COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. O˜ nate and E. Dvorkin (Eds.) c
CIMNE, Barcelona, Spain 1998
A GRADIENT ENHANCEMENT WITH APPLICATION TO ANISOTROPIC CONTINUUM DAMAGE Ellen Kuhl? , Ren´ e de Borst†, and Ekkehard Ramm? ?
Institute of Structural Mechanics, University of Stuttgart Pfaffenwaldring 7, D-70550 Stuttgart, Germany e-mail: kuhl/
[email protected] web page: http://www.uni-stuttgart.de/ibs
† Koiter Insitute Delft, Faculty of Civil Engineering, TU Delft P.O. Box 5048, 2600 GA Delft, Netherlands e-mail:
[email protected] web page: http://gcl.ct.tudelft.nl:80/comp.html
Key words: gradient enhanced continuum, continuum damage mechanics, failure induced anisotropy, microplane model, concrete modelling Abstract.The paper focuses on the modelling of the quasi-brittle failure of materials with a heterogeneous microstructure. Failure induced anisotropy is incorporated in terms of the microplane theory. This theory is based on scalar damage laws formulated on several individual material planes. Anisotropy of the overall constitutive law is introduced in a natural fashion by transforming the scalar valued damage operators into a damage tensor of fourth order. To account for microstructural interaction, strain gradients are included in the constitutive equations to preserve the well-posedness of the underlying boundary value problem in the postcritical regime. The gradient enhancement of the continuum model results in an additional partial differential equation defining an averaged quantity, the nonlocal strain tensor. This additional gradient equation is satisfied in a weak sense which leads to a modified finite element formulation with the nonlocal strains as additional nodal degrees of freedom. The governing equations are linearized consistently and solved within an incremental iterative Newton Raphson solution procedure. The gradient enhanced finite element formulation for anisotropic damage will be illustrated by means of several numerical examples.
1
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
1
INTRODUCTION
The failure mechanism of heterogeneous materials such as concrete is of a complex nature. Microcracks develop at the interface between the grains and the cement matrix which leads to failure induced anisotropy. Growth and coalescence of these microcracks result in stiffness degradation and a typical localized macroscopic failure mechanism. The general idea of modelling anisotropy by considering the material behavior on several independent planes, which is the scope of this study, goes back to the early ideas of Taylor [29] motivated by the crystallographic slip on several independent slip planes. This reduction to scalar constitutive laws on individual material planes was modified by Baˇzant & Gambarova [2] who introduced the so-called microplane model in the context of the quasi-brittle failure of concrete. This model has been enhanced by Baˇzant & Prat [4], Carol, Baˇzant & Prat [11], Baˇzant & Oˇzbolt [5] and recently by Kuhl & Ramm [17]. The stiffness degradation on the individual microplanes is described by the concept of continuum damage mechanics, see e.g. Kachanov [15], Lemaitre & Chaboche [19] and Simo & Ju [27]. Including different uniaxial constitutive laws for tension, compression and shear, the model takes into account mode I as well as mode II failure in a natural fashion. Material damage leads to stiffness degradation which eventually results in a mathematically ill-posed problem caused by a local loss of ellipticity. This deficiency of local continuum models can be avoided by introducing an internal length scale e.g. through a nonlocal continuum enhancement. Physically, the introduction of nonlocal terms can be interpreted by taking into account the heterogeneous substructure of the material leading to characteristic microscopic mechanisms such as dislocation motion in metal plasticity, e.g. Aifantis [1], or microcrack interaction in materials like concrete, see Baˇzant [6]. Since classical continua are unable to describe this interaction at the material point level, the enhancement of a continuum by nonlocal terms either through an integral equation or through an additional gradient equation has become popular. The incorporation of nonlocal quantities through integral equations has already been proposed by Kr¨ oner [16] as well as by Eringen & Edelen [13] for elastic material models. Later, their ideas have been extended to continuum damage mechanics by Pijaudier-Cabot & Baˇzant [3, 26]. They introduced either the energy release rate or the damage variable as a scalar valued nonlocal quantity. Nonlocal anisotropic damage formulations based on a tensorial nonlocal variable, namely the strain field, have been discussed, see Baˇzant & Oˇzbolt [5] and Oˇzbolt [21]. However, the application of nonlocal integral models is not efficient from a computational point of view. A global averaging procedure is required and the resulting equations cannot be linearized easily, see Peerlings, de Borst, Brekelmans, de Vree & Spee [24]. Consequently, nonlocal integral models are not considered promising for large scale computations. An alternative to include nonlocal terms was proposed among others by Lasry & Belytschko [18] and by M¨ uhlhaus & Aifantis [20]. In their gradient continuum models, 2
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
nonlocality is incorporated through a partial differential equation including a gradient term. This additional equation is usually fulfilled in a weak sense and the nonlocal quantity is introduced as independent variable. The constitutive equations thus remain local in a finite element sense and the linearization is straightforward. The introduction of gradients in combination with softening plasticity models is already well established, see M¨ uhlhaus & Aifantis [20], de Borst & M¨ uhlhaus [7] and Pamin [22] among others. The application of gradients in combination with continuum damage mechanics, however, has been presented only recently by Peerlings, de Borst, Brekelmans & de Vree [23] in the context of isotropic damage. Again, the equivalent strain, see Peerlings, de Borst, Brekelmans & de Vree [23] and Geers [14], or the damage variable itself, see de Borst, Benallal & Heeres [9], can be chosen as the nonlocal quantity. The application of the gradient continuum to anisotropic damage evolution, which is the scope of this study, has not been analyzed hitherto. After briefly summarizing the constitutive equations of the microplane model, we will introduce the ideas of the gradient enhancement of a continuum. The influence of the strain gradient term on the microplane model will be elaborated. Afterwards, the finite element implementation of the gradient enhanced microplane model will be derived. Finally, some examples will be given to demonstrate the features of the model by simulating the complex failure mechanisms of quasi-brittle materials.
2
ANISOTROPIC DAMAGE MODEL
The following anisotropic damage description is based on the ideas of the microplane model as described in detail by Baˇzant & Prat [4] or more recently by Carol & Baˇzant [12]. Restricting the constitutive model to small strains, the macroscopic strain tensor is given as the symmetric part of the displacement gradient ∇u. = ∇sym u
(1)
The scalar valued strain components for every microplane I, namely V , ID and RI T , can be determined by projecting the macroscopic strain tensor onto the individual material planes. RI V = : V ID = : D I RI (2) T = : T Herein, V ,D I and T RI represent the volumetric, deviatoric and tangential projection tensor, respectively, which are defined by the geometry of the I-th microplane through the normal of the plane nI , as depicted in figure 1. V NI DI T RI
:= := := :=
1 1 3
nI ⊗ nI n ⊗ nI − 13 1 1 I I I I I n 1 + n 1 − 2n n n KR JR J K J K R 2 I
3
(3)
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
nI ID V nI
t 1I
1I T
IT
t 2I 2I T
model of concrete microstructure
imaginary sphere in each integration point
kinematics of microplane I
Figure 1: Microplane kinematics Note, that the volumetric strain component is identical for each microplane. For the general three-dimensional case, the projection tensor T RI is of third order, reducing to a second order tensor T I = nI ⊗ tI with nI · tI = 0 for a two-dimensional setting. ConseI quently, the vector of tangential strains RI T reduces to a scalar T for the two-dimensional I case. The corresponding microplane stresses σV , σD and σTRI can be determined through the one-dimensional damage laws, see Lemaitre & Chaboche [19], which are formulated individually on every microplane in terms of the damage variables dV ,dID and dIT and the 0 initial constitutive moduli CV0 ,CD and CT0 . σV = (1 − dV ) CV0 V I 0 σD = (1 − dID ) CD ID σTRI = (1 − dIT ) CT0 RI T
(4)
Note, that the tangential damage law is based on the so-called ’parallel tangential hypothesis’ assuming that the tangential stress vector σTRI always remains parallel to the corresponding strain vector RI zant & Prat [4] T . This assumption was first proposed by Baˇ to preserve the one-dimensional character of the constitutive microplane laws even for the tangential law. The macroscopic stress tensor is related to the microplane stress components by the equivalence of virtual work performed on the surface of a unit sphere. ˆ macro (σ) = W ˆ micro (σV , σ I , σ I ) W D T
(5)
The overall stress tensor σ can thus be expressed as a weighted summation of the individual microplane stress components. σ = σV 1 +
nmp X I=1
I σD N I wI
4
+
nmp X I=1
σTRI T RI w I .
(6)
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
Herein, nmp represents the number of microplanes and w I are the corresponding weight coefficients. In a two-dimensional setting, twelve microplanes, nmp = 12, of equal weight w I = 1/12 have been shown to yield reasonable results, see for example Baˇzant & Prat [4]. For a three-dimensional model problem, at least nmp = 21 microplanes should be taken into account, see Stroud [28] for their geometry and the corresponding weight coefficients. ˜ which can be interpreted as a summation of the weighted The modified elasticity tensor C, microplane moduli projected by the individual projection tensors, yields the constitutive relation between the macroscopic strain tensor and the overall stress tensor σ. ˜ = C (1 − dV )CV0 1 ⊗ V Pnmp 0 N I ⊗ DI wI + I=1 (1 − dID )CD P + nmp (1 − dIT )CT0 T RI ⊗ T RI w I I=1
(7)
For the originally undamaged material, the damage moduli dV , dID and dIT are equal to ˜ is equal to the elasticity tensor C el . The microplane moduli C 0 ,C 0 and C 0 zero and C V D T can thus be expressed in terms of the Lam´e constants µ and λ. ˜ C
= 2µ14 + λ1 ⊗ 1 =
C el
(8)
Applying the effective stress concept together with the principle of strain equivalence, see Lemaitre & Chaboche [19], the overall stress strain relation of anisotropic damage can be expressed in terms of the tensor of initial elastic moduli C el and a fourth-order damage tensor D. ˜ : = (1 − D) : C el : σ=C (9) ˜ which was defined in equation (7) can be Consequently, the modified elasticity tensor C cast into an overall fourth-order damage tensor. ˜ : C el−1 D =1−C
(10)
Thus, the present model fits into the framework of tensorial damage, see Carol, Prat & Baˇzant [10]. However, there is no need to explicitly calculate this damage tensor D. The evolution of the microplane damage parameters dV ,dID and dIT is governed by the individual history parameters κV , κID and κIT , representing the most severe deformation in the history of the microplane I. dV = dˆV (κV )
dID = dˆID (κID )
dIT = dˆIT (κIT )
(11)
Considering concrete behavior, different laws for tension and compression need to be applied. Baˇzant & Prat [4] have chosen exponential damage laws except for volumetric compression. They are summarized in table 1.The resulting stress strain relations on the individual microplanes are depicted in figure 2. The non-decreasing values of κ,
5
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
volumetric
deviatoric
tangential
1 − dV
1−
tension
κV ≥ 0
compression
κV < 0
tension
κID ≥ 0
compression
κID < 0
tension
κIT ≥ 0
compression
κIT < 0
dID
1 − dIT
κV p1 exp − a " 1 q # κV −p κV 1− + − a b " " # # I p1 κD exp − " " a1 #p2 # κI exp − D a2 " " #p # κIT 3 exp − " " a3 #p3 # −κIT exp − a3
Table 1: Microplane damage moduli (1 − d) for concrete according to Baˇzant & Prat which only grow if the corresponding damage loading function Φ is equal to zero, can be determined through the Kuhn Tucker conditions. κ˙V ≥ 0 κ˙ID ≥ 0 κ˙IT ≥ 0
ΦV = V − κV ≤ 0 ΦID = ID − κID ≤ 0 ΦIT = γTI − κIT ≤ 0
κ˙V ΦV = 0 κ˙ID ΦID = 0 κ˙IT ΦIT = 0
(12)
Note, that according to the parallel tangential hypothesis, the tangential loading function is governed by the norm of the tangential strain vector, γT I =
q
RI RI T T
(13)
to preserve the one-dimensional format of the microplane laws. sV
sD
eV
eD
sT
eT
normal volumetric
normal deviatoric
tangential
Figure 2: Individual constitutive laws on a microplane 6
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
3 3.1
GRADIENT ENHANCED CONTINUUM MODELS Nonlocal continuum models
Motivated by the long range microstructural interaction, in nonlocal continuum models the stress response σ of a material point not only depends on the state of the point itself, but also on the state of its neighborhood. Based on the early nonlocal elasticity models by Kr¨oner [16] and Eringen & Edelen [13], the definition of any nonlocal quantity Y¯ can be expressed as the weighted average of its local counterpart Y over a surrounding volume V, Z 1 Y¯ = g(ξ)Y (x + ξ)dV (14) V V whereby g(ξ) represents a certain weight function, for example the Gaussian function.
exp g(ξ) =
−ξ 2 2l2
with
2πl2
1 V
Z V
g(ξ)dV = 1
(15)
Herein, ξ denotes the distance from the point x to a point in its neighborhood. For isotropic material behavior, it is generally sufficient to apply the averaging equation (14) to a scalar-valued quantity, for example the equivalent strain or the damage variable. For anisotropic models, however, the nonlocal generalization should be used for a tensorial quantity.
3.2
Introduction of strain gradients
In the following, the averaging equation (14) will be applied to the strain tensor , leading to the definition of its nonlocal counterpart ¯ in the following form. ¯ =
1 V
Z V
g(ξ)(x + ξ)dV
(16)
Following the ideas of Lasry & Belytschko [18] and M¨ uhlhaus & Aifantis [20] we will approximate the nonlocal integral equation by a differential equation. Consequently, we replace the local strain tensor of equation (16) by its Taylor expansion at ξ = 0. 1 (2) ∇ (x)ξ(2) 2! 1 (3) 1 ∇ (x)ξ(3) + ∇(4) (x)ξ(4) +... 3! 4!
(x, ξ) = (x) + ∇(x)ξ + +
(17)
Assuming isotropy and truncating the Taylor series after the quadratic term leads to a definition of the nonlocal strain tensor, which can be expressed through the following 7
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
partial differential equation. ¯ = + c∇2
(18)
Herein, the integrations constant c is proportional to a length squared. For sake of simplicity, we will apply only one integration constant c weighting each component of the gradient term equally. Nevertheless, for the case of anisotropic damage evolution, an extension to a different weighting of the individual coefficients which results in several different integration constants, c11 , c22 , c12 , etc. is straightforward and considered sensible. Note, that within this contribution, a constant gradient damage model is adopted. In contrast, Geers [14] has proposed to replace the constant gradient parameter c by a varying quantity. When embedding equation (18) in a finite element analysis, the discretized displacement field u would have to fulfill C 1 continuity requirements. Therefore, we will approximate equation (18) by the following expression, = ¯ − c∇2 ¯
(19)
which is still of second-order accuracy although requiring only a C 0 continuous displacement approximation, see Peerlings, de Borst, Brekelmans & de Vree [23]. Additional boundary conditions, either of the Dirichlet or the Neumann type, ¯ = ¯p
or
∇¯ · n = ¯pn
on Γ
(20)
are required in order to solve the averaging partial differential equation (19). The application of additional boundary conditions has been discussed in detail by Lasry & Belytschko [18] as well as by M¨ uhlhaus & Aifantis [20]. Herein, we will adopt the natural boundary condition of a vanishing gradient ∇¯ ·n=0
(21)
at every point of the boundary. Nevertheless, the physical interpretation of the additional boundary condition is still an unsolved issue.
3.3
Gradient enhanced microplane model
The nonlocal strain terms are included in the constitutive equations of the microplane ¯ In analogy to gradient plasmodel through the individual damage loading functions Φ. ticity, see de Borst and M¨ uhlhaus [7] or de Borst, Pamin, Peerlings & Sluys [8] among others, the loading functions introduced in equation (12) are assumed to depend not only on the local strains and on the local history variables κ, but also on the strain gradients ∇2 . ˆ¯ ∇2 , κ) ¯ = Φ(, Φ (22) With the help of equations (19), the individual damage loading functions of the classical microplane model can thus be rewritten in terms of the nonlocal strains ¯ and the history 8
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
variable κ.
¯ V = ¯V − κV ≤ 0 Φ ¯ I = ¯D I − κI ≤ 0 Φ D D ¯ IT = γ¯T I − κIT ≤ 0 Φ
(23)
Again, the tangential law is governed by the norm of the corresponding nonlocal tangential strain vector. q γ¯T I = ¯RI RI (24) T ¯ T The nonlocal strain components of equations (23) and (24) are defined analogously to their local counterparts, compare with equation (2). ¯D I = ¯ : D I
¯V = ¯ : V
¯T RI = ¯ : T RI
(25)
Remarkably, the other microplane equations are not effected by the gradient continuum enhancement.
4 4.1
FINITE ELEMENT IMPLEMENTATION Governing equations
The boundary value problem of gradient damage is thus governed by the equilibrium equation and by the implicit definition of the nonlocal strains, ∇·σ+f =0
(26)
¯ − c∇2 ¯ =
(27)
whereby f represents an external load vector. The governing equations can be solved by applying an independent finite element discretization of the displacement field u and the nonlocal strain field ¯. Integrating equations (26) and (27) over the domain Ω and weighting with w u and w ¯ , respectively, yields the weak forms given as follows. Z Z Ω
Ω
w Tu (∇ · σ + f )dΩ = 0
w T¯ (¯ −
Z 2
c∇ ¯)dΩ =
Ω
(28)
w T¯ dΩ
(29)
Integrating equations (28) and (29) by parts, applying the divergence theorem and including the boundary conditions results in the following expressions. Z Ω
Z Ω
Z
∇wTu σdΩ
w T¯ ¯dΩ +
=
Z Ω
Ω
wTu f dΩ
Z
+
∇w T¯ c∇¯ dΩ = 9
Γ
w Tu tdΓ
Z
Ω
w T¯ dΩ
(30) (31)
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
4.2
Discretization
The finite element discretization of the displacement field u and the nonlocal strain field ¯ is given as follows. u = N u du ∇u = B u du (32) ¯ = N ¯d¯ ∇¯ = B ¯ d¯ Herein, du and d¯ denote the nodal degrees of freedom. N u and N ¯ represent the shape functions and B u and B ¯ include their partial derivatives with respect to the coordinates. According to the Bubnov Galerkin method, an analogous discretization can be applied to the weight functions w u and w ¯ . ∇w u = B u dwu ∇w ¯ = B ¯ dw¯
w u = N u dwu w ¯ = N ¯ dw¯
(33)
The resulting equations have to be satisfied for all admissible nodal weights dwu and dw¯ . The discretized equilibrium equation thus reduces to Z Ω
B Tu σdΩ
Z
= Ω
N Tu f dΩ
Z
+ Γ
N Tu tdΓ
(34)
whereas the discretized implicit definition of the nonlocal strains can be written as follows. Z Ω
4.3
(N T¯ N ¯ +
B T¯ cB ¯ )d¯dΩ
Z
= Ω
N T¯ dΩ
(35)
Linearization
In order to construct a consistent incremental iterative Newton Raphson solution procedure, equations (34) and (35) have to be linearized. The main advantage of the gradient continuum equations compared to the nonlocal integral equations is, that the linearization is straightforward. At the nodal level, the linearization at iteration i with respect to the previous iteration i − 1 is given by du,i = du,i−1 + ∆du d¯,i = d¯,i−1 + ∆d¯
(36)
resulting in the following linearization of the stress tensor σ at the integration point level. σ i = σ i−1 + ∆σ
(37)
Herein, the incremental stresses ∆σ are given by
∂σ ∂σ ∆σ = : ∆ + : ∆¯ ∂ i−1 ∂¯ i−1 10
(38)
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
whereby the linearized local and nonlocal strain tensor given in the following form. i = i−1 + ∆ ¯i = ¯i−1 + ∆¯
∆ = B u ∆du ∆¯ = N ¯∆d¯
(39)
The partial derivative of the stress tensor σ with respect to the local strain tensor yields ˜ which was already defined in equation (7). the modified elasticity tensor C
∂σ ∂ i−1
˜ i−1 = := C (1 − dV,i−1 )CV0 1 ⊗ V Pnmp I I 0 + (1 − dD,i−1)CD N ⊗ DI wI PI=1 nmp + (1 − dIT,i−1)CT0 T RI ⊗ T RI w I I=1
(40)
The partial derivative of the stresses with respect to the nonlocal strains ¯ results in a ˜ fourth-order tensor which we will denote by ∆C.
∂σ ∂ ¯ i−1
˜ = := ∆C ∆CV 1 ⊗ V Pnmp I I ∆CD N ⊗ DI wI + PI=1 nmp + ∆CTRS,I T RI ⊗ T SI w I I=1
(41)
I Herein, ∆CV , ∆CD and ∆CTRS,I denote the linearization of the individual constitutive moduli on the microplane level,
"
∆CV I ∆CD
∆CTRS,I
#
"
#
∂(1 − dV ) ∂κV = V,i−1 CV0 ∂κ ∂ ¯ V " #i−1 " V #i−1 ∂(1 − dID ) ∂κID 0 = ID,i−1 CD I ∂κID ∂¯ " # i−1 " D # i−1 RI ¯SI ∂(1 − dIT ) ∂κIT T,i−1 T,i−1 = CT0 I I I ∂κT ∂¯ γ γ ¯ T i−1 T i−1
(42)
whereby the partial derivatives of the damage moduli (1 − d) according to table 1 with respect to the evolution variables κ are given in table 2. Note, that the individual ∆CV , I ∆CD and ∆CTRS,I of equation (42) reduce to zero for un- and reloading. "
#
(
¯ V 6= 0 ∂κV 0 for Φ = ¯V = 0 1 for Φ ∂¯ " V #i−1 ( ¯ I 6= 0 ∂κID 0 for Φ D = I ¯I = 0 1 for Φ ∂¯ D " D # i−1 ( ¯ IT 6= 0 ∂κIT 0 for Φ = ¯I = 0 1 for Φ ∂¯ γTI i−1 T
11
(unloading) (loading) (unloading) (loading) (unloading) (loading)
(43)
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
"
"
"
∂(1 − dV ) ∂κV ∂(1 − dID ) ∂κID ∂(1 − dIT ) ∂κIT
tension
κV ≥ 0
compression
κV < 0
tension
κID ≥ 0
compression
κID < 0
tension
κIT ≥ 0
compression
κIT < 0
# i−1
# i−1
# i−1
κV p1 1 p1 p1 −1 exp − −p1 κV a1 " # " a1 # −p−1 −p κV κV q−1 q 1− + − a a b b " " # # p1 I p1 κD 1 p1 −1 exp − −p1 κD a1 " " a1 #p2 # p2 κI 1 exp − D −p2 κpD2 −1 a2 a2 " " # # p3 I p3 κT 1 p3 −1 exp − −p3 κT a3 3 " " #pa # p 3 −κIT 3 1 exp − −p3 [−κT ]p3 −1 a3 a3
Table 2: Linearization of microplane damage moduli (1 − d) Substitution of equation (38) into (34) yields the linearized form of the equilibrium equation which takes the familiar form of the purely displacement based finite element R ˜ ¯∆d¯ dΩ which only occurs for virgin formulation except for the coupling term Ω B Tu ∆CN loading. Z Z ˜ ¯ ∆d¯ dΩ ˜ i−1 B u ∆du dΩ + B T ∆CN B Tu C u Ω Z Z Ω Z (44) = N Tu f dΩ + N Tu tdΓ − B Tu σ i−1 dΩ Ω
Γ
Ω
The linearized averaging equation obtained from equation (35) by substituting (39) is given in the following form. − =
Z
N T¯ B u ∆du dΩ +
ZΩ Ω
Z Ω
(N T¯ N ¯ + B T¯ cB ¯)∆d¯ dΩ
(45)
(N T¯ N ¯ + B T¯ cB ¯)d¯,i−1 − N T¯ B u du,i−1 dΩ
The governing equations (44) and (45) can be combined in a system of equations including the enhanced element stiffness matrix, the vector of the unknowns and the external and internal element forces. "
K uu,i−1 K u¯,i−1 K ¯u,i−1 K ¯¯,i−1
#"
∆du ∆d¯
12
#
"
=
f ext u f ext ¯
#
"
−
f int u,i−1 f int ¯,i−1
#
(46)
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
Herein, the submatrices and subvectors are defined as follows. Z
K uu,i−1 = Z
Ω
Z
Ω
Z
Ω
Z
Ω
Z
Ω
K u¯,i−1 = K ¯u,i−1 = − K ¯¯,i−1
=
f int u,i−1
=
f ext u
=
f int ¯,i−1
= −
f ext ¯
=
Ω
˜ i−1 B u dΩ B Tu C ˜ ¯dΩ B Tu ∆CN N T¯ B u dΩ (N T¯ N ¯ + B T¯ cB ¯ )dΩ B Tu σ i−1 dΩ N Tu f dΩ
Z Ω
Z
+ Γ
N Tu tdΓ
(N T¯ N ¯ + B T¯ cB ¯)d¯,i−1 − N T¯ B u du,i−1 dΩ
0
Note, that the element stiffness matrix of gradient microplane damage becomes nonsymmetric since K u¯ 6= K T¯u . After assembling the global system of equations, which is of course nonsymmetric as well, a Newton Raphson method can be applied to determine the incremental update of the unknowns ∆du and ∆d¯. Remark The special case of a local continuum can be obtained by setting the internal length equal to zero, c = 0. Consequently, the nonlocal strain tensor ¯ becomes equal to the local ˜ and ∆C ˜ defines the consistent strain tensor . For this particular case, the sum of C lin ˜ of the classical continuum. tangent operator C ˜ + ∆C ˜ ˜ lin = C C
(47)
The consistent tangent operator can be applied to a localization analysis, see Kuhl & Ramm [17], to determine the local loss of ellipticity of the governing equations of the non-enhanced local microplane model. The classical continuum yields a finite number of solutions as long as the determinant of the acoustic tensor q, given by the double ˜ lin with all normals n, remains positive. As soon as det q becomes zero, contraction of C
˜ lin n = 0 det q = det n C the proposed gradient enhancement becomes necessary. 13
(48)
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
5
EXAMPLES
The following examples are based on a set of microplane parameters which have been chosen according to Baˇzant & Oˇzbolt [5]. The ten microplane parameters as well as the two elastic parameters E and ν are summarized in table 3. To avoid stress oscillations, it is important that the interpolations of the displacement field and the nonlocal strain field are balanced. Therefore, we will apply serendipity shape functions for the interpolation of the displacement field u whereas the interpolation of the nonlocal strain field ¯ will be bilinear, see Peerlings, de Borst, Brekelmans & de Vree [23]. In the following examples, both fields will be integrated numerically by a 2x2 point Gauss integration. a 0.05 b 0.035 a1 0.00006 p1 1.2 p 1.0 q 1.85 a2 0.0004 p2 1.1 E 20000 ν 0.18 a3 0.0004 p3 1.1 Table 3: Parameters for the microplane model
5.1
Tension versus compression loading
Concrete specimens show different failure modes under tensile and compressive loading. Subjected to a tensile loading, a concrete specimen fails due to decohesion whereas under compressive loading, frictional slip dominates the failure mechanism. The microplane model covers the two individual failure modes and a combination of both in a natural fashion. Figure 3 reflects the simulation of a bar with an imperfection subjected to tensile as well as compressive loading. Strain localization is triggered by reducing Young’s modulus of four elements in the middle of the bar by 5 %. Obviously, the two different failure modes are covered by the microplane model. The strains tend to localize in a fracture zone normal to the loading axis under tensile loading (figure 3, left). When the specimen is subjected to compression, however, a zone of localized shear can be observed under an angle of 450 (figure 3, right).
Figure 3: Strain distribution in bar with imperfection under tension and compression loading
14
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
5.2
Influence of the gradient enhancement
In the second example, the influence of the gradient term will be investigated. A bar of the length L=100mm will be analyzed under tensile loading. In order to trigger localization, the Young’s modulus has been reduced by 5 % in a l=10mm wide zone in the middle of the bar , see figure 4. The bar has been discretized with 40, 80 and 160 elements, respectively.
E = 19000 N/mm2 l = 10 mm h = 10 mm E = 20000 N/mm2 L = 100 mm Figure 4: One-dimensional bar with imperfection Figures 5 show the load deflection curves of the three different discretizations (left) as well as the influence of varying the gradient parameter for the discretization with 80 elements (right). The specimen response is almost identical for the three different discretizations. Due to the incorporation of strain gradients in the loading functions, a mesh independent solution with finite energy dissipation has been found. Obviously, the gradient parameter c influences the brittleness of the response, as can be seen in figure 5, right. For an increasing value of the gradient parameter, the response becomes more ductile and the load carrying capacity increases. The spreading of the localization zone for larger gradient parameters obviously delays the beginning of localization. The evolution of the strain distribution
1.5
1.0
0.5
80 elements stress [N/mm2]
stress [N/mm2]
c=5mm
40 el. 80 el. 160 el.
1.5
1.0
0.5 c=1 c=5 c=15 mm 0.000 0.002 0.004 0.006 0.008 elongation [mm]
0.000 0.002 0.004 0.006 0.008 elongation [mm]
Figure 5: One-dimensional bar with imperfection - load deflection curves 15
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
for the discretization with 80 elements is given in figure 6, which depicts five different loading stages, from top to bottom. A zone of concentrated straining can already be observed after a few load steps (figure 6, top). Upon further loading, a clear narrowing of the localization zone can be observed. Physically, this behavior can be explained by the coalescence of several microcracks in one single macroscopic crack pattern (figure 6, bottom). Nevertheless, the evolution of the nonlocal strains ¯ given in figure 7 is completely different. The width of concentrated straining remains unchanged throughout all steps. Although the amount of the nonlocal strains increases upon further loading, the width of the affected zone is already fixed upon its initiation. When comparing the first and the second loading stage of figures 6 and 7, both distributions are almost identical. Obviously, the influence of the strain gradients is not yet visible at this early stage of loading.
Figure 6: Evolution of strain distribution in bar with imperfection
5.3
Localization within a compression panel
The last example concerns a compression problem of a square panel clamped between two rigid platens. The friction coefficients between the loading platens and the specimen are assumed as infinite thus constraining the lateral movement of those boundaries. The gradient parameter has been chosen to c = 15mm2 . Only one quarter of the system is modelled by 5x5, 10x10 and 15x15 elements, respectively. The corresponding load deflection curves are given in figure 8. A slight difference between the curve of the coarsest mesh and the two other curves can be observed since the 5x5 element discretization only represents a very rough approximation of the problem. The two finer discretizations show almost identical behavior. The influence of the gradient enhancement can be analyzed by comparing the three different deformed configurations of figure 9 as well as the strain distributions given in figure 10. The shear band deformation pattern with an almost 16
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
mesh independent width highlights once more the regularizing influence of the gradient approach.
6
CONCLUSION
An anisotropic damage model motivated by the microplane theory has been presented. The examples have demonstrated the ability of the model to capture tension as well as compression failure. To account for microstructural effects, a gradient enhancement has been suggested. Due to the incorporation of gradient terms, which can be interpreted physically by microcrack interaction, the loss of ellipticity of the boundary value problem can be avoided. Consequently, the solution has been shown to remain mesh-independent. The presented model is considered a powerful approach to model the localized failure of quasi-brittle materials.
ACKNOWLEDGEMENT The present study is supported by the Von Humboldt Foundation and the Max Planck Society through the Max Planck Research Award 1996 as well as by grants of the German National Science Foundation (DFG) within the graduate program ’Modellierung und Diskretisierungsmethoden f¨ ur Kontinua und Str¨omungen’. This support is gratefully acknowledged.
17
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
200 mm
load factor
Figure 7: Evolution of nonlocal strain distribution ¯ in bar with imperfection
4.0
5x5 elements 10x10 elements 15x15 elements
3.0 2.0 1.0 0.0
200 mm
0.00
–0.02
–0.04
–0.06 –0.08 –0.10 elongation [mm]
Figure 8: Geometry and load deflection curve of compression panel
18
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
Figure 9: Deformed configurations of compression panel
Figure 10: Strain distribution of compression panel
19
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
References [1] E. Aifantis, On the microstructural origin of certain inelastic models, Journal of Engineering Materials and Technology 106 (1984) 326–330. [2] P.Z. Baˇzant and P.G. Gambarova, Crack shear in concrete: Crack band microplane model, Journal of Structural Engineering 110 (1984) 2015–2036. [3] P.Z. Baˇzant and G. Pijaudier-Cabot, Nonlocal damage, localization instability and convergence, Journal of Applied Mechanics 55 (1988) 287–293. [4] P.Z. Baˇzant and P. Prat, Microplane model for brittle plastic material, I. theory & II. verification, Journal of Engineering Mechanics 114 (1988) 1672–1702. [5] P.Z. Baˇzant and J. Oˇzbolt, Nonlocal microplane model for fracture, damage and size effects in structures, Journal of Engineering Mechanics 116 (1990) 2485–2505. [6] P.Z. Baˇzant, Why continuum damage is nonlocal: micromechanics arguments, Journal of Engineering Mechanics 117 (1991) 1070–1087. [7] R. de Borst and H.B. M¨ uhlhaus, Gradient-dependent plasticity: Formulation and algorithmic aspects, International Journal for Numerical Methods in Engineering 35 (1992) 521–539. [8] R. de Borst, J. Pamin, R.H.J. Peerlings and L.J. Sluys, On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials, Computational Mechanics 17 (1995) 130–141. [9] R. de Borst, A. Benallal and O. Heeres, A gradient-enhanced damage approach to fracture, Journal de Physique IV 6 (1996) 491–502. [10] I. Carol, P.Z. Baˇzant and P. Prat, Geometric damage tensor based on microplane model, Journal of Engineering Mechanics 117 (1991) 2429–2448. [11] I. Carol, P. Prat and P.Z. Baˇzant, New explicit microplane model for concrete: theoretical aspects and numerical implementation, International Journal of Solids and Structures 29 (1992) 1173–1191. [12] I. Carol and P.Z. Baˇzant, Damage and plasticity in microplane theory, International Journal of Solids and Structures 34 (1997) 3807–3835. [13] A.C. Eringen and D.G.B. Edelen, On nonlocal elasticity, International Journal for Engineering Science 10 (1972) 233–248. [14] M.G.D. Geers, Experimental analysis and computational modelling of damage and fracture (PhD Thesis, Eindhoven University of Technology, the Netherlands, 1997). [15] L.M. Kachanov, On the creep rupture time, Izv. Akad. Nauk SSR, Otd. Tekhn. Nauk 8 (1958) 26–31. [16] E. Kr¨oner, Elasticity theory of materials with long range cohesive forces, International Journal of Solids and Structures 3 (1967) 731–742. [17] E. Kuhl and E. Ramm, On the linearization of the microplane model, Mechancis of Cohesive Frictional Materials accepted for publication (1998). [18] D. Lasry and T. Belytschko, Localization limiters in transient problems, International Journal of Solids and Structures 24 (1988) 581–597. 20
Ellen Kuhl Ren´e de Borst and Ekkehard Ramm
[19] J. Lemaitre and J.L. Chaboche, Mechanics of Solid Materials (Cambridge University Press, Cambridge, 1990). [20] H.B. M¨ uhlhaus and E.C. Aifantis, A variational principle for gradient plasticity, International Journal of Solids and Structures 28 (1991) 845–857. [21] J. Oˇzbolt, Microplane model for quasi-brittle materials, Report Institut f¨ ur Werkstoffe im Bauwesen 96-1 (Universit¨at Stuttgart, Germany, 1996). [22] J. Pamin, Gradient-dependent plasticity in numerical simulation of localization phenomena (PhD Thesis, TU Delft, the Netherland, 1994). [23] R.H.J. Peerlings, R. de Borst, W.A.M. Brekelmans and J.H.P. de Vree, Gradientenhanced damage for quasi-brittle materials, International Journal for Numerical Methods in Engineering 39 (1996) 3391–3403. [24] R.H.J. Peerlings, R. de Borst, W.A.M. Brekelmans, J.H.P. de Vree and I. Spee, Some observations on localisation in non-local and gradient damage models, European Journal of Mechanics A 15 (1996) 937–953. [25] R.H.J. Peerlings, R. de Borst, W.A.M. Brekelmans and M.G.D. Geers, Gradient enhanced modelling of concrete fracture, Mechanics of Cohesive Frictional Materials accepted for publication (1998). [26] G. Pijaudier-Cabot and P.Z. Baˇzant, Nonlocal damage theory, Journal of Engineering Mechanics 113 (1987) 1512–1533. [27] J.C. Simo and J.W. Ju, Strain- and stress-based continuum damage models, International Journal of Solids and Structures 23 (1987) 821–869. [28] A.H. Stroud, Approximate calculation of multiple integrals, (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971). [29] G.I. Taylor, Plastic strain in metals, Journal of Inst. Metals 62 (1938) 307–324.
21
