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1008 Part E Modeling and Simulation Methods

section, where the diffusion in liquid and glassy phases

are discussed.

17.4.2 Diffusion in Liquid and Glassy Phases

The diffusion mechanism in the liquid phase or the

glassy phase is not clearly understood. So, in this sec-

tion, we try to clarify the mechanism in those phases

by MD simulation for a model system. The qualitative

change in diffusion property during the liquid quench

process found in the simulations is discussed in relation

to conventional theories of glass transitions.

The physics behind the glass transition is a long-

standing problem in materials science. The discov-

ery [17.45, 46] of bulk metallic glasses has given us

a good opportunity to investigate the glass transition

in a nearly ideal situation because of the broad stable

supercooled liquid region in the system. Indeed, experi-

mental observations [17.28, 47–49]ofatomictransport

phenomena such as diffusivity and viscosity in metal-

lic glasses have gradually revealed the dynamical nature

of the glass transition. Especially, recent nuclear mag-

netic resonance (NMR) measurements [17.50]have

shed light on the microscopic transport mechanism in

both supercooled liquid and glassy phases. Moreover,

many tasks using MD simulations also give [17.51–54]

us useful information about the intrinsic nature of the

glass transition from an atomic point of view. In this

context, we shall describe a series of MD simula-

tions while focusing on transport phenomena such as

atomic diffusion in both liquid and glassy states by us-

ing the same model of binary alloys as discussed in

Sect. 17.3.3.

MD calculations for an idealized alloy system

interacting via the 8–4 LJ potential are performed

under periodic boundary conditions. The pressure of

the system is kept at zero by using the Parinello–

Rahman method and the temperature of the system is

controlled by the momentum-scaling method. In or-

der to investigate transport properties in supercooled

liquid and glassy phases, we need to prevent nucle-

ation of the crystalline phase during the observation.

In other words, we should select the potential pa-

rameters such that the system should have a high

glass-forming ability, or equivalently, a stable super-

cooled liquid region. The glass-forming ability of

the system under rapid solidiﬁcation has been in-

vestigated in Sect. 17.3.3 by another series of MD

simulations and it was shown that a large atomic size

difference plays a decisive role. Hence we take the

atomic size ratio as r

= 0.8 and vary the concentra-

tion x

of the element 2. For simplicity, we assume

an equi-chemical interaction, e

= e

= 1. We

also assume the interaction distance between different

elements is r

= (r

)/2. All physical quanti-

ties are expressed in this units in this section as in

Sect. 17.3.3.

The melting temperature T

of the pure system is

around 0.67 in these reduced units. We prepare the bi-

nary mixture with x

=0.2, 0.5, and 0.7atT =0.8and

slowly cool the system down. The cooling rate 1× 10

−4

corresponds to about 10

−11

K/s for a typical metal-

lic system. The liquid-to-glass transition is observed

in every case and the nucleation of crystalline phases

has not been observed in any case. The glass-transition

temperature T

is determined by the discontinuity of

the speciﬁc heat, as discussed in Sect. 17.3.1. The

atomic self-diffusion coefﬁcients D are calculated by

the mean square displacement, as shown in the preced-

ing section.

Figure 17.44 shows the time evolution of the to-

tal diffusion coefﬁcient D in the equimolar (x

=0.5)

system in a cooling procedure. The temperature de-

pendence goes down remarkably beyond some crit-

ical temperature T

and then changes to another

behavior below the glass-transition temperature T

as indicated in the ﬁgure. According to the tem-

perature dependence of the diffusivities, we can

classify the phases during the liquid-to-glass tran-

sition into three stages. That is, ordinary liquid

or weakly supercooled liquid phases (region I),

strongly supercooled liquid phases (region II), and

0.5 1 1.5 2 2.5 3 3.5

–1

–2

–3

–4

–5

Diffusivity

Liquid I Liquid II

Glassy

Fig. 17.44 Three stages of the temperature dependence of

atomic diffusivity during rapid solidiﬁcation of a binary

system with an atomic size ratio of 1 : 0.8 and a concen-

tration of 50%

Part E 17.4

Molecular Dynamics 17.4 Diffusion 1009

glassy phases (region III), as shown schematically in

Fig. 17.44.

The transition from region I to II looks a transi-

tion from viscous ﬂow to restricted diffusion caused by

structural arrest or inﬁnitely slow relaxation processes,

while the transition from II to III is understood as the

so-called glass transition because it coincides with the

determined calorimetrically. In other words, T

cor-

responds to the beginning of the structural arrest or

structural slowing down and T

corresponds to the com-

pletion of the structural arrest or structural freezing.

Note that the latter change of the diffusion rate at T

has

been detected in experimental observations [17.48–50]

for metallic glasses.

On the other hand, the former change has not been

experimentally observed yet. The Arrhenius behavior of

the temperature dependence in region II suggests that

the atomic diffusion needs some activation processes

such as cooperative motion. The activation energies es-

timated from the Arrhenius plots are 6.0inregionIIand

1.6 in region III in the LJ units. These correspond to 3.0

and 0.8 eV for a typical metallic system, respectively,

both of which are consistent with the experimental re-

sults [17.28,49].

We have also plotted the temperature dependence of

the diffusivity in a crystalline phase in Fig. 17.44 by the

closed triangles, which are obtained during a solidiﬁca-

Fig. 17.45 Trajectories of chain-like collective motions in

a glassy phase at T = 0.15. The balls denote the ﬁnal pos-

itions of atoms and the sticks connect them to the initial

positions

tion process of the pure system with x

=0. Compared

to the diffusivity in glassy phases, this indicates that the

atomic mobility would be much higher in glassy phases

than in crystalline phases.

For the systems with other compositions (x

=0.2

and 0.7), we can also recognize three regions I–III in

both cases, although T

as well as T

depends on the so-

lute composition. Therefore, the three-stage feature of

diffusivity found in the simulations seems to be a uni-

versal character in glass transitions.

To investigate the microscopic nature of atomic dif-

fusion processes in the supercooled liquid and glassy

phases, we calculate the atom trajectories, as discussed

in the previous section. We ﬁrst pick the atoms that have

moved more than half of the nearest neighbor distance

in a time period and then draw lines from the initial to

the ﬁnal position of such atoms. Figure 17.45 shows the

trajectories of such atoms in a glassy phase at T =0.15

of the equimolar system in the interval of 10

time

steps. Many chain-like trajectories are observed. Al-

though these chain-like trajectories look very similar to

those found in crystalline phases depicted in Figs. 17.42

and 17.43, the time analysis reveals the following differ-

ent features in these motions.

Firstly, the motion of atoms in the same chain oc-

curs almost simultaneously (within a few picoseconds),

while the sequence of motions found in crystalline

phases takes place in a one after another way and ranges

over a longer period. This indicates that collective mo-

tion should exist in the glassy phases. Another feature is

that this collective motion is not exactly simultaneous,

but seems to start from the tail and end with the head of

the chain, as in diffusion via interstitials.

Similar analyses have been done for the liquid

phases (regions I and II). We also found the chain-like

cooperative motion in the supercooled liquid phases.

However, the atomic motion in the liquid phases is com-

plicated and other types of collective motion seem to

exist, so more extended analysis is needed; this is one

direction for future work.

We have found that three independent regimes ex-

ist during the liquid-to-glass transition, which show the

different diffusion characteristics: ordinary (or weakly

undercooled) liquid phases (region I), strongly super-

cooled liquid phase (region II), and glassy phase

(region III). We believe that this will be a universal fea-

ture in glass-transition phenomena. For bulk metallic

glasses, the transition from II to III has been ob-

served [17.28], while that from I to II has not yet.

However, for ionic glasses, a sign of transition from

I to II has been observed in the relaxation property by

Part E 17.4

1010 Part E Modeling and Simulation Methods

neutron scattering experiments [17.55]. The landscape

theory [17.56] as well as the free-volume theory [17.57]

and the mode-coupling theory [17.58] might provide

a key to understanding the nature of this three-stage

behavior. From this viewpoint, this peculiar behavior

in atomic transport might have its microscopic origin

in atomic cooperative motions or the redistribution of

atomic free-volumes. Since MD simulation is a con-

venient method to clarify the microscopic nature of

atomic transport phenomena, it is well worth trying to

continue MD study for the glass transition along this

line.

17.5 Summary

Due to the increasing power of modern computers,

computer simulations will become the third class of

investigation method along side the experimental and

theoretical method. Among these simulation methods,

MD simulation is a powerful tool to investigate the mi-

croscopic mechanism in various types of phenomena,

since all information about the atoms in the simula-

tion system can be available. A main drawback of this

method is the limit in both length scale and time scale

due to a lack of sufﬁcient computational power. Even

under such limitations, we can apply this method to the

investigations of various types of phenomena in mater-

ials, as discussed in this chapter

•

martensitic transformation in bulk and nano-clus-

ters,

•

solid-state amorphization,

•

glass-formation by rapid solidiﬁcation,

•

structural relaxation and crystallization of amorph-

ous alloys,

•

atomic diffusion in crystalline phases, glassy

phases, and liquid phases.

Of course, there will be an inﬁnite number of other

applications for MD simulation in materials science.

Therefore, it is very promising to cultivate this method

in this ﬁeld.

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Part E 17

1012 Part E Modeling and Simulation Methods

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Part E 17

1013

Continuum Co

18. Continuum Constitutive Modeling

Constitutive models play an important role when

characterizing structural materials in order to

evaluate their thermomechanical behavior. The

experimental characterization of materials (using

techniques discussed in Part C of this Handbook)

involves measuring and controlling macroscopic

variables such as force, displacement and temper-

ature. Concise models are also of great use when

characterizing the continuous media used to create

structural materials, because phenomenological

modeling can be carried out regardless of the in-

ternal material structure. This continuum modeling

usually successfully describes the behavior of var-

ious classes of material under complex boundary

conditions.

This chapter presents phenomenological con-

stitutive models from both macroscopic and

microscopic viewpoints:

•

Starting from viscoplasticity models, model

performance is reviewed in order to predict the

mechanical response under creep–plasticity

interaction conditions, taking into account

internal state variables.

•

Material anisotropy is discussed; mathemati-

cal modeling of initial anisotropy and induced

anisotropy based on the representation the-

orem for higher order isotropic tensors is

presented.

•

Thermomechanical coupling phenomena

involving phase transformations predominate

18.1 Phenomenological Viscoplasticity...........1013

18.1.1 General Models of Viscoplasticity....1013

18.1.2 Inelasticity Models .......................1015

18.1.3 Model Performance ......................1016

18.2 Material Anisotropy ..............................1018

18.2.1 Description of Material Anisotropy..1018

18.2.2 Initial Anisotropy .........................1019

18.2.3 Induced Anisotropy ......................1021

18.3 Metallothermomechanical Coupling .......1023

18.3.1 Phase Changes.............................1023

18.3.2 Numerical Methodology ................1025

18.3.3 Applications to Heat Treatment

and Metal Forming .......................1025

18.4 Crystal Plasticity ...................................1026

18.4.1 Single-Crystal Model.....................1026

18.4.2 Grain Boundary Sliding .................1029

18.4.3 Inhomogeneous Deformation ........1029

References ..................................................1030

in engineering applications of heat treat-

ment and material processing. A continuum

model is presented that takes into account

the way structural rearrangement evolves in

materials.

•

Finally, microscopic analysis based on crystal

plasticity, which relates the resolved shear

stress to crystal slip, is applied to describe

the inhomogeneous deformation process in

polycrystalline materials.

In the context of Part E of this Handbook, which deals

with various methods of modeling and simulation, the

subject of this chapter – continuum modeling – sits be-

tween the molecular modeling methods explored in the

previous chapter and the ﬁnite element modeling meth-

ods discussed in the next chapter.

18.1 Phenomenological Viscoplasticity

18.1.1 General Models of Viscoplasticity

Materials exhibit a wide variety of phenomena in their

thermal and mechanical behavior. If we consider struc-

tural materials speciﬁcally, they are usually composed

of large numbers of crystals and they show clear yield

points that occur when the stress exceeds a certain limit.

The mechanical behavior after yield is termed as the

Part E 18

1014 Part E Modeling and Simulation Methods

Limit surface

Static yield surface

Viscos (excess) stress

Static yield stress

Back stress

Origin

point

Applied stress

Inelastic strain rate

α'

Fig. 18.1 Schematic illustration of uniﬁed inelasticity model based

on excess stress hypothesis

plasticity regime and is believed to be independent of

time. However, the materials even exhibit permanent de-

formation below the yieldpoint, particularly at elevated

temperature. This deformation is called creep and it is

believed that different mechanisms apply to this low-

stress creep compared to those that occur during metal

plasticity. On the other hand, when the stress is high, ma-

terials exhibit complicated behavior arising from a mix-

ture of the two components (plasticity and creep).

The equations used to investigate the thermome-

chanical behavior of materials are derived from (a) con-

servation laws for mass, linear momentum, angular mo-

mentum and energy, (b) kinematic relationships relating

displacement to strain, and (c) constitutive equations.

Among these, the constitutive equations reﬂect the spe-

ciﬁc response of a material from mechanical and thermal

viewpoints, while the other equations are independent

of the material under consideration and so they all are

common to any kind of boundary value problems. The

linear elasticity model is a typical example of a constitu-

tive equation, where the Young’s modulus and Poisson’s

ratio (or in another form, Lame’s constants) specify the

mechanical response of the material. The Fourier law

of heat conduction is another one, where the heat ﬂux

is proportional to the gradient of temperature. The co-

efﬁcient in the Fourier law simply corresponds to the

elasticity constant. For convenience we limit our discus-

sion to isothermal processes in this section; nonisother-

mal problems are discussed in Sect. 18.3 (including heat

treatment processes).

Viscoplasticity models have been proposed in or-

der to deal with rate-dependent behavior. Among these,

the so-called superposition model is the model that sim-

ply combines the conventional plastic strain ε

with the

creep strain ε

=ε

+ε

. (18.1)

Here we tacitly assume the independence of the elas-

tic strain ε

. In contrast, the uniﬁed constitutive model

gives the rate-dependent viscoplastic (or inelastic) strain



= ε

+ε



regardless of the plasticity and creep.

The latter formulation is sometimes advantageous be-

cause it can be hard to distinguish between plasticity

and creep deformation. We employ the Cartesian coordi-

nate system throughout this section and denote the strain

tensor ε by ε

; the index notation merely indicates the

components of the tensors within a particular Cartesian

coordinate system. We also write equations in direct no-

tation where it does not cause confusion.

The phenomenological modeling and the thermo-

dynamic considerations of uniﬁed constitutive mod-

els were ﬁrst discussed by Malvern [18.1]andPer-

zyna [18.2] in the early 1960s, and more sophisticated

models have since been proposed that include internal

state variables, which allow us to predict the creep-

plasticity interaction and cyclic hardening/softening be-

havior. A modern treatment of uniﬁed constitutive mod-

els can be found in monographs by Miller [18.3]and

Krausz [18.4], and these models are applied to ﬁnite el-

ement codes in order to analyze the structural responses

of real machine components.

When modeling a uniﬁed inelastic constitutive equa-

tion, the following assumptions are widely employed:

the stress–strain curve is static independent of the time

scale used, the stress is explicitly dependent on the evo-

lution of the viscoplastic strain rate, and the normality

rule applies to the ﬂow potential F. A schematic illustra-

tion is shown in Fig. 18.1, where kinematic back stress

is introduced at the center of the yield surface. The vis-

coplastic strain rate is expressed as

=Φ(σ, α)

∂F

∂σ

, (18.2)

where the function Φ describes the magnitude of the

strain rate, while the second term describes the direction

of the strain rate. The function Φ is chosen such that the

steady creep rate can be expressed in terms of the stress σ

and the internal state variables α (scalar or tensor). The

internal state variables (or simply the internal variables)

are introduced from a phenomenological point of view

in such a way that the model reﬂects hardening and/or

softening behavior depending on the prior deformation

history of the material. They may correspond to a phys-

ical property such as back stress or drag stress in some

cases, while the dislocation density can also chosen as

an internal variable in other cases.

Based on the previous discussion and the illustra-

tion in Fig. 18.1, a typical evolution law for viscoplastic

Part E 18.1

Continuum Constitutive Modeling 18.1 Phenomenological Viscoplasticity 1015

strain can be derived as follows [18.5]. Many materials

obey a power law for creep and kinematic hardening

can be idealized using the back stress α. Introducing the

static yield stress r as the threshold for the onset of yield,

we can write the viscoplastic strain rate as



F (σ −α) −r



∂F

∂σ

, (18.3)

where F stands for the effective stress given by

F =







−α







−α







1/2

. (18.4)

Here the prime (



) denotes the deviatoric part. If we ne-

glect the back stress and yield stress, (18.3) is identical

to the well-known creep equation obtained by following

the Norton rule. In contrast, the solution of (18.3) for uni-

axial stress leads to

σ =r +D





1/n

+α. (18.5)

When a large value is employed for the power factor n,

the second term of the right-hand side becomes negligi-

ble and so we obtain an almost rate-independent stress–

strain curve. Consequently, this constitutive model is ex-

pected to cover a wide range of stress–strain responses,

from steady creep to rate-independent plastic behavior.

Internal state variables play a key role when model-

ing the uniﬁed inelasticity model, through which com-

plicated loading histories can be predicted. The internal

state variables reﬂect macroscopic and/or microscopic

changes in internal structures [18.6]. The back stress,

for example, can be measured in such a way that the

Bauschinger effect is produced by the variation, while

the back stress cannot be controlled during the deforma-

tion process. Therefore, the evolution laws may involve

the external variables χ, their rates, and the internal state

variables a themselves

α = A(χ, α) :

χ + B(χ, α) . (18.6)

Any evolution equation in a constitutive model follows

the above principle. Identifying speciﬁc parameters is,

however, another issue. Material parameters are usually

speciﬁed using stress–strain diagrams, by trial and error.

18.1.2 Inelasticity Models

Some of typical inelastic constitutive models are sum-

marized here. We assume that the total strain rate is

decomposed into an elastic part and an inelastic part.

The elastic part is expressed by Hooke’s law, and

the temperature and other effects are neglected for

simplicity.

(a) The Conventional Superposition Model

This type of constitutive model has been widely used

and is still often used in real structural analysis due to its

simplicity. The model employs the same basic assumed

decomposition of strain as (18.1), and the plasticity and

creep models are introduced independently. For exam-

ple, an isotropic plasticity model can be written in terms

of the von Mises yield criterion as

F =



−α







−α





−

=0 , (18.7)

where the evolution of the back stress α follows Prager’s

law,



. (18.8)

Here a, b and c are material parameters. For the creep

model, Norton’s law is introduced with the form

MAσ

N−1

M−1



, (18.9)

where A, N,andM are material parameters and t is the

time. Other types of plasticity and creep laws are also

available using this model.

(b) The Chaboche Model [18.5]

Chaboche proposed a uniﬁed constitutive model with

the type of power law used in (18.3) through (18.5).

The key point of the model is the decomposition of back

stress. In the original model, three kinds of back stress

(namely α

(1)

, α

(2)

and α

(3)

) are introduced via

α =



k=1

(k)

. (18.10)

It is also assumed that the static yield stress r is de-

composed into a constant part r

and a development R.

Taking the nonlinear hardening laws into account, the

evolutions are given by

(1)



−



(1)



−h

m−1

(1)

(18.11a)



(2)



−



(2)



, (18.11b)



(3)

, (18.11c)

R =b(Q − R)

−γ R

. (18.12)

Part E 18.1

1016 Part E Modeling and Simulation Methods

Here

is the equivalent viscoplastic strain rate and

α the effective back stress. There are many material

parameters here and they need to be identiﬁed from

conventional test data.

Miller applied Garofalo’s macroscopic creep equation

to a uniﬁed constitutive model. The function Φ in (18.2)

is expressed as

Φ = Bθ



sinh





F(σ −α)

√



1.5



. (18.13)

Here B, θ, n and D

are constants. The temperature

dependence is expected to depend on the parameter θ

the most. Since the function sinh(.) shows a steep

change with variations in F, the model describes quasi-

rate-independent yielding without a static yield stress

component. The evolution equations for back stress α

and drag stress D should suitably be deﬁned.

(d) The Krempl Model [18.8]

This model can be regarded as an extension of vis-

coelasticity based on a generalized form of the Maxwell

and Voigt models. Using nonlinear functions for the co-

efﬁcients of viscoelasticity, a simple form can be written



σ −g[ε]



ε +g[ε]=σ +k



σ −g[ε]



σ. (18.14)

The function g[ε] describes a static stress–strain dia-

gram. Since the following relationship holds

E =

m[σ −g[ε]]

k[σ −g[ε]]

(18.15)

around the stress origin (E stands for the elasticity con-

stant), the total strain rate is expressed as

ε =

σ −g[ε]



σ −g[ε]



. (18.16)

If we regard the second term as the viscoplastic

strain rate, choosing g[ε] and k[σ −g[ε]] appropri-

ately enables us to describe real nonlinear stress–strain

responses. The substantial part contributing to the vis-

coplastic strain rate σ −g[ε] is called the over-stress.

(e) The Bodner Model [18.9]

This model was developed in order to cover a wide range

of deformations at high strain rate. The ﬂow rule of the

viscoplastic strain rate is subject to the isotropic Levy–

Mises condition. The second invariant of the viscoplastic

strain rate D

, which is the evolution, is deﬁned by

= D

exp



−







n +1







. (18.17)

Here D

and n are material constants and J

is the second

invariant of the deviatoric part of the stress; Z is the inter-

nal state variable related to the visoplastic work W

via

Z = Z

(

−Z

)

exp



−mW



(18.18a)

m = m

exp



−αW



(18.18b)

, Z

, m

and α are material parameters. Es-

trin [18.10] reﬁned this model, taking into account met-

allurgical considerations, and applied it to various defor-

mation histories involving plasticity and creep.

(f) The Endochronic Model [18.11]

Valanis assumes the existence of an intrinsic time mea-

sure ζ particular to each material using

dζ

=α

dξ

+β

. (18.19)

Here ξ is a parameter related to the strain accumula-

tion, while t means the real time scale. This means that

ξ reﬂects a combined history of deformation and time.

He further assumes an intrinsic time scale z(ζ), which

increases monotonically with ζ.Letz be, for example,

expressed as

z =

log

(1 +βζ) . (18.20)

Using proper functions for μ(ζ )andK(ζ), the stress

deviator and the mean stress are integrated over the

intrinsic time as

s =2



μ(z −z



)

∂ε

∂z



, (18.21a)



K(z −z



)

∂e

∂z



. (18.21b)

It should be noted that these material functions im-

ply tangent moduli not under the real time scale, but

under the intrinsic time scale.

18.1.3 Model Performance

As described in the previous section, a large number

of inelasticity models have been proposed [18.12–15].

One is based on a macroscopic creep law, another one

Part E 18.1

Continuum Constitutive Modeling 18.1 Phenomenological Viscoplasticity 1017

300

Number of cycles N

Superposition (A), (C), (D)

Modified superposition (A)

Mróz

Chaboche (A), (B), (C)

Miller

Krempl

Fraction

Murakami–Ohno

NLMOS

Ohno

Experimental

Ratcheting strain ε (%)

10 20

10 c

10 s 250 MPa

275 MPa

10 c 10 c

= 50 MPa/s, 5 MPa/s, 50 MPa/s

Fig. 18.2 Accumulation of ratcheting strain under various

stress levels

reﬂects the physical deformation mechanism. In order

to clarify the performance of each model, benchmark

tests [18.16–18] have been carried out by the Subcom-

mittee on Inelastic Analysis and Life Prediction under

the support of the Committee on High Temperature

Strength of Materials, The Society of Materials Science,

Japan. A few examples are demonstrated here. The test

material employed was 2.1/4Cr–1Mo ferrite steel, and

the experiments were performed at an elevated temper-

ature of 600

◦

Metallic materials reveal a dependence of the me-

chanical behavior on rate at elevated temperatures, due

to creep deformation, and the test material showed re-

markable rate dependence at this temperature. Each

model was used to predict the strain accumulated when

the tensile stress is loaded at different rates and various

holding times are used. Figure 18.2 shows the accu-

mulation of strain for various stress rates and stress

levels [18.17]. Since the strain always increases and ac-

cumulates with the number of cycles, this accumulation

is called ratcheting. As shown in the ﬁgure, at least ten

models are compared. It should be noted that different

results are predicted, even by the same model, when

the set of material parameters used is changed. This

means that the speciﬁcation of material parameters is

not unique for a model and that even small differences

in basic test data for monotonic tension, cyclic loading

and monotonic creep seriously inﬂuence the ratcheting

behavior. It has been reported that the responses under

Experiment

Superposition model

Mróz model

–0.4

–300

σ (MPa)

0.4

–0.4

0.4

–0.4

0.4

–0.4 0.4

Δε = 0.8%

–0.4

N = 10 cycles

300

0.4

ε (%)

–0.4

–300

(MPa)

300

0.4

Chaboche model (B)

Endochronic model

Ohno–Murakami (A)

τ/√3 (%)

√3τ/

√3τ

√3τ/

Fig. 18.3 Stress–strain responses under an out-of-phase strain path with square form

Part E 18.1