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Directional Distortional Hardening in Plasticity

within Thermodynamics

Yannis F. Dafalias

and Heidi P. Feigenbaum

Department of Mechanics, Faculty of Applied Mathematical and Physical Science,

National Technical University of Athens, Zographou, Hellas, and Department of Civil and

Environmental Engineering, University of California, Davis CA, USA

yfdafalias@central.ntua.gr

Department of Mechanical Engineering, Northern Arizona University, Flagstaff AZ, USA

Heidi.Feigenbaum@nau.edu

Abstract. This paper presents a complete theory for metal plasticity that includes

directional distortional hardening, supplemented by the classical kinematic and

isotropic hardenings. Starting from an isotropic yield surface, the distortional hard-

ening will be modeled either by fourth-order tensor-valued internal variable mul-

tiplied by a scalar, a scalar-valued internal variable in conjunction with the back

stress, or a second-order tensor-valued internal variable. These models are unique

because the rate equations for all internal variables, including the fourth order ten-

sor, are derived strictly on the basis of sufﬁcient conditions for the satisfaction of the

second law of thermodynamics for positive dissipation, in conjunction with a few

simple and plausible assumptions about free energy storage and release in the mate-

rial. The models are shown to ﬁt experimentally found yield surfaces rather well, in

particular the model with the fourth-order tensor. Furthermore, this model is shown

to simulate stress controlled biaxial ratchetting better than the same model without

distortion of the yield surface.

1 Introduction

In the ﬁeld of continuum mechanics it has long been recognized that plastic defor-

mations may induce anisotropy in materials that are initially isotropic. For metals

such anisotropy is due to the development of internal stresses and texture formation

because of preferred orientations of grains in a polycrystal, among other reasons.

The macroscopic manifestation of metal anisotropy takes the form of translation

and shape distortion of the yield surface in stress space, modeled by kinematic and

distortional hardening, respectively. Isotropic hardening or softening is an additional

feature that is modeled by a uniform increase or decrease of the size of the yield sur-

face and does not reﬂect anisotropic development but rather it is related to change

of the density of dislocations.

In particular, “directional distortion” is a distortion of the shape of the yield

surface such that a region of high curvature (sharpening) develops roughly in the

62 Y.F. Dafalias and H.P. Feigenbaum

direction of loading while a region of lower curvature (ﬂattening) develops in the

opposite direction. This observation has been seen in numerous experiments on vari-

ous types of metals including, but not limited to, those by Phillips et al. [14], Naghdi

et al. [12], McComb [11], Wu and Yeh [18], and Boucher et al. [3]. In general, di-

rectional distortion of the yield surface is observed when a relatively sensitive def-

inition of yield is used. For example, Phillips et al. [14], Wu and Yeh [18], and

Boucher et al. [3] deﬁne yield by offset strains of 3×10

−6

,5×10

−6

,and4×10

−5

correspondingly.

Fig. 1 illustrates the directional distortion by showing a sample yield surface in a

two-dimensional stress space of a normal and a shear stress component, often used

in plotting experimental data. The factor

√

3 multiplying the shear stress compo-

nent aims at representing the Mises-type yield surface as a circle, had it not been

distortion, and such circular yield surface is also shown by dashed line for com-

parison with the distorted one. Also, anticipating the associative ﬂow rule, the unit

deviatoric tensor n, normal to the yield surface, is drawn at a current stress point

symbolized by s. Notice that s is not the tension stress that induced the yield surface

conﬁgurations of Fig. 1, but it was chosen for illustration purposes at an arbitrary

location on the distorted yield surface. In general the “directions” of the back-stress

α, unit radial expressed by the n

, and outward unit normal n, are different tensors,

as shown in the ﬁgure, where the direction of α

α is along the tension axis for this

ﬁgure. In particular, the unit tensors n

and n are different because of the distortion,

since without it the yield surface remains a circle and the direction of the radius

along n

coincides with that of the normal along n.

The modeling of directional distortion has been addressed at various degrees

of success by Ortiz and Popov [13], Voyiadjis and Foroozesh [16], Kurtyka and

Zyczkowski [10], Yeh et al. [17], and Franc¸ois [7], among others. Most recently

Feigenbaum an Dafalias [5, 6] developed models for directional distortional hard-

ening. The Feigenbaum and Dafalias 2007 uses a fourth order tensor multiplied by

a scalar to capture directional distortional hardening. In Feigenbaum and Dafalias

2008 two simpler models are proposed which capture directional distortional hard-

ening. The ﬁrst uses a scalar-valued internal variable in conjunction with the back-

stress to model the directional distortion, and the second model uses a new second

order tensor to capture directional distortion, and thus distortion and kinematic hard-

ening are completely decoupled. In all the Feigenbaum and Dafalias models [5, 6]

the rate equations for all internal variables, including the fourth order tensor, are

derived strictly on the basis of sufﬁcient conditions for the satisfaction of the second

law of thermodynamics for positive dissipation, in conjunction with a few simple

plausible assumptions of free energy storage and release in the material. One aspect

of any distortional hardening model, including directional distortion, is the preser-

vation of convexity of the yield surface during its change of shape process, an issue

of great importance in plasticity theory for reasons of uniqueness and stability. In

Plesek et al. [15] conditions to ensure convexity of the Feigenbaum and Dafalias

models [5, 6] were derived.

The current paper examines the models by Feigenbaum and Dafalias [5, 6] in-

cluding the mathematical formulation of the models and model performance. The

Directional Distortional Hardening in Plasticity within Thermodynamics 63

Fig. 1. An example of directional distortion of the yield surface for loading in pure ten-

sion. Note the two subsequent yield surfaces have the same ﬁnal stress point in tension, but

the center of the directionally distorted yield surface is moved less than that with kinematic

hardening.

performance of the models will be evaluated against experimentally found yield sur-

faces as well as experimental results from cyclic loading in the plastic range, a.k.a.

ratchetting. Modeling ratchetting behavior is extremely difﬁcult because any small

error in plastic strain during a single cycle will add to become a large error after

many cycles. As is typical with metals, most constitutive models used to ﬁt ratchet-

ing use the associative ﬂow rule which states that the plastic strain increment is in

the direction normal to the yield surface. When the associative ﬂow rule is used, it is

important to have the shape of the yield surface modeled accurately because small

deviations in shape may result in large deviations in the normal to the yield surface

and thus the plastic strain increment. During cyclic plastic loading these deviations

will accumulate and may result in large errors for the predicted strains.

In terms of notation, all second order tensors will be denoted, henceforth, by bold

face in direct notation, e.g. m

m, and all fourth order tensors will be capitalized in bold

calligraphy, e.g. M

M. No bold face symbols will be used when indexed components of

tensors are used. The proposed constitutive model is conﬁned to small deformations.

The stress tensor is denoted by σ

σ and the linearized strain tensor is denoted by ε

ε.

As usual, the strain tensor is decomposed into elastic and plastic parts (ε

ε = ε

+ε

)

and the elastic constitutive law will be assumed linear and isotropic.

64 Y.F. Dafalias and H.P. Feigenbaum

2 Mathematical Formulation of Models

The Feigenbaum and Dafalias 2007 model uses the following yield function:

f =(s − α

α):[I

I +(n

: α

α)A

A](s− α

α)−k

= 0. (1)

where s is the deviatoric component of the stress tensor, α

α is the deviatoric

back-stress tensor that represents the “center” of the yield surface, k represents

the size of the yield surface, A

A is the fourth-order evolving anisotropic ten-

sor, I

I is the fourth order isotropic unit tensor given by the expression I

ijkl





+ δ



−



, with δ

the Kronecker delta, and n

is the ra-

dial, in regards to the center of the yield surface, deviatoric unit tensor illustrated in

Fig. 1 and deﬁned by

s− α

(2)

with the symbol | ·| representing the norm of a tensor deﬁned as |m| =

√

m : m,and

where : denote the trace operation of the product of two adjacent tensors. The fourth

order anisotropic tensor, A

A, is responsible for distortional hardening in general, and

the trace-type scalar multiplier, n

:α

α, is responsible for the directionality of distor-

tion. This can be intuitively understood if one observes that the trace n

: α

α is an

inner product between the tensors n

and α

α, thus, it actually takes the “projection”

of α

α along the different “unit” directions n

. This product can vary from |α

α| to −|α

α|

passing through zero, and such variation effects the role of the distortional tensor A

which is multiplied by it. Since the fourth-order tensor A

A is the corner stone of this

model, it will henceforth be referred to as the “A

A-model.”

In Feigenbaum and Dafalias 2008 two yield functions are proposed, the ﬁrst of

which can been understood as a special case of (1) where A

A is constantly aligned

with the fourth order unit isotropic tensor I

I by means of a possibly varying propor-

tionality factor, which gives rise to the following simpler expression for the yield

function:

f =

[

1 − c(n

: α

α)

]

(s− α

α):(s− α

α)−k

= 0(3)

where c is a scalar-valued internal variable (c can be also a constant) whose magni-

tude is directly associated with the amount of distortion of the yield surface. Since

the back stress α

α is the entity which dictates the directional distortion, this model

will be referred to as the “α

α-model.” One criticism one might raise for the models in

Eqs. (1) and (3) is that distortional hardening is coupled with kinematic hardening

due to the role the back stress plays in the distortion scheme, while the underlying

physics would imply an uncoupled consideration. To alleviate this problem a second

yield function was proposed in Feigenbaum and Dafalias 2008,

f =

[

1 −(n

: r)

]

(s− α

α):(s− α

α)−k

= 0(4)

where r is a second order deviatoric tensor-valued directional distortional hardening

internal variable. By introducing r in Eq. (4) the scalar quantity n

: r is completely

Directional Distortional Hardening in Plasticity within Thermodynamics 65

responsible for directional distortion, therefore, kinematic hardening has been de-

coupled from distortional hardening. Since r is the entity which dictates the direc-

tional distortion, this model will be referred to as the “r-model.”

All models in consideration use the associative ﬂow rule, as is typical with metal

plasticity:

= λ

∂f

∂σ

= λ



∂f

∂σ



n (5)

where

is the rate of plastic strain, λ is the loading index (alias plastic multiplier),

which can be determined from the consistency condition and the hardening rules for

the internal variables, and n is the unit tensor (n : n = 1) normal to the yield surface

as shown in Fig. 1.

The hardening rules for these models are obtained from conditions sufﬁcient for

satisfaction of the second law of thermodynamics, in conjunction with a few simple

and plausible assumptions about energy storage and release in the material. Details

of these thermodynamic assumptions and how the hardening rules are derived in a

way which is sufﬁcient to satisfy the second law of thermodynamics can be found in

Feigenbaum and Dafalias [5, 6]. For now only the hardening rules themselves along

with some thermodynamically necessary restrictions will be presented.

All three models give rise to the same form of isotropic and kinematic hardening,

namely:

k = λκ

k(1 − κ

k) (6)

α = λ



∂f

∂σ



(n− a

α) (7)

where κ

, κ

, a

and a

are non-negative material constants whose exact values

depend on which model is used, i.e. the values for these constants will not be the

same in the A

A-model as in the α

α-model or in the r-model, since the form for the

plastic modulus depends on which model is used.

The hardening rules that arise for the distortional parameters for the three models

are as follows:

A =−λA

s− α



(

: α

)

⊗n



(8)

c =

λc

|s− α

α|

[(n

: α

α)−c

c] (9)

r =

λρ

|s− α

α|

− ρ

r) (10)

where A

, A

, c

, ρ

and ρ

are non-negative material constants. These hard-

ening rules are sufﬁcient for the satisfaction of the second law of thermodynamics

provided that

 1,c

 1, ρ

|r|

 1 (11)

for all time, where |A

= A

A :: A

A and |r|

= r : r are Euclidean norms.

66 Y.F. Dafalias and H.P. Feigenbaum

Notice that all hardening rules for all models are of the evanescent memory type,

therefore the internal variables all reach ﬁnite limits. These limits can be found by

setting the rate equations in (6)-(10) equal to zero, which yields:

(12)

(13)

=−

⊗n

(14)

(15)

(16)

where the superscript l denotes the limit values of the quantities involved, and the

fact that n

= n

at the limit, as shown in Feigenbaum and Dafalias [5, 6], has been

evoked. Assuming that the maximum of the inequalities in (11) occurs at the limit,

the thermodynamic requirements in (11) give the following restrictions on material

constants for the A

A-model, α

α-model and r-model, correspondingly:



, a

 1,ρ

 1 (17)

The assumption of associative ﬂow in conjunction with Drucker’s or Il’iushin’s pos-

tulates makes it necessary to explore the issue of convexity for these models. In

Plesek et al. [15] conditions for convexity were proven. For the A

A-, α

α-, and r mod-

els, respectively,

> 0.55 , a

> 1,ρ

> 1 (18)

is required for the yield surface to remain convex. Note that as explained in Plesek

et al. [15], for the A

A-model it is assumed that the most distorted yield surface occurs

at the limit state. Also notice for the α

α-andr-models the requirement for convexity

is the same as the requirement for the satisfaction of thermodynamic requirements.

3 Model Performance

In order to show the effectiveness of the model, its simulations will be compared

to experimentally found yield points. Figs. 2 - 7 show experimental data from Wu

and Yeh [18]. These experiments were performed on thin-walled tubes of annealed

304 stainless steel. The data in Figs. 2 - 4 were obtained using a strain controlled

procedure and loading in torsion (shear). For Figs. 5 - 7 the data was obtained us-

ing stress controlled procedure with loading in combination of tension and torsion.

Directional Distortional Hardening in Plasticity within Thermodynamics 67

Table 1. Material constants used to ﬁt Wu and Yeh [18] experimental data on stainless steel

304 found in Figs. 2 - 7.

A-model α

α-model r-model

E 196687 MPa 196687 MPa 196687 MPa

ν 0.28 0.28 0.28

128 MPa 128 MPa 128 MPa

6000 MPa

0.008 MPa

−1

0.012 MPa

−1

0.012 MPa

−1

40000 MPa 17000 MPa 18000 MPa

0.0065 MPa

−1

0.012 MPa

−1

0.01 MPa

−1

0.5 MPa

−4

20000 MPa

-0.01MPa

−3

- 10001 MPa

--1.9MPa

−1

--1.3

Fig. 5 shows both loading, unloading and negative loading in this path. Wu and Yeh

[18] deﬁned yield as an offset equal to 5 ×10

−6

Figs. 8 - 9 show experimental data from Boucher et al. [3]. These experiments

were performed on thin-walled tubes of aluminum alloy AU4G T4 (2024). The tubes

were loaded using stress control in combination of tension and torsion (shear) and

yield was deﬁned by an offset equal to 4 ×10

−5

. Figs. 8 - 9 shows loading and

unloading in non-proportional path (path is given by O-A-B-C as shown in the ﬁg-

ures). Notice that there is a scale difference in the plotings of Figs. 8 and 9, hence,

the same data appear to be different at ﬁrst impression.

To ﬁt these experiments, the three versions of the model were implemented nu-

merically using the procedure outlined by Bardet and Choucair [1]. This procedure

essentially takes loading increments of stress, strain, or combinations of the two and

converts them into increments of stress only. Using this procedure, the numerical

loading conditions exactly matched those from the experiment, and each increment

of load is converted into stress increments. Once the stress increment is obtained,

incremental changes of strain and internal variables are calculated using the asso-

ciative ﬂow rule and the hardening rules, along with the standard calculations for

loading index and plastic modulus. The numerical procedure solves the rate equa-

tions by an explicit method, and thus small steps were used to ensure convergence

and good accuracy.

Table 1 shows the material parameters used to obtain the model ﬁtting in

Figs. 2 - 7. For Fig. 8 the kinematic and distortional hardening constants are given

by: a

= 2 ×10

MPa, a

= 0.015 MPa

−1

, A

= 20 MPa

−4

,andA

= 2.5 ×10

MPa

. To determine the plastic material parameters, ﬁrst experimental data were

ﬁt using only Armstrong-Frederick kinematic hardening along with isotropic hard-

ening, then the amount of kinematic hardening was reduced (the limit value was

68 Y.F. Dafalias and H.P. Feigenbaum

Fig. 2. A

A-model compared to torsion experiments by Wu and Yeh [18]. The ﬁrst, second,

third, and fourth subsequent yield surfaces resulted from strain controlled loading with ﬁ-

nal values γ = 0.19%,0.38%,0.49% and 1.03%, respectively. Lines represent the proposed

constitutive model and discrete points represent experimental data.

increased and the rate of saturation slowed) to allow for distortion to be added.

Lastly, both the parameters associated with kinematic hardening (a

and a

)aswell

as those associated with distortion (A

and A

,orc

and c

,orρ

and ρ

)were

adjusted, using the constraint in Eqs. (17) and (18) as a guide. Thus the material pa-

rameters are not necessarily optimal for the material, but rather a relatively good ﬁt

for the data shown. It was also observed that with the α

α-model the use of a constant

value of c gave very good results compared to the choice of a variable c (this can

also be detected from the large difference of the numerical values of c

and c

Table 1). The constant value of c still allows for the directional distortional feature

of the α-model and can be viewed as a special case that is even simpler than those

presented in this paper. Fig. 9 shows the ﬁtting of the Boucher et al. [3] data using

the α-model and a constant c=0.019MPa

−1

Clearly, there are some ﬂaws in the way the models ﬁt the data. Since the Wu

and Yeh [18] data in Figs. 2 - 4 were ﬁt using strain control, it was much more

difﬁcult to match the stress points, which can be seen in these ﬁgures. With both

the α

α and r-models, the simulations ﬁt the region of high curvature relatively well,

but they do not show sufﬁcient ﬂattening in the direction opposite, whereas the

experiments do. The same could be said for the A

A-model, however, overall that

model ﬁts the data better than either of the other two. Figs. 8 - 9 show that the