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Identification of the parameters of an elasto-plastic
model with strain-softening by inverse analysis of
pressuremeter tests
Christophe Dano, Pierre Yves Hicher
To cite this version:
Christophe Dano, Pierre Yves Hicher. Identification of the parameters of an elasto-plastic model with strain-softening by inverse analysis of pressuremeter tests. 3rd International Conference on Identification in Engineering Systems, 2002, Swansea, United Kingdom. �hal-01008276�
IDENTIFICATION OF THE PARAMETERS OF AN ELASTO-PLASTIC MODEL WITH STRAIN-SOFTENING BY INVERSE ANALYSIS OF
PRESSUREMETER TESTS Christophe DANO
Civil Engineering Laboratory of Nantes Saint-Nazaire, France (E-mail : Christophe.Dano@ec-nantes.fr)
Pierre-Yves HICHER
Civil Engineering Laboratory of Nantes Saint-Nazaire, France
Abstract : In this paper, we present a procedure for the identification of the parameters of constitutive models from in situ pressuremeter tests. We first develop a semi-analytical solution of the pressuremeter curve using a linear elastic perfectly plastic model with a post-peak strain-softening and a small strain hypothesis. More usual expressions of the pressuremeter curve are then deduced. These expressions have been implemented in a commercial software. Finally, we show the effect of the softening and of the strain level hypothesis on the values of the friction angle obtained by optimisation computations using a Gauss-Newton algorithm.
1. - INTRODUCTION
The pressuremeter test is an in-place test widely used in soil mechanics in order to assess soil properties (Baguelin et al. 1978 ; Cassan 1978 ; Wroth 1984). The pressuremeter device (Fig. 1) is composed of a radially expandable cylindrical probe either inserted into a pre-bored hole (Ménard pressuremeter test) or directly driven into the soil (Self-boring pressuremeter test) at the desired depth. Application of water pressure increments, monitored from the pressure – volume control unit located at the surface, causes the probe to be inflated. A standard procedure (French Standard NF P 94-110 or US Standard ASTM D 4719-87) then allows to deduce an
experimental stress – deformation curve, named Sexp. Its shape (Fig. 1) depends on
the mechanical properties of the soil in which the probe has been inserted.
A theoretical pressuremeter curve Sth can also be obtained by solving the
mechanical boundary problem posed by the pressuremeter test, either analytically or by finite elements computations, depending on the degree of complexity of the constitutive model adopted for representing the behaviour of the tested soil. Then, it becomes possible to identify the parameters of the constitutive model by inverse analysis. The purpose of this mathematical tool is to fit the estimates of the theoretical pressuremeter curve Sth to field data Sexp by iterative computations on the
values of the parameters of the constitutive model.
In this paper, we analyse some pressuremeter tests carried out in uncemented soils. The soil behaviour is represented using the linear isotropic elastic plastic model shown in Figure 2. The elastic part is defined by the elastic modulus E and the Poisson’s ratio ν. The shear modulus G is equal to E/(2(1+ν)). The
Mohr-Coulomb failure criterion (Eq. 1), characterised by the friction angle ϕ’, determines the value of the peak strength :
( )
,k(
)
k( )
sin '(
)
0 F r p d r + ε × ϕ×σ +σ = σ − σ = σ θ θ (Eq. 1)where σr and σθ are respectively the radial stress and the orthoradial stress.
Volumetric strains in the plastic domain are calculated assuming a non-standard flow rule (Eq. 2) :
( )
(
r) (
g sin) (
r)
Gσ = σθ −σ + χ ψ × σθ +σ (Eq. 2)
where ψ is the dilation angle and χ a value between 0 and 1.
0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 V a ri a ti o n o f v o lu m e ∆ V /V 0 Pressure p Pseudo-elastic domain Large deformations domain p 0
Figure 1. The pressuremeter device and the resulting pressuremeter curve.
(Carter et al. 1985) have noted that the post-peak behaviour has a great effect on the stress distribution around the cavity, that is why we consider a post-peak strain softening law that depends on the deviatoric plastic strain εdp such as :
( )
p d p d 1 kε = +α×ε (Eq. 3) p p r p d =ε −εθ ε (Eq. 4)where α represents the strain softening rate. The function k is equal to 1 at the peak strength and is set to the value β < 1 at the critical state at which the strength remains constant. We model therefore a gradual decrease of the friction angle between the peak and the critical state.
-0.5 0 0.5 1 1.5 2 2.5 3 -0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 S tr e s s d e v ia to r q V o lu m e tr ic s tr a in ε v Axial strain ε 1 1 E G = E / (2x(1+ν)) f(ν) EPP Model ESP Model g 1(ψ) g 1(ψ) g 2(ψ) β f(α) p
p0 Elastic domain Strain softening plastic domain r = a r = b r = c Critical state domain
EPP : Elastic perfectly plastic model ESP : Elastic plastic model with strain softening
Fig. 2. The constitutive model and the corresponding zone into the soil.
The semi-analytical expression of the pressuremeter curve developed from this constitutive model is then compared to relationships found in the literature (Hughes et al. 1977 ; Yu et al. 1991 ; Monnet et al. 1994). We show the effect of the different assumptions on the values of the friction angle obtained by optimisation computations.
2 – EXPRESSIONS OF THE PRESSUREMETER CURVE
2.1 – Resolution of the mechanical problem
The pressuremeter test is formally a problem of expansion of a cylindrical cavity, with an initial radius a, into an infinite medium (Fig. 2). We assume that : - the initial stress state is isotropic and is defined by the earth pressure at rest p0 ;
- the medium is homogeneous and isotropic ; - the problem is axisymmetric ;
- the vertical stress is always the intermediate principal stress.
The assumption of plane strain is also adopted. It is justified by the use of guard cells placed on both sides of the probe. Finally, the elastic strains εe and the plastic strains εp are linked to the total strain ε by the partition rule (Eq. 5) :
p e+ε
ε =
ε (Eq. 5)
To determine the relation between the deformation u(a)/a and the pressure p applied by the probe at the cavity wall, we have to combine equations of the constitutive model, equilibrium equations in the horizontal plane (Eq. 6), compatibility equations (Eq. 7) that link the strains to the displacements, initial and boundary conditions (Eq. 8).
( )
( )
( )
0 r r r dr r d r r = σ − σ − σ θ (Eq. 6)( )
( )
dr r du r r = ε( ) ( )
r r u r = εθ (Eq. 7)(
r a)
p r = =− σ σr(
r→∞)
=−p0 (Eq. 8)The following notations are also adopted :
ψ χ + ψ χ − = χ sin 1 sin 1 n n1 =nχ=1 G 2 ' sin p c k 0 2 3 = ϕ ' sin 1 ' sin 1 kp ϕ + ϕ − = 2 3 10 c k 2 1 k = + α 23 11 c k 2 k =− α
(
)
[
]
1 n 1 1 11 lim lim 1 1k b c R = + β− − + =( )
(1 n )(1 k sin ') 2 n 1 11 10 1 10 1 ' sin 1 X ' sin k ' sin k 1 X f ϕ + + + ϕ + ϕ + ϕ + =2.2 – Pressuremeter curve equation using the strain softening model
Initial pressure increments are small enough to cause small and reversible strains. We can therefore consider that the soil behaves elastically as long as p is less than ppl such as :
(
1 sin ')
p
ppl = 0× + ϕ (Eq. 9)
The equation of the pressuremeter curve in the elastic domain is then :
( )
G 2 p p a a u − 0 = (Eq. 10)As soon as p reaches the value ppl, a plastic annulus develops concentrically
around the borehole. The soil between the radius a and the radius c (Fig. 2) undergoes strain softening. Then :
(
)
p a c a c f ' sin 1 p 1 k sin ' ' sin k 2 0 10 10 = × × ϕ + + ϕ ϕ (Eq. 11a) and( )
1 n1 2 3 1 2 3 1 1 a c c k n 1 2 c k 1 n 1 n a a u + + + + − = (Eq. 11b)Equation (11a) is a polynomial function of the variable (c/a) raised to two distinct powers that depend on the values of the friction angle and the dilation angle.
The determination of this variable requires to calculate the deformation at the cavity wall (eq. 11b). It can be done by optimisation using a Gauss-Newton algorithm.
These equations (11a,b) are suitable as long as the pressure p into the probe is less than ppa for which the soil reaches the critical state. The pressure ppa mainly
depends on the parameters α and β :
(
) (
)
1 k sin ' ' sin k 2 lim lim pl pa 10 10 R R f p p + ϕ ϕ × × = (Eq. 12)When p > ppa, a third domain (a < r < b) coexists with the two previous zones
(Fig. 2) : the elastic zone (r > c) and the strain softening zone (b < r < c). Assuming that the area of the strain softening zone is set when the critical state is reached, that is to say that the ratio (c/b) is set to the value Rlim, the pressuremeter curve (Eq. 13)
can be explicitly calculated :
( )
( )
(
)
(
)
χ χ + χ χ θ + − + + + − + − ε = χ n 1 c k 1 n k a b n 1 c k 1 n k b a a u 2 3 1 n 1 2 3 1 (Eq. 13) with( )
(
)
1 n1 lim 2 3 1 2 3 1 1 R c k n 1 2 c k 1 n 1 n b + θ + + + − = ε and(
) (
)
(
)
( )( ) ( )2 10 10 ' sin k 4 ' sin 1 ' sin k 1 lim ' sin 2 ' sin 1 lim 0 R R f ' sin 1 p p a b ϕ β ϕ β + ϕ + ϕ β ϕ β + × × ϕ + =2.3 – Pressuremeter curve equation using the small strain assumption
The previous developments constitute a generalisation of the pressuremeter curves proposed in the literature. Indeed, if we assume :
- a linear elastic perfectly plastic constitutive model ; - a small strain hypothesis ;
- α = 0, β = 1,
then ppl = ppa and (Eq. 13) becomes :
( )
(
)
( ) ψ − × ψ + ϕ = ϕ + ψ ϕ + sin p p sin 1 G 2 ' sin p a a u sin '1 sin ' sin 1 pl 0 (Eq. 14)Equation (14) is similar to the relation proposed by (Monnet et al. 1994) if the elastic strain is set to a constant value in the plastic domain. On the contrary, if the elastic strain is supposed negligible in the plastic domain, we obtain the relation proposed by (Hughes et al. 1977) :
( )
( ) ϕ = ϕ + ψ ϕ + sin 1 ' sin ' sin 1 pl 0 p p G 2 ' sin p a a u (Eq. 15)The equation (15) leads to a stiffer pressuremeter curve than the equation (14).
2.4 – Pressuremeter curve equation using the large strain assumption
(Yu et al. 1991) have put the validity of the small strain hypothesis into question since it tends to give too high values of the limit pressure. With the linear elastic perfectly plastic model, they have developed an expression of the pressuremeter curve, using a large strain hypothesis and new definition of the compatibility rules (Eq. 16) :
( )
= ε 0 r dr dr Ln r( )
= εθ 0 r r Ln r (Eq. 16)(Yu et al. 1991) then stated that :
( )
(
)
( ) 1 p ' sin 1 p a a u 21 sin ' ' sin 0 sin 1 2 − ϕ + × Γ = + ϕ ϕ ψ + (Eq. 17)where Γ is a function of the parameters of the constitutive model. Γ requires the calculation of a serie whose first five terms provide a good estimation.
3 – OPTIMISATION PROCEDURE
3.1 – Minimisation algorithm
The different equations of the pressuremeter curve (Eqs. 10, 11, 13, 14, 15, 17) have been implemented in a commercial software that also provides an optimisation algorithm, based on either the Gauss-Newton algorithm or the conjugate gradient method. The Gauss-Newton algorithm minimises the difference, raised to the power 2, between the experimental pressuremeter data Sexp(pi) and the theoretical one
Sth(pi, J) where pi is the pressure into the probe and J is the set of parameters :
( )
( )
[
]
∑
− = N 1 2 i th i exp p S p ,J S L( )
( )
k k f InfL J J L ℜ ∈ = (Eq. 18)( )
( )
∂ ∂ ∂ ∂ + = − + k 1 k 2 2 k 1 k J J L J J L J JHowever, the simultaneous optimisation of all the parameters by the Gauss-Newton algorithm leads to divergent computations or erroneous estimates because of the identical weight assigned to the parameters. This problem is emphasised by
the coupling that links some of the parameters, the friction angle and the dilation angle for instance.
Therefore, in order to reduce the number of parameters that have to be simultaneously optimised, we establish an identification procedure based on a sensitivity study and on laboratory experiments.
3.2 – Identification procedure
The linear elastic perfectly plastic model is defined by 4 parameters : the earth pressure at rest p0, the shear modulus G, the friction angle ϕ’ and the dilation angle
ψ.
The value of the earth pressure at rest measured at the beginning of the pseudo-elastic stage of the pressuremeter curve, often integrates experimental errors (removal of the load during sounding, partial collapse of the borehole, …). So, without reliable geotechnical investigations or information about the historical record of the site to estimate the earth pressure, we calculate p0 from the relation :
' K
p0 = 0σv (Eq. 19)
where K0 is the coefficient of earth pressure and σv’ the vertical effective stress at
the depth z at which the pressuremeter test is carried out (σ’v = γh.z – u where γh is
the unit weight of the soil and u the pore pressure). The unknown disturbance due to the boring stage also affects the beginning of the pressuremeter curve. Consequently, the optimisation procedure is applied on the portion of the curve which corresponds to u(a)/a > 1,5 %, as proposed by (Cambou et al. 1993).
The shear modulus G is determined on an unload / reload cycle on which the effect of the initial disturbance is negligible :
( )
cycle cyc a a u 2 p G ∆ ∆ = (Eq. 20)But this value Gcyc has to be corrected since the elastic properties of soils are highly
dependent of the stress and strain states. (Bellotti et al. 1989) proposed the relation :
(
)
5 , 0 0 c 0 0 cyc ' p ' p 2 , 0 ' p ' p G G − × + × = (Eq. 21)where pc’ is the pressure into the probe at the time of unloading.
For natural uncemented sands, the dilation angle ψ is correlated to the friction angle ϕ’ by : ° − ϕ = ψ ' 30 (Eq. 22)
Consequently, only the friction angle ϕ’ has to be optimised when considering the elastic perfectly plastic model.
The strain softening constitutive model adds 3 parameters : the softening rate
state χ. The parameter α is optimised simultaneously with the friction angle ϕ’. The value of χ is set to 0 in the critical state since it is characterised by a constant volume deformation in the case of uncemented sands. Finally, the parameter β is the ratio between sin(ϕcv) and sin(ϕ’) where ϕcv and ϕ’ are respectively the critical state
friction angle and the peak friction angle. The two angles are highly correlated. The parameter β mainly depends on the density of the soil and on the mean stress. (Bolton 1986) proposed :
(
)
' sin 8 , 0 ' sin ϕ ψ − ϕ = β (Eq. 23)In order to avoid convergence towards values which do not present any physical meaning, we also add initial constraints on the parameters. For instance, the friction angle must be comprised between 20 and 55°, as it is the case for most of the uncemented soils.
The optimisation computation involves a change in the value of the friction angle, and, consequently, a change in the values of the domain bounds ppl and ppa.
Then, we have to finally control that the expression of the pressuremeter curve used in the optimisation process corresponds to the pressure level into the probe.
4 – APPLICATIONS
In order to show the effect of the strain softening and the effect of the definition of the compatibility rules on the value of the friction angle determined by optimisation, we present two pressuremeter tests, named SP1-8 and SP2-8. They were carried out in a clean fine sand formation at 8 meters depth (σv’ = 128 kPa, K0
= 1).
The value of the shear modulus G, after being corrected (Eq. 21), is 33 MPa. The dilation angle is correlated to the friction angle according to (Eq. 22). χ is set to 1 in the strain softening domain and to 0 in the critical state. The friction angle at critical state is 33°, then β is close to 0,8.
According to the procedure previously described, computations with the large strain hypothesis model by (Yu et al. 1991) (Eq. 17) lead to a value of the friction angle equal to 39,7° for the test SP1-8 and 51,0° for the test SP2-8. These values enclose the value of the friction angle ϕ’triax (ϕ’triax = 43°) obtained by means of
laboratory triaxial tests if we also take into account the difference due to the stress path. The field data and the theoretical pressuremeter curves are compared in Figure (3a).
Considering the elastic perfectly plastic model with the small strain assumption (Eq. 14), the optimised value of the friction angle is respectively 34,1° for the test SP1-8 and 41,0° for the test SP2-8.
Finally, with our strain-softening model, the pair of optimised values (ϕ’, α) is respectively (ϕ’ = 43°, α = 30) for the test SP1-8 and (ϕ’ = 43°, α = 1) for the test SP2-8. The value α = 30 means that the decrease of the strength is relatively fast after the peak whereas the value α = 1 corresponds to a slow decrease from the peak to the critical state.
The difference between the two tests could be attributed to different site conditions (local cementation between grains, different shear moduli, …). In particular, the determination of the earth pressure at rest is an usual problem in geomechanics. If we assume, for example, that K0 = 0,5 instead of K0 = 1,
optimisation computations lead to a value of the friction angle equal to 48° for the test SP1-8 and the elastic perfectly plastic model using a large strain hypothesis (Eq. 17). We note that the result of the computations is very sensitive to the value of the earth pressure at rest that also depends on the quality of the borehole.
In all cases, the theoretical pressuremeter curves accurately fit the field data, as shown in Figure (3b) for the test SP2-8.
0 0.5 1 1.5 2 2.5 3 0 0.05 0.1 0.15 0.2 0.25 0.3 P ( M P a) U a/a Essai SP2-8 Essai SP1-8 Continuous lines : experiments Dotted lines : optimisation computations
Fig. 3a 0.5 1 1.5 2 2.5 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Expérience Mod. EPP Mod. EPE P ( M P a) U a/a
EPP : Elastic perfectly plastic model (Eq. 17) EPE : Elastic plastic model with strain softening
Fig. 3b
Figure 3. Comparison of field data and computations.
Model Equation Optimised parameters
Elastic perfectly plastic model – Large strains 17 ϕ’ = 45,3° Elastic perfectly plastic model – Small strains 14 ϕ’ = 37,5°
Elastic plastic model with strain softening 11, 13 ϕ’ = 43°, α = 30 Test SP1-8
ϕ’ = 43°, α = 1 Test SP2-8
Table 1. Recapitulative table.
5 – CONCLUSIONS
We present a semi-analytical expression of the pressuremeter curve assuming a linear elastic plastic model with a post-peak strain softening. We confront this model with the expressions of the pressuremeter curve obtained considering a linear elastic perfectly plastic model, with a small strain hypothesis or a large strain one. These expressions are then implemented in a commercial software that also provides an optimisation tool.
Using the procedure developed to identify the friction angle of uncemented sands from field tests, we show the great effect of the strain softening and of the
definition of the compatibility rules (small or large strain hypothesis) on the values of the friction angle determined by optimisation. The strain softening model can also provide information about the state of the soil in which the pressuremeter tests are carried out. But, the analytical resolution of the mechanical problem do not allow the simultaneous identification of many parameters. Further developments will be necessary to achieve this goal.
6 – REFERENCES
Baguelin, F., Jézéquel, J-F. and Shields, D.H. 1978, The pressuremeter and foundation engineering. Trans Tech Publications.
Bellotti, R., Ghionna, V., Jamiolkowski, M., Robertson, P.K. and Peterson, R.W., 1989, “Interpretation of moduli from self-boring pressuremeter tests in sands”, Geotechnique, Vol. 39, No. 2, pp 269–292.
Bolton, M.D., 1986, “The strength and dilatancy of sands”, Geotechnique, Vol. 36, No. 1, pp 68-78.
Cambou, B. and Bahar, R., 1993, “Utilisation de l’essai pressiométrique pour l’identification de paramètres intrinsèques du comportement d’un sol”, Revue Française de Géotechnique, No. 63, pp 39–50.
Carter, J.P. and Yeung, S.K., 1985, “Analysis of cylindrical cavity expansion in a strain weakening material”, Computers and Geotechnics, No. 1, pp 161-180.
Cassan, M., 1978, Essais in situ en mécanique des sols – Tome I : réalisation et interprétation. Editions Eyrolles, pp 153–268.
Hughes, J.M.O., Wroth, C.P. and Windle, D., 1977, “Pressuremeter tests in sands”, Geotechnique, Vol. 27, No. 4, pp 455-477.
Monnet, J. and Khlif, J., 1994, “Etude théorique de l’équilibre élasto-plastique d’un sol pulvérulent autour du pressiomètre”, Revue Française de Géotechnique, No. 67, pp 3-12.
Wroth, C.P., 1984, “The interpretation of in situ test”, Geotechnique, Vol. 34, No. 4, pp 449-489.
Yu, H.S. and Houlsby, G.T., 1991, “Finite cavity expansion in dilatant soils : loading analysis”, Geotechnique, Vol. 41, No. 2, pp 173–183.
