Nonexistence and Non-decoupling of the Dissipative Potential for Geo- materials

Liu Yuan-xue

Logistical Engineering University, Chongqing, China Contents

0. Introduction

1. Thermodynamic foundation

2. Dissipative potential

3. Decoupling

4. Conclusions

2 Introduction Damage mechanics is an important branch of solid mechanics, and the deterioration process of geo- materials is always described by the constitutive model involving damage. The phenomenological method which integrates the irreversible thermodynamics with continuum theory is used to describe the deterioration process of the material. .

particle void bond A distinctive property of the theory is that the constitutive relation could be derived from two specific functions, the free energy function and the dissipation potential function. Of particular importance is the problem of constructing the dissipation potential function, because both plasticity and damage mechanisms involve energy dissipation. Currently, the dissipation potential function is dealt with along two principal approaches. One approach is that unique dissipation potential function is used for plasticity and damage which is deemed that these two dissipation processes occur simultaneously and correlate with each other. Another approach is that the energy dissipated due to plasticity and damage are independent of each other. The damage theory for geo-material is not so perfect as classical plastic theory. The main reason may be that some fundamental theoretical problems have not been solved in damage mechanics for geo-material. For example, the existence or nonexistence of the dissipative potential, and whether the dissipative potential could be decomposed into a damage potential and a plastic one or not, namely, whether these two mechanisms can be decoupled or not. In this paper, sufficient and necessary conditions are given for the existence of dissipation potential in total relation and incremental relation for geo-material, and the sufficient and necessary conditions are provided for the decoupling of dissipation potential in total and incremental forms for geo- material also. 1. Thermodynamic foundation

Based on the First Law and the Second Law of thermodynamics. It is assumed there is no local heat source, which means, the thermal dissipation is zero, local entropy production inequality take following forms p d  Kdk YdD  0 Where  p , k , D are increments for plastic strain, plastic internal variable, and damage variable, and  ， K ， Y are the corresponding thermodynamics force respectively. 2. Dissipative potentialon

Except the free energy function, the dissipative function is also needed for establishing of constitutive model based on thermodynamics. However, the misunderstanding, which takes the dissipative function as the result of the second law of thermodynamics, exists in the determining of dissipative function. In fact, only the dissipative inequality Eq. (1) could be obtained according to the second law of thermodynamics. The dissipative potential would be developed under a set of additional hypothesis 2.1 Total relation The total relation for constitutive model indicates that arbitrary value of state variables corresponding to a certain value of the corresponding thermodynamics forces, and the dissipative potential  for total relation expression is defined as function of state variables

p p   ( , , p ,k, D) v s  p p   p Where v， s，  ， are plastic volumetric strain， generalized plastic shear strain，and Lode angle of plastic strain respectively. Theorem 1:

p p   ( , , p ,k, D) There exists a dissipative potential v s  , such that the following equations should hold in the thermodynamics strain field. p q p Y  p q p K p p      p p p s v  p  D   v k v v

q K q q q Y q K   p   p  k  s p k  p  p  s   D  s

q Y K Y    D  p  D k A point to note for the analysis is that these ten equations have to be hold in the thermodynamics strain field, if  is dissipative potential.

p q p Y  p q p K p p      p p p s v  p  D   v k v v

q K q q q Y q K   p   p  k  s p k  p  p  s   D  s

q Y K Y    D  p  D k In the pseudo triaxial condition, the three principal stresses satisfy  1   2   3, and the stress Lode angle (   ) keeps constant, so the computation of plastic strain could be simplified as

p d v  Adp  Bdq  p d s  Cdp  Ddq A B   The matrix C D  for geo-materials here, is different from that for the metal material with B  C  0 . Generally . Under the negative dilatancy deformation stage (shear leads to volume shrinking) of geo-material, B  0 C  0 can be derived, i.e. B  C . Hence, the matrix should not be a singular one, and the following transformation exists

dp 1  D  Bd p  v      p  dq AD  BC  C A d     s   p p  Namely dp  p  p d p      v s  v  q q p dq  d s   p p  v  s  For the negative dilatancy deformation stage of geo- materials, B  C . So

p  B  C q p    p  s AD  BC AD  BC v The main import of the above result is that equation p q  p p can not hold at all the deformation stage of  s v geo-materials. According to the necessary and sufficient condition for the dissipation potential of geo-materials, it could be concluded that there is no dissipative potential with total relation expression for geo-materials. And the total relation for constitutive model cannot be constructed by the dissipative potential. 2.2 Incremental relation Different from the total relation for constitutive model, the incremental relation indicates that arbitrary increment of state variables corresponding to a certain increment of the thermodynamics forces, and the dissipative potential  with incremental expression is defined as function of state variable increment p p   (d ,d ,d p ,dk,dD) v s  p d p Where d  , s, d  p , d k and d D are the increments for v  plastic volumetric strain, generalized plastic shear strain, Lode angle of plastic strain, plastic internal variable, and damage variable respectively. Theorem 2: p p Q  Q(d ,d ,d p ,dk,dD) There exists a dissipative potential v s  , such that the following equations should hold in the thermodynamics strain field.

p q p K p Y p q     p  p p p p (d p ) (d ) (dk) (dv ) (d s ) (dv )  v (dD) (dv )

q q q K q Y  p q K  p  p (d p ) (d s )   (dk) (d s ) (dD) (d s ) (dk) (d p ) 

q Y K Y    (dD) (d p )  (dD) (dk) Making use of the basic mechanical characteristics of geo-materials to verify whether the dissipative potential function with incremental expression exists or not.

(a) Stress increment (b) Strain increment (%) Figure 1. The strain increment influenced by stress increment

q  0 p p ( q keeps constant withd  vincreasing), while (dv ) p  0 p d p  p (d p ) ( increasing with s ( ) decreasing) under the s q p  action of No. 8 stress increment path, i.e. p p . (dv ) (d s ) The main import of the above result is that the equation could not be satisfied. According to theorem 2, it could be obtained that there is no dissipative potential with incremental expression for geo-materials, and the incremental relation for constitutive model cannot be constructed by the dissipative potential with incremental expression 3. Decoupling

In order to avoid the difficulty of constructing the complete dissipative potential, another attempt is to establish the dissipative potentials for plasticity and damage separately. In this form decoupling is required for these two mechanisms. The qualification for the decoupling of dissipative potential of geo-materials is presented in the following. 3.1 Decoupling for the dissipative potential with total relation expression Theorem 3: There exists a dissipative potential with total relation expression which could be decoupled into a plastic dissipative potential  p and a damage one  d , whose necessary and sufficient condition is that these equations should hold in the thermodynamics strain field.

p q q K  0  0   0  0 D D D D

Y  0 Y Y p  0  0 Y v p   p  0 s  k Thus, qualifications for the decoupling of dissipative potential of geo-materials are:

p q p q   p K p Y p p  p  p  p  s v  p v  k v D  v

q K q q q K q Y   p  p  k  p  p  p   s k  s D  s q Y K Y    D  p D k  p q q K  0  0   0  0 D D D D

Y  0 Y Y p  0  0 Y v p   p  0 s  k That is to say eight new requirements should be satisfied .

p q q K  0  0   0  0 D D D D

Y  0 Y Y p  0  0 Y v p   p  0 s  k p q  That in part 2.1 has shown that equation p p can  s v not hold at all the deformation stage of geo-materials. So the dissipative potential for geo-materials with total relation expression could not be decoupled into the dissipative potentials for plasticity and damage separately. 3.2 Decoupling for the dissipative potential with incremental expression Theorem 4: There exists a dissipative potential with with incremental expression which could be decoupled into  a plastic dissipative potential  p and a damage one d, whose necessary and sufficient condition is that these equations should hold in the thermodynamics strain field. p q q Y  0  0   0  0 (dD) (dD) p (dD) (dv )

Y Y K  0 Y p  0  0  0 (d p ) (d s )  (dD) (dk) Thus, qualifications for the decoupling of dissipative potential of geo-materials with incremental expression are: p q p K p Y p q     p  p p p p (d p ) (d ) (dk) (dv ) (d s ) (dv )  v (dD) (dv )

q q q K q Y  p q K  p  p (d p ) (d s )   (dk) (d s ) (dD) (d s ) (dk) (d p ) 

q Y K Y    (dD) (d p )  (dD) (dk)

p q q Y  0  0   0  0 (dD) (dD) p (dD) (dv )

Y Y K  0 Y p  0  0  0 (d p ) (d s )  (dD) (dk) That is to say eight new requirements should be satisfied .

p q q Y  0  0   0  0 (dD) (dD) p (dD) (dv )

Y Y K  0 Y p  0  0  0 (d p ) (d s )  (dD) (dk) q p  That in part 2.2 has shown that equation p p (dv ) (d s ) can not hold at all the deformation stage of geo- materials. So the dissipative potential with incremental expression could not be decoupled into the dissipative potentials for plasticity and damage respectively. Conclusion

1. There is no dissipative potential with total relation expression for geo-materials, and the total relation for constitutive model cannot be constructed by the dissipative potential with total relation expression. 2. There is no dissipative potential with incremental expression for geo-materials, and the incremental relation for constitutive model cannot be constructed by the dissipative potential with incremental expression. 3. The dissipative potential for geo-materials with total relation expression could not be decoupled into the dissipative potentials for plasticity and damage separately. 4. The dissipative potential with incremental expression could not be decoupled into a dissipative potential for plasticity and that for damage respectively. Acknowledgments

We gratefully acknowledge the support from National Natural Science Foundation of China under grant No. 50979112, and Chongqing Natural Science Foundation under grant No. CSTC2008BB6144. Thank you for your attention!