1-Lab1 (+++)

Geodynamics,Spring2010 Lab#1 Due:April7,2010

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G eodynam ic s G 456- 556

Lab #1: Elasticity

Introduction:

Often the Earth behaves as an elastic material. in many ways we can understand how the earth deforms by understanding how a spring deforms.

Elasticreboundofupperplateofsubductionzone.Downgoingplateappliesaforcetotheupperplate,forcingittobend.Whentheloadreleases(earthquake),theupperplatereturnstoit’soriginalshape.

For an elastic material, two things are true:

1) the strain (deformation) is proportional to stress (load), This relationship is Hooke’s Law::

σ=Ε ε Where σ is stress, and ε is strain, and Ε is Young’s modulus (a material property

that relates to how “stiff” a material is) 2) When the load is released, the material returns to its original shape.


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Young’s modulus (quoted in either N/m2, or Pa), a physical property that relates to the “stiffness” of a material. Different kinds of rock have different Young’s moduli. For example,

Material Young’s Modulus Steel 200 GPa Concrete 20 GPa Sandstone 20 GPa Granite 50 Gpa Basalt 50 GPa

In this lab we will be studying elastic behaviour of a spring by applying a load (F) and

measuring the change in length (ΔL) to determine the spring constant, k.

k=F/ΔL The spring constant is related to Young’s modulus (by the original length and the cross-

section area; k =Ε∗Α/L. . . but we won’t worry about this…). By plotting the applied loads vs the changes in lengths, you can use the above equation to

determine the spring constant.

When a material is over-loaded, it no longer behaves as an elastic material, and may either break, or deform plastically, or viscously. Once a material exceeds it’s elastic strength, it does not return to it’s original shape.

To do:

Determine the spring constant for the two springs provided.

S u ppl i e s: 2 different springs collection of masses

P r oc e du r e :

As a class: starting with light weights, load each spring and measure the displacement. Remove the weight and measure the displacement. Use at least 6 weights.

Using Matlab, write an M-File to analyze and plot your data. Your M-File will need the following:

First, you must enter your data . Then, you will want to find the slope of the line to determine the spring constant. Matlab has a built-in function to curve-fit data, p=po l y fi t (st r e ss, st r a i n , n ). Where stress and strain are your load and change-in-length, and n=1 (for a straight line). In this case, the po l y fi t function returns the two variables for a straight line

Y=mx+b, or in this case, stress =p(1)* strain + p(2)


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p(1) is the slope (spring constant), and p(2) is the y-intercept (p(2) should be close to zero). HINT: Since the springs do not behave elastically for all the loads, be sure to use polyfit to find the spring constant for only the loads that cause an elastic change in length.

Now, write the commands to plot your data and your curve-fit. To plot your data and your curve-fit, you will need to use a variation of the following commands; yi=linspace(0,loadmax,10) %you will need to input an appropriate value of loadmax xi=(polyval(p, yi); plot (change_in_length, load, ‘ro’, xi, yi,’-‘); legend (‘data’, ‘elastic curve-fit’); title(['stress-strain curve for spring #1. Spring constant= ', num2str(p(1)), ‘ N/cm’]);

Your plot must be complete, i.e., title, labeled axes, etc; In the title, include the value of the spring constant.

You may find this web page a helpful resource http://www.physics.ualr.edu/Open_Lab/plotdata.html .

On your plot, indicate (by hand, or using matlab) where material stops behaving as an elastic medium.

To h a n d i n :

1- Your M-Flie 2- Your 2 plots, including data, curve fit, legend, title, labeled axes, spring

constant, and indicate where the spring ceased to behave elastically Your plot should look something like the plot below. Note that the values in this

example are completely made-up have nothing to do with reality!

3. Now, apply the ideas of elastic behaviour to a real earth problem. Briefly (just a few

sentences) describe a geologic setting that deforms elastically. In your description, be sure to discuss what is the applied load, how the deformation is measured, and what happens when (if) the elastic limit is exceeded.

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1-Lab1 (+++)