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Thermodynamically consistent nonlocal theory of ductile damage
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 31 (2004) 355–363 www.elsevier.com/locate/mechrescom
Thermodynamically consistent nonlocal theory of ductile damage H. Stumpf a
a,*
, J. Makowski a, J. Gorski b, K. Hackl
a
Lehrstuhl f € ur Allgemeine Mechanik, IA 3/1126, Ruhr-Universit€at Bochum, Universit€atsstrasse 150, D-44780 Bochum, Germany b Lehrstuhl f €ur Grundbau und Bodenmechanik, Ruhr-Universit€at Bochum, D-44780 Bochum, Germany Received 20 November 2003
Abstract In this paper a thermodynamically consistent, weakly nonlocal theory of ductile damage is presented. The theory is based on the classical dynamical balance laws of forces and couples in the physical space and dynamical balance laws of material forces on evolving defects and on the first and second law of thermodynamics formulated for physical and material space. Assuming general constitutive equations their frame-invariant and thermodynamically admissible form is determined. It is shown that physical and material forces and stresses consist of two parts, a nondissipative part derivable from a free energy potential, and a dissipative part, which can be obtained from a dissipation pseudopotential, if such a pseudo-potential exists. The theory can be considered as a framework with gradient elastoplasticity, isotropic and anisotropic brittle and ductile gradient damage at finite strain as special cases. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Microstructure; Nonlocal damage; Brittle damage; Ductile damage; Gradient–elastoplasticity; Finite elastoplasticity; Size effects
1. Introduction It is well-known that local theories of finite elastoplasticity and damage suffer from essential drawbacks: local theories are not able to simulate appropriately length-scale dependent problems (Bazant, 1991; Bazant and Pijaudier-Cabot, 1988), and in the case of strain and damage localization FE solutions based on local theories suffer from strong mesh-dependency (de Borst and M€ uhlhaus, 1992; de Borst et al., 1996). To overcome these difficulties various regularization techniques applying integral and gradient enhancements were proposed in the literature (M€ uhlhaus and Aifantis, 1991; Sluys et al., 1993; de Borst et al., 1995). During the last decade essential research efforts on the field of elastoplasticity and damage were devoted to the development of nonlocal theories. Nonlocal models of elastoplasticity were proposed in the literature by taking into account the dislocation density and torsion tensor, respectively, (e.g. Le and Stumpf, *
Corresponding author. Tel.: +49-32-2-7385; fax: +49-234-32-14154. E-mail address: [email protected] (H. Stumpf).
0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2003.11.012
356
H. Stumpf et al. / Mechanics Research Communications 31 (2004) 355–363
1996a,b; Shizawa and Zbib, 1999; Cleja-Tigoiu, 2002a,b) or torsion and curvature tensor (Saczuk et al., 2001; Rakotomanana, 2003), where the latter corresponds to the disclination density tensor. Gradient damage models were considered for isotropic brittle damage by Fremond and Nedjar (1996) and Steinmann (1999), for isotropic damage coupled with small strain plasticity by Nedjar (2001), and for anisotropic damage of viscoelastic materials by Stumpf and Hackl (2003). The aim of this paper is to develop a thermodynamically consistent objective gradient theory of brittle and ductile damage at finite strain based on balance laws of physical and material forces and first and second law of thermodynamics for physical and material space. Material forces on defects were first investigated by Leibfried (1949); Peach and Koehler (1950), and Eshelby (1951). The paper is organized as follows. In Section 2 the physical deformation gradient F, a symmetric Lagrangian damage tensor D and the plastic deformation ‘‘gradient’’ Fp , which is a locally defined linear map, and the temperature h are introduced as independent thermokinematical variables of the theory. In Section 3 the dynamical balance laws for physical and material forces and couples are formulated, where it is assumed that dynamically evolving defects as dislocations, the bearer of plastic deformations, microcracks and microvoids causing the material degradation, have inertia and kinetic energy. In Section 4 the first and second law of thermodynamics for motion in the physical space and defect and temperature evolution in the material space are formulated. In Section 5 general constitutive equations are assumed and their thermodynamically admissible and Lagrangian objective forms are determined. It is shown that they can be expressed by two potentials, the free energy density and a dissipation pseudo-potential, if the latter exists. Introducing the constitutive equations into the dynamical balance laws of physical and material forces, the nonlocal governing equations of dynamically deforming elastic–plastic structures with damage evolution are obtained. The general theory presented in this paper can be considered as a framework, which enables the derivation of various nonlocal and local theories of finite elastoplasticity and isotropic and anisotropic brittle and ductile damage by introducing simplifying assumptions.
2. Independent thermokinematical variables in physical and material space For an appropriate nonlocal damage analysis of engineering structures with brittle and ductile material behavior the thermomechanics of motion and defect evolution in physical and material space has to be formulated. As point of departure we have to decide about the independent thermokinematical variables to describe the motion of material points in the physical space and defect and temperature evolution in the material space. For continuous media the motion of material points can be defined by a vector-valued function, the actual position vector xðX; tÞ and the displacement vector uðX; tÞ, respectively, where X is the position vector in the reference configuration and t the time parameter, and for granular media by the actual position vector xðX; tÞ and a rotation tensor QðX; tÞ taking into account the rotational degrees of freedom. Less obvious is how to describe the evolution of defects as microcracks, microvoids and dislocations within the material space. The simplest assumption is that of isotropic elastic damage, where the damage evolution is defined by a scalar-valued function DðX; tÞ with 0 6 D 6 1, where D ¼ 0 denotes no damage and D ¼ 1 total damage. The physical interpretation may be that the scalar D represents the amount of microcracks and/or microvoids in a representative volume element. Since microdefects are orientation dependent, the simplest assumption of isotropic damage is in general not sufficient and has to be replaced by an anisotropic damage description using a tensor-valued function DðX; tÞ. In this paper we chose D as a symmetric tensor referred to the undeformed and homogeneous reference configuration. Most local theories of finite elastoplasticity proposed in the literature so far, are based on the multiplicative decomposition of the total deformation gradient F into elastic, Fe , and plastic, Fp , parts, F ¼ Fe Fp ,
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where Fe and Fp are locally defined linear maps and not gradients, because there exists no associated vector field. In this paper we do not use the multiplicative decomposition formula, instead, for a Lagrangian description, we define the motion in the physical space by the actual vector field xðX; tÞ and the physical deformation gradient F ¼ ox=oX, respectively, and the plastic evolution in the material space by the locally defined linear plastic map Fp ðX; tÞ. We have to point out that for poly-crystalline materials Fp may represent the plastic deformation based on dislocation motion, for other materials as e.g. concrete, rocks and soil, Fp may be interpreted as the irreversible deformation in the material space. For an appropriate description of the thermomechanical process of deformation and material degradation we have to introduce, besides F, D and Fp , also the temperature field hðX; tÞ as fourth independent thermokinematical variable. 3. Local balance laws of forces and couples in the physical-material space In local theories of damage and elastoplasticity published in the literature the kinematical variables D and Fp are considered as internal variables, for which evolution laws have to be given. Since numerical investigations have shown that FE solutions based on local theories exhibit a strong mesh-dependency, if localization in the material space occurs, and that they are not able to simulate appropriately sizedependent effects, we propose in this paper an alternative nonlocal and weakly nonlocal, respectively, approach based on balance laws for material forces and material stresses, respectively, acting on evolving defects. Besides the classical global balance laws of physical forces and physical couples, we formulate global balance laws of material forces in a representative volume element. Passing to the limit of a material point, the following local balance laws of physical and material forces and couples can be obtained: Balance of physical forces €; Div T þ b ¼ .0 x
ð3:1Þ
where for simplicity it is assumed that the referential mass density .0 ðXÞ is independent of time, Balance of physical couples TFT FTT ¼ 0
ð3:2Þ
with the denotations TðX; tÞ physical first Piola–Kirchhoff stress tensor bðX; tÞ external physical body force FðX; tÞ physical deformation gradient Balance of tensor-valued material forces associated with the evolution of defects as microcracks and microvoids _ ; DivHd Hd þ Gd ¼ .0 ðJd DÞ ð3:3Þ Balance of tensor-valued material forces associated with plastic evolution and dislocation motion, respectively, DivHp Tp þ Gp ¼ .0 ðJp F_ p Þ
with the denotations Hd ðX; tÞ symmetric second-order tensor of material force power-conjugate to the damage tensor D Hd ðX; tÞ third-order material stress tensor power-conjugate to rD
ð3:4Þ
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Gd ðX; tÞ external influence tensor representing e.g. chemical reactions breaking internal material bonds associated with defect evolution Jd ðX; tÞ inertia tensor associated with dynamically moving defects as microcracks and microvoids Tp ðX; tÞ second-order tensor of material force power-conjugate to Fp Hp ðX; tÞ third-order stress tensor power-conjugate to rFp Gp ðX; tÞ external influence tensor breaking internal material bonds associated with plastic evolution and dislocations motion, respectively Jp ðX; tÞ inertia tensor associated with dynamically moving dislocations. The inertia terms on the right-hand side of the balance laws of material forces (3.3) and (3.4) are due to the assumption that dynamically moving defects and dislocations have kinetic energy. Then the total kinetic energy j consists of two parts j ¼ jph þ jm ;
ð3:5Þ
where jph is the kinetic energy of moving material points in the physical space 1 jph ¼ .0 x_ x_ 2
ð3:6Þ
and jm is the kinetic energy due to moving defects as microcracks and microvoids, jd , and due to moving dislocations, jp , assumed in the form 1 _ D _ þ 1 .0 Jp F_ p F_ p : jm ¼ jd þ jp ¼ .0 Jd D 2 2
ð3:7Þ
In this paper, we do not present the boundary and initial conditions associated with the balance laws (3.1)–(3.4). These balance laws of forces and couples were derived under the assumption that there are no singular or discontinuous forces in the representative volume element. If such forces exist, corresponding singularity or jump conditions have to be added to (3.1)–(3.4).
4. First and second law of thermomechanics for motion in the physical-material space Besides the global and local, respectively, balance laws of forces and couples in the physical and material space, we have to formulate the global form of the first and second law of thermodynamics. Their localization leads to the local laws. Balance of energy e_ ¼ r þ r Div q;
ð4:1Þ
dissipation inequality g_ ðh1 r Divðh1 qÞÞ P 0 with the denotations eðX; tÞ rðX; tÞ rðX; tÞ qðX; tÞ gðX; tÞ
specific internal energy stress power external body heating referential heat flux vector entropy
ð4:2Þ
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It should be noted that in the form (4.2) the local dissipation inequality representing the second law of thermodynamics is independent of the laws of mechanics and the law of energy balance (4.1). Making use of (4.1) and introducing the free energy WðX; tÞ (measured per unit volume of the reference configuration) defined by W e hg;
ð4:3Þ
the dissipation inequality (4.2) can be rewritten in an equivalent form as _ gh_ h1 q rh P 0; DrW
ð4:4Þ
referred to as the reduced dissipation inequality. It represents the second law of thermodynamics under the assumption that the balance laws of physical and material forces and the balance of energy hold. Corresponding to (3.1)–(3.4) the stress power rðX; tÞ consists of two parts, r ¼ rph þ rm ;
ð4:5Þ
the stress power in the physical space, rph , _ rph ¼ T F;
ð4:6Þ
and the stress power in the material space, rm ðX; tÞ, decomposed as rm ¼ rd þ rp ;
ð4:7Þ
with the stress power due to defect evolution, _ þ Hd rD; _ rd ¼ Hd D
ð4:8Þ
and the stress power due to dislocation motion, rp ¼ Tp F_ p þ Hp rF_ p :
ð4:9Þ
On the right-hand side of (4.8) and (4.9), respectively, the first term corresponds to a local theory and the second term due to the nonlocal contribution.
5. Constitutive modeling 5.1. General form of the constitutive equations The independent thermokinematical variables of the theory presented in this paper are ðF; D; Fp ; hÞ, where F ¼ rx is the deformation gradient of the motion in the physical space, D the symmetric Lagrangian damage tensor, Fp the plastic deformation gradient, and h the absolute temperature. Thus, the variables ðF; D; Fp ; hÞ must be determined as solution of the problem defined by the field equations of Sections 3 and 4, supplemented by appropriate initial and boundary conditions and suitable constitutive equations. Let us denote with e the set of kinematical variables e ðF; D; rD; Fp ; rFp Þ
ð5:1Þ
and with e_ their rates _ D; _ rD; _ F_ p ; rF_ p Þ: e_ ðF;
ð5:2Þ
Then the set of physical and material stress tensors power-conjugate to (5.1) and (5.2), respectively, is r ¼ ðT; Hd ; Hd ; Tp ; Hp Þ;
ð5:3Þ
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which enables to write the stress power (4.5) in the compact form r ¼ r e_ :
ð5:4Þ
To formulate the constitutive equations, we have to express the free energy (4.3) and the stress tensors (5.3) together with the entropy g and the heat flux vector q as functions of e, e_ , h, and rh, b e_ ; h; rhÞ; W ¼ Wðe; ^ðe; e_ ; h; rhÞ; r¼r ^ðe; e_ ; h; rhÞ; g¼g ^ðe; e_ ; h; rhÞ: q¼q
ð5:5Þ
The general constitutive equations (5.5) have to satisfy the restrictions of thermodynamical admissibility and objectivity. 5.2. Thermodynamical admissibility The form of the constitutive equations (5.5) is restricted by the second law of thermodynamics. They have to satisfy the dissipation inequality (4.4). Introducing the stress power (5.4) with (5.3) and (5.2) into (4.4) leads to b e_ ; h; rhÞ e_ oh Wðe; b e_ ; h; rhÞ þ ^gðe; e_ ; h; rhÞ h_ ^ðe; e_ ; h; rhÞ oe Wðe; D¼ r b e_ ; h; rhÞ rh_ P 0: h1 ^ qðe; e_ ; h; rhÞ rh orh Wðe; ð5:6Þ From (5.6) the following constitutive restrictions are obtained b ¼ 0; oe_ W
b ¼ 0; orh W
ð5:7Þ
stating that the free energy function has the constitutive form b hÞ: W ¼ Wðe;
ð5:8Þ
Moreover, (5.6) leads to the constitutive equation for the entropy determined by the free energy, b hÞ: g¼^ gðe; hÞ ¼ oh Wðe; In view of (5.8) and (5.9) the dissipation inequality (5.6) reduces to the form b hÞ e_ h1 ^ ^ðe; h; e_ ; rhÞ oe Wðe; qðe; h; e_ ; rhÞ rh P 0: D¼ r
ð5:9Þ
ð5:10Þ
The inspection of inequality (5.10) leads to the result that the physical and material stresses (5.3) consist b hÞ, and a of two parts, a nondissipative part, which can be derived from the free energy potential, oe Wðe; dissipative part, which is indicated by a lower asterisk, b hÞ þ r ^ ðe; h; e_ ; rhÞ; ^ðe; h; e_ ; rhÞ ¼ oe Wðe; r¼r
ð5:11Þ
where the dissipative parts of the physical and material stresses (5.3) r ¼ ðT ; Hd ; Hd ; Tp ; Hp Þ
ð5:12Þ
have to satisfy the dissipation inequality (5.10). Introducing (5.11) into (5.10) leads to the entropy production inequality ^ðe; h; e_ ; rhÞ e_ h1 ^ D¼r qðe; h; e_ ; rhÞ rh P 0:
ð5:13Þ
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In (5.13) the first term is the entropy production of the driving forces in the physical and material space and the second term is the entropy production due to heat flux in the material space. The form of the entropy production inequality (5.13) suggests to assume the existence of a dissipation pseudo-potential U of the form U ¼ Uðe; h; e_ ; rhÞ;
ð5:14Þ
where e and h can be considered as parameters. If such a dissipation pseudo-potential exists, the dissipative driving stresses (5.12) and the heat flux vector q can be derived from U as r ¼ oe_ Uðe; h; e_ ; rhÞ; h1 q ¼ orh Uðe; h; e_ ; rhÞ:
ð5:15Þ
Introducing (5.15)1 into (5.11) leads to the constitutive equations expressed by two potentials, the free energy W and the dissipation pseudo-potential U, b hÞ þ oe_ Uðe; h; e_ ; rhÞ: r ¼ oe Wðe;
ð5:16Þ
Furthermore, with (5.15) the dissipation inequality (5.13) takes the form D ¼ oe_ Uðe; h; e_ ; rhÞ e_ þ orh Uðe; h; e_ ; rhÞ rh P 0;
ð5:17Þ
where the first term is the entropy production due to the dissipative driving forces in the physical-material space and the second term is the entropy production due to the heat flux in the material space. 5.3. Objectivity requirements Besides the postulate of thermodynamical admissibility, the constitutive equations have to be objective. The nonobjective set of kinematical variables (5.1), their rates (5.2) and the power-conjugate stresses (5.3) have to be replaced by their objective counterparts. In this paper, we restrict our considerations to the Lagrangian objective form of all variables. Their Eulerian objective form can then be obtained by pushforward of all variables to the actual configuration. 5.3.1. Lagrangian objective physical space variables Considering the stress power of the physical stresses (4.6), we can pull-back both power-conjugate variables to the undeformed reference configuration leading to 1 rph ¼ T F_ ¼ S C_ 2
ð5:18Þ
with the objective second Piola–Kirchhoff stress tensor S and the Lagrangian objective Cauchy–Green deformation tensor C defined by S ¼ F1 T;
C ¼ FT F:
ð5:19Þ
5.3.2. Lagrangian objective damage variables The damage tensor D is introduced as symmetric tensor referred to the homogeneous undeformed reference configuration and is therefore Lagrangian objective. Since r denotes the gradient operator with respect to the undeformed reference configuration, the term rD is Lagrangian objective as well. It follows that the power-conjugate stresses in (4.8), Hd and Hd , are Lagrangian objective.
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5.3.3. Lagrangian objective plastic deformation variables The plastic deformation ‘‘gradient’’ Fp ðX; tÞ is a locally defined linear map from the undeformed reference configuration without inhomogeneities into an inhomogenous state obtained by Fp ðX; tÞ. Therefore, Fp ðX; tÞ and the power-conjugate stresses Tp ðX; tÞ as well as rFp ðX; tÞ and the power-conjugate third-order material stress tensor Hp ðX; tÞ are not objective. In Le and Stumpf (1996a) the Lagrangian objective deformation measures Cp ¼ FpT Fp ;
Tp ¼ Fp1 rFp ;
ð5:20Þ
p
were introduced, where C can be considered as Lagrangian metric, and the skew-symmetric part of Tp is the so-called torsion tensor representing physically the dislocation density referred to the undeformed reference configuration. In Le and Stumpf (1996b) it was shown that from (5.20) the plastic rotation tensor Rp , following from the polar decomposition theorem, Fp ¼ Rp Up with Up the plastic stretch tensor, can be determined uniquely up to a rigid body rotation. Expressing in the plastic stress power (4.9) the rates F_ p and rF_ p by the objective Lagrangian rates C_ p and p T_ , respectively, the plastic stress power rp is obtained as 1 rp ¼ Sp C_ p þ Sp T_ p 2
ð5:21Þ
with the Lagrangian objective stress tensors Sp ¼ Fp1 Tp ;
Sp ¼ FpT Hp ;
ð5:22Þ
power-conjugate to C_ and T_ , respectively. In deriving (5.21) we assumed that a term of the form T HpT is small compared with Tp and can be neglected. The thermodynamically admissible objective constitutive equations are obtained by replacing in (5.16) the set e according to (5.1), the rates e_ according to (5.2) and the set r according to (5.3) by their objective forms: e ðC; D; rD; Cp ; Tp Þ; ð5:23Þ p
p
_ D; _ rD; _ C_ p ; T_ p Þ; e_ ðC; 1 1 S; Hd ; Hd ; Sp ; Sp : r 2 2
p
ð5:24Þ ð5:25Þ
6. Dynamic governing equations of deformation and defect and plastic evolution Introducing the thermodynamically admissible and objective constitutive equations derived above into the balance laws of physical and material forces and stresses, respectively, obtained in Section 3, the following dynamical governing equations of deformation in the physical space and defect and plastic evolution in the material space are obtained: €; DivðFSðe; h; e_ ; rhÞÞ þ b ¼ .0 x ð5:26Þ _ ; Div Hd ðe; h; e_ ; rhÞ Hd ðe; h; e_ ; rhÞ þ Gd ¼ .0 ðJd DÞ T Div ðFp1 Þ Sp ðe; h; e_ ; rhÞ Fp Sp ðe; h; e_ ; rhÞ þ Gp ¼ .0 ðJp F_ p Þ :
ð5:27Þ ð5:28Þ
These equations together with corresponding boundary and initial conditions define a weakly nonlocal and gradient, respectively, model of ductile damage.
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7. Conclusions The thermodynamically consistent and objective gradient theory of ductile damage presented in this paper can be considered as a framework, which enables the derivation of special theories by introducing simplifying assumptions including: consistent gradient theories of finite elastoplasticity, isotropic brittle damage, anisotropic brittle damage, isotropic ductile damage, anisotropic ductile damage, simplified gradient and local theories by assuming that for the problem under consideration some gradient terms can be neglected, while others have to be taken into account.
Acknowledgements The financial support, provided by Deutsche Forschungsgemeinschaft (DFG) under Grant SFB 398-A7, is gratefully acknowledged.
References Bazant, Z.P., Pijaudier-Cabot, G., 1988. J. Appl. Mech. 55, 287. Bazant, Z.P., 1991. Int. J. Eng. Mech. 117 (5), 1070. Cleja-Tigoiu, S., 2002a. Theor. Appl. Mech 28, 93. Cleja-Tigoiu, S., 2002b. Z. Angew. Math. Phys. 53, 996. de Borst, R., M€ uhlhaus, H.-B., 1992. Int. J. Numer. Meths Eng. 35, 521. de Borst, R., Pamin, J., Peerlings, R.H.J., Sluys, L.J., 1995. Comp. Mech. 17, 130. de Borst, R., Benallal, A., Heeres, O.M., 1996. J. Phys. IV C6 C6, 491. Eshelby, J.D., 1951. Philos. Trans. R. Soc. London A 244, 87. Fremond, M., Nedjar, B., 1996. Int. J. Solids Struct. 33, 1083. Le, K.C., Stumpf, H., 1996a. Int. J. Plast. 12, 611. Le, K.C., Stumpf, H., 1996b. Proc. R. Soc. Lond. A 425, 359. Leibfried, G., 1949. Z. Phys. 126, 781. M€ uhlhaus, H.-B., Aifantis, E.C., 1991. Int. J. Solids Struct. 28, 845. Nedjar, B., 2001. Int. J. Solids Struct. 38, 5421. Peach, M.O., Koehler, J.S., 1950. Phys. Rev. 80, 436. Rakotomanana, L.R., 2003. A geometric approach to thermo-mechanics of dissipative continua. Progress in Mathematical Physics Series, volume 31, Birkh€auser, Boston. Saczuk, J., Stumpf, H., Vallee, C., 2001. Int. J. Solids Struct. 38, 9545. Shizawa, K., Zbib, H.M., 1999. Int. J. Plast. 15, 899. Sluys, L.J., de Borst, R., M€ uhlhaus, H.-B., 1993. Int. J. Solids Struct. 30, 1153. Steinmann, P., 1999. Int. J. Numer. Meth. Eng. 46, 757. Stumpf, H., Hackl, K., 2003. Int. J. Solids Struct. 40, 1567.
Mechanics Research Communications 31 (2004) 355–363 www.elsevier.com/locate/mechrescom
Thermodynamically consistent nonlocal theory of ductile damage H. Stumpf a
a,*
, J. Makowski a, J. Gorski b, K. Hackl
a
Lehrstuhl f € ur Allgemeine Mechanik, IA 3/1126, Ruhr-Universit€at Bochum, Universit€atsstrasse 150, D-44780 Bochum, Germany b Lehrstuhl f €ur Grundbau und Bodenmechanik, Ruhr-Universit€at Bochum, D-44780 Bochum, Germany Received 20 November 2003
Abstract In this paper a thermodynamically consistent, weakly nonlocal theory of ductile damage is presented. The theory is based on the classical dynamical balance laws of forces and couples in the physical space and dynamical balance laws of material forces on evolving defects and on the first and second law of thermodynamics formulated for physical and material space. Assuming general constitutive equations their frame-invariant and thermodynamically admissible form is determined. It is shown that physical and material forces and stresses consist of two parts, a nondissipative part derivable from a free energy potential, and a dissipative part, which can be obtained from a dissipation pseudopotential, if such a pseudo-potential exists. The theory can be considered as a framework with gradient elastoplasticity, isotropic and anisotropic brittle and ductile gradient damage at finite strain as special cases. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Microstructure; Nonlocal damage; Brittle damage; Ductile damage; Gradient–elastoplasticity; Finite elastoplasticity; Size effects
1. Introduction It is well-known that local theories of finite elastoplasticity and damage suffer from essential drawbacks: local theories are not able to simulate appropriately length-scale dependent problems (Bazant, 1991; Bazant and Pijaudier-Cabot, 1988), and in the case of strain and damage localization FE solutions based on local theories suffer from strong mesh-dependency (de Borst and M€ uhlhaus, 1992; de Borst et al., 1996). To overcome these difficulties various regularization techniques applying integral and gradient enhancements were proposed in the literature (M€ uhlhaus and Aifantis, 1991; Sluys et al., 1993; de Borst et al., 1995). During the last decade essential research efforts on the field of elastoplasticity and damage were devoted to the development of nonlocal theories. Nonlocal models of elastoplasticity were proposed in the literature by taking into account the dislocation density and torsion tensor, respectively, (e.g. Le and Stumpf, *
Corresponding author. Tel.: +49-32-2-7385; fax: +49-234-32-14154. E-mail address: [email protected] (H. Stumpf).
0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2003.11.012
356
H. Stumpf et al. / Mechanics Research Communications 31 (2004) 355–363
1996a,b; Shizawa and Zbib, 1999; Cleja-Tigoiu, 2002a,b) or torsion and curvature tensor (Saczuk et al., 2001; Rakotomanana, 2003), where the latter corresponds to the disclination density tensor. Gradient damage models were considered for isotropic brittle damage by Fremond and Nedjar (1996) and Steinmann (1999), for isotropic damage coupled with small strain plasticity by Nedjar (2001), and for anisotropic damage of viscoelastic materials by Stumpf and Hackl (2003). The aim of this paper is to develop a thermodynamically consistent objective gradient theory of brittle and ductile damage at finite strain based on balance laws of physical and material forces and first and second law of thermodynamics for physical and material space. Material forces on defects were first investigated by Leibfried (1949); Peach and Koehler (1950), and Eshelby (1951). The paper is organized as follows. In Section 2 the physical deformation gradient F, a symmetric Lagrangian damage tensor D and the plastic deformation ‘‘gradient’’ Fp , which is a locally defined linear map, and the temperature h are introduced as independent thermokinematical variables of the theory. In Section 3 the dynamical balance laws for physical and material forces and couples are formulated, where it is assumed that dynamically evolving defects as dislocations, the bearer of plastic deformations, microcracks and microvoids causing the material degradation, have inertia and kinetic energy. In Section 4 the first and second law of thermodynamics for motion in the physical space and defect and temperature evolution in the material space are formulated. In Section 5 general constitutive equations are assumed and their thermodynamically admissible and Lagrangian objective forms are determined. It is shown that they can be expressed by two potentials, the free energy density and a dissipation pseudo-potential, if the latter exists. Introducing the constitutive equations into the dynamical balance laws of physical and material forces, the nonlocal governing equations of dynamically deforming elastic–plastic structures with damage evolution are obtained. The general theory presented in this paper can be considered as a framework, which enables the derivation of various nonlocal and local theories of finite elastoplasticity and isotropic and anisotropic brittle and ductile damage by introducing simplifying assumptions.
2. Independent thermokinematical variables in physical and material space For an appropriate nonlocal damage analysis of engineering structures with brittle and ductile material behavior the thermomechanics of motion and defect evolution in physical and material space has to be formulated. As point of departure we have to decide about the independent thermokinematical variables to describe the motion of material points in the physical space and defect and temperature evolution in the material space. For continuous media the motion of material points can be defined by a vector-valued function, the actual position vector xðX; tÞ and the displacement vector uðX; tÞ, respectively, where X is the position vector in the reference configuration and t the time parameter, and for granular media by the actual position vector xðX; tÞ and a rotation tensor QðX; tÞ taking into account the rotational degrees of freedom. Less obvious is how to describe the evolution of defects as microcracks, microvoids and dislocations within the material space. The simplest assumption is that of isotropic elastic damage, where the damage evolution is defined by a scalar-valued function DðX; tÞ with 0 6 D 6 1, where D ¼ 0 denotes no damage and D ¼ 1 total damage. The physical interpretation may be that the scalar D represents the amount of microcracks and/or microvoids in a representative volume element. Since microdefects are orientation dependent, the simplest assumption of isotropic damage is in general not sufficient and has to be replaced by an anisotropic damage description using a tensor-valued function DðX; tÞ. In this paper we chose D as a symmetric tensor referred to the undeformed and homogeneous reference configuration. Most local theories of finite elastoplasticity proposed in the literature so far, are based on the multiplicative decomposition of the total deformation gradient F into elastic, Fe , and plastic, Fp , parts, F ¼ Fe Fp ,
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where Fe and Fp are locally defined linear maps and not gradients, because there exists no associated vector field. In this paper we do not use the multiplicative decomposition formula, instead, for a Lagrangian description, we define the motion in the physical space by the actual vector field xðX; tÞ and the physical deformation gradient F ¼ ox=oX, respectively, and the plastic evolution in the material space by the locally defined linear plastic map Fp ðX; tÞ. We have to point out that for poly-crystalline materials Fp may represent the plastic deformation based on dislocation motion, for other materials as e.g. concrete, rocks and soil, Fp may be interpreted as the irreversible deformation in the material space. For an appropriate description of the thermomechanical process of deformation and material degradation we have to introduce, besides F, D and Fp , also the temperature field hðX; tÞ as fourth independent thermokinematical variable. 3. Local balance laws of forces and couples in the physical-material space In local theories of damage and elastoplasticity published in the literature the kinematical variables D and Fp are considered as internal variables, for which evolution laws have to be given. Since numerical investigations have shown that FE solutions based on local theories exhibit a strong mesh-dependency, if localization in the material space occurs, and that they are not able to simulate appropriately sizedependent effects, we propose in this paper an alternative nonlocal and weakly nonlocal, respectively, approach based on balance laws for material forces and material stresses, respectively, acting on evolving defects. Besides the classical global balance laws of physical forces and physical couples, we formulate global balance laws of material forces in a representative volume element. Passing to the limit of a material point, the following local balance laws of physical and material forces and couples can be obtained: Balance of physical forces €; Div T þ b ¼ .0 x
ð3:1Þ
where for simplicity it is assumed that the referential mass density .0 ðXÞ is independent of time, Balance of physical couples TFT FTT ¼ 0
ð3:2Þ
with the denotations TðX; tÞ physical first Piola–Kirchhoff stress tensor bðX; tÞ external physical body force FðX; tÞ physical deformation gradient Balance of tensor-valued material forces associated with the evolution of defects as microcracks and microvoids _ ; DivHd Hd þ Gd ¼ .0 ðJd DÞ ð3:3Þ Balance of tensor-valued material forces associated with plastic evolution and dislocation motion, respectively, DivHp Tp þ Gp ¼ .0 ðJp F_ p Þ
with the denotations Hd ðX; tÞ symmetric second-order tensor of material force power-conjugate to the damage tensor D Hd ðX; tÞ third-order material stress tensor power-conjugate to rD
ð3:4Þ
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Gd ðX; tÞ external influence tensor representing e.g. chemical reactions breaking internal material bonds associated with defect evolution Jd ðX; tÞ inertia tensor associated with dynamically moving defects as microcracks and microvoids Tp ðX; tÞ second-order tensor of material force power-conjugate to Fp Hp ðX; tÞ third-order stress tensor power-conjugate to rFp Gp ðX; tÞ external influence tensor breaking internal material bonds associated with plastic evolution and dislocations motion, respectively Jp ðX; tÞ inertia tensor associated with dynamically moving dislocations. The inertia terms on the right-hand side of the balance laws of material forces (3.3) and (3.4) are due to the assumption that dynamically moving defects and dislocations have kinetic energy. Then the total kinetic energy j consists of two parts j ¼ jph þ jm ;
ð3:5Þ
where jph is the kinetic energy of moving material points in the physical space 1 jph ¼ .0 x_ x_ 2
ð3:6Þ
and jm is the kinetic energy due to moving defects as microcracks and microvoids, jd , and due to moving dislocations, jp , assumed in the form 1 _ D _ þ 1 .0 Jp F_ p F_ p : jm ¼ jd þ jp ¼ .0 Jd D 2 2
ð3:7Þ
In this paper, we do not present the boundary and initial conditions associated with the balance laws (3.1)–(3.4). These balance laws of forces and couples were derived under the assumption that there are no singular or discontinuous forces in the representative volume element. If such forces exist, corresponding singularity or jump conditions have to be added to (3.1)–(3.4).
4. First and second law of thermomechanics for motion in the physical-material space Besides the global and local, respectively, balance laws of forces and couples in the physical and material space, we have to formulate the global form of the first and second law of thermodynamics. Their localization leads to the local laws. Balance of energy e_ ¼ r þ r Div q;
ð4:1Þ
dissipation inequality g_ ðh1 r Divðh1 qÞÞ P 0 with the denotations eðX; tÞ rðX; tÞ rðX; tÞ qðX; tÞ gðX; tÞ
specific internal energy stress power external body heating referential heat flux vector entropy
ð4:2Þ
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It should be noted that in the form (4.2) the local dissipation inequality representing the second law of thermodynamics is independent of the laws of mechanics and the law of energy balance (4.1). Making use of (4.1) and introducing the free energy WðX; tÞ (measured per unit volume of the reference configuration) defined by W e hg;
ð4:3Þ
the dissipation inequality (4.2) can be rewritten in an equivalent form as _ gh_ h1 q rh P 0; DrW
ð4:4Þ
referred to as the reduced dissipation inequality. It represents the second law of thermodynamics under the assumption that the balance laws of physical and material forces and the balance of energy hold. Corresponding to (3.1)–(3.4) the stress power rðX; tÞ consists of two parts, r ¼ rph þ rm ;
ð4:5Þ
the stress power in the physical space, rph , _ rph ¼ T F;
ð4:6Þ
and the stress power in the material space, rm ðX; tÞ, decomposed as rm ¼ rd þ rp ;
ð4:7Þ
with the stress power due to defect evolution, _ þ Hd rD; _ rd ¼ Hd D
ð4:8Þ
and the stress power due to dislocation motion, rp ¼ Tp F_ p þ Hp rF_ p :
ð4:9Þ
On the right-hand side of (4.8) and (4.9), respectively, the first term corresponds to a local theory and the second term due to the nonlocal contribution.
5. Constitutive modeling 5.1. General form of the constitutive equations The independent thermokinematical variables of the theory presented in this paper are ðF; D; Fp ; hÞ, where F ¼ rx is the deformation gradient of the motion in the physical space, D the symmetric Lagrangian damage tensor, Fp the plastic deformation gradient, and h the absolute temperature. Thus, the variables ðF; D; Fp ; hÞ must be determined as solution of the problem defined by the field equations of Sections 3 and 4, supplemented by appropriate initial and boundary conditions and suitable constitutive equations. Let us denote with e the set of kinematical variables e ðF; D; rD; Fp ; rFp Þ
ð5:1Þ
and with e_ their rates _ D; _ rD; _ F_ p ; rF_ p Þ: e_ ðF;
ð5:2Þ
Then the set of physical and material stress tensors power-conjugate to (5.1) and (5.2), respectively, is r ¼ ðT; Hd ; Hd ; Tp ; Hp Þ;
ð5:3Þ
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which enables to write the stress power (4.5) in the compact form r ¼ r e_ :
ð5:4Þ
To formulate the constitutive equations, we have to express the free energy (4.3) and the stress tensors (5.3) together with the entropy g and the heat flux vector q as functions of e, e_ , h, and rh, b e_ ; h; rhÞ; W ¼ Wðe; ^ðe; e_ ; h; rhÞ; r¼r ^ðe; e_ ; h; rhÞ; g¼g ^ðe; e_ ; h; rhÞ: q¼q
ð5:5Þ
The general constitutive equations (5.5) have to satisfy the restrictions of thermodynamical admissibility and objectivity. 5.2. Thermodynamical admissibility The form of the constitutive equations (5.5) is restricted by the second law of thermodynamics. They have to satisfy the dissipation inequality (4.4). Introducing the stress power (5.4) with (5.3) and (5.2) into (4.4) leads to b e_ ; h; rhÞ e_ oh Wðe; b e_ ; h; rhÞ þ ^gðe; e_ ; h; rhÞ h_ ^ðe; e_ ; h; rhÞ oe Wðe; D¼ r b e_ ; h; rhÞ rh_ P 0: h1 ^ qðe; e_ ; h; rhÞ rh orh Wðe; ð5:6Þ From (5.6) the following constitutive restrictions are obtained b ¼ 0; oe_ W
b ¼ 0; orh W
ð5:7Þ
stating that the free energy function has the constitutive form b hÞ: W ¼ Wðe;
ð5:8Þ
Moreover, (5.6) leads to the constitutive equation for the entropy determined by the free energy, b hÞ: g¼^ gðe; hÞ ¼ oh Wðe; In view of (5.8) and (5.9) the dissipation inequality (5.6) reduces to the form b hÞ e_ h1 ^ ^ðe; h; e_ ; rhÞ oe Wðe; qðe; h; e_ ; rhÞ rh P 0: D¼ r
ð5:9Þ
ð5:10Þ
The inspection of inequality (5.10) leads to the result that the physical and material stresses (5.3) consist b hÞ, and a of two parts, a nondissipative part, which can be derived from the free energy potential, oe Wðe; dissipative part, which is indicated by a lower asterisk, b hÞ þ r ^ ðe; h; e_ ; rhÞ; ^ðe; h; e_ ; rhÞ ¼ oe Wðe; r¼r
ð5:11Þ
where the dissipative parts of the physical and material stresses (5.3) r ¼ ðT ; Hd ; Hd ; Tp ; Hp Þ
ð5:12Þ
have to satisfy the dissipation inequality (5.10). Introducing (5.11) into (5.10) leads to the entropy production inequality ^ðe; h; e_ ; rhÞ e_ h1 ^ D¼r qðe; h; e_ ; rhÞ rh P 0:
ð5:13Þ
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In (5.13) the first term is the entropy production of the driving forces in the physical and material space and the second term is the entropy production due to heat flux in the material space. The form of the entropy production inequality (5.13) suggests to assume the existence of a dissipation pseudo-potential U of the form U ¼ Uðe; h; e_ ; rhÞ;
ð5:14Þ
where e and h can be considered as parameters. If such a dissipation pseudo-potential exists, the dissipative driving stresses (5.12) and the heat flux vector q can be derived from U as r ¼ oe_ Uðe; h; e_ ; rhÞ; h1 q ¼ orh Uðe; h; e_ ; rhÞ:
ð5:15Þ
Introducing (5.15)1 into (5.11) leads to the constitutive equations expressed by two potentials, the free energy W and the dissipation pseudo-potential U, b hÞ þ oe_ Uðe; h; e_ ; rhÞ: r ¼ oe Wðe;
ð5:16Þ
Furthermore, with (5.15) the dissipation inequality (5.13) takes the form D ¼ oe_ Uðe; h; e_ ; rhÞ e_ þ orh Uðe; h; e_ ; rhÞ rh P 0;
ð5:17Þ
where the first term is the entropy production due to the dissipative driving forces in the physical-material space and the second term is the entropy production due to the heat flux in the material space. 5.3. Objectivity requirements Besides the postulate of thermodynamical admissibility, the constitutive equations have to be objective. The nonobjective set of kinematical variables (5.1), their rates (5.2) and the power-conjugate stresses (5.3) have to be replaced by their objective counterparts. In this paper, we restrict our considerations to the Lagrangian objective form of all variables. Their Eulerian objective form can then be obtained by pushforward of all variables to the actual configuration. 5.3.1. Lagrangian objective physical space variables Considering the stress power of the physical stresses (4.6), we can pull-back both power-conjugate variables to the undeformed reference configuration leading to 1 rph ¼ T F_ ¼ S C_ 2
ð5:18Þ
with the objective second Piola–Kirchhoff stress tensor S and the Lagrangian objective Cauchy–Green deformation tensor C defined by S ¼ F1 T;
C ¼ FT F:
ð5:19Þ
5.3.2. Lagrangian objective damage variables The damage tensor D is introduced as symmetric tensor referred to the homogeneous undeformed reference configuration and is therefore Lagrangian objective. Since r denotes the gradient operator with respect to the undeformed reference configuration, the term rD is Lagrangian objective as well. It follows that the power-conjugate stresses in (4.8), Hd and Hd , are Lagrangian objective.
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5.3.3. Lagrangian objective plastic deformation variables The plastic deformation ‘‘gradient’’ Fp ðX; tÞ is a locally defined linear map from the undeformed reference configuration without inhomogeneities into an inhomogenous state obtained by Fp ðX; tÞ. Therefore, Fp ðX; tÞ and the power-conjugate stresses Tp ðX; tÞ as well as rFp ðX; tÞ and the power-conjugate third-order material stress tensor Hp ðX; tÞ are not objective. In Le and Stumpf (1996a) the Lagrangian objective deformation measures Cp ¼ FpT Fp ;
Tp ¼ Fp1 rFp ;
ð5:20Þ
p
were introduced, where C can be considered as Lagrangian metric, and the skew-symmetric part of Tp is the so-called torsion tensor representing physically the dislocation density referred to the undeformed reference configuration. In Le and Stumpf (1996b) it was shown that from (5.20) the plastic rotation tensor Rp , following from the polar decomposition theorem, Fp ¼ Rp Up with Up the plastic stretch tensor, can be determined uniquely up to a rigid body rotation. Expressing in the plastic stress power (4.9) the rates F_ p and rF_ p by the objective Lagrangian rates C_ p and p T_ , respectively, the plastic stress power rp is obtained as 1 rp ¼ Sp C_ p þ Sp T_ p 2
ð5:21Þ
with the Lagrangian objective stress tensors Sp ¼ Fp1 Tp ;
Sp ¼ FpT Hp ;
ð5:22Þ
power-conjugate to C_ and T_ , respectively. In deriving (5.21) we assumed that a term of the form T HpT is small compared with Tp and can be neglected. The thermodynamically admissible objective constitutive equations are obtained by replacing in (5.16) the set e according to (5.1), the rates e_ according to (5.2) and the set r according to (5.3) by their objective forms: e ðC; D; rD; Cp ; Tp Þ; ð5:23Þ p
p
_ D; _ rD; _ C_ p ; T_ p Þ; e_ ðC; 1 1 S; Hd ; Hd ; Sp ; Sp : r 2 2
p
ð5:24Þ ð5:25Þ
6. Dynamic governing equations of deformation and defect and plastic evolution Introducing the thermodynamically admissible and objective constitutive equations derived above into the balance laws of physical and material forces and stresses, respectively, obtained in Section 3, the following dynamical governing equations of deformation in the physical space and defect and plastic evolution in the material space are obtained: €; DivðFSðe; h; e_ ; rhÞÞ þ b ¼ .0 x ð5:26Þ _ ; Div Hd ðe; h; e_ ; rhÞ Hd ðe; h; e_ ; rhÞ þ Gd ¼ .0 ðJd DÞ T Div ðFp1 Þ Sp ðe; h; e_ ; rhÞ Fp Sp ðe; h; e_ ; rhÞ þ Gp ¼ .0 ðJp F_ p Þ :
ð5:27Þ ð5:28Þ
These equations together with corresponding boundary and initial conditions define a weakly nonlocal and gradient, respectively, model of ductile damage.
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7. Conclusions The thermodynamically consistent and objective gradient theory of ductile damage presented in this paper can be considered as a framework, which enables the derivation of special theories by introducing simplifying assumptions including: consistent gradient theories of finite elastoplasticity, isotropic brittle damage, anisotropic brittle damage, isotropic ductile damage, anisotropic ductile damage, simplified gradient and local theories by assuming that for the problem under consideration some gradient terms can be neglected, while others have to be taken into account.
Acknowledgements The financial support, provided by Deutsche Forschungsgemeinschaft (DFG) under Grant SFB 398-A7, is gratefully acknowledged.
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