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Multi-scale non-local approach for geomaterials
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 29 (2002) 121–129 www.elsevier.com/locate/mechrescom
Multi-scale non-local approach for geomaterials George Z. Voyiadjis a
a,*
, Chung R. Song
b
Department of Civil and Environmental Engineering, Louisiana State University, 3502 CEBA Bldg., Baton Rouge, LA 70808, USA b Geotechnical Division, Sambo Engineering Co. Ltd., Seoul 138-724, South Korea Received 10 October 2001; accepted 9 February 2002
Abstract A thermodynamically consistent multi-scale, rate dependent, non local approach is developed in this work for geo-materials in conjunction with the anisotropic modified Cam Clay model. The gradient for the micro-structure is incorporated through the micro level gradient of the back-stress and volumetric strain while the gradient for macrostructure is incorporated through the macro level gradient of back-stress and volumetric plastic strain. Gradient results in the regularization of the local behavior. Visco-plasticity is also incorporated for an additional regularization of the local behavior. Therefore, the effects of two separate regularizations are naturally separated. The plastic spin is incorporated to separate the effect of micro-structural rotation from the gradient effect. The flow characteristics of the soil is also incorporated in order to separate the viscosity effect from the flow effect. Through this multi scale non local approach, a more realistic simulation of large strain problems such as shear band formation can be achieved. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The behavior of soils is governed by many factors such as stiffness, strain rate and stress history, etc. The effects due to micro-structural changes are especially important for soils that are subjected to large strains and rotations. This study incorporates the effects of micro-structural changes by utilizing the plastic spin and the gradient theory. The plastic spin is used to incorporate the non-coaxiality of micro-structural rotation in reference to the macro-structural rotation. The gradient theory is used here in order to capture the micro-structural interaction between the micro-scale material particles. The plastic spin is known to consider the non-coaxiality between the substructure rotation and the macro-rotation (Zbib and Aifantis, 1988). For small strain conditions, this non-coaxiality is minor. However, it cannot be ignored at high strain conditions. To obtain the tensor quantity of the plastic spin, the rotation of the micro-structure needs to be addressed. Since the quantification of the micro-structure rotation is not possible in the continuum
*
Corresponding author. Tel.: +1-225-388-8668; fax: +1-225-578-9176. E-mail addresses: [email protected] (G.Z. Voyiadjis), [email protected] (C.R. Song).
0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 2 3 3 - 1
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G.Z. Voyiadjis, C.R. Song / Mechanics Research Communications 29 (2002) 121–129
mechanics platform, a certain relationship between the plastic spin and the embedded stress (back-stress) is used. In geomaterials, substantial portion of the external load is carried by inter-granular frictional resistance. This frictional resistance is very irregular. Some grains are subjected to high friction while some other grains are subject to low friction. The grains that are subjected to high friction will undergo smaller grain rotation for the same applied energy than the grains that are subjected to low friction. This phenomenon will result in a different plastic spin even for the same back stress. The expression of the plastic spin with respect to the back stress should include effects such as the inter-granular frictional force. The gradient theory allows one to incorporate the inter-granular stress into the continuum mechanics. When materials with a simple cubic array are subjected to deformation as shown in Fig. 1(a), grains will fall into the void space and will reach the compacted condition as shown in Fig. 1(b). This process will reduce the volume and nullify or reduce the contact stress for the adjacent layer of grains on the top. When materials with a cubic tetrahedral array are subjected to deformation as shown in Fig. 1(b), grains will climb over the adjacent grains and reach the loose packing condition as shown in Fig. 1(a). This process increases the volume. If the volume expansion is restricted, this process increases the contact stresses for the adjacent layer of grains at the top. This behavior indicates that whenever one has uneven stress (strain) distribution, it develops stress (strain) transfer to the neighboring grains. This kind of stress (strain) transfer is important when one has a locally concentrated very high strain region, such as a shear band. Along the shear band, micro-structural stress redistribution is needed for modeling the accurate behavior of the material. This is the fundamental concept of the gradient theory. In this study, the gradient theory is coupled with the plastic spin for separation of the micro-mechanical particle rotation and the inter-particle interaction. The gradient theory is also formulated in two length scales: the meso-scale and the macro-scale. The meso-scale gradient is formulated in terms of micro-level back stress gradient and volumetric strain gradient, while the macro-scale gradient is formulated in terms of the macro-level back stress gradient and the (corresponding) volumetric strain gradient. For this study, an anisotropic modified Cam Clay model (AMCCM) is used. The triaxial test and the cone penetration test maybe used to check the validity of the proposed model. Numerically obtained results for the proposed model using the plastic spin and the gradient maybe used to validate the model by comparing them with those obtained experimentally. The effects of the micro-structural changes (plastic spin) need to be incorporated in the analysis when the strains are high. The effects of the micro-structural interaction (gradient) are also significant in the case of large strains. The effects of the gradient are especially important in localization problems, since the gradient results in well-posedness of the numerical solution and smoothening of the strain distribution in the shear band.
Fig. 1. Grain arrangement during shear: (a) simple cubic array, (b) cubic tetrahedral array.
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2. Formulation of the thermodynamics Voyiadjis and Deliktas (2000) derived a multi-scale gradient theory based on a consistent thermodynamic formulation for metals and metal matrix composites. In this paper, similar approach is used for the gradient theory in geomaterials using an AMCCM. The Helmholtz free energy is expressed in terms of the internal state variables such that: ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ W ¼ W ee ; T ; AðkÞ ; rAðkÞ ; r2 AðkÞ ; r^ A^ðkÞ ; r^2 A^ðkÞ ð1Þ ðpÞ
where, ee is the elastic strain, T is the temperature, superscript ‘p’ represents plasticity, AðkÞ is a macro-scale internal state variable such as for the isotropic hardening or the kinematic hardening in plasticity. r ðpÞ represents the first order gradient, r2 represents the second order gradient, and A^ðkÞ represents the microscale internal state variable in a representative volume element (RVE). When one assumes symmetrical distribution of the gradients, the odd order gradient terms vanish, and Eq. (1) becomes: ðpÞ e 2 ðpÞ ^2 ^ðpÞ W ¼ W e ; T ; AðkÞ ; r AðkÞ ; r AðkÞ ð2Þ The link between macro-scale internal variables and micro-scale internal variables is obtained by averaging the micro-scale internal variables in the RVE. Z 1 ðpÞ ðpÞ AðkÞ ¼ ð3Þ A^ dVRVE VRVE VRVE ðkÞ where, VRVE is the volume of the RVE. The first and second order gradients of the macro- and micro-internal variables are defined as follows: 2 o Aij 2 ðpÞ r AðkÞ ¼ ð4Þ oXk oXk ðpÞ r^2 A^ðkÞ ¼
1 VRVE
Z VRVE
ðpÞ o2 A^ðkÞ
oXk oXk
ð5Þ
dVRVE
The integration maybe performed over a sub-volume of the RVE. One can express the time derivative of Eq. (2) in terms of its higher order state variables. By substituting into the Clausius–Duhem inequality one obtains: oW oW oW oW ðpÞ ðpÞ ^ 2 A^ðpÞ P 0 r q e : e_e þ r : e_p q ðpÞ : A_ ðkÞ q : r2 A_ ðkÞ q :r ð6Þ ðkÞ ðpÞ ðpÞ oe 2 ^ ^ oAðkÞ or2 AðkÞ or AðkÞ ðpÞ
ðpÞ
where, the meso-scale gradient terms r2 A^ðkÞ are dependent on the macro-scale variables, AðkÞ . However, the random periodic boundary condition ensures that there is no net flux of meso-scale gradients across the RVE boundary. Such a constraint would effectively prevent coupling between macro-scale and meso-scale gradient terms. The first term in Eq. (6) gives the relation between the stress, r and its associated variable, elastic strain, e_e . The remaining terms represent the total dissipation process due to plasticity. This can be expressed as the sum of the plastic dissipation as follows: ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ _ ðpÞ P ¼ r : e_p VðkÞ : A_ ðkÞ WðkÞ : r2 A_ ðkÞ XðkÞ : r2 A^ðkÞ
ð7Þ
124
G.Z. Voyiadjis, C.R. Song / Mechanics Research Communications 29 (2002) 121–129 ðrÞ
ðrÞ
ðrÞ
where, VðkÞ , WðkÞ , and XðkÞ are the thermodynamic force conjugates and are expressed as follows: oW
ðpÞ
VðkÞ ¼ q
oW
ðpÞ
WðkÞ ¼ q
ð8bÞ
ðpÞ
or2 AðkÞ oW
ðpÞ
XðkÞ ¼ q
ð8aÞ
ðpÞ
oAðkÞ
ð8cÞ
ðpÞ or^2 A^ðkÞ
One can now expresses the analytical form of the Helmholtz free energy as the quadratic form of its internal state variables 1 1 ðpÞ ðpÞ ðpÞ 1 ðpÞ 1 ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ qW ¼ ee : Eð/Þ : ee þ aðkÞ AðkÞ : AðkÞ þ bðkÞ r2 AðkÞ : r2 AðkÞ þ cðkÞ r^2 A^ðkÞ : r^2 A^ðkÞ 2 2 2 2
ð9Þ
Using Eqs. (8a)–(9) the following definitions can be obtained for the thermodynamic forces ðpÞ
ðpÞ
ðpÞ
VðkÞ ¼ aðkÞ AðkÞ ðpÞ
ð10aÞ
ðpÞ
ðpÞ
WðkÞ ¼ bðkÞ r2 AðkÞ
ð10bÞ
ðpÞ ðpÞ ðpÞ ZðkÞ ¼ cðkÞ r2 A^ðkÞ
ð10cÞ
The value of the thermodynamic forces can be obtained through the evolution relations of the internal state variables. However, it should be pointed out that there are two classes of evolution equations that need to be developed, normally one at the macro-scale and the other at the meso-scale level. The former can be obtained by assuming the physical existence of the dissipation potential at the macro-scale. The later one can be obtained by a micro-mechanical or phenomenological approach. ðpÞ One may also consider that the evolution equations of the internal state variables AðkÞ , can be obtained ðpÞ by integrating the evolution equations of the local internal state variables at the meso-scale, that is A^ðkÞ over ðpÞ ðpÞ the domain of the RVE. However, integration of the A^ðkÞ is a cumbersome task since at the meso-scale, A^ðkÞ is a function of many different aspects of the material inhomogeneities such as interaction of defects, size of defects, spacing between them, and distribution of defects within the sub RVE. Therefore, in this work the evolution equations of the meso-scale internal state variables are obtained through the use of the generalized normality rule of thermodynamics. In this regard the macro-scale dissipation potential is defined only in terms of the macro-scale flux variables as follows ðpÞ ðpÞ _ ðpÞ H ¼ H e_p ; A_ ðkÞ ; r2 A_ ðkÞ ; r2 A^ðkÞ ð11Þ By using the Legendre–Fenchel transformation of the dissipation potential ðHÞ, one can obtain complementary laws in the form of the evolution laws of flux variables as function of the dual variables ðpÞ ðpÞ ðpÞ H ¼ H r_ ; B ; r2 B ; r2 B^ ð12Þ ðkÞ
ðkÞ
ðkÞ
Eq. (12) is also expressed as follows: ! N N X X _ H ¼ F r_ ; p_ 0;i ; ai i¼1
i¼1
ð13Þ
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where (13) p0;i and ai are the isotropic and kinematic hardening forces respectively. The evolution equation of p0;i and ai are obtained from the normality rule as follows: oW ðpÞ ðpÞ B_ ðkÞ ¼ C A_ ðkÞ ¼ C ðpÞ oBðkÞ
ð14Þ
where, C is a constant. Using Eq. (14) and Perzyna’s elastic/viscoplasticity (1966) and applying the normality rule, one obtains the following evolution equations: oF ¼ ch/i tr of ð15aÞ e_pv ¼ ch/i tr orij orij a_ ij ¼ C1 ch/i
oF of ¼ C1 ch/i o aij o a ij
ð15bÞ
oF of ^ 2_a ^ij ¼ C2 ch/i ¼ C2 ch/i r _ ^ 2a ^ 2_a ^ij ^ij or r
ð15cÞ
oF of ^ 2_ ^ p0 ¼ C3 ch/i ¼ C3 ch/i r _ ^ 2 ^ 2_^ ^p0 p0 or or
ð15dÞ
where, F is the plastic potential, aij represent the back stress, ch/i represents the viscoplastic multiplier. h i is the MaCauley bracket, c is the viscosity. When one uses the AMCCM expressed by Voyiadjis and Song (2000), the following expression is obtained for the yield criterion: f ¼ p2 pp0 þ
3 fðsij paij Þðsij paij Þ þ ðp0 pÞpaij aij g ¼ 0 2M 2
ð16Þ
where, p is the mean principal stress, p0 is the initial mean principal stress, M is the slope of the critical state line (CSL) in p vs. q space, sij is the deviatoric stress, aij is the rotation of the yield surface with respect to the p-axis (dimensionless back stress factor). When one incorporates the gradient to the isotropic hardening factor p0 and the anisotropic hardening factor aij the following expressions are obtained: ^ 2_ p^0 p_ 0 ¼ p_ 0 a1 r2 p_ 0 b1 r
ð17aÞ
^ 2_^ a~_ ij ¼ a_ ij a2 r2 a_ ij b2 r aij_
ð17bÞ
where, p_ 0 , p_ 0 are global and local mean principal stresses for the isotropic hardening factor and a~_ ij , a_ ij are global and local dimensionless back stresses for the kinematic hardening factor, respectively. In Eqs. (17a) and (17b), the parameters a1 , b1 , a2 , b2 , also take into account the dimensional discrepancies between a quantity and the corresponding second order gradient of that quantity. One obtains the evolution equation for a_ ij from Eq. (15b) (note that a_ ij ¼ a_ ij =p, p ¼ mean principle stress). One also obtains the evolution equation for r2 a^_ ij from Eq. (15c). In this study, the evolution equation of a^_ ij was used to obtain the evolution equation of r2 a^_ ij . The partial derivatives of the yield function with respect to aij and r2 a^_ ij are as follows: of 3 ^ 2a ^ij sij ¼ 2 p p0 aij a2 r2 aij b2 r ð18aÞ oaij M
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of 3 2 2 2a 2p ^ ^ ^ ^ ¼ p s p a a r a b a r p b r r b p 2 ij ij 2 ij 2 ij 0 1 0 1 0 pb2 aij ^ij fM 2 fr^2 a 2 2 ^ ^ a2 r aij b2 r aij
ð18bÞ
where, f accounts for the dimensional discrepancy problem. b2 in Eq. (17b) accounts for both the dimensional discrepancy problem and the material characteristics. One also obtains similarly the evolution equation for p_ 0 from Eq. (15a) as follows: p_ 0 ¼
1 þ e0 p0 ðp0 þ DpÞ p e_v ¼ K e_pv Dp kj
In the same way as Eq. (18b), the following equation is obtained. of 1 3 2 2 ^2 a^ij ^2 a^ij pb1 þ ¼ pb a r a b a r a b r r a a 1 ij 2 ij 2 ij 2 ij 2 2M 2 nr^2 p^0 n
ð19Þ
ð20Þ
where, n is a similar parameter as to that of f in Eq. (18b). 3. Visco-plasticity Rate dependency of the material is due to the viscous property of the material. For localized large strain problems, such as shear band problems, the strain rate at the shear band is much larger than that outside the shear band. For example, the strain rate at the shear band may be 100 times that outside the shear band. When the overall strain level of the material is not large, the shear bands typically do not develop, therefore, severe strain rate differences in the material do not develop. However, in situations of localized large strain problems such as the cone penetration problem, slope stability problem etc., the strains are large, and therefore, the incorporation of the rate dependency in the constitutive relations is required. Typically a material shows higher strength at higher strain rates. This results in the mobilization of relatively higher shear strength of the material at the shear band than outside the shear band. Therefore the incorporation of rate dependency reduces the strain level at the shear band. This strain reduction is called regularization or homogenization. When one incorporates the gradients in dense material, the gradients usually reduce the stress and strain level at the shear band, since the stresses and strains are transferred to the neighboring particles. Both the rate dependency of the material and the stress transfer by gradients act as homogenization mechanisms. When one has only one homogenization mechanism in the model, one may obtain reasonably good results. However, the model calibration may not be correct for this case, and the extension capability of the model shall be very limited. The homogenization effects by rate dependency is obtained by applying Perzyna’s (1966) overstress approach. In this work, the creep is not addressed. The additive decomposition of total strain rate into elastic and visco-plastic parts gives the following relation. e_ij ¼ e_eij þ e_vp ij
ð21Þ
where, e_ij is the total strain rate, e_eij is the elastic strain rate, and e_vp ij is the visco-plastic strain rate. Making use of the AMCCM, and Perzyna’s (1966) elasto-visco-plastic theory, one obtains the following expression for the visco-plastic strain. e_vp ij ¼ chUðF Þi
ofd orij
ð22Þ
G.Z. Voyiadjis, C.R. Song / Mechanics Research Communications 29 (2002) 121–129
127
where, c is the viscosity, h i is the McCauly brackets, fd is a dynamic yield function incorporated with rate dependency, and UðF Þ is a visco-plastic multiplier, that is a function of F which is defined as follows: F ¼
fd js js
ð23Þ
where, js is a hardening factor in static loading. Eq. (23) is rewritten as follows: fd ¼ ð1 þ F Þjs
ð24Þ
When one uses an AMCCM (Dafalias, 1987; Voyiadjis and Song, 2000), the yield criterion for rate dependent condition appears as follows assuming isotropic rate dependency: fs ¼ p2 p~ p0 þ
3 fðsij paij Þðsij paij Þ þ ð~ p0 pÞpaij aij g ¼ 0 2M 2
ð25Þ
where, p is the mean principal stress, p~0 is the initial mean principal stress for rate dependent loading, M is the slope of the CSL, in p vs. q space, sij is the deviatoric stress, and aij is the rotation of the yield surface with respect to the p-axis (dimensionless back stress factor). The functional form of U is assumed similar to Oka (1981) as follows: p~ UðF Þ ¼ C exp m 0 ð26Þ p0 where, C and m are the material constants, and p0 is the initial principal stress for static condition. In Eqs. (25) and (26), the gradients are included in aij , p0 and p~0 . Therefore Eqs. (21)–(26) are for the rate dependent gradient theory. Through this approach, the incorporation of both homogenization mechanisms is achieved. 4. Rate dependency due to flow in porous media The rate dependency of porous media depends not only on the viscous material properties but also on the pore fluid flow characteristics of the material during deformation. For time dependent loading for saturated soils, the pore water in the soil is pressurized and tends to escape from the pressurized zone to the unpressurized zone. This is a natural procedure of entropy. Due to the limited hydraulic conductivity of the soil, this pore water escape procedure (so called ‘‘dissipation’’) takes time. Therefore another time dependent behavior needs to be considered here. The coupled theory of mixtures deals with this kind of flow problems in the porous media. Here, Prevost’s (1980) coupling formulation for saturated granular material is adopted as follows: div½ðnw =qw ÞK ws ðgrad Pw qw b þ qw aw Þ þ div vs ¼ 0
ð27Þ
where, qs is the mass density of the soil, qw is the mass density of the water, aw is the acceleration of water, K ws is the permeability tensor, qw is the density of water, vs is the solid velocity, Pw is the pore water pressure, and b is the body force vector. In the case when the acceleration is negligible, Eq. (27) reduces to div½ðnw =qw ÞK ws ðgrad Pw qw bÞ þ div vs ¼ 0
ð28Þ
For an updated Lagrangian reference frame, Eq. (28) can be expressed as follows (Voyiadjis and AbuFarsakh, 1997): o nw ws s oPw s J s Cijs e_ij JCijs1 Cijs1 XD;a KAB Xa;A q w Bb ¼0 ð29Þ oXp qw oXB s s s s where, Cijs ¼ XK;I XK;I , e_ij is the strain rate tensor, Xa;A ¼ onþ1 Xa =on XA , Bb ¼ bb =Xb;B and J is the Jacobian.
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5. Separation of gradient effects from inelastic micro-structural rotation Gradient terms consider the micro-structural interaction. In this gradient theory, another micro-structural activity––the inelastic rotation of the micro-structure is not separated. Therefore, the effect of inelastic micro-structural rotations is taken into account here through the implementation of the plastic spin. The plastic spin is a tensor quantity, and is related to the micro-structural behavior. It is related to the back stress tensor. The relation between the plastic spin and the backstress tensor is given by Lee et al. (1983), and Voyiadjis and Kattan (1989) as shown below: 00
00
00
W s ¼ nð ad s d s aÞ n is a function of the plastic strain expressed as follows (Paulun and Pecherski, 1985): rffiffiffi 3 3e2eq e_eq n¼ 2 1 þ 3e2eq
ð30Þ
ð31Þ
where, eeq and e_eq are the equivalent plastic strain and its rate, respectively. However, in this study n is assumed 0.1. Through mathematical manipulations, Eq. (30) is expressed as follows: 00
s W s ¼ nNabcd dcd
ð32Þ
0s where, Nabcd is defined as ð aam Mmbcd Mancd anb Þ, Mijmn is defined as ðcEklmn nkl nij =H Þ, and c is defined as 0s c ¼ H =ðH þ Eabcd nab ncd Þ.
6. Conclusions A thermodynamically consistent multi-scale, rate dependent, non-local approach is developed in this work for geomaterials in conjunction with the AMCCM. The gradient for the micro-structure is incorporated through the micro-level gradient of the back-stress and volumetric strain while the gradient for macro-structure is incorporated through the macro-level gradient of back-stress and volumetric plastic strain. Gradient results in the regularization of the local behavior. Visco-plasticity is also incorporated for an additional regularization of the local behavior. Therefore, the effects of two separate regularizations are naturally separated. The plastic spin is incorporated to separate the effect of micro-structural rotation from the gradient effect. The flow characteristics of the soil is also incorporated in order to separate the viscosity effect from the flow effect. Through this multi-scale non-local approach, a more realistic simulation of large strain problems such as shear band formation can be achieved. The formulation for rate dependent gradient theory is presented here. Therefore two homogenization mechanisms are rationally coupled. For the application to saturated geomaterials, the coupled theory of mixtures is also incorporated to another homogenization mechanism called the pore pressure dissipation. Throughout the coupling of all the three homogenization mechanisms, a complete formulation for the homogenization mechanism is obtained. To investigate the performance of the proposed formulations, numerical simulations of the triaxial test will be carried out in forthcoming work and compared with the test results. Therefore a more rational simulation for large strain behaviors such as the shear band analysis can be achieved.
References Dafalias, Y.F., 1987. An anisotropic critical state clay plasticity model. In: Desai, C.S. et al. (Eds.), Constitutive Laws for Engineering Materials: Theory and Applications, pp. 513–521.
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Lee, E.H., Mallett, R.L., Wertheimer, T.B., 1983. Stress analysis for anisotropic hardening in finite-deformation plasticity. Transactions of the ASME, Journal of Applied Mechanics 50, 554–560. Oka, F., 1981. Prediction of time-dependent behavior of clay, 10th Proceedings of International Conference on Soil Mechanics and Foundation Engineering. pp. 215–218. Paulun, J.E., Pecherski, R.B., 1985. Study of corotational rates for kinematic hardening in finite deformation plasticity. Archives of Mechanics 37 (6), 661–677. Perzyna, P., 1966. Fundamental problems in viscoplasticity. Advances in Applied Mechanics 9, 243–377. Prevost, J.H., 1980. Mechanics of continuous porous media. International Journal of Engineering Science 18, 787–800. Voyiadjis, G.Z., Abu-Farsakh, Y.M., 1997. Coupled theory of mixtures for clayey soils. Computer and Geotechnics 20 (3–4), 195–222. Voyiadjis, G.Z., Deliktas, B., 2000. Multi-scale analysis of multiple damage mechanisms coupled with inelastic behavior of composite materials. Mechanics Research Communications Journal 27 (3), 295–300. Voyiadjis, G.Z., Kattan, P.I., 1989. Eulerian constitutive model for finite deformation plasticity with anisotropic hardening. Mechanics of Materials 7 (4), 279–293. Voyiadjis, G.Z., Song, C.R., 2000. Anisotropic modified cam clay model with plastic spin for finite strains. Journal of EM Division, ASCE 126 (10), 1012–1019. Zbib, H.M., Aifantis, E.C., 1988. On the concept of relative and plastic spins and its implications to large deformation theories. Part II. Anisotropic hardening plasticity. Acta Mechanica 75, 35–66.
Mechanics Research Communications 29 (2002) 121–129 www.elsevier.com/locate/mechrescom
Multi-scale non-local approach for geomaterials George Z. Voyiadjis a
a,*
, Chung R. Song
b
Department of Civil and Environmental Engineering, Louisiana State University, 3502 CEBA Bldg., Baton Rouge, LA 70808, USA b Geotechnical Division, Sambo Engineering Co. Ltd., Seoul 138-724, South Korea Received 10 October 2001; accepted 9 February 2002
Abstract A thermodynamically consistent multi-scale, rate dependent, non local approach is developed in this work for geo-materials in conjunction with the anisotropic modified Cam Clay model. The gradient for the micro-structure is incorporated through the micro level gradient of the back-stress and volumetric strain while the gradient for macrostructure is incorporated through the macro level gradient of back-stress and volumetric plastic strain. Gradient results in the regularization of the local behavior. Visco-plasticity is also incorporated for an additional regularization of the local behavior. Therefore, the effects of two separate regularizations are naturally separated. The plastic spin is incorporated to separate the effect of micro-structural rotation from the gradient effect. The flow characteristics of the soil is also incorporated in order to separate the viscosity effect from the flow effect. Through this multi scale non local approach, a more realistic simulation of large strain problems such as shear band formation can be achieved. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The behavior of soils is governed by many factors such as stiffness, strain rate and stress history, etc. The effects due to micro-structural changes are especially important for soils that are subjected to large strains and rotations. This study incorporates the effects of micro-structural changes by utilizing the plastic spin and the gradient theory. The plastic spin is used to incorporate the non-coaxiality of micro-structural rotation in reference to the macro-structural rotation. The gradient theory is used here in order to capture the micro-structural interaction between the micro-scale material particles. The plastic spin is known to consider the non-coaxiality between the substructure rotation and the macro-rotation (Zbib and Aifantis, 1988). For small strain conditions, this non-coaxiality is minor. However, it cannot be ignored at high strain conditions. To obtain the tensor quantity of the plastic spin, the rotation of the micro-structure needs to be addressed. Since the quantification of the micro-structure rotation is not possible in the continuum
*
Corresponding author. Tel.: +1-225-388-8668; fax: +1-225-578-9176. E-mail addresses: [email protected] (G.Z. Voyiadjis), [email protected] (C.R. Song).
0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 2 3 3 - 1
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mechanics platform, a certain relationship between the plastic spin and the embedded stress (back-stress) is used. In geomaterials, substantial portion of the external load is carried by inter-granular frictional resistance. This frictional resistance is very irregular. Some grains are subjected to high friction while some other grains are subject to low friction. The grains that are subjected to high friction will undergo smaller grain rotation for the same applied energy than the grains that are subjected to low friction. This phenomenon will result in a different plastic spin even for the same back stress. The expression of the plastic spin with respect to the back stress should include effects such as the inter-granular frictional force. The gradient theory allows one to incorporate the inter-granular stress into the continuum mechanics. When materials with a simple cubic array are subjected to deformation as shown in Fig. 1(a), grains will fall into the void space and will reach the compacted condition as shown in Fig. 1(b). This process will reduce the volume and nullify or reduce the contact stress for the adjacent layer of grains on the top. When materials with a cubic tetrahedral array are subjected to deformation as shown in Fig. 1(b), grains will climb over the adjacent grains and reach the loose packing condition as shown in Fig. 1(a). This process increases the volume. If the volume expansion is restricted, this process increases the contact stresses for the adjacent layer of grains at the top. This behavior indicates that whenever one has uneven stress (strain) distribution, it develops stress (strain) transfer to the neighboring grains. This kind of stress (strain) transfer is important when one has a locally concentrated very high strain region, such as a shear band. Along the shear band, micro-structural stress redistribution is needed for modeling the accurate behavior of the material. This is the fundamental concept of the gradient theory. In this study, the gradient theory is coupled with the plastic spin for separation of the micro-mechanical particle rotation and the inter-particle interaction. The gradient theory is also formulated in two length scales: the meso-scale and the macro-scale. The meso-scale gradient is formulated in terms of micro-level back stress gradient and volumetric strain gradient, while the macro-scale gradient is formulated in terms of the macro-level back stress gradient and the (corresponding) volumetric strain gradient. For this study, an anisotropic modified Cam Clay model (AMCCM) is used. The triaxial test and the cone penetration test maybe used to check the validity of the proposed model. Numerically obtained results for the proposed model using the plastic spin and the gradient maybe used to validate the model by comparing them with those obtained experimentally. The effects of the micro-structural changes (plastic spin) need to be incorporated in the analysis when the strains are high. The effects of the micro-structural interaction (gradient) are also significant in the case of large strains. The effects of the gradient are especially important in localization problems, since the gradient results in well-posedness of the numerical solution and smoothening of the strain distribution in the shear band.
Fig. 1. Grain arrangement during shear: (a) simple cubic array, (b) cubic tetrahedral array.
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2. Formulation of the thermodynamics Voyiadjis and Deliktas (2000) derived a multi-scale gradient theory based on a consistent thermodynamic formulation for metals and metal matrix composites. In this paper, similar approach is used for the gradient theory in geomaterials using an AMCCM. The Helmholtz free energy is expressed in terms of the internal state variables such that: ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ W ¼ W ee ; T ; AðkÞ ; rAðkÞ ; r2 AðkÞ ; r^ A^ðkÞ ; r^2 A^ðkÞ ð1Þ ðpÞ
where, ee is the elastic strain, T is the temperature, superscript ‘p’ represents plasticity, AðkÞ is a macro-scale internal state variable such as for the isotropic hardening or the kinematic hardening in plasticity. r ðpÞ represents the first order gradient, r2 represents the second order gradient, and A^ðkÞ represents the microscale internal state variable in a representative volume element (RVE). When one assumes symmetrical distribution of the gradients, the odd order gradient terms vanish, and Eq. (1) becomes: ðpÞ e 2 ðpÞ ^2 ^ðpÞ W ¼ W e ; T ; AðkÞ ; r AðkÞ ; r AðkÞ ð2Þ The link between macro-scale internal variables and micro-scale internal variables is obtained by averaging the micro-scale internal variables in the RVE. Z 1 ðpÞ ðpÞ AðkÞ ¼ ð3Þ A^ dVRVE VRVE VRVE ðkÞ where, VRVE is the volume of the RVE. The first and second order gradients of the macro- and micro-internal variables are defined as follows: 2 o Aij 2 ðpÞ r AðkÞ ¼ ð4Þ oXk oXk ðpÞ r^2 A^ðkÞ ¼
1 VRVE
Z VRVE
ðpÞ o2 A^ðkÞ
oXk oXk
ð5Þ
dVRVE
The integration maybe performed over a sub-volume of the RVE. One can express the time derivative of Eq. (2) in terms of its higher order state variables. By substituting into the Clausius–Duhem inequality one obtains: oW oW oW oW ðpÞ ðpÞ ^ 2 A^ðpÞ P 0 r q e : e_e þ r : e_p q ðpÞ : A_ ðkÞ q : r2 A_ ðkÞ q :r ð6Þ ðkÞ ðpÞ ðpÞ oe 2 ^ ^ oAðkÞ or2 AðkÞ or AðkÞ ðpÞ
ðpÞ
where, the meso-scale gradient terms r2 A^ðkÞ are dependent on the macro-scale variables, AðkÞ . However, the random periodic boundary condition ensures that there is no net flux of meso-scale gradients across the RVE boundary. Such a constraint would effectively prevent coupling between macro-scale and meso-scale gradient terms. The first term in Eq. (6) gives the relation between the stress, r and its associated variable, elastic strain, e_e . The remaining terms represent the total dissipation process due to plasticity. This can be expressed as the sum of the plastic dissipation as follows: ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ _ ðpÞ P ¼ r : e_p VðkÞ : A_ ðkÞ WðkÞ : r2 A_ ðkÞ XðkÞ : r2 A^ðkÞ
ð7Þ
124
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ðrÞ
ðrÞ
where, VðkÞ , WðkÞ , and XðkÞ are the thermodynamic force conjugates and are expressed as follows: oW
ðpÞ
VðkÞ ¼ q
oW
ðpÞ
WðkÞ ¼ q
ð8bÞ
ðpÞ
or2 AðkÞ oW
ðpÞ
XðkÞ ¼ q
ð8aÞ
ðpÞ
oAðkÞ
ð8cÞ
ðpÞ or^2 A^ðkÞ
One can now expresses the analytical form of the Helmholtz free energy as the quadratic form of its internal state variables 1 1 ðpÞ ðpÞ ðpÞ 1 ðpÞ 1 ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ qW ¼ ee : Eð/Þ : ee þ aðkÞ AðkÞ : AðkÞ þ bðkÞ r2 AðkÞ : r2 AðkÞ þ cðkÞ r^2 A^ðkÞ : r^2 A^ðkÞ 2 2 2 2
ð9Þ
Using Eqs. (8a)–(9) the following definitions can be obtained for the thermodynamic forces ðpÞ
ðpÞ
ðpÞ
VðkÞ ¼ aðkÞ AðkÞ ðpÞ
ð10aÞ
ðpÞ
ðpÞ
WðkÞ ¼ bðkÞ r2 AðkÞ
ð10bÞ
ðpÞ ðpÞ ðpÞ ZðkÞ ¼ cðkÞ r2 A^ðkÞ
ð10cÞ
The value of the thermodynamic forces can be obtained through the evolution relations of the internal state variables. However, it should be pointed out that there are two classes of evolution equations that need to be developed, normally one at the macro-scale and the other at the meso-scale level. The former can be obtained by assuming the physical existence of the dissipation potential at the macro-scale. The later one can be obtained by a micro-mechanical or phenomenological approach. ðpÞ One may also consider that the evolution equations of the internal state variables AðkÞ , can be obtained ðpÞ by integrating the evolution equations of the local internal state variables at the meso-scale, that is A^ðkÞ over ðpÞ ðpÞ the domain of the RVE. However, integration of the A^ðkÞ is a cumbersome task since at the meso-scale, A^ðkÞ is a function of many different aspects of the material inhomogeneities such as interaction of defects, size of defects, spacing between them, and distribution of defects within the sub RVE. Therefore, in this work the evolution equations of the meso-scale internal state variables are obtained through the use of the generalized normality rule of thermodynamics. In this regard the macro-scale dissipation potential is defined only in terms of the macro-scale flux variables as follows ðpÞ ðpÞ _ ðpÞ H ¼ H e_p ; A_ ðkÞ ; r2 A_ ðkÞ ; r2 A^ðkÞ ð11Þ By using the Legendre–Fenchel transformation of the dissipation potential ðHÞ, one can obtain complementary laws in the form of the evolution laws of flux variables as function of the dual variables ðpÞ ðpÞ ðpÞ H ¼ H r_ ; B ; r2 B ; r2 B^ ð12Þ ðkÞ
ðkÞ
ðkÞ
Eq. (12) is also expressed as follows: ! N N X X _ H ¼ F r_ ; p_ 0;i ; ai i¼1
i¼1
ð13Þ
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where (13) p0;i and ai are the isotropic and kinematic hardening forces respectively. The evolution equation of p0;i and ai are obtained from the normality rule as follows: oW ðpÞ ðpÞ B_ ðkÞ ¼ C A_ ðkÞ ¼ C ðpÞ oBðkÞ
ð14Þ
where, C is a constant. Using Eq. (14) and Perzyna’s elastic/viscoplasticity (1966) and applying the normality rule, one obtains the following evolution equations: oF ¼ ch/i tr of ð15aÞ e_pv ¼ ch/i tr orij orij a_ ij ¼ C1 ch/i
oF of ¼ C1 ch/i o aij o a ij
ð15bÞ
oF of ^ 2_a ^ij ¼ C2 ch/i ¼ C2 ch/i r _ ^ 2a ^ 2_a ^ij ^ij or r
ð15cÞ
oF of ^ 2_ ^ p0 ¼ C3 ch/i ¼ C3 ch/i r _ ^ 2 ^ 2_^ ^p0 p0 or or
ð15dÞ
where, F is the plastic potential, aij represent the back stress, ch/i represents the viscoplastic multiplier. h i is the MaCauley bracket, c is the viscosity. When one uses the AMCCM expressed by Voyiadjis and Song (2000), the following expression is obtained for the yield criterion: f ¼ p2 pp0 þ
3 fðsij paij Þðsij paij Þ þ ðp0 pÞpaij aij g ¼ 0 2M 2
ð16Þ
where, p is the mean principal stress, p0 is the initial mean principal stress, M is the slope of the critical state line (CSL) in p vs. q space, sij is the deviatoric stress, aij is the rotation of the yield surface with respect to the p-axis (dimensionless back stress factor). When one incorporates the gradient to the isotropic hardening factor p0 and the anisotropic hardening factor aij the following expressions are obtained: ^ 2_ p^0 p_ 0 ¼ p_ 0 a1 r2 p_ 0 b1 r
ð17aÞ
^ 2_^ a~_ ij ¼ a_ ij a2 r2 a_ ij b2 r aij_
ð17bÞ
where, p_ 0 , p_ 0 are global and local mean principal stresses for the isotropic hardening factor and a~_ ij , a_ ij are global and local dimensionless back stresses for the kinematic hardening factor, respectively. In Eqs. (17a) and (17b), the parameters a1 , b1 , a2 , b2 , also take into account the dimensional discrepancies between a quantity and the corresponding second order gradient of that quantity. One obtains the evolution equation for a_ ij from Eq. (15b) (note that a_ ij ¼ a_ ij =p, p ¼ mean principle stress). One also obtains the evolution equation for r2 a^_ ij from Eq. (15c). In this study, the evolution equation of a^_ ij was used to obtain the evolution equation of r2 a^_ ij . The partial derivatives of the yield function with respect to aij and r2 a^_ ij are as follows: of 3 ^ 2a ^ij sij ¼ 2 p p0 aij a2 r2 aij b2 r ð18aÞ oaij M
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of 3 2 2 2a 2p ^ ^ ^ ^ ¼ p s p a a r a b a r p b r r b p 2 ij ij 2 ij 2 ij 0 1 0 1 0 pb2 aij ^ij fM 2 fr^2 a 2 2 ^ ^ a2 r aij b2 r aij
ð18bÞ
where, f accounts for the dimensional discrepancy problem. b2 in Eq. (17b) accounts for both the dimensional discrepancy problem and the material characteristics. One also obtains similarly the evolution equation for p_ 0 from Eq. (15a) as follows: p_ 0 ¼
1 þ e0 p0 ðp0 þ DpÞ p e_v ¼ K e_pv Dp kj
In the same way as Eq. (18b), the following equation is obtained. of 1 3 2 2 ^2 a^ij ^2 a^ij pb1 þ ¼ pb a r a b a r a b r r a a 1 ij 2 ij 2 ij 2 ij 2 2M 2 nr^2 p^0 n
ð19Þ
ð20Þ
where, n is a similar parameter as to that of f in Eq. (18b). 3. Visco-plasticity Rate dependency of the material is due to the viscous property of the material. For localized large strain problems, such as shear band problems, the strain rate at the shear band is much larger than that outside the shear band. For example, the strain rate at the shear band may be 100 times that outside the shear band. When the overall strain level of the material is not large, the shear bands typically do not develop, therefore, severe strain rate differences in the material do not develop. However, in situations of localized large strain problems such as the cone penetration problem, slope stability problem etc., the strains are large, and therefore, the incorporation of the rate dependency in the constitutive relations is required. Typically a material shows higher strength at higher strain rates. This results in the mobilization of relatively higher shear strength of the material at the shear band than outside the shear band. Therefore the incorporation of rate dependency reduces the strain level at the shear band. This strain reduction is called regularization or homogenization. When one incorporates the gradients in dense material, the gradients usually reduce the stress and strain level at the shear band, since the stresses and strains are transferred to the neighboring particles. Both the rate dependency of the material and the stress transfer by gradients act as homogenization mechanisms. When one has only one homogenization mechanism in the model, one may obtain reasonably good results. However, the model calibration may not be correct for this case, and the extension capability of the model shall be very limited. The homogenization effects by rate dependency is obtained by applying Perzyna’s (1966) overstress approach. In this work, the creep is not addressed. The additive decomposition of total strain rate into elastic and visco-plastic parts gives the following relation. e_ij ¼ e_eij þ e_vp ij
ð21Þ
where, e_ij is the total strain rate, e_eij is the elastic strain rate, and e_vp ij is the visco-plastic strain rate. Making use of the AMCCM, and Perzyna’s (1966) elasto-visco-plastic theory, one obtains the following expression for the visco-plastic strain. e_vp ij ¼ chUðF Þi
ofd orij
ð22Þ
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where, c is the viscosity, h i is the McCauly brackets, fd is a dynamic yield function incorporated with rate dependency, and UðF Þ is a visco-plastic multiplier, that is a function of F which is defined as follows: F ¼
fd js js
ð23Þ
where, js is a hardening factor in static loading. Eq. (23) is rewritten as follows: fd ¼ ð1 þ F Þjs
ð24Þ
When one uses an AMCCM (Dafalias, 1987; Voyiadjis and Song, 2000), the yield criterion for rate dependent condition appears as follows assuming isotropic rate dependency: fs ¼ p2 p~ p0 þ
3 fðsij paij Þðsij paij Þ þ ð~ p0 pÞpaij aij g ¼ 0 2M 2
ð25Þ
where, p is the mean principal stress, p~0 is the initial mean principal stress for rate dependent loading, M is the slope of the CSL, in p vs. q space, sij is the deviatoric stress, and aij is the rotation of the yield surface with respect to the p-axis (dimensionless back stress factor). The functional form of U is assumed similar to Oka (1981) as follows: p~ UðF Þ ¼ C exp m 0 ð26Þ p0 where, C and m are the material constants, and p0 is the initial principal stress for static condition. In Eqs. (25) and (26), the gradients are included in aij , p0 and p~0 . Therefore Eqs. (21)–(26) are for the rate dependent gradient theory. Through this approach, the incorporation of both homogenization mechanisms is achieved. 4. Rate dependency due to flow in porous media The rate dependency of porous media depends not only on the viscous material properties but also on the pore fluid flow characteristics of the material during deformation. For time dependent loading for saturated soils, the pore water in the soil is pressurized and tends to escape from the pressurized zone to the unpressurized zone. This is a natural procedure of entropy. Due to the limited hydraulic conductivity of the soil, this pore water escape procedure (so called ‘‘dissipation’’) takes time. Therefore another time dependent behavior needs to be considered here. The coupled theory of mixtures deals with this kind of flow problems in the porous media. Here, Prevost’s (1980) coupling formulation for saturated granular material is adopted as follows: div½ðnw =qw ÞK ws ðgrad Pw qw b þ qw aw Þ þ div vs ¼ 0
ð27Þ
where, qs is the mass density of the soil, qw is the mass density of the water, aw is the acceleration of water, K ws is the permeability tensor, qw is the density of water, vs is the solid velocity, Pw is the pore water pressure, and b is the body force vector. In the case when the acceleration is negligible, Eq. (27) reduces to div½ðnw =qw ÞK ws ðgrad Pw qw bÞ þ div vs ¼ 0
ð28Þ
For an updated Lagrangian reference frame, Eq. (28) can be expressed as follows (Voyiadjis and AbuFarsakh, 1997): o nw ws s oPw s J s Cijs e_ij JCijs1 Cijs1 XD;a KAB Xa;A q w Bb ¼0 ð29Þ oXp qw oXB s s s s where, Cijs ¼ XK;I XK;I , e_ij is the strain rate tensor, Xa;A ¼ onþ1 Xa =on XA , Bb ¼ bb =Xb;B and J is the Jacobian.
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5. Separation of gradient effects from inelastic micro-structural rotation Gradient terms consider the micro-structural interaction. In this gradient theory, another micro-structural activity––the inelastic rotation of the micro-structure is not separated. Therefore, the effect of inelastic micro-structural rotations is taken into account here through the implementation of the plastic spin. The plastic spin is a tensor quantity, and is related to the micro-structural behavior. It is related to the back stress tensor. The relation between the plastic spin and the backstress tensor is given by Lee et al. (1983), and Voyiadjis and Kattan (1989) as shown below: 00
00
00
W s ¼ nð ad s d s aÞ n is a function of the plastic strain expressed as follows (Paulun and Pecherski, 1985): rffiffiffi 3 3e2eq e_eq n¼ 2 1 þ 3e2eq
ð30Þ
ð31Þ
where, eeq and e_eq are the equivalent plastic strain and its rate, respectively. However, in this study n is assumed 0.1. Through mathematical manipulations, Eq. (30) is expressed as follows: 00
s W s ¼ nNabcd dcd
ð32Þ
0s where, Nabcd is defined as ð aam Mmbcd Mancd anb Þ, Mijmn is defined as ðcEklmn nkl nij =H Þ, and c is defined as 0s c ¼ H =ðH þ Eabcd nab ncd Þ.
6. Conclusions A thermodynamically consistent multi-scale, rate dependent, non-local approach is developed in this work for geomaterials in conjunction with the AMCCM. The gradient for the micro-structure is incorporated through the micro-level gradient of the back-stress and volumetric strain while the gradient for macro-structure is incorporated through the macro-level gradient of back-stress and volumetric plastic strain. Gradient results in the regularization of the local behavior. Visco-plasticity is also incorporated for an additional regularization of the local behavior. Therefore, the effects of two separate regularizations are naturally separated. The plastic spin is incorporated to separate the effect of micro-structural rotation from the gradient effect. The flow characteristics of the soil is also incorporated in order to separate the viscosity effect from the flow effect. Through this multi-scale non-local approach, a more realistic simulation of large strain problems such as shear band formation can be achieved. The formulation for rate dependent gradient theory is presented here. Therefore two homogenization mechanisms are rationally coupled. For the application to saturated geomaterials, the coupled theory of mixtures is also incorporated to another homogenization mechanism called the pore pressure dissipation. Throughout the coupling of all the three homogenization mechanisms, a complete formulation for the homogenization mechanism is obtained. To investigate the performance of the proposed formulations, numerical simulations of the triaxial test will be carried out in forthcoming work and compared with the test results. Therefore a more rational simulation for large strain behaviors such as the shear band analysis can be achieved.
References Dafalias, Y.F., 1987. An anisotropic critical state clay plasticity model. In: Desai, C.S. et al. (Eds.), Constitutive Laws for Engineering Materials: Theory and Applications, pp. 513–521.
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Lee, E.H., Mallett, R.L., Wertheimer, T.B., 1983. Stress analysis for anisotropic hardening in finite-deformation plasticity. Transactions of the ASME, Journal of Applied Mechanics 50, 554–560. Oka, F., 1981. Prediction of time-dependent behavior of clay, 10th Proceedings of International Conference on Soil Mechanics and Foundation Engineering. pp. 215–218. Paulun, J.E., Pecherski, R.B., 1985. Study of corotational rates for kinematic hardening in finite deformation plasticity. Archives of Mechanics 37 (6), 661–677. Perzyna, P., 1966. Fundamental problems in viscoplasticity. Advances in Applied Mechanics 9, 243–377. Prevost, J.H., 1980. Mechanics of continuous porous media. International Journal of Engineering Science 18, 787–800. Voyiadjis, G.Z., Abu-Farsakh, Y.M., 1997. Coupled theory of mixtures for clayey soils. Computer and Geotechnics 20 (3–4), 195–222. Voyiadjis, G.Z., Deliktas, B., 2000. Multi-scale analysis of multiple damage mechanisms coupled with inelastic behavior of composite materials. Mechanics Research Communications Journal 27 (3), 295–300. Voyiadjis, G.Z., Kattan, P.I., 1989. Eulerian constitutive model for finite deformation plasticity with anisotropic hardening. Mechanics of Materials 7 (4), 279–293. Voyiadjis, G.Z., Song, C.R., 2000. Anisotropic modified cam clay model with plastic spin for finite strains. Journal of EM Division, ASCE 126 (10), 1012–1019. Zbib, H.M., Aifantis, E.C., 1988. On the concept of relative and plastic spins and its implications to large deformation theories. Part II. Anisotropic hardening plasticity. Acta Mechanica 75, 35–66.