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Geodynamics

Basics of elasticity Lecture 5.4 - Uniaxial strain

Lecturer: David Whipp david.whipp@helsinki.fi

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Goals of this lecture

• Present an example of uniaxial strain in the context of sedimentation

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Stresses as a result of burial

• How does elastic stress change in sedimentary rocks as a result of burial?

• What stress/strain conditions are appropriate for this scenario?

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Uniaxial strain

• Uniaxial strain occurs when only one component of the principal strains is nonzero (𝜀1 in this example)

• In this case, if we consider 𝜀2 = 𝜀3 = 0, the equations for linear elasticity reduce to

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�2 = �3 =⌫

(1� ⌫)�1

�1 =(1� ⌫)E"1

(1 + ⌫)(1� 2⌫)

Uniaxial strain

• Let’s consider a parcel of rock initially at the surface that now has been buried by sediments of density 𝜌 to a depth ℎ

• In this case, we can assume 𝜎1 is vertical and equal to the weight of the overburden, 𝜎1 = 𝜌𝑔ℎ

• From the equations on the previous slide, we find

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�2 = �3 =⌫

(1� ⌫)⇢gh

p =1

3(�1 + �2 + �3) =

(1 + ⌫)

3(1� ⌫)⇢gh

Uniaxial strain

• Let’s now consider the effects on the deviatoric stress, the principal stresses minus pressure 𝑝

• Which results in deviatoric stresses

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�01 = �1 � p =

2(1� 2⌫)

3(1� ⌫)⇢gh

�02 = �2 � p = �0

3 = �3 � p = � (1� 2⌫)

3(1� ⌫)⇢gh

Uniaxial strain

• Let’s now consider the effects on the deviatoric stress, the principal stresses minus pressure 𝑝

• Which results in deviatoric stresses

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p =1

3(�1 + �2 + �3) =

(1� ⌫)

3(1� ⌫)⇢gh

�01 = �1 � p =

2(1� 2⌫)

3(1� ⌫)⇢gh

�02 = �2 � p = �0

3 = �3 � p = � (1� 2⌫)

3(1� ⌫)⇢ghUnder tension!

Let’s see what you’ve learned…

• If you’re watching this lecture in Moodle, you will now be automatically directed to the quiz!

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