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The Simplified Elastic Plastic CJS Model for 2D Materials
Appendix 1 The Simplified Elastic Plastic CJS Model for 2D Materials
In this work, a simple form of the elastic plastic CJS model [CAM 88a, MAL 00, BAG 11, DUR 15] is used for 2D granular materials. As presented in equation [5.1], the total strain is separated into an elastic part and a plastic part. For the sake of simplicity, a linear elasticity is considered for the modeling of elastic strains. The incremental elastic strain tensor is calculated by Hooke’s law, using a Young’s modulus, E, and a Poisson’s ratio, ν. Plasticity is generated when the stress state reaches the deviatoric yield criterion. Nevertheless, the stress state must be contained within a valid domain of stresses whose boundary is defined by the failure surface, which is isotropic and defined as: f (σ) = sII − Rfail I1 ,
[A1.1]
where Rfail is the radius of the failure surface, sII is the second invariant of the deviatoric part s of the stress tensor √ (sII = sij sij ) and I1 is the first invariant of the stress tensor σ (I1 = σkk ).
142
Granular Materials at the Meso-scale
The deviatoric plastic mechanism is defined by the following yield criterion, which exhibits the same shape as the failure surface (Figure A1.1): [A1.2]
f (σ, X) = qII − Re I1 , where qII is the norm of tensor q defined by:
[A1.3]
q = s − XI1 .
Hydrostatic axis σ1 = σ2
Center of the yield surface Xij
Fai lu
re s
urf
ace
σ1
Yield surface ace urf s e lur Fai σ2 Figure A1.1. Definition of the failure surface and of the yield surface of the CJS model.
In relations [A1.2] and [A1.3], X is a deviatoric tensor associated with the center of the deviatoric yield surface: it defines the anisotropy state of the considered material, Re is the elastic radius which defines the amplitude of the elastic domain and is a model parameter. In this study, this elastic parameter was considered very small since experiments
Appendix 1
143
showed that granular materials exhibit a very small actual elastic domain. The center of the yield surface may change according to the kinematic hardening mechanism: X˙ = aα ˙
with
d
α ˙ = λ I1
!
q˙ − φX˙ qII
"!
I1 2pa
"m
, [A1.4]
where: – a is the model parameter ruling the velocity of the kinematic hardening; – α is the tensor related to the kinematic hardening variable X; – φ is a function which limits the evolution of the kinematic hardening; ! " I1 m – the term reduces the influence of the mean 2pa pressure in the mechanical behavior (pa is a reference pressure set to 100 kPa and m = 1.2). Function φ is defined as: φ = φ0 qII
with
φ0 =
1 . Rfail
[A1.5]
The increment of the plastic volumetric strain related to the plastic deviatoric mechanism is calculated by: ε˙dp v =β
!
"
sII −1 schar II
|s : e˙ dp | , sII
[A1.6]
where β is the dilatancy parameter, edp is the deviatoric part of the strain tensor and schar is the value of sII at the II characteristic state for the current mean pressure. The
144
Granular Materials at the Meso-scale
characteristic surface, which is also an isotropic surface, with the same shape as the failure surface, is written as: f char (σ) = schar II − Rchar I1 ,
[A1.7]
where Rchar is the radius of the characteristic state and is a model parameter. Considering relation [A1.7] contractancy occurs if sII < schar II and dilatancy occurs otherwise. In the initial CJS model the stress ratio curve at the beginning of loading and unloading is not well described. So we have improved the model by considering that the kinematic hardening parameter a at a given state depends on the induced anisotropy X at this state. The parameter a is then defined using the following exponential relationship: a = bec(X1 − X1 ini) ,
[A1.8]
where X1 is the current anisotropy, X1 ini is the initial anisotropy for the considered loading path (e.g loading or unloading) and b and c are two parameters of the kinematic hardening.
In this work, a simple form of the elastic plastic CJS model [CAM 88a, MAL 00, BAG 11, DUR 15] is used for 2D granular materials. As presented in equation [5.1], the total strain is separated into an elastic part and a plastic part. For the sake of simplicity, a linear elasticity is considered for the modeling of elastic strains. The incremental elastic strain tensor is calculated by Hooke’s law, using a Young’s modulus, E, and a Poisson’s ratio, ν. Plasticity is generated when the stress state reaches the deviatoric yield criterion. Nevertheless, the stress state must be contained within a valid domain of stresses whose boundary is defined by the failure surface, which is isotropic and defined as: f (σ) = sII − Rfail I1 ,
[A1.1]
where Rfail is the radius of the failure surface, sII is the second invariant of the deviatoric part s of the stress tensor √ (sII = sij sij ) and I1 is the first invariant of the stress tensor σ (I1 = σkk ).
142
Granular Materials at the Meso-scale
The deviatoric plastic mechanism is defined by the following yield criterion, which exhibits the same shape as the failure surface (Figure A1.1): [A1.2]
f (σ, X) = qII − Re I1 , where qII is the norm of tensor q defined by:
[A1.3]
q = s − XI1 .
Hydrostatic axis σ1 = σ2
Center of the yield surface Xij
Fai lu
re s
urf
ace
σ1
Yield surface ace urf s e lur Fai σ2 Figure A1.1. Definition of the failure surface and of the yield surface of the CJS model.
In relations [A1.2] and [A1.3], X is a deviatoric tensor associated with the center of the deviatoric yield surface: it defines the anisotropy state of the considered material, Re is the elastic radius which defines the amplitude of the elastic domain and is a model parameter. In this study, this elastic parameter was considered very small since experiments
Appendix 1
143
showed that granular materials exhibit a very small actual elastic domain. The center of the yield surface may change according to the kinematic hardening mechanism: X˙ = aα ˙
with
d
α ˙ = λ I1
!
q˙ − φX˙ qII
"!
I1 2pa
"m
, [A1.4]
where: – a is the model parameter ruling the velocity of the kinematic hardening; – α is the tensor related to the kinematic hardening variable X; – φ is a function which limits the evolution of the kinematic hardening; ! " I1 m – the term reduces the influence of the mean 2pa pressure in the mechanical behavior (pa is a reference pressure set to 100 kPa and m = 1.2). Function φ is defined as: φ = φ0 qII
with
φ0 =
1 . Rfail
[A1.5]
The increment of the plastic volumetric strain related to the plastic deviatoric mechanism is calculated by: ε˙dp v =β
!
"
sII −1 schar II
|s : e˙ dp | , sII
[A1.6]
where β is the dilatancy parameter, edp is the deviatoric part of the strain tensor and schar is the value of sII at the II characteristic state for the current mean pressure. The
144
Granular Materials at the Meso-scale
characteristic surface, which is also an isotropic surface, with the same shape as the failure surface, is written as: f char (σ) = schar II − Rchar I1 ,
[A1.7]
where Rchar is the radius of the characteristic state and is a model parameter. Considering relation [A1.7] contractancy occurs if sII < schar II and dilatancy occurs otherwise. In the initial CJS model the stress ratio curve at the beginning of loading and unloading is not well described. So we have improved the model by considering that the kinematic hardening parameter a at a given state depends on the induced anisotropy X at this state. The parameter a is then defined using the following exponential relationship: a = bec(X1 − X1 ini) ,
[A1.8]
where X1 is the current anisotropy, X1 ini is the initial anisotropy for the considered loading path (e.g loading or unloading) and b and c are two parameters of the kinematic hardening.