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Generalized Kelvin model for micro-cracked viscoelastic materials
Engineering Fracture Mechanics 127 (2014) 226–234
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Generalized Kelvin model for micro-cracked viscoelastic materials S.T. Nguyen ⇑ Duy Tan University, Danang, Viet Nam
a r t i c l e
i n f o
Article history: Received 22 March 2014 Received in revised form 10 June 2014 Accepted 16 June 2014 Available online 25 June 2014 Keywords: Damage Viscoelasticity Micro-crack Generalized Kelvin model
a b s t r a c t The aim of this paper is to model the viscoelastic properties of micro-cracked materials based on the homogenization micro–macro approach. The isotropic case with random orientation distribution of micro-crack in Burgers nonageing linear viscoelastic solid was previously modeled. This study develops an alternative generalized Kelvin viscoelastic model (GKM) for fractured viscoelastic materials. The use of the same GKM of the noncracked materials to model the viscoelastic properties of the micro-cracked materials is an approximation. This is a new technique to avoid the complexity of the inverse Laplace–Carson (LC) transform. This approximation is carried out in short and long term behavior in the LC space and is validated in transient situation with exact solution obtained from the inverse LC transform for simple loading condition. This paper focuses on the isotropic case with random orientation distribution of open cracks. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In many materials like concrete or rocks, the presence of micro-cracks is permanent and affects the behavior of the materials [1,4,6,8,14,15]. For the case of concrete, the impact of micro-cracks on viscoelastic properties is attracting a good deal of attention nowadays. The micro–macro approach is largely developed to deal with the case of heterogeneous composite comprising micro-fractured materials [3,16]. The coupling between the micro–macro technique and the LC transform [5,12] allows modeling the evolution of the viscoelastic properties of linear nonageing viscoelastic heterogeneous materials [7,9,10]. The effective behavior of the heterogeneous media is firstly obtained in the LC space with help of the homogenization techniques and the inverse LC transform is usually employed to return to the real space [9]. The inverse LC is analytically complex and sometimes impossible. In such a case the numerical or semi-numerical methodologies are needed. The latter is also complex and consumes huge computational simulation time. This paper presents a very useful technique to avoid the complexity of the inverse LC transform by developing a set of explicit formulas for effective viscoelastic properties of heterogeneous media. For the case of isotropic damage viscoelastic materials with random distribution of micro-cracks, based on LC transform and on the linear relationships between the jumps of displacements across the cracks and the macroscopic stress, Nguyen et al. [11] developed explicit formulas for the evolution of eight viscoelastic parameters of the Burgers model as functions of the crack density parameter.
⇑ Tel.: +33 683041780. E-mail address: [email protected] http://dx.doi.org/10.1016/j.engfracmech.2014.06.010 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
227
Nomenclature fourth order stiffness tensor second order macroscopic stress tensor second order macroscopic strain tensor second order unit tensor Laplace–Carson variable time characteristic time bulk modulus shear modulus Poisson ratio viscosity crack density parameter exponent for the variables in Laplace–Carson space index for the Kelvin part of the viscoelastic rheological model exponent for the spherical part exponent for the deviatoric part index/exponent for short term behavior index/exponent for long term behavior exponent for homogenized values
C R E 1 p t
s k
l m g K s d o 1 hom
However, in many cases the viscoelastic behavior of the materials is modeled by the GKM. The alternative development of [11] for the more complicated viscoelastic behaviors such as the GKM is therefore necessary. As shown in [11], the LC transform yields the state equation of linear nonageing viscoelastic materials which relates the stress and the strain tensor into the linear form r ¼ C ðpÞ : e where the exponent stands for the LC transform and C ðpÞ is the apparent stiffness tensor in the LC space. For the case of isotropic viscoelastic solid, C ðpÞ depends on two scalars: appar ent bulk modulus k ðpÞ and apparent shear modulus l ðpÞ. The apparent Poisson coefficient m ðpÞ is commonly defined as:
3k 2l 6k þ 2l
m ¼
ð1Þ
Considering the so-called stress-based dilute scheme [2] of a single penny-shaped crack in an infinite elastic medium of stiffness C ðpÞ with a remote stress state R . Nguyen et al. [11] developed the homogenized bulk and shear moduli of the micro-cracked viscoelastic materials in LC space:
k
hom
k ¼ ð1 þ Q Þ
lhom ¼
l ð1 þ M Þ
with Q ¼
2 16 1 m
9ð1 2m Þ 32ð1 m Þð5 m Þ with M ¼ 45ð2 m Þ
ð2Þ
where k ; l and m are respectively the apparent bulk, shear moduli and the apparent Poisson ratio (see (1)) in LC space of the solid phase (non-cracked); is the crack density parameter which is defined by: ¼ Na3 with a is the radius of the cracks (the cracks are supposed to have the same radius) and N is the number of the cracks per unit volume of the micro-cracked material. Note that this solution is for the case of open cracks which is studied in this paper. To avoid the complexity due to the inverse LC transform, the idea shown in [11] is to approach the effective viscoelastic hom behavior of the micro-cracked material by the same viscoelastic model of the solid phase and by approaching k and lhom hom hom in the LC space by carrying out the short and long term behaviors. The idea is to develop k and l in polynomial series of the LC variable p in the vicinity of p ¼ 0 (long term) and of p ¼ 1 (short term) (see also [17,18]). This development depends on the viscoelastic behavior of the solid phase. The development in [11] is limited to the case of micro-cracked Burgers materials. To develop this concept for more complex viscoelastic behavior such as the GKM, the main idea is to consider (1) and (2) with k and l are functions of the viscoelastic properties of the solid phase that depend on the viscoelastic model chosen. Firstly, Section 2 provides a simple three elements rheological model (standard model for solid, Fig. 1). This model is then generalized in Section 3 for 2n þ 1 elements of n Kelvin’s systems (a spring in continue with a dash-pot) in parallel with a spring (Fig. 2). 2. Three elements model First we consider a three elements rheological model (two springs and one dash-pot) as shown in Fig. 1. This is a spring in continue with a Kelvin system (a second spring in parallel with a dash-pot). Considering the isotropic behavior, each spring is
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S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
Fig. 1. Rheological solid standard three elements model.
Fig. 2. Rheological generalized Kelvin model.
characterized by two elastic moduli (the bulk and the shear moduli) and the dash-pot is characterized by two viscosities (the bulk and the shear viscosities). The first spring defines the instantaneous elastic behavior of the material. The elastic moduli of this spring is then denoted by ko and lo . The bulk and the shear elastic moduli of the spring of the Kelvin system are denoted respectively by kK and lK . The bulk and the shear viscosities of the dash-pot of the Kelvin system are denoted 1 1 respectively by gs and gd . We note also by k1 ¼ k1K þ k1o and l1 ¼ l1 þ l1 the long term elastic bulk and shear moduli K
o
(when time tends to infinity). For this rheological three elements model, in LC space, the apparent bulk and the apparent shear moduli of the material are functions of the LC variable p and of the viscoelastic properties as:
1 1 1 þ ¼ kK þ pgs =3 ko k 1 1 1 ¼ þ l lK þ pgd =2 lo
ð3Þ
The series expansions of 1=k and 1=l in the vicinity of p ¼ 0 and of p ¼ 1 are:
1 1 1 1 þ OðpÞ; ¼ þ OðpÞ ¼ k1 l l1 k 1 1 1 3 1 1 1 1 2 1 þ þO 2 ; ¼ þ þO 2 p¼1: ¼ s d ko p g p p l lo p g k
p¼0:
ð4Þ
By introducing (3) in (1) and then in (2), we get Q and M . The series expansions of Q and of M in the vicinity of p ¼ 0 and of p ¼ 1 are:
p¼0: p ¼ 1;
Q ¼ Q oo þ OðpÞ;
M ¼ M oo þ OðpÞ 1 1 1 1 1 1 1 ; M Q M þ þ O ¼ M þ þ O Q ¼ Q1 o o p 1 p2 p 1 p2
ð5Þ
1 o 1 1 where the parameters Q oo ; Q 1 1 ; Q o , M o ; M 1 and M o are calculated by:
16 1 m21 3k1 2l1 ; m1 ¼ 6k1 þ 2l1 9ð1 2m1 Þ 2 2 4ko 4lo þ 6lo ko þ 9ko 1 1 ¼ Q1 1 2 sd ss 3lo ðlo þ 3ko Þ 2 16 1 mo 3ko 2lo Q1 ; mo ¼ o ¼ 6ko þ 2lo 9ð1 2mo Þ 32ð1 m1 Þð5 m1 Þ o Mo ¼ 45ð2 m1 Þ 2 48ko lo 16l2o þ 60lo ko þ 63ko 1 1 M1 2 2 1 ¼ sd ss 45 lo þ 3ko 2lo þ 3ko 32ð1 mo Þð5 mo Þ M1 o ¼ 45ð2 mo Þ
Q oo ¼
s
d
g With ss ¼ 3k and sd ¼ 2gl are the characteristic times. o o hom Combining (4) and (5) together with (2), the series expansions of 1=k and 1=lhom in the vicinity of p ¼ 0 are:
ð6Þ
ð7Þ ð8Þ ð9Þ
ð10Þ ð11Þ
S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
1 hom
k
1
lhom
1 1 þ Q oo þ OðpÞ k1 1 ¼ 1 þ M oo þ OðpÞ
229
¼
ð12Þ
l1
And in the vicinity of p ¼ 1 are:
1
hom
k
1 hom
l
1 1 3 1 1 1 1 þ O ð1 þ Q 1 ð1 þ Q Þ þ ð Q Þ o Þþ o 1 ko p gs ko p2 1 1 2 1 1 ¼ ð1 þ M 1 ð1 þ M1 ðM1 o Þþ o Þþ 1 Þ þ O d p g p2 lo lo
¼
ð13Þ
Now we assume that the effective viscoelastic behavior of the micro-cracked material is similar to that of the solid phase (non-cracked material) defined by the rheological model in Fig. 1. This assumption is an approximation and will be validated after by comparing with the exact solution (Fig. 6). The apparent effective bulk and shear moduli in the LC are then determined by (see (3)):
1 hom
k
1
lhom
1 1 þ kK ðÞ þ pgs ðÞ=3 ko ðÞ 1 1 ¼ þ lK ðÞ þ pgd ðÞ=2 lo ðÞ
¼
ð14Þ
where the viscoelastic properties kK ðÞ, gs ðÞ; ko ðÞ, lK ðÞ; gd ðÞ and lo ðÞ are functions of the crack density and of the vis hom coelastic properties of the solid phase. The series expansions of 1=k and 1=lhom determined by (14) in the vicinity of p ¼ 0 are:
1 hom
k
1
lhom
1 þ OðpÞ k1 ðÞ 1 ¼ þ OðpÞ l1 ðÞ
¼
ð15Þ
And in the vicinity of p ¼ 1 are:
1
hom
k
1
lhom
1 1 3 1 þ þ O ko ðÞ p gs ðÞ p2 1 1 2 1 ¼ þ þ O p2 lo ðÞ p gd ðÞ
¼
ð16Þ
where k11ðÞ ¼ ko1ðÞ þ kK1ðÞ and l 1ðÞ ¼ l 1ðÞ þ l 1ðÞ. The damaged stiffness and viscosity parameters, that are ka ðÞ; la ðÞ (a ¼ o or o
1
K
1Þ; gs ðÞ and gd ðÞ must ensure the compatibility between (12) and (15), as well as between (13) and (16). These conditions yield:
1 1 1 1 1 þ Q 1 1 þ Q oo ¼ ¼ o ; ko ðÞ ko k1 ðÞ k1 1 1 1 1 1 1 s ; ð1 þ Q 1 ¼ ¼ o Þ þ s Q 1 kK ðÞ k1 ðÞ ko ðÞ gs ðÞ gs
ð17Þ
And
1
lo ðÞ
¼
1
lo
1 þ M 1 o ;
1 1 1 ; ¼ lK ðÞ l1 ðÞ lo ðÞ
1
l1 ð Þ
¼
1
l1
1 þ M oo
1 1 1 d ð1 þ M 1 ¼ o Þ þ s M 1 gd ðÞ gd
ð18Þ
Note that the solutions ko ðÞ and lo ðÞ of the short term elastic behavior and the solutions k1 ðÞ and l1 ðÞ of the long term elastic behavior given by (17) and (18) are exactly the same results obtained for the fractured elastic media with random orientation distribution of open cracks by using the classical Mori–Tanaka approach (see [3,2] for example). The results in (17) and (18) will be developed in the next section for GKM and will be validated with exact solution obtained from direct inverse LC transform for a specific loading case (Fig. 6). 3. Generalized Kelvin model The solution developed in Section 2 will be generalized for the case of 2n þ 1 elements of with a spring in continue with n Kelvin systems (Fig. 2 presents for example the case of n ¼ 2Þ.
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S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
Similar to (3), the apparent bulk and shear moduli of the material in the LC space are determined by:
1 1 X 1 þ ¼ ko k þ p gsi =3 k i i 1 1 X 1 ¼ þ l lo li þ pgdi =2 i
ð19Þ
where ka and la (a ¼ i or oÞ denote the bulk and shear moduli of the Kelvin system i and of the first spring which defines the instantaneous elastic behavior, whereas gsi and gdi represent the bulk and the viscosity of the Kelvin system i, respectively. Similar to (4), the series expansions of 1=k and of 1=l in the vicinity of p ¼ 0 and in the vicinity of p ¼ 1 are:
1 1 1 1 þ OðpÞ; ¼ þ OðpÞ ¼ k1 l l1 k 1 1 1X 3 1 1 1 1X 2 1 ; þ þ O ¼ þ þ O p¼1: ¼ ko p i gsi p2 p2 l lo p i gdi k
p¼0:
where
1 k1
¼ k1o þ
P
1 i ki
and l1 ¼ l1 þ 1
P
o
i
1
li
ð20Þ
are respectively the asymptotic bulk and the asymptotic shear elastic moduli of the
materials when time tends to infinity. By introducing (20) in (1) and then in (2), we get Q and M . The series expansions of Q and of M in the vicinity of p ¼ 0 1 o 1 1 and of p ¼ 1 takes the same form of (5) where the parameters Q oo ; Q 1 1 ; Q o , M o ; M 1 and M o are calculated by (6)–(11). Note P 1 P 1 1 1 and sd ¼ 2l1 . that the characteristic times appear in (7) and (10) are recalculated by: ss ¼ 3k1o i gs i gd o
i
hom
Combining (20) and (5) together with (2), the series expansions of 1=k
1
i
and 1=lhom in the vicinity of p ¼ 0 are:
1 1 þ Q oo þ OðpÞ k1 1 ¼ 1 þ M oo þ OðpÞ
¼
hom
k
1
lhom
ð21Þ
l1
And in the vicinity of p ¼ 1 are:
hom
k
1
lhom
!
1 þO 2 p ! 1 1 X2 1 1 1 1 ¼ ð1 þ M 1 Þ þ ð1 þ M Þ þ M þ O o o p p2 lo lo 1 gdi i
1 1 X3 1 ¼ ð1 þ Q 1 ð1 þ Q 1 Q 1 o Þþ o Þþ ko p ko 1 gsi i
1
ð22Þ
As the case of three elements developed in Section 2, we suppose that the effective viscoelastic behavior of the cracked material is also modeled by the GKM as the behavior of the solid phase. The apparent bulk and shear moduli of the cracked material in the LC space are determined by (see (19)):
1
X 1 1 þ ko ðÞ k ð Þ þ pgsi ðÞ=3 i i X 1 1 ¼ þ lo ðÞ i li ðÞ þ pgdi ðÞ=2
¼
hom
k
1
lhom
hom
The series expansions of 1=k
1
¼
hom
k
1 þ OðpÞ; k1 ðÞ
1
lhom
ð23Þ
and 1=lhom determined by (23) in the vicinity of p ¼ 0 are:
¼
1
l1 ðÞ
þ OðpÞ
ð24Þ
And in the vicinity of p ¼ 1 are:
1 hom
k
1 1X 3 1 þ þ O ko ðÞ p i gsi ðÞ p2 1 1X 2 1 þ O ¼ þ p2 lo ðÞ p i gdi ðÞ
¼
1
lhom
P and l 1ðÞ ¼ l 1ðÞ þ i l 1ðÞ. The damaged stiffness and viscosity parameters, that are ka ðÞ; la ðÞ 1 o i (a ¼ o or 1Þ; gsi ðÞ and gdi ðÞ must ensure the compatibility between (21) and (24), as well as between (22) and (25). These conditions yield: where
1 k1 ðÞ
¼ ko1ðÞ þ
P
ð25Þ
1 i ki ðÞ
S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
1 1 1 1 1 þ Q 1 1 þ Q oo ¼ ¼ o ; ko ðÞ ko k1 ðÞ k1 ! ! X1 X 1 X1 X 1 AðÞ; BðÞ ¼ ¼ ki ðÞ ki gsi ðÞ gsi i i i i
231
ð26Þ
And for the deviatoric part:
1 1 1 þ M 1 1 þ M oo ¼ o ; l1 ð Þ l1 ! ! X 1 X1 X 1 X1 ¼ ¼ CðÞ; DðÞ li ðÞ li gdi ðÞ gdi i i i i 1
lo ðÞ
¼
1
lo
ð27Þ
where to simplify the formulas, we noted:
AðÞ ¼ 1 þ Q oo þ a Q oo Q 1 o 1 s BðÞ ¼ ð1 þ Q 1 o Þ þ s Q 1 o o CðÞ ¼ 1 þ Mo þ b Mo M1 o 1 d DðÞ ¼ ð1 þ M 1 o Þ þ s M 1
ð28Þ
With
a¼
1 X1 ko ki i
!1 ;
b¼
X1
1
lo
i
!1 ð29Þ
li
As concluded for the case of three elements, the solutions ko ðÞ and lo ðÞ of the short term elastic behavior and the solutions k1 ðÞ and l1 ðÞ of the long term elastic behavior given by (26) and (27) are exactly the same results obtained for the fractured elastic media with random orientation distribution of open cracks by using the classical Mori–Tanaka approach [3,2]. The solutions in (26) and (27) do not give the explicit estimation of ki ðÞ; gsi ðÞ; li ðÞ and gdi ðÞ but the sum of them over all the Kelvin systems. To estimate the effective viscoelastic parameter of each single Kelvin system, we assume the following approximations:
ki ðÞ ¼
ki ; AðÞ
gsi ðÞ ¼
gsi BðÞ
;
li ðÞ ¼
gd li ; gdi ðÞ ¼ i CðÞ DðÞ
ð30Þ
where the functions AðÞ; BðÞ; CðÞ, DðÞ are detailed in (28). Finally, Eqs. (26), (27) and (30) allow us to write the following explicit dimensionless formulas for the determination of the effective viscoelastic properties of the micro-cracked media:
ko ðÞ 1 ; ¼ ko 1 þ Q 1 o k i ð Þ 1 ¼ ; ki AðÞ
lo ðÞ 1 ¼ lo 1 þ M 1 o
li ðÞ 1 ¼ CðÞ li gsi ðÞ gdi ðÞ 1 1 ¼ ; ¼ BðÞ DðÞ gsi gdi
ð31Þ ð32Þ ð33Þ
Fig. 3. Approximation of the effective behavior of a micro-cracked medium by a generalized Kelvin model – the case of five elements: k0 ¼ 24:42 GPa, lo ¼ 13:27 GPa, k1 ¼ 11:22 GPa, l1 ¼ 6:03 GPa, k2 ¼ 11:22 GPa, l2 ¼ 3:015 GPa, gs1 ¼ 91:2 106 GPa s, gd1 ¼ 10:16 106 GPa s, gs2 ¼ 60:8 106 GPa s, gd2 ¼ 15:24 106 GPa s.
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S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
Fig. 3 presents the dependence of ten effective viscoelastic properties (normalized by the viscoelastic properties of the solid phase as shown in Eqs. (31)–(33)) of a five elements model (n ¼ 2) on the damage parameter . To validate the developed model on transient situation, we consider the creep problem under constant isotropic stress traction:
R ¼ Ro HðtÞ1
ð34Þ
where Ro is a given constant stress, HðtÞ is the Heaviside function and 1 is the second order unit tensor. The LC transform of this stress takes the form:
R ¼ Ro 1
ð35Þ
The macroscopic behavior in the LC space is written as:
E ðpÞ ¼
Ro 3k
hom
1
ð36Þ
The macroscopic strain is then isotropic and takes the form: E ðpÞ ¼ E ðpÞ1 with
E ðpÞ ¼
Ro 3k
ð37Þ
hom
hom
where k is calculated from the effective viscoelastic model (23). The inverse LC transform of E ðpÞ in (37) gives the macroscopic strain component EðtÞ in real space.
EðtÞ ¼
Ro 3
1 1 t 1 t Exp Exp k1 ðÞ k1 ðÞ k2 ðÞ s1 ðÞ s2 ð Þ
ð38Þ
With
s1 ðÞ ¼
gs1 ðÞ 3k1 ðÞ
¼
gs1 AðÞ 3k1 BðÞ
¼ s1
AðÞ BðÞ
ð39Þ
and
s2 ðÞ ¼
gs2 ðÞ 3k2 ðÞ
¼
gs2 AðÞ 3k2 BðÞ
¼
s1 ðÞ h
ð40Þ
are the effective characteristic times. Note that h ¼ s1 ðÞ=s2 ðÞ is a dimensionless parameter that iss independent of the crack g density parameter . s1 is the first characteristic time of the material without micro-crack s1 ¼ 3k11 . At t ¼ 0, the solution (38) becomes:
Eo ¼ Eðt ¼ 0Þ ¼
Ro 3ko ðÞ
ð41Þ
This allows us to write (38) under the dimensionless form:
EðtÞ ko ðÞ t ko ðÞ t þ ¼1þ 1 Exp 1 Exp Eo k1 ðÞ k2 ðÞ s1 ðÞ s2 ðÞ
ð42Þ
Fig. 4. Effect of the crack density parameter on the macroscopic strain – example of 5 elements model: k0 ¼ 24:42 GPa;lo ¼ 13:27 GPa, k1 ¼ 11:22 GPa, l1 ¼ 6:03 GPa, k2 ¼ 11:22 GPa, l2 ¼ 3:015 GPa, gs1 ¼ 91:2 106 GPa s, gd1 ¼ 10:16 106 GPa s, gs2 ¼ 60:8 106 GPa s, gd2 ¼ 15:24 106 GPa s, Ro ¼ 1 Mpa.
S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
Fig. 5. Effect of the ratio k2 =k1 on the macroscopic strain – example of 5 elements model: k0 ¼ 24:42 GPa, lo ¼ 13:27 GPa, k1 ¼ 11:22 GPa, l2 ¼ 3:015 GPa, gs1 ¼ 91:2 106 GPa s, gd1 ¼ 10:16 106 GPa s, gs2 ¼ 60:8 106 GPa s, gd2 ¼ 15:24 106 GPa s, Ro ¼ 1 Mpa.
233
l1 ¼ 6:03 GPa,
Fig. 6. Validation of the effective generalized Kelvin model in transient situation – example of 5 elements model: k0 ¼ 24:42 GPa, lo ¼ 13:27 GPa, k1 ¼ 11:22 GPa, l1 ¼ 6:03 GPa, k2 ¼ 11:22 GPa, l2 ¼ 3:015 GPa, gs1 ¼ 91:2 106 GPa s, gd1 ¼ 10:16 106 GPa s, gs2 ¼ 60:8 106 GPa s, gd2 ¼ 15:24 106 GPa s, Ro ¼ 1 Mpa.
By introducing (31)–(33) into (42) we obtains:
Eðt Þ AðÞ ko BðÞ t ko BðÞ t þ ¼1þ 1 Exp 1 Exp h Eo AðÞ s1 AðÞ s1 k1 k2 1 þ Q 1 o
ð43Þ
For long term behavior when t ! 1, the macroscopic strain (normalized by the instantaneous macroscopic strain when t ¼ 0Þ is reduced to:
E1 AðÞ ko ko ¼1þ þ Eo k1 k2 1 þ Q 1 o
ð44Þ
Figs. 4 and 5 show the evolution of the macroscopic strain (normalized by the instantaneous macroscopic strain) versus time (normalized by the characteristic time s1 Þ with effects of crack density and of viscoelastic properties of non-cracked material (Fig. 5 is an example of effect of k2 Þ. It is interesting to observe that more the material is damaged (high crack density) more the long term macroscopic strain is high compared with the instantaneous macroscopic strain. Similar observation is shown in Fig. 5, higher elastic property k2 gives high ratio EEðotÞ. Note that, for the simple loading condition (34), the analytical solution of the macroscopic strain can be obtained exactly by employing the transverse LC transform. The solution (38) is compared with the exact solution obtained by considering the
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S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234 hom
inverse LC transform of E ðpÞ in (37) with k calculated directly from (2). Fig. 6 shows an excellent validation of the effective generalized Kelvin model (5 elements model) in short, long terms and transient situation. 4. Conclusions The effective viscoelastic models established previously allows us to determine explicitly the effective viscoelastic properties of micro-cracked generalized Kelvin material given the damage parameter and the viscoelastic moduli of the nondamaged material. These results allow us to easily and fast determine the effective behavior of a micro-cracked viscoelastic behavior, avoiding the complexity and the computational effort of the inverse LC transform. The approximations are derived from short and long terms behavior. The short and the long terms solutions are identical to that estimated for the elastic case based on Mori–Tanaka approach [3,2]. The comparison with the exact solution of a creep problem under constant isotropic traction stress shows excellent validations of the effective viscoelastic models on transient situation. This study is limited to the cases of random orientation distribution of open cracks in linear nonageing viscoelastic materials. Farther development taking into account of the complex micro-crack distribution, crack closure, complex ageing viscoelastic behavior or viscoplastic behavior [13,19,20] and of the crack propagation [11b,c] could be the perspective of this study. References [1] Budiansky B, O’connell RJ. Elastic moduli of a cracked solid. Int J Solids Struct 1976;12(2):81–97. [2] Dormieux L, Kondo D. Stress-based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects. Comput Mater Sci 2009;46(1):173–9. [3] Dormieux L, Kondo D, Ulm FJ. Microporomechanics. John Wiley & Sons; 2006. [4] Evans RH, Marathe MS. Microcracking and stress–strain curves for concrete in tension. Mater Struct 1968;1(1):61–4. [5] Haddad YM. Viscoelasticity of engineering materials. Springer; 1995. [6] Hillerborg A, Modéer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 1976;6(6):773–81. [7] Hoang-Duc H, Bonnet G, Meftah F. Generalized self-consistent scheme for the effective behavior of viscoelastic heterogeneous media: a simple approximate solution. Eur J Mech – A/Solids 2013;39:35–49. [8] Horii H, Nemat-Nasser S. Compression-induced microcrack growth in brittle solids: axial splitting and shear failure. J Geophys Res: Solid Earth (1978– 2012) 1985;90(B4):3105–25. [9] Le QV, Meftah F, He QC, Le Pape Y. Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori–Tanaka homogenization scheme and a discrete microscopic retardation spectrum. Mech Time-Depen Mater 2007;11(3–4):309–31. [10] Lévesque M, Gilchrist MD, Bouleau N, Derrien K, Baptiste D. Numerical inversion of the Laplace–Carson transform applied to homogenization of randomly reinforced linear viscoelastic media. Comput Mech 2007;40(4):771–89. [11] Nguyen ST, Dormieux L, Le Pape Y, Sanahuja J. A Burger model for the effective behavior of a microcracked viscoelastic solid. Int J Damage Mech 2011;20(8):1116–29. [11b] Nguyen ST, Dormieux L, Pape YL, Sanahuja J. Crack propagation in viscoelastic structures: theoretical and numerical analyses. Comput Mater Sci 2010;50(1):83–91. [11c] Nguyen ST, Jeannin L, Dormieux L, Renard F. Fracturing of viscoelastic geomaterials and application to sedimentary layered rocks. Mech Res Commun 2013;49(0):50–6. [12] Salençon J. Viscosité. Paris: Presses de l’ENPC; 1983. [13] Sanahuja J. Effective behaviour of ageing linear viscoelastic composites: homogenization approach. Int J Solids Struct 2013;50(19):2846–56. [14] Sayers CM, Kachanov M. Microcrack-induced elastic wave anisotropy of brittle rocks. J Geophys Res: Solid Earth (1978–2012) 1995;100(B3):4149–56. [15] Zoback MD, Byerlee JD. The effect of microcrack dilatancy on the permeability of Westerly granite. J Geophys Res 1975;80(5):752–5. [16] Zhu QZ, Kondo D, Shao JF. Homogenization-based analysis of anisotropic damage in brittle materials with unilateral effect and interactions between microcracks. Int J Numer Anal Methods Geomech 2009;33(6):749–72. [17] Brenner R, Suquet P. Overall response of viscoelastic composites and polycrystals: exact asymptotic relations and approximate estimates. Int J Solids Struct 2013;50(10):1824–38. [18] Suquet P. Four exact relations for the effective relaxation function of linear viscoelastic composites. CR Méc 2012;340(4):387–99. [19] Masson R, Brenner R, Castelnau O. Incremental homogenization approach for ageing viscoelastic polycrystals. CR Méc 2012;340(4):378–86. [20] Miled B, Doghri I, Brassart L, Delannay L. Micromechanical modeling of coupled viscoelastic–viscoplastic composites based on an incrementally affine formulation. Int J Solids Struct 2013;50(10):1755–69.
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Generalized Kelvin model for micro-cracked viscoelastic materials S.T. Nguyen ⇑ Duy Tan University, Danang, Viet Nam
a r t i c l e
i n f o
Article history: Received 22 March 2014 Received in revised form 10 June 2014 Accepted 16 June 2014 Available online 25 June 2014 Keywords: Damage Viscoelasticity Micro-crack Generalized Kelvin model
a b s t r a c t The aim of this paper is to model the viscoelastic properties of micro-cracked materials based on the homogenization micro–macro approach. The isotropic case with random orientation distribution of micro-crack in Burgers nonageing linear viscoelastic solid was previously modeled. This study develops an alternative generalized Kelvin viscoelastic model (GKM) for fractured viscoelastic materials. The use of the same GKM of the noncracked materials to model the viscoelastic properties of the micro-cracked materials is an approximation. This is a new technique to avoid the complexity of the inverse Laplace–Carson (LC) transform. This approximation is carried out in short and long term behavior in the LC space and is validated in transient situation with exact solution obtained from the inverse LC transform for simple loading condition. This paper focuses on the isotropic case with random orientation distribution of open cracks. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In many materials like concrete or rocks, the presence of micro-cracks is permanent and affects the behavior of the materials [1,4,6,8,14,15]. For the case of concrete, the impact of micro-cracks on viscoelastic properties is attracting a good deal of attention nowadays. The micro–macro approach is largely developed to deal with the case of heterogeneous composite comprising micro-fractured materials [3,16]. The coupling between the micro–macro technique and the LC transform [5,12] allows modeling the evolution of the viscoelastic properties of linear nonageing viscoelastic heterogeneous materials [7,9,10]. The effective behavior of the heterogeneous media is firstly obtained in the LC space with help of the homogenization techniques and the inverse LC transform is usually employed to return to the real space [9]. The inverse LC is analytically complex and sometimes impossible. In such a case the numerical or semi-numerical methodologies are needed. The latter is also complex and consumes huge computational simulation time. This paper presents a very useful technique to avoid the complexity of the inverse LC transform by developing a set of explicit formulas for effective viscoelastic properties of heterogeneous media. For the case of isotropic damage viscoelastic materials with random distribution of micro-cracks, based on LC transform and on the linear relationships between the jumps of displacements across the cracks and the macroscopic stress, Nguyen et al. [11] developed explicit formulas for the evolution of eight viscoelastic parameters of the Burgers model as functions of the crack density parameter.
⇑ Tel.: +33 683041780. E-mail address: [email protected] http://dx.doi.org/10.1016/j.engfracmech.2014.06.010 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.
S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
227
Nomenclature fourth order stiffness tensor second order macroscopic stress tensor second order macroscopic strain tensor second order unit tensor Laplace–Carson variable time characteristic time bulk modulus shear modulus Poisson ratio viscosity crack density parameter exponent for the variables in Laplace–Carson space index for the Kelvin part of the viscoelastic rheological model exponent for the spherical part exponent for the deviatoric part index/exponent for short term behavior index/exponent for long term behavior exponent for homogenized values
C R E 1 p t
s k
l m g K s d o 1 hom
However, in many cases the viscoelastic behavior of the materials is modeled by the GKM. The alternative development of [11] for the more complicated viscoelastic behaviors such as the GKM is therefore necessary. As shown in [11], the LC transform yields the state equation of linear nonageing viscoelastic materials which relates the stress and the strain tensor into the linear form r ¼ C ðpÞ : e where the exponent stands for the LC transform and C ðpÞ is the apparent stiffness tensor in the LC space. For the case of isotropic viscoelastic solid, C ðpÞ depends on two scalars: appar ent bulk modulus k ðpÞ and apparent shear modulus l ðpÞ. The apparent Poisson coefficient m ðpÞ is commonly defined as:
3k 2l 6k þ 2l
m ¼
ð1Þ
Considering the so-called stress-based dilute scheme [2] of a single penny-shaped crack in an infinite elastic medium of stiffness C ðpÞ with a remote stress state R . Nguyen et al. [11] developed the homogenized bulk and shear moduli of the micro-cracked viscoelastic materials in LC space:
k
hom
k ¼ ð1 þ Q Þ
lhom ¼
l ð1 þ M Þ
with Q ¼
2 16 1 m
9ð1 2m Þ 32ð1 m Þð5 m Þ with M ¼ 45ð2 m Þ
ð2Þ
where k ; l and m are respectively the apparent bulk, shear moduli and the apparent Poisson ratio (see (1)) in LC space of the solid phase (non-cracked); is the crack density parameter which is defined by: ¼ Na3 with a is the radius of the cracks (the cracks are supposed to have the same radius) and N is the number of the cracks per unit volume of the micro-cracked material. Note that this solution is for the case of open cracks which is studied in this paper. To avoid the complexity due to the inverse LC transform, the idea shown in [11] is to approach the effective viscoelastic hom behavior of the micro-cracked material by the same viscoelastic model of the solid phase and by approaching k and lhom hom hom in the LC space by carrying out the short and long term behaviors. The idea is to develop k and l in polynomial series of the LC variable p in the vicinity of p ¼ 0 (long term) and of p ¼ 1 (short term) (see also [17,18]). This development depends on the viscoelastic behavior of the solid phase. The development in [11] is limited to the case of micro-cracked Burgers materials. To develop this concept for more complex viscoelastic behavior such as the GKM, the main idea is to consider (1) and (2) with k and l are functions of the viscoelastic properties of the solid phase that depend on the viscoelastic model chosen. Firstly, Section 2 provides a simple three elements rheological model (standard model for solid, Fig. 1). This model is then generalized in Section 3 for 2n þ 1 elements of n Kelvin’s systems (a spring in continue with a dash-pot) in parallel with a spring (Fig. 2). 2. Three elements model First we consider a three elements rheological model (two springs and one dash-pot) as shown in Fig. 1. This is a spring in continue with a Kelvin system (a second spring in parallel with a dash-pot). Considering the isotropic behavior, each spring is
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S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
Fig. 1. Rheological solid standard three elements model.
Fig. 2. Rheological generalized Kelvin model.
characterized by two elastic moduli (the bulk and the shear moduli) and the dash-pot is characterized by two viscosities (the bulk and the shear viscosities). The first spring defines the instantaneous elastic behavior of the material. The elastic moduli of this spring is then denoted by ko and lo . The bulk and the shear elastic moduli of the spring of the Kelvin system are denoted respectively by kK and lK . The bulk and the shear viscosities of the dash-pot of the Kelvin system are denoted 1 1 respectively by gs and gd . We note also by k1 ¼ k1K þ k1o and l1 ¼ l1 þ l1 the long term elastic bulk and shear moduli K
o
(when time tends to infinity). For this rheological three elements model, in LC space, the apparent bulk and the apparent shear moduli of the material are functions of the LC variable p and of the viscoelastic properties as:
1 1 1 þ ¼ kK þ pgs =3 ko k 1 1 1 ¼ þ l lK þ pgd =2 lo
ð3Þ
The series expansions of 1=k and 1=l in the vicinity of p ¼ 0 and of p ¼ 1 are:
1 1 1 1 þ OðpÞ; ¼ þ OðpÞ ¼ k1 l l1 k 1 1 1 3 1 1 1 1 2 1 þ þO 2 ; ¼ þ þO 2 p¼1: ¼ s d ko p g p p l lo p g k
p¼0:
ð4Þ
By introducing (3) in (1) and then in (2), we get Q and M . The series expansions of Q and of M in the vicinity of p ¼ 0 and of p ¼ 1 are:
p¼0: p ¼ 1;
Q ¼ Q oo þ OðpÞ;
M ¼ M oo þ OðpÞ 1 1 1 1 1 1 1 ; M Q M þ þ O ¼ M þ þ O Q ¼ Q1 o o p 1 p2 p 1 p2
ð5Þ
1 o 1 1 where the parameters Q oo ; Q 1 1 ; Q o , M o ; M 1 and M o are calculated by:
16 1 m21 3k1 2l1 ; m1 ¼ 6k1 þ 2l1 9ð1 2m1 Þ 2 2 4ko 4lo þ 6lo ko þ 9ko 1 1 ¼ Q1 1 2 sd ss 3lo ðlo þ 3ko Þ 2 16 1 mo 3ko 2lo Q1 ; mo ¼ o ¼ 6ko þ 2lo 9ð1 2mo Þ 32ð1 m1 Þð5 m1 Þ o Mo ¼ 45ð2 m1 Þ 2 48ko lo 16l2o þ 60lo ko þ 63ko 1 1 M1 2 2 1 ¼ sd ss 45 lo þ 3ko 2lo þ 3ko 32ð1 mo Þð5 mo Þ M1 o ¼ 45ð2 mo Þ
Q oo ¼
s
d
g With ss ¼ 3k and sd ¼ 2gl are the characteristic times. o o hom Combining (4) and (5) together with (2), the series expansions of 1=k and 1=lhom in the vicinity of p ¼ 0 are:
ð6Þ
ð7Þ ð8Þ ð9Þ
ð10Þ ð11Þ
S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
1 hom
k
1
lhom
1 1 þ Q oo þ OðpÞ k1 1 ¼ 1 þ M oo þ OðpÞ
229
¼
ð12Þ
l1
And in the vicinity of p ¼ 1 are:
1
hom
k
1 hom
l
1 1 3 1 1 1 1 þ O ð1 þ Q 1 ð1 þ Q Þ þ ð Q Þ o Þþ o 1 ko p gs ko p2 1 1 2 1 1 ¼ ð1 þ M 1 ð1 þ M1 ðM1 o Þþ o Þþ 1 Þ þ O d p g p2 lo lo
¼
ð13Þ
Now we assume that the effective viscoelastic behavior of the micro-cracked material is similar to that of the solid phase (non-cracked material) defined by the rheological model in Fig. 1. This assumption is an approximation and will be validated after by comparing with the exact solution (Fig. 6). The apparent effective bulk and shear moduli in the LC are then determined by (see (3)):
1 hom
k
1
lhom
1 1 þ kK ðÞ þ pgs ðÞ=3 ko ðÞ 1 1 ¼ þ lK ðÞ þ pgd ðÞ=2 lo ðÞ
¼
ð14Þ
where the viscoelastic properties kK ðÞ, gs ðÞ; ko ðÞ, lK ðÞ; gd ðÞ and lo ðÞ are functions of the crack density and of the vis hom coelastic properties of the solid phase. The series expansions of 1=k and 1=lhom determined by (14) in the vicinity of p ¼ 0 are:
1 hom
k
1
lhom
1 þ OðpÞ k1 ðÞ 1 ¼ þ OðpÞ l1 ðÞ
¼
ð15Þ
And in the vicinity of p ¼ 1 are:
1
hom
k
1
lhom
1 1 3 1 þ þ O ko ðÞ p gs ðÞ p2 1 1 2 1 ¼ þ þ O p2 lo ðÞ p gd ðÞ
¼
ð16Þ
where k11ðÞ ¼ ko1ðÞ þ kK1ðÞ and l 1ðÞ ¼ l 1ðÞ þ l 1ðÞ. The damaged stiffness and viscosity parameters, that are ka ðÞ; la ðÞ (a ¼ o or o
1
K
1Þ; gs ðÞ and gd ðÞ must ensure the compatibility between (12) and (15), as well as between (13) and (16). These conditions yield:
1 1 1 1 1 þ Q 1 1 þ Q oo ¼ ¼ o ; ko ðÞ ko k1 ðÞ k1 1 1 1 1 1 1 s ; ð1 þ Q 1 ¼ ¼ o Þ þ s Q 1 kK ðÞ k1 ðÞ ko ðÞ gs ðÞ gs
ð17Þ
And
1
lo ðÞ
¼
1
lo
1 þ M 1 o ;
1 1 1 ; ¼ lK ðÞ l1 ðÞ lo ðÞ
1
l1 ð Þ
¼
1
l1
1 þ M oo
1 1 1 d ð1 þ M 1 ¼ o Þ þ s M 1 gd ðÞ gd
ð18Þ
Note that the solutions ko ðÞ and lo ðÞ of the short term elastic behavior and the solutions k1 ðÞ and l1 ðÞ of the long term elastic behavior given by (17) and (18) are exactly the same results obtained for the fractured elastic media with random orientation distribution of open cracks by using the classical Mori–Tanaka approach (see [3,2] for example). The results in (17) and (18) will be developed in the next section for GKM and will be validated with exact solution obtained from direct inverse LC transform for a specific loading case (Fig. 6). 3. Generalized Kelvin model The solution developed in Section 2 will be generalized for the case of 2n þ 1 elements of with a spring in continue with n Kelvin systems (Fig. 2 presents for example the case of n ¼ 2Þ.
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S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
Similar to (3), the apparent bulk and shear moduli of the material in the LC space are determined by:
1 1 X 1 þ ¼ ko k þ p gsi =3 k i i 1 1 X 1 ¼ þ l lo li þ pgdi =2 i
ð19Þ
where ka and la (a ¼ i or oÞ denote the bulk and shear moduli of the Kelvin system i and of the first spring which defines the instantaneous elastic behavior, whereas gsi and gdi represent the bulk and the viscosity of the Kelvin system i, respectively. Similar to (4), the series expansions of 1=k and of 1=l in the vicinity of p ¼ 0 and in the vicinity of p ¼ 1 are:
1 1 1 1 þ OðpÞ; ¼ þ OðpÞ ¼ k1 l l1 k 1 1 1X 3 1 1 1 1X 2 1 ; þ þ O ¼ þ þ O p¼1: ¼ ko p i gsi p2 p2 l lo p i gdi k
p¼0:
where
1 k1
¼ k1o þ
P
1 i ki
and l1 ¼ l1 þ 1
P
o
i
1
li
ð20Þ
are respectively the asymptotic bulk and the asymptotic shear elastic moduli of the
materials when time tends to infinity. By introducing (20) in (1) and then in (2), we get Q and M . The series expansions of Q and of M in the vicinity of p ¼ 0 1 o 1 1 and of p ¼ 1 takes the same form of (5) where the parameters Q oo ; Q 1 1 ; Q o , M o ; M 1 and M o are calculated by (6)–(11). Note P 1 P 1 1 1 and sd ¼ 2l1 . that the characteristic times appear in (7) and (10) are recalculated by: ss ¼ 3k1o i gs i gd o
i
hom
Combining (20) and (5) together with (2), the series expansions of 1=k
1
i
and 1=lhom in the vicinity of p ¼ 0 are:
1 1 þ Q oo þ OðpÞ k1 1 ¼ 1 þ M oo þ OðpÞ
¼
hom
k
1
lhom
ð21Þ
l1
And in the vicinity of p ¼ 1 are:
hom
k
1
lhom
!
1 þO 2 p ! 1 1 X2 1 1 1 1 ¼ ð1 þ M 1 Þ þ ð1 þ M Þ þ M þ O o o p p2 lo lo 1 gdi i
1 1 X3 1 ¼ ð1 þ Q 1 ð1 þ Q 1 Q 1 o Þþ o Þþ ko p ko 1 gsi i
1
ð22Þ
As the case of three elements developed in Section 2, we suppose that the effective viscoelastic behavior of the cracked material is also modeled by the GKM as the behavior of the solid phase. The apparent bulk and shear moduli of the cracked material in the LC space are determined by (see (19)):
1
X 1 1 þ ko ðÞ k ð Þ þ pgsi ðÞ=3 i i X 1 1 ¼ þ lo ðÞ i li ðÞ þ pgdi ðÞ=2
¼
hom
k
1
lhom
hom
The series expansions of 1=k
1
¼
hom
k
1 þ OðpÞ; k1 ðÞ
1
lhom
ð23Þ
and 1=lhom determined by (23) in the vicinity of p ¼ 0 are:
¼
1
l1 ðÞ
þ OðpÞ
ð24Þ
And in the vicinity of p ¼ 1 are:
1 hom
k
1 1X 3 1 þ þ O ko ðÞ p i gsi ðÞ p2 1 1X 2 1 þ O ¼ þ p2 lo ðÞ p i gdi ðÞ
¼
1
lhom
P and l 1ðÞ ¼ l 1ðÞ þ i l 1ðÞ. The damaged stiffness and viscosity parameters, that are ka ðÞ; la ðÞ 1 o i (a ¼ o or 1Þ; gsi ðÞ and gdi ðÞ must ensure the compatibility between (21) and (24), as well as between (22) and (25). These conditions yield: where
1 k1 ðÞ
¼ ko1ðÞ þ
P
ð25Þ
1 i ki ðÞ
S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
1 1 1 1 1 þ Q 1 1 þ Q oo ¼ ¼ o ; ko ðÞ ko k1 ðÞ k1 ! ! X1 X 1 X1 X 1 AðÞ; BðÞ ¼ ¼ ki ðÞ ki gsi ðÞ gsi i i i i
231
ð26Þ
And for the deviatoric part:
1 1 1 þ M 1 1 þ M oo ¼ o ; l1 ð Þ l1 ! ! X 1 X1 X 1 X1 ¼ ¼ CðÞ; DðÞ li ðÞ li gdi ðÞ gdi i i i i 1
lo ðÞ
¼
1
lo
ð27Þ
where to simplify the formulas, we noted:
AðÞ ¼ 1 þ Q oo þ a Q oo Q 1 o 1 s BðÞ ¼ ð1 þ Q 1 o Þ þ s Q 1 o o CðÞ ¼ 1 þ Mo þ b Mo M1 o 1 d DðÞ ¼ ð1 þ M 1 o Þ þ s M 1
ð28Þ
With
a¼
1 X1 ko ki i
!1 ;
b¼
X1
1
lo
i
!1 ð29Þ
li
As concluded for the case of three elements, the solutions ko ðÞ and lo ðÞ of the short term elastic behavior and the solutions k1 ðÞ and l1 ðÞ of the long term elastic behavior given by (26) and (27) are exactly the same results obtained for the fractured elastic media with random orientation distribution of open cracks by using the classical Mori–Tanaka approach [3,2]. The solutions in (26) and (27) do not give the explicit estimation of ki ðÞ; gsi ðÞ; li ðÞ and gdi ðÞ but the sum of them over all the Kelvin systems. To estimate the effective viscoelastic parameter of each single Kelvin system, we assume the following approximations:
ki ðÞ ¼
ki ; AðÞ
gsi ðÞ ¼
gsi BðÞ
;
li ðÞ ¼
gd li ; gdi ðÞ ¼ i CðÞ DðÞ
ð30Þ
where the functions AðÞ; BðÞ; CðÞ, DðÞ are detailed in (28). Finally, Eqs. (26), (27) and (30) allow us to write the following explicit dimensionless formulas for the determination of the effective viscoelastic properties of the micro-cracked media:
ko ðÞ 1 ; ¼ ko 1 þ Q 1 o k i ð Þ 1 ¼ ; ki AðÞ
lo ðÞ 1 ¼ lo 1 þ M 1 o
li ðÞ 1 ¼ CðÞ li gsi ðÞ gdi ðÞ 1 1 ¼ ; ¼ BðÞ DðÞ gsi gdi
ð31Þ ð32Þ ð33Þ
Fig. 3. Approximation of the effective behavior of a micro-cracked medium by a generalized Kelvin model – the case of five elements: k0 ¼ 24:42 GPa, lo ¼ 13:27 GPa, k1 ¼ 11:22 GPa, l1 ¼ 6:03 GPa, k2 ¼ 11:22 GPa, l2 ¼ 3:015 GPa, gs1 ¼ 91:2 106 GPa s, gd1 ¼ 10:16 106 GPa s, gs2 ¼ 60:8 106 GPa s, gd2 ¼ 15:24 106 GPa s.
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S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
Fig. 3 presents the dependence of ten effective viscoelastic properties (normalized by the viscoelastic properties of the solid phase as shown in Eqs. (31)–(33)) of a five elements model (n ¼ 2) on the damage parameter . To validate the developed model on transient situation, we consider the creep problem under constant isotropic stress traction:
R ¼ Ro HðtÞ1
ð34Þ
where Ro is a given constant stress, HðtÞ is the Heaviside function and 1 is the second order unit tensor. The LC transform of this stress takes the form:
R ¼ Ro 1
ð35Þ
The macroscopic behavior in the LC space is written as:
E ðpÞ ¼
Ro 3k
hom
1
ð36Þ
The macroscopic strain is then isotropic and takes the form: E ðpÞ ¼ E ðpÞ1 with
E ðpÞ ¼
Ro 3k
ð37Þ
hom
hom
where k is calculated from the effective viscoelastic model (23). The inverse LC transform of E ðpÞ in (37) gives the macroscopic strain component EðtÞ in real space.
EðtÞ ¼
Ro 3
1 1 t 1 t Exp Exp k1 ðÞ k1 ðÞ k2 ðÞ s1 ðÞ s2 ð Þ
ð38Þ
With
s1 ðÞ ¼
gs1 ðÞ 3k1 ðÞ
¼
gs1 AðÞ 3k1 BðÞ
¼ s1
AðÞ BðÞ
ð39Þ
and
s2 ðÞ ¼
gs2 ðÞ 3k2 ðÞ
¼
gs2 AðÞ 3k2 BðÞ
¼
s1 ðÞ h
ð40Þ
are the effective characteristic times. Note that h ¼ s1 ðÞ=s2 ðÞ is a dimensionless parameter that iss independent of the crack g density parameter . s1 is the first characteristic time of the material without micro-crack s1 ¼ 3k11 . At t ¼ 0, the solution (38) becomes:
Eo ¼ Eðt ¼ 0Þ ¼
Ro 3ko ðÞ
ð41Þ
This allows us to write (38) under the dimensionless form:
EðtÞ ko ðÞ t ko ðÞ t þ ¼1þ 1 Exp 1 Exp Eo k1 ðÞ k2 ðÞ s1 ðÞ s2 ðÞ
ð42Þ
Fig. 4. Effect of the crack density parameter on the macroscopic strain – example of 5 elements model: k0 ¼ 24:42 GPa;lo ¼ 13:27 GPa, k1 ¼ 11:22 GPa, l1 ¼ 6:03 GPa, k2 ¼ 11:22 GPa, l2 ¼ 3:015 GPa, gs1 ¼ 91:2 106 GPa s, gd1 ¼ 10:16 106 GPa s, gs2 ¼ 60:8 106 GPa s, gd2 ¼ 15:24 106 GPa s, Ro ¼ 1 Mpa.
S.T. Nguyen / Engineering Fracture Mechanics 127 (2014) 226–234
Fig. 5. Effect of the ratio k2 =k1 on the macroscopic strain – example of 5 elements model: k0 ¼ 24:42 GPa, lo ¼ 13:27 GPa, k1 ¼ 11:22 GPa, l2 ¼ 3:015 GPa, gs1 ¼ 91:2 106 GPa s, gd1 ¼ 10:16 106 GPa s, gs2 ¼ 60:8 106 GPa s, gd2 ¼ 15:24 106 GPa s, Ro ¼ 1 Mpa.
233
l1 ¼ 6:03 GPa,
Fig. 6. Validation of the effective generalized Kelvin model in transient situation – example of 5 elements model: k0 ¼ 24:42 GPa, lo ¼ 13:27 GPa, k1 ¼ 11:22 GPa, l1 ¼ 6:03 GPa, k2 ¼ 11:22 GPa, l2 ¼ 3:015 GPa, gs1 ¼ 91:2 106 GPa s, gd1 ¼ 10:16 106 GPa s, gs2 ¼ 60:8 106 GPa s, gd2 ¼ 15:24 106 GPa s, Ro ¼ 1 Mpa.
By introducing (31)–(33) into (42) we obtains:
Eðt Þ AðÞ ko BðÞ t ko BðÞ t þ ¼1þ 1 Exp 1 Exp h Eo AðÞ s1 AðÞ s1 k1 k2 1 þ Q 1 o
ð43Þ
For long term behavior when t ! 1, the macroscopic strain (normalized by the instantaneous macroscopic strain when t ¼ 0Þ is reduced to:
E1 AðÞ ko ko ¼1þ þ Eo k1 k2 1 þ Q 1 o
ð44Þ
Figs. 4 and 5 show the evolution of the macroscopic strain (normalized by the instantaneous macroscopic strain) versus time (normalized by the characteristic time s1 Þ with effects of crack density and of viscoelastic properties of non-cracked material (Fig. 5 is an example of effect of k2 Þ. It is interesting to observe that more the material is damaged (high crack density) more the long term macroscopic strain is high compared with the instantaneous macroscopic strain. Similar observation is shown in Fig. 5, higher elastic property k2 gives high ratio EEðotÞ. Note that, for the simple loading condition (34), the analytical solution of the macroscopic strain can be obtained exactly by employing the transverse LC transform. The solution (38) is compared with the exact solution obtained by considering the
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inverse LC transform of E ðpÞ in (37) with k calculated directly from (2). Fig. 6 shows an excellent validation of the effective generalized Kelvin model (5 elements model) in short, long terms and transient situation. 4. Conclusions The effective viscoelastic models established previously allows us to determine explicitly the effective viscoelastic properties of micro-cracked generalized Kelvin material given the damage parameter and the viscoelastic moduli of the nondamaged material. These results allow us to easily and fast determine the effective behavior of a micro-cracked viscoelastic behavior, avoiding the complexity and the computational effort of the inverse LC transform. The approximations are derived from short and long terms behavior. The short and the long terms solutions are identical to that estimated for the elastic case based on Mori–Tanaka approach [3,2]. The comparison with the exact solution of a creep problem under constant isotropic traction stress shows excellent validations of the effective viscoelastic models on transient situation. This study is limited to the cases of random orientation distribution of open cracks in linear nonageing viscoelastic materials. Farther development taking into account of the complex micro-crack distribution, crack closure, complex ageing viscoelastic behavior or viscoplastic behavior [13,19,20] and of the crack propagation [11b,c] could be the perspective of this study. References [1] Budiansky B, O’connell RJ. Elastic moduli of a cracked solid. Int J Solids Struct 1976;12(2):81–97. [2] Dormieux L, Kondo D. Stress-based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects. Comput Mater Sci 2009;46(1):173–9. [3] Dormieux L, Kondo D, Ulm FJ. Microporomechanics. John Wiley & Sons; 2006. [4] Evans RH, Marathe MS. Microcracking and stress–strain curves for concrete in tension. Mater Struct 1968;1(1):61–4. [5] Haddad YM. Viscoelasticity of engineering materials. Springer; 1995. [6] Hillerborg A, Modéer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 1976;6(6):773–81. [7] Hoang-Duc H, Bonnet G, Meftah F. Generalized self-consistent scheme for the effective behavior of viscoelastic heterogeneous media: a simple approximate solution. Eur J Mech – A/Solids 2013;39:35–49. [8] Horii H, Nemat-Nasser S. Compression-induced microcrack growth in brittle solids: axial splitting and shear failure. J Geophys Res: Solid Earth (1978– 2012) 1985;90(B4):3105–25. [9] Le QV, Meftah F, He QC, Le Pape Y. Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori–Tanaka homogenization scheme and a discrete microscopic retardation spectrum. Mech Time-Depen Mater 2007;11(3–4):309–31. [10] Lévesque M, Gilchrist MD, Bouleau N, Derrien K, Baptiste D. Numerical inversion of the Laplace–Carson transform applied to homogenization of randomly reinforced linear viscoelastic media. Comput Mech 2007;40(4):771–89. [11] Nguyen ST, Dormieux L, Le Pape Y, Sanahuja J. A Burger model for the effective behavior of a microcracked viscoelastic solid. Int J Damage Mech 2011;20(8):1116–29. [11b] Nguyen ST, Dormieux L, Pape YL, Sanahuja J. Crack propagation in viscoelastic structures: theoretical and numerical analyses. Comput Mater Sci 2010;50(1):83–91. [11c] Nguyen ST, Jeannin L, Dormieux L, Renard F. Fracturing of viscoelastic geomaterials and application to sedimentary layered rocks. Mech Res Commun 2013;49(0):50–6. [12] Salençon J. Viscosité. Paris: Presses de l’ENPC; 1983. [13] Sanahuja J. Effective behaviour of ageing linear viscoelastic composites: homogenization approach. Int J Solids Struct 2013;50(19):2846–56. [14] Sayers CM, Kachanov M. Microcrack-induced elastic wave anisotropy of brittle rocks. J Geophys Res: Solid Earth (1978–2012) 1995;100(B3):4149–56. [15] Zoback MD, Byerlee JD. The effect of microcrack dilatancy on the permeability of Westerly granite. J Geophys Res 1975;80(5):752–5. [16] Zhu QZ, Kondo D, Shao JF. Homogenization-based analysis of anisotropic damage in brittle materials with unilateral effect and interactions between microcracks. Int J Numer Anal Methods Geomech 2009;33(6):749–72. [17] Brenner R, Suquet P. Overall response of viscoelastic composites and polycrystals: exact asymptotic relations and approximate estimates. Int J Solids Struct 2013;50(10):1824–38. [18] Suquet P. Four exact relations for the effective relaxation function of linear viscoelastic composites. CR Méc 2012;340(4):387–99. [19] Masson R, Brenner R, Castelnau O. Incremental homogenization approach for ageing viscoelastic polycrystals. CR Méc 2012;340(4):378–86. [20] Miled B, Doghri I, Brassart L, Delannay L. Micromechanical modeling of coupled viscoelastic–viscoplastic composites based on an incrementally affine formulation. Int J Solids Struct 2013;50(10):1755–69.