GEO E050 Finite Element Method Mohr-Coulomb and other constitutive models Wojciech Sołowski
To learn today. Reminder elasticity 2. Elastic perfectly plastic theory: concept 3. Specific elastic-perfectly plastic models: - Mohr Coulomb (important) - Hoek Brown (important for rock mechanics) - Tresca - von Mises More derivations, Boundary Element Method, Finite Difference Method moved to next lecture: we need models for exercises Department of Civil Engineering Finite Element Method. W. Sołowski 2
Philosophy Geomaterials behaviour COMPLEX Constitutive models huge simplifications usually the simplification designer is making is mainly about choice of model and its parameters Model and choice of its parameters must be clearly selected for given problem. No constitutive model for geomaterials is good for solving all problems. (these which are trying to be good for everything, like the MIT constitutive model for geomaterials are extremely complex and difficult to use, validate and calibrate) Need to balance simplicity, accuracy and problem needs Department of Civil Engineering Finite Element Method. W. Sołowski 3
Philosophy Challenge: - understand the simplifications - use them to your benefit - get accurate results from crude tools - no results is right - each result is because of the assumptions made - if you do good job with modelling, you will get result which is close to reality also because you have chosen the right model yet primarily because you have chosen the right set of parameters for the model and the problem Department of Civil Engineering Finite Element Method. W. Sołowski 4
Department of Civil Engineering 5 Finite Element Method. W. Sołowski Elasticity This lecture covers the very basics of constitutive modelling You know elasticity, we will build on that More on modelling in Advanced Soil Mechanics Even more in the Numerical Methods in Geotechnics ( )( ) ε D + = 2 22 2 33 22 2 0 0 0 0 0 2 ε ε ε v v v v v v v v v E
Outline. Constitutive modelling: Elasticity a. Linear elasticity b. Non-linear elasticity c. Hypoelasticity 2. Constitutive modelling: Elasto-plasticity a. What is elasto-plasticity, basic concepts and theory b. Mohr-Coulomb c. Other elastic-perfectly plastic constitutive models More on modelling in Advanced Soil Mechanics Even more in the Numerical Methods in Geotechnics Department of Civil Engineering Finite Element Method. W. Sołowski 6
Elasticity Idea: - one of the oldest approaches to describe material behaviour - assumes that after removing the stress the material returns back to the initial state - behaviour is reversible ) no matter the level of stress, the material will not break 2) no matter the level of stress, after removing it, the material will be back in the original state before the loading, no change Department of Civil Engineering Finite Element Method. W. Sołowski 7
Elasticity Linear elasticity useful? Real soil behaviour Department of Civil Engineering Finite Element Method. W. Sołowski 8
Elasticity Can be useful as long as you know the stress level, deformations and have data on how the soil behaves at that stress level and deformations level - than you just choose the exactly right elastic parameter for given circumstances and calculate the strains from the stresses or stresses from strains - used for example in calculations of settlements we know the stress levels from some simple solutions than we compute strains using right set of elastic parameters for given small layer of soil Can be pretty accurate Department of Civil Engineering Finite Element Method. W. Sołowski 9
Department of Civil Engineering 0 Finite Element Method. W. Sołowski Elasticity ( )( ) ε D + = 23 3 2 33 22 23 3 2 33 22 2. 0 2 0 0 2 0 0 0 0 0 0 0 0 0 2 ε ε ε ε ε ε el v v v v v v v v v v v v v v E ε D = el
Non-linear elasticity Real soil behaviour Department of Civil Engineering Finite Element Method. W. Sołowski
Non-linear elasticity We can only match single line as no dependency on stress level Real soil behaviour Department of Civil Engineering Finite Element Method. W. Sołowski 2
Non-linear elasticity el = D ) ε ( 0 Department of Civil Engineering Finite Element Method. W. Sołowski 3
Hypoelasticity Introduces dependence on stress level etc; adds complexity but does not solve all problems; still there are some successful models based on these concepts Non-linear elasticity: Real soil behaviour Department of Civil Engineering Finite Element Method. W. Sołowski 4
Elasto-plasticity Idea: - let s divide strains into these which are elastic (which are fully recoverable) and plastic (which are generally not) - assumes that after removing the stress the material does not necessarily returns back to the initial state (only the elastic part of strains is reversible) - plastic behaviour (in simpler models) unrecoverable - in more advanced models, the plastic strains are connected to the model parameter called hardening parameter - in such case if we have hardening parameter decrease, the plastic strain will decrease too Department of Civil Engineering Finite Element Method. W. Sołowski 5
Basic Concepts of Perfect Plasticity
Plastic Behaviour Example of elasto-plastic behaviour: traction test (D) in metals F A O A F L F A Yielding B A C D Y A L 0 C Elastic behaviour e L L 0 L O B D L 0 s p ~Y (logscale )
Plastic Behaviour F A Idealization of elasto-plastic behaviour A C Y P O B D L L 0 Yield point, the stress cannot be higher than this value Elastic behaviour Plastic behaviour: unrestricted plastic flow takes place at this stress level. p ε ε e ε ε ε = ε e + ε p
Plastic Behaviour Idealization of elasto-plastic behaviour, different models Yield limit depends on (effective) stresses ε ε ε ε Rigid Perfectly Plastic Elasto-plastic perfect plasticity Elasto-plastic hardening Elasto-plastic softening dε = dε e + dε p total elastic plastic
Some Basic Concepts Plastic models allow to determine in a direct way the ultimate states and failure to model irrecoverable strains to model changes in material behaviour to model a more proper way the behaviour of fragile or quasi-fragile materials
Some Basic Concepts Strains (ε) Total strains Elastic strains (recoverable on unloading) Plastic strains (not recoverable on unloading) Total strains = Elastic strains + Plastic strains Stresses () Total stresses = Effective stresses + Pore Pressures
Yield Surface Used to delimit the elastic domain PLASTIC On the surface IMPOSSIBLE STATE Outside It is a generalization of the D case Yield limit (-D) Yield surface (2D- 3D) 3 ELASTIC Inside yield surface 2 F(, ξ ) = 0 F( )=0 ij i
Flow Rule In one-dimensional problem, it is clear that plastic strains take place along the direction of applied stress, ε p ε p In 2D or 3D we need to make a hypothesis regarding the direction of plastic flow (relative magnitude of plastic strain increments) 3, ε 3 p
Plastic Potential and Flow Rule Plastic Deformations In general, dε p ij control the magnitude of plastic deformation g = dλ ij control the direction of the plastic deformations: the vector of the plastic deformations is normal to the g = constant surfaces g
Plastic Potential and Flow Rule Yield Surface (f) and Plastic Potential (g) are generally different functions If f g => associated plasticity The components of the plastic deformations are related, i.e. there is a coupling, which is defined by the flow rule The plastic deformations depend on the stress state rather than the increment of the stresses applied
Plastic Potential and Flow Rule FLOW RULE ASSOCIATED FLOW RULE NON ASSOCIATED The flow rule defines direction of plastic strain increment So, we know the plastic-strain direction, but how we can determine the magnitude?
Mohr Coulomb Model
3 3 2 = 3 Mohr-Coulomb Idealisation of Geomaterials E 3 3 ε
Elasto-plasticity: Mohr-Coulomb Reasonably good for: - determining soil strength - especially when soil is *not* overconsolidated Bad for: - deformations - inaccuarate both in elastic and plastic regions F = ( ) ( + )sinφ 2c cosφ = 0 3 3 Problem: cohesion is not constant. Friction angle is also not constant (but changes usually little in the usable range) Department of Civil Engineering Finite Element Method. W. Sołowski 29
Elasto-plasticity: Mohr-Coulomb Engineers are used to MC model so it is widely used Many modifications to improve its prediction of deformations F = ( ) ( + )sinφ 2c cosφ = 0 3 3 Problem: cohesion is not constant. Friction angle is also not constant (but changes usually little in the usable range) Department of Civil Engineering Finite Element Method. W. Sołowski 30
Elasto-plasticity: Mohr-Coulomb F = ( ) ( + )sinφ 2c cosφ = 0 3 3 Department of Civil Engineering Finite Element Method. W. Sołowski 3
Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes place in the soil mass when mobilised (actual) shear stress at any plane (τ m ) becomes equal to shear strength (τ f ) which is given by: τ m = c + n tanφ = τ f where c and φ are strength parameters. f( )= τ - n tanφ c = 0
τ Failure criterion 3 2 n > > 2 3 Note that the value of intermediate stress ( 2 ) does not influence failure
τ τ = c + tanφ f n φ τ f C D c A 3 90 φ nf B 2θ n 45 φ 2 45 φ 2 > 3 3 3 F 3 3 FAILURE PLANES = ( ) ( + )sinφ 2c cosφ = 0
Mohr-Coulomb in Principal Stress Space 3 2 = 2 = 3 Mohr Coulomb failure surface is a irregular hexagon in the principal stress space
Mohr-Coulomb in Principal Stress Space Mohr-Coulomb dε p It has corners that may sometimes create problems in computations 3 2
Flow Rule for Mohr Coulomb For Mohr-Coulomb flow rule is defined through the dilatancy angle of the soil. τ Plastic potential function Yield function φ G( )= τ - n tanψ const.= 0 ψ where ψ is the dilatancy angle and ψ φ. 3 n > > 2 3
Associated and Non-Associated Flow Rule τ, γ τ, dγ p F=0 G=0 n, dε n p, ε n p n
How to understand dilatancy i.e., why do we get volume changes when applying shear stresses? ϕ = ψ + ϕ i The apparent externally mobilized angle of friction on horizontal planes (ϕ) is larger than the angle of friction resisting sliding on the inclined planes (ϕ i ) strength = friction + dilatancy
How to understand dilatancy When dense sands or overconsolidated clays are sheared they dilate Larger the particle size, greater the dilation Mohr-Coulomb idealisation implies dilation at a constant rate when soil is sheared. This is unrealistic.
Mohr Coulomb Model some more advanced subjects
MC model p'- q- space F = ( ) ( + )sinφ 2c cosφ = 0 3 3 ' = 3p' 2 ' q ' 3 3 = ' ' = 3p' 2 ' p' = ( ' + 2 ' 3 ) q = ' ' 3 3 3 + 2q = 3p' + 2q ' 3 = 3p' 2 ' 3 ' + ' 3 6 p ' + q = 3 q = 3p' 3 q
MC model p'- q- space 6 p + q or q = sinφ + 2ccosφ 3 3q= 6 p sinφ + qsinφ + 6ccosφ q 6sinφ 6ccosφ = p + 3 sinφ 3 sinφ q = η p + c where η * 6sinφ 6ccosφ 3 sinφ 3 sinφ * =, c = F = q η p c = * 0 MC - Model formulated in p' - q
MC model p'-q- space p' q = K' 0 e 0 ε v e 3G' ε q Assuming associated flow rule and ideal plasticity ( ) F p q = q p c = *, η 0 F F dp + dq = 0 : consistency condition condition p q
Formulation of D ep for MC p ( εv εv ) p ( εq εq ) dp = K d d dq = 3G d d dε dε p v p q = = F dλ p F dλ q Substituting into consistency condition leads to: F F F F F F Kdεv Kdλ + 3Gdεq 3Gdλ = 0 p p p Q q q
Formulation of D ep for MC F F F F K 0 dεv Kdε 3, v + Gdεq p Q p q 0 3G q dε q dλ = = F F F F F K + 3G p p q q F F K 0 p, p q 0 3G F q
Formulation of D ep for MC F F ηkdεv + 3Gdεq = η, =, dλ = 2 p q η K + 3G p p dε = dλη, dε = dλ v dp p K 0 dε dε v v = p dq 0 3G dε dε q q q T { dε } K { η } dλ K 0 0 v = 0 3G dε 0 3G q { dε } { η } [ ] { dε } η F q η p c K 0 K 0 v K 3G = G dε G K G 2 0 3 0 3 η + 3 q * = = dε v q 0
Formulation of D ep for MC 2 K 0 K 0 η K 3Gη 2 0 3G η K + 3G 0 3G η K 3G = 2 2 K 0 η K 3GKη 2 2 = 0 3G η K + 3G 3GKη 9G = K η K 2 2 3GKη 2 2 η K + 3G η K + 3G 2 3GKη 9G 3G 2 2 η K + 3G η K + 3G { dε } dε v q { dε } dε v q { dε } dε v q
Formulation of D ep for MC D ep 2 2 η K 3GKη K 2 2 η K + 3G η K + 3G 3GKη 9G 3G 2 2 η K + 3G η K + 3G = 2 ep det D = 0 perfect plasticity In general form : for non associated flow: Q D ep = T e e F F e T D D D T F e F D
MC model for element tests
Limitations of MC model ()
Limitations of MC model (2)
Warning for dense sands
Tips for fine-grained soils
Drawbacks of MC Mohr-Coulomb failure criterion is well proven through experiments for most geomaterials, but data for clays is still contradictory! An associated flow rule implies continuous dilation at a constant rate upon shearing; this is unrealistic and leads to negative pore pressures in undrained conditions. In an non-associated flow rule with ψ < φ, the rate of dilation is less, but it is still constant. If ψ = 0 then the rate of dilation is zero. Care must be taken in applying the model for undrained loading.
Drawbacks of MC Soils on shearing exhibit variable volume change characteristics depending on pre-consolidation pressure which cannot be accounted for with MC In soft soils volumetric plastic strains on shearing are compressive (negative dilation) whilst Mohr-Coulomb model will predict continuous dilation
To summarize the limitations of MC are: bi-linearity (const. E ) unlimited dilation isotropy elastic response far from the limit state more advanced approximation of soil behavior: Hardening Soil Model (sand), Soft Soil Model (clay)
Other elastic-perfectly plastic models
Elasto-plasticity: Hoek - Brown Used in rock mechanics Department of Civil Engineering Finite Element Method. W. Sołowski 59
GSI Geological strength index Department of Civil Engineering Finite Element Method. W. Sołowski 60
Elasto-plasticity: Hoek - Brown Used in rock mechanics Department of Civil Engineering Finite Element Method. W. Sołowski 6
Elasto-plasticity: Hoek - Brown Designed for: - relatively lowheight walls (generally 0-20m, less than 50m) - tunnels - generally rocks If you use it outside of the intended use you have problems. Hoek-Brown into Mohr- Coulomb very tricky Department of Civil Engineering Finite Element Method. W. Sołowski 62
Elasto-plasticity: Tresca Simplest: - perfect plasticity (concept dated back to Roman times) yield criterion: anything in compression is safe - currently used for soils Tresca model (elastic perfectly plastic) Size of the yield surface does not change is independent from the mean stress level: Department of Civil Engineering Finite Element Method. W. Sołowski 63
Elasto-plasticity: Tresca Tresca Useful? Department of Civil Engineering Finite Element Method. W. Sołowski 64
Elasto-plasticity: Tresca Useful when we assume that soil has undrained shear strength s u which is independent from the stress state Used (when strain rate dependency added e.g. for simulation of pile installation) Tresca Department of Civil Engineering Finite Element Method. W. Sołowski 65
Tresca F = Y 3 0 = 0 in invariant formulation F = 2 cosθ Y0 = 0 > dependent on Lode angle
Tresca in 3D F = 2 cos θ Y = 0 0 > dependent on Lode angle
Tresca Applications in geotechnics Total stress analysis for undrained behaviour (ϕ = 0) c u.. undrained shear strength Y0 = 2 c u No volume change > ν = 0.5 (for numerical reasons 0.49) Plastic volumetric strains? assume Lode angle = 0 F 2 2 c = = u 0 Assume associated flow rule dε P M F = dλ M = 0 F dε P = dλ = 2dλ d ε P vol = d ε P M = 0
Tresca Applications in geotechnics d ε P vol = d ε P M = 0
Stress invariants ( ) ( ) z y x M + + = + + = 3 3 3 2 J 2 = ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] 2 3 2 2 2 2 3 2 3 2 2 2 2 2 6 M M M J + + = + + = ( )( )( ) M M M J = 3 2 3 = 3 3 2 3 3 sin 3 θ J Mean effective stress Deviatoric stress (in general form) Lode angle -30 < θ < 30
Meaning of stress invariants ON = 3 M PN 2 = = = OP 2 2 ( ) + ( ) + ( ) 2J 2 2 ON M 2 = 2 M 3 M 2 = PN = J = 2 2 2
Von Mises F = 3 Y = 0 0 > not dependent on Lode angle Y 0.. yield stress for axial compression or tension r = 2 Y 3 0 Von Mises criterion in m space Von Mises criterion in π - plane
Von Mises in 3D F = 3 Y = 0 0 > not dependent on Lode angle Y 0.. yield stress for axial compression or tension
Drucker-Prager
MC Invariant formulation F = M sin ϕ + cosθ sinϕ sinθ c cosϕ = 3 0
Drucker-Prager vs MC Drucker-Prager and Mohr-Coulomb criteria in π - plane
Thank you
Literature Hard to find a single book Check: Briaud JL. Geotechnical Engineering : Unsaturated and Saturated Soils. (linked in the materials section, good introduction) Potts & Zdravkovic, Finite element analysis in geotechnical engineering. Theory. Thomas Telford, 999. Muir Wood D., Soil Behaviour and Critical State Soil Mechanics. Cambridge University Press. 990. Books on theory of plasticity can be helpful Department of Civil Engineering Finite Element Method. W. Sołowski 78