All materials deform and behave differently under the influence of load/Stress. This is important in rock deformation in structural analysis in geology. For Example if stresses acts towards each other in the rock layer, that rock under such influence may become deformed to form folds as geological structures. In geophysics this is important as some physical properties such as rock density may vary with deformation intensity, as rock density may influence propagation of seismic waves in rock materials. Hence it is important to have the basic idea on elastic properties to understand fields of seismology and geodynamics.
Here down I have tried to give you basic on the elasticity theory in solids,
Before proceeding it is important to be aware with these terminologies as mostly used in describing elastic properties
Stress : Is defined as applied force (F) per unit cross section area (A) of a material.
Stress = F/A
SI unit is the same as that of pressure, N/m2.
Strain : Is the measure of degree of deformation of a material.
Strain = change in parameter / Original value of that parameter.
Stress always varies as directly proportional to strain of a material if the elastic limit is not exceeded. This is hooke's law
Ss = KSn
Then, K is the constant term , known as Elastic Modulus. It describes the Linear relationship between stress and strain within an elastic field.
K = Ss /Sn
Where Ss and Sn are stress and strain respectively.
The following are the three (3) common elastic properties in solids,
1.young's modulus
Tensile stress: Is the applied force (F) per unit cross section area (A) of a material. It's SI unit is the same as that of pressure, N/m2.
Since Tensile stress = F/A
Tensile strain: Is the ratio of change in length (∆L) to original length (L) of a material.
Tensile strain = ∆L/L
Consider the figure 1, here below,
Figure 1: Linear elasticity
Young's Modulus (Y) = Tensile stress per tensile strain
Y = (F/A)/(∆L/L)
Y = FL/A∆L
The SI unit of Young's Modulus is the same as that of stress, N/m2.
It is elasticity related to change in length of a material.
2. bulk modulus: Is the volume stress per volume strain
Bulk Modulus (B): volume stress / volume strain.
Volume stress : Applied force (F) per unit cross section area (A)
Volume strain : Is the ratio of change in volume (∆V) to original volume of a material (V)
If material is squeezed, change of volume will be taken as negative, - ∆V as shown in figure 2,
Figure 2: Volume elasticity
Then if we assume our material is compressed (Squeezed), consider the diagram here below
Bulk Modulus (B) = (F/A)/(- ∆V/V)
B = - (FV/A∆V)
SI unit of Bulk Modulus (B), is the same as that of stress, N/m2.
It is elasticity related to change in volume of a material.
3. shear modulus: Is the shear stress per shear strain
Shear stress: Is the applied force (F) per unit cross section area (A) related to angular deformation.
Shear strain : Is the given as change in angle
Such as shear strain = tan Q = ∆x/h,
Consider the figure 3 below
Figure 3: Angular elasticity
Shear Modulus (S) : (F/A)/(∆x/h)
S = Fh/A∆x
SI unit of shear Modulus (S) is the same as that of stress, N/m2.
It is elasticity related to rigidity such that change in shape of a materials.
Let us now try to expand and explaining these terms, in an understandable manner and how they are related to elasticity theory.
Basic Principles of Stress
As we have already seen that Forces acting on a rock during deformation are expressed in terms of force per unit area are known as STRESS.
Representation of stresses
In general, position and orientation of a line in space are expressed in terms of three perpendicular axes referred to as x, y and z (see figure below)
Each of these axes are normal to one of three principal planes (x, y, and z) which are perpendicular to each other. Each plane is referred to by a letter (number) similar to that of its corresponding axis. Stress acting on a point can be expressed in terms of the following nine - vectorial components.
σxx σyx σxz
σyx σyy σyz
σzx σzy σzz
Since the conditions of stress at a point can be expressed as a unit cube in a rock mass is given by magnitude and orientation of these three perpendicular axes normal to each face of the unit cube together with the 6 independent stress arrays which acts parallel to the faces of the unit cube.
The three (3) axes on a cube with the first letter (number) in each of the components of stress indicates the principal plane in which the component is acting. The second letter (number) indicates the geometrical axis which is parallel to the component of stress (see figure above).
The three (3) components σxx, σyy and σzz are Principal (Normal) stresses since they are acting perpendicular (normal) to the principal planes. On the other hand, six (6) components of stress σyx , σxz , σyx , σyz , σzx , σzy are Shear (Rotational) stresses since they are acting at parallel (at an angle) to the principal plane.
However in a homogeneous stress field there are always 3 perpendicular planes with no shear components of stresses but with only normal components of stresses. These planes are referred to as Principal Planes of stresses and their poles (normal axes) are referred to as Principal Axes of stress denoted as 1, 2 and 3 for x, y, and z respectively at which 1 is greater than 2 in magnitude of force and 2 is greater than 3 such as σ1 > σ2 > σ3 . Compressive stresses are always taken as Positive stresses whereas tensional stresses are taken as Negative stresses.
Basic Principles of Strain
Since a rock can be strained with or without a change in its total volume. Therefore we have either a volume-change or a no volume-change strain regime.
Since Strain involves a certain amount of either extension, compaction or rotational towards a certain direction.
Representation of Strain
Therefore, we can represent the strain in space, just like stress in vectorial form with magnitude and direction. In a given unit cube, the strain that exists as extensional or compaction that is perpendicular to the faces of the cube is referred to as Normal strain while the strain which are rotational or shear parallel to the faces of the cube are referred to as shear strains.
The normal strain also can be expressed in terms of 3 perpendicular axes known as Principal strain Axes 1, 2 and 3 for x, y, and z respectively while their relative magnitude and orientation can be expressed by a strain ellipsoid at which 1 is greater than 2 and 3 is the least in magnitude. Extension strain is taken as Positive strain whereas compaction strain is taken as Negative strain.
However If a material under stress is purely isotropic such as with similar elasticity in all directions, then 1, 2 and 3 (x, y, z) strain axes will coincide with 3, 2 and 1 (z, y, x) stress axes, respectively. However, most of the rock-constituents are anisotropic such that their elasticity is directional therefore this coincidence of stress/strain axes is often not present in deformed rocks.
Formula of Elastic Theory
For deformation of materials that shows Linear relationship between stress and strain, usually conforms to the Hooke's law of elasticity at which linear straining the elasticity is a function of a constant E known as Young's modulus.
So these materials can be regarded to possess Hookean Elastic behavior (Elasticity), while undergo Elastic Deformation. This mean that if the applied stress is removed then materials can recover to their original (initial) state.
Although there is a constant term known as Viscosity which is usually used to describe the permanent straining that occur in viscous materials.
The stress - strain relationship is well described in the figure. Let us start by describing this graph, but we have to note that nearly all materials are elastic to some extent during their initial stage of straining (within elastic range) before reaching the Elastic Limit.
This is telling us that some materials after reaching their elastic limits break instantaneously into fragments. It is because the applied stress on the material exceeds the resisting stress of the material itself. Within that point these materials are said to undergo Brittle Deformation. Fault is one of the common resulting geology structures due to Brittle rock deformation.
Other materials undergo Newtonian (Ductile) Deformation after exceeding their elastic limits. This means that materials do not break but still deform and behave as a coherent (ductile) state even if the applied stress is acting on it. Fold is a common example of geology structure resulting due to ductile deformation.
The more general term that can be used to describe ductile and brittle deformation is Plastic Deformation, that means the material does not recover to its original (initial) state after the applied stress is removed.
However, the term ductility and brittleness of materials is a function of temperature. Materials become more ductile and therefore easier to deform at higher temperatures and more brittle (difficult to deform) at lower temperatures.
Lamé constants
What are Lame' Constants?
Lamé constants (Lamé moduli) are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships. They are then referred to as Lamé first parameter, λ denoted as Lambda and μ as Lamé second parameter denoted by Rigidity (shear modulus).
And there is no doubt that these constants are not new as they were directly already mentioned by French mathematician Gabriel Lamé since 1795.
bulk modulus (K) in terms of poisson ratio (V) and Young's modulus (E)
Since total Longitudinal strain on x - direction is the result of two transverse longitudinal strains (y and z - directions) and the normal longitudinal strain (x - direction)
εxx = εxx + (- εyy) + (- εzz)......................(1)
y and z are negative as these are transverse longitudinal strains act in opposite direction with respect to x
But E = σ /ε, then ε = σ / E
Where E = Bulk modulus, ε = strain, σ = stress.
Recall poisson ratio (V) is the ratio of transverse longitudinal strain to normal longitudinal strain.
Such that V = - εyy/εxx (in y - direction) and V = - εzz/εxx (in z - direction)
But if we put εxx = σxx/E in the above poisson relation as it is described from the definition of Young's modulus.
Then, εxx = - Vσxx/E.
By putting εxx = - Vσxx/E into equation 1 above.
εxx = σxx/E + (- Vσyy/E) + (- Vσzz/E)..........................(1a)
Rearranging, and multiplying by E on both sides from equation 1a
Eεxx = σxx + (- Vσyy) + (- Vσzz).
Eεxx = σxx - Vσyy - Vσzz........................(1b)
Also if we consider total longitudinal strain in y - direction using the same procedures as described in x - direction,
Eεyy = σyy - Vσzz - Vσxx.......................(1c)
Likewise in z - direction
Eεzz = σzz - Vσxx - Vσyy......................(1d)
Adding total longitudinal strain in all three directions as given in equation (1b), (1c) and (1d) above.
E[εxx + εyy + εzz] = [σxx + σyy + σzz] (1 - 2V)..............(2)
But εxx + εyy + εzz = D
Where D = Dilation
If σxx = σyy = σzz = - P which is hydrostatic pressure.
Then adding three the same - P, we get - 3P
Then the equation 2 reduces to ED = (1 - 2V)(- 3P}
Let us make E the subject in the equation above,
E = (1 - 2V) - 3P/D
But the term - P/D is the bulk modulus (K), since D = dilation which is a fractional change in volume (∆V/V).
E = (1 - 2V)( 3K),
Then on making K, the subject of the equation above.
K = E / 3(1 - 2V)......................(3)
The equation 3 above indicates that bulk modulus (K) is expressed in terms of Poisson ratio (V) and Young's modulus (E).
Recall, equation 2, Then ED = [σxx + σyy + σzz] (1 - 2V).
Then, σxx + σyy + σzz = ED/(1 - 2V)...................(2a)
Also from equation 1b, Eεxx = σxx - Vσyy - Vσzz.
Rearranging, Eεxx = (1 + V)σxx - V (σxx + σyy + σzz)
Making the subject σxx
Eεxx = σxx - Vσyy - Vσzz.
Eεxx + V (σxx + σyy + σzz)/1 + V = σxx ...................(2b)
Inserting equation 2a into equation 2b above,
σxx = Eεxx + V (ED/ 1 - 2V)/1 + V
σxx = [Eεxx /(1 + V)] + [VED/(1 - 2V)(1 + V)]
σxx = [VED/(1 - 2V)(1 + V)] + [Eεxx /(1 + V)]...........(2c)
Then if 2μ = [E/(1 + V)] and λ = [VE/(1 - 2V)(1 + V)]
Where λ and μ (Rigidity) are Lamé constants (Lamé Moduli)
Then, σxx = λD + 2μεxx ....................(4)
Bulk modulus (K), Young's modulus (E) and Poisson ratio (V) with Lamé constants
Then Let us express Bulk modulus (K), Young's modulus (E) and Poisson ratio (V) in terms of Lamé constants
Bulk Modulus (K)
Recall, equation 4 above, if we consider all three directions, x, y and z.
σxx = λD + 2μεxx ............................(4a)
σyy = λD + 2μεyy ............................(4b)
σzz = λD + 2μεzz ............................(4c)
Then add three sub - equation 4 above such as 4a, 4b & 4c.
σxx + σyy + σzz = 3λD + 2μεxx + 2μεyy + 2μεzz
σxx + σyy + σzz = 3λD + 2μ(εxx + εyy + εzz) ................(4d)
σxx = σyy = σzz = hydrostatic pressure ,-P and
εxx + εyy + εzz = D (see equation 2)
-3P = 3λD + 2μD
Dividing by D throughout, -3P/D = 3λ + 2μ
The term - P/D on the left hand side defines the Bulk modulus (K).
3K = 3λ + 2μ, While K is made a subject
K = λ + 2μ/3.....................(5)
Equation 5 above relates Bulk modulus (K) with λ and μ
Young's modulus (E)
Since Young's modulus involve only horizontal direction (x - direction)
When we recall equation 4d, σxx + σyy + σzz = 3λD + 2μ(εxx + εyy + εzz) and put σyy = σzz = 0
σxx = 3λD + 2μD
Then D = σxx / (3λ + 2μ)....................(4e)
Now, we have to put equation 4e above into equation 4a, so that to get the final relationship expression
σxx = λD + 2μεxx, but D = σxx / (3λ + 2μ)
Then, σxx = λσxx / (3λ + 2μ)+ 2μεxx
σxx - λσxx / (3λ + 2μ) = 2μεxx
σxx [1 - λ / (3λ + 2μ)] = 2μεxx
After that we divide by εxx throughout at which this term will cancel on right hand side of the equation above, while another term σxx /εxx defined as Young's modulus (E)
(σxx /εxx)[1 - λ / (3λ + 2μ)] = 2μ
E[1 - λ / (3λ + 2μ)] = 2μ
After making E the subject of the expression above, we get
E = (3λ + 2μ)μ/λ + μ
E = μ(3λ + 2μ)/λ + μ......................(6)
Equation 6 above relates Young's modulus (E) with λ and μ
Poisson Ratio (V)
Poisson ratio is defined as the ratio of vertical longitudinal (transverse) strain (εyy or εzz ) to horizontal longitudinal strain (εxx).
Poisson ratio (V) = -εyy (εzz )/εxx
However poisson ratio (V) is related with Young's modulus (E) and Bulk modulus (K) as shown in the equation 3
K = E / 3(1 - 2V)
By inserting expressions for K from equation 5 and E from equation 6 in the above equation.
λ + 2μ/3 = μ(3λ + 2μ)/3(λ + μ)(1 - 2V)
On the way to get V, we multiplying by 3 both sides, then exchange the positions of (3λ + 2μ) with (1 - 2V)
(1 - 2V) = μ(3λ + 2μ)/(3λ + 2μ)(λ + μ) while 3λ + 2μ cancelled out.
1 - 2V = μ/λ + μ
Then making V the subject of the above equation to get the final relationship.
V = λ/2(λ + μ).......................(7)
The equation 7 above relates poisson ratio (V) with λ and μ
Values for Poisson Ratio
Values for poisson ratios in the Earth's interior varies at a range from 0 to 0.5 depending on the various factors such as rock types, compressibility. Such that hard rocks like granite their poisson ratio approach towards 0, example 0.05 while unconsolidated sediments and soft rocks have values approaching towards 0.5 such as as 0.45 However If we consider equation 7 above and set λ = μ
Then V = μ/2(μ + μ)
V = μ/2(2μ)
V = μ/4μ = 1/4
V = 0.25, This is considered as an ideal value for poisson ratio.
Poisson Ratio and Seismic waves (P and S) velocities
Poisson ratio can be used to relate the seismic waves velocities (Vp and Vs), since the seismic propagating velocity depends on appropriate elastic modulus and the density of materials through which it passes.
Seismic velocity = (elastic modulus/density)1/2. This means that as the rock becomes denser its ability to pass seismic velocity fast increases also and vice versa is true.
But when we consider Primary wave velocity (Vp) = [(K + 4μ/3)/P]1/2
Where K - Bulk Modulus, μ - Shear Modulus, and P - Density.
Also for secondary waves (Vs) = (μ/P)1/2
While taking the ratio for Vp to Vs, such that Vp/Vs = [(K + 4μ/3)/P]1/2/ (μ/P)1/2
Vp/Vs = [(K + 4μ/3)/μ)]1/2
Why is elasticity theory important?
From the above description I hope that now you have already seen that the elasticity theory involves interrelation of all rock deformation. Therefore elasticity theory is important as it gives us an idea and theoretical background regarding many geology and geophysical phenomena related to deformations and rock and material movements within the fields such as plate tectonics, rock physics, seismic waves propagation in the field of seismology just to mention a few.
There are many to mention regarding the importance of elasticity theory in the world of geoscience, However, when it comes to geophysics such as in earthquake seismology we can apply elasticity theory when it comes to issues of Solving Fault plane problems.
Also in seismic inversion, elasticity theory give us the chance of knowing and calculating inversions parameters P-wave impedance (ρVp) and S-wave impedance (ρVs). Through Lame constants and elastic modulus relationships such as Poisson’s ratio (ν), shear modulus (μ), bulk modulus (K) and density (ρ) which again can be related to Vp and Vs. This is because parameter inversion might determine impedance (Zp, Zs) and density (ρ), at which Zp, Vp /Vs and ρ would contain the same information.
Thank you for your time!
SHARE with your friends and colleagues!
Is the content useful and helpful to you? Want to support the Effort! You can support the Effort via >"Support the Effort Page"<
Follow this blog on Telegram group to gain more benefits.
Don't hesitate to check out with Online Scientist Assistance for any enquiry of your geoscience Assignment.
You can check here below the simple lists of equipment and tools that can suit your Project!
BASIC RESISTIVITY EQUIPMENTS and TOOLS
- Electrical resistivity meter
- Geological Hammer (The friend of Geologists)
NEED METAL DETECTORS
- Lightweight, High accuracy Metal Detector.
- Gold Detector for Mineral Exploration.
- Professional underground Gold digger - Detector.
SHARE! With your friends and colleagues if you found it interesting!
Get Android Apps for Free.
Online learning App
Uacademy Learner App
Social media App
YoWhatapp
Arab social media platform - Baaz App
Improve your Android Performance
Viruses Cleaner
File Downloader
Grow Your Email Marketing.
Try Moosend Now
0 Comments