The Constitutive of Binary Medium for Soils Based on Voigt Model and Reuss Model

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Procedia Earth Planetary Science 00 (2011) Procedia Earth and and Planetary Science 5 (2012) 218 –000–000 221

Procedia Earth and Planetary Science www.elsevier.com/locate/procedia

2012 International Conference on Structural Computation and Geotechnical Mechanics

The Constitutive of Binary Medium for Soils Based on Voigt Model and Reuss Model YANG Ruimina, DING Jianwen, JI Feng, a* Institute of Geotechnical Engineering, Southeast University, Nanjing and 210096, China

Abstract In order to study the structure of natural sedimental clays, the theoretical framework of breakage mechanics for soils and the linary medium model are proposed by Shen Zhujiang based on compatibility of deformation. However, the deformation between structural blocks and soft bands is compatible, and the stress between them is continuous. In this paper, two new constitutive equations are derived based on Voigt model and Reuss model respectively so as to provide theoretical bases for FEM and the test of mechanical properties for soils. At last, a numerical example is given to make clear the breakage laws of structural blocks.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for © 2011 Published by Elsevier Ltd. Resources, Environment and Engineering. Keywords:Voigt model; Ruess model; linary medium; breakage constitutive equation

1. Introduction There is a characteristics for natural sedimental clays that is structural, Shen Zhujiang pointed out that the structural damage during deformation must be considered when building the constitutive models for structural soils [1]. For this, Shen Zhujiang proposed a new soil mechanical theory which was called breakage mechanics [2-4]. The breakage mechanics is a macroscopic analysis mechanical theory based on the quasi-continuum concept whose objects of study are seriously broken rocks and soils, and the structural rocks and soils are considered to be the binary medium composed by structural blocks with strong bond strength and soft bands with weak bond strength, the structural blocks breakage are converted into soft bands during deformation [2-4]. Shen Zhujiang derived the breakage mechanics constitutive equation based on compatibility of deformation between the structural blocks and soft bands [3]. However, * Corresponding author: YANG Ruimin. Tel.: 0086-025-83792220. E-mail address: [email protected].

1878-5220 © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Resources, Environment and Engineering. doi:10.1016/j.proeps.2012.01.038

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the deformation between structural blocks and soft bands is compatible, and the stress between them is continuous. There is another method to build the constitutive equation based on stress continuity between structural blocks and soft bands. Here, two constitutive equations are derived respectively based on Viogt model and Reuss model in order to provide theoretical bases for FEM and the test of mechanical properties for soils. 2. Derivation for constitutive equations 2.1. The constitutive equation based on Voigt model Similar to composite materials, the uniformity theory can be used to analyse the rock and soil mass[5]. A representative volume element (RVE) is taken out from the binary medium, and it is infinite small at the macroscopic scale so that it can be taken to be a point, and it is also infinite big in microcosmic scale so as to include enough mechanical and geometrical statistic information, and the RVE is called binary structural body. Let V, VI, VF are respectively on behalf of the volumes for binary structural body, structural blocks and soft bonds. Mean stress and strain for them are defined as follows: σI 

1 1  σdVI , σF   σdVF VI VF

(1)

1 1  εdVI , ε F   εdVF VI VF

(2)

εI 

Let   VF /V ,  is called breakage rate for volume. From formula (1) and (2) we can get mean stress and strain for binary structural body as follows: σ  (1  )σI  σF , ε  (1  )εI  εF

(3)

The mean strain εI , ε F and ε satisfy the following relations: εI  C : ε , ε F 

1



[I  (1  )C] : ε

(4)

In the formula (4-1), C is called localized strain conductivity tensor, and I is fourth order unit tensor in formula (4-2). The Mean stress and strain for structural blocks and soft bonds satisfy the following relations respectively: σI  DI : εI , σF  DF : εF

In formula (5), DI , DF are elastic stiffness of the tensors for structural and soft bonds respectively. Substitute formula (4-1),(4-2) into (5-1) and (5-2) respectively, and consider (3-1), we can get:

(5)

σ （ [ 1  ）DI : C  DF : (I  (1  )C)] : ε  D : ε

(6) : I  B） DF : B is defined as the elastic stiffness of the tensor for binary structural In formula (6), D  DI （ body, B  I  (1  )C is called breakage parameter tensor. Voigt model[6] assumes that the deformation between structural blocks and soft bonds is compatible, that is ε I  ε F  ε .And the strains ε I , ε F and σI , σF of structural blocks and soft bonds satisfy: ε I  DI 1 : σ I , ε F  DF1 : σ F

(7) Last, substitute formula (7) into (6), we can get the constitutive equation based on Viogt model as follows: σ  DI （ : I  B）: DI 1 : σ I  DF : B : DF1 : σ F

It's worth noting that σ I , σ F in formula (8) are no more than σI , σF in formula (1).

(8)

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2.2. The constitutive equation based on Reuss model Similar to the derivation process above, let the mean stress of structural blocks and soft bonds σ I , σF and σ satisfy follow relations: ~ ~ 1 σ I  C : σ , σ F  (I  (1  )C) : σ



(9)

~ In the formula (9), C is called localized stress coefficient tensor. And considering the formula (5) and (3-

2), we can get:

~ ~ ε  [(1  )DI 1 : C  DF1 : (I  (1 )C)] : σ  D1 : σ (10) ~ ~ In the formula (10), D1  DI 1 : (I  B)  DF1 : B is defined as the flexibility tensor for binary structural body, ~ and B is breakage parameter tensor.

Reuss model[7] assumes that the stress between structural blocks and soft bonds is continuous, that is σ  σ I  σ F .And considering formula (7), we can get from formula (10): ~ ~ ε  DI 1 : (I  B) : DI : ε I  DF1 : B : DF : ε F

It's also worth noting that ε I , ε F in formula (11) are no more than ε I , εI in formula (2).

(11)

3. Discussion If tensor C is degenerated into scalar form c , B  I  (1  )C can be degenerated into scalar form b  1  (1  )c . when unidirectional compression and directly sheared ,we can get the breakage parameters respectively as follow: bE 

  I E  EI b  ， EF  EI  F  I

(12)

E, EI , EF and , I , F are moduluses of elasticity and shear moduluses for binary structural body, structur

-al blocks and soft bonds respectively. For the case of isotropic materials, when    I   F ,we can get: σ  (1  b)σ I  bσ F

(13) And b  bE  b in formula (13) which is same as Shen Zhujiang’s result.However,it is not satisfy with the stress continuity between structural blocks and soft bonds. ~ Similarly, when unidirectional compression and directly sheared, the breakage parameter tensor B can be degenerated into scalar form respectively as follows: ~ ~  E bE  F bE , b  F b  E

For the case of isotropic materials, when    I   F , we can get: ~ ~ ε  (1  b )ε I  b ε F

~ ~ ~ And b  bE  b in formula (15) which is not satisfy with the compatibility of deformation.

(14) (15)

4. A numerical example Because the structural blocks’ breakage has much influence on constitutive tensors, a numerical example is given to discuss the breakage laws for binary medium. Let EI  8200 Kpa , E  4600 Kpa ,  I  0.22 ,   0.26 , GF  0.36, FF  0.6 , 1   3  f  600 kpa , EF  10 3 ,then we can get the relations for components of Bijkl vs.  3 and s as follows (fig.1, fig2, fig3). Here, we only give the relations between breakage parameter ~ tensor Bijkl and  3 and s , however the relations of Bijkl vs.  3 and s which are contrary to the former are

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YANG Ruimin et al./ Procedia / ProcediaEarth Earthand andPlanetary PlanetaryScience Science005 (2011) (2012) 000–000 218 – 221 YANG Ruimin

not given because of space constraints.

Fig. 1 Relationship between Biiii and s under σ3 (left) Fig. 2 Relationship between Biijj and s under σ3 (right)

Fig. 3 Relationship between Bijij and s under σ3

5. Conclusion The constitutive equations based on Viogt model and Reuss model are derived with tensor forms. Compared with Shen Zhujiang’s results which are only applicative to the case that structural blocks and soft bonds are isotropic, the constitutive equations we derived can be applicative to the case those structural blocks and soft bonds are anisotropic. And they can provide theoretical bases for FEM and the test of mechanical properties for soils. A numerical example is given to discuss the breakage laws for binary medium, and the calculated results show that the breakage tensors are related to confining pressure and stress level. References [1] Shen Zhujiang. Theory of soil mechanics [M]. Beijing: China Water Power Press, 2000. [2] Shen Zhujiang. Breakage mechanics and binary medium model for geological materials [J]. Hydro-Science and Engineering, 2002, (4): 1-6. [3] Shen Zhujiang. Breakage mechanics for geological materials: an ideal brittle-elastic-plastic model [J]. Chinese Journal of Geotechnical Engineering, 2003, 25 (3): 253-257. [4] Shen Zhujiang. Binary medium modeling of geological material [C]//International Conference of Heterogeneous Materials Mechanics. Chongqing: [s.n.], 2004: 581-584. [5] Wang J G, Leung C F, Ichicawa Y.A. Simplified homogenization method for composite soils [J]. Computers and Geotechnics, 2002, 10: 32-40. [6] Voigt, W. Űber die Beziehung zwischen den beiden Elastizi ä tskonstanten isotroper Körper. Wied.Ann, 1889, 38: 573-587. [7] Reuss,A. Berchung der Fiessgrenze von Mischkristallen auf Grund der Plastizi ä tsbedingung für Einkristalle. Z. angew. Math. mech, 1929, 9: 49-58.

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