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Equivalence Approach
Mattias Olsson, Matti Ristinmaa
To cite this version:
the Strain Equivalence Approach
MATTIASOLSSON*ANDMATTIRISTINMAADivision of Solid Mechanics Lund University, Box 118
S-221 00 Lund, Sweden
ABSTRACT: A phenomenological thermodynamic approach used for describing elasto-plasticity coupled with isotropic damage is considered. Using this approach it was recently shown by the authors that certain unwanted features exist in the plasticity-induced damage theories. This paper gives a suggestion on how to solve these issues.
Based on physical arguments a simple modification of the postulate of strain equivalence is suggested. To show the effects of the modification, the von Mises plasticity model coupled with isotropic damage is used as a prototype model in the derivations. Both analytical considerations and simulations of a strain-controlled uniaxial model are performed. The results reveal that the modification of the postulate of strain equivalence gives promising results. In contrast to the postulates of strain equivalence and (complementary) energy equivalence, the present formulation predicts the elastic strain to vanish when failure takes place.
KEY WORDS: isotropic damage, damage evolution, plasticity-induced damage, elasto-plasticity, strain equivalence, thermodynamics.
INTRODUCTION
A
MATERIAL THATis exposed to loading will at a certain load level start to deteriorate and small voids, cracks and cavities of other forms will appear. The collective effect of these voids is called damage. Within the field of continuum damage mechanics, internal variables are introduced to enable modelling of the material as a continuum.In this study, continuum damage mechanics will be adopted and the coupling between elasto-plasticity and damage will be investigated. The critical phase in formulating damage models is the introduction of damage
International Journal ofDAMAGEMECHANICS, Vol. 14—January 2005 25
1056-7895/05/01 0025–26 $10.00/0 DOI: 10.1177/1056789505044286
ß 2005 Sage Publications
effects without abandoning the properties of well-accepted formulations for elasticity and elasto-plasticity. With this in mind, different concepts and postulates dealing with the physical behaviour of the material have been introduced in the literature for the purpose of taking the effects of damage into account. Here, the concept of effective stress (Odqvist and Hult, 1961; Rabotnov, 1968) together with the postulate of strain equivalence (Lemaıˆtre and Chaboche, 1978; Sidoroff, 1980) will be employed.
Recently it was shown by Olsson and Ristinmaa (2003) that some unwanted consequences will arise when the postulate of strain equivalence or the postulate of (complementary) energy equivalence is used. One of the troublesome issues is that if a split of the total strain e into an elastic part ee and a plastic part ep, which also includes damage effects, is adopted,
i.e. e ¼ eeþep, the two postulates will predict that ee6¼0 at final fracture.
Evidently, if ee6¼0 at final fracture the common interpretation of elastic
strain is lost, which yields that quantities based upon the elastic strain, for example elastic strain energy, lose their meaning. A simple and physically motivated solution to this problematic issue will be the focus in this work. In addition, in (Olsson and Ristinmaa, 2003) it was shown that for the postulate of energy equivalence the final value for the damage variable D ¼1 is an asymptotic value, i.e. the damage rate _DDwill at some instance begin to decrease when continuing loading.
A different approach than assuming that the total strain is decomposed as e ¼ eeþep is to assume that a damage strain can be identified, i.e. e ¼ eeþepþed. This approach was taken by for example, Armero and Oller (2000) where the damage strain is given by a number of damage mechanisms. Here, the decomposition e ¼ eeþepwill be adopted and it will
be assumed that ee is the true elastic strain. Then, evidently, ep is the
collection of plasticity and damage. The important implication is that ed is
not explicitly identified, since it is believed that this cannot be done for all models. As an example, consider Gurson’s model (Gurson, 1977) and its extension to non-spherical voids (Gologanu et al., 1993). In this model, ep (assuming, for simplicity, small strains) is the contribution from both
plasticity as well as damage, however they cannot be separated.
of voids. Using that for the voids both the shear modulus and the bulk modulus are zero. Davison et al. (1977) found that
Kd ¼K 1 3 1 ð Þ 2 1 2ð Þ! Gd ¼G 1 15 1 ð Þ 7 5 ! ð1Þ
where is the Poisson’s ratio. In (1), ! is the damage variable representing the volume fraction that is occupied by voids.
According to the postulate of strain equivalence associated with the concept of effective stress, the stiffness of the damaged material is found to be Ed ¼Eð1 DÞ, where Ed and E are the Young’s modulus for the
damaged and the undamaged materials respectively. Moreover, D is the classical damage variable introduced by Kachanov (1958), which Odqvist and Hult (1961) identified as the fraction between the cross-sectional area occupied by voids and the total cross-sectional area. The effective stress is defined as ~rr ¼ r= 1 Dð Þ, where r is the stress tensor. A modification of the postulate of strain equivalence similar to Davison et al. (1977) in (1) could be the assumption that Ed ¼Eð1 qDÞfor the uniaxial case, which
for the general case yields
Ld¼L 1 qDð Þ ð2Þ
where Ld and L are the stiffness tensors for the damaged and the undamaged materials respectively. Here, for simplicity, only one q-value will be considered. Within continuum damage mechanics when using Kachanov’s damage variable it is assumed that q ¼ 1, but this leads to certain undesirable properties which will be discussed later. Adopting the Poisson’s ratio to obtain the values 0 < 0:5 (the factor in front of ! can be compared with the q-value) the factor in front of ! in (1) is always larger than 3=2. However, it is noted that the damage variable in (1) is different than that in (2). Here, the classical damage variable D will be used and the influence of different values of q > 0 on the constitutive equations will be studied. However, as will be discussed later on, these two models can be related to each other.
plasticity is assumed, since this leads to the ‘classical’ continuum damage model based on the postulate of strain equivalence (Lemaıˆtre and Chaboche, 1990). For simplicity, as already indicated, isotropic damage described with only one damage parameter D will be employed; representing the simplest possible isotropic formulation.
THEORETICAL FRAMEWORK
The theory of thermodynamics with internal variables, introduced by Onsager (1931a,b) and Coleman and Gurtin (1967), is adopted. On the basis of the principles of thermodynamics, i.e. the conservation of energy and the rate of entropy production, the theory provides constitutive relations described as relations between the internal variables and the thermodynamic conjugated forces. These relations are summarised here.
Small displacement gradients are assumed, which makes it possible to make a decomposition of the strain tensor e ¼ eeþep, where e is the strain
tensor and the elastic part and the plastic part are denoted as ee and ep
respectively. Assuming isothermal conditions (Truesdell and Toupin, 1960), the mechanical dissipation inequality becomes
g ¼ r : _ee _ 0 ð3Þ where ¼ eð e, , DÞis Helmholtz’s free energy per unit mass, r is the stress
tensor and is the mass density. Moreover, denotes the internal variable associated with the plastic behaviour and D the damage variable. The following form of Helmholtz’s free energy is chosen:
eð e, , DÞ ¼1 2e e:Ld : eeþ1 2h d2þ dð ÞD ð4Þ
It is assumed that the stiffness tensorLdand the hardening parameter hdare
functions of the damage variable D. Taking advantage of (4) in (3), the standard arguments of Coleman and Gurtin (1967) provides the stress–strain relation
r ¼ @ @ee¼L
d : ee ð5Þ
the mechanical dissipation inequality in (3) then becomes
¼ 1þ20 ð7Þ
Here the mechanical dissipation has been split into two parts, both fulfilling the above inequality, i.e.
1 ¼r : _eepR_ 0 and 2¼ Y _DD 0 ð8Þ
For non-associative formulation and plasticity-induced damage it is assumed that a potential function g r, R, Y , Dð Þexists, and that the evolution laws are given as
_e ep¼ _@g @r, ¼ __ @g @R, DD ¼ __ @g @Y ð9Þ
The plastic multiplier _and the yield function f r, R, Dð Þare assumed to fulfil the following conditions
_
0, f _ ¼0, and f 0 ð10Þ Note that the yield function f r, R, Dð Þ 0 defines the elastic region. From these assumptions it follows that plastic loading can only take place when f ¼ 0, and for plastic loading to continue it is required that _ff ¼0 holds. The relation _ff ¼0 is known as the consistency relation. For plasticity-induced damage following for example Lemaıˆtre (1985a,b), it is assumed that the potential function can be divided into one part associated with plasticity and one part associated with damage evolution, i.e.
g r, R, Y , Dð Þ ¼gpðr, R, DÞ þgdðY, DÞ ð11Þ To fulfil the dissipation inequality in the first equation of (8) it is assumed that gpfor a given D is a convex function in r and K, and that gpðr, R, DÞ
gpð0, 0, DÞ 0. Further, to fulfil the second equation of (8) it is also assumed
that Y 0, and that _DD 0. Later on the choice gp¼f is utilized leading to
the condition that the evolution laws for _eep and _ can, for the considered model, be derived by considering an associative formulation.
THE POSTULATES OF STRAIN AND ENERGY EQUIVALENCE In the concept of effective stress (Odqvist and Hult, 1961; Rabotnov, 1968), a mapping between stress r and effective stress ~rr is introduced as
~r
r ¼ r
Odqvist and Hult (1961) identified D as the fraction between the cross-sectional area occupied by voids and cracks and the total cross-cross-sectional area. It has also been shown (Chaboche, 1998; Olsson and Ristinmaa, 2003), that a mapping of the hardening variable similar to that for the stress is a natural choice. The mapping of the hardening variable (first introduced by Chaboche (1977)) is defined as
~ R
R ¼ R
1 D ð13Þ
Two different postulates exist, which are connected to the concept of effective stress; the postulate of strain equivalence and the postulate of (complementary) energy equivalence.
Lemaıˆtre and Chaboche (1978) and Sidoroff (1980) introduced the postulate of strain equivalence. Sometimes it is referred to as the extended postulate of strain equivalence (see Edlund and Klarbring, 1993 for example), if also mapping of the hardening variable is utilized. The postulate can be written as
eeðr, DÞ ¼eeðr, 0~r Þ R, Dð Þ ¼ ~RR, 0
ð14Þ
From (5) and (6) using (12)–(14) the following constitutive equations are obtained r ¼Ld: ee where Ld ¼ð1 DÞL R ¼ hd where hd ¼ð1 DÞh Y ¼ 1 2e e:L : ee1 2h 2þ@ d @D ð15Þ
whereL and h are the undamaged stiffness tensor and hardening parameter respectively.
For completeness, the postulate of energy equivalence introduced by Cordebois and Sidoroff (1979) and Sidoroff (1980) is also discussed here. The extension of the postulate to include mapping of the hardening variable was introduced by Saanouni et al. (1994). The (extended) postulate of energy equivalence can be written as
where Saanouni et al. (1994) defined the mappings of the elastic strain and of the internal variable as
~e
ee¼ð1 DÞee ~
¼ð1 DÞ ð17Þ
From (5) and (6) using (12), (13), (16) and (17) the following constitutive equations are obtained
r ¼Ld : ee where Ld ¼ð1 DÞ2L
R ¼ hd where hd ¼ð1 DÞ2h
Y ¼ ð1 DÞee:L : eeð1 DÞh2
ð18Þ
For both the postulates considered above, to generalise the potential function gp and the yield function f to account for damage evolution, it is
assumed that
gpðr, R, DÞ ¼gpr, ~~r RR, 0 f r, R, Dð Þ ¼fr, ~~r RR, 0
ð19Þ
(See Chow and Wang, 1987; Ju, 1989; Lemaıˆtre and Chaboche, 1990; Hansen and Schreyer, 1992, 1994; Zhu and Cescotto, 1995 for example). The postulates do not give any information about the potential function gd.
As discussed in the introduction, it was shown by Olsson and Ristinmaa (2003) that shortcomings exist for both the (extended) postulate of strain equivalence and the (extended) postulate of energy equivalence. Olsson and Ristinmaa (2003) considered plasticity-induced damage for the von Mises plasticity model coupled with isotropic damage. One shortcoming is that, irrespective of the postulate employed, the elastic strain will not equal zero when failure takes place. This is evidently not consistent with the interpretation of elastic strain. It was also shown for the postulate of energy equivalence that the damage rate at some point will decrease with increasing loading and, as a consequence, D will approach 1 when the strain approaches infinity, i.e. D ¼ 1 is an asymptotic value. This result is not physically acceptable. Here, a modification of the postulate of strain equivalence will be discussed that do not have the above short-comings.
is the power of 1 Dð Þ. A tempting idea is to study a more general choice of Helmholtz’s free energy. Assume
eð e, , DÞ ¼1 2ð1 DÞ
mee:L : eeþ1
2ð1 DÞ
nh 2þ dð ÞD
where m 0 and n 0. Note, if m ¼ n ¼ 1 and m ¼ n ¼ 2 are chosen the postulates of strain equivalence and energy equivalence are obtained, respectively. It can be shown, see Appendix, that even for this more general choice of Helmholtz’s free energy the same shortcomings exist as for the postulates of strain equivalence and energy equivalence. Different choices of
dð ÞD and gdðY, DÞwill not change this conclusion, again see Appendix.
A MODIFICATION OF THE POSTULATE OF STRAIN EQUIVALENCE
In order to find a formulation for elasto-plasticity coupled with damage which does not have the shortcomings that the postulates of strain equiva-lence and energy equivaequiva-lence imply (discussed in the previous section), a modification of the postulate of strain equivalence is assumed. Thus, a formulation is sought for that is consistent with the interpretation of elastic strain and where the damage evolution does not have the asymptotic value
_ D
D ¼0 at failure.
Using the surface definition of Odqvist and Hult (1961), quantities depending on the cross-sectional area should be mapped using 1 Dð Þ between the fictitious undamaged configuration and the (true) damaged configuration. However, quantities that are not directly related to the cross-sectional area should be mapped differently, for example quantities depending on the volume. This can easily be understood by considering a simple example with a spherical void with radius r in a circular cylinder with radius R and height 2R. Considering uniaxial loading along the cylinder axis and identifying the damage variable as D ¼ r=Rð Þ2, if the cross-sectional definition is used, it follows that complete loss of stress carrying capacity is obtained when r ¼ R, i.e. D ¼ 1. In terms of void volume fraction !, i.e. the fraction between the volume of the void and the total volume or ! ¼ 2=3ðr=RÞ3, complete loss of stress carrying capacity is found when ! ¼ 2=3, see also the discussion in (Rabier, 1989). The damage variable can be related as ! ¼ 2=3D3=2, indicating that for the simple example discussed here, there
exists a simple relationship between the two damage variables. However, a simple relationship is not expected in the general situation. Therefore, in this study the implications of adopting two different mappings 1 Dð Þand
1 qD
By considering the results of Budiansky (1970) and Davison et al. (1977) it is physically natural to introduce the material parameter q as described in (2), i.e. the stiffness is assumed to decrease with increasing damage according toLd¼L 1 qDð Þ. Adopting this, the constant q is studied for all positive
values, i.e. q > 0. Note that it is possible to introduce two constants q1and q2
in line with (1) for isotropic damage. However, for the purpose of this study it is sufficient to use one q to show the essential results derived.
Helmholtz’s free energy defined for the postulate of strain equivalence in (4) and (15) will have to be modified to include the constant q as in (2). Here, it will be assumed that Helmholtz’s free energy is given by
eð e, , DÞ ¼1
2ð1 qDÞe
e:L : eeþ1
2ð1 DÞh
2þ dð ÞD ð20Þ
where h > 0 is assumed. To reduce the complexity and not to deviate too much from the ‘classical’ continuum damage model based on the postulate of strain equivalence, the damage part of Helmholtz’s free energy d is
assumed to obey
@ d
@D ¼0 ð21Þ
Evidently, the postulate of strain equivalence is obtained when choosing q ¼1 in (20). From (5) and (6) using (20) and (21) the following constitutive equations are obtained
r ¼ 1 qDð ÞL : ee R ¼ð1 DÞh Y ¼ 1 2qe e:L : ee1 2h 2 ð22Þ
It is noted that from the first equation of (22) it follows thatLd ¼ð1 qDÞL,
which is equivalent to (2).
VON MISES PLASTICITY
For illustration purposes an isotropic hardening von Mises plasticity model coupled with damage is studied for a strain-controlled loading situation. The isotropic hardening von Mises model, when modified to include damage according to (19), becomes
gpðr, R, DÞ ¼f r, R, Dð Þ ¼ e 1 D
R
where the effective von Mises stress e, defined by the deviatoric stress s, is defined as e¼ 3 2s : s 1=2 , s ¼ r 1 3tr rð ÞI ð24Þ
Moreover, I is the second-order unit tensor and yo is the constant initial
yield stress. The trace of a tensor is denoted by tr ð Þ. Note that in (23) the mapping introduced in (19) has been accepted. However, if (2) is accepted for the elastic part it is not evident that ð1 DÞ should be introduced in (23) and instead 1 rDð Þis a tempting choice. But as will be shown later on, if 1 Dð Þ is accepted complete failure is obtained when D ¼1, which is in line with the definition of Kachanov (1958), see (Odqvist and Hult, 1961).
The damage potential function is chosen to be
gdðY, DÞ ¼S 2 1 1 D Y S 2 ð25Þ
where S is a positive constant. This choice of damage potential function is commonly used in the literature (Lemaıˆtre, 1985b; Lemaıˆtre and Chaboche, 1990).
From (9) using the potential function g ¼ gpþgd chosen in (23) and (25)
the evolution laws are obtained as
_e ep¼ _ 1 Dss where ss ¼ 3s2e _ ¼ _ 1 D _ D D ¼ Y_ where Y ¼ 1 1 D Y S ð26Þ
It turns out that the relations in (22) can be further reduced by noting that the termL : eecan for isotropy be written as
L : ee¼2GeeþKtr eð ÞI ð27Þ
where advantage was taken of tr eð Þ ¼tr eð eÞ. Further, ee is the deviatoric
part of the elastic strain tensor defined as ee¼ee1=3tr eð eÞI. The material
found using (22), the second equation of (24) and (27): s ¼ 2G 1 qDð Þee; tr rð Þ ¼3K 1 qDð Þtr eð Þ R ¼ hð1 DÞ Y ¼ 1 2q 3G e e eff 2 þK tr eð ð ÞÞ2 h i 1 2h 2 ð28Þ
where the effective elastic strain was defined as ee eff¼ 23e
e: ee
1=2
.
The consistency relation _ff ¼0 can while using (23), (26) and (28) be rewritten as 2G 1 qDð Þss : _ee ¼ A _ ð29Þ where A ¼2G1 qD 1 D þh 1 q 1 qD ð Þð1 DÞeY ð30Þ
The constitutive model is now defined by (26), and (28)–(30). Note that ss : _ee is always positive when plastic loading takes place, except for the situation s ¼ 0 for which it is undefined and for the trivial situation _ee ¼ 0.
The task, now, is to study the behaviour of the thermodynamic conjugate forces and the internal variables close to failure, i.e. when D approaches 1. Since a strain-controlled loading case is considered, the internal variable described via the evolution law in the second equation of (26) is bounded. This follows directly from the second equation of (26) if fracture takes place for a finite value of e. The value of at failure is denoted as f, i.e. when
D !1. Using that f ¼ 0 for plasticity to continue, together with (23), (24) and (28) leads to the following limits at failure
Obviously, the case q ¼ 1 (i.e. the postulate of strain equivalence) is not preferable since the interpretation of elastic strain is lost, and the same holds for the case when q > 1. Though, a choice of q to be 0 < q < 1 seems promising as the elastic strain will vanish at failure.
Now the attention turns to studying the behaviour of the damage rate close to failure. First, the case q ¼ 1 is discussed. Using the third equation of (28), (29) and (30) in the third equation of (26) yields
_ D D ¼ G Sð2G þ hÞ 3G e e eff 2 þK tr eð ð ÞÞ2þh2 n o ss : _ee ð33Þ
Introducing tr eð ð ÞÞ2! when D ! 1, where is a positive constant and using this together with (32) and (33) yields
_ D D ! G Sð2G þ hÞ hf þyo 2 3G þK þ h 2 f ( ) ss : _ee when D ! 1 ð34Þ
In conclusion, for the case q ¼ 1, the damage rate is always positive, i.e. _
D
D >0. This result and the result in (31) for the case q ¼ 1 are in line with the findings by Olsson and Ristinmaa (2003).
Consider now the case 0 < q < 1. Taking advantage of the third equation of (26), (29) and (30) the damage rate is found as
_ D D ¼2G S 2G Y 1 D 1 qD h Y 1 q 1 qD ð Þ2ð1 DÞ e S 1 ss : _ee ð35Þ To approach the above some preliminary results are needed. Using the third equation of (28) and (32) lead to
2G Y ! 4G qK þ h2 f when D ! 1 ð36Þ
For the second term within the brackets in (35), it directly follows that 1 D
1 qD h
Y!0 when D ! 1 ð37Þ
Finally, using that f ¼ 0 for plasticity, together with (23) yields
Thus, after insertion of (36)–(38) into (35), it is possible to conclude that the limit for the damage rate close to failure is
_ D D !2G S 4G qK þ h2 f 1 1 q hf þyo S ( )1 ss : _ee when D ! 1 ð39Þ
Evidently, the case ¼ 0 and h ¼ 0 is troublesome, since the damage rate will approach zero when the damage approaches 1. However, this is a very special situation that has almost no practical use as it is assumed that all materials here are described by h > 0. The case when the parenthesis is changing sign to be negative will not cause any trouble, as the parenthesis will be zero and the damage rate will reach infinity before this takes place, i.e. D ¼ 1 will be reached. Thus, for the case 0 < q < 1 the damage rate is always positive _DD >0, except if ¼ 0 and h ¼ 0.
Finally, for completeness the case q > 1 is studied. Using the third equation of (26), (29) and (30) the damage rate is found to be identical to (35). To evaluate this, some preliminary results are needed. Using that f ¼ 0 for plasticity together with (23) the limit for the last term within the parenthesis in (35) can be found:
1 q
1 qD
ð Þ2ð1 DÞ e
S ! 1 when D ! 1=q ð40Þ
Using the third equation of (28) and (32) yields 2G Y !0 when D ! 1=q 1 D 1 qD h Y!0 when D ! 1=q ð41Þ
The limit of the damage rate can now be found, by inserting (40) and (41) into (35):
_ D
D !0 when D ! 1=q ð42Þ
This is troublesome, since for the case q > 1 the damage rate will approach zero before failure. This means that at some point the damage evolution will decrease with increasing load.
To summarise, it was shown that for the choice 0 < q < 1, a model was obtained which has the property that ee
elastic strain vanishes at failure. Moreover, it was also concluded that for continuous loading DD >_ 0 is satisfied, i.e. failure will take place at a finite value of e. Thus the shortcomings that exists in the strain equivalence formulation is not present in this formulation, i.e. when 0 < q < 1.
NUMERICAL RESULTS FOR THE UNIAXIAL LOADING CASE To illustrate the conclusions in the previous section, numerical simula-tions for the strain-controlled uniaxial loading case are performed. For the uniaxial case, the isotropic hardening von Mises model in (23) becomes
gpð, R, DÞ ¼f ð , R, DÞ ¼ 1 D
R
1 Dyo ð43Þ The damage potential function is chosen to be the one in (25). Using the potential function g ¼ gpþgd the following evolution laws are obtained
from (26) _""p¼ _ 1 D; ¼_ _ 1 D; DD ¼ Y_ _ ð44Þ where Y¼ 1 1 D Y S ð45Þ
For the proposed model the following expressions for the thermodynamic conjugated forces are found from (22):
¼ð1 qDÞE " "ð pÞ R ¼ð1 DÞh Y ¼ 1 2qE " " p ð Þ21 2h 2 ð46Þ
where E is the elastic stiffness. Note, the damage strain energy release rate Y <0 and Y >0.
yield function f chosen in (43) into the consistency relation _ff ¼0 leads to 1 qD ð ÞE _"" ¼ A _ where A ¼ E1 qD 1 D þh 1 q 1 qD ð Þð1 DÞY ð47Þ
Equations (44)–(47) will be used in the numerical calculations for different values of q > 0. Further derivations similar to those in (31)–(42) can easily be conducted also for the uniaxial case using (44)–(47) and will therefore not be repeated here. A strain-controlled test is considered for which all the variables are zero at the beginning, i.e., ¼ R ¼ Y ¼ "p¼ ¼ D ¼0
when " ¼ 0. The material constants are set to E ¼ 200 GPa, yo¼0:6 GPa,
h ¼1 GPa and S ¼ 0:1 MPa.
The results from the numerical calculations are shown in Figures 1–3 as functions of strain for q ¼ 0:75, q ¼ 1 and q ¼ 1:25 respectively. Note, the case q ¼ 1 is identical with the postulate of strain equivalence. As a comparison, calculations for a model without damage evolution, i.e. D 0, were made and included in Figures 1–3.
Both for the case q ¼ 0:75 and the case q ¼ 1 the Figures 1(a) and 2(a) show that the stress and the isotropic hardening variable vanish at failure, i.e. ¼ R ¼ 0 when D ! 1. On the other hand, Figure 3(a) shows that for the case q ¼ 1:25 the stress and the isotropic hardening variable have the asymptotic values ! 1 1=qð Þðhf þyoÞ and R ! 1 1=qð Þhf
respec-tively, when " ! 1. This emphasises the conclusion made in the last section that the case q > 1 does not yield satisfying results close to failure. As the material is completely deteriorated, both the stress and the isotropic hardening variable must vanish.
Figures 1(b) and 2(b) show for the cases q ¼ 0:75 and q ¼ 1 the damage evolution to be increasing with increasing load, i.e. _DD >0. Again the case q ¼1:25 is not appealing as an asymptotic value D ! 1=q when " ! 1 is obtained, see Figure 3(b). The assumption of q ¼ 1:25 implies that at some point the damage rate will decrease with increasing loading. This emphasises again the conclusion that the assumption q > 1 does not yield satisfying results close to failure.
In Figures 4–6 the strain is plotted as a function of damage evolution. In all the three figures the plastic strain at failure, "pf, is shown. For the case q ¼1 in Figure 5 the elastic strain at failure is ðhf þyoÞ=E, and for the
0 1 5 0 0.5 ε (%) σ , R (GPa) (a) σ for D ≡ 0 R for D≡ 0 σ R 0 1 5 0 0.5 1 ε (%) Damage ( − ) (b)
Figure 1. A simulation of a uniaxial strain-controlled test for the modified postulate of strain equivalence, where Ed¼E 1 0:75Dð Þwas assumed: (a) stress and isotropic hardening
0 1 5 0 0.5 ε (%) σ , R (GPa) (a) σ for D ≡ 0 R for D≡ 0 σ R 0 1 5 0 0.5 1 ε (%) Damage (–) (b)
Figure 2. A simulation of a uniaxial strain-controlled test for the modified postulate of strain equivalence, where Ed¼E 1 Dð Þwas assumed: (a) stress and isotropic hardening
0 1 5 0 0.5 ε (%) σ , R (GPa) (a) →0.2(hκf+σ yo) when ε → ∞ →0.2hκf when ε → ∞ σ for D ≡ 0 R for D≡ 0 σ R 0 1 5 0 0.5 1 ε (%) Damage (–) (b) →0.8 when ε → ∞
Figure 3. A simulation of a uniaxial strain-controlled test for the modified postulate of strain equivalence, where Ed¼E 1 1:25Dð Þwas assumed: (a) stress and isotropic hardening
0 0.5 1 0 1 5 Damage (–) Strain (%) = ε f p ε εp εe
Figure 4. A simulation of a uniaxial strain-controlled test for the modified postulate of strain equivalence, Ed¼E 1 0:75Dð Þwas assumed. Total strain ", elastic strain "e and plastic
strain "pas functions of damage.
0 0.5 1 0 1 5 Damage (–) Strain (%) = ε f p = (hκ f+σyo)/E ε εp εe
Figure 5. A simulation of a uniaxial strain-controlled test for the modified postulate of strain equivalence, Ed¼E 1 Dð Þwas assumed. Total strain ", elastic strain "eand plastic strain "p
conclusion made in the last section that the case 0 < q < 1 yields a formulation that is consistent with the interpretation of elastic strain.
USE OF DIFFERENT DAMAGE VARIABLES
When using Kachanov’s classical damage variable D, it was found that a suitable choice for Young’s modulus is Ed¼Eð1 qDÞ, where 0 < q < 1 is
a constant. ‘Generalising’ the findings to hold for the bulk modulus, it is assumed that Kd ¼Kð1 qDÞ. In the model by Budiansky (1970) and
Davison et al. (1977) another damage variable ! was chosen and the bulk modulus was found to follow Kd ¼Kð1 Q!Þ, see (1), where Q depends
on Poisson’s ratio and fulfils the condition Q > 3=2. Rabier (1989) showed that failure occurs for ! < 1. Especially it was shown that for the assumption of spherical voids, the structure fails at ! 0:5. By introducing a scaling between the two damage variables it is possible to assure the same suitable results for either of the damage variables. The scaling is obtained by comparing the expressions for Kd, which yields qD ¼ Q!. For this model,
failure occurs when ! ! q=Q. Further, by adopting the scaling between the
0 0.5 1 0 1 5 Damage (–) Strain (%) ∞ when D →0.8 → → εfp when D → 0.8 ε εp εe
Figure 6. A simulation of a uniaxial strain-controlled test for the modified postulate of strain equivalence, Ed¼E 1 1:25Dð Þ was assumed. Total strain ", elastic strain "e and plastic
two damage variables the governing equations discussed in this study for the damage variable D can all be converted to instead include the damage variable !. The fundamental assumptions are given here. By making use of the scaling between the damage variables, Helmholtz’s free energy in (20) is changed to eð e, , !Þ ¼1 2ð1 Q!Þe e:L : eeþ1 2 1 Q q! h2þ !ð Þ! ð48Þ
The damage strain energy release rate Y connected to ! is defined as in the second equation of (6), i.e.
Y ¼ @
@! ð49Þ
and the evolution law for ! is obtained from the third equation of (9) giving _
!
! ¼ _@g
@Y ð50Þ
Also the potential functions in (23) and (25) are transformed according to gpðr, R, !Þ ¼f r, R, !ð Þ ¼ e 1 ðQ=qÞ! R 1 ðQ=qÞ!yo ð51Þ and gdðY, !Þ ¼S 2 1 1 ðQ=qÞ! Y S 2 ð52Þ Thus, the relations (48)–(52) define a model in the damage variable ! that is equivalent to the model discussed previously. Hereby, it is possible to perform the same derivations for the model using ! (the volume fraction) as was done in this study for Kachanov’s damage variable D. The same benefits as was found in the last sections are also obtained for this formulation.
CONCLUSIONS
that the damage rate at some point will decrease with increasing loading and, as a consequence D ¼ 1 is an asymptotic value when the strain approaches infinity. The purpose of this work was to give a suggestion to a modification of the postulate of strain equivalence that do not have these shortcomings.
In this work, it was suggested that the damage stiffness for the postu-late of strain equivalence should be modified to include a constant q according toLd ¼L 1 qDð Þ, see (2). The postulate of strain equivalence is
obtained by assuming q ¼ 1. To evaluate the modification, the von Mises plasticity model coupled with plasticity-induced isotropic damage was considered.
The proposed formulation was studied for q > 0 both analytically and by numerical simulations, shown in Figures 1–6. The results revealed once more the shortcomings of the postulate of strain equivalence, i.e. q ¼ 1, see Figures 2 and 5. For the case q ¼ 1, it was found that the elastic strain is not equal to zero when failure takes place.
It was found that for the case q > 1 the strain will approach infinity before failure takes place, and thereby the damage rate at some point will decrease with increasing load, see Figure 3. Also for the case q > 1 the elastic strain is not equal to zero when failure takes place, see Figure 6. Thus, the case q > 1 yields non-appealing results.
Satisfying results were obtained for the modified strain equivalence formulation for the case 0 < q < 1. Both the stress, the isotropic hardening variable and the elastic strain vanish at failure, see Figures 1 and 4. It can thus be concluded, for the case 0 < q < 1, that ee is the true elastic strain and the formulation is consistent with the natural interpretation of elastic strain. To conclude, the modification of the postulate of strain equivalence with the choice 0 < q < 1 would be a good platform for attacking different applications with continuum damage mechanics.
APPENDIX
Generalisation of the Postulate of Strain Equivalence
The purpose here is to consider the implications of a model that is a generalisation of the postulates of strain equivalence and energy equiva-lence. The concept of effective stress, for which ~rr ¼ r= 1 Dð Þis defined, is considered in this model. Assume Helmholtz’s free energy to follow
where m 0 and n 0. Note, if m ¼ n ¼ 1 and m ¼ n ¼ 2 are chosen the postulate of strain equivalence and energy equivalence are obtained respectively. The damage part of Helmholtz’s free energy is assumed to be
dð Þ ¼ F DD ð Þ ðA2Þ
where F is an arbitrary function of D. Using the relations in (5) and (6) on this choice of Helmholtz’s free energy, i.e. (A1) and (A2), the following expressions for the thermodynamic conjugate forces follow
r ¼ 1 Dð ÞmL : ee R ¼ð1 DÞnh Y ¼ m 2ð1 DÞ m1 ee:L : een 2ð1 DÞ n1 h2þ@F @D ðA3Þ
von Mises plasticity is assumed, i.e. the plastic potential function and the yield function to be as in (23). The damage potential function is assumed to be given as
gdðY, DÞ ¼ G Y , Dð Þ ðA4Þ
where G is an arbitrary function of Y and D. The evolution laws are then defined by (26), except that
Y¼ @G
@Y ðA5Þ
Taking advantage of (27), (A3) can be rewritten as s ¼ 2G 1 Dð Þmee; tr rð Þ ¼3K 1 Dð Þmtr eð Þ R ¼ hð1 DÞn Y ¼ m 2ð1 DÞ m1 3G ee eff 2 þK tr eð ð ÞÞ2 h i n 2ð1 DÞ n1h2þ@F @D ðA6Þ
Now consider the limit of the stress and the isotropic hardening variable when D ! 1, i.e. close to failure. Using (23) and (A6) it is found that
R ¼ hð1 DÞn !0 when D ! 1 e¼R þð1 DÞyo!0 when D ! 1
ðA9Þ
where it was used that is bounded. Moreover, using (24), (A3) and (A9) yields the following expression for the effective elastic strain:
eeeff¼h
3Gð1 DÞ
nmþyo
3Gð1 DÞ
1m ðA10Þ
which has the following limits depending on the choice of the constants mand n: eeeff! 0 for 0 m < 1 and m < n hf 3G for 0 m ¼ n < 1 yo 3G for m ¼ 1 < n hf þyo 3G for m ¼ n ¼ 1 1 for 0 n < m 1 or m > 1 when D ! 1 8 > > > > > > > > > > < > > > > > > > > > > : ðA11Þ
where f ¼limD!1. The only preferable case is when 0 m < 1 and
m < n, as the elastic strain will vanish close to failure. For all the other cases the interpretation of elastic strain is lost. Especially, this can be observed for m ¼ n ¼1, i.e. for the postulate of strain equivalence, and for m ¼ n ¼ 2, i.e. for the postulate of energy equivalence.
Finally, consider the damage rate, which is obtained by insertion of (A7) in the third equation of (26), i.e.
_ D
D ¼2G 1 Dð ÞmY
A ss : _ee ðA12Þ
To be able to determine the behaviour of the damage rate close to failure the term 1 Dð ÞmY=Aneeds to be studied closely. Using (A5) and (A8)
It is possible to choose G Y , Dð Þ such that the first term within the parenthesis in (A13) is bounded and positive. Thus, the second term within the parenthesis in (A13) will control the damage rate. Taking advantage of (A9) it follows that
m 1
ð Þeðn 1ÞR
1 D
ð Þmþ1 !
0 for m ¼ n ¼ 1
1 for m 6¼ 1 and n 6¼ 1 when D ! 1
ðA14Þ To conclude, for m 6¼ 1 and n 6¼ 1, _DD !0 follows and this is evidently not appealing. Thus, the only preferable case is when m ¼ n ¼ 1, for which the damage rate will not vanish at failure. This is though not compatible with the restrictions for m and n found earlier (0 m < 1 and m < n). No suffi-cient choice for m, n, F and G can thus be found. To conclude this generalisation of the postulate strain equivalence has the same type of drawbacks as the postulates of strain equivalence and energy equivalence.
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