Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 / pp. 80-89 UDC: 59.74:004.0 Stress Integration for the Drucker-rager Material Model Without Hardening Using the Incremental lasticity heory D. Rakić, M. Živković, R. Slavković, M. Kojić 4,5,6 Faculty of Mechanical ngineering, University of Kragujevac drakic@kg.ac.yu zile@kg.ac.yu radovan@kg.ac.yu 4 Harvard School of ublic Health, Boston, USA mkojic@hsph.harvard.edu 5 Research and Development Center for Bioengineering BIOIRC, Kragujevac, Serbia 6 Department of Biomedical ngineering, University of exas Medical Center at Houston, USA Abstract his paper presents the formulation of the stress integration procedure for the Drucker-rager material model with associative yielding condition by using the incremental plasticity theory. he idea of this method is to reach the solution by calculating the plastic matrix according to the method of incremental plasticity (used for elastic matrix corrections), and with use of the total strain increment. he computational procedure is implemented within the program package AK. Results of this procedure were compared with the solutions obtained by the governing parameter method (Kojic and Bathe 005), as well as with the results obtained by the program package NX-Nastran. Keywords: Drucker-rager model, FM, Geomechanics, Incremental plasticity. Introduction Stress integration represents calculation of stress change during an incremental step, corresponding to strain increments in the step. It is in essence the incremental integration of inelastic constitutive relationships to trace the history of material deformation. he stress integration is an important ingredient in the overall finite element inelastic analysis of structures. It is important that the integration algorithm accurately reproduces the material behavior since the mechanical response of the entire structure is directly dependent on this accuracy. he algorithm should be also computationally efficient because the stress integration is performed at all integration points. For general applications, this computational procedure should be robust, providing reliable results under all possible loading (straining) conditions (see e.g. discussion of these issues in e.g. Kojic and Bathe 005). In this paper we present a formulation of the computational algorithm for the Drucker- rager material model without hardening using an incremental plasticity approach. he results using this approach were compared with the results obtained using the implicit integrations
Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 8 scheme termed as the governing parameter method (Kojic, 996), as well as with results obtained by using other software packages. In the next section we present formulation the Drucker-rager material model, followed by the derivation of the elastic-plastic constitutive matrix in general associated plasticity. hen, the general relations are implemented on of the Drucker-rager material model.. Formulation of the Drucker-rager model his material model is defined by the yield surface called the Drucker-rager yield surface. In the space of main stresses, this surface represents a cone, whose axis matches the space diagonal of the coordinate system of principal stresses σ, σ, σ (Figure ). his material model is a special case of the Drucker-rager material model with a cap (Kojic and Bathe 005). Fig.. Drucker-rager failure surface. he model can be represented in the I J coordinate system, as shown in Fig., where Fig.. Drucker-rager model in space I J. is the first stress invariant, and I σ m σδ ij ij σ ii ()
8 D. Rakić et al. : Stress Integration for the Drucker-rager Material Model J SS () ij ij is the second invariant of the deviatoric stress. Deviatoric stress S ij is defined as: Sij σ ij σmδij () he elastic domain is bounded by the Drucker-rager surface and by the maximum tension line with the tension cutoff stress. his representation of the model is suitable for development of the computational procedure. If the stress point of a trial solution is above the Drucker-rager line, the stresses at the end of step have to be reduced to satisfy the following yield condition: f I J k (4) tδ t tδ t tδt D α 0 where α and k are material constants; and the upper left index t Δ t denotes the end of step. If the stress exceeds the allowed tension stress value, the stress reduction is performed by equalizing all three components of normal stress to one third of the maximum tension stress, i. e.: σ σ σ (5) while the shear stress components are set to zero: σ ij 0 i j (6). lastic-plastic matrix For an elastic-plastic material, the stress increment { dσ } can be expressed by the total strain increment { de } as { dσ} C { de} (7) where C is the elastic-plastic constitutive matrix defined below. In the case of small strain material deformation, the total strain increment can be divided into elastic and plastic part: from which follows { } { de de } { de } (8) { de } { de} { de } (9) he stress and elastic strain increments are related by the elastic constitutive law, { } dσ C { de } (0) where C is elastic constitutive matrix. Substituting (9) into (0), we obtain
Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 8 { dσ} C ( { de} { de }) () We next include into this derivation a yield function F, which in general has a form: ( ) ( ) A differential of the yield the function can be expressed as F F σ F σ ij () F F F F F F df dσ dσ dσ dσ dσ dσ σ σ σ σ σ σ xx yy zz xy yz zx xx yy zz xy yz zx () or in matrix form, df { dσ} In the incremental theory of plasticity we have that the yield function during plastic deformation must be equal to zero (when F 0 we have elastic unloading), hence F 0 and consequently df 0 is satisfied over the load step. he last condition can be written in the following form: (4) df { dσ} 0 (5) where is whereas { dσ } is F F F F F F σxx σyy σzz σxy σyz σzx (6) { dσ} dσxx dσ yy dσzz dσxy dσ yz dσzx (7) Further, we assume a non-associative plasticity, therefore the increment of plastic strain is { de } G dλ where: G ( σ ) is the plastic potential function, and dλ is a positive scalar. (8) Substituting the plastic strain increment (8) into (), we obtain the stress increment: { σ} { } G d C de dλ C σ Now, we substitute this stress increment into (5): (9) or F df C de d C G σ { } λ 0 (0)
84 D. Rakić et al. : Stress Integration for the Drucker-rager Material Model hen, the parameter dλ is df C de d C F F G σ σ { } λ 0 () { de} C σ dλ F G C Finally, substituting dλ from () into (9), obtain the relationship between { dσ } and the total strain increment { de } : () { dσ} C { de} G C C de σ σ G C By comparing this and equation (7) it follows: { } () G C C σ σ C C G C his is a general expression for the elastic-plastic constitutive matrix. Also, by substituting () into (8) we can calculate the plastic strain increment{ de } : (4) { de } G F C { de} σ σ G C In the case of associative plasticity the yield function F is at the same time the plastic potential function. In our implementation of the above general expressions to the Drucker- rager model we will adopt the associative plasticity assumption. (5) 4. Implementation of the general associative plasticity relations to the Drucker-rager material model We further use the Drucker-rager yielding surface defined as: F α I J k (6) and assume the associative plasticity, hence we have that the plastic potential is defined by the same function:
Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 85 G α I J k (7) dil he first invariant I is defined by equation (), while the second stress invariant J is defined by (): J ( σ σ ) ( σ σ ) ( σ σ ) σ σ σ 6 4 5 6 where the one-index notation is used (as in the subsequent text). Also, we can express J as J I I (9) where the second stress invariant I is: 4 5 6 (8) I σ σ σ σ σ σ σ σ σ (0) Most frequently used material constants of geological materials (which are experimentally evaluated) are the internal friction angle φ and the cohesion c. hey are the characteristic of the Mohr-Coulomb material model. here are direct relations between these parameters and parameters k and α of the Ducker-rager model: α sinφ ( φ ) ( sinφ ) sin k 6c cosφ () hese relations are obtained by coinciding the Drucker-rager cone with the outer apexes of the Mohr-Coulomb hexagon locus (compressive cone). he derivatives of the Drucker-rager yield function with respect to stresses can be calculated by using the chain rule: J F I F () I J σ Note that this expression is applicable to all models where the yield function is expressed in terms the stress invariants I and J (which is the case for a large number of plasticity models). Also we have the following relations from (6): F α () I and F J J he derivatives of the stress invariants with respect to stresses are: (4) and I [ 0 0 0] (5)
86 D. Rakić et al. : Stress Integration for the Drucker-rager Material Model J ( ) ( ) D σ σ σ σ σ σ ( σ σ σ) σ4 σ5 σ6 (6) σ Now, substituting () to (6) into () we obtain: σ σ σ α 6 J σ σ σ α 6 J σ σ σ α 6 J σ 4 J σ 5 J σ 6 J (7) he above relations for the stress integration over a load (time) step are summarized in able. he left upper indices t and t Δ t denote start and end of a time step. tδ Known quantities: { t e}, { t e }, { t t p σ }, { e } A. rial (elastic) results: tδt t { dσ} [ Ce]{ de } [ Ce] ({ e} { e} ) { t Δ t σ} { t σ} { dσ} I σ σ σ J ( σ σ ) ( σ σ ) ( σ σ ) σ σ σ 6 4 5 6 F α I J k B. Yielding condition check: IF ( F < 0 ) trial result correct (go to ) IF ( F 0 ) elastic-plastic result (CONINU) J F I F I J σ
Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 87 Associative yielding assumption: G [ Ce ]{ de} σ dλ F G [ Ce ] C. Stress correction (solution by bisection - bisection loop): { de } D. IF ( ABS ( F ) G dλ { de } { de} { de } { dσ } [ C ]{ e de } { t Δ t σ} { t σ} { dσ} Calculation of new invariants: I σ σ σ J ( σ σ ) ( σ σ ) ( σ σ ) σ σ σ 6 4 5 6 F α I J k tδt t { e } { e } { de } tδ. ND: { t tδt p σ},{ e } OL ) go back to C with new dλ from bisection able. Algorithm for incremental stress integration 5. Verification of the computational procedure Verification of the proposed computational procedure was done by solving elastic-plastic deformation of a sand box specimen, as specified in the literature (Kojic and Bathe 005). he loading is modeled by prescribed displacement on the top, while only vertical sliding is allowed on the sides. he bottom model side is completely restricted. A half of the specimen is modeled with the use of the appropriate symmetry boundary conditions (Figure ). ight-node D finite elements were used. Displacement is applied at the nodes on the model upper side.
88 D. Rakić et al. : Stress Integration for the Drucker-rager Material Model [] step [] displacement u [] 0 [4] 0.0 [5] 0 [6] 0.0 [7] 0 [8] 0.05 Material data: Young s modulus, 00 oisson s ratio, ν 0. Yield function parameter, α 0.05 Yield function parameter, k 0. ension Cutoff limit, 0.0 Dimension, D.0 Dimension, H 0.5 Fig.. Sand specimen geometry, material data and load. he problem was solved using the program package AK (Kojic et al. 998), by employing the described method and the governing parameter method. Also the program NX NASRAN was used. Figures 4 and 5 show the results. e xx 0.0-0.07-0.06-0.05-0.04-0.0-0.0-0.0 0 -.0 -.0 -.0-4.0 Nas tran AK -5.0-6.0 σ xx Fig. 4. he calculated axial stress as a function of the axial strain.
Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 89 Failure surface AK 0.9 0.8 0.7 0.6 0.5 0.4 0. 0. 0. sqrt(j ) -5 I - - -9-7 -5 - - 0 Fig. 5. he second invariant deviatoric stress as function of first stress invariant. It can be seen from the presented results that all three solutions are practically the same. 6. Conclusions he presented methodology gives very accurate results when compared with other methods, such as the governing parameter method. he advantage of the presented computational procedure is that it is formulated in general form so that it can be applied to various yield functions. Also, this procedure can be implemented in an explicit computationally efficient incremental scheme, where yield condition check D (able ) is not performed, but then very small load steps are necessary. References Bathe K. J. (98), Finite lement rocedures in ngineering Analysis, rentice-hall, Inc., nglewood Cliffs, New Jersey. Geo-Slope Office (00), Sigma W-for finite element stress and deformation analysis V5, User s guide. Kojic M. (996), he governing parameter method for implicit integration of viscoplastic constitutive relations for isotropic and orthotropic metals, Computational Mechanics, Vol. 9, No., pp. 49-57. Kojic M., Bathe K. J. (005), Inelastic Analysis of Solids and Structures, Springer Berlin Heidelberg New York. Kojic M., Slavkovic R., Zivkovic M., Grujovic N. (998), AK-finite element program for linear and nonlinear structural analysis and heat transfer, Faculty for Mechanical ngineering, Kragujevac, University of Kragujevac. Woods R., Rahim A. (007), CRIS Geotechnical Finite lement Analysis Software, echnical Reference Manual.