# 7.5: Stress and Strain

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

The earth’s crust is constantly subjected to forces that push, pull, or twist it. These forces are called stress. In response to stress, the rocks of the earth undergo strain, also known as deformation. Strain is any change in volume or shape.There are four general types of stress. One type of stress is uniform, which means the force applies equally on all sides of a body of rock. The other three types of stress, tension, compression and shear, are non-uniform, or directed, stresses.All rocks in the earth experience a uniform stress at all times. This uniform stress is called lithostatic pressure and it comes from the weight of rock above a given point in the earth. Lithostatic pressure is also called hydrostatic pressure. (Included in lithostatic pressure are the weight of the atmosphere and, if beneath an ocean or lake, the weight of the column of water above that point in the earth. However, compared to the pressure caused by the weight of rocks above, the amount of pressure due to the weight of water and air above a rock is negligible, except at the earth’s surface.) The only way for lithostatic pressure on a rock to change is for the rock’s depth within the earth to change.Because lithostatic pressure is a uniform stress, a change in lithostatic pressure does not cause fracturing and slippage along faults. Nevertheless, it may be the cause of certain types of earthquakes. In subducting tectonic plates, the increased pressure of greater depth within the earth may cause the minerals in the plate to metamorphose spontaneously into a new set of denser minerals that are stable at the higher pressure. This is thought to be the likely cause of certain types of deep earthquakes in subduction zones, including the deepest earthquakes ever recorded.

Rocks are also subjected to the three types of directed (non-uniform) stress – tension, compression, and shear.

• Tension is a directed (non-uniform) stress that pulls rock apart in opposite directions. The tensional (also called extensional) forces pull away from each other.
• Compression is a directed (non-uniform) stress that pushes rocks together. The compressional forces push towards each other.
• Shear is a directed (non-uniform) stress that pushes one side of a body of rock in one direction, and the opposite side of the body of rock in the opposite direction. The shear forces are pushing in opposite ways.

In response to stress, rock may undergo three different types of strain – elastic strain, ductile strain, or fracture.

• Elastic strain is reversible. Rock that has undergone only elastic strain will go back to its original shape if the stress is released.
• Ductile strain is irreversible. A rock that has undergone ductile strain will remain deformed even if the stress stops. Another term for ductile strain is plastic deformation.
• Fracture is also called rupture. A rock that has ruptured has abruptly broken into distinct pieces. If the pieces are offset—shifted in opposite directions from each other—the fracture is a fault.

## DUCTILE AND BRITTLE STRAIN

Earth’s rocks are composed of a variety of minerals and exist in a variety of conditions. In different situations, rocks may act either as ductile materials that are able to undergo an extensive amount of ductile strain in response to stress, or as brittle materials, which will only undergo a little or no ductile strain before they fracture. The factors that determine whether a rock is ductile or brittle include:

• Composition—Some minerals, such as quartz, tend to be brittle and are thus more likely to break under stress. Other minerals, such as calcite, clay, and mica, tend to be ductile and can undergo much plastic deformation. In addition, the presence of water in rock tends to make it more ductile and less brittle.
• Temperature—Rocks become softer (more ductile) at higher temperature. Rocks at mantle and core temperatures are ductile and will not fracture under the stresses that occur deep within the earth. The crust, and to some extent the lithosphere, are cold enough to fracture if the stress is high enough.
• Lithostatic pressure—The deeper in the earth a rock is, the higher the lithostatic pressure it is subjected to. High lithostatic pressure reduces the possibility of fracture because the high pressure closes fractures before they can form or spread. The high lithostatic pressures of the earth’s sub-lithospheric mantle and solid inner core, along with the high temperatures, are why there are no earthquakes deep in the earth.
• Strain rate—The faster a rock is being strained, the greater its chance of fracturing. Even brittle rocks and minerals, such as quartz, or a layer of cold basalt at the earth’s surface, can undergo ductile deformation if the strain rate is slow enough.

Most earthquakes occur in the earth’s crust. A smaller number of earthquakes occur in the uppermost mantle (to about 700 km deep) where subduction is taking place. Rocks in the deeper parts of the earth do not undergo fracturing and do not produce earthquakes because the temperatures and pressures there are high enough to make all strain ductile. No earthquakes originate from below the the earth’s upper mantle.

## STRESS AND FAULT TYPES

The following correlations can be made between types of stress in the earth, and the type of fault that is likely to result:

• Tension leads to normal faults.
• Compression leads to reverse or thrust faults.
• Horizontal shear leads to strike-slip faults.

Correlations between type of stress and type of fault can have exceptions. For example, zones of horizontal stress will likely have strike-slip faults as the predominant fault type. However there may be active normal and thrust faults in such zones as well, particularly where there are bends or gaps in the major strike-slip faults.

To give another example, in a region of compression stress in the crust, where sheets of rock are stacked on active thrust faults, strike-slip faults commonly connect some of the thrust faults together.