Mohr–Coulomb Theory
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Mohr–Coulomb theory is a
mathematical model A mathematical model is an abstract and concrete, abstract description of a concrete system using mathematics, mathematical concepts and language of mathematics, language. The process of developing a mathematical model is termed ''mathematical m ...
(see
yield surface A yield surface is a five-dimensional surface in the six-dimensional space of Stress (mechanics), stresses. The yield surface is usually convex polytope, convex and the state of stress of ''inside'' the yield surface is elastic. When the stress ...
) describing the response of brittle materials such as
concrete Concrete is a composite material composed of aggregate bound together with a fluid cement that cures to a solid over time. It is the second-most-used substance (after water), the most–widely used building material, and the most-manufactur ...
, or rubble piles, to
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
as well as normal stress. Most of the classical engineering materials follow this rule in at least a portion of their shear failure envelope. Generally the theory applies to materials for which the
compressive strength In mechanics, compressive strength (or compression strength) is the capacity of a material or Structural system, structure to withstand Structural load, loads tending to reduce size (Compression (physics), compression). It is opposed to ''tensil ...
far exceeds the
tensile strength Ultimate tensile strength (also called UTS, tensile strength, TS, ultimate strength or F_\text in notation) is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials, the ultimate ...
. In
geotechnical engineering Geotechnical engineering, also known as geotechnics, is the branch of civil engineering concerned with the engineering behavior of earth materials. It uses the principles of soil mechanics and rock mechanics to solve its engineering problems. I ...
it is used to define shear strength of soils and rocks at different
effective stress Effective stress is a fundamental concept in soil mechanics and geotechnical engineering that describes the portion of total stress in a soil mass that is carried by the solid soil skeleton, rather than the pore water. It is crucial for understan ...
es. In
structural engineering Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and joints' that create the form and shape of human-made Structure#Load-bearing, structures. Structural engineers also ...
it is used to determine failure load as well as the angle of
fracture Fracture is the appearance of a crack or complete separation of an object or material into two or more pieces under the action of stress (mechanics), stress. The fracture of a solid usually occurs due to the development of certain displacemen ...
of a displacement fracture in concrete and similar materials.
Coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). It is defined to be equal to the electric charge delivered by a 1 ampere current in 1 second, with the elementary charge ''e'' as a defining c ...
's
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
hypothesis is used to determine the combination of shear and
normal stress In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to ''tensile'' stress and may undergo elongati ...
that will cause a fracture of the material.
Mohr's circle Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor. Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineer ...
is used to determine which principal stresses will produce this combination of shear and normal stress, and the angle of the plane in which this will occur. According to the principle of normality the stress introduced at failure will be perpendicular to the line describing the fracture condition. It can be shown that a material failing according to Coulomb's friction hypothesis will show the displacement introduced at failure forming an angle to the line of fracture equal to the angle of friction. This makes the strength of the material determinable by comparing the external
mechanical work In science, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stre ...
introduced by the displacement and the external load with the internal mechanical work introduced by the strain and stress at the line of failure. By
conservation of energy The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
the sum of these must be zero and this will make it possible to calculate the failure load of the construction. A common improvement of this model is to combine Coulomb's friction hypothesis with Rankine's principal stress hypothesis to describe a separation fracture.Coulomb, C. A. (1776). ''Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture.'' Mem. Acad. Roy. Div. Sav., vol. 7, pp. 343–387. An alternative view derives the Mohr-Coulomb criterion as extension failure.


History of the development

The Mohr–Coulomb theory is named in honour of
Charles-Augustin de Coulomb Charles-Augustin de Coulomb ( ; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of att ...
and Christian Otto Mohr. Coulomb's contribution was a 1776 essay entitled "''Essai sur une application des règles des maximis et minimis à quelques problèmes de statique relatifs à l'architecture''" . Mohr developed a generalised form of the theory around the end of the 19th century. As the generalised form affected the interpretation of the criterion, but not the substance of it, some texts continue to refer to the criterion as simply the 'Coulomb criterion'.


Mohr–Coulomb failure criterion

The Mohr–CoulombCoulomb, C. A. (1776). ''Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture.'' Mem. Acad. Roy. Div. Sav., vol. 7, pp. 343–387. failure criterion represents the linear envelope that is obtained from a plot of the shear strength of a material versus the applied normal stress. This relation is expressed as : \tau = \sigma~\tan(\phi) + c where \tau is the shear strength, \sigma is the normal stress, c is the intercept of the failure envelope with the \tau axis, and \tan(\phi) is the slope of the failure envelope. The quantity c is often called the cohesion and the angle \phi is called the angle of internal friction. Compression is assumed to be positive in the following discussion. If compression is assumed to be negative then \sigma should be replaced with -\sigma. If \phi = 0, the Mohr–Coulomb criterion reduces to the Tresca criterion. On the other hand, if \phi = 90^\circ the Mohr–Coulomb model is equivalent to the Rankine model. Higher values of \phi are not allowed. From
Mohr's circle Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor. Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineer ...
we have \sigma = \sigma_m - \tau_m \sin\phi ~;~~ \tau = \tau_m \cos\phi where \tau_m = \cfrac ~;~~ \sigma_m = \cfrac and \sigma_1 is the maximum principal stress and \sigma_3 is the minimum principal stress. Therefore, the Mohr–Coulomb criterion may also be expressed as \tau_m = \sigma_m \sin\phi + c \cos\phi ~. This form of the Mohr–Coulomb criterion is applicable to failure on a plane that is parallel to the \sigma_2 direction.


Mohr–Coulomb failure criterion in three dimensions

The Mohr–Coulomb criterion in three dimensions is often expressed as : \left\{\begin{align} \pm\cfrac{\sigma_1 - \sigma_2}{2} & = \left cfrac{\sigma_1 + \sigma_2}{2}\rightsin(\phi) + c\cos(\phi) \\ \pm\cfrac{\sigma_2 - \sigma_3}{2} & = \left cfrac{\sigma_2 + \sigma_3}{2}\rightsin(\phi) + c\cos(\phi)\\ \pm\cfrac{\sigma_3 - \sigma_1}{2} & = \left cfrac{\sigma_3 + \sigma_1}{2}\rightsin(\phi) + c\cos(\phi). \end{align}\right. The Mohr–Coulomb failure surface is a cone with a hexagonal cross section in deviatoric stress space. The expressions for \tau and \sigma can be generalized to three dimensions by developing expressions for the normal stress and the resolved shear stress on a plane of arbitrary orientation with respect to the coordinate axes (basis vectors). If the unit normal to the plane of interest is : \mathbf{n} = n_1~\mathbf{e}_1 + n_2~\mathbf{e}_2 + n_3~\mathbf{e}_3 where \mathbf{e}_i,~~ i=1,2,3 are three orthonormal unit basis vectors, and if the principal stresses \sigma_1, \sigma_2, \sigma_3 are aligned with the basis vectors \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, then the expressions for \sigma,\tau are : \begin{align} \sigma & = n_1^2 \sigma_{1} + n_2^2 \sigma_{2} + n_3^2 \sigma_{3} \\ \tau & = \sqrt{(n_1\sigma_{1})^2 + (n_2\sigma_{2})^2 + (n_3\sigma_{3})^2 - \sigma^2} \\ & = \sqrt{n_1^2 n_2^2 (\sigma_1-\sigma_2)^2 + n_2^2 n_3^2 (\sigma_2-\sigma_3)^2 + n_3^2 n_1^2 (\sigma_3 - \sigma_1)^2}. \end{align} The Mohr–Coulomb failure criterion can then be evaluated using the usual expression \tau = \sigma~\tan(\phi) + c for the six planes of maximum shear stress. :{, class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of normal and shear stress on a plane , - , Let the unit normal to the plane of interest be : \mathbf{n} = n_1~\mathbf{e}_1 + n_2~\mathbf{e}_2 + n_3~\mathbf{e}_3 where \mathbf{e}_i,~~ i=1,2,3 are three orthonormal unit basis vectors. Then the traction vector on the plane is given by : \mathbf{t} = n_i~\sigma_{ij}~\mathbf{e}_j ~~~\text{(repeated indices indicate summation)} The magnitude of the traction vector is given by : , \mathbf{t}, = \sqrt{ (n_j~\sigma_{1j})^2 + (n_k~\sigma_{2k})^2 + (n_l~\sigma_{3l})^2} ~~~\text{(repeated indices indicate summation)} Then the magnitude of the stress normal to the plane is given by : \sigma = \mathbf{t}\cdot\mathbf{n} = n_i~\sigma_{ij}~n_j ~~\text{(repeated indices indicate summation)} The magnitude of the resolved shear stress on the plane is given by \tau = \sqrt{, \mathbf{t}, ^2 - \sigma^2} In terms of components, we have : \begin{align} \sigma & = n_1^2 \sigma_{11} + n_2^2 \sigma_{22} + n_3^2 \sigma_{33} + 2(n_1 n_2 \sigma_{12} + n_2 n_3 \sigma_{23} + n_3 n_1 \sigma_{31}) \\ \tau & = \sqrt{(n_1\sigma_{11} + n_2\sigma_{12} + n_3\sigma_{31})^2 + (n_1\sigma_{12} + n_2\sigma_{22} + n_3\sigma_{23})^2 + (n_1\sigma_{31} + n_2\sigma_{23} + n_3\sigma_{33})^2 - \sigma^2} \end{align} If the principal stresses \sigma_1, \sigma_2, \sigma_3 are aligned with the basis vectors \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, then the expressions for \sigma,\tau are : \begin{align} \sigma & = n_1^2 \sigma_{1} + n_2^2 \sigma_{2} + n_3^2 \sigma_{3} \\ \tau & = \sqrt{(n_1\sigma_{1})^2 + (n_2\sigma_{2})^2 + (n_3\sigma_{3})^2 - \sigma^2} \\ & = \sqrt{n_1^2 n_2^2 (\sigma_1-\sigma_2)^2 + n_2^2 n_3^2 (\sigma_2-\sigma_3)^2 + n_3^2 n_1^2 (\sigma_3 - \sigma_1)^2} \end{align} {, border="0" , - , valign="bottom", , , , valign="bottom", , -


Mohr–Coulomb failure surface in Haigh–Westergaard space

The Mohr–Coulomb failure (yield) surface is often expressed in Haigh–Westergaad coordinates. For example, the function \cfrac{\sigma_1-\sigma_3}{2} = \cfrac{\sigma_1+\sigma_3}{2}~\sin\phi + c\cos\phi can be expressed as : \left sqrt{3}~\sin\left(\theta+\cfrac{\pi}{3}\right) + \sin\phi\cos\left(\theta+\cfrac{\pi}{3}\right)\rightrho - \sqrt{2}\sin(\phi)\xi = \sqrt{6} c \cos\phi. Alternatively, in terms of the invariants p, q, r we can write : \left cfrac{1}{\sqrt{3}~\cos\phi}~\sin\left(\theta+\cfrac{\pi}{3}\right) + \cfrac{1}{3}\tan\phi~\cos\left(\theta+\cfrac{\pi}{3}\right)\right - p~\tan\phi = c where \theta = \cfrac{1}{3}\arccos\left left(\cfrac{r}{q}\right)^3\right~. :{, class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Derivation of alternative forms of Mohr–Coulomb yield function , - , We can express the yield function \cfrac{\sigma_1-\sigma_3}{2} = \cfrac{\sigma_1+\sigma_3}{2}~\sin\phi + c\cos\phi as : \sigma_1~\cfrac{(1-\sin\phi)}{2~c~\cos\phi} - \sigma_3~\cfrac{(1+\sin\phi)}{2~c~\cos\phi} = 1 ~. The Haigh–Westergaard invariants are related to the principal stresses by : \sigma_1 = \cfrac{1}{\sqrt{3~\xi + \sqrt{\cfrac{2}{3~\rho~\cos\theta ~;~~ \sigma_3 = \cfrac{1}{\sqrt{3~\xi + \sqrt{\cfrac{2}{3~\rho~\cos\left(\theta+\cfrac{2\pi}{3}\right) ~. Plugging into the expression for the Mohr–Coulomb yield function gives us : -\sqrt{2}~\xi~\sin\phi + \rho cos\theta - \cos(\theta+2\pi/3)- \rho\sin\phi cos\theta+\cos(\theta+2\pi/3)= \sqrt{6}~c~\cos\phi Using trigonometric identities for the sum and difference of cosines and rearrangement gives us the expression of the Mohr–Coulomb yield function in terms of \xi, \rho, \theta. We can express the yield function in terms of p,q by using the relations \xi = \sqrt{3}~p ~;~~ \rho = \sqrt{\cfrac{2}{3~q and straightforward substitution.


Mohr–Coulomb yield and plasticity

The Mohr–Coulomb yield surface is often used to model the plastic flow of geomaterials (and other cohesive-frictional materials). Many such materials show dilatational behavior under triaxial states of stress which the Mohr–Coulomb model does not include. Also, since the yield surface has corners, it may be inconvenient to use the original Mohr–Coulomb model to determine the direction of plastic flow (in the flow theory of plasticity). A common approach is to use a non-associated plastic flow potential that is smooth. An example of such a potential is the function g:= \sqrt{(\alpha c_\mathrm{y} \tan\psi)^2 + G^2(\phi, \theta)~ q^2} - p \tan\phi where \alpha is a parameter, c_\mathrm{y} is the value of c when the plastic strain is zero (also called the initial cohesion yield stress), \psi is the angle made by the yield surface in the Rendulic plane at high values of p (this angle is also called the dilation angle), and G(\phi,\theta) is an appropriate function that is also smooth in the deviatoric stress plane.


Typical values of cohesion and angle of internal friction

Cohesion (alternatively called the cohesive strength) and friction angle values for rocks and some common soils are listed in the tables below. {, class="wikitable" style="text-align:center" , + Cohesive strength (c) for some materials !Material !Cohesive strength in kPa !Cohesive strength in
psi Psi, PSI or Ψ may refer to: Alphabetic letters * Psi (Greek) (Ψ or ψ), the twenty-third letter of the Greek alphabet * Psi (Cyrillic), letter of the early Cyrillic alphabet, adopted from Greek Arts and entertainment * "Psi" as an abbreviat ...
, - , align=left,
Rock Rock most often refers to: * Rock (geology), a naturally occurring solid aggregate of minerals or mineraloids * Rock music, a genre of popular music Rock or Rocks may also refer to: Places United Kingdom * Rock, Caerphilly, a location in Wale ...
, , , - , align=left,
Silt Silt is granular material of a size between sand and clay and composed mostly of broken grains of quartz. Silt may occur as a soil (often mixed with sand or clay) or as sediment mixed in suspension (chemistry), suspension with water. Silt usually ...
, , , - , align=left,
Clay Clay is a type of fine-grained natural soil material containing clay minerals (hydrous aluminium phyllosilicates, e.g. kaolinite, ). Most pure clay minerals are white or light-coloured, but natural clays show a variety of colours from impuriti ...
, to , to , - , align=left, Very soft clay , to , to , - , align=left, Soft clay , to , to , - , align=left, Medium clay , to , to , - , align=left, Stiff clay , to , to , - , align=left, Very stiff clay , to , to , - , align=left, Hard clay , > , > {, class="wikitable" style="text-align:center" , + Angle of internal friction (\phi) for some materials !Material !Friction angle in degrees , - , align=left,
Rock Rock most often refers to: * Rock (geology), a naturally occurring solid aggregate of minerals or mineraloids * Rock music, a genre of popular music Rock or Rocks may also refer to: Places United Kingdom * Rock, Caerphilly, a location in Wale ...
, ° , - , align=left,
Sand Sand is a granular material composed of finely divided mineral particles. Sand has various compositions but is usually defined by its grain size. Sand grains are smaller than gravel and coarser than silt. Sand can also refer to a textural ...
, ° to ° , - , align=left,
Gravel Gravel () is a loose aggregation of rock fragments. Gravel occurs naturally on Earth as a result of sedimentation, sedimentary and erosion, erosive geological processes; it is also produced in large quantities commercially as crushed stone. Gr ...
, ° , - , align=left,
Silt Silt is granular material of a size between sand and clay and composed mostly of broken grains of quartz. Silt may occur as a soil (often mixed with sand or clay) or as sediment mixed in suspension (chemistry), suspension with water. Silt usually ...
, ° to ° , - , align=left,
Clay Clay is a type of fine-grained natural soil material containing clay minerals (hydrous aluminium phyllosilicates, e.g. kaolinite, ). Most pure clay minerals are white or light-coloured, but natural clays show a variety of colours from impuriti ...
, ° , - , align=left, Loose sand , ° to ° , - , align=left, Medium sand , ° , - , align=left, Dense sand , ° to ° , - , align=left, Sandy gravel , > ° to °


See also

*
3-D elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
* Hoek–Brown failure criterion * Byerlee's law *
Lateral earth pressure The lateral earth pressure is the pressure that soil exerts in the horizontal direction. It is important because it affects the consolidation behavior and strength of the soil and because it is considered in the design of geotechnical engineering ...
* von Mises stress *
Yield (engineering) In materials science and engineering, the yield point is the point on a stress–strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and w ...
* Drucker Prager yield criterion — a smooth version of the M–C yield criterion * Lode coordinates * Bigoni–Piccolroaz yield criterion


References

* https://web.archive.org/web/20061008230404/http://fbe.uwe.ac.uk/public/geocal/SoilMech/basic/soilbasi.htm * http://www.civil.usyd.edu.au/courses/civl2410/earth_pressures_rankine.doc {{DEFAULTSORT:Mohr-Coulomb Theory Shear strength Solid mechanics Soil mechanics Plasticity (physics) Materials science Applied mathematics Yield criteria