A Simple And Efficient FEM-Implementation Of The Modified Mohr-Coulomb Criterion Clausen, Johan Christian; Damkilde, Lars

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1 Aalborg Universitet A Simle And Efficient FEM-Imlementation Of The Modified Mohr-Coulomb Criterion Clausen, Johan Christian; Damkilde, Lars Published in: Proceedings of the 9th Nordic Seminar on Comutational Mechanics Publication date: 2006 Document Version Publisher's PDF, also known as Version of record Link to ublication from Aalborg University Citation for ublished version (APA): Clausen, J. C., & Damkilde, L. (2006). A Simle And Efficient FEM-Imlementation Of The Modified Mohr- Coulomb Criterion. In O. Dahlblom, L. Fuchs, K. Persson, M. Ristinmaa, G. Sandberg, & I. Svensson (Eds.), Proceedings of the 9th Nordic Seminar on Comutational Mechanics ( ). Lund Universitet. General rights Coyright and moral rights for the ublications made accessible in the ublic ortal are retained by the authors and/or other coyright owners and it is a condition of accessing ublications that users recognise and abide by the legal requirements associated with these rights.? Users may download and rint one coy of any ublication from the ublic ortal for the urose of rivate study or research.? You may not further distribute the material or use it for any rofit-making activity or commercial gain? You may freely distribute the URL identifying the ublication in the ublic ortal? Take down olicy If you believe that this document breaches coyright lease contact us at vbn@aub.aau.dk roviding details, and we will remove access to the work immediately and investigate your claim. Downloaded from vbn.aau.dk on: aril 02, 208

2 A simle and efficient FEM-imlementation of the Modified Mohr-Coulomb criterion Johan Clausen and Lars Damkilde Esbjerg Institute of Technology Aalborg University Esbjerg, Esbjerg, Denmark e mail: jc@aaue.dk and ld@aaue.dk Summary This aer resents a concetually simle finite element imlementation of the combined elasto-lastic Mohr-Coulomb and Rankine material models, also known as Modified Mohr-Coulomb lasticity. The stress udate is based on a return maing scheme where all maniulations are carried out in rincial stress sace which simlifies the calculations. The model suorts both associated and non-associated erfect lasticity. Introduction Materials such as sand and concrete show ressure deendent strength roerties. The simlest material model which incororates this ressure deendency is the Mohr-Coulomb material model. The yield criterion uses the well known arameters friction angle, ϕ, and cohesion, c f MC = kσ σ c =0, with k = +sinϕ sin ϕ and σ c =2c k () where σ, and are the rincial stresses. In this aer tension is taken as ositive. When c>0the Mohr-Coulomb model redicts a tensile strength which is larger than the tensile strength observed exerimentally, see e.g. references [], [2] and [3]. This discreancy can be mended by the introduction of the Rankine or tension cut-off criterion f R = σ σ t =0 (2) where σ t is the tension cut-off value, which is the highest tensile stress allowed in the material. The combination of these criteria is usually referred to as the Modified Mohr-Coulomb Criterion, cf. [2]. On Fig. this criterion can be seen in the rincial stress sace and in the σ lane. The Rankine art of the criterion, Eq. (2), is taken to be associated whereas the Mohr-Coulomb art is non-associated. σ3 a) b) f>0 Rankine f<0 σ Mohr-Coulomb f =0 aex σ = σ t σ k = +sin ϕ sin ϕ = σ c Figure : The Modified Mohr-Coulomb Criterion in a) rincial stress sace and b) the section through the σ lane.

3 Plastic stress udate for yield lanes in rincial stress sace From Eqs. (), (2) and Fig. it can be seen that the criterion consists of intersecting lanes in the rincial stress sace. As will be shown later the Modified Mohr-Coulomb criterion leads to nine different tyes of stress return, which must be roerly identified. This can be a cumbersome task in general stress sace. Therefore the method of [4] and [5] is very well suited for carrying out the lastic integration and formation of the constitutive matrix. In the following a short summary of the method will be given. The stress udate and formation of the consistent constitutive matrix require the derivative of the yield function and the first and second derivatives of the lastic otential. As only isotroic material models are considered the maniulations can be carried out with resect to any set of coordinate axes. Therefore the redictor stress is transformed into rincial stress sace and returned to the yield surface. Considering the fact that the stress return reserves the rincial directions, the udated stress can then be transformed back into the original coordinate system. This simlifies the maniulations of the return maing scheme considerably comared to the standard formulation, see, e.g. [6]. There are two reasons for this. Firstly the dimension of the roblem reduces from six to three, and secondly, in the three-dimensional stress sace the stress states can be visualized grahically, making it ossible to aly geometric arguments. Linear yield criteria in the rincial stresses are visualized as lanes in rincial stress sace. These lanes intersect in lines and oints, making three tyes of stress returns and constitutive matrices necessary: Return to a yield lane, return to a line, e.g. intersection of two yield lanes and finally return to a oint, e.g. intersection of three or more yield lanes. The three tyes of return are visualized on Fig. 2. The formulae for the different returns will be established in the following. The conditions for determining which return is needed will also be established by dividing the stress sace into different stress regions. Vectors and matrices are exressed with resect to the rincial axes. This means that the last three comonents of vectors are always zero and are not be shown as a matter of convenience. Even so all matrices and vectors are six-dimensional. σ Δσ σ B Return to lane σ C σ C σ C Δσ Δσ Return to oint σ B Return to line σ B Region II n rl r II I σ II I =0 l Region I f =0 Figure 2: Three intersecting yield lanes in rincial stress sace with three tyes of return shown. Figure 3: Boundary lane II I =0with normal n II I, which searates the stress regions I and II.

4 The task is to determine the udated stress, σ C, in the equation σ C = σ B Δσ = σ B ΔλD g σ (3) where σ B is the redictor stress state found by the solution of the global system of FEM equations, Δσ is the lastic corrector stress, Δλ is a lastic multilier, D is the elastic constitutive matrix and g is the lastic otential. Stress return to a lane The equation of a yield lane and a lastic otential in the rincial stress sace can be written as f(σ) =a T (σ σ f )=0 and g(σ) =b T σ (4) where σ f is a oint on the lane and a and b are the gradients, a = f σ and b = g σ (5) Both a and b are constant. The lastic corrector stress can be comuted as Δσ = f(σb ) b T Da Db= f(σb ) r with r = Db b T (6) Da where r is the scaled direction of the lastic corrector in rincial stress sace, i.e. r is at an angle with the lastic strain direction, b. Stress return to a line A line, l, in rincial stress sace has the equation l : σ = t r l + σ l (7) where t is a arameter with the unit of stress, σ l is a oint on the line and r l is the direction vector. The arameter t can be found as t = (r r 2 )T (σ B σ l ) (r r 2 )T r l (8) where r and r 2 are the lastic corrector vectors from Eq. (6) for the two yield lanes intersecting at the line. Stress return to a oint If the stress is to be returned to a singularity oint, σ a, e.g. an aex oint, see Figure 2, there is no need for calculations, as the returned stress is simly σ C = σ a (9)

5 Stress regions In this section it will be outlined how to determine to which lane, line or oint the stress should be returned. In order to do this the concet of stress regions is introduced, and the boundary lanes that searate them are defined. Each yield lane, line and oint is associated with a articular stress region. When the redictor stress is located in a given region it must be returned to the corresonding lane, line or oint. Two stress regions, I and II, searated by a boundary lane, II I =0are illustrated on Figure 3. When the yield functions and lastic otentials are linear in the rincial stresses, the boundary lanes are also linear. The direction of the lastic corrector, r, c.f. (6), and the direction vector of the line, r l, define the orientation of the lane, and so the equation of a boundary lane can be found as: II I (σ) =(r r l ) T (σ σ l )=n T II I (σ σ l)=0 (0) where n II I is the normal of the lane. The indices indicate that the normal oints into region II from region I. The oint on the lane is σ l, which can be taken as a oint that also belongs to l, see Fig. 3 and Eq. (7). If two stress regions are located as seen on Fig. 3, the following is valid for a given redictor stress, σ B located outside the yield locus, i.e. f(σ B ) > 0: II I (σ B ) 0 II I (σ B ) > 0 Region I Region II Return to f =0 Return to l () Constitutive matrix The consistent constitutive matrix is also formed in rincial stress sace. Details are given in refs. [4] and [5]. Modified Mohr-Coulomb lasticity The rincial stresses are ordered in descending order, i.e. σ. This means that the Modified Mohr-Coulomb criterion consists of only two lanes in the rincial stress sace, see Fig. 4. As can be seen on the figure the geometry of the yield lanes is bounded by five lines which a) b) σ a l MC 2 r MC 2 f =0 r MC σ σ R 2 l R 3 r R 2 l R 2 r R 3 r R σ R l R l MC σ Figure 4: a) The Modified Mohr-Coulomb criterion in rincial stress sace. b) Detail of the tension cut-off lane, f R. The line,, is the hydrostatic axis.

6 intersect at three oints. With reference to Fig. 4 the equations for the lines and their direction vectors are l MC : σ = tr MC + σ MC a, r MC =[ k] T (2) l2 MC : σ = tr MC 2 + σ MC a, r MC 2 =[ k k] T (3) l R : σ = tr R + σ a, r R = [0 0 ] T (4) l3 R : σ = tr R 3 + σ R, r R 3 = [0 0] T (5) where t is a arameter with the dimension of stress, and σ MC a, σ a and σ R are the Mohr-Coulomb aex (not shown on Fig. 4), the Modified Mohr-Coulomb aex and the intersection between lines l MC and l3 R, resectively. A fourth oint, denoted σr. These oints have the coordinates l R 3 σ MC a = σ c k, σ a = σ t σ t σ t, is the intersection between lines lmc 2 and σ t σ t, σr = σ t, σr 2 = kσ t σ c (6) kσ t σ c kσ t σ c The boundary lanes that searate the nine stress regions can be seen on Fig. 5. The equations of the boundary lanes will not be given here but can be found from the Eqs. (0) and (2-6). a) IX b) VIII IX VIII III I IV VII σ VI V IV II III I σ II Figure 5: a) Stress regions, denoted by roman numerals. b) Detail. Numerical examle A finite element calculation is carried out on a rigid smooth footing resting on a frictional cohesive soil. Two material models are emloyed. The first is a erfectly lastic Mohr-Coulomb model with ϕ =20, ψ =5 and c =20kPa. The second is the Modified Mohr-Coulomb material model with the same arameters and also a tension cut-off, σ t =0. A mesh of six-noded triangular linear strain elements is created, and can be seen on Fig. 6. This element mesh has a total of 347 elements with 500 degrees of freedom. The radius/halfwidth of the footing is r and the domain has a width of 2r and a height of 0r. A forced dislacement is alied to the footing in stes, and the footing ressure q is found from the reaction forces. The soil has a selfweight of γ =20kN/m 3, a modulus of elasticity of E =20MPa and a Poisson s ratio of ν =0.26. An initial earth ressure coefficient of k 0 =is used.

7 C L r q 0.8 q/q u Plane strain Axisymmetry 0r Modified Mohr-Coulomb Mohr-Coulomb 2r u/r Figure 6: Geometry, boundary conditions and element mesh for the comutational examle. Figure 7: Normalized load-dislacement curves. On Fig. 7 the load-dislacement curves can be seen. The dislacement has been normalized with resect to the footing radius and the load has been normalized according to Terzaghi s suerosition equation for the bearing caacity of surface footings q u = cn c + γrn γ (7) Fig. 7 shows that the Mohr-Coulomb and the Modified Mohr-Coulomb model redict almost the same bearing caacity with the Mohr-Coulomb bearing caacity being slightly larger. In a roblem with an eccentric load the difference would be more ronounced, as ositive normal strains could develo between the soil and a art of the footing without the develoment of tensile stresses. Conclusion A simle and efficient method of erforming the lastic stress udate for a Modified Mohr- Coulomb material is resented. In the method all maniulations are carried out in the rincial stress sace which simlifies these considerably comared to the equivalent maniulations in general six-dimensional stress sace. A numerical examle shows the erformance of the method. References [] Mogens Peter Nielsen. Limit analysis and concrete lasticity. CRC ress, 999. [2] Niels Saabye Ottosen and Matti Ristinmaa. The Mechanics of Constitutive Modelling. Elsevier, [3] R.B.J. Brinkgreve and P.A. Vermeer. Plaxis, Finite Element Code for Soil and Rock Analyses, Version 7. A.A. Balkema, Rotterdam, 998. [4] Johan Clausen, Lars Damkilde, and Lars Andersen. Efficient return algorithms for associated lasticity with multile yield lanes. International Journal for Numerical Methods in Engineering, 66(6): , [5] J. Clausen, L. Damkilde, and L. Andersen. An efficient return algorithm for non-associated mohrcoulomb lasticity. In B. H. V. Toing, editor, Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Comuting, Stirling, United Kingdom, Civil- Com Press. aer 44. [6] M.A. Crisfield. Non-Linear Finite Element Analysis of Solids and Structures, volume 2: Advanced Toics. John Wiley & Sons, 997.