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12.1: Stress and Strain - Geosciences


Rocks are subject to stress—mostly related to plate tectonics but also to the weight of overlying rocks—and their response to that stress is strain (deformation). At transform plate boundaries, where plates are moving side by side there is sideways or shear stress—meaning that there are forces in opposite directions parallel to a plane. Rocks have highly varying strain responses to stress because of their different compositions and physical properties, and because temperature is a big factor and rock temperatures within the crust can vary greatly.

We can describe the stress applied to a rock by breaking it down into three dimensions—all at right angles to one-another (Figure (PageIndex{1})). If the rock is subject only to the pressure of burial, the stresses in all three directions will likely be the same. If it is subject to both burial and tectonic forces, the pressures will be different in different directions.

Rock can respond to stress in three ways: it can deform elastically, it can deform plastically, and it can break or fracture. Elastic strain is reversible; if the stress is removed, the rock will return to its original shape just like a rubber band that is stretched and released. Plastic strain is not reversible. As already noted, different rocks at different temperatures will behave in different ways to stress. Higher temperatures lead to more plastic behaviour. Some rocks or sediments are also more plastic when they are wet. Another factor is the rate at which the stress is applied. If the stress is applied quickly (for example, because of an extraterrestrial impact or an earthquake), there will be an increased tendency for the rock to fracture. Some different types of strain response are illustrated in Figure (PageIndex{2}).

The outcomes of placing rock under stress are highly variable, but they include fracturing, tilting and folding, stretching and squeezing, and faulting. A fracture is a simple break that does not involve significant movement of the rock on either side. Fracturing is particularly common in volcanic rock, which shrinks as it cools. The basalt columns in Figure (PageIndex{3})a are a good example of fracture. Beds are sometimes tilted by tectonic forces, as shown in Figure (PageIndex{3})b, or folded as shown in Figure (PageIndex{1}).

When a body of rock is compressed in one direction it is typically extended (or stretched) in another. This is an important concept because some geological structures only form under compression, while others only form under tension. Most of the rock in Figure (PageIndex{3})c is limestone, which is relatively easily deformed when heated. The dark rock is chert, which remains brittle. As the limestone stretched (parallel to the hammer handle) the brittle chert was forced to break into fragments to accommodate the change in shape of the body of rock. A fault is a rock boundary along which the rocks on either side have been displaced relative to each other (Figure (PageIndex{3})d).

Media Attributions

  • Figures 12.1.1, Figure (PageIndex{2}), Figure (PageIndex{3}): © Steven Earle. CC BY.

12.4: Stress, Strain, and Elastic Modulus (Part 1)

  • Contributed by OpenStax
  • General Physics at OpenStax CNX
  • Explain the concepts of stress and strain in describing elastic deformations of materials
  • Describe the types of elastic deformation of objects and materials

A model of a rigid body is an idealized example of an object that does not deform under the actions of external forces. It is very useful when analyzing mechanical systems&mdashand many physical objects are indeed rigid to a great extent. The extent to which an object can be perceived as rigid depends on the physical properties of the material from which it is made. For example, a ping-pong ball made of plastic is brittle, and a tennis ball made of rubber is elastic when acted upon by squashing forces. However, under other circumstances, both a ping-pong ball and a tennis ball may bounce well as rigid bodies. Similarly, someone who designs prosthetic limbs may be able to approximate the mechanics of human limbs by modeling them as rigid bodies however, the actual combination of bones and tissues is an elastic medium.

For the remainder of this section, we move from consideration of forces that affect the motion of an object to those that affect an object&rsquos shape. A change in shape due to the application of a force is known as a deformation. Even very small forces are known to cause some deformation. Deformation is experienced by objects or physical media under the action of external forces&mdashfor example, this may be squashing, squeezing, ripping, twisting, shearing, or pulling the objects apart. In the language of physics, two terms describe the forces on objects undergoing deformation: stress and strain.

Stress is a quantity that describes the magnitude of forces that cause deformation. Stress is generally defined as force per unit area. When forces pull on an object and cause its elongation, like the stretching of an elastic band, we call such stress a tensile stress. When forces cause a compression of an object, we call it a compressive stress. When an object is being squeezed from all sides, like a submarine in the depths of an ocean, we call this kind of stress a bulk stress (or volume stress). In other situations, the acting forces may be neither tensile nor compressive, and still produce a noticeable deformation. For example, suppose you hold a book tightly between the palms of your hands, then with one hand you press-and-pull on the front cover away from you, while with the other hand you press-and-pull on the back cover toward you. In such a case, when deforming forces act tangentially to the object&rsquos surface, we call them &lsquoshear&rsquo forces and the stress they cause is called shear stress.

The SI unit of stress is the pascal (Pa). When one newton of force presses on a unit surface area of one meter squared, the resulting stress is one pascal:

In the British system of units, the unit of stress is &lsquopsi,&rsquo which stands for &lsquopound per square inch&rsquo (lb/in 2 ). Another unit that is often used for bulk stress is the atm (atmosphere). Conversion factors are

[1 psi = 6895 Pa and 1 Pa = 1.450 imes 10^<-4> psi]

[1 atm = 1.013 imes 10^<5> Pa = 14.7 psi ldotp]

An object or medium under stress becomes deformed. The quantity that describes this deformation is called strain. Strain is given as a fractional change in either length (under tensile stress) or volume (under bulk stress) or geometry (under shear stress). Therefore, strain is a dimensionless number. Strain under a tensile stress is called tensile strain, strain under bulk stress is called bulk strain (or volume strain), and that caused by shear stress is called shear strain.

The greater the stress, the greater the strain however, the relation between strain and stress does not need to be linear. Only when stress is sufficiently low is the deformation it causes in direct proportion to the stress value. The proportionality constant in this relation is called the elastic modulus. In the linear limit of low stress values, the general relation between stress and strain is

[stress = (elastic modulus) imes strain ldotp label<12.33>]

As we can see from dimensional analysis of this relation, the elastic modulus has the same physical unit as stress because strain is dimensionless.

We can also see from Equation ef <12.33>that when an object is characterized by a large value of elastic modulus, the effect of stress is small. On the other hand, a small elastic modulus means that stress produces large strain and noticeable deformation. For example, a stress on a rubber band produces larger strain (deformation) than the same stress on a steel band of the same dimensions because the elastic modulus for rubber is two orders of magnitude smaller than the elastic modulus for steel.

The elastic modulus for tensile stress is called Young&rsquos modulus that for the bulk stress is called the bulk modulus and that for shear stress is called the shear modulus. Note that the relation between stress and strain is an observed relation, measured in the laboratory. Elastic moduli for various materials are measured under various physical conditions, such as varying temperature, and collected in engineering data tables for reference (Table (PageIndex<1>)). These tables are valuable references for industry and for anyone involved in engineering or construction. In the next section, we discuss strain-stress relations beyond the linear limit represented by Equation ef<12.33>, in the full range of stress values up to a fracture point. In the remainder of this section, we study the linear limit expressed by Equation ef<12.33>.

Table (PageIndex<1>): Approximate Elastic Moduli for Selected Materials
Material Young&rsquos modulus × 10 10 Pa Bulk modulus × 10 10 Pa Shear modulus × 10 10 Pa
Aluminum 7.0 7.5 2.5
Bone (tension) 1.6 0.8 8.0
Bone (compression) 0.9
Brass 9.0 6.0 3.5
Brick 1.5
Concrete 2.0
Copper 11.0 14.0 4.4
Crown glass 6.0 5.0 2.5
Granite 4.5 4.5 2.0
Hair (human) 1.0
Hardwood 1.5 1.0
Iron 21.0 16.0 7.7
Lead 1.6 4.1 0.6
Marble 6.0 7.0 2.0
Nickel 21.0 17.0 7.8
Polystyrene 3.0
Silk 6.0
Spider thread 3.0
Steel 20.0 16.0 7.5
Acetone 0.07
Ethanol 0.09
Glycerin 0.45
Mercury 2.5
Water 0.22


Rupture Surfaces Are Where the Action Happens

Images like 12.3 are useful for illustrating elastic deformation and rupture, but they can be misleading. The rupture that happens doesn’t occur as in 12.3, with the block being ruptured through and through. The rupture and displacement only happen along a subsection of a fault, called the rupture surface. In Figure 12.4, the rupture surface is the dark pink patch. It takes up only a part of the fault plane (lighter pink). The fault plane represents the surface where the fault exists, and where ruptures have happened in the past. Although the fault plane is drawn as being flat in Figure 12.4, faults are not actually perfectly flat.

The location on the fault plane where the rupture happens is called the hypocentre or focus of the earthquake (Figure 12.4, right). The location on Earth’s surface immediately above the hypocentre is the epicentre of the earthquake.

Figure 12.4 Rupture surface (dark pink), on a fault plane (light pink). The diagram represents a part of the crust that may be tens or hundreds of kilometres long. The rupture surface is the part of the fault plane along which displacement occurred. Left: In this example, the near side of the fault is moving to the left, and the lengths of the arrows within the rupture surface represent relative amounts of displacement. Coloured arrows represent propagation of failure on a rupture surface. In this case, the failure starts at the dark blue heavy arrow and propagates outward, reaching the left side first (green arrows) and the right side last (yellow arrows). Right: An earthquake’s location can be described in terms of its hypocentre (or focus), the location on the fault plane where the rupture happens, or in terms of its epicentre (red star), the location above the hypocentre. Source: Left: Steven Earle (2015) CC BY 4.0 view source. Right: Karla Panchuk (2017) CC BY 4.0.

Within the rupture surface, the amount of displacement varies. In Figure 12.4, the larger arrows indicate where there has been more displacement, and the smaller arrows where there has been less. Beyond the edge of the rupture surface there is no displacement at all. Notice that this particular rupture surface doesn’t even extend to the land surface of the diagram.

The size of a rupture surface and the amount of displacement along it will depend on a number of factors, including the type and strength of the rock, and the degree to which the rock was stressed beforehand. The magnitude of an earthquake will depend on the size of the rupture surface and the amount of displacement.

A rupture doesn’t occur all at once along a rupture surface. It starts at a single point and spreads rapidly from there. Figure 12.4 illustrates a case where rupturing starts at the heavy blue arrow in the middle, then continues through the lighter blue arrows. The rupture spreads to the left side (green arrows), then the right (yellow arrows).

Depending on the extent of the rupture surface, the propagation of failures (incremental ruptures contributing to making the final rupture surface) from the point of initiation is typically completed within seconds to several tens of seconds. The initiation point isn’t necessarily in the centre of the rupture surface it may be close to one end, near the top, or near the bottom.


Introduction

Many physical properties of crystalline materials are direction dependent because the arrangement of the atoms in the crystal lattice are different in different directions. If one heats a block of glass it will expand by the same amount in each direction, but the expansion of a crystal will differ depending on whether one is measuring parallel to the a-axis or the b-axis. For this reason properties such as the elasticity and thermal expansivity cannot be expressed as scalars. We use tensors as a tool to deal with more this complex situation and because single crystal properties are important for understanding the bulk behavior of rocks (and Earth), we wind up dealing with tensors fairly often in mineral physics.

A tensor is a multi-dimensional array of numerical values that can be used to describe the physical state or properties of a material. A simple example of a geophysically relevant tensor is stress. Stress, like pressure is defined as force per unit area. Pressure is isotropic, but if a material has finite strength, it can support different forces applied in different directions. Figure 1 below, illustrates a unit cube of material with forces acting on it in three dimensions. By dividing by the surface area over which the forces are acting, the stresses on the cube can be obtained. Any arbitrary stress state can be decomposed into 9 components (labeled σij). These components form a second rank tensor the stress tensor (Figure 1).


Tensor math allows us to solve problems that involve tensors. For example, let's say you measure the forces imposed on a single crystal in a deformation apparatus. It is easy to calculate the values in the stress tensor in the coordinate system tied to the apparatus. However you may be really interested in understanding the stresses acting on various crystallographic planes, which are best viewed in terms of the crystallographic coordinates. Tensor math allows you to calculate the stresses acting on the crystallographic planes by transforming the stress tensor from one coordinate system to another. Another familiar tensor property is electrical permittivity which gives rise to birefringence in polarized light microscopy. You are probably familiar with the optical indicatrix which is an ellipsoid constructed on the three principle refractive indices. The refractive index in any given direction through the crystal is governed by the dielectric constant Kij which is a tensor. The dielectric constants "maps" the electric field Ej into the electric displacement Di :


Were k0 is the permitivity of a vacuum. Di can be calculated from Ej as follows:

D1 = koK11E1 + koK12E2 + koK13E3

D2 = koK21E1 + koK22E2 + koK23E3

D3 = koK31E1 + koK32E2 + koK33E3

So you can see that even if E1 is the only non-zero value in the electric field, all the components of Di may be non-zero.

Tensors are referred to by their "rank" which is a description of the tensor's dimension. A zero rank tensor is a scalar, a first rank tensor is a vector a one-dimensional array of numbers. A second rank tensor looks like a typical square matrix. Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all second rank tensors. A third rank tensor would look like a three-dimensional matrix a cube of numbers. Piezoelectricity is described by a third rank tensor. A fourth rank tensor is a four-dimensional array of numbers. The elasticity of single crystals is described by a fourth rank tensor.

As mentioned above, it is often desirable to know the value of a tensor property in a new coordinate system, so the tensor needs to be "transformed" from the original coordinate system to the new one. As an example we will consider the transformation of a first rank tensor which is a vector. If we have a vector P with components p1, p2, p3 along the coordinate axes X1, X2, X3 and we want to write P in terms of p′1, p′2, p′3 along new coordinate axes Z1, Z2, Z3, we first need to describe how the coordinate systems are related to each other. This can be done by noting the angle between each axis of the new coordinate system and each axis of the new coordinate system altogether there will be 9 angles, three of which are illustrated in Figure 2:

Figure 2


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