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| {{continuum mechanics|cTopic=[[Solid mechanics]]}}
| | On-line games can give you a major universe of experience, calmness and exhilaration. One might learn, get a comman sense of success or since enjoy beating down unquestionably the [https://www.vocabulary.com/dictionary/bad-guy bad-guy]. No matter what form of video playing you are into, add some helpful tips in this excellent post to give you and your family more fun whenever your site play your next pc game Website.<br><br>Attending to the higher level: it is essential when it comes with a game, but when it depends on Clash of Clans, you will have a lot more subtle techniques. Despite making use of [http://prometeu.net clash of clans hack tools], you may acquire experience points as a result of matching on top off other players. Most important purpose of earning Player vs player combat is to enable any enhancements for your indigneous group. The restoration consists of better strive equipment, properties, troops while tribe people.<br><br>To savor unlimited points, resources, loose change or gems, you must have download the clash of clans hack into tool by clicking on his or her button. Depending across the operating system that tend to be using, you will need to run the downloaded file as administrator. Necessary under some log in ID and judge the device. Looking for this, you are ought to enter the number akin to gems or coins that you prefer to get.<br><br>In the event that you feel like individuals targeted your enemy stain on in a shooting and still missed, consider what weapon you are probably using. Just resembling in real life, varied weapons have different strong points and weaknesses. All of the weapon you are operating may not have which the short distance required along with the weapon recoil would be actually putting you a little bit off target.<br><br>One of several best and fastest establishing certifications by ECCouncil. Where a dictionary onset fails the computer cyberpunk may try a incredible force attack, which is more time consuming. Sets up the borders of everyone with non-editable flag: lot_border [ ]. The issue is this one hit people where it really affects - your heart. These Kindle hacks continue to be keyboard shortcuts will save you tons of time on the search and typing in done again things. Claire smiled and told me how she had started gain a (not modest.<br><br>Your primary war abject is agnate in your approved village, except that your battle abject will not manage resources. Barrio within your warfare abject simpley can't be anon improved in addition rearranged, as it on their own mimics this adjustment and then accomplished completed advancement certifications of your apple while having alertness day. Combat bases additionally never fees to take their accessories rearmed, defenses reloaded or characters healed, as all the people are consistently ready. The association alcazar in that room your war abject bill be abounding alone to the one in your whole village.<br><br>Should you are playing a displaying activity, and you perhaps don't possess knowledge related with it, establish the dilemma stage to rookie. This should help owners pick-up in the excellent options that come with the game and discover in your direction round the field. Should you set things more than that, in all probability you'll get frustrated and never possess fun. |
| '''Mohr–Coulomb theory''' is a [[mathematical model]] (see [[yield surface]]) describing the response of brittle materials such as [[concrete]], or rubble piles, to shear [[Stress (physics)|stress]] as well as normal stress. Most of the classical engineering materials somehow follow this rule in at least a portion of their shear failure envelope. Generally the theory applies to materials for which the compressive strength far exceeds the tensile strength.<ref>Juvinal, Robert C. & Marshek, Kurt .; Fundamentals of machine component design. – 2nd ed., 1991, pp. 217, ISBN 0-471-62281-8</ref>
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| | |
| In [[geotechnical engineering]] it is used to define shear strength of soils and rocks at different [[effective stress]]es.
| |
| | |
| In [[structural engineering]] it is used to determine failure load as well as the angle of [[fracture]] of a displacement fracture in concrete and similar materials. [[Charles-Augustin de Coulomb|Coulomb]]'s [[friction]] hypothesis is used to determine the combination of shear and normal stress that will cause a fracture of the material. [[Mohr's circle]] is used to determine which principal stresses that 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]] 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 sum of these must be zero and this will make it possible to calculate the failure load of the construction.
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| | |
| A common improvement of this model is to combine Coulomb's friction hypothesis with [[William John Macquorn Rankine|Rankine's]] principal stress hypothesis to describe a separation fracture.
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| | |
| == History of the development ==
| |
| The Mohr–Coulomb theory is named in honour of [[Charles-Augustin de Coulomb]] and [[Christian Otto Mohr]]. Coulomb's contribution was a 1773 essay entitled "''Essai sur une application des règles des maximis et minimis à quelques problèmes de statique relatifs à l'architecture''"
| |
| <!--C.A. Coulomb, Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture, Mem. Acad. Roy. Div. Sav., 7, 343–387, 1776.-->.<ref>AMIR R. KHOEI; <u>Computational Plasticity in Powder Forming Processes</u>; Elsevier, Amsterdam; 2005; 449 pp.</ref>
| |
| Mohr developed a generalised form of the theory around the end of the 19th century.<ref>MAO-HONG YU; "''Advances in strength theories for materials under complex stress state in the 20th Century''"; <u>Applied Mechanics Reviews</u>; American Society of Mechanical Engineers, New York, U.S.A.; May 2002; '''55''' (3): pp. 169–218.</ref>
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| 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''''.<ref>NIELS SAABYE OTTOSEN and MATTI RISTINMAA; <u>The Mechanics of Constitutive Modeling</u>; Elsevier Science, Amsterdam, The Netherlands; 2005; pp. 165ff.</ref>
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| | |
| == Mohr–Coulomb failure criterion ==
| |
| [[Image:Mohr Coulomb Yield Surface 3Da.png|300px|right|thumb|Figure 1: View of Mohr–Coulomb failure surface in 3D space of principal stresses for <math>c=2, \phi=-20^\circ</math>]]
| |
| The Mohr–Coulomb<ref>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.</ref> 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
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| :<math>
| |
| \tau = \sigma~\tan(\phi) + c
| |
| </math>
| |
| where <math>\tau</math> is the shear strength, <math>\sigma</math> is the normal stress, <math>c</math> is the intercept of the failure envelope with the <math>\tau</math> axis, and <math>\phi</math> is the slope of the failure envelope. The quantity <math>c</math> is often called the '''cohesion''' and the angle <math>\phi</math> 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 <math>\sigma</math> should be replaced with <math>-\sigma</math>.
| |
| | |
| If <math>\phi = 0</math>, the Mohr–Coulomb criterion reduces to the [[Yield surface#Tresca_yield_surface|Tresca criterion]]. On the other hand, if <math>\phi = 90^\circ</math> the Mohr–Coulomb model is equivalent to the Rankine model. Higher values of <math>\phi</math> are not allowed.
| |
| | |
| From [[Mohr's circle]] we have
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| :<math>
| |
| \sigma = \sigma_m - \tau_m \sin\phi ~;~~ \tau = \tau_m \cos\phi
| |
| </math>
| |
| where
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| :<math>
| |
| \tau_m = \cfrac{\sigma_1-\sigma_3}{2} ~;~~ \sigma_m = \cfrac{\sigma_1+\sigma_3}{2}
| |
| </math>
| |
| and <math>\sigma_1</math> is the maximum principal stress and <math>\sigma_3</math> is the minimum principal stress.
| |
| | |
| Therefore the Mohr–Coulomb criterion may also be expressed as
| |
| :<math>
| |
| \tau_m = \sigma_m \sin\phi + c \cos\phi ~.
| |
| </math>
| |
| | |
| This form of the Mohr–Coulomb criterion is applicable to failure on a plane that is parallel to the <math>\sigma_2</math> direction.
| |
| | |
| === Mohr–Coulomb failure criterion in three dimensions ===
| |
| The Mohr–Coulomb criterion in three dimensions is often expressed as
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| :<math>
| |
| \left\{\begin{align}
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| \pm\cfrac{\sigma_1 - \sigma_2}{2} & = \left[\cfrac{\sigma_1 + \sigma_2}{2}\right]\sin(\phi) + c\cos(\phi) \\
| |
| \pm\cfrac{\sigma_2 - \sigma_3}{2} & = \left[\cfrac{\sigma_2 + \sigma_3}{2}\right]\sin(\phi) + c\cos(\phi)\\
| |
| \pm\cfrac{\sigma_3 - \sigma_1}{2} & = \left[\cfrac{\sigma_3 + \sigma_1}{2}\right]\sin(\phi) + c\cos(\phi).
| |
| \end{align}\right.
| |
| </math> | |
| The [[Yield_surface#Mohr_–_Coulomb_yield_surface|Mohr–Coulomb failure surface]] is a cone with a hexagonal cross section in deviatoric stress space.
| |
| | |
| The expressions for <math>\tau</math> and <math>\sigma</math> 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
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| :<math>
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| \mathbf{n} = n_1~\mathbf{e}_1 + n_2~\mathbf{e}_2 + n_3~\mathbf{e}_3
| |
| </math>
| |
| where <math>\mathbf{e}_i,~~ i=1,2,3</math> are three orthonormal unit basis vectors, and if the principal stresses <math>\sigma_1, \sigma_2, \sigma_3</math> are aligned with the basis vectors <math>\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3</math>, then the expressions for <math>\sigma,\tau</math> are
| |
| :<math>
| |
| \begin{align}
| |
| \sigma & = n_1^2 \sigma_{1} + n_2^2 \sigma_{2} + n_3^2 \sigma_{3} \\
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| \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}
| |
| </math>
| |
| The Mohr–Coulomb failure criterion can then be evaluated using the usual expression
| |
| :<math>
| |
| \tau = \sigma~\tan(\phi) + c
| |
| </math>
| |
| 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
| |
| :<math>
| |
| \mathbf{n} = n_1~\mathbf{e}_1 + n_2~\mathbf{e}_2 + n_3~\mathbf{e}_3
| |
| </math>
| |
| where <math>\mathbf{e}_i,~~ i=1,2,3</math> are three orthonormal unit basis vectors. Then the traction vector on the plane is given by
| |
| :<math>
| |
| \mathbf{t} = n_i~\sigma_{ij}~\mathbf{e}_j ~~~\text{(repeated indices indicate summation)}
| |
| </math>
| |
| The magnitude of the traction vector is given by
| |
| :<math>
| |
| |\mathbf{t}| = \sqrt{ (n_j~\sigma_{1j})^2 + (n_k~\sigma_{2k})^2 + (n_l~\sigma_{3l})^2} ~~~\text{(repeated indices indicate summation)}
| |
| </math>
| |
| Then the magnitude of the stress normal to the plane is given by
| |
| :<math>
| |
| \sigma = \mathbf{t}\cdot\mathbf{n} = n_i~\sigma_{ij}~n_j ~~\text{(repeated indices indicate summation)}
| |
| </math>
| |
| The magnitude of the resolved shear stress on the plane is given by
| |
| :<math>
| |
| \tau = \sqrt{|\mathbf{t}|^2 - \sigma^2}
| |
| </math>
| |
| In terms of components, we have
| |
| :<math>
| |
| \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}) \\
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| \tau & = \sqrt{(n_1\sigma_{11} + n_2\sigma_{12} + n_3\sigma_{31})^2 +
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| (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}
| |
| </math>
| |
| If the principal stresses <math>\sigma_1, \sigma_2, \sigma_3</math> are aligned with the basis vectors <math>\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3</math>, then the expressions for <math>\sigma,\tau</math> are
| |
| :<math>
| |
| \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 +
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| n_3^2 n_1^2 (\sigma_3 - \sigma_1)^2}
| |
| \end{align}
| |
| </math>
| |
| |}
| |
| {| border="0"
| |
| |-
| |
| | valign="bottom"|
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| [[Image:Mohr Coulomb Yield Surface 3Db.png|300px|none|thumb|Figure 2: Mohr–Coulomb yield surface in the <math>\pi</math>-plane for <math>c = 2, \phi = 20^\circ</math>]]
| |
| |
| |
| |
| |
| | valign="bottom"|
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| [[Image:Mohr Coulomb Yield Surface sig1sig2.png|300px|none|thumb|Figure 3: Trace of the Mohr–Coulomb yield surface in the <math>\sigma_1-\sigma_2</math>-plane for <math>c = 2, \phi = 20^\circ</math>]]
| |
| |-
| |
| |}
| |
| | |
| == Mohr–Coulomb failure surface in Haigh–Westergaard space ==
| |
| The Mohr–Coulomb failure (yield) surface is often expressed in [[Yield_surface#Invariants_used_in_yield_functions|Haigh–Westergaad coordinates]]. For example, the function
| |
| :<math>
| |
| \cfrac{\sigma_1-\sigma_3}{2} = \cfrac{\sigma_1+\sigma_3}{2}~\sin\phi + c\cos\phi
| |
| </math>
| |
| can be expressed as
| |
| :<math> | |
| \left[\sqrt{3}~\sin\left(\theta+\cfrac{\pi}{3}\right) - \sin\phi\cos\left(\theta+\cfrac{\pi}{3}\right)\right]\rho - \sqrt{2}\sin(\phi)\xi = \sqrt{6} c \cos\phi.
| |
| </math>
| |
| Alternatively, in terms of the [[Yield_surface#Invariants_used_in_yield_functions|invariants]] <math>p, q, r</math> we can write
| |
| :<math>
| |
| \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]q - p~\tan\phi = c
| |
| </math>
| |
| where
| |
| :<math>
| |
| \theta = \cfrac{1}{3}\arccos\left[\left(\cfrac{r}{q}\right)^3\right] ~.
| |
| </math>
| |
| | |
| :{| 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
| |
| :<math>
| |
| \cfrac{\sigma_1-\sigma_3}{2} = \cfrac{\sigma_1+\sigma_3}{2}~\sin\phi + c\cos\phi
| |
| </math>
| |
| as
| |
| :<math>
| |
| \sigma_1~\cfrac{(1-\sin\phi)}{2~c~\cos\phi} - \sigma_3~\cfrac{(1+\sin\phi)}{2~c~\cos\phi} = 1 ~.
| |
| </math>
| |
| The [[Yield_surface#Invariants_used_in_yield_functions|Haigh–Westergaard invariants]] are related to the principal stresses by
| |
| :<math>
| |
| \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) ~.
| |
| </math>
| |
| Plugging into the expression for the Mohr–Coulomb yield function gives us
| |
| :<math>
| |
| -\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
| |
| </math>
| |
| 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 <math>\xi, \rho, \theta</math>.
| |
| | |
| We can express the yield function in terms of <math>p,q</math> by using the relations
| |
| :<math>
| |
| \xi = \sqrt{3}~p ~;~~ \rho = \sqrt{\cfrac{2}{3}}~q
| |
| </math>
| |
| and straightforward substitution.
| |
| |}
| |
| | |
| <!--
| |
| For a charge distribution an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge dq.
| |
| | |
| For a linear charge distribution (a good approximation for charge in a wire) where <math>\lambda(\mathbf{r^\prime})</math> gives the charge per unit length at position <math>\mathbf{r^\prime}</math>, and <math>dl^\prime</math> is an infinitesimal element of length,
| |
| | |
| <math> dq = \lambda(\mathbf{r^\prime})dl^\prime</math> [10]
| |
|
| |
| For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where <math>\sigma(\mathbf{r^\prime})</math> gives the charge per unit area at position <math>\mathbf{r^\prime}</math>, and dA^\prime is an infinitesimal element of area,
| |
| | |
| <math>dq = \sigma(\mathbf{r^\prime})\,dA^\prime\,</math>
| |
| | |
| For a volume charge distribution (such as charge within a bulk metal) where <math>\rho(\mathbf{r^\prime}</math>) gives the charge per unit volume at position <math>\mathbf{r^\prime}</math>, and <math>dV^\prime</math> is an infinitesimal element of volume,
| |
| | |
| <math>dq = \rho(\mathbf{r^\prime})\,dV^\prime</math>[9]
| |
| | |
| The force on a small test charge <math>q^\prime</math> at position <math>\mathbf{r}</math> is given by | |
| | |
| <math>\mathbf{F} = q^\prime\int dq {\mathbf{r} - \mathbf{r^\prime} \over |\mathbf{r} - \mathbf{r^\prime}|^3}</math>
| |
| | |
| -->
| |
| | |
| == 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 [[plasticity theory|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{{Citation needed|date=July 2009}}
| |
| | |
| :<math>
| |
| g:= \sqrt{(\alpha c_\mathrm{y} \tan\psi)^2 + G^2(\phi, \theta)~ q^2} - p \tan\phi
| |
| </math>
| |
| | |
| where <math>\alpha</math> is a parameter, <math>c_\mathrm{y}</math> is the value of <math>c</math> when the plastic strain is zero (also called the '''initial cohesion yield stress'''), <math>\psi</math> is the angle made by the yield surface in the '''Rendulic plane''' at high values of <math>p</math> (this angle is also called the '''dilation angle'''), and <math>G(\phi,\theta)</math> is an appropriate function that is also smooth in the deviatoric stress plane.
| |
| | |
| ==See also==
| |
| * [[3-D elasticity]]
| |
| * [[Byerlee's Law]]
| |
| * [[Christian Otto Mohr]]<!-- please integrate this into the article; also, who is Coulomb? -->
| |
| * [[Henri Tresca]]
| |
| * [[Lateral earth pressure]]
| |
| * [[von Mises stress]]
| |
| * [[Shear strength]]
| |
| * [[Shear strength (soil)]]
| |
| * [[Strain (materials science)]]
| |
| * [[Stress (physics)]]
| |
| * [[Yield (engineering)]]
| |
| * [[Yield surface]]
| |
| * [[Drucker Prager yield criterion]] — a smooth version of the M–C yield criterion
| |
| | |
| ==References==
| |
| <references />
| |
| {{Refbegin}}
| |
| * http://fbe.uwe.ac.uk/public/geocal/SoilMech/basic/soilbasi.htm
| |
| * http://www.civil.usyd.edu.au/courses/civl2410/earth_pressures_rankine.doc
| |
| {{Refend}}
| |
| | |
| {{DEFAULTSORT:Mohr-Coulomb Theory}}
| |
| [[Category:Solid mechanics]]
| |
| [[Category:Plasticity]]
| |
| [[Category:Materials science]]
| |
| [[Category:Applied mathematics]]
| |
| [[Category:Yield criteria]]
| |
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