LEC-6 CL 601 Failure criteria and plasticity for geomaterials.pptx

Failure criteria;
plasticity for geomaterials;
Invariant Conventions;
Failure Criteria for Metals;
Independent Criteria for Soils;
Mohr-­‐Coulomb Criterion;
Drucker-­‐Prager Criterion;
Matsuoka-­‐Nakai criterion;
Lade-­‐Duncan Criterion;
Hoek-­‐Brown Criterion;
Soil plasticity;
plastic flow;
Plastic strains;
Consistency condition;
Flow rule;
Taylor & Quinney experiments;
Hardening rules;
Stress Control;
Constitutive modelling of soil;
M Tech Geotech Engineering;

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: 6 : Failure criteria and plasticity for Geomaterials

2. Content of the presentation 2

3. 5. Failure criteria and plasticity for geomaterials

4. The first criterion used to limit the allowable stresses in materials is usually referred to as the Galileo-­‐Rankine-­‐Navier criterion. It imposes that maximum and minimum normal stresses applied to a solid must be limited by two threshold values, that reflect the strength in tension and compression. It can accommodate ductile and brittle materials. Galileo Galilei (1564–1642) Claude Navier (1785–1836) William Rankine (1820–1872) The first criterion for material failure

5. These criteria can be written as: where: which are particularly convenient for triaxial conditions (axisymmetric loading) : Invariant Conventions The failure criteria outlined hereafter are based on the following conventions: p   I 1 3 q  3 J 2 cos3   3 3 J 3 2 J 3 2 2

6. Invariant Conventions by considering that σ 1 > σ 2 > σ 3  1  3   1 2 3 J 2 cos   1 2 J 2 cos     3   2  3   3  3  3  1 2 3 J 2 cos     3 

7. Failure Criteria for Metals One of the first failure criteria to be used for metals is the Tresca criterion . This criterion postulates that failure takes place when the difference between the maximum and the minimum principal stresses achieve a limit value. Henri Tresca (1814–1885) This can be written in terms of invariants : f  J 2 ,    2 J 2 sin     3    y  where, k   y 2

8. When the Tresca criterion is plotted in the stress space the criterion generates a hexagonal prism (i.e., six independent planes). Failure Criteria for Metals A very common representation of this criterion is given for plane stress states ( σ 3 =0)

9. Failure Criteria for Metals Another major criterion for metals is associated with the name of Von Mises. The criterion is simply written in terms of the second invariant: G i ve n t h e defi n i ti on , t h e cr i ter io n in tr odu ces dependence on all the principal stresses, as embedded in J2. The representation of the Mises criterion in the stress space is a cylinder with circular section: Richard von Mises (1883–1953)

10. Failure Criteria for Metals

11. Failure Criteria for Metals The two criteria are often considered to be equivalent. They are both pressure independent and provide the same strength prediction for compression and extension states. However, their accuracy for more general stress conditions can be assessed by comparing them with actual experiments. For the majority of metals, the Mises criterion provides more accurate predictions.

12. Use of Pressure-­‐Independent Criteria for Soils The most typical case in which pressure-­‐independent criteria are used for geomaterials is to model the undrained strength of soils by using a total stress approach. This hypothesis is often very effective, but experiments show that undrained strength is dramatically dependent on the mode of shear.

13. The most simple criteria accounting for pressure-­‐dependent behavior of soils are frictional failure criteria. Charles Coulomb (1736-­‐1806) Otto Mohr (1835-­‐1918) Purely Frictional Cohesive-­‐Frictional Pressure-­‐Dependent Criteria: Frictional Criteria for Soils

14. In other words, the Mohr-Coulomb criterion belongs to a class of criteria that exhibit dependence on all the three invariants. f  I 1 , J 2 , J 3   Pressure-­‐Dependent Criteria: Mohr-­‐Coulomb Criterion By writing the criterion in analytical form, it reads as:   c   N tan  Similar to the Tresca criterion, it does not depend on the intermediate principal stress, having that an alternative way to write it is:   1   3  s i n     1   3   2 c c o s   The lack of dependence on the intermediate principal stress generates a particular dependence on the Lode’s angle, written by:

15. The graphical representation is a cone with a distorted hexagonal section. This particular shape suggests that the maximum deviator sustained under compression and extension do not coincide. Pressure-­‐Dependent Criteria: Mohr-­‐Coulomb Criterion

16. The distorsion of the section depends on the values of friction and cohesion. For zero friction and non-zero cohesion, the criterion degenerates in the usual Tresca surface. Pressure-­‐Dependent Criteria: Mohr-­‐Coulomb Criterion

17. Simila r t o th e Mohr-Coulom b criterion , another well-know plastic failur e criterion strength properties is the Drucker-Prager introducing a pressure-dependence of the criterion, defined as: Daniel Drucker (1918-2001) William Prager (1903-1980) f  I 1 , J 2    I 1  J 2  Y  Pressure-­‐Dependent Criteria: Drucker-­‐Prager Criterion

18. Given the difference in shape, the Drucker-Prager model provides prediction that coincide with those of the Mohr-Coulomb criterion only for specific values of its parameters.   60     c   2 sin  3 3  3 sin  2 sin   e  3 3  3 sin  Y  6 c cos  3  3  sin   Y  6 c cos  3  3  sin   Pressure-­‐Dependent Criteria: Drucker-­‐ Prager Criterion

19. Pressure-­‐Dependent Criteria The accuracy of the two shapes in the deviatoric plane must be assessed through experiments. Typically, experiments show that the shape associated with the Mohr- Coulomb criterion is much more accurate than other simple pressure-dependent criteria such as the Drucker-Prager.

20. Pressure-­‐Dependent Criteria: Improved shapes In order to provide better accuracy for capturing the strength of soils, several criteria have been proposed, which incorporate a curved shape that resembles that of the Mohr- Coulomb criterion. The key ingredient of such approaches was based on the incorporation of some dependence of the mobilized friction angle on on the intermediate principal stress. M     6sin  3  sin  sin 3  Zienkiewicz & Pande Criterion (1977) Gudheus Criterion (1971)   2 c M c M    1  c    1  c  sin 3  c M e c   0.77 M

21. A widely used approach, which enjoys a particularly good match with the experimental evidence was proposed by Matsuoka and Nakai, and involves the definition of three different mobilized angles based on the three principal stress values.   1 2 1 2 3 I I I 3 f I , I , I   k  1 2 tan  *  tan 2   tan 2   tan 2  13 12 23 Pressure-­‐Dependent Criteria: Matsuoka-­‐Nakai

22.     Another widely used expression is the one proposed by Lade and Duncan, which also involves the third invariant. 3 1 1 3 I I 3 f I , I   k  At variance with the MN criterion, LD predicts a larger mobilized friction angle for failure in extension. Pressure-­‐Dependent Criteria: Lade-­‐Duncan

23. Both MN and LD criteria are particularly accurate in reproducing soil strength. Their relative accuracy therefore depends on the specific soil. Pressure-­‐Dependent Criteria: Frictional Criteria for Soils

24. A very popular failure criterion used for rocks is the Hoek-Brown criterion. Also this is a cohesive-frictional criterion. Similar to Mohr-Coulomb, it is defined in the t-s plane, and it does not depend on the intermediate principal stress (it imposes a limitation on the maximum shear stress). At variance with MC, it includes a dependence of the mobilized friction angle with the applied isotropic stress. Pressure-­‐Dependent Criteria: Hoek -­‐Brown Criterion

25. Failure criteria are also referred to as “limits of elasticity”. Indeed, in the simplified assumption that the transition from elastic response to failure is characterized by a sharp transition, they are the first violation of the elasticity postulates. Even within the frame of this assumption, describing failure requires more than just a limit “threshold”. The exit from the elastic regime requires to describe inelastic deformation effects. The most common theory to reproduce material inelasticity is the so-called Theory of Plasticity. The essence of the plasticity theory is to model how a material can “flow” plastically under applied stresses. To capture this process we will need to introduce the concepts of plastic flow, dilatancy, hardening and softening. Plasticity: More than a ‘‘threshold’’

26. Non linearity imples that the tangent stiffness changes throughout the loading process. Irreversibility of the response is associated with the development of permanent (plastic) strains. Incremental non linearity implies di f ferences between loading and unloading response. History dependency implies that the stresses at which significant plastic strains can take place depend on the previous deformation history (moving threshold). Plastic Strains Evolvin g T angent Stiffness Different stiffness in unloading Movin g theshol d fo r Plastic Strains Soil plasticity

27. d  ij  D ijhk d  hk d  ij  C ijhk d  hk or tenso r Mor e specificall y , th e constitutiv e must take the following form:  hk p hk i j   , d  ,  C ijhk  C ijhk Non Linearity Incremental Non Linearity History Dependency It is therefore impossible to reproduce these properties by means of elastic constitutive laws. In particular, it is impossible to identity a unique correspondence between stresses and strains (i.e., multiple strains may correspond to a given state of stress and vice versa). In order to overcome this limitation it is necessary to write the constitutive law in incremental form, by relating increments of stresses to increments of strains: Incremental constitutive relation

28. In mechanics the term plasticity indicates permanent deformations, i.e. strains that cannot be recovered upon unloading. For general imposed stress-strain conditions, it is necessary to quantify such inelastic effects. This task requires two notions that define the plastic behavior together with the notion of “threshold”. These notions are those of plastic flow and hardening. What are plastic strains?    p Strength Threshold   Hardening SoBening Perfect Plasticity

29. Mohr-­‐Coulomb failure and plastic flow In terms of shear and normal stress   c   N tan  In terms of stress invariants In terms of volumetric and distortional stress F  p  , q   q  Mp   c   F   ij  q  M * p   c dil ation

30. Plastic strains The theory of plasticity postulates that the deformation of a plastic solid consists of two contributions:    e   p ij ij ij Loading criteria dF  i j     F   i j d  ij  unloading criteria dF  i j     F   i j d  ij 

31. Consistency condition Stress states before and after plastic loading should stay on the failure surface. This feature is also known as consistency condition. T he plastic loading occurs if the incremental stress vector is perpendicular to the outward normal to the failure surface

32. Plastic work and stability postulate Mechanical work increment for uniaxial system Second order work done > 0 Stable < 0 Unstable Drucker stability postulate

33. Flow rule Flow rule is the necessary kinematic assumption postulated for plastic deformation  ij g   ij    p ij Analogy Elastic strain can be derived from Elastic (strain energy) potential/complimentary elastic potential i j e    V   i j 1928 Von Mises proposed the concept of plastic potential ( g ) which is a scalar function of stresses Plastic strain from plastic potential i j p  g      i j  Positive scalar factor of proportionality Associated flow: F = g Non-­‐associated flow rule: F ≠ g

34. Normality and associated flow rule Convexity Normality Mohr-­‐coulomb plasticity ???

35. Assumptions of incremental stress-­‐strain response at plastic loading Decomposition of strain increment d   d  e  d  p ij ij ij Elastic relationship d   D e d  e i j ij k l k l The failure surface F   ij   Consistency condition   i j i j dF   F d   Flow rule i j d  p     g   i j

36. Elastic-­‐plastic tangent stiffness tensor d  ij  D ijkl d  kl

37. 5. Failure criteria and plasticity for geomaterials

38. Pure tension F Pure torsion Combined Loading The yield surface provides information on WHEN the plastic strains take place (i.e., defines the stress threshold separating elastic states from inelastic states). To complete the description of inelastic deformation it is necessary to incorporate a criterion for calculating the amount of plastic strains due to loading. In other words it is still necessary to define HOW plastic strains develop. Some significant advances on this aspect derive from the mechanics of metallic materials. Taylor and Quinney (1936) for instance studied the features of plastic strains from tests on steel cylinders subject to combined states of tension and torsion. Plasticity of Metals: Taylor & Quinney experiments Yielding Points T

39. A clear identification of yielding was associated with the appearance of irreversible elongations of the bars even under an external loading imposing only torsional moments. Plasticity of Metals: Taylor & Quinney experiments

40. The mechanical response of plastic materials as observed in laboratory experiments is characterized by: (i) marked non linearity; (ii) irreversibility and (iii) history/path dependency. Plasticity of metals and soils

41. Plastic yielding Plastic flow starts long before the failure. This phenomenon is called the plastic yielding . Because yielding is not the failure, there is no restriction on the direction of the incremental stress vector -­‐ all stress states are statically admissible -­‐ inside, outside and on the current yield surface

42. Y   i  j ,  l   are the Internal Variables (i.e., a set of variables defining size and position of the yield locus). Whenever f<0 the state of the material is ELASTI C (i.e. , n o plasti c strain s are generated). I f th e stres s stat e achieve d th e surface, further loading is possible only by “moving” the location of the surface, i.e., either by inducing a translation of the yield locus (kinematic hardening) or by expanding the surface (isotropic hardening). Where  l Example of isotropic hardening AB: elastic loading CD: elastic unloading CE: plastic loading The movement of the stress threshold for plastic yielding is described by a set of “internal variables”, which evolve with plastic strains. Hardening rules

43. The consistency condition is given by: i j l   dY   ,    Y    i j i j d     Y   l l d   When,  Y d    the surface has to expand so that  Y l d   Hardening When,   i  j  Y d    the surface has to contract so that   l  Y   l d  l  Softening When,   i  j  Y   i  j d  i  j   Y   l d  l  Elastic unloading Hardening rules

44. Hardening rules Isotropic hardening Kinematic hardening Mixed hardening Perfect plasticity Y   i  j ,  l   Y   i  j    i j   d   l   Y   i  j ,  l   Y   i  j    i j   l    d  Y   i  j ,  l   Y   i  j    i j   l    d   l   Y   i  j ,  l   Y   i  j    ij   d 

45. Let us assume to apply a stress increment d s to a soil specimen. If the stress lays on the plastic surface: f  σ  , Ψ   If the stress does not imply elastic unloading, in order not to violate the plasticity constraints the stress must either remain on the fixed surface (perfect plasticity), or it must cause its expansion (hardening). In either cases: T T ⎝ ⎠ ⎝ ⎠ df  ⎛  f ⎞ d σ   ⎛  f ⎞ d Ψ  ⎜  σ  ⎟ ⎜  Ψ ⎟ d σ  The evolution of the internal variables will depend on the hardening laws: Ψ  Ψ  ε p  Incremental Constitutive Equations: Stress Control

46. Incremental Constitutive Equations: Stress Control The increment in the internal variables is given by: By using the plastic flow rule: the consistency condition becomes: d ε p    g  σ  where: is the Hardening Modulus (or Plastic Modulus ). In perfect plasticity H=0.

47. Incremental Constitutive Equations: Stress Control The previous set of equations allows one to quantify the plastic multiplier due to an incremental loading process (and thus, allow to quantify the plastic strains associated with the process. From the consistency condition, in fact: It is finally possible to reconstruct the incremental strains due to the application of a stress increment: Thus: With: d ε  C e d σ   C p d σ   C ep d σ 

48. It is useful to describe the series of steps to derive the tangent elastoplastic stiffness consistent with an elastoplastic soil model. This is the case of strain controlled integration, i.e. the approach used in displacement-based FEM codes. The constitutive equations are intergrated in selected points (Gauss points) in order to calculate the stresses across the domain. Nodal Displacements are converted 1 into imposed strains across the elements Nodal Displacements are converted 2 into imposed strains across the elements 3 d σ   D ep d ε Incremental Constitutive Equations: Strain Control

49. Let us assume to impose an incremental strain. The purpose of the constitutive equations is to: (i) evaluate the resulting stress state and (ii) distinguish how much of the imposed strain is stored as elastic deformation and how much is inducing plasticity. If the initial state of stress is on the yield surface and the incremental loading causes plasticity the consistency condition is again given by: In this case the incremental stress is not known. Therefore it is possible to express it as: d σ   D e d ε e  D e  d ε  d ε p  Even if we do not know yet the amount plastic strains, it is possible to state that: d ε p    g  σ  Incremental Constitutive Equations: Strain Control

50. By using again the hardening rules: The consistency condition can be now rewritten as: That, by using the plastic flow rule, becomes: and Incremental Constitutive Equations: Strain Control

51. Incremental Constitutive Equations: Strain Control By introducing: It is possible to evaluate the plastic multiplier as a function of the imposed strain: It is possible now to reconstruct the entire incremental response:

52. d σ   D e d ε e  D e  d ε  d ε p  with: c 1 H  H D e D e d ε d σ   D e d ε  D e d ε p  D e d ε   D e  g  D e d ε   σ  That provides: d σ   D ep d ε with The tangent elastoplastic stiffness is instead given by: Incremental Constitutive Equations: Strain Control

53. References

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