How to realize volume conservation during finite plastic deformation He-Ling Wang, Dong-Jie Jiang, Li-Yuan Zhang, Bin Liu Heling Wang1 AML, CNMM, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL, 60208, USA e-mail:
[email protected] Dong-Jie Jiang AML, CNMM, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China e-mail:
[email protected]
Li-Yuan Zhang AML, CNMM, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China e-mail:
[email protected] Bin Liu1 AML, CNMM, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China e-mail:
[email protected]
1
Corresponding authors
Abstract Volume conservation during plastic deformation is the most important feature, and should be realized in elastoplastic theories. However, it is found in this paper that an elastoplastic theory is not volume conserved if it improperly sets an arbitrary plastic strain rate tensor to be deviatoric. We discuss how to rigorously realize volume conservation in finite strain regime, especially when the unloading stress free configuration is not adopted in the elastoplastic theories. An accurate condition of volume conservation is first clarified and used in this paper that the density of a volume element after the applied loads are completely removed should be identical to that of the initial stress free states. For the elastoplastic theories that adopt the unloading stress free configuration (i.e. the intermediate configuration), the accurate condition of volume conservation is satisfied only if specific definitions of the plastic strain rate are used among many other different definitions. For the elastoplastic theories that do not adopt the unloading stress free configuration, it is even more difficult to realize volume conservation as the information of the stress free configuration lacks. To find a universal approach of realizing volume conservation for elastoplastic theories whether or not adopt the unloading stress free configuration, we propose a single assumption that the density of material only depends on the trace of the Cauchy stress by using their objectivities. Two strategies are further discussed to satisfy the accurate condition of volume conservation: directly and slightly revising the tangential stiffness tensor or using a properly chosen stress/strain measure and elastic compliance tensor. They are implemented into existing elastoplastic theories, and the volume conservation is demonstrated by both theoretical proof and numerical examples. The potential application of the proposed theories is a better simulation of manufacture process such as metal forming.
Keywords: Volume conservation; elastic-plastic material; finite strain; constitutive behavior
2
Table of nomenclature Deformation gradient F Elastic deformation gradient: deformation gradient from the unloading Fe stress free configuration to the current configuration p Plastic deformation gradient: deformation gradient from the initial stress F free configuration to the unloading stress free configuration p Plastic right Cauchy–Green tensor C Green strain E ln Logarithmic strain E ε Strain of small deformation Strain by taking the unloading stress free configuration as the reference E configuration Strain rate E Strain rate by taking the unloading stress free configuration as the reference E configuration d Deformation rate SO E Plastic strain rate suggested by the Simo–Ortiz theory
E E E E E
p
RH
MOS
SO
Plastic strain rate suggested by the Rice–Hill theory
p
p
Elastic strain rate suggested by the Simo–Ortiz theory
e
RH
Elastic strain rate suggested by the Rice–Hill theory
e
MOS
p
σ nominal σ Cauchy , σ
τ
σ ln σ p
A V
i i 1,2,3 J E ,
0, ini cur sf
Plastic strain rate suggested by the Moran–Ortiz–Shih theory
Elastic strain rate suggested by the Moran–Ortiz–Shih theory Nominal stress Cauchy stress (true stress) Kirchhoff stress Logarithmic stress (work conjugate stress to the logarithmic strain rate) Plastic stress rate density Area of the cross section volume Stretch ratio Volume ratio The material derivative of a variable with respect to time Quantities in a certain configuration Initial stress free configuration Current configuration Unloading stress free configuration (intermediate configuration)
3
1.
Introduction In elastoplastic constitutive theories, one of the most fundamental and important basis is
that the plastic deformation does not change the volume of material. However, in the regime of finite deformation, this volume conservation is not rigorously realized by some theories, although various elastoplastic theories have been developed from different standpoints. If an elastoplastic theory cannot realize volume conservation, it not only just predicts a changed density after unloading but also may lead to significant errors in the stress. For example, supposing that a sample has been deformed inhomogeneously. After unloading, all material elements cannot return to the stress free configuration due to geometrical compatibility and therefore give rise to residual stress. If the elastoplastic theory predicts an inaccurate volume (or density) of the material element in its unloading stress free configuration, it also predicts an inaccurate residual stress. Therefore it is important for the elastoplastic theory to be volume conserved. First of all, the definition of the volume change due to plastic deformation (or the plastic volume deformation) should be expressed clearly and easy to check. In the infinitesimal deformation regime, the plastic volume deformation can be unambiguously defined as the trace of the plastic strain tensor, since all strain definitions converge. But in the finite deformation regime, because the strain and the plastic strain (or their rates) may have different definitions in different elastoplastic theories, it can be expected that the plastic volume deformation has many different definitions correspondingly if it is still directly defined by the plastic strain. Some literatures regarding to the elastoplastic theories of finite deformation are listed here to show the various definitions of the plastic strain or the plastic strain rate [1–18]. There are also plenty of books introducing elastoplastic theories [19–22]. The above papers and books only cover a very small number of existing literatures, and reviews such as [23, 24] are recommended for a more comprehensive classification and discussion of representative theories. Recently elastoplastic constitutive models are also developed for some advanced materials, such as ferroelectric ceramics [25–30] shape memory alloys and polymers [31–35], magnetic shape memory alloys [36], proteins in biomaterials 4
[37], polymers [38, 39], hydrogels [40] and oxides [41]. In the elastoplastic theories that adopt the multiplicative decomposition and the unloading stress free configuration, the volume conservation condition during plastic deformation can be stated as det C p 1 where C p is the plastic right Cauchy–Green
tensor [11, 42]. In rate form, the condition is tr F p F p
1
0 , where
F p is the plastic
deformation gradient. Accordingly, volume conserved elastoplastic theories are established and in numerical sense the exponential map algorithm is developed to properly realize volume conservation [43–48]. However whether it is proper to adopt the unloading stress free configuration in elastoplastic theories is still in question, because this configuration sometimes is (1) not unique for different unloading paths; (2) unreachable when the plastic deformation occurs during unloading [49, 50], and (3) causes incompatibility when the deformation is not homogeneous [24]. Even if a theory adopts the unloading stress free configuration, it still may not realize the volume conservation since there are many different ways to define the plastic strain or plastic strain rate. Therefore the question remains that how to rigorously and properly realize plastic volume conservation with extended applicability to theories whether or not adopt the unloading stress free configuration. This paper is aimed at the above question. In Section 2, we first clarify an accurate condition of volume conservation that is clear and unambiguous as the benchmark throughout this paper. In Section 3, we show some examples to evaluate the volume conservation of elastoplastic theories. Some classical elastoplastic theories cannot realize volume conservation and lead to significant error, even if some of them adopt the unloading stress free configuration. Then in Section 4, we discuss how to realize the volume conservation without using the unloading stress free configuration, and two strategies with numerical implements and demonstrating examples. The strategies are compared and discussed in Section 5. Finally Section 6 concludes the paper by summarizing the major points.
5
2.
The accurate volume conservation condition An elastoplastic theory is volume conserved if it predicts no volume change when the
applied loads are removed. There are two possible complexities during the unloading process of a structure or a solid: residual stress due to non-uniform deformation and non-unique unloading configurations arising from different unloading plastic deformation paths. Therefore, to clarify the meaning of volume conservation in the regime of finite deformation, we first distinguish three levels of volume conservation condition here. Level 1: For a volume element subject to uniform deformation, its density change between the unique unloading stress free configuration (without any reverse plastic deformation) and the initial stress free configuration is zero. At this level, it is assumed that for a loaded current configuration, the corresponding unloading stress free configuration is unique and serves as a base to realize volume conservation. Level 2: Sometimes different unloading paths and the reverse plastic deformation can lead to many different unloading stress free configurations corresponding to the same current loaded configuration. Therefore a more strict condition should be stated that for a volume element subject to uniform deformation, all of its unloading stress free configurations should have the same density as the initial stress free configuration. Level 3: An even more strict volume conservation condition can be stated that for a structure instead of a volume element. Under arbitrary deformation, after the applied loads are totally removed through any unloading path, the volume (or overall average density) of the structure should be the same as the volume (or density) before any load has been applied. In this condition, we do not rule out the residual stresses that can arise from the non-uniform deformation after unloading. The level 1 condition has been realized and satisfied by some elastoplastic theories that adopt the unloading stress free configuration. Usually these theories assume that the unloading stress free configuration is unique for a current configuration, which is sometimes too strong. The level 2 condition has much wider applicability without the strong assumptions on the unloading process. The level 1 condition is automatically satisfied if the level 2 6
condition is satisfied. The level 3 condition can only be satisfied by satisfying level 2 condition and assuming a linear relationship between the density change and the stress, because the average stress and the average density change are zero after the applied loads are totally removed. The level 3 condition is stringent and usually does not apply for the material whose elastic volume change is not linearly proportional to the stress.
Stress
Current configuration(cur)
Density 0
Density sf
Volume V 0 Initial stress free configuration (0)
Volume V sf
Plastic volume deformation: sf 0 V 0 Vsf 0 V sf
Unloading Stress free configuration (sf) Strain
Fig. 1. Illustration of the accurate volume conservation condition and the definition of plastic volume deformation. The configuration whose plastic volume deformation is to be defined is denoted by a blue square, and its corresponding unloading stress free configuration is denoted by a red circle. The accurate volume conservation condition is sf 0 .
Therefore in this paper we focus on how to satisfy level 2 condition, and it is what we mean by the term accurate volume conservation condition unless explicitly stated otherwise. We also suggest using / 0 to measure the volume change due to the plastic deformation after the loads are removed, where 0 is the density of the initial stress free configuration, is the difference between the density of configuration when the loads are totally removed and 0 , as schematically shown in Fig. 1. In this paper, we only concern 7
volume conservation when the loads are removed (or “volume conservation” hereafter for simplicity). For the configuration under loading as denoted by the blue square in Fig. 1, the volume conservation is checked only on its corresponding unloading configuration denoted by the red circle.
3.
Evaluate the volume conservation based on sf 0 In this section, we show some examples to illustrate how to use the accurate volume
conservation condition to evaluate an elastoplastic theory and show how much error can be caused if an elastoplastic theory is not volume conserved. The traditional way in most theories to realize volume conservation is setting the plastic strain rate to be a deviatoric tensor. Considering that the plastic strain rate has many different definitions for finite deformation, obviously not all theories can realize the volume conservation, as discussed in more details in Section 3.1.
(a)
(b)
Rice–Hill theory
Stress
Stress
E t
Simo–Ortiz theory E t
E
p
C
A
t
Strain
B
E
D p
Strain
t
Fig. 2. The definitions of the plastic strain rate suggested by (a) the Rice–Hill theory and (b) the Simo–Ortiz theory.
3.1.
Various plastic strain rates defined by different theories In establishing an elastoplastic theory at finite deformation, one usually has to make the
following choices at least: (1) The manner of strain rate decomposition. Among the various candidates, three ways 8
suggested by three classical theories are considered in this paper, namely the Rice– Hill theory [2–5, 51], the Simo–Ortiz theory [49, 52, 53] and the Moran–Ortiz–Shih theory [6]. The plastic strain rate (or increment) of the first two theories are illustrated by the red segments in Fig. 2. (2) The reference configuration that is used to define the strain and the stress. Three configurations are usually used, which are the initial stress free configuration, the current configuration and the unloading stress free configuration, as shown in Fig 1. (3) The stress/strain measure (Appendix A can be referred to).
#1 Stress measure #1 unloading
# 2 Stress measure #2 unloading (b)
Stress measure #2
Stress measure #1
(a)
#1
#2
Strain measure #1
#1
#2
Strain measure #2
Fig. 3. An illustration that the strain rate decomposition suggested by the Rice–Hill theory is dependent on the stress/strain measure.
The above three choices can lead to many plastic strain rates. To illustrate how the arbitrary choice of the stress measure can affect the strain rate decomposition, the Rice–Hill theory with different stress unloading manners is discussed briefly here. Figure 3 schematically shows two stress–strain curves with two different stress/strain measures. Point
in Fig. 3(a) and Fig. 3(b) represent the same configuration, named as configuration , and the corresponding stresses are denoted by σ#1 and σ#2 for two stress measures respectively. After incremental deformation, the configuration becomes configuration , denoted by Point . In the strain rate decomposition suggested by the Rice–Hill theory, 9
defining the plastic strain increment from configuration to configuration requires the third configuration . This configuration is achieved by unloading the stress to the same value of configuration . The plastic part of the deformation increment can then be defined by the difference of strain between configuration and configuration . However, with different stress measures, e.g. stress measure #1 and #2, unloading the stress to the same value of configuration will lead to different configuration , such as #1 and # 2 in Fig. 3, and different plastic deformation increments (or plastic strain increments). It should be pointed out that the relative difference between these different plastic deformation increments does not vanish when the strain increment becomes infinitesimal, as demonstrated in the following example.
L Configuration (reference)
x3 x2
Cauchy 11 0 11nominal 11
x1
L L Configuration (loading)
Cauchy 11 L / L 11nominal , 11
Configuration (Cauchy stress unloading) #2
Configuration (Nominal stress unloading) #1
11nominal 11nominal Cauchy Cauchy 11 11 #1
11 p
#1
11 p
#2
11nominal 11nominal Cauchy Cauchy 11 11 #2
#2
#1
Fig. 4. A uniaxial tension example to illustrate that the strain decomposition suggested by the Rice–Hill theory depends on the stress/strain measure.
As shown in Fig. 4, a bar under uniaxial tension is investigated. Stress measure #1 and #2 are chosen to be the nominal stress and the Cauchy stress, respectively. For convenience 10
we chose configuration as the reference configuration so that the nominal stress and Cauchy Cauchy stress at this moment are the same, i.e. 11nominal 11 . Next we stretch the bar to
configuration , changing its length from L to L L , and the strain is
11
L L
(1)
The stress increment can be determined from elastoplastic constitutive relation. Suppose the nominal stress increment during loading process is related to strain increment by
11nominal Cep11
(2)
11nominal Ce11
(3)
and during unloading process
where Cep and Ce are the tangential loading and unloading stiffness, respectively. So the nominal stress in configuration is nominal 11nominal 11 Cep11
(4)
and the Cauchy stress (or true stress) in configuration is determined as
11Cauchy
A A
nominal 11nominal 11 1 11
(5)
where A indicates the area of the cross section in a certain configuration and volume conservation of deformation is assumed in obtained Eq. (5). If we unload the nominal stress to
11nominal its previous value, namely 11nominal , we obtain the nominal stress unloading #1
configuration #1 and the plastic strain
11 p 11 11 #1
nominal 11nominal 11
#1
Ce
Ce Cep Ce
11
(6)
and the Cauchy stress in configuration #1 is
1 #1 nominal 1 11Cauchy 11nominal 11 p 11 #1
#1
Ce Cep Ce
nominal 11 11Cauchy 11
(7)
It is obvious that the Cauchy stress does not come back to its original value of configuration 11
when the nominal stress does. On the other hand, if we unload the Cauchy stress, namely nominal #2 11Cauchy 11Cauchy and 11 , we obtain the Cauchy stress unloading configuration #2
1
1
Cauchy 11nominal Cauchy nominal 11nominal 11 1 # 2 11 1 # 2 11 11 p 11 p #2
#2
(8)
Therefore from nominal 11nominal Ce 11 11 11nominal 11 Cep11 and Eq. (6), we obtain #2
11 p
#2
#2
11 #2
It is noted from Eq. (10) that
11 p
#1
Ce Cep Ce +
11 =
nominal 11
and 11 p
#2
Ce Ce +
nominal 11
11 p
#1
(9)
(10)
are unequal even though 11 0 . Taking
Cauchy C 0.1Ce , numerical result shows that the relative error of 11nominal 11 0.1Ce and ep
these two plastic strain increments (or rates) could be about 10%. In summary, because there are so many different definitions of the plastic strain rate, setting all these different plastic strain rates to be deviatoric tensors cannot always realize volume conservation. A rigorous theoretical analysis is presented in the next subsection.
3.2.
Theoretical evaluation of the volume conservation Among the large number of elastoplastic theories in literature, we choose the Simo–Ortiz
theory and the Moran–Ortiz–Shih theory as examples to be evaluated by the accurate condition of volume conservation proposed in Section 2 ( / 0 0 ). With different configurations chosen as the reference configuration, the means to define the plastic strain rate adopted in these two theories are also used in many other theories, so we evaluate their volume conservations theoretically in Section 3.2 and numerically in Section 3.3. We first demonstrate that the accurate condition of volume conservation is equivalent to tr F p ( F p )1 0 in rate form. We denote the deformation gradient of the loading process
(from the initial stress free configuration to the current configuration) and the unloading 12
process (from the current configuration to the unloading stress free configuration) by F and
F
e 1
respectively, so the plastic deformation gradient from the initial stress free
configuration to the unloading stress free configuration is
F p F e F 1
(11)
where the dot represents the dot product of two tensors, namely A B Aik Bkj . The accurate condition of volume conservation can be written as
0 sf
Vsf V 0
J p det( F p ) 1
(12)
where subscript (sf) and (0) indicate that the quantities are in the unloading stress free configuration and the initial stress free configuration, respectively; sf 0 is the density of material in the unloading stress free configuration; Vsf and V0 are the volumes of a material element; J p is the volume ratio of the unloading stress free configuration. Taking derivatives on Eq. (12), we have
Jp
det(F p ) p : F J p ( F p )T : F p J p tr ( F p ) 1 F p J p tr F p ( F p ) 1 0 F p
(13)
where the double dot represents the scalar product, namely A : B Aij Bij , det( F p ) is the third invariant of F p and tr A A11 A22 A33 is the first invariant and called the trace of a second order tensor A . The accurate volume conservation condition is then reduced to tr F p ( F p )1 0
(14)
Equation (14) is referred to in this paper as the unloading stress free form of volume conservation condition, since it needs the information of the unloading stress free configuration. As mentioned in Section 3.1, there are at least three different choices on the reference configuration, giving rise to three plastic strain rates of the Simo-Ortiz strain rate decomposition and the Moran–Ortiz–Shih strain rate decomposition respectively. The 13
elastoplastic theories usually set one of these plastic strain rates to be a deviatoric tensor, which will be checked if Eq. (14) is rigorously satisfied. The plastic strain rates of the Simo–Ortiz strain rate decomposition are
1 pT p SO pT p E p 2 F F F F 1 SO ( F p ) T E SO ( F p ) 1 ( F p ) T F pT F p ( F p ) 1 E p p 2 SO 1 T SO 1 e T p T pT e 1 e T p p 1 e 1 d p F E p F 2 ( F ) ( F ) F ( F ) ( F ) F ( F ) ( F )
(15a-c) where E , E and d denote the strain rate when taking the initial stress free configuration, the unloading stress free configuration and the current configuration as the reference configuration, respectively. Here the strain measure is chosen to be the Green strain E . A superscript is added adjacent to the symbol to distinguish the manner of the strain rate decomposition, with SO standing for Simo–Ortiz and MOS standing for Moran–Ortiz–Shih. The subscript “p” outside the bracket indicates the term is the plastic part, while subscript “e” indicates that the term is the elastic part. A combination of the choice on the manner of strain rate decomposition and the reference configuration gives rise to different elastoplastic theories. In abbreviation, the theories are named as SO-ini, SO-cur and SO-sf for the Simo– Ortiz strain rate decomposition with the initial stress free, current and the unloading stress free configuration as the reference configuration, respectively. SO can be replaced by MOS to denote theories using the other strain rate decomposition. The traces of the strain rates in Eq.(15) are
tr E SO p tr F pT F p (F p )1 F p SO tr F p ( F p ) 1 tr E p SO e T p p 1 e 1 tr d p tr F F F F
For the Moran–Ortiz–Shih theory, we have
14
(16a-c)
1 pT eT e p MOS pT eT e p E p 2 F F F F F F F F 1 MOS ( F p ) T E MOS ( F p ) 1 F eT F e F p ( F p ) 1 ( F p ) T F pT F eT F e E p p 2 MOS 1 F T E MOS F 1 F e F p (F p )1 (F e ) 1 ( F e ) T ( F p ) T F pT F eT d p p 2
(17a-c)
1 tr E MOS tr ( F p )T F eT F e F p F p F p p 1 MOS tr F e F p F p F eT tr E p 1 MOS p tr F p F p tr d
(18a-c)
Table 1. Volume conservation evaluation of the elastoplastic theories using different strain decompositions suggested by the Simo–Ortiz and Moran–Ortiz–Shih theory with three different choices of the reference configurations. The symbol √ indicates that the volume conservation condition is satisfied by setting this term to be a deviatoric tensor, and × indicates the condition cannot be satisfied in this way. Reference configuration Initial stress free
Unloading stress Current configuration
configuration
Simo– Ortiz
Moran– Ortiz– Shih
tr E
SO
p
tr d SO
F pT F p × tr p 1 p ( F ) F
tr E MOS
p
free configuration
( F p )T F eT F e × tr p 1 F F p F p
p
F F p × tr p 1 e 1 F F e T
tr F
tr d
MOS
p
15
p
F p
√ 1
tr E SO
p
tr F ( F ) p
p 1
√
tr E MOS p Fe Fp × tr 1 F p F eT
The traces of the plastic strain rates derived above are also summarized in Table 1. It can
0
be concluded by observing Eq. (16) and Eq. (18) that only tr E SO
tr d MOS
p
0
and
p
satisfy the accurate volume conservation condition Eq. (14), namely only
the SO-sf theory
and the MOS-cur theory are capable of rigorously realizing volume
conservation, although the other theories also adopt the unloading stress free configuration. Therefore even the theories that adopt the unloading stress free configuration should be very careful to choose the proper plastic strain tensor to satisfy the accurate condition of volume conservation.
3.3.
Numerical evaluation of the volume conservation A uniaxial loading example (Fig. 5(a)) is presented in this subsection to illustrate that
when the accurate condition of volume conservation is not satisfied, significant errors will arise. Besides the two manners of strain rate decomposition discussed in Section 3.2, we also evaluate the Rice-Hill strain rate decomposition using the initial stress-free (RH-ini) and current (RH-cur) configuration as the reference configuration respectively. The material is assumed to be isotropic linear strain hardening expressed by the relationship between the logarithmic strain and the Cauchy stress. The Young’s modulus, Poisson’s ratio and hardening coefficient of the material are denoted by Ce , and Cep respectively. Under uniaxial loading in the x1 direction, the Cauchy stress 11 and the logarithmic strain E11ln are assumed to have the following relationship (Fig. 5(b))
Ce E11ln 0 E11ln Ecr 11 ln E11ln Ecr cr Cep E11 Ecr Ce E11ln Ecr E11ln 0 11 ln E11ln Ecr cr Cep E11 Ecr
for tension
for compression
(19)
(20)
where cr is the initial yielding stress and Ecr cr / Ce is the initial yielding strain. 16
Cartesian coordinate system is used here, and 1, 2, 3 stand for x1 , x2 , x3 , respectively. The elastic property of the material is assumed to be uncoupled with the plastic deformation, so under elastic uniaxial loading or unloading E11ln
1 11 , E22ln E33ln 11 Ce Ce
(21)
It should be pointed out that the stress–strain relations Eq. (19) and Eq. (20) are expressed in terms of the Cauchy stress and the logarithmic strain, as required by the commercial software ABAQUS (version 6.14), since the plastic volume deformation predicted by ABAQUS is also evaluated in this paper. In the following numerical examples, the material parameters are set as Cep 0.1Ce , cr 0.001Ce and 0.3
3.3.1. Plastic volume deformation The plastic volume deformation is calculated as
0
sf 0 0
V 0 V sf Vsf
1 sf 2sf 3sf
1
1
(22)
where isf is the stretch ratio in the unloading stress free configuration. The detailed formulae for calculating the plastic volume deformation in this uniaxial loading example are presented in Appendix B. The volume change of the unloading stress free configuration as a function of the stretch ratio of the loading configuration 1 is shown in Fig. 6 for different choices on the reference configuration and different manners of strain rate decompositions suggested by the Rice–Hill, Simo–Ortiz and Moran–Ortiz–Shih theories respectively. Only in two cases, namely the SO-sf theory and the MOS-cur theory, the volume conservation is realized.
17
11 (a)
x1 x x3
1
3
2
11
1
1sf
2
3sf 2sf
1
1
(b)
11 Current configuration (cur)
Ecr , cr
1
Cep
Ce 1
Ce 1
E11ln
Unloading Stress free configuration (sf)
Fig. 5. (a) Illustration of the uniaxial loading example; (b) the stress–strain curve used in this example.
18
(b) Moran–Ortiz–Shih strain rate decomposition
Volume change of the stress free configuration / 0
Volume change of the stress free configuration / 0
(a) Simo–Ortiz strain rate decomposition 0.5
SO-ini SO-sf 0.0
SO-cur -0.5
0.2
1
2
3
Stretch ratio
1
4
5
Volume change of the stress free configuration / 0
(c) Rice–Hill strain rate decomposition RH-ini
0.4
0.5
0.3 0.2
0.4
MOS-ini
0.3
MOS-sf
0.2 0.1 0.0 -0.1
0.2
MOS-cur 1
2
3
Stretch ratio
4
1
5
Reference configuration: Initial stress free Current Unloading stress free
1.0
0.5
0.5
0.0
0.1
RH-cur
0.0 -0.1 0.2
-0.5 1
2
3
0.5
Stretch ratio 1
4
1.0
5
1.5
2.0
2.5
Fig. 6. The volume change of the unloading stress free configuration predicted by various elastoplastic theories.
The predicted plastic volume deformations for other theories presented in Fig. 6 (SO-ini, SO-cur, MOS-ini, MOS-sf, RH-ini and RH-cur) are not zero, and can be very large for the theories taking the initial configuration as the reference configuration (SO-ini, MOS-ini and RH-ini) in both compression and tension. The plastic volume deformations predicted by SO-ini, MOS-ini and RH-ini exceed 50% when the stretch ratio 1 2 (stretch to 2 times) or
1 0.2 (compress to 1/5). Among the evaluated theories that are not volume conserved, RH-cur has the best performance, but still predicts a plastic volume deformation of 3% when
1 5 or 1 0.2 . The above results are obtained through numerical integral whose convergence is guaranteed by using a very small increment. Taking the SO-cur theory for example, the plastic volume deformation predicted by using different increments in the 19
numerical integral is shown in Fig. 7. The calculation is convergent when the increment is reduced to 1t t / 1t e 0.001 , so we use increment smaller than this value in the numerical calculation. 0.0
/ 0
Volume change of the stress free confiuguraiton
-0.1 SO-cur with step increment:
1t t e0.1 1t
-0.2 0.0
e0.01
-0.3
-0.1
e0.001 , e0.0001
-0.4
-0.2
-0.5
2
-0.3
4
6
8
10
-0.4 -0.5
2
4
6
8
10
Stretch ratio 1
Fig. 7. An illustration of the numerical convergence in the uniaxial loading example
Finite element simulations are also carried out using commercial software ABAQUS (standard, version 6.14) , COMSOL (version 5.0) and ANSYS (version 16.0) respectively, with an 8-node brick element. Because the problem is nonlinear, the simulation is divided into many increments. The amount of stretch 1 applied in each increment affects the result of ABAQUS. Using a small 1 , ABAQUS predicts almost zero plastic volume deformation but the simulation does not converge when the total applied stretch 1 is large (the black solid line in Fig. 8(a)). The convergence can be improved by increasing the increment 1 , but the predicted plastic volume deformation also increases (the red dash-dotted line and the blue short dashed line in Fig. 8(a)). In brief the elastoplastic simulation of ABAQUS is not always volume conserved. As shown by Fig. 8(b), COMSOL and ANSYS predict large plastic volume deformation 20
even for quite small applied stretch ratio. Therefore they also do not predict volume conserved result for the elastoplastic problem of finite deformation.
Predicted by ABAQUS Increment: 1 0.01 1 0.1 1 1.0
0.08 0.08 0.06 0.06 0.04 0.04
Inconvergent 0.02 0.02
Inconvergent
0.00 0.00 0.1
1
0.1
1
0.08
(b)
10
10
Stretch ratio 1
Volume change of the stress free configuration / 0
Volume change of the stress free configuration / 0
(a)
0.20
0.20
0.06
Predicted by: COMSOL ANSYS
0.15
0.15
0.04 0.10
0.10
0.02 0.05
0.05
0.00 0.00 0.00
0.4 0.1
0.4
0.6
0.6
0.8
0.8
1.0
1.0
11.2
1.2
1.4
1.4
1.6
10
1.6
Stretch ratio 1
Fig. 8. The plastic volume deformation predicted by commercial software (a) ABAQUS; (b) COMSOL and ANSYS.
3.3.2. Error on the loading stress When an elastoplastic theory that cannot rigorously realize volume conservation is used, significant error could arise in the prediction of stress. Similar to the predictions on the plastic volume deformation, only SO-sf and MOS-cur theories predict the accurate stress. The theories using the initial configuration as the reference configuration give rise to a relative error larger than 50% when the required stretch ratio 2sf 1.3 or 2sf 0.7 . The relative error of SO-cur and MOS-sf exceeds 20% when 2sf 3 or 2sf 0.5 . SO-cur has the best performance, with a relative error about 1% for 0.5 2sf 3 . The above results are shown in Fig. C1 in Appendix C.
4.
Strategies to realize volume conservation without referring to the unloading stress free configuration
4.1.
A volume conservation condition without referring to the unloading stress free configuration 21
The evaluation of volume conservation in Section 3 is all conducted on the unloading stress free configuration, whether or not the theory adopts this configuration. The unloading stress free configuration has many assumptions on the unloading process, such as (1) No reverse plastic deformation occurs when the material is unloaded to the stress free configuration. This assumption is not always valid for materials showing kinetic hardening behavior. (2) Usually the elastic moduli during unloading is assumed to be constant and the same as those during the initial loading process before any plastic deformation occurs. However this assumption is probably improper in large deformation because the elastic moduli might be intrinsically changed or extrinsically changed due to the measure dependence. Above assumptions are sometimes too strong and cannot always be met, making the adoption of the unloading stress free configuration in elastoplastic theories a controversial issue. Therefore we should seek alternate strategies to establish elastoplastic theories which do not need to use the unloading stress free configuration. In this way most above assumptions can be discarded and the applicability is expanded. First of all, a condition equivalent to the accurate volume conservation condition should be proposed and expressed in the current configuration, without explicitly referring to the unloading stress free configuration. Here a volume conservation condition is proposed based on only one assumption that the density of material is a function of the trace of the Cauchy stress, namely
0
1 1 g tr σ J
(23)
where function g x satisfies g 0 0 and g ' x 0 . It is not difficult to determine
g x from experiment [54] such as uniaxial tests. If Eq. (23) is satisfied, the volume conservation is realized because in the unloading stress free configuration σ 0 , giving rise to 0 . Cauchy stress has a straightforward physical meaning without requiring a reference configuration and then has a unique position superior to other stress measures. 22
Therefore it is chosen in the above volume conservation condition. The following discussion is mainly based on the Cauchy stress if no explicit indication is given. Similarly, the density of the current configuration also has a clear physical meaning independent of any arbitrary choice. Hereafter Eq. (23) is referred to as the current deformation form of the volume conservation condition, and g tr σ is called the volume constitutive function. Equation (23) provides a volume conservation condition in deformation form to constrain the elastoplastic theory. As most elastoplastic theories are in rate form, next we derive a rate form of Eq. (23). Taking derivatives on Eq. (23) yields
J g ' tr σ tr σ J2 On the other hand, by noticing that J det F , we can derive
(24)
J Jtr d
(25)
Substituting Eq. (25) into Eq. (24) yields another volume conservation condition in rate form
tr d Jg ' tr σ tr σ tr σ / K where K
1 is a function of the trace of the Cauchy stress Jg ' tr σ
(26)
σ or the volume
ratio J . Noticing that this condition is in the current configuration, the unloading stress free configuration is not needed. We refer to Eq. (26) as the current rate form of the volume conservation condition. 4.2.
Evaluate an elastoplastic theory based on Eq. (26) Then we discuss how to evaluate and revise an existing elastoplastic theory. For an
arbitrary elastoplastic theory, its constitutive relationship can be expressed by σ obj L : d
(27)
where L is the tangential stiffness tensor and σ obj represents one type of objective rate of the Cauchy stress σ . If originally the constitutive relationship is expressed in other stress/strain rates than the Cauchy stress rate σ obj and the deformation rate d , it can be 23
easily transformed to the form of Eq. (27), because the stress and the strain of different measures are related. Using Eq. (27), we can calculate tr σ and tr d , and check if Eq. (26) is satisfied or not. 4.3.
Strategies to realize volume conservation
4.3.1. Strategy 1: directly and slightly revise the tangential stiffness tensor If an elastoplastic theory does not satisfy Eq. (26), a revision strategy might be used as a remedy. To realize volume conservation, we revise L to Lrev L L so the constitutive relationship Eq. (27) becomes
σ obj L L : d
(28)
Two conditions should be met in this revision: (1) after revision, Eq. (23) or its rate form Eq. (26) is satisfied; (2) the revision term L is the most minor one in the sense that its norm L L L Lijkl Lijkl is the minimum.
The revision term of the stiffness tensor L depends on L and the choice on the objective stress rate. There are many different choices on the objective stress rate. Here we first use the Jaumann rate σ Jau σ σ w w σ as an example, where w is the spin tensor. As demonstrated in Appendix D L is 3 3 3 5 1 1 L K L K L K Lkkqq iiii kkii kkpp 9 k 1 k 1 k 1 9 9 3 3 3 2 2 1 Liijj Ljjii K Lkkii K Lkkjj K Lkkll 9 k 1 k 1 k 1 9 9 3 3 Liikl Lklii Liilk Llkii 1 Lppkl Lpplk 6 p 1 p 1 Lijkl L jikl Lijlk Lklij 0
i 1, 2,3; p i, q i, p q i, j 1, 2,3; i j , l i, l j i, k , l 1, 2,3; k l i, j , k , l 1, 2,3; i j , k l (29)
There is no dummy summation in Eq. (29). If an objective stress rate other than the Jaumann rate is used, the revision term of the stiffness tensor L can also be derived easily as presented in Appendix D. In finite element simulations, we suggest the following procedure to carry out the 24
revision and eliminate the possible accumulated numerical error that arises from the incremental algorithm. Step 1: Supposing that at the current time, we have balanced stresses and compatible strains, then keep the strain unchanged, revise the stress according to Eq. (23) so that the deviation from the volume conservation condition is eliminated. Step 2: the unbalanced stress caused by Step 1 is added to the total unbalanced force. Finite element simulation is conducted using the incremental algorithm with the revised tangential stiffness tensor Lrev to obtain new balanced stresses and compatible strains. Then check if Eq. (23) is satisfied with an acceptable error; if not, return to Step 1. We should notice that the choice of the objective stress rate is also an important issue in constitutive theories. An improper objective stress rate may cause energy conservation problem or large accumulate errors, and use of the corotational rate may solve the problem for moderate large deformation cases. A discussion of the choice of the objective stress rate in constitutive theory is not in the scope of the current paper and can be found in many literatures such as [55–65]. We only emphasize that even though the elastoplastic theory chooses a proper objective rate, it still can have the volume conservation problem. Fortunately for an elastoplastic theory using an arbitrary objective stress, the revision strategy Section 4.3.1 can still be carried out, as demonstrated in Appendix D.
4.3.2. Strategy 2: use the logarithmic stress/strain measure Through the procedure presented in Section 4.3.1, an arbitrary elastoplastic theory (including existing theories) can be revised so that the volume conservation is realized. After revision the plastic strain rate is usually no longer a deviatoric tensor. Because most elastoplastic theories are based on the assumption that the plastic strain rate is a deviatoric tensor, it will be convenient and consistent if the volume conservation condition can be satisfied while keeping the plastic strain rate a deviatoric tensor. In this subsection, we discuss a strategy that: (1) does not need the unloading stress free configuration; (2) automatically 25
satisfies the volume conservation without posterior revision and (3) keeps the plastic strain rate a deviatoric tensor. As the first step, a stress/strain measure needs to be chosen. According to Eq. (23) or Eq. (26), the most convenient stress measure seems to be the Cauchy stress because the density of volume is assumed to be a function of the trace of Cauchy stress. However, the Cauchy stress is defined in the current configuration so an objective stress rate also needs to be chosen among various candidates. To avoid this issue, the logarithmic strain E ln and the logarithmic stress σ ln , which is work-conjugated to E ln , may be a good choice, because the traces of the logarithmic stress σ ln and the Cauchy stress σ are closely related by
tr σ ln Jtr σ . Besides the logarithmic stress is defined by taking the initial stress free configuration as the reference configuration so that the objective stress rate is avoided. Elastoplastic theories using the logarithmic stress/strain measure have been developed in quite a few literatures, such as [33, 58, 59, 66–69]. Taking the combined hardening material as an example, an elastoplastic constitutive theory has the following form during elastoplastic loading
E ln
p
9
σ ln ' σ b ' σ ln ' σ b ' : σ ln 4Cp eq2
E ln
e
(30)
Me : σ ln
(31)
where Cp is the plastic modulus, M e is the elastic compliance tensor, σ b is the back stress, the superscript ' indicates the deviatoric part of a second order tensor, for example
σ ' σ ln
ln
1 3 σ ln ' σ b ' : σ ln ' σ b ' ; I is the second order tr σ ln I ; eq 3 2
identity tensor. From Eq (30) we find that the plastic strain rate tr E ln
p
0 . In many of the
above theories, a constant elastic compliance tensor M e is usually assumed, i.e.
M e m 1 I I I where m is a constant and I
(32)
is the fourth order identity tensor. We can easily prove that 26
the current rate form of the volume conservation condition Eq. (26) is satisfied, if the volume constitutive function in Eq. (23) is assumed to satisfy
g 'tr σ 1 g m 1 2 g' 1 g
(33)
where tr σ ln tr Jσ , tr σ ln ' σ b ' σ ln ' σ b ' : σ ln 0 , Eq. (24) and Eq. (32) are used in the derivations. However assuming a linear constant elastic compliance tensor M e may not be very proper, as we discuss below.
When the elastic compliance tensor M e is constant, the relationship between the volume change vs. the trace of Cauchy stress is not linear. In our opinion, it may be more proper to assume a linear relationship between the volume change vs. the trace of Cauchy stress, namely
0
tr σ 1 1 g tr σ J KV
(34)
where KV is the constant volume modulus, as for many materials like metals, the volume deformation is still modest even though the plastic deformation is large. Then to accurately satisfy the current rate form of the volume conservation condition Eq. (26), a stress dependent elastic compliance tensor M e is derived as
KV tr σ Me 2 1 I I I KV 1 2
(35)
From Eq. (30), Eq. (31) and Eq. (35), it is demonstrated that
KV tr σ J tr d tr E ln tr σ ln tr σ Jg ' tr σ tr σ 2 KV KV
(36)
Therefore the volume conservation condition Eq. (26) is automatically satisfied. From Eq. (34) and Eq. (35), we can see that when the relation of the volume change vs. trace of Cauchy stress is linear, the elastic compliance tensor M e is not a constant. It is interesting to point out that the constant elastic compliance tensor M e and the linear relation of the volume change vs. trace of Cauchy stress cannot be satisfied at the same time, because 27
the relation between the volume and the strain is not linear for finite deformation. In our opinion, the linear relation of the volume change vs. trace of Cauchy stress is more reasonable and should have the priority to be satisfied. In a more general case when the elastic compliance tensor M e is not constant while the relationship of the volume change vs. trace of Cauchy stress is also nonlinear, the elastic compliance tensor M e is then derived as
g ' tr σ 1 g tr σ
1 I I I 1 g tr σ g ' tr σ tr σ 1 2 It is then obtained that Me
tr d tr E ln
g ' tr σ 1 g tr σ
(37)
tr σ ln
1 g tr σ g ' tr σ tr σ g ' tr σ 1 g tr σ Jg ' tr σ tr σ 1 Jtr σ 1 g tr σ g ' tr σ tr σ Jg ' tr σ tr σ
(38)
Therefore the volume conservation condition Eq. (26) is automatically satisfied. In this Strategy 2, no information of the unloading process to zero stress is needed, so the elastoplastic theory avoids the unloading stress free configuration. The theory also avoids the choice of the objective stress rate, as the logarithmic strain and its work-conjugate stress is defined by taking the initial stress free configuration as the reference configuration.
4.3.3. Numerical examples to illustrate the volume conservation of Strategy 1 and 2 based theory The elastoplastic theory based on Strategy 1 and 2 are evaluated by a numerical example where a material element is subjected to subsequent biaxial loadings ( 11 is first applied and then fixed during 22 loading process) as shown in Fig. 9(a). The material property is still expressed by Eq. (19) and Eq. (20) (shown by Fig. 5(b)). The RH-ini theory is revised by Strategy 1 presented in Section 4.2. The predicted plastic volume deformation is almost zero, with a very small numerical error less than 10 8 for an 28
increment 1t t / 1t e0.1 in each step, while the RH-ini theory without revision predicts a large plastic volume deformation even with a much smaller increment 1t t / 1t e0.001 as shown by the red dashed line in Fig. 9. The prediction using the elastoplastic theory based on Strategy 2 also indicates almost zero plastic volume deformation, with a numerical error less than 1010 for an increment 1t t / 1t e0.1 .
11
11
(a)
1
1
22 2
22
1 1
11
1.0
(b)
11
3
x1
Volume change of the stress free configuration / 0
0.8 1.0
0.6
1.0
0.8
0.4
0.8
0.6
0.2
0.6
RH-Ini revised by strategy 1 Theory based on strategy 2
0.4
0.0
0.4
0.1 1 10
RH-Ini
RH-Ini
0.2
0.6
0.8 0.2
1.2
Revised theory
0.0
0.0 0.6
1.0
0.8
0.6 1.0
1 1.1
x3 x2
0.8 1.2
1.0
1.2
Stretch ratio 2
Fig. 9. The volume change of the stress free configuration of a material element subjected to subsequent biaxial loading, predicted by the RH-Ini theories (the red dashed line) and the theories revised by Strategy 1 and 2 (the black solid line).
5.
Discussion
5.1. Applicability of the strategies to realize volume conservation 5.1.1. Strategy 0: use theories such as SO-sf and MOS-cur when the unloading stress free configuration can be uniquely determined 29
If an elastoplastic theory satisfies the unloading stress free form of volume conservation condition Eq. (14), it is volume conserved. Among the theories evaluated in Section 3.2 and Section 3.3, by setting the plastic strain rate to be a deviatoric tensor the SO-sf theory and the MOS-cur theory realize volume conservation. A major drawback for adopting this strategy is that the unloading stress free configuration cannot be avoided. For example in the SO-sf theory and the MOS-cur theory the plastic strain rates are defined through the plastic deformation gradient F p explicitly as illustrated by Eq. (15b) and Eq. (17c). 5.1.2. The two strategies when the unloading stress free configuration is avoided Two strategies to realize volume conservation without referring to the unloading stress free configuration are discussed in this paper. Strategy 1 realizes the volume conservation by slightly revising the tangential stiffness tensor according to the relation between the density of material and the trace of the Cauchy stress tensor. This revision can be conducted on an arbitrary elastoplastic theory easily with just a little change to the original theory. After revision, the theory becomes strictly volume conserved for any stress free configurations. The revision might lead to slight incompatibility in some aspects, such as the plastic strain rate is not a deviatoric tensor anymore. However this cost is acceptable, as the volume conservation is fundamental in elastoplastic theories and some existing theories have serious problems in realizing volume conservation as demonstrated by the numerical examples in Section 3.3. The most advantageous aspect of this strategy is that it can be adopted for an arbitrary elastoplastic theory easily, no matter it depends on the unloading stress free configuration or not. Some of the elastoplastic theories that are not volume conserved may have advantages in other aspects, such as a good prediction of the relationship between stresses and strains in certain cases. Therefore it may not be a good idea to abandon all the theories. Strategy 1 can provide a remedy for these theories so that they can realize volume conservation. A more rigorous treatment is provided by Strategy 2. By the logarithmic stress/strain measure and the properly determined elastic compliance tensor, the elastoplastic theory can then ensure the volume conservation condition while keeping the plastic strain rate still a deviatoric tensor. Neither the unloading stress free configuration nor the objective stress rate 30
is needed in this strategy, because the logarithmic strain/stress used is defined by taking the initial stress free configuration as the reference configuration. The comparison of the advantages and disadvantages among the three strategies are summarized in Table 2. Table 2. Comparison among the strategies to realize volume conservation
Strategy 0
Strategy 1
Strategy 2
Description
Advantage
Satisfy the unloading stress free form of the volume conservation condition Eq. (14) by setting the proper plastic strain rate to be deviatoric Slightly revise the stiffness tensor to satisfy the current form of the volume conservation condition Eq. (23) or Eq. (26) in the most minor way
(1) without posterior revision; (2) keeps the plastic strain rate a deviatoric tensor; (3) directly applies for anisotropic media.
Use the strain rate decomposition consistent with the volume conservation condition, the logarithmic strain/stress and the properly determined elastic compliance tensor to satisfy Eq. (23) or Eq. (26) automatically
(1)
(1)
(2)
(3)
(2)
(3) (4)
31
Disadvantage
(1) needs the unloading stress free configuration; (2) does not apply for materials with reversal plastic deformation. does not need the (1) is posterior unloading stress revision free configuration; (2) the plastic strain applies for rate after revision materials with may not be reversal plastic deviatoric; deformation; (3) cannot directly easy to implement apply for for an arbitrary anisotropic elastoplastic media. theory. does not need the (1) cannot directly unloading stress apply for free configuration; anisotropic applies for media. materials with reversal plastic deformation; without posterior revision; keeps the plastic strain rate a deviatoric tensor.
0.03 0.02
(b) SO-cur ABAQUS Theory revised by Strategy 1 and theory based on strategy 2
/ 0
Volume change of the stress free configuration / 0
(a)
0.01
1.0
SO-ini
0.8 0.6 0.4 0.2
0.00
0.0 -0.01 0.1
1
Stretch ratio
10
1
0.4
0.6
0.8
1.0
Stretch ratio
1.2
1
1.4
Fig. 10. The predicted volume change of the stress free configuration under uniaxial loading for elastic-perfectly plastic material with small elastic deformation.
5.2. Small elastic deformation Previous studies do not presume the degree of deformations. Considering that in some practical applications the elastic deformation is small, a brief discussion is presented below. If the elastic deformation gradient F e is close to the identity tensor, then from Eq. (16) and Eq. (18) it seems that SO-cur and MOS-sf theories are close to satisfy the volume conservation condition. However, we will demonstrate that the elastoplastic theories still cannot realize volume conservation within an acceptable error tolerance. We still investigate the example that a material element is subjected to uniaxial loading, but the material property is elastic-perfectly plastic now ( Cep 0 in Eq. (19), Eq. (20) and Fig. 5). The maximum elastic strain (or yielding strain) here is about 0.1%. The volume changes of the unloading stress free configuration predicted by SO-cur theory, SO-ini theory, ABAQUS and the theories based on the strategies proposed in this paper are presented in Fig. 10. We can see that with sufficient small increment, the SO-cur theory still predicts a plastic volume deformation of about 1.5% for 1 0.1 or 1 10.0 , while ABAQUS predicts a plastic volume deformation of about 2.5% for 1 0.1 and 0.5% for 1 10.0 , and the SO-ini theory has the worst prediction. Noticing that the error of the volume deformation is larger than the elastic deformation (on the order of 0.1%), which will lead to a significant error on the predicted stress. By contrast, the elastoplastic theory that implements the strategy proposed by this paper predicts almost 32
zero plastic volume deformation (with a numerical error less than 10 5 for 0.1 1 10.0 when implemented into ABAQUS).
5.3. Anisotropic media For anisotropic media if no reversal plastic deformation occurs and the unloading stress free configuration can be uniquely determined, strategy 0 can still be used, i.e. setting the proper plastic strain rate to be a deviatoric tensor. To avoid referring to the unloading stress free configuration explicitly, the relation of the volume change vs. the Cauchy stress that satisfies the symmetry of the media should be provided as the volume conservation condition. Then the two Strategies proposed in Section 4 can be adopted. The general relation for anisotropic media can be expressed as
0
1 1 ga σ J
(39)
where the function ga σ 0 when σ 0 . As we have discussed in the isotropic media case, for many materials it is reasonable to assume a linear relationship of the volume change vs. the Cauchy stress. Under this assumption and given the symmetry of the media, this relationship can be determined as we show next by taking the transversely isotropic media and the orthotropic media as examples Transversely isotropic media:
0
tr σ k σ k 1 1 J KV 1 KV 2
(40)
where k is a vector characterizing the preferred direction. Orthotropic media:
0
tr σ tr N σ tr N N σ 1 1 J KV 1 KV 2 KV 3
(41)
where N ii jj , i and j are vectors characterizing the orthogonal directions. In general, the volume conservation condition can be written as
33
0
1 1 D : σ J
(42)
where D is a second order tensor. With the relationship of the volume change vs. the Cauchy stress given, we can revise an existing theory (Strategy 1) or using the logarithmic stress/strain and chosen the proper elastic compliance tenser (Strategy 2) to satisfy Eq. (42) or its rate form.
5.4. Zero trace of the plastic stress rate vs. zero trace of the plastic strain rate As mentioned previously that elastoplastic theories usually assume the trace of the plastic strain rate to be zero, and fail most times in ensuring the volume conservation. In the following, we will demonstrate that the volume conservation, however, corresponds to the
Stress
zero trace of the plastic stress rate.
σ p t
Deformation gradient
Fig. 11. Illustration of the stress rate decomposition and the plastic stress rate
Figure 11 schematically shows the definition of the plastic stress rate
σ p . The material
first deforms from the current configuration with stress σ to a nearby configuration
with stress σ . Then it is unloaded to configuration whose deformation gradient
F is the same as that of configuration . The stress difference between configuration and configuration is defined as the plastic stress increment plastic stress rate is defined as
σ p lim σ σ / t . t 0 34
σ p t σ σ , and the In general an objective rate
should be used when taking the derivative of the Cauchy stress with respect to time, but the plastic stress rate defined above is already an objective tensor as configuration and have the same deformation gradient so all the objective rates converge to the material rate. It should be emphasized that the strain unloading configuration is unique, because the identical deformation gradient has unambiguous meaning. In comparison the stress unloading configuration has many different choices corresponding to different stress measures as shown in Section 3.1 (Fig. 3 and Fig. 4). Therefore the plastic stress rate can be defined uniquely but the plastic strain rate cannot. From Eq.(23), we obtain
0
g ' tr σ tr σ t
g tr σ tr σ p t g tr σ
p
(43)
The deformations of configuration and are the same, so and a very concise volume conservation condition in rate form is
tr σ p 0
(44)
It is interesting to note that the zero trace of the plastic stress rate tensor, not the plastic strain rate tensor, is equivalent to the accurate condition of volume conservation.
6.
Conclusion In this paper, we focus on the issue of rigorously realizing volume conservation during
finite elastoplastic deformation. The following conclusions can be drawn. (1) An unambiguous and accurate condition of volume conservation is clarified and used as benchmark, requiring that the density of any unloading states should be the same to that of the initial stress free configuration. (2) Some classical elastoplastic theories are evaluated both theoretically and numerically. Numerical results indicate that the error can be significant if an elastoplastic theory is not volume conserved. (3) A volume conservation condition without referring to the unloading stress free 35
configuration is proposed that the density of material only depends on the trace of the Cauchy stress. Both quantities are defined for the current configuration, and have clear physical meanings independent of any arbitrary choice, such as reference configurations or strain measures. Therefore, the volume conservation condition is objective as well. Based on it two strategies for volume conservation are thus discussed: directly and slightly revising the tangential stiffness tensor or using a properly chosen stress/strain measure and elastic compliance tensor. They are implemented in different elastoplastic theories, and the volume conservation is demonstrated by numerical examples. The established theories do not need the unloading stress free configuration and thus have expanded applicability. Their potential applications are an improved simulation of manufacture process such as metal forming. In the numerical examples given in this paper, only the static deformation is considered. We should emphasize that the volume conservation issue may be more significant in the case of dynamic applications. For example in high speed impact, both the strain and the strain rate can be very large. Then if an elastoplastic theory has an inaccurate stiffness tensor due to volume conservation issue, the large error of its prediction in each time step may accumulate fast. A study on the volume conservation of the elastoplastic theory in the context of dynamic application may be an interesting future direction of the current work.
Acknowledgement The authors acknowledge the support from National Natural Science Foundation of China (Grant Nos. 11425208, 51232004 and 11372158) and Tsinghua University Initiative Scientific Research Program (No. 2011Z02173).
36
Reference [1] Mandel, J., 1974, "Director vectors and constitutive equations for plastic and visco-plastic media," Problems of Plasticity, A. Sawczwk, ed, Leyden, Noordhoff, pp. 135–143. [2] Rice, J.R., 1971, "Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity," Journal of the Mechanics and Physics of Solids, 19, pp. 433–455. DOI: 10.1016/0022-5096(71)90010-X [3] Rice, J.R., 1975, "Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanics," Constitutive Equations in Plasticity, A.S. Argon, ed, MIT Press, Cambridge. Chapter 2: 23–79. [4] Hill, R., and Rice, J.R., 1972, "Constitutive analysis of elastic-plastic crystals at arbitrary strain," Journal of the Mechanics and Physics of Solids, 20, pp. 401–413. DOI: 10.1016/0022-5096(72)90017-8 [5] Hill, R., and Rice, J.R., 1973, "Elastic potential and the structure of inelastic constitutive laws," SIAM Journal on Applied Mathematics, 25(3), pp. 448–461. DOI: 10.1137/0125045 [6] Moran, B., Ortiz, M., and Shih, C.F., 1990, "Formulation of implicit finite element methods for multiplicative finite deformation plasticity," International Journal for Numerical Methods in Engineering, 29, pp. 483–514. DOI: 10.1002/nme.1620290304 [7] Besseling, J.F., and Van der Giessen, E., 1994, Mathematical Modelling of Inelastic Deformation, Chapman & Hall, London, UK. ISBN: 0-412-45280-4 [8] Yang, Q., Mota, A., and Ortiz, M., 2006, "A finite-deformation constitutive model of bulk metallic
glass
plasticity,"
Computational
Mechanics
37(2),
pp.
194–204.
DOI:
10.1007/s00466-005-0690-5 [9] Lele, Suvrat P., Anand, and Lallit, 2009, "A large-deformation strain-gradient theory for isotropic viscoplastic materials," International Journal of Plasticity, 25(3), pp. 420–453. DOI: 10.1016/j.ijplas.2008.04.003 [10] Rubin, M.B., and Ichihara, M., 2010, "Rheological models for large deformations of elastic-viscoplastic materials," International Journal of Engineering Science, 48(11), pp. 37
1534–1543. DOI: 10.1016/j.ijengsci.2010.06.029 [11] Vladimirov, I.N., Pietryga, M.P., and Reese, S., 2010, "Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to sheet metal forming," International Journal of Plasticity, 26, pp. 659–687. DOI: 10.1016/j.ijplas.2009.09.008 [12] Volokh, K.Y., 2013, "An approach to elastoplasticity at large deformations," European Journal of Mechanics A-Solids, 39, pp. 153–162. DOI: 10.1016/j.euromechsol.2012.11.002 [13] Altenbach, H., and Eremeyev V. A., 2014, "Strain rate tensors and constitutive equations of inelastic micropolar materials," International Journal of Plasticity, 63, pp. 3–7. DOI: 10.1016/j.ijplas.2014.05.009 [14] Shutov, A.V., and Ihlemann, J., 2014, "Analysis of some basic approaches to finite strain elasto-plasticity in view of reference change," International Journal of Plasticity, 63, pp. 183– 197. DOI: 10.1016/j.ijplas.2014.07.004 [15] Rajagopal, K.R., and Srinivasa, A.R., 2015, "Inelastic response of solids described by implicit constitutive relations with nonlinear small strain elastic response," International Journal of Plasticity, 71, pp. 1–9. DOI: 10.1016/j.ijplas.2015.02.007 [16] Chaouki, H., Picard, D., Ziegler, D., Azari, K., Alamdari, H., and Fafard, M., 2016, "Viscoplastic modeling of the green anode paste compaction process," ASME Journal of Applied Mechanics, 83(2), 021002. DOI: 10.1115/1.4031857 [17] Zhu, Y., Kang, G., Kan, Q., Bruhns, O.T., and Liu, Y., 2016, "Thermo-mechanically coupled cyclic elasto-viscoplastic constitutive model of metals: theory and application," International Journal of Plasticity, 79, pp. 111–152. DOI: 10.1016/j.ijplas.2015.12.005 [18] Chowdhury, S.R., Kar, G., Roy, D., and Reddy, J. N., 2017, "Two-Temperature Thermodynamics
for Metal
Viscoplasticity:
Continuum Modeling and Numerical
Experiments," Journal of Applied Mechanics, 84(1), 011002. DOI: 10.1115/1.4034726 [19] Bertram, A., 2005, Elasticity and Plasticity of Large Deformations: an Introduction, Springer, Berlin, Germany. ISBN: 978-3-540-24033-4 [20] Dunne, F., and Petrinic N., 2005, Introduction to computational plasticity, Oxford 38
University Press, New York, USA. ISBN: 978-0-19-856826-1 [21] Hashiguchi, K., 2009, Elastoplasticity theory, Springer, Berlin, Germany. ISBN: 978-3-642-35848-7 [22] Lubarda, V., A., 2001, Elastoplasticity theory, CRC Press, Florida, USA. ISBN: 0-8493-1138-1 [23] Naghdi, P.M., 1990, "A critical-review of the state of finite plasticity," ASME Journal of Applied Mathematics and Physics, 41, pp. 315–394. DOI: 10.1007/BF00959986 [24] Xiao, H., Bruhns, O.T., and Meyers, A., 2006, "Elastoplasticity beyond small deformations," Acta Mechanica, 182, pp. 31–111. DOI: 10.1007/s00707-005-0282-7 [25] Chen, H.S., Ma, J., Pei, Y., and Fang, D.N., 2013, "Anti-plane Yoffe-type crack in ferroelectric materials," International Journal of Fracture, 179, pp. 35–43. DOI: 10.1007/s10704-012-9767-2 [26] Chen, H.S., Pei, Y., Liu, J., and Fang, D.N., 2013, "Moving polarization saturation crack in ferroelectric solids," European Journal of Mechanics-A/Solids, 41, pp. 43–49. DOI: 10.1016/j.euromechsol.2013.02.002 [27] Chen, H.S., Pei, Y.M., Liu, B., and Fang, D.N., 2013, "Rate dependant heat generation in single cycle of domain switching of lead zirconate titanate via in-situ spontaneous temperature measurement," Applied Physics Letters, 102, 242912. DOI: 10.1063/1.4811702 [28] Chen, H.S., Wang, H.L., Pei, Y.M., Wei, Y.J, Liu B., and Fang D.N., 2015, "Crack instability of ferroelectric solids under alternative electric loading," Journal of the Mechanics and Physics of Solids, 81, pp. 75–90. DOI: 10.1016/j.jmps.2015.04.014 [29] Wojnar, C.S., and Kochmann, D.M., 2017, "Linking Internal Dissipation Mechanisms to the Effective Complex Viscoelastic Moduli of Ferroelectrics," ASME Journal of Applied Mechanics, 84(2), 021006. DOI: 10.1115/1.4035033 [30] Zhang, H., Yu, Z., Pei, Y., and Fang, D., 2017, "Butterfly Change in Electric Field-Dependent Young's Modulus: Bulge Test and Phase Field Model," ASME Journal of Applied Mechanics, 84(5), 051009. DOI: 10.1115/1.4036298 39
[31] Ziolkowski, A., 2007, "Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains," Continuum Mechanics and Thermodynamics, 19, pp. 379–398. DOI: 10.1007/s00161-007-0060-y [32] Thamburaja, P., 2010, "A finite-deformation-based phenomenological theory for shape-memory alloys," International Journal of Plasticity, 26(8), pp. 1195–1219. DOI: 10.1016/j.ijplas.2009.12.004 [33] Arghavani, J., Auricchio, F., and Naghdabadi, R., 2011, "A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys," International Journal of Plasticity, 27(6), pp. 940– 961. DOI: 10.1016/j.ijplas.2010.10.006 [34] Kelly, A., Stebner, A.P., and Bhattacharya, K., 2014, "A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation," Journal of the Mechanics and Physics of Solids, 97, pp. 197–224. DOI: 10.1016/j.jmps.2016.02.007 [35] Moon, S., Rao, I.J., and Chester, S.A., 2016, "Triple shape memory polymers: constitutive modeling and numerical simulation," ASME Journal of Applied Mechanics, 83(7), 071008. DOI: 10.1115/1.4033380 [36] LaMaster, D.H., Feigenbaum, H.P., Nelson, I.D., and Ciocanel, C, 2014, "A Full Two-Dimensional Thermodynamic-Based Model for Magnetic Shape Memory Alloys," ASME Journal of Applied Mechanics, 81(6), 061003. DOI: 10.1115/1.4026483 [37] Tang, H., Barthelat, F., and Espinosa, H.D., 2007, "An elasto-viscoplastic interface model for investigating the constitutive behavior of nacre," Journal of the Mechanics and Physics of Solids, 55(7), pp. 1410–1438. DOI: 10.1016/j.jmps.2006.12.009 [38] Chen, K., Kang, G., Yu, C., Lu, F., and Jiang, H., 2016, "Time-Dependent Uniaxial Ratchetting of Ultrahigh Molecular Weight Polyethylene Polymer: Viscoelastic–Viscoplastic Constitutive Model," ASME Journal of Applied Mechanics, 83(10), 101003. DOI: 10.1115/1.4034120 40
[39] Yu, C., Kang, G., Lu, F., Zhu, Y., and Chen, K., 2016, "Viscoelastic–viscoplastic cyclic deformation of polycarbonate polymer: experiment and constitutive model," ASME Journal of Applied Mechanics, 83(4), 041002. DOI: 10.1115/1.4032374 [40] Liu, Y., Zhang, H., and Zheng, Y., 2016, "A Micromechanically Based Constitutive Model for the Inelastic and Swelling Behaviors in Double Network Hydrogels," ASME Journal of Applied Mechanics, 83(2), 021008. DOI: 10.1115/1.4031897 [41] Wang, H., and Shen, S., 2016, "A chemomechanical model for stress evolution and distribution in the viscoplastic oxide scale during oxidation," ASME Journal of Applied Mechanics, 83(5), 051008. DOI: 10.1115/1.4032796 [42] Miehe, C., Apel, N., and Lambrecht, M., 2002, "Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials," Computer Methods in Applied Mechanics and Engineering, 191, pp. 5383–5425. DOI: 10.1016/S0045-7825(02)00438-3 [43] Weber, G., and Anand, L., 1990, "Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic–viscoplastic solids," Computer Methods in Applied Mechanics and Engineering, 79, pp. 173–202. DOI: 10.1016/0045-7825(90)90131-5 [44] Simo, J.C., 1992, "Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory," Computer Methods
in
Applied
Mechanics
and
Engineering,
99,
pp.
61–112.
DOI:
10.1016/0045-7825(92)90123-2 [45] Reese, S., and Govindjee, S., 1998, "A theory of finite viscoelasticity and numerical aspects," International Journal of Solids and Structures, 35, pp. 3455–3482. DOI: 10.1016/S0020-7683(97)00217-5 [46] Dettmer, W., and Reese, S., 2004, "On the theoretical and numerical modelling of Armstrong–Frederick kinematic hardening in the finite strain regime," Computer Methods in Applied Mechanics and Engineering, 193, pp. 87–116. DOI: 10.1016/j.cma.2003.09.005 [47] Reese, S., and Christ, D., 2008, "Finite deformation pseudo-elasticity of shape memory 41
alloys– constitutive modelling and finite element implementation," International Journal of Plasticity, 24, pp. 455–482. DOI: 10.1016/j.ijplas.2007.05.005 [48] Vladimirov, I.N., Pietryga, M.P., and Reese, S., 2008, "On the modelling of non-linear kinematic hardening at finite strains with application to springback–comparison of time integration algorithms," International Journal for Numerical Methods in Engineering, 75, pp. 1–28. DOI: 10.1002/nme.2234 [49] Green A.E., and Naghdi, P.M., 1971, "Some remarks on elastic-plastic deformation at finite strain," International Journal of Engineering Science, 9, pp. 1219–1229. DOI: 10.1016/0020-7225(71)90086-3 [50] Bruhns, O.T., 2015, "The multiplicative decomposition of the deformation gradient in plasticity—origin and limitations," H. Altenbach et al., eds, From Creep Damage Mechanics to Homogenization Methods, Springer, pp. 37–66. ISBN: 1869-8433 [51] Hill, R., 1978. "Aspects of invariance in solids mechanics, " C.S. Yih, ed, Advances in Applied Mechanics, Academic Press, New York, USA, 18, pp. 1–75. ISBN: 0-12-002027-0 [52] Green A.E., and Naghdi, P.M., 1965, "A general theory of an elastic-plastic continuum," Archive for Rational Mechanics and Analysis, 18, pp. 251–281. DOI: 10.1007/BF00251666 [53] Simo, J.C., and Ortiz, M., 1985, "A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations," Computer Methods in Applied Mechanics and Engineering, 49, pp. 221–245. DOI: 10.1016/0045-7825(85)90061-1 [54] Gross, A. J., and Ravi-Chandar, K., 2015. "On the extraction of elastic–plastic constitutive properties from three-dimensional deformation measurements," ASME Journal of Applied Mechanics, 82(7), 071013. DOI: 10.1115/1.4030322 [55] Bažant, Z.P., 1971, "A correlation study of formulations of incremental deformation and stability of continuous bodies," ASME Journal of Applied Mechanics, 38(4), pp. 919–928. DOI: 10.1115/1.3408976 [56] Lee, E. H., Mallett, R.L., and Wertheimer, T.B., 1983, "Stress analysis for anisotropic hardening in finite-deformation plasticity," ASME Journal of Applied Mechanics, 50(3), pp. 42
554–560. DOI: 10.1115/1.3167090 [57] Atluri, S.N., 1984, "On constitutive relations at finite strain: hypo-elasticity and elasto-plasticity with isotropic or kinematic hardening," Computer Methods in Applied Mechanics and Engineering, 43, pp. 137–171. DOI: 10.1016/0045-7825(84)90002-1 [58] Xiao, H., Bruhns, O.T., and Meyers, A., 1997, "Logarithmic strain, logarithmic spin and logarithmic rate," Acta Mechanica, 124, pp. 89–105. DOI: 10.1007/BF01213020 [59] Xiao, H., Bruhns, O.T., and Meyers, A., 1997, "Hypoelasticity model based upon the logarithmic stress rate," Journal of Elasticity, 47(1), pp. 51–68. DOI: 10.1023/A: 1007356925912 [60] Xiao, H., Bruhns, O.T., Meyers, A., 1998. On objective corotational rates and their defining spin tensors. International Journal of Solids and Structures 35(30), 4001–4014. DOI: 10.1016/S0020-7683(97)00267-9 [61] Bažant, Z.P., Gattu, M., and Vorel, J., 2012, "Work conjugacy error in commercial finite-element codes: its magnitude and how to compensate for it," Proceedings of the Royal Society A, 468, pp. 3047–3058. DOI: 10.1098/rspa.2012.0167 [62] Ji, W., Waas, A.M., Bazant, Z.P., 2013, "On the importance of work-conjugacy and objective stress rates in finite deformation incremental finite element analysis," ASME Journal of Applied Mechanics, 80(4), 041024. DOI: 10.1115/1.4007828 [63] Vorel, J., Bažant, Z.P., and Gattu, M., 2013, "Elastic soft-core sandwich plates: critical loads and energy errors in commercial codes due to choice of objective stress rate," ASME Journal of Applied Mechanics, 80(4), 041034. DOI: 10.1115/1.4023024 [64] Bažant, Z.P., and Vorel, J., 2014, "Energy-conservation error due to use of green–naghdi objective stress rate in commercial finite-element codes and its compensation," ASME Journal of Applied Mechanics, 81(2), 021008. DOI: 10.1115/1.4024411 [65] Vorel, J., and Bažant, Z.P., 2014, "Review of energy conservation errors in finite element softwares caused by using energy-inconsistent objective stress rates," Advances in Engineering Software, 72, pp. 3–7. DOI: 10.1016/j.advengsoft.2013.06.005 43
[66] Petric, D., Owen, D.R.J., and Honnor, M.E., 1992, "A model for finite strain elasto-plasticity based on logarithmic strains: Computational issues," Computer Methods in Applied Mechanics and Engineering, 94, pp. 35–61. DOI: 10.1016/0045-7825(92)90156-E [67] Bruhns, O.T., Xiao, H., and Meyers, A., 1999, "Self-consistent eulerian rate type elasto-plasticity models based upon the logarithmic stress rate," International Journal of Plasticity, 15, pp. 479–520. DOI: 10.1016/S0749-6419(99)00003-0 [68] Xiao, H., Bruhns, O.T., and Meyers, A., 2000, "A consistent finite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and the deformation gradient,"
International
Journal
of
Plasticity,
16,
pp.
143–177.
DOI:
10.1016/S0749-6419(99)00045-5 [69] Xiao, H., Bruhns, O.T., and Meyers, A., 2001, "Large strain responses of elastic-perfect plasticity and kinematic hardening plasticity with the logarithmic rate: swift effect in torsion, " International Journal of Plasticity, 17, pp. 211–235. DOI: 10.1016/S0749-6419(00)00054-1
44
Appendix A. The stress/strain measure A class of strain measures can be constructed as 3
E i N i N i
(A1)
i 1
where E is the strain using an arbitrary strain measure, N i is the base vector of the
Lagrange frame and is the measure function expressed by
1 2n 1 n 0 n 2n ln n0 where
(A2)
n is an arbitrary real number. The stress σ is then defined according to the
work-conjugation, i.e. W σ : E
(A3)
where W is the rate of work per unit volume.
Appendix B. Derivation of the plastic volume deformation of the uniaxial loading example Considering the loading-unloading path from the initial stress free configuration through the current configuration and finally to the unloading stress free configuration, numerical calculations using increment algorithm are conducted to obtain the stress ratio isf of the unloading stress free configuration and then the plastic volume deformation is obtained from its definition
0
1
1sf 2sf 3sf
1 .
B1. RH-ini theory Supposing that in the current time point, the strain E ln and the stress
σ are obtained,
we derive the components of the strain rate E22ln E33ln as a function of the component E11ln . Then as we move forward along the loading–unloading path, the strain and the stress can be updated. The reference configuration is chosen to be the initial configuration first. We consider 45
the Rice–Hill strain rate decomposition using an arbitrary Seth’s stress/strain measure σ n / E n here and by taking n 1 it degenerates to the RH-ini theory of the main text. By
taking n 0 it degenerates to the theory we discuss in Section 5.2. In the uniaxial loading example, the Seth’s stress is related to the Cauchy stress by
11 n 23
11 ln 11 exp 2n 1 E11ln +2exp E22 12 n -1
(B1)
Now we derive the plastic and the elastic strain rate defined by the Rice–Hill theory during elastoplastic loading. The condition that the stress rate during elastoplastic loading equals to that of the elastic unloading leads to
C 2n 1 exp 2n 1 E 2C exp E E 2 exp E E C 2n 1 exp 2n 1 E 2 C exp E E ep
e
E
ln 22 e
ln 11
11
ln 11
11
E11ln
e
ln 22
ep
e
ln 11
ln 22
11
ln 22
11
ln 22
(B2)
ln 11 e
is used in the above derivation. The plastic strain rate E n is a deviatoric
tensor, so that
E n 22
p
1 n E11 2
(B3) p
Rewriting Eq. (B3) using the logarithmic strain and strain rate, we have
E
ln 22 p
1 ln exp 2nE11ln 2nE22 E11ln p 2
Equation (B2), Eq. (B4) and
E C ln 11 e
e
E
ln 22 e
E11ln
e
(B4)
give rise to
ln ln 2Ce exp E22 11 2n 1 exp 2n 1 E11ln 11 exp 2nE11ln 2n 1 E22
ln ln 2Cep exp E22 Cep 11 2n 1 exp 2n 1 E11ln 11 exp 2nE11ln 2n 1 E22 E11ln
(B5)
E
ln 11 p
C
E11ln E11ln
e
ln ln 2Ce exp E22 Ce 11 2n 1 exp 2n 1 E11ln 11 exp 2nE11ln 2n 1 E22 e
1
ln Cep exp 2n 1 E11ln 2 Ce Cep exp E22 E11ln
(B6) 46
1
Therefore we have the desired relationship between the components of the strain rate, which are 1 ln ln ln E22 E33 exp 2nE11ln 2nE22 E11ln E11ln p e 2
(B7)
during elastoplastic loading and ln ln E22 E33 E11ln
(B8)
during elastic loading or unloading.
Integrating Eq. (B7) and Eq. (B8) along the path indicated by the arrows in Fig. 5, we can obtain the logarithmic strain in the unloading stress free configuration. ln ln ln ln ln ln E33 sf E22sf E22 d1 E22 exp E11 dE11
(B9)
The stretch ratio is then calculated by isf =exp Eiilnsf , with no dummy summation on index i.
B2. RH-cur theory When the current configuration is taken to be the reference configuration, the strain rate is the deformation rate d independent of the measure function, and the stress conjugate to
d is the Kirchhoff stress τ Jσ . Noticing that in uniaxial loading, the principal component of the deformation rate and the Kirchhoff stress is equal to the principal component of the logarithmic strain and the logarithmic stress respectively, i.e.
ii iiln , dii Eiiln
i 1,2,3 no summation on i
Therefore the derivation in Appendix B1 applies for the RH-cur theory by taking n 0 .
47
(B10)
B3. Theories using Simo–Ortiz and Moran–Ortiz–Shih strain rate decompositions Supposing that in the current time point, the strain E ln and the plastic deformation gradient F p are obtained, we derive the components of the rate of the plastic deformation rate F11p and F22p F33p as a function of E11ln . F11p is equal to the stretch ratio of the unloading stress free configuration
F11p 1sf exp E11lnsf
(B11)
so
F11p E11ln sf exp E11ln sf
Ce Cep Ce
F11p E11ln
(B12)
during elastoplastic loading and F11p 0 during elastic loading or elastic unloading. The other two principal components of F p are obtained from the zero trace of the plastic strain rates (Eq. (16) and Eq. (18)) as strain rate decomposition refrence configuration F F / 2 F F11p F22p / 2 F11p F11p F22p F22e2 / 2 F11p F11e2 p p F22 F33 p p e2 p e2 F11 F11 F11 / 2 F22 F22 p p e2 p e2 F11 F22 F11 / 2 F11 F22 p p p F11 F22 / 2 F11 p p 11 11
p 22
Simo Ortiz
Initial stress free
Simo Ortiz
Unloading stress free
Simo Ortiz
Current
(B13)
Moran Ortiz Shih Initial stress free Moran Ortiz Shih Unloading stress free Moran Ortiz Shih Current
during elastoplastic loading and F22p F33p 0 during elastic loading or elastic unloading, where F11 1 exp E11ln , F11e F11 / F11p
(B14)
Integrating Eq. (B12) and Eq. (B13) along the path indicated by the arrows in Fig. 5, we can obtain F p in the unloading stress free configuration and the stretch ratios in the unloading stress free configuration are i sf Fiip where i 1,2,3 with no dummy summation on the index. 48
Appendix C Numerical example of the incorrect stress predicted by some elastoplastic theories If we attempt to shape a material element of unit size 111 to a bar or plate whose width is 2 sf , then from the stress–strain relation Eq. (19), Eq. (20) and the plastic volume conservation, we can derive that the applied stress in the current configuration should be
accurate cr 2
CeCep Ce Cep
ln 2sf
(C1)
when 2sf 1 (tension) and
accurate cr 2
CeCep Ce Cep
ln 2sf
(C2)
when 2sf 1 (compression). But the elastoplastic theories and software that are not volume conserved predict a different applied stress, as shown by Fig. C1. (b) Moran–Ortiz–Shih strain rate decomposition
Relative error / accurate
Relative error / accurate
(a) Simo–Ortiz strain rate decomposition 0.4
SO-ini 0.2
SO-sf
0.0 -0.2
SO-cur
-0.4 0.5
1.0
1.5
2.0
2.5
3.0
0.6
MOS-ini
0.4
MOS-sf
0.2
MOS-cur
0.0 -0.2 -0.4 -0.6 0.5
Stretch ratio 2sf
Relative error / accurate
2.0
2.5
3.0
Reference configuration: Initial stress free Current Unloading stress free
1.0 0.4
RH-ini
1.5
Stretch ratio 2sf
(c) Rice–Hill strain rate decomposition
0.2
1.0
0.5
RH-cur 0.0 0.0 -0.2 -0.5
-0.4 0.5
1.0
1.5
2.0
0.5
2.5
1.0
3.0
1.5
2.0
2.5
Stretch ratio 2sf
Fig. C1. The relative error of the predicted stress by some elastoplastic theories. 49
Appendix D Derivation of the revision term in Eq. (28). In this appendix we derive the revision term L to revise an elastoplastic theory which is originally not volume conserved. Adding the revision term to the original tangential stiffness tensor L , we have
σ obj L L : d
(D1)
There are various choices of the objective stress rates σ obj , and here we derive L for some
typical choices.
Jaumann objective stress rate For the Jaumann rate σ Jau σ σ w w σ , we have 3
i 1
ii
3 3 3 3 3 Liikk Liikk d kk Liikl Liikl d kl k 1 i 1 i 1 i 1 k ,l 1,3;k l i 1
(D2)
Noticing that tr σ w 0 . The current rate form of the volume conservation condition requires that for any deformation rate d Eq. (26) is satisfied, so the revision term L is constrained by the following equations 3
L i 1
ii11
3
3
3
3
3
3
3
i 1
i 1
i 1
i 1
i 1
i 1
i 1
Lii11 Lii 22 Lii 22 Lii 33 Lii 33 K , Liikl Liikl 0
(D3)
L is also assumed to have the symmetric properties of an elastic stiffness tensor, namely
Lijkl Ljikl Lijlk Lklij . Then we seek for the most minor revision term L by minimizing its norm defined by L L L
i , j , k ,l 1,3
Lijkl Lijkl , and the components presented in Eq. (29)
are obtained. Work-conjugate objective stress rate A set of work-conjugate objective stree rate is proposed in (Bažant 1971) as σ
m
σ d w σ σ d w tr d σ
1 2 m σ d d σ 2
(D4)
For m 2 , Eq. (D4) reduces to the Truesdell objective stress rate. From Eq. (D1) and Eq. 50
(D4) we have 3
i 1
ii
3 3 3 3 Liikk ii m kk Liikk d kk k 1 i 1 i 1 i 1 3 3 Liikl m kl Liikl d kl k ,l 1,3; k l i 1 i 1
(D5)
The same as in the case of Jaumann rate, for any deformation rate d Eq. (26) must be satisfied, L has the symmetry of an elastic stiffness tensor and is the most minor revision, so we derive that
3 3 5 1 L K L m K Lkkpp m pp iiii kkii ii 9 9 k 1 k 1 3 3 1 1 K Lkkqq m qq kk 9 k 1 3 i 1 3 3 Liijj Ljjii 2 K Lkkii m ii 2 K Lkkjj m jj 9 k 1 k 1 9 3 3 1 1 K Lkkll m ll kk 9 k 1 3 i 1 3 3 L L L L 1 L L m m klii iilk lkii ppkl pplk kl lk iikl 6 p 1 p 1 L L L L 0 jikl ijlk klij ijkl
i 1, 2,3; p i, q i, p q
i, j 1, 2,3; i j , l i, l j
i, k , l 1, 2,3; k l i, j , k , l 1, 2,3; i j , k l
(D6)
51
