- Email: [email protected]
A multiscale mechanical model for materials based on virtual internal bond theory
Acta Mechanica Solida Sinica, Vol. 19, No. 3, September, 2006 Published by HUST, Wuhan, China. DOI: 10.1007/s10338-006-0624-6
ISSN 0894-9166
A MULTISCALE MECHANICAL MODEL FOR MATERIALS BASED ON VIRTUAL INTERNAL BOND THEORY Zhang Zhennan1,2
Ge Xiurun3,4
Li Yonghe1
1
( Department of Civil Engineering, Shanghai University, Shanghai 200072, China) (2 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China) (3 Institute of Geotechnical Engineering, Shanghai Jiaotong University, Shanghai 200030, China) (4 Institute of Rock and Soil Mechanics, the Chinese Academy of Sciences, Wuhan 430071, China)
Received 29 December 2005; revision received 25 May 2006
ABSTRACT Only two macroscopic parameters are needed to describe the mechanical properties of linear elastic solids, i.e. the Poisson’s ratio and Young’s modulus. Correspondingly, there should be two microscopic parameters to determine the mechanical properties of material if the macroscopic mechanical properties of linear elastic solids are derived from the microscopic level. Enlightened by this idea, a multiscale mechanical model for material, the virtual multi-dimensional internal bonds (VMIB) model, is proposed by incorporating a shear bond into the virtual internal bond (VIB) model. By this modification, the VMIB model associates the macro mechanical properties of material with the microscopic mechanical properties of discrete structure and the corresponding relationship between micro and macro parameters is derived. The tensor quality of the energy density function, which contains coordinate vector, is mathematically proved. From the point of view of VMIB, the macroscopic nonlinear behaviors of material could be attributed to the evolution of virtual bond distribution density induced by the imposed deformation. With this theoretical hypothesis, as an application example, a uniaxial compressive failure of brittle material is simulated. Good agreement between the experimental results and the simulated ones is found.
KEY WORDS virtual multi-dimensional internal bond, material property dimensionality, multiscale modeling, molecular dynamics, virtual internal bond
I. INTRODUCTION Material exhibits its properties differently in different size-scales. It appears to be discrete in microscopic properties while continuous in macroscopic. The continuum mechanics theory is a theory system based on the hypothesis of the continuous-field. With this hypothesis, continuum mechanics derives the macroscopic constitutive relationship of solid materials at the continuous level without considering the freedom of each ‘atom’ below the continuous level. The continuum mechanics has been a dominant theory for analyzing mechanical engineering problems. However, the continuous-field hypothesis cannot always be satisfied in the failure process of materials. For example, a material would transform from the continuous medium at the initial state to the discontinuous medium during the failure process, where the continuous-field would be violated. Consequently, the constitutive relationship previously
Corresponding author. E-mail: [email protected], [email protected] Project supported by the National Basic Research Program of China (973 Project) (No. 2002CB412704).
Vol. 19, No. 3
Zhang Zhennan et al.: A Multiscale Mechanical Model of Materials
· 197 ·
derived from the continuous-field hypothesis will not hold. So, there are some limitations to the continuum mechanics theory in analyzing the failure problems. If the macroscopic mechanical properties of material are not derived from the continuous level with continuous-field hypothesis, but from the microscopic discrete level, the failure problem of materials could be better addressed. For example, the molecular dynamics method, which can simulate the behaviors of material at the atom level through atom potential. Undoubtedly, molecular dynamics is an effective method of studying small-scale failure problems. However, this method usually involves numerous atoms, leading to an extremely large amount of internal freedom, occupying enormous storing space and consuming increditably plentiful CPU computing time, even making the problem computationally unsolvable for large-scale problems. As for reducing internal freedom, continuum mechanics is superior. It determines the freedom of each ‘atom’ by a field function with the continuous-field hypothesis. If the molecular dynamic method could be combined with continuum mechanics, it would be more valuable. To simulate the fracture behaviors of isotropic solids, Gao and Klein[1] proposed the virtual internal bond (VIB) model, which considers solid material as consisting of micro-particles with mass. These particles are connected with virtual bonds. According to the Cauchy-Born rule[2−−5] , the location of each particle is associated with the deformation gradient. Based on the hypo-elastic theory, the macroscopic constitutive relationship is derived from the cohesive law between particles. Inheriting the advantages of the molecular dynamics method and the continuum mechanics method, the VIB model with its great advantages in simulating fracture behaviors, provides a good idea for multiscale modeling of materials. Later, Zhang and Ge[6] assigned a shear stiffness to the normal bond of VIB and derived a corresponding relationship between bond stiffness (shear and normal stiffness) and the macro material constants. The tensile failure behaviors are analyzed with this modified VIB. Different from Refs.[6,7] introduced a shear bond to the VIB and re-derived the corresponding relationship between bond stiffness and material constants. In Refs.[6,7], the theory of virtual multi-dimensional internal bond (VMIB) has been preliminarily established. However, some theoretical problems are still to be mathematically proved. For instance, in Ref.[7], as the energy density function contains coordinate vectors, the tensor quality of the energy density function has to be further strictly proved. Moreover, we discussed the VMIB from the point of view of material property dimensionality and then applied VMIB to the simulation of uniaxial compressive failure.
II. MATERIAL PROPERTY DIMENSIONALITY Only two parameters are needed to describe the mechanical properties of isotropic linear elastic solids on microscopic scale, i.e. the Young’s modulus and the Poisson’s ratio. They can be treated as two dimensionalities of material properties. These two parameters are necessary and sufficient macroscopically for the isotropic linear elastic solids. Correspondingly, two micro parameters are needed to determine the mechanical properties of material if the macro mechanical properties of material are derived from the micro discrete level for the isotropic linear elastic solids. In the molecular dynamics method, the interaction between atoms is usually described by the so-called atom potential, which cannot reflect the detailed interaction mechanism of atoms. Those potentials which are only used to describe the radial interaction of pairwise atoms actually only describe the mechanical properties in one dimension, i.e. the radial interaction. If the radial interaction between two atoms were degraded to the case of linear elastic interaction, the interaction of atoms can be described with a parameter called ‘bond stiffness’. Consequently, there is only one characteristic dimension in microscopic parameter, i.e. the bond stiffness. As it is, there are two characteristic dimension in macroscopic parameter, i.e. the Poisson’s ratio and the Young’s modulus, which are necessary and sufficient. Only one characteristic dimension in microscopic is not sufficient to determine the mechanical properties of an isotropic linear elastic solid. Therefore, another characteristic dimension of material in microscopic parameter is needed. For a pairwise atom, as shown in Fig.1, there also exists relative rotation freedom β besides the relative radial displacement . To restrict the rotation freedom, a shear bond is introduced, as shown in Fig.2. The mechanical property of the shear bond can be treated as another microscopic characteristic dimension. If the characteristic dimensions constructed by the normal and shear bond stiffness were sufficient, there should be a corresponding relationship between the two bond stiffness and the material constants (Young’s modulus and Poisson’s ratio). Fortunately, there does exist a corresponding relationship between bond stiffness and material constants, suggested as Refs.[7,8]. So, it is reasonable to
· 198 ·
ACTA MECHANICA SOLIDA SINICA
Fig. 1. Decomposition of pairwise particle displacement ( L-bond is the normal bond).
2006
Fig. 2. Illustration of shear bond ( R-bond is shear bond; L-bond is normal bond).
employ the two bond stiffnesses (the normal and shear bond stiffness) as the characteristic dimension of material in microscopic properties. In VMIB, the interaction between atoms is described by the bond stiffness, however, in the molecular dynamics method, the interaction between atoms is described by the atom potential. This is the biggest difference between VMIB and molecular dynamics in interaction description. In the molecular dynamics method, many-body potentials, e.g. Tersoff potential[9] and StillingerWeber potential[10] , are also extensively employed besides the two-body potential. The many-body potential is dependent on both the relative displacement of atoms and the included angles between bonds. Owing to the motion of atoms, the included angle between bonds varies from time to time. To compute the included angle between bonds, besides the considered pairwise atom connected with a bond another atom is needed as the reference. It is the fact that the many-body potential computes the bond included angle in a local coordinate system. In VMIB, the strain energy potential is also related to the angles between bonds, but the approach to accounting for the bond angle contribution is different. The VMIB takes the bond rotation angle relative to the general coordinate system as a parameter of energy potential. Though it doesn’t explicitly account for the included angles between bonds, it actually implicitly considers the contribution of the bond angle by the bond rotation relative to the general coordinate system. Moreover, the magnitude of contribution of the bond rotation angle is determined by the stiffness of shear bond.
III. ESTABLISHMENT AND PROOF OF VMIB According to Cauchy-Born rule[2−−5] , the particle moves to where the deformation gradient requires. So, the strain energy stored in the normal bond shown in Fig.2 is: UL =
1 2 k (0 ξi εij ξj ) 2
Fig. 3. Illustration of L-bond in sphere coordinate.
(1)
Vol. 19, No. 3
Zhang Zhennan et al.: A Multiscale Mechanical Model of Materials
· 199 ·
where k is the stiffness of normal bond; ξ = sin θ cos φ, sin θ sin φ, cos θ is the unit orientation vector in the sphere coordinate system shown in Fig.3; εij the strain tensor; 0 the original length of normal bond, here let 0 be unit[11] . The strain energy stored in the shear bond is: 2 2 2 1 1 1 UR1 = r ξi εij ηj , UR2 = r ξi εij ηj , UR3 = r ξi εij ηj (2) 2 2 2 where r is the stiffness of shear bond; URi (i = 1, 2, 3) the strain energy stored in the shear bond related to xi coordinate; η = ξ × x 1 × ξ , η = ξ × x 2 × ξ , η = ξ × x 3 × ξ . By Eqs.(1) and (2), the total strain energy stored in a pairwise particle IJ is:
WIJ = UL + UR1 + UR2 + UR3 2 1 2 1 1 1 (3) = k (εij ξi ξj )2 + r εij ξi ηj + r εij ξi ηj + r εij ξi ηj 2 2 2 2 As the parameter η in Eq.(3) contains coordinate vectors, it is necessary to prove the tensor quality 2 of Eq.(3). It is obvious that the part of the normal bond, i.e. k (εij ξi ξj ) /2, possesses tensor quality. The proof for the shear bond part of Eq.(3) is as follows: Proof: As shown in Fig.4, let ε and ξ respectively denote the stress tensor and orientation vector in a coordinate system x1 x2 x3 ; the energy of an arbitrary shear bond is URi = 0.5r (εij ξi ηj )2 ; the unit vectors corresponding to three coordinate axes are X , X and X , respectively. Correspondingly, let ε¯ and ξ¯ respectively denote the stress tensor and vector in another orientation ¯1 x ¯2 x ¯3 ; the energy of an arcoordinate system x ¯Ri = 0.5r ε¯ij ξ¯i η¯j 2 ; the unit bitrary shear bond is U Fig. 4 A pairwise particle IJ in different coordinate sysvectors corresponding to three coordinate axes are tem. X¯ , X¯ and X¯ , respectively. 2 Take UR1 = 0.5r εij ξi ηj as an example: η = ξ × (X × ξ) = (ξ · ξ) X − (ξ · X ) ξ εij ξi ηj = εij ξi (ξ · ξ) Xj − (ξ · X ) ξj = (ξ · ξ) εij ξi Xj − (ξ · X ) εij ξi ξj ε¯ij ξ¯i η¯ = ε¯ij ξ¯i ξ¯ · ξ¯ X¯ j − ξ¯ · X¯ ξ¯j = ξ¯ · ξ¯ ε¯ij ξ¯i X¯ j − ξ¯ · X¯ ε¯ij ξ¯i ξ¯j j
The transformation relationship between two coordinate systems is: ¯ = Qij X , ξ¯i = Qij ξj , ε¯ij = Qim Qjn εmn X i j
(4) (5) (6) (7)
where Qij is the transformation tensor. By substitute Eq.(7) into Eq.(6), the following can be obtained: ε¯ij ξ¯i η¯j = ε¯ij ξ¯i ξ¯ · ξ¯ X¯ j − ξ¯ · X¯ ξ¯j = Qim Qjn εmn Qip ξp ξ¯ · ξ¯ Qjq Xq − (Qkf ξf Qkl Xl ) Qjs ξs = ξ¯ · ξ¯ ξm Qjn εmn Qjq X − (ξ · X ) ξm Qjn εmn Qjs ξs q
= (ξ · ξ) εmn ξm Xn − (ξ · X ) εmn ξm ξn Comparing Eq.(8) with Eq.(5) yields ε¯ij ξ¯i η¯j = εij ξi ηj . ¯R1 By Eq.(2), UR1 = U ¯R2 , UR3 = U ¯R3 . By the same way: UR2 = U So, the energy expression Eq.(3) possesses tensor quality. By Eqs.(1) and (2), the strain energy stored in a unit volume is: 2π π 2π π W = UL DL (θ, φ) sin θdθdφ + UR1 DR1 (θ, φ) sin θdθdφ 0
0 2π π
+ 0
0
0
UR2 DR2 (θ, φ) sin θdθdφ +
0 2π
0
0
π
UR3 DR3 (θ, φ) sin θdθdφ
(8)
(9)
· 200 ·
ACTA MECHANICA SOLIDA SINICA
2006
where DL (θ, φ) is the distribution function of the normal bond; DL (θ, φ)sin(θ)dθdφ is the number of normal bonds in the unit volume between θ, θ + dθ and φ, φ + dφ in the sphere coordinate system; DRi (θ, φ) (i = 1, 2, 3) is the distribution function of the shear bond related to xi coordinate. Based on the hypoelastic theory[12, 13] , the elastic tensor can be obtained from Eq.(9): 2π π ∂W Cijmn = = kξi ξj ξm ξn DL (θ, φ) sin θdθdφ ∂εij ∂εmn 0 0 2π π rξi ηj ξm ηn DR1 (θ, φ) sin θdθdφ + 0
2π
0
π
+ 0
2π
0
+ 0
0
π
rξi ηj ξm ηn DR2 (θ, φ) sin θdθdφ rξi ηj ξm ηn DR3 (θ, φ) sin θdθdφ
(10)
Let the distribution function be unit, i.e. DL (θ, φ) = DR1 (θ, φ) = DR2 (θ, φ) = DR3 (θ, φ) = 1, then the corresponding relationship[7, 8] between bond stiffness and material constants can be derived as: k=
3E , 4π (1 − 2ν)
r=
3 (1 − 4ν) E 4π (1 + ν) (1 − 2ν)
(11)
where E and ν are the Young’s modulus and the Poisson’s ratio of material, respectively. From Eq.(11), it can be found that on the left-hand side of equation is the bond stiffness while at the right-hand side is the expression of material constants, which suggests that the macro material constant is determined by the mechanical properties of virtual bonds on microscopic scale. The damage evolution process of material on the macroscopic scale can be considered to result from the internal bond evolution with the external factor, e.g. imposed stress, etc.. Zhang and Ge[6, 7] have applied VMIB to the analysis and simulation of uniaxial tensile failure of brittle materials and the result is inspiring.
IV. APPLICATION EXAMPLE As a typical application of the VMIB model, here the VMIB model is used to analyze the uniaxial compressive failure of brittle materials. Usually the shear failure is dominant in the uniaxial compressive failure for brittle materials. From the point of view of VMIB, the bond distribution density would attenuate with the shear deformation increasing and microcrack initiating, as shown in Fig.5. The shear deformation vertical to ξ can be described with the rotation angle of normal bond in direction ξ. To quantitatively describe the relationship between the bond distribution density and bond
Fig. 5. Illustration of local shear break and bond density reduction.
Vol. 19, No. 3
Zhang Zhennan et al.: A Multiscale Mechanical Model of Materials
· 201 ·
deformation, the following phenomenological relationship is proposed: nc1
nc2
− 2β exp −c2 DL (θ, φ) = exp −c1 εc
(1 + ν) εc nc1 (12) 2β DR (θ, φ) = exp −c1 (1 + ν) εc i where β = max β1, β2, β3 , βi = ξm εmn ηn ; εc is the uniaxial compressive strain strength; = ξi εij ξj ; c1 , c2 , nc1 , nc2 are model parameters, which govern the characteristics of complete stress-strain curves. Different material corresponds to different set of model parameters. Substitute Eq.(12) into Eq.(10), the macro constitutive relationship containing damage evolution information will be obtained and then can use it to simulate the uniaxial compressive test of concrete in Ref.[14]. Both the experimental data and the parameters of experimental samples in this paper are taken from Ref.[14], which are listed in Table 1. The experimental and simulated results are shown in Fig.6. Table 1. The experimental sample parameters and the corresponding model parameters
Sample number HS06 HS10 HS15
E (GPa) 40.68 59.99 64.12
Sample parameters ν εc (×10−3 ) 0.16 1.9 0.15 2.2 0.15 2.0
fc (MPa) 47.23 71.09 107.29
Model parameters c1 c2 nc1 nc2 1.05 0.067 2.0 2.0 1.2 0.2 2.0 3.0 0.2 0.3 2.0 10.0
Fig. 6. Comparison between the predicted and the experimental results (Experiment results from Ref.[14], HS15, HS10 and HS06 are the experimental sample numbers).
From Fig.6, it can be found that there is good agreement between the experimental and predicted results, showing that VMIB is applicable to simulating the uniaxial compressive failure of brittle material. To apply VMIB to more extensive practical engineering, further researches are needed.
V. CONCLUSION By discussion on the characteristic dimension of material, it is concluded that two micro characteristic dimensions are needed to describe the mechanical properties of isotropic linear elastic solids. To supply another characteristic dimension, a shear bond is introduced into the original VIB model, which is used to restrict the relative rotation freedom of a pairwise particle. The normal bond and the shear bond construct the micro characteristic dimension of material. There exists a corresponding relationship between the bond stiffness and the material constants. The theoretical strictness of VMIB has been further mathematically proved. The macro behaviors of material can be attributed to the evolution of micro virtual bond distribution density subjected to imposed deformation. With this hypothesis, a uniaxial compressive failure of brittle material has been simulated in this paper, showing the advantages of VMIB in simulating the failure process of materials.
· 202 ·
ACTA MECHANICA SOLIDA SINICA
2006
Acknowledgement The authors would like to show their appreciation to Prof. Feng X.Q. for his valuable comments and advice on the present work.
References [1] Gao,H.J. and Klein,P., Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bond. Journal of the Mechanics and Physics of Solids, 1998, Vol.46, No.2: 187-218. [2] Tadmor,E.B., Ortiz,M. and Phillips,R., Quasicontinuum analysis of defects in solids. Phil. Mag. A, 1996, Vol.73, No.6: 1529-1593. [3] Milstein,F., Review: Theoretical elastic behavior at large strains. J. Mat. Sci., 1980, Vol.15: 1071-1084. [4] Milstein,F., Crystal elasticity. Hopkins,H.G. and Sewell,M.J. ed, //Mechanics of Solidsedit. Oxford: Pergamon Press, 1982. [5] Ericksen,J.L., Phase Transformations and Material Instabilities in Solids. Gurtin,M. ed. New York: Academic Press, 1984. [6] Zhang,Z.N. and Ge,X.R., A new quasi-continuum constitutive model for crack growth in an isotropic solid. European Journal of Mechanics: A/Solids, 2005, Vol.24, No.2: 243-252. [7] Zhang,Z.N. and Ge,X.R., Micromechanical consideration of tensile crack behavior based on virtual internal bond in contrast to cohesive stress. Theoretical and Applied Fracture Mechanics, 2005, Vol.43, No.3: 342-359. [8] Zhang,Z.N., Virtual multi-dimensional internal bond model (VMIB) and its application to the failure of brittle material and fibre-reinforced composite. Ph.D. Thesis, Shanghai Jiaotong University, 2005 (in Chinese). [9] Tersoff,J., New empirical model for the structural properties of silicon. Phys. Rev. Lett., 1986, Vol.56: 632-635. [10] Stillinger,F.H. and Weber,T.A., Computer simulation of local order in condensed phases of silicon. Phys. Rev. B, 1985, 31: 5262-5271. [11] Thiagarajan,G., Hsia,K.J. and Huang,Y.G., Finite element implementation of virtual internal spring model for simulating crack behavior. Engineering Fracture Mechanics, 2004, Vol.71, No.3: 401-423. [12] Ogden,R.W., Nonlinear Elastic Deformations. New York: John Wiley and Sons, 1984. [13] Marsden,J.E. and Hughes,T.J.R., Mathematical Foundations of Elasticity. Englewood Cliffs, NJ: PrenticeHall, 1983. [14] Li,Q.B. and Farhad,A., Mechanics of damage and constitutive relationships for high-strength concrete in triaxial compression. Journal of Engineering Mechanics, 1999, Vol.125, No.1: 1-10.
ISSN 0894-9166
A MULTISCALE MECHANICAL MODEL FOR MATERIALS BASED ON VIRTUAL INTERNAL BOND THEORY Zhang Zhennan1,2
Ge Xiurun3,4
Li Yonghe1
1
( Department of Civil Engineering, Shanghai University, Shanghai 200072, China) (2 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China) (3 Institute of Geotechnical Engineering, Shanghai Jiaotong University, Shanghai 200030, China) (4 Institute of Rock and Soil Mechanics, the Chinese Academy of Sciences, Wuhan 430071, China)
Received 29 December 2005; revision received 25 May 2006
ABSTRACT Only two macroscopic parameters are needed to describe the mechanical properties of linear elastic solids, i.e. the Poisson’s ratio and Young’s modulus. Correspondingly, there should be two microscopic parameters to determine the mechanical properties of material if the macroscopic mechanical properties of linear elastic solids are derived from the microscopic level. Enlightened by this idea, a multiscale mechanical model for material, the virtual multi-dimensional internal bonds (VMIB) model, is proposed by incorporating a shear bond into the virtual internal bond (VIB) model. By this modification, the VMIB model associates the macro mechanical properties of material with the microscopic mechanical properties of discrete structure and the corresponding relationship between micro and macro parameters is derived. The tensor quality of the energy density function, which contains coordinate vector, is mathematically proved. From the point of view of VMIB, the macroscopic nonlinear behaviors of material could be attributed to the evolution of virtual bond distribution density induced by the imposed deformation. With this theoretical hypothesis, as an application example, a uniaxial compressive failure of brittle material is simulated. Good agreement between the experimental results and the simulated ones is found.
KEY WORDS virtual multi-dimensional internal bond, material property dimensionality, multiscale modeling, molecular dynamics, virtual internal bond
I. INTRODUCTION Material exhibits its properties differently in different size-scales. It appears to be discrete in microscopic properties while continuous in macroscopic. The continuum mechanics theory is a theory system based on the hypothesis of the continuous-field. With this hypothesis, continuum mechanics derives the macroscopic constitutive relationship of solid materials at the continuous level without considering the freedom of each ‘atom’ below the continuous level. The continuum mechanics has been a dominant theory for analyzing mechanical engineering problems. However, the continuous-field hypothesis cannot always be satisfied in the failure process of materials. For example, a material would transform from the continuous medium at the initial state to the discontinuous medium during the failure process, where the continuous-field would be violated. Consequently, the constitutive relationship previously
Corresponding author. E-mail: [email protected], [email protected] Project supported by the National Basic Research Program of China (973 Project) (No. 2002CB412704).
Vol. 19, No. 3
Zhang Zhennan et al.: A Multiscale Mechanical Model of Materials
· 197 ·
derived from the continuous-field hypothesis will not hold. So, there are some limitations to the continuum mechanics theory in analyzing the failure problems. If the macroscopic mechanical properties of material are not derived from the continuous level with continuous-field hypothesis, but from the microscopic discrete level, the failure problem of materials could be better addressed. For example, the molecular dynamics method, which can simulate the behaviors of material at the atom level through atom potential. Undoubtedly, molecular dynamics is an effective method of studying small-scale failure problems. However, this method usually involves numerous atoms, leading to an extremely large amount of internal freedom, occupying enormous storing space and consuming increditably plentiful CPU computing time, even making the problem computationally unsolvable for large-scale problems. As for reducing internal freedom, continuum mechanics is superior. It determines the freedom of each ‘atom’ by a field function with the continuous-field hypothesis. If the molecular dynamic method could be combined with continuum mechanics, it would be more valuable. To simulate the fracture behaviors of isotropic solids, Gao and Klein[1] proposed the virtual internal bond (VIB) model, which considers solid material as consisting of micro-particles with mass. These particles are connected with virtual bonds. According to the Cauchy-Born rule[2−−5] , the location of each particle is associated with the deformation gradient. Based on the hypo-elastic theory, the macroscopic constitutive relationship is derived from the cohesive law between particles. Inheriting the advantages of the molecular dynamics method and the continuum mechanics method, the VIB model with its great advantages in simulating fracture behaviors, provides a good idea for multiscale modeling of materials. Later, Zhang and Ge[6] assigned a shear stiffness to the normal bond of VIB and derived a corresponding relationship between bond stiffness (shear and normal stiffness) and the macro material constants. The tensile failure behaviors are analyzed with this modified VIB. Different from Refs.[6,7] introduced a shear bond to the VIB and re-derived the corresponding relationship between bond stiffness and material constants. In Refs.[6,7], the theory of virtual multi-dimensional internal bond (VMIB) has been preliminarily established. However, some theoretical problems are still to be mathematically proved. For instance, in Ref.[7], as the energy density function contains coordinate vectors, the tensor quality of the energy density function has to be further strictly proved. Moreover, we discussed the VMIB from the point of view of material property dimensionality and then applied VMIB to the simulation of uniaxial compressive failure.
II. MATERIAL PROPERTY DIMENSIONALITY Only two parameters are needed to describe the mechanical properties of isotropic linear elastic solids on microscopic scale, i.e. the Young’s modulus and the Poisson’s ratio. They can be treated as two dimensionalities of material properties. These two parameters are necessary and sufficient macroscopically for the isotropic linear elastic solids. Correspondingly, two micro parameters are needed to determine the mechanical properties of material if the macro mechanical properties of material are derived from the micro discrete level for the isotropic linear elastic solids. In the molecular dynamics method, the interaction between atoms is usually described by the so-called atom potential, which cannot reflect the detailed interaction mechanism of atoms. Those potentials which are only used to describe the radial interaction of pairwise atoms actually only describe the mechanical properties in one dimension, i.e. the radial interaction. If the radial interaction between two atoms were degraded to the case of linear elastic interaction, the interaction of atoms can be described with a parameter called ‘bond stiffness’. Consequently, there is only one characteristic dimension in microscopic parameter, i.e. the bond stiffness. As it is, there are two characteristic dimension in macroscopic parameter, i.e. the Poisson’s ratio and the Young’s modulus, which are necessary and sufficient. Only one characteristic dimension in microscopic is not sufficient to determine the mechanical properties of an isotropic linear elastic solid. Therefore, another characteristic dimension of material in microscopic parameter is needed. For a pairwise atom, as shown in Fig.1, there also exists relative rotation freedom β besides the relative radial displacement . To restrict the rotation freedom, a shear bond is introduced, as shown in Fig.2. The mechanical property of the shear bond can be treated as another microscopic characteristic dimension. If the characteristic dimensions constructed by the normal and shear bond stiffness were sufficient, there should be a corresponding relationship between the two bond stiffness and the material constants (Young’s modulus and Poisson’s ratio). Fortunately, there does exist a corresponding relationship between bond stiffness and material constants, suggested as Refs.[7,8]. So, it is reasonable to
· 198 ·
ACTA MECHANICA SOLIDA SINICA
Fig. 1. Decomposition of pairwise particle displacement ( L-bond is the normal bond).
2006
Fig. 2. Illustration of shear bond ( R-bond is shear bond; L-bond is normal bond).
employ the two bond stiffnesses (the normal and shear bond stiffness) as the characteristic dimension of material in microscopic properties. In VMIB, the interaction between atoms is described by the bond stiffness, however, in the molecular dynamics method, the interaction between atoms is described by the atom potential. This is the biggest difference between VMIB and molecular dynamics in interaction description. In the molecular dynamics method, many-body potentials, e.g. Tersoff potential[9] and StillingerWeber potential[10] , are also extensively employed besides the two-body potential. The many-body potential is dependent on both the relative displacement of atoms and the included angles between bonds. Owing to the motion of atoms, the included angle between bonds varies from time to time. To compute the included angle between bonds, besides the considered pairwise atom connected with a bond another atom is needed as the reference. It is the fact that the many-body potential computes the bond included angle in a local coordinate system. In VMIB, the strain energy potential is also related to the angles between bonds, but the approach to accounting for the bond angle contribution is different. The VMIB takes the bond rotation angle relative to the general coordinate system as a parameter of energy potential. Though it doesn’t explicitly account for the included angles between bonds, it actually implicitly considers the contribution of the bond angle by the bond rotation relative to the general coordinate system. Moreover, the magnitude of contribution of the bond rotation angle is determined by the stiffness of shear bond.
III. ESTABLISHMENT AND PROOF OF VMIB According to Cauchy-Born rule[2−−5] , the particle moves to where the deformation gradient requires. So, the strain energy stored in the normal bond shown in Fig.2 is: UL =
1 2 k (0 ξi εij ξj ) 2
Fig. 3. Illustration of L-bond in sphere coordinate.
(1)
Vol. 19, No. 3
Zhang Zhennan et al.: A Multiscale Mechanical Model of Materials
· 199 ·
where k is the stiffness of normal bond; ξ = sin θ cos φ, sin θ sin φ, cos θ is the unit orientation vector in the sphere coordinate system shown in Fig.3; εij the strain tensor; 0 the original length of normal bond, here let 0 be unit[11] . The strain energy stored in the shear bond is: 2 2 2 1 1 1 UR1 = r ξi εij ηj , UR2 = r ξi εij ηj , UR3 = r ξi εij ηj (2) 2 2 2 where r is the stiffness of shear bond; URi (i = 1, 2, 3) the strain energy stored in the shear bond related to xi coordinate; η = ξ × x 1 × ξ , η = ξ × x 2 × ξ , η = ξ × x 3 × ξ . By Eqs.(1) and (2), the total strain energy stored in a pairwise particle IJ is:
WIJ = UL + UR1 + UR2 + UR3 2 1 2 1 1 1 (3) = k (εij ξi ξj )2 + r εij ξi ηj + r εij ξi ηj + r εij ξi ηj 2 2 2 2 As the parameter η in Eq.(3) contains coordinate vectors, it is necessary to prove the tensor quality 2 of Eq.(3). It is obvious that the part of the normal bond, i.e. k (εij ξi ξj ) /2, possesses tensor quality. The proof for the shear bond part of Eq.(3) is as follows: Proof: As shown in Fig.4, let ε and ξ respectively denote the stress tensor and orientation vector in a coordinate system x1 x2 x3 ; the energy of an arbitrary shear bond is URi = 0.5r (εij ξi ηj )2 ; the unit vectors corresponding to three coordinate axes are X , X and X , respectively. Correspondingly, let ε¯ and ξ¯ respectively denote the stress tensor and vector in another orientation ¯1 x ¯2 x ¯3 ; the energy of an arcoordinate system x ¯Ri = 0.5r ε¯ij ξ¯i η¯j 2 ; the unit bitrary shear bond is U Fig. 4 A pairwise particle IJ in different coordinate sysvectors corresponding to three coordinate axes are tem. X¯ , X¯ and X¯ , respectively. 2 Take UR1 = 0.5r εij ξi ηj as an example: η = ξ × (X × ξ) = (ξ · ξ) X − (ξ · X ) ξ εij ξi ηj = εij ξi (ξ · ξ) Xj − (ξ · X ) ξj = (ξ · ξ) εij ξi Xj − (ξ · X ) εij ξi ξj ε¯ij ξ¯i η¯ = ε¯ij ξ¯i ξ¯ · ξ¯ X¯ j − ξ¯ · X¯ ξ¯j = ξ¯ · ξ¯ ε¯ij ξ¯i X¯ j − ξ¯ · X¯ ε¯ij ξ¯i ξ¯j j
The transformation relationship between two coordinate systems is: ¯ = Qij X , ξ¯i = Qij ξj , ε¯ij = Qim Qjn εmn X i j
(4) (5) (6) (7)
where Qij is the transformation tensor. By substitute Eq.(7) into Eq.(6), the following can be obtained: ε¯ij ξ¯i η¯j = ε¯ij ξ¯i ξ¯ · ξ¯ X¯ j − ξ¯ · X¯ ξ¯j = Qim Qjn εmn Qip ξp ξ¯ · ξ¯ Qjq Xq − (Qkf ξf Qkl Xl ) Qjs ξs = ξ¯ · ξ¯ ξm Qjn εmn Qjq X − (ξ · X ) ξm Qjn εmn Qjs ξs q
= (ξ · ξ) εmn ξm Xn − (ξ · X ) εmn ξm ξn Comparing Eq.(8) with Eq.(5) yields ε¯ij ξ¯i η¯j = εij ξi ηj . ¯R1 By Eq.(2), UR1 = U ¯R2 , UR3 = U ¯R3 . By the same way: UR2 = U So, the energy expression Eq.(3) possesses tensor quality. By Eqs.(1) and (2), the strain energy stored in a unit volume is: 2π π 2π π W = UL DL (θ, φ) sin θdθdφ + UR1 DR1 (θ, φ) sin θdθdφ 0
0 2π π
+ 0
0
0
UR2 DR2 (θ, φ) sin θdθdφ +
0 2π
0
0
π
UR3 DR3 (θ, φ) sin θdθdφ
(8)
(9)
· 200 ·
ACTA MECHANICA SOLIDA SINICA
2006
where DL (θ, φ) is the distribution function of the normal bond; DL (θ, φ)sin(θ)dθdφ is the number of normal bonds in the unit volume between θ, θ + dθ and φ, φ + dφ in the sphere coordinate system; DRi (θ, φ) (i = 1, 2, 3) is the distribution function of the shear bond related to xi coordinate. Based on the hypoelastic theory[12, 13] , the elastic tensor can be obtained from Eq.(9): 2π π ∂W Cijmn = = kξi ξj ξm ξn DL (θ, φ) sin θdθdφ ∂εij ∂εmn 0 0 2π π rξi ηj ξm ηn DR1 (θ, φ) sin θdθdφ + 0
2π
0
π
+ 0
2π
0
+ 0
0
π
rξi ηj ξm ηn DR2 (θ, φ) sin θdθdφ rξi ηj ξm ηn DR3 (θ, φ) sin θdθdφ
(10)
Let the distribution function be unit, i.e. DL (θ, φ) = DR1 (θ, φ) = DR2 (θ, φ) = DR3 (θ, φ) = 1, then the corresponding relationship[7, 8] between bond stiffness and material constants can be derived as: k=
3E , 4π (1 − 2ν)
r=
3 (1 − 4ν) E 4π (1 + ν) (1 − 2ν)
(11)
where E and ν are the Young’s modulus and the Poisson’s ratio of material, respectively. From Eq.(11), it can be found that on the left-hand side of equation is the bond stiffness while at the right-hand side is the expression of material constants, which suggests that the macro material constant is determined by the mechanical properties of virtual bonds on microscopic scale. The damage evolution process of material on the macroscopic scale can be considered to result from the internal bond evolution with the external factor, e.g. imposed stress, etc.. Zhang and Ge[6, 7] have applied VMIB to the analysis and simulation of uniaxial tensile failure of brittle materials and the result is inspiring.
IV. APPLICATION EXAMPLE As a typical application of the VMIB model, here the VMIB model is used to analyze the uniaxial compressive failure of brittle materials. Usually the shear failure is dominant in the uniaxial compressive failure for brittle materials. From the point of view of VMIB, the bond distribution density would attenuate with the shear deformation increasing and microcrack initiating, as shown in Fig.5. The shear deformation vertical to ξ can be described with the rotation angle of normal bond in direction ξ. To quantitatively describe the relationship between the bond distribution density and bond
Fig. 5. Illustration of local shear break and bond density reduction.
Vol. 19, No. 3
Zhang Zhennan et al.: A Multiscale Mechanical Model of Materials
· 201 ·
deformation, the following phenomenological relationship is proposed: nc1
nc2
− 2β exp −c2 DL (θ, φ) = exp −c1 εc
(1 + ν) εc nc1 (12) 2β DR (θ, φ) = exp −c1 (1 + ν) εc i where β = max β1, β2, β3 , βi = ξm εmn ηn ; εc is the uniaxial compressive strain strength; = ξi εij ξj ; c1 , c2 , nc1 , nc2 are model parameters, which govern the characteristics of complete stress-strain curves. Different material corresponds to different set of model parameters. Substitute Eq.(12) into Eq.(10), the macro constitutive relationship containing damage evolution information will be obtained and then can use it to simulate the uniaxial compressive test of concrete in Ref.[14]. Both the experimental data and the parameters of experimental samples in this paper are taken from Ref.[14], which are listed in Table 1. The experimental and simulated results are shown in Fig.6. Table 1. The experimental sample parameters and the corresponding model parameters
Sample number HS06 HS10 HS15
E (GPa) 40.68 59.99 64.12
Sample parameters ν εc (×10−3 ) 0.16 1.9 0.15 2.2 0.15 2.0
fc (MPa) 47.23 71.09 107.29
Model parameters c1 c2 nc1 nc2 1.05 0.067 2.0 2.0 1.2 0.2 2.0 3.0 0.2 0.3 2.0 10.0
Fig. 6. Comparison between the predicted and the experimental results (Experiment results from Ref.[14], HS15, HS10 and HS06 are the experimental sample numbers).
From Fig.6, it can be found that there is good agreement between the experimental and predicted results, showing that VMIB is applicable to simulating the uniaxial compressive failure of brittle material. To apply VMIB to more extensive practical engineering, further researches are needed.
V. CONCLUSION By discussion on the characteristic dimension of material, it is concluded that two micro characteristic dimensions are needed to describe the mechanical properties of isotropic linear elastic solids. To supply another characteristic dimension, a shear bond is introduced into the original VIB model, which is used to restrict the relative rotation freedom of a pairwise particle. The normal bond and the shear bond construct the micro characteristic dimension of material. There exists a corresponding relationship between the bond stiffness and the material constants. The theoretical strictness of VMIB has been further mathematically proved. The macro behaviors of material can be attributed to the evolution of micro virtual bond distribution density subjected to imposed deformation. With this hypothesis, a uniaxial compressive failure of brittle material has been simulated in this paper, showing the advantages of VMIB in simulating the failure process of materials.
· 202 ·
ACTA MECHANICA SOLIDA SINICA
2006
Acknowledgement The authors would like to show their appreciation to Prof. Feng X.Q. for his valuable comments and advice on the present work.
References [1] Gao,H.J. and Klein,P., Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bond. Journal of the Mechanics and Physics of Solids, 1998, Vol.46, No.2: 187-218. [2] Tadmor,E.B., Ortiz,M. and Phillips,R., Quasicontinuum analysis of defects in solids. Phil. Mag. A, 1996, Vol.73, No.6: 1529-1593. [3] Milstein,F., Review: Theoretical elastic behavior at large strains. J. Mat. Sci., 1980, Vol.15: 1071-1084. [4] Milstein,F., Crystal elasticity. Hopkins,H.G. and Sewell,M.J. ed, //Mechanics of Solidsedit. Oxford: Pergamon Press, 1982. [5] Ericksen,J.L., Phase Transformations and Material Instabilities in Solids. Gurtin,M. ed. New York: Academic Press, 1984. [6] Zhang,Z.N. and Ge,X.R., A new quasi-continuum constitutive model for crack growth in an isotropic solid. European Journal of Mechanics: A/Solids, 2005, Vol.24, No.2: 243-252. [7] Zhang,Z.N. and Ge,X.R., Micromechanical consideration of tensile crack behavior based on virtual internal bond in contrast to cohesive stress. Theoretical and Applied Fracture Mechanics, 2005, Vol.43, No.3: 342-359. [8] Zhang,Z.N., Virtual multi-dimensional internal bond model (VMIB) and its application to the failure of brittle material and fibre-reinforced composite. Ph.D. Thesis, Shanghai Jiaotong University, 2005 (in Chinese). [9] Tersoff,J., New empirical model for the structural properties of silicon. Phys. Rev. Lett., 1986, Vol.56: 632-635. [10] Stillinger,F.H. and Weber,T.A., Computer simulation of local order in condensed phases of silicon. Phys. Rev. B, 1985, 31: 5262-5271. [11] Thiagarajan,G., Hsia,K.J. and Huang,Y.G., Finite element implementation of virtual internal spring model for simulating crack behavior. Engineering Fracture Mechanics, 2004, Vol.71, No.3: 401-423. [12] Ogden,R.W., Nonlinear Elastic Deformations. New York: John Wiley and Sons, 1984. [13] Marsden,J.E. and Hughes,T.J.R., Mathematical Foundations of Elasticity. Englewood Cliffs, NJ: PrenticeHall, 1983. [14] Li,Q.B. and Farhad,A., Mechanics of damage and constitutive relationships for high-strength concrete in triaxial compression. Journal of Engineering Mechanics, 1999, Vol.125, No.1: 1-10.