Lesson: Stress, Strain and Hooke's Law

Contributed by: VU Bioengineering RET Program, School of Engineering, Vanderbilt University
 SummaryThis lesson offers an introduction to Hooke's Law as well as stress-strain relationships. Students will first learn the governing equations. Then students will work through several example problems first individually, then as a class. In addition, the lesson includes a two-part associated activity. In the first part, students explore Hooke's law by experimentally determining an unknown spring constant. In the second part, students will apply what they've learned to create a strain graph depicting a tumor using Microsoft Excel. Finally, the lesson includes an attached stress-strain quiz to assess each student's knowledge following the activities.
 Engineering Connection Relating science and/or math concept(s) to engineering Over three hundred years ago, Robert Hooke identified a proportionality which has remained a fundamental concept to physicists and engineers today. Though his law was established for the case of springs alone, it has since been related to all materials of known surface area. The relationship used most readily today is the direct proportionality between stress and strain. Together, civil engineers, mechanical engineers and material scientists, must carefully select structural materials which are able to safely endure everyday stress while remaining in the elastic region of the stress-strain curve, otherwise permanent deformation will ensue. Architects who once chose stone for its aesthetic appeal are now choosing steel for its long term endurance. For biomedical engineers, titanium is the current material of choice for its biocompatibility but most importantly, it's capability to withstand the tensile and compressive stress of the body's weight. In the attached problem set, students explore applications of Hooke's Law and stress-strain relationships. In problem 7 specifically, students apply these relationships to the case of body tissue, like a biomedical engineer would.

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 Grade Level: 11 (10-12) Lessons in this Unit: 123 Time Required: 75 minutes Lesson Dependency :Lesson 1: Detecting Breast Cancer My Rating: Avg Rating: Not Yet Rated.Teacher Experiences  |  Share your experience!

Related Curriculum

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Educational Standards :

•   Common Core State Standards for Mathematics: Math
•   International Technology and Engineering Educators Association: Technology
•   Maryland: Science

A basic understanding of algebra and the ability to solve simple algebraic expressions. In addition, an understanding of the cancer detection challenge as introduced in the previous lesson.

After this lesson, students should be able to:

1. Understand stress and strain concepts and the relationship between them.
2. Understand Hooke's Law and apply it to analyze springs.
3. Be able to use Excel to make a simple strain plot.
4. Relate stress and strain to the engineering challenge.

In today's lesson, we will begin to learn about Hooke's law and then we will learn how to apply this proportionality to body tissue. We will learn exactly what the terms stress and strain describe as well as the relationship between them. After going through the lesson's material, I will pass out a handout with sample problems. I would like these to be worked to the best of your abilities independently first; then we will review the problems as a class. After growing familiar with using the new equations, we will explore Hooke's law in an associated activity (Applying Hooke's Law to Cancer Detection) by experimentally determining an unknown proportionality constant. After exploring Hooke's law, in the second portion of the activity, we will begin to apply what we've learned to develop a means of imaging body tissues and we will soon be able to detect malignant tumors! You will practice graphing prepared data to depict cancerous tissue. After we have mastered this material, we will have a quiz on stress, strain and Hooke's law. Please take careful notes and be sure to ask any questions you may have about the example problems we will be working through.
Referring back to the legacy cycle which we discussed in the previous lesson, today's lesson will constitute the research and revise phase. Refer back to your initial thoughts notes and record any new information which will apply to solving the challenge. Your goal today is to review, revise and expand you current knowledge! Now, let's learn how to detect cancer.

Legacy cycle information:

This lesson falls into the Research and Revise phase of the legacy cycle. Students will begin to learn the basic concepts required for creating a strain graph to depict cancerous tissue. Following this lesson, students should revise their initial thoughts and at the conclusion of the associated activity, students should have the skills necessary to Go Public with a solution. But before Going Public, students will complete the attached handout- Quiz- Stress, Strain and Hooke's Law as part of the Test Your Mettle phase of the legacy cycle. This quiz will offer formative assessment while the next lesson's Go Public phase will offer summative assessment.

Lecture Information:

In the late 1600s, Robert Hooke stated that "The power of any springy body is in the same proportion with the extension." Though Hooke's law has remained valid today, the wording has been corrected, replacing power with force. The law is explained by a direct proportionality between a spring's compression or expansion and the restoring force which ensues. The relationship is given by F= -k * Δx where Δx is the distance a spring has been stretch, F is the restoring force exerted by the spring and k is the spring constant which characterizes elastic properties of the spring's material. This law is valid within the elastic limit of a linear spring, when acting along a frictionless surface.
Extending Hooke's exploration of springs, it becomes apparent that most materials act like springs with force being directly proportional to displacement. But as compared to springs, other materials possess an area which must be accounted for. Replacing force with a measure of stress and displacement with a measure of strain, the following expression may be obtained, σ = E* ε. We will now explore the measures of stress and strain.
Stress is a measure of average force per unit area, given by σ = F/A where average stress, σ, equals force, F, acting over area, A. The SI unit for stress is pascals (Pa) which is equal to 1 Newton per square meter. The Psi is an alternative unit which expresses pounds per square inch. The units of stress are equal to the units of pressure which is also a measure of force per unit area.
Stress cannot be measured directly and is therefore inferred from a measure of strain and a constant known as Young's modulus of elasticity. The relationship is given by σ = E* ε, where σ represents stress, ε represents strain and E represents Young's modulus of elasticity. Using this means of inferring stress, strain is a geometrical measure of deformation and Young's modulus is a measure used to characterize the stiffness of an elastic material. Strain does not carry a unit but the units of Young's modulus are Pa.
Strain is characterized by the ratio of total deformation or change in length to the initial length. This relationship is given by ε = Δl/l 0 where strain, ε, is change in l divided by initial length , l 0 .
The following problems may be worked independently and reviewed as a class, encouraging students to become more familiar with using the equations given above. Each student should receive a copy of the Stress, Strain and Hooke's Law Problem Set (pdf).
You will need to SHOW ALL WORK. Useful constants that you will need to know are in a table below. (assume given constants have 3 Significant Figures (SF). Please also note the relationships we've just discussed given below.
Material
1. Steel
• Young's Module: 200x109 E(Pa)
1. Cast Iron
• Young's Module: 100x109 E(Pa)
1. Concrete
• Young's Module: 20.0x109 E(Pa)
F=m*a σ=F/A ε = Δl/l0 σ = E* ε F= -k * Δx
1. A 3340 N ball is supported vertically by a 1.90 cm diameter steel cable. Assuming the cable has a length of 10.3 m, determine the stress and the strain in the cable.
2. Consider an iron rod with a cross-sectional area of 3.81 cm2 that has a force of 66,700 N applied to it. Find the stress in the rod.
3. A concrete post with a 50.8 cm diameter is supporting a compressive load of 8910 Newtons. Determine the stress the post is bearing.
4. The concrete post in the previous problem has an initial height of 0.55 m. How much shorter is the post once the load is applied (in mm)?
5. A construction crane with a 1.90 cm diameter cable has a maximum functioning stress of 138 MPa. Find the maximum load that the crane can endure.
6. Consider Hooke's Law as a simple proportionality where F is directly proportional to Δx. Therefore, we know the force stretching a spring is directly proportional to the distance the spring stretches. If 223 N stretches a spring 12.7 cm, how much stretch can we expect to result from a of 534 N?
7. The figure below shows a column of fatty tissue, determine the strain in each of the three regions.

 Stress: 1. The physical pressure, pull, or other force exerted on a system by another. 2. A load, force, or system of forces producing a strain. 3. The ratio of force to area. Strain: 1. Deformation of a body or structure as a result of an applied force. 2. Stretch beyond the proper point or limit. Radiologist: 1. A medical specialist who examines photographs of tissues, organs, bones for use in the treatment of disease.

Embedded Assessment:

The attached problem set is to be completed in class and may be used to gauge student comprehension. The final question of the problem set and the application questions of Activity 2 offer an assessment of the students' understanding of the challenge. These questions should be used as a means of testing whether the students are applying their acquired knowledge toward solving the engineering challenge.

Post Lesson Assessment:

The attached quiz will offer a formative post-lesson assessment as part of the Test your Mettle phase of the legacy cycle.

Dictionary.com. Lexico Publishing Group,LLC. Accessed December 28, 2008. (Source of vocabulary definitions, with some adaptation) http://www.dictionary.com

Contributors

Luke Diamond, Primary Author, Meghan Murphy