LEC-7 CL 601 Constitutive modelling- Cam-clay models.pptx

Constitutive modelling of soil;
Constitutive modelling of Geomaterials;
Cam-clay model;
Modified cam-clay model;
strain controlled loading;
Stress controlled loading;
M Tech Geotechnical Engineering; Cam Clay modelling of soil;
Critical State parameters;

1. Constitutive Modelling of Geomaterials Prof. Samirsinh P Parmar Mail: [email protected] Asst. Prof. Department of Civil Engineering, Faculty of Technology, Dharmasinh Desai University, Nadiad , Gujarat, INDIA Lecture: 7 : C a m – c l a y models

2. The constraints of plasticity ( L >0) can be therefore satisfied even if H=0 or H<0, provided that H-Hc>0 and then Hc<H. The quantity Hc is the critical hardening modulus for strain control: This is the tangent constitutive operator that governs the response of the material under strain control. The tendency to undergo plastic strains is governed by the elastic trial stress (stress increment that would occur if the material were elastic): e d σ tr  D e d ε Elastoplastic Constitutive Equations: Strain Controlled Loading The tangent elastoplastic stiffness is given by: which appears in:

3. A new paradigm for soil behavior: the Cam Clay Model Cap model is obviously elementary and unable to describe the complex soil behaviour. At Cambridge various authors tried to refine that model to describe the behaviour of kaolin, on which many axisymmetric tests (isotropic, oedometric, drained and undrained triaxial) had been performed, both for normally consolidated and overconsolidated conditions. In particular, Schofield e Wroth (1968) proposed the Cam Clay model, that was a real change of paradigm in the world panorama of Soil Mechanics. The main hypothesis of the Cam Clay model are the following: There exists a surface in the stress space, delimiting a region within which soil behavior can be considered as elastic (yield surface). Plastic strains occur when the stress state lays on this surface. The direction of the vector giving the plastic strain increment is that of the gradient to the yield locus (normality rule). Within the elastic domain the response is governed by a non-linear elastic law.

4. A new paradigm for soil behavior: the Cam Clay Model The starting assumption of the Cam Clay model is an energy assumption. The dissipated (plastic) energy is basically governed by friction. dW p  p  d  p  qd  p  Mp  d  p v s s At a material level this is equivalent to state that the dissipation is proportional to the “confinement stress” acting over a sliding surface, to the sliding between the two contacting surfaces and to a material parameter related to the internal friction. dW p  Ndv p  Tdu p   Ndu p

5. A new paradigm for soil behavior: the Cam Clay Model By defining the “ dilatanc y ” as d d=d e p v / e p , it is possible to work out the energy assumption by obtaining the so-called stress-dilatancy relationship : d  M   d  v d  p s d  p   f  p   f  q p   f  q  f  Mp   f  p   q  q dq  q  M d p  p  q  p   C  M ln p   df   f dp    f dq   p   q Vector Normal to Yield Surface Vector Parallel to Yield Surface Energy Dissipation The stress-dilatancy relationship is a condition about the plastic flow (dep vector) and thus on the yield surface (given the assumption of assoiated flow rule): + Differential Equation for the stress states laying on the yield locus Expression of the Yield Surface

6. A new paradigm for soil behavior: the Cam Clay Model The yield surface can be finally rewritten in terms of an internal variable, p c , which is associated with its size.

7. Examples of Simulations: Lightly Overconsolidated Clay

8. PC: Borja, R. I., and Aydin, A. (2004). “Computational modeling of deformation bands in granular media. I. Geological and mathematical framework.” Computer Methods in Applied Mechanics and Engineering, 193(27-­‐29), 2667–2698.  1  2  3 q p ' Wiki.. A new paradigm for soil behaviour: the Cam Clay Model

9. A new paradigm for soil behavior: the Cam Clay Model ln p u  ln p c  1 f  q  Mp  ln p   p c Critical State Conditions Critical State The specific volume is therefore linked to the state of stress: Although many concepts come from from the plasticity theory, the initial set up of the model does not introduce any hardening laws. To quantify the amount of elastic/plastic strains the concept of critical state was used.

10. Finally, the plastic strains were calculated as: s d  p  v d  d  p d  p v M   s It is important to note that when η = M , d ε p  infinite. Therefore, η = M coincides with the intuitive notion of failure (unlimited shear strains). A new paradigm for soil behavior : the Cam Clay Model

11. The constraints of plasticity ( L >0) can be therefore satisfied even if H=0 or H<0, provided that H-Hc>0 and then Hc<H. The quantity Hc is the critical hardening modulus for strain control: This is the tangent constitutive operator that governs the response of the material under strain control. The tendency to undergo plastic strains is governed by the elastic trial stress (stress increment that would occur if the material were elastic): e d σ tr  D e d ε Elastoplastic Constitutive Equations: Strain Controlled Loading The tangent elastoplastic stiffness is given by: which appears in:

12. A new paradigm for soil behavior: the Cam Clay Model Cap model is obviously elementary and unable to describe the complex soil behaviour. At Cambridge various authors tried to refine that model to describe the behaviour of kaolin, on which many axisymmetric tests (isotropic, oedometric, drained and undrained triaxial) had been performed, both for normally consolidated and overconsolidated conditions. In particular, Schofield e Wroth (1968) proposed the Cam Clay model, that was a real change of paradigm in the world panorama of Soil Mechanics. The main hypothesis of the Cam Clay model are the following: There exists a surface in the stress space, delimiting a region within which soil behavior can be considered as elastic (yield surface). Plastic strains occur when the stress state lays on this surface. The direction of the vector giving the plastic strain increment is that of the gradient to the yield locus (normality rule). Within the elastic domain the response is governed by a non-linear elastic law.

13. d  p v   p v  dv      ⎛ dp  d  ⎞ v ⎝ ⎜ p   M ⎠ ⎟ p' q v p c1 p c2 p c3 p c1 p c2 p c3 A new paradigm for soil behavior: the Cam Clay Model Hardening rule Isotropic compression c dp   vp c       v d  p

14. A new paradigm for soil behavior: the Cam Clay Model The Cam Clay model can be reformulated in accordance with the standard formalism of elasto-plasticity. Consistency Condition for Incremental Loading It is therefore necessary to include a hardening law for p c . f ( p  , q , p c )   f ( p   dp  , q  dq , p c  dp c )  c c df   f dp    f dq   f dp   p   q  p c dp   p c  p v p d    p c v s  p d  p s p c  p c (  p ,  p ) v s By using the plastic flow rule and the consistency condition, it follows:

15. dW p  p  d  p v 2 q d  p s     2 s  Mp  d  p   2 2 c f  q  M p p  p  The modified Cam Clay Model Roscoe and Burland (1968) Dry side Wet side

16. The modified Cam Clay Model Stress dilatancy relationship M 2  2 p   p  d  c  2 q M 2   2 2  d  v d  p s d  p   f  p   f  q Incremental plastic stress strain relationship

17. The modified Cam Clay Model Drained triaxial test response (lightly over consolidated) A A

18. A B C D q p' O A B C D E E q ε s v A A B C v D E Drained triaxial test response (lightly over consolidated) A B C D E

19. The modified Cam Clay Model Drained triaxial test response (Normally consolidated)

20. The modified Cam Clay Model Drained triaxial test response (Heavily over consolidated)

21. A B C D q p' E O A B C D E q ε s v A A B C v D E B C A D E Drained triaxial test response (Heavily over consolidated)

22. Undrained triaxial test response (Normally -­‐ consolidated) Elastic d   d  e  v v p   p i  , q  Elasto-­‐plastic d  e  d  p  v v v p  v p c    p         p c   p    M 2   2    p        2   The modified Cam Clay Model

23. The modified Cam Clay Model Drained triaxial test response (Lightly overconsolidated)

24. A C D q E O A B C D q B A A,B D C E E p' v Drained triaxial test response (Lightly overconsolidated)

25. The modified Cam Clay Model Drained triaxial test response (Heavily overconsolidated)

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