Lesson: Stress, Strain and Hooke's LawContributed by: VU Bioengineering RET Program, School of Engineering, Vanderbilt University
Educational Standards :
Pre-Req Knowledge (Return to Contents)
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.
Learning Objectives (Return to Contents)
After this lesson, students should be able to:
Introduction/Motivation (Return to Contents)
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.
Lesson Background & Concepts for Teachers (Return to Contents)
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.
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.
F=m*a σ=F/A ε = Δl/l0 σ = E* ε F= -k * Δx
Vocabulary/Definitions (Return to Contents)
Associated Activities (Return to Contents)
Attachments (Return to Contents)
Assessment (Return to Contents)
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.
References (Return to Contents)
Dictionary.com. Lexico Publishing Group,LLC. Accessed December 28, 2008. (Source of vocabulary definitions, with some adaptation) http://www.dictionary.com
ContributorsLuke Diamond, Primary Author, Meghan Murphy
Copyright© 2007 by Vanderbilt University
Including copyrighted works from other educational institutions and/or U.S. government agencies; all rights reserved. The contents of this digital library curriculum were developed under a grant from the National Science Foundation RET grants no. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government.
Supporting Program (Return to Contents)VU Bioengineering RET Program, School of Engineering, Vanderbilt University
Last Modified: August 1, 2014