General method for simulating laboratory tests with constitutive models for geomechanics

Transcription

1 General method for simulating laboratory tests with constitutive models for geomechanics Tomáš Janda 1 and David Mašín 2 1 Czech Technical University in Prague, Faculty of Civil Engineering, Czech Republic 2 Charles University in Prague, Faculty of Science, Czech Republic August 18, Abstract The paper describes an approach to the simulation of arbitrary laboratory tests with homogeneous stress and strain fields applicable to most constitutive models common in geomechanics. The method by Bardet and Choucair [Int. J. Numer. Anal. Meth. Geomech. 15(1):119, 1991] is generalized for an arbitrary number of controlling and controlled variables. The approach is illustrated for time-dependent thermo-hydro-mechanical constitutive models. The purpose of the method is to define an endless variety of different laboratory tests declaratively by means of two matrices E and S, which define the constraints on the controlled quantities such as stress, strains, suction or temperature Introduction Most of the constitutive models for mechanical, thermal or coupled analysis can be written in the general rate form σ g = g σ ( ε g, σ g, q) (1) q = g q ( ε g, σ g, q) (2) where ε g denotes the controlling variables vector containing all independent (prescribed) quantities and σ g is the controlled variables vector containing all dependent (unknown) quantities. Vector q represents state variables and ( ) = 1

2 d() dt denotes the time derivative. For time independent models, t is the pseudo- time (integration variable). For both time-dependent or time-independent thermo-hydro-mechanical con- stitutive models the above quantities can be written as σ g = {σ 11, σ 22, σ 33, σ 12, σ 13, σ 23, S r } T (3) ε g = {ε 11, ε 22, ε 33, ε 12, ε 13, ε 23, s, T } T (4) where σ ij are components of the stress tensor (in the present hydro-mechanical case σ ij represents a net stress, calculated as the total stress minus the pore air pressure), S r is the degree of saturation, ε ij are components of the strain tensor, s is suction and T is temperature. In general, the controlling and controlled variables vectors can have any number of components. Number of components of the controlling variables tensor will be denoted as n ɛ (in Eq. (7), n ɛ = 8) and number of components of the controlled variables tensor will be denoted as n σ (in Eq. (7), n σ = 7). It is to be pointed out that the vectors ε g and σ g are not classical strain and stress vectors which would have to be work-conjugated, instead they are variables chosen purely based on convenience. As an example, let us consider partially saturated soil at constant temperature. For such a soil, Houlsby [2] derived the following equations for work-conjugated generalised stress and strain tensors: σ g = {σ 11, σ 22, σ 33, σ 12, σ 13, σ 23, ns} T (5) ε g = {ε 11, ε 22, ε 33, ε 12, ε 13, ε 23, S r } T (6) 36 where n is porosity and σ is so-called Bishop stress calculated as σ = σ tot 1 (u a S r s) (7) where σ tot is total stress, u a is pore air pressure and 1 is second-order identity tensor. Still, as discussed for example by Sheng et al. [6], in conventional finite element codes the displacements, pore pressures and temperatures are first solved at nodal points. The strains at integration points are then solved from the displacements. Then, purely for the model implementation, it is most convenient to consider strains, suction and temperature as controlling variables and stresses 2

3 and degree of saturation as controlled variables. To obtain the model s response to the prescribed increment of the controlling variables we need to integrate the governing rate equations (1) and (2). The simplest technique to achieve this is employing the forward Euler integration scheme, but more advanced methods such as the adaptive Runge-Kutta methods can also be utilized. For a finite increment of the controlling variables vactor and an initial values of the controlled variables vector and state variables we formally write the resulting increment of stress and state variables in the form σ g = g σ ( ε g, σ g, q) = q = g q ( ε g, σ g, q) = t2 t 1 g σ ( ε g (t), σ g, q)dt (8) t2 t 1 g q ( ε g (t), σ g, q)dt (9) where each of the functions g σ and g q covers the integration of both equations (1) and (2) over a finite increment of the controlling variables vector ε g, t 1 and t 2 are times at the beginning and at the end of the current step. In the case of rate-dependent models, t 1 and t 2 represent real times and g σ and g q are defined in terms of real rates, whereas in rate-independent materials time is merely an integration variable. The above formulation is sufficient for the simulation of laboratory tests in which only the components of the controlling variables vector are prescribed. When the laboratory test requires to prescribe or constrain also the components of the controlled variables vector we exploit an approach proposed Bardet and Couchair [1]. The change of the controlling variables vector, controlled variables vector and state variables due to a single loading step has to satisfy n ɛ, possibly nonlinear, equations S σ g + E ε g = y (10) where E is a matrix of n ɛ rows and columns and S is a matrix of n ɛ rows and n σ columns. The vector y has n ɛ components. It has only the first of its components non-zero, thus: y = { y, 0,..., 0} T (11) where y controls the size of the load increment. The first row of matrix S resp. E assign the load increment y to one or more of the components of 3

4 vector σ g or ε g. The rest of the rows of S and E define the constraints applied on vectors σ g or ε g during a particular test. Note that this technique requires the laboratory test having only one degree of freedom, i.e. all quantities controlled during the test are in fact expressed in terms of a single scalar quantity y. It will be shown later that most laboratory tests satisfy this criterion and, if not, the vector y can be made more general. Obviously, this technique is valid only for laboratory tests in which we can assume homogeneous stress and strain fields within the sample Nonlinear solution using Newton Raphson method The function g σ ( ε g, σ g, q) is nonlinear for most of the nontrivial constitutive models. Therefore for laboratory tests that prescribe the dependent variables σ g the whole system of equations (10) becomes nonlinear and has to be solved iteratively. A standard approach to the solution of such a system is the Newton Raphson method which searches for the root of vector-valued function that in our case becomes f( ε g ) = Sg σ ( ε g, σ g, q) + E ε g y = 0 (12) We solve the equations for a prescribed finite load increment y. From the implementation point of view, we need apart form the ability to evaluate the function f( ε g ) to follow three steps to solve the above equation by means of the Newton Raphson method: Provide a Jacobian matrix, i.e. matrix of first derivatives, of f( ε g ) J ij ( ε g ) = f i( ε g ) ε g,j, (13) where f i ( ε g ) is the i-th component of the vector-valued function f( ε g ) and ε g,j is the j-th component of its argument vector ε g. For models, where function f( ε g ) is not differentiable, such as the models based on the theory of hypoplasticity, we need to provide an appropriate approximation of Jacobian matrix. Set the initial value of the strain increment ε g. It is reasonable to chose 4

5 the initial value ε j=0 g = 0. Formulate a condition upon which the iterative algorithm stops. The stop- condition requires special attention. In principle, we can use the Euclidean norm of f and stop the iterations when it is below a certain tolerance, i.e. f f T OL. This approach is, however, not recommended here since the rows of equation (12) are the residua of potentially different quantities. For such a setup it is difficult to come up with an appropriate tolerance as it is applied jointly to stress components, strain components and other quantities in vectors ε g and σ g. We therefore prefer to formulate the stop condition solely in terms of ε g and use the form e e T OL, (14) 107 where e is a vector of the normalized residua defined as e = RJ 1 f( ε g ), (15) with R = diag(1,..., 1, 1/r s, 1/r T ) being a diagonal matrix which merely normalizes the last two non-strain components corresponding to suction and temperature by factors r s and r T, respectively. It should be noted that only the error of expression RJ 1 f is checked to be small instead of the original requirement of f to be small. This should be considered in cases where the components of J are large, such as in very stiff materials. The algorithm for the Newton Raphson method is summarized as 5

6 i = 0; ε i g = ε init g ; σ i g = σ init g ; q i = q init ; r i = 0; // Loop over loading steps while i < numloadsteps do J 1 = J 1 (σ i, q i ); f( ε g ) = f( ε g, σ i g, q i ) + r i ; j = 0; ε j g = 0; // Loop over Newton Raphson iterations while e e > T OL do = ε j g J 1 f( ε j g); ε j+1 g j = j + 1; end r i+1 σ = f( εj g) σ i+1 g = σ i g + f σ ( ε j g, σ i g, q i ); q i+1 = q i + f q ( ε j g, σ i g, q i ); ε i+1 g = ε i g + ε j g; i = i + 1 end Algorithm 1: Forward integration of the constitutive model. Index i corresponds to a loading step, index j denotes the iteration of the Newton Raphson method. Note that after the stop condition is satisfied, in the case of controlled stress components, there still remain residua r of the function f which differ from the required zero values. Accuracy of the algorithm is significantly improved (by several orders of magnitude), if these residua increments are considered as targets in the subsequent load step. How they are introduced is clear from the algorithm above. The introduction of residua into the function f( ε g ) prevent the inaccuracies of the strain increment ε g to be accumulated along multiple loading steps as illustrated in Fig. 1. Also note that the corrections using residua, as suggested and implemented in this paper, is only necessary when the Newton Raphson algorithm targets load increments without reflecting the global level of loading. Checking the globally defined quantities is more standard approach, which is usually em- 6

7 ployed within the nonlinear finite element method. This more conventional approach can also be utilized in the presented context but it would require more complicated setup for load paths in which the load controlled values are not proportional, for example the stress paths with a constant direction starting from the non-zero stress state. Figure 1: Schema of the Newton Raphson iterations in the first load step resulting in residua r 1 (left) and the iterations in i-th step with function f shifted by r i (right) Examples of S and E matrices In this section, for the sake of illustration, we define the matrices S and E for two basic laboratory tests, namely the oedometric test and drained triaxial shear test. Matrices S and E for various other laboratory tests are defined in Appendix Oedometric test For a strain controlled oedometer test with a time independent material the matrices read S = 8 7 0, (16) E = 8 8 1, (17) where 0 is a matrix with zero elements and 1 is the identity matrix. This way the load increment y corresponds to the prescribed increment of strain ε xx while the remaining components of ε g are kept zero. The system of equations does 7

8 not impose any other constrains to components of the controlled variables vector σ g. In this special type of laboratory test we control only the independent variables ε g and the dependent variables σ g are just calculated by the model via equation (8) Drained triaxial test The shearing phase of the strain controlled drained triaxial test is characterized by an increasing vertical strain while the lateral and shear components of stress are kept constant. The nonzero elements of the matrices E and S read E 11 = E 77 = E 88 = 1, (18) S 22 = S 33 = S 44 = S 55 = S 66 = 1. (19) With these matrices the first equation of the system (10) prescribes the load increment y to the first component of the controlling variables vector ε g, i.e. the normal strain component ε xx. All components of stress apart from the vertical one are prescribed as constants together with the last two components of the controlling variables vector ε g, i.e. the suction s and temperature T Demonstration of the algorithm performance The proposed approach will be demonstrated by simulating three different experiments using an advanced thermo-hydro-mechanical double structure hypoplastic model for expansive clays. The model, proposed by Mašín [5, 4], is an evolution of the hydro-mechanical double structure model proposed in Ref. [3]. This model has been selected as it is highly non-linear, and due to its complex nature only a crude estimate of the Jacobian J is available. To be more specific, in the present implementation, mechanical part of the Jacobian has been given by 3f s L, where f s is hypoplastic scalar factor and L is hypoplastic fourth-order tensor (more in [5, 4]). The multiplier 3 has been selected by a trial-and-error procedure to make sure that non-linearity induced by the non-linear components of the hypoplastic model does not imply the actual stiffness to be higher then the Jacobian. The non-mechanical components of the Jacobian have been found by direct perturbation. Figure 2 shows the results of one of the experiments simulated (heatingcooling cyclic test at constant net stress and suction), compared with experi- 8

9 80 70 exp., s=110 MPa exp., s=39 MPa exp., s=9 MPa model, s=110 MPa model, s=39 MPa model, s=9 MPa Temperature [ C] p=0.1 MPa ε v [%] Figure 2: Simulations of heating-cooling test at constant mean net stress p and suction s using thermo-hydro-mechanical double structure hypoplastic model, compared with experimental data on MX80 bentonite (figure from [5, 4], data from [7]) mental data by Tang et al. [7]. Details of the model and parameters adopted are in [5, 4]. The proposed algorithm has been implemented into an in-house element test driver, which adopts the forward Euler scheme for the model integration. The following three experiments have been simulated, which all have at least one controlled variables vector component prescribed, so Newton Raphson iterations are needed to solve Equation (12). Drained triaxial test: axisymmetric test with controlled axial strain increment. Horizontal stresses, suction and temperature are held constant. Water retention wetting test: Suction-controlled experiment with constant net stress (equal to 0) and temperature. Heating-cooling cyclic test: Temperature-controlled experiment with constant net stress and suction. The simulations have been run at sufficiently small forward Euler step size (such that the step size has minor effect on predictions). Figures 3 to 5 show the dependency of the controlled stress component (a) and number of iterations (b) on the controlling variables vector components for three values of the tolerance T OL. Clearly, even relatively high tolerance T OL=10 2 leads to minor fluctuations in the prescribed stress component, and practically perfect match 9

10 σ r [kpa] TOL=1e TOL=1e-3 TOL=1e ε a [-] Number of iterations [-] TOL=1e-4 TOL=1e-3 TOL=1e ε a [-] (a) (b) Figure 3: Simulations of drained triaxial test at constant suction and temperature, dependency on tolerance T OL. (a) evolution of horizontal stresses, (b) number of iterations. p net [kpa] TOL=1e-4 TOL=1e-3 TOL=1e suction [kpa] (a) Number of iterations [-] suction [kpa] (b) TOL=1e-4 TOL=1e-3 TOL=1e-2 Figure 4: Simulations of water retention wetting test at zero net stress and constant temperature, dependency on tolerance T OL. (a) evolution of mean net stress, (b) number of iterations is reached for lower tolerance of T OL=10 4. This good performance is tightly linked to the consideration of residual vector r in the algorithm: two to three orders of magnitude lower values of T OL would be needed to reach similar level of accuracy without consideration of r. Due to the crude estimation of the Jacobian J, a relatively high number of iterations is required to reach the specified tolerance. As expected, the number of iterations increases with a decreasing tolerance T OL. The simulations have been run with r s = and r T = 100, but it is noted that in the present case these values have no effect on the predictions. 10

11 p net [kpa] TOL=1e-4 TOL=1e-3 TOL=1e-2 Number of iterations [-] TOL=1e-4 TOL=1e-3 TOL=1e (a) T [K] (b) T [K] Figure 5: Simulations of heating-cooling test at constant net stress and suction, dependency on tolerance T OL. (a) evolution of mean net stress, (b) number of iterations Conclusions In this paper, a general algorithm was introduced for predicting various laboratory experiments using element-test driver. Using the method, it is possible to define an endless variety of different laboratory tests declaratively by the means of two matrices E and S. Numerical solution of the system of equations, adopting the Newton Raphson integration scheme and considering the effect of residual vector r, has also been introduced. The scheme yields accurate results for quite large tolerance values T OL, even for highly non-linear models adopting an inaccurate estimation of the Jacobian J Acknowledgment The authors are grateful for the financial support by the grant TACR TA The second author also acknowledges support by the grant GACR S References [1] J. P. Bardet and W. Choucair. A linearized integration technique for incremental constitutive equations. Int. J. Numer. Anal. Meth. Geomech., 15(1):1 19, [2] G. T. Houlsby. The work input to an unsaturated granular material. Géotechnique, 47(1): ,

12 [3] D. Mašín. Double structure hydromechanical coupling formalism and a model for unsaturated expansive clays. Engineering Geology, 165:73 88, [4] D. Mašín. Coupled thermo-hydro-mechanical double structure model for expansive soils. Journal of Engineering Mechanics (under review), [5] D. Mašín. Development of a coupled thermo-hydro-mechanical double structure model for expansive soils. In Delage, P. et al., editor, 3 rd European Conference on Unsaturated Soils, Paris, France (in print), [6] Sloan S. Sheng, D. and A. Gens. A constitutive model for unsaturated soils: thermomechanical and computational aspects. Computational Mechanics, 6: , [7] A.-M. Tang, Y.-J. Cui, and N. Barnel. Thermo-mechanical behaviour of a compacted swelling clay. Géotechnique, 58(1):45 54, Appendix In this Appendix matrices E and S are defined for various laboratory tests. In the following, we define non-zero components of the matrices E and S in the index notation (row number, column number, counted from 1) (all the other components are zero). Axial strain controlled undrained triaxial shear stress (constant volume). E 11 = E 31 = E 32 = E 33 = E 77 = E 88 = 1 (20) S 22 = S 44 = S 55 = S 66 = 1 (21) S 23 = 1 (22) 238 Stress-controlled isotropic compression test. E 77 = E 88 = 1 (23) S 11 = S 21 = S 31 = S 44 = S 55 = S 66 = 1 (24) S 22 = S 33 = 1 (25) 12

13 Axial strain controlled constant deviatoric stress test at constant suction and temperature. E 11 = E 77 = E 88 = 1 (26) S 21 = S 31 = S 44 = S 55 = S 66 = 1 (27) S 22 = S 33 = 1 (28) Axial strain controlled constant mean stress test at constant suction and temperature. E 11 = E 77 = E 88 = 1 (29) S 21 = S 22 = S 23 = S 32 = S 44 = S 55 = S 66 = 1 (30) S 33 = 1 (31) ε 12 controlled drained (constant vertical stress) simple shear test at con- stant suction and temperature. E 14 = E 22 = E 33 = E 55 = E 66 = E 77 = E 88 = 1 (32) S 41 = = 1 (33) ε 12 controlled undrained (constant volume) simple shear test at constant suction and temperature. E 14 = E 22 = E 33 = E 41 = E 55 = E 66 = E 77 = E 88 = 1 (34) 247 Suction-controlled constant net stress and temperature test. E 17 = E 88 = 1 (35) S 22 = S 33 = S 44 = S 55 = S 66 = S 71 = 1 (36) 248 Temperature-controlled constant net stress and suction test. E 18 = E 77 = 1 (37) S 22 = S 33 = S 44 = S 55 = S 66 = S 81 = 1 (38) 249 Suction-controlled swelling pressure (constant volume) test at constant 13

14 250 temperature. E 17 = E 22 = E 33 = E 44 = E 55 = E 66 = E 71 = E 88 = 1 (39) Suction-controlled constant vertical stress oedometric test at constant temperature (swelling strain test). E 17 = E 22 = E 33 = E 44 = E 55 = E 66 = E 88 = 1 (40) S 71 = 1 (41) Temperature-controlled heating pressure (constant volume) test at con- stant suction. E 18 = E 22 = E 33 = E 44 = E 55 = E 66 = E 77 = E 81 = 1 (42) 14