Geology
Fall 2013