Published in: Proceedings of the Twenty Second Nordic Seminar on Computational Mechanics

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1 Aaborg Universitet An Efficient Formuation of the Easto-pastic Constitutive Matrix on Yied Surface Corners Causen, Johan Christian; Andersen, Lars Vabbersgaard; Damkide, Lars Pubished in: Proceedings of the Twenty Second Nordic Seminar on Computationa Mechanics Pubication date: 2009 Document Version Pubisher's PDF, aso known as Version of record Link to pubication from Aaborg University Citation for pubished version (APA): Causen, J., Andersen, L., & Damkide, L. (2009). An Efficient Formuation of the Easto-pastic Constitutive Matrix on Yied Surface Corners. In L. Damkide, L. Andersen, A. S. Kristensen, & E. Lund (Eds.), Proceedings of the Twenty Second Nordic Seminar on Computationa Mechanics (pp ). Department of Civi Engineering, Aaborg University. DCE Technica Memorandum, No. 11 Genera rights Copyright and mora rights for the pubications made accessibe in the pubic porta are retained by the authors and/or other copyright owners and it is a condition of accessing pubications that users recognise and abide by the ega requirements associated with these rights.? Users may downoad and print one copy of any pubication from the pubic porta for the purpose of private study or research.? You may not further distribute the materia or use it for any profit-making activity or commercia gain? You may freey distribute the URL identifying the pubication in the pubic porta? Take down poicy If you beieve that this document breaches copyright pease contact us at vbn@aub.aau.dk providing detais, and we wi remove access to the work immediatey and investigate your caim. Downoaded from vbn.aau.dk on: september 14, 2018

2 Proceedings of the Twenty Second Nordic Seminar on Computationa Mechanics Aaborg University 2009 ISSN DCE Technica Memorandum No. 11 An efficient formuation of the easto-pastic constitutive matrix on yied surface corners Johan Causen, Lars Andersen and Lars Damkide Department of Civi Engineering Aaborg University, Aaborg, Denmark e mai: jc@civi.aau.dk Summary A formuation for the easto-pastic constitutive matrices on discontinuities on yied surfaces is presented, for use in finite eement cacuations. The formuation entais no rounding of the yied surface or the pastic potentia, as it is done in most other formuations, and therefore exact anaytica soutions can be approached. Computationa exampes are given with the Mohr-Couomb, the Modified Mohr-Couomb and the Hoek-Brown modes. Introduction In practica geotechnica engineering most design cacuations on soi structures are carried out with the Mohr-Couomb materia mode, with the we-known hexagona shaped yied criterion in principa stress space, see Fig. 1b. For cays the Tresca criterion, Fig. 1a, is used, and for rocks and concrete the Modified Mohr-Couomb and the Hoek-Brown criteria are often used, see Fig. 1c and d. As can be seen from the figure these criteria possess corners and apices, which expicity have to be taken into account when formuating the constitutive matrices used for formuating the goba stiffness matrix. This is especiay true for 3D-cacuations where a the different corner and apex discontinuities may come into pay. One option of deaing with these discontinuities is to perform a oca rounding of the yied criterion and/or the pastic potentia, see e.g. [1, 2]. This option seems to work but the obtained numerica resuts do no onger converge towards the exact anaytica resuts. a) b) Tresca Mohr-Couomb c) d) Rankine Hoek-Brown Modified Mohr-Couomb Figure 1: Exampes of yied criteria with corners in principa stress space: a) The Tresca criterion. b) The Mohr-Couomb Criterion. c) The Modified Mohr-Couomb criterion. d) The Hoek-Brown criterion. 135

3 In this paper a formuation is presented that does not incude a rounding of the corners or apices. It is aso shown that the numerica soution for a footing on a Mohr-Couomb soi converges towards the exact anaytica soution. Constitutive matrix on a surface When a stress point is ocated on a yied surface the easto-pastic constitutive matrix is found as D ep = D DbT ad a T (1) Db where a = f / σ, b = g/ σ and D is the eastic constitutive matrix. f and g is the yied function and the pastic potentia, respectivey. Note that D ep is singuar with respect to b,i.e.d ep b = 0. Constitutive matrix on a corner and an apex When the stress point is ocated on a corner the constitutive matrix must be singuar with respect to both b 1 and b 2. In Fig. 3 a direction vector of a yied surface corner, is shown. This can be regarded as a direction vector of any of the ines defining the yied criteria in Fig. 1. In Fig. 3 the direction vector of the pastic potentia corner, g, is aso shown. From these direction vectors it is shown in [3] that the douby singuar constitutive matrix on a ine in principa stress space can be expressed as: D ep = ( g ) T T D 1 g = (ā 1 ā 2 )( b 1 b 2 ) T (ā 1 ā 2 ) T D 1 (2) ( b 1 b 2 ) The overbar indicates the the vectors and matrices are expressed with respect to the principa coordinates without the shear component terms, i.e. the vectors have three components and the matrices three by three. The symbo indicates the cross product. The shear part is added, [ D ē ˆD ep ep ] = (3) Ḡ and the matrix is transformed from the principa stress coordinate system into the xyz-coordinate system. In the above equation the hat, ˆ, signifies that the matrix incudes a six by six components and is expressed in the principa coordinate system. The matrix Ḡ is the shear part of the eastic constitutive matrix. There are two different forms of constitutive matrix on an apex. If the stress point is ocated on an apex on the hydrostatic ine the constitutive matrix must be singuar with respect to a stress directions, i.e. ˆD ep a,1 = Dep a,1 = 0 (4) b 1 g Figure 2: A direction vector,, of an intersection ine in principa stress space. The corresponding potentia curve direction vector is denoted g.an eastic strain direction vector is denoted ē.the vectors b 1 and b 2 are perpendicuar to the direction vector of the pastic potentia intersection ine, g. This is the case on the Mohr-Couomb apex, the Hoek-Brown apex and one of the Modified Mohr- Couomb apices, see Fig. 1. If, on the other hand, the stress point is ocated on an apex not on the hydrostatic ine it is singuar ony in the norma directions, i.e. its composition in the principa coordinates is [ ] ˆD ep 0 a,2 = (5) Ḡ b 2 136

4 This is the case for stress points ocated on the Modified Mohr-Couomb apices outside the hydrostatic ine, see Fig 1c. Improved formuation The formuations for the constitutive matrices given above works we for two-dimensiona modes where the (instant) friction ange is not too high, see e.g. [3, 4]. But for high friction anges and/or three dimensiona probems the above formuations can cause the goba stiffness matrix to become i-conditioned. This is due to many stress points ocated on either corners or apices which add many singuarities to the goba stiffness matrix. This probem can be mended by adding a sma stiffness in appropriate directions. Improved formuation on the apex When the easticity of the materia is inear the impicit stress integration can written in the return mapping formuation, σ C = σ B Δ σ p, with σ B = σ A + DΔε and Δ σ p = DΔ ε p (6) Here σ C is the updated stress point on the yied surface, σ B is the eastic predictor stress and Δ σ p is the pastic corrector stress, a three expressed in the principa stress space as indicated by the overbar. The tota strain increment is denoted Δε and the pastic strain increment in principa coordinates is Δ ε p. A key point of the easto-pastic constitutive matrix is that it must be singuar in the direction of the pastic strain increment. A simpe method to add a itte stiffness in the formuation of D ep on the apex is given as D ep a,mod = 1 ( α D D T Δ ε p (Δ ε p ) T ) D (Δ ε p ) T DΔ ε p This matrix is singuar in the pastic strain direction and depending on the vaue of α posesses a sma stiffness in other directions. In the presentation a study on the optima vaue of α wi be given. Improved formuation on a corner When the updated stress point is ocated on a corner the basic formuation for the constitutive matrix is given by Eq. (2). Again a simpe formuation that adds a itte stiffness is (7) D epc = ( g ) T T ( D c ) 1 g + 1 β c c T c T ( D c ) 1 c (8) The direction vector c is the direction perpendicuar to the pastic strain direction, Δ ε p,andthe ine defining the corner,, see Fig. 2. In the presentation different resuts indicating the optima vaue of the scaar β wi be given. β contros the amount of stiffness that wi be added. The Ref. [5]. Computationa exampe To assess the vaidity of the formuation a cacuation is carried out with a rough circuar footing resting on a cohesioness Mohr-Couomb soi with a friction ange of ϕ = 30, and a sefweight of γ = 20 kn/m 3. For symmetry reasons ony a quarter of the footing is modeed, see Fig. 3a. 137

p FEM p exact 1[%] a) b) 40 35 30 25 20 15 n dof 10 000 30 000 100 000 Figure 3: a) A quarter of a circuar footing and an exampe of the eement mesh with 7425 degrees of freedom.

5 p FEM p exact 1[%] a) b) n dof Figure 3: a) A quarter of a circuar footing and an exampe of the eement mesh with 7425 degrees of freedom. b) Resuts from the bearing capacity cacuations compared to the exact vaue. The eements are standard ten-node tetrahedrons. A vertica forced dispacement is appied to the footing in steps and the bearing capacity is cacuated from the sum of the maximum reaction forces at the footing nodes divided by the footing area. The exact bearing capacity is found in Ref. [6]. The resut of the cacuations can be seen in Fig. 3 for different mesh densities. It is seen that the cacuated vaues converge toward the exact vaue. In the presentation resuts for the other materia modes shown in Fig. 1 wi be given. Concusion A formuation for easto-pastic constitutive matrices on corner and apex singuarities is given. The initia formuation is improved in order to make fu 3D-cacuations stabe. It is shown that finite eement cacuations based on the formuation converge towards the exact soution. References [1] A. J. Abbo and S. W. Soan. A smooth hyperboic approximation to the mohr-couomb yied criterion. Computers & Structures, 54(3): , [2] H. A. Taiebat and J. P. Carter. Fow rue effects in the tresca mode. Computers and Geotechnics, 35: , [3] Johan Causen, Lars Damkide, and Lars Andersen. An efficient return agorithm for non-associated pasticity with inear yied criteria in principa stress space. Computers & Structures, 85: , [4] Johan Causen, Lars Damkide, and Lars Andersen. Efficient return agorithms for associated pasticity with mutipe yied panes. Internationa Journa for Numerica Methods in Engineering, 66(6): , [5] Johan Causen. Efficient Non-Linear Finite Eement Impementation of Easto-Pasticity for Geotechnica Probems. PhD thesis, Esbjerg Institute of Technoogy, Aaborg University, [6] C. M. Martin. User guide for ABC - anaysis of bearing capacity, version 1.0. OUEL Report No. 2261/03, University of Oxford,

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