MSEG302 HW7 Solutions Fall2019 PDF

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Download MSEG302 HW7 Solutions Fall2019 PDF

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MSEG 302: Materials Science for Engineers, Fall 2019 Homework 7-SOLUTIONS assigned 10/24 due 10/31 at 11:00 AM Please neatly hand write or type your answers to the following questions. Submit a pdf of your solutions to Canvas by 11:00 AM on Thursday October 31.

1

Stress/Strain Diagrams [2 pt]

Consider a cylindrical titanium wire 3.0 mm in diameter and 2.5 × 104 mm long. Calculate its elongation when a load of 500 N is applied. Assume the deformation is totally elastic, with an elastic modulus of 107 GPa. Solution: We are solving for the elongation, or ∆l. We’ll use the equations for axial stress, strain and modulus to solve for ∆l. F A0 ∆l ǫ= l0 σ E= ǫ

σ=

Combining these equations and solving the ∆l leads to

∆l = ∆l =

F lo EAo

(500N )(25m) (107 × 109 Pa)(π ∗ (0.003/2)2 ) ∆l = 0.0165m ∆l = 16.5 mm

2

Design Safety Factor [3 pt]

A large tower is to be supported by a series of steel wires; it is estimated that the load on each wire will be 11.1 kN. Determine the minimum required wire diameter, assuming a factor of safety of 2.0 and a yield strength of 1030 MPa for the steel. Solution: We will use the stress equation σ = F/A to solve for the minimum area (and diameter) such that the stress does not exceed our working yield stress for the given force. The working yield stress is given by σw = σy /2, so σw = 515MPa. Using this stress to solve for the area gives:

F A F A= σ 11100N A= 515E6P a A = 2.15 × 10−5 m2 σ=

The diameter then, is r

A π r 2.15 × 10−5 m2 d =2∗ π d = 5.2 × 10−3 m d =2∗

d = 5.2 mm

3

CRSS [3pt]

Consider a single crystal of silver oriented such that a tensile stress is applied along a [0 0 1] direction. If slip occurs on a (1 1 1) plane and in a [1 0 1] direction and is initiated at an applied tensile stress of 1.1 MPa, compute the critical resolved shear stress. Consider a single crystal of silver oriented such that a tensile stress is applied along a [0 0 1] direction. If slip occurs on a (1 1 1) plane and in a [1 0 1] direction and is initiated at an applied tensile stress of 1.1 MPa, compute the critical resolved shear stress. Solution: Since we know that slip is initiated at an applied tensile stress of 1.1 MPa, then σy = 1.1 MPa. Then the τR = σy cos(φ)cos(λ). To find λ we need to find the angle between [0 0 1] and [1 0 1]. The left figure below shows the angle λ. Solving for lambda, gives a cos(λ) = √ 2a 1 λ = cos−1 √ 2 λ = 45◦

To find φ we need to find the angle between [0 0 1] and [1 1 1]. The right figure below shows the angle φ. Remember that for the slip plane (1 1 1) the normal to the plane is the vector [1 1 1].

a cos(φ) = √ 3a 1 φ = cos−1 √ 3 φ = 54.7◦

We can now solve for the critical resolved shear stress τR . τR = σy cos(φ)cos(λ) = 1.1cos(54.7)cos(45) = 0.45 MPa

4

Stress-Strain Plots [12 pt]

In class we generated data for steel, brass and aluminum test specimens. You will analyze this data to determine the elastic modulus and yield strength for each specimen. Find the text file posted on Canvas under “HW and Solutions” labeled “HW6MechPropData2 Fall2019”. This text file should have six columns of data: position (m) and force (N) data for aluminum (columns 1 and 2), brass (columns 3 and 4) and steel (columns 5 and 6). For each metal complete the following: a. Convert the data to an engineering stress-strain plot. The cylindrical specimen diameter is given as 3.3 mm and the initial length is 39.3 mm. Label the axes with stress (GPa) and strain and include a title. Solution: The figure below shows the stress-strain curves for aluminum, brass and steel. The strain values have been scaled by 0.01. That is, multiply the plotted values by 0.01 to get the actual strain values. b. Using your data, estimate the elastic modulus (in GPa) . You may have to replot the linear portion of the data and do a linear fit to determine the modulus. Briefly (1-2 sentences) state how you determined the elastic modulus. Solution: In order to find the elastic modulus, you must find the slope of the linear region. This is best done by replotting the linear portion, and fitting a trendline to that data. Defining the region

to plot can be up to some interpretation. For each metal I chose a set of points below the yield strength. The trendline was plotted with a set intercept at 0 for brass and steel. The data and general allowable range are noted in the table below. Please note: There’s some interpretation to trendline fitting, and so the process used to determine the modulus is more important here than the actual number determined. The possible range takes into account literature values as well as possible interpretations of the data.

Metal Exp. Elastic Modulus (GPa) Possible Range Aluminum 69 50-75 Brass 105 80-125 Steel 280 200-300 c. Estimate the yield strength using the ǫp = 0.002 method. You can determine this using your plotting software or you can carefully estimate visually. Briefly (1-2 sentences) state your process to determine the yield strength. Solution: In order to find the yield strength using the ǫp = 0.002 method, you can draw a line with slope equal to your modulus that crosses the strain axis at 0.002. These are the dotted lines shown on the figure below. The corresponding stress value at the interesection of this line with the stress-strain curve is the yield strength. Again, your description and data will be more important here than the value you determined.

Metal Aluminum Brass Steel

Exp. Yield Strength (GPa) 0.36 0.29 0.73

Range 0.3-0.4 0.2-0.35 0.7-0.8

CATME As part of our adaptive technology unit, you will be placed into groups. To do so, our class will use the CATME Team-Maker online resource. You should have received an email from CATME asking you to create an account and password. Please do so using your @udel email address as your CATME name. The password is created by you, not the instructor or the CATME program. At 12:00AM on Oct 25, you will receive a survey from CATME asking you a few demographic and personality questions. Your answers to these questions will be used to place you into groups by CATME. These answers will ONLY by used for group formation, and nothing else. Answer the questions honestly, as it will be helpful in creating functional teams. Please complete the survey by Oct 31 at 11:59 PM. The survey will close after this time. Do not wait until the last minute to complete the survey. Completing the survey will count as your first points earned on the Adaptive Tech. project (which will be handed out Thursday Oct 31)....