ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY &
ENVIRONMENT

AZOJETE June 2020. Vol. 16(2):423-434
Published by the Faculty of Engineering, University of Maiduguri, Maiduguri, Nigeria.

Print ISSN: 1596-2490, Electronic ISSN: 2545-5818
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ORIGINAL RESEARCH ARTICLE
BASIC CONSTITUTIVE EQUATIONS IN ELASTO-PLASTICITY – A REVIEW

O. H. Aliyu
(Department of Building, Faculty of Environmental Sciences: The Federal University Birnin Kebbi, P.M.B. 1157, Birnin Kebbi, Nigeria)

*Corresponding author’s email address: ohunene.hafsa@fubk.edu.ng

1.0 Introduction
Constitutive equations tend to characterize and establish the macroscopic behaviour of
materials when loaded. They are ideally mathematical equations that substantially approximate
the physical observations of real materials responses over limited range of parameters for
example, temperature and deformations. Real materials, generally behave in a complex manner
under various parameters (of temperature and deformation) as such separate equations of
elasticity, viscosity, visco-elasticity, visco-plasticity as well as plasticity are used to describe the
different observed material responses (Marlvern 1969; Stein 1993).

Here we will be looking at the elastoplastic constitutive equations for materials such as metals,
concrete, soils and so on which exhibit elastic-plastic material behaviour. These materials are
classified as either rate-independent or rate-dependent (Belytschko et al., 2014) with their
constitutive equations grouped into infinitesimal and finite elastoplasticity. The finite
elastoplasticity constitutive equations are further classified as either hyperelastic-base plasticity
or hypoelastic-base plasticity; this has been widely studied by Hashiguchi, (2017) and Hashiguchi,
(2019). These elasto-plastic models are however based on a combination of the following
plasticity theories namely: decomposition of the strain increments into elastic and plastic parts;
yield functions which control the inception and continuation of plastic straining; flow principles
of plasticity which governs plastic flow; as well as evolution equations (Grassl and Jirasek, 2006;
Marlvern 1969; Belytschko et al., 2014).

2.0 Constitutive Equations for Rate-independent and Rate-dependent Elastic-
plastic Material Behaviour
When the stress-strain curve of a material is independent of its rate of deformation, such a
material is said to be rate independent; else it is rate dependent. The stress-strain responses of
rate-independent and rate-dependent materials in one dimension are illustrated in Figure 1.

ARTICLE
INFORMATION

ABSTRACT
The present paper reviews the constitutive equations of engineering materials in
the elasto-plastic regime. Here the plastic flow rules, strain rate, yield function,
overstress function as well as evolution equations for the internal parameters are
discussed. Similarly, the implementation of the elasto-viscoplastic constitutive
model in a standard finite element code is also discussed and it has been
concluded that an appropriate choice of material model is key in accurately
simulating the behaviour of an engineering material to applied loading. It is also
recommended that the appropriateness of the selected constitutive response is
crucial for mimicking precise the material behaviour.

© 2020 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved.

Submitted 05 May, 2020
Revised 21 May, 2020
Accepted 24 May, 2020

Keywords:
Elasto-plasticity
finite strain
rate-of-deformation
rate independent
rate dependent

Figure 1: Engineering stress-strain curve a) rate-independent b) rate-dependent adapted from
Belytschko et al. 2014.

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Aliyu: Basic Constitutive Equations in Elasto-Plasticity – A Review. AZOJETE,16(2):423-434. ISSN 1596-2490; e-ISSN 2545-5818,
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In the following sections, the basic frame work for the rate-independent and rate-dependent
elastoplastic constitutive equations are described concisely.

2.1 Rate-independent Plasticity
Rate-independent plasticity constitutive equations are grouped under the infinitesimal
elastoplasticity such as:

A. Here the infinitesimal strains �� in Equation (1) are additively decomposed into elastic
and plastic parts.

��� � ���� � ���� ���

Hence, in terms of strain rates it is expressed as shown in Equation (2):

�� � ��� � ��� �t�

B. The stress rate is related by the elastic modulus to the strain rate as given in Equation
(3):

�� � ���� � ���� � ���� ���

Here the elastic strain (���) is calculated by subtracting the plastic strain from the effective strain
that is: (��� � ��� � ����) while the stress is calculated by substituting the elastic strain into the
hyperelastic equation as given in Equation (4).

�� � ���� �t�
where:
� � is the Elastic (Young’s) modulus (Figure 2).

C. The plastic strain rate is given by the plastic flow rule as expressed in Equation (5):

��� � ���
�Ψ
��

� ��� � ��� � � � � � ��� Ψ �/ σ /� ���

D. The evolution equation for the back stress (� also known as the kinematic hardening
variable) is given as in Equation (6):

�� � t��� ���
E. The yield condition (�) with isotropic and kinematic hardening is expressed as given in

Equation (7) :
� �/� � �/ � ������ � h �t�

F. The loading and unloading condition is expressed as in Equation (8):

�� � h� � � h� ��� � h ���
where:

�� � h � is the condition for plastic loading

�� � h � is the condition for elastic loading
� � is the internal variable called the back stress
��� � is the effective plastic strain rate
�� � is the effective plastic strain
t � is the kinematic hardening variable
��� � is the plastic strain rate

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Ψ� is the plastic flow potential

For plastic flow to occur the yield condition � � h must be met.

G. The tangent modulus (����) during the plastic loading is given as:

���� � �� �
�t

� � �d� t�
�t�

where:

� � is the plastic switch parameter having a value of 1 for plastic loading and zero for purely
elastic loading or unloading.
d� is the plastic modulus.
���� � is the elastic-plastic tangent modulus, (Belytschko et al., 2014; Hashiguchi and Ueno,
2017; Hashiguchi 2019).

Figure 2: Stress-strain curve for a typical elastic–plastic material adapted from Belytschko et al.
(2014)

2.2 Rate-dependent Plasticity
For rate-dependant materials:

A. The plastic response depends on the rate of loading which can be described by
an over stress model given as:

��� �
�������

� � �� ��h�

where:
� � is the overstress
� � is the viscosity
��� is the effective plastic strain
��� � is called the effective plastic strain rate and it is equivalent to the plastic rate

parameter (�� �.

B. The plastic flow rule and the evolution equation is given by:

�� � ������t�� t� � ��� ����

C. The above plastic deformation in rate-dependant materials occur when the yield
condition giving in Equation (12)

� �/� � �/ � ������ � h ��t�

is met or exceeded. (Belytschko et al., 2014; Hashiguchi and Ueno, 2017; Hashiguchi 2019).

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Aliyu: Basic Constitutive Equations in Elasto-Plasticity – A Review. AZOJETE,16(2):423-434. ISSN 1596-2490; e-ISSN 2545-5818,
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D. The evolution equation is given by Belytschko et al. (2014) as:

�� � t��� ���⹉ �σ� � ����

3 Elasto-viscoplastic Constitutive Model
For the above rate dependent model in Equation 10, the plastic deformation (that is the
viscoplastic deformation which is unrecoverable) is described by an overstress model
(Hashiguchi 2014). In this section, the original overstress model called the Bingham model is
modified so that it is applicable for use in loading rates ranging from quasi-static to impact loads.

3.1 Elasto-viscoplastic Constitutive Equations
Generally, constitutive equation adopted in describing rate dependence at a stress level below
yield is called the viscoelastic constitutive equation also known as the Maxwell model. Here,
the strain rate ε� is additively decomposed into the elastic and viscous strain rates.

�� � ��� � ��� ��t�
The above viscoelastic constitutive equation is applicable with rate dependent deformation at
stress below the yield stress (Hashiguchi, 2014).

On the other hand, when the stress level is at yield stress (Figure 2) to describe the plastic
deformation, the Prandtl elasto plastic constitutive models are used where the strain rate is
additively composed of the elastic and plastic parts.

�� � ��� �t� � � ��

��� � ��� �t� � � �� ����

With rate dependent models where the stress is above the yield stress (Figure 3) the model
that describes plastic deformation is called the elasto-viscoplastic model; here the strain rate
ε� is given by:

�� � ��� � ���� �t� � � �� (16)

The above original overstress model in Equation (16); called the Bingham model was first
proposed by Bingham (1922) and subsequently extended by Prager (1961) for Von Mises metal
and later generalised by Perzyna, (1963) for all materials expressed as given in Equation (17):

� � �� � ��� ��t�

(Hashiguchi, 2014; Fincato and Tsutsumi, 2019).
where:

The strain rate � is additively decomposed into the elastic strain rates �� and viscoplastic strain
rates ��� with the viscoplastic strain rate expressed as given in Equation (18):

��� �
�
μ�
f�σ�
F�H�

� �
n

N ����

μ� � Viscoplastic coefficient

d� Is the hardening variable

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Figure 3: Rate dependent deformation of solids adapted from Hashiguchi, (2014)

The deformation ��� as in Equation (17) above is expressed as:

� � ���� �� �
�
μ�
f�σ�
F�H�

� �
n

N ��t�

�� � �� � �
�
μ�

f�σ�
F�H�

� �
n

E�N �th�

(Hashiguchi, 2014; Hashiguchi and Ueno 2017)

Equations 14-20 above give a brief history of the advancement in the rate dependent
viscoplastic constitutive models.

With infinitesimal rates of deformation, where �� � h and � � h� the yield condition f�σ� �
F�H� (Hashiguchi, 2014; Hashiguchi and Ueno, 2017) is satisfied that is:

f�σ�
F�H�

� � � h �t��

Equation (21) gives the elastoplastic response in a quasi-static deformation (Figure 4).

Figure 4: Response past over stress model adapted from Hashiguchi, (2014)
For infinite rate of deformations here �� � t and � � t Equation (20) above becomes

�� � ���� h �tt�
Suggesting the material behaves elastically under extreme loading which is not realistic. Implying
that Equation 19 and 20 as it is, are not applicable for use in low speeds (high period) impact
loading. For use with such loading conditions, Equation (19) is modified as follows:

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Aliyu: Basic Constitutive Equations in Elasto-Plasticity – A Review. AZOJETE,16(2):423-434. ISSN 1596-2490; e-ISSN 2545-5818,
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Corresponding author’s e-mail address: ohunene.hafsa@fubk.edu.ng 428

� � �� � ���� � � ���� �� �
�
μ�
R � � nN �t��

where:

R �
f���
F�H�

�tt�

So that when � � ��; R > 1

Equation (23) is better able to model the exact material behaviour under low speeds (high
period) impact loading by restricting the dynamic ratio as shown in Figure 5 to a limiting value
of ��� � �� (Hashiguchi, 2014; Fincato and Tsutsumi, 2019).

where:

R- is the dynamic loading ratio

��- is the limiting dynamic loading ratio

Figure 5: stress-strain cure predicted by the modified overstress model adapted from
Hashiguchi, (2014)

4 Hypoelastic-based Plasticity
The constitutive models here relate the rates of stress to the rate-of-deformation. They are
appropriate when the elastic strains (ε�e ) are small in comparison to the plastic deformations
�ε�p� (Belytschko et al., 2014; Hashiguchi 2019). Here the use of Cauchy (true) stress mainly
results in constitutive equations with non-symmetric tangent modulus. Using the Kirchhoff
stress on the other hand, leads to constitutive equations with symmetric tangent modulus.

4.1 Hypoelastic-plastic Constitutive Equations (based on the Cauchy stress
formulations)

A. Here the rate-of-deformation tensor ��� is additively decomposed into elastic and
plastic parts.

D � D� � D� �t��

B. The (Jaumann) Cauchy stress rate is related by the constant elastic moduli (��뼘
�� ) to the

Elastic part of the rate-of-deformation expressed as:

�∇� � ��뼘
���D� � ��뼘

���D � D�� �t��

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Although, the elastic response is hypo-elastic, the work done in a closed cycle of deformation
does not vanish hence cannot be expressed precisely (Belytschko et al., 2014; Kim, 2015;
Hashiguchi, 2017; Hashiguchi, 2018).

C. The plastic strain rate is given by the plastic flow rule as:

D� � ��� ���t�� t� � ��� ���t� �tt�
where:

�� �is the plastic rate parameter

D. The yield condition (�) expressed as in Equations (28) is required to be isotropic:
����t� � h �t��

E. The loading and unloading condition is expressed as given in Equations (29):
�� � h� � � h� ��� � h �tt�

F. The relationship between the (Jaumann) Cauchy stress rate and the total rate-of-
deformation tensor for plastic loading is expressed as:

�∇� � ����� ��h�

So that with elastic loading or unloading, ��� � ��뼘
��

where:

��� � Is the continuum elasto-plastic tangent modulus which is a fourth order tensor
(Belytschko et al., 2014; Kim, 2015).

In formulating the above constitutive equations in terms Kirchhoff stress ��� (which is also
called the weighted Cauchy stress expressed as � � t� ) that is using the Kirchhoff stress in
the plastic flow equation, will result in a symmetric tangent moduli (Belytschko et al., 2014).
The above hypoelastic- based plasticity constitutive equations have been expressed under the
limitations of the infinitesimal elastic deformations (Hashiguchi 2017; Hashiguchi 2019).

4.2 Hyperelastic-plastic Constitutive Equations
Hyperelastic constitutive equations are designed to eliminate some of the restrictions on the
hypoelastic equations such as the isotropy of its elastic moduli as well as its yield function
(Belytschko et al., 2014). Here:

A. The deformation gradient (F = �� �� ) is decomposed multiplicatively into elastic

�� and plastic �� parts as in Equation 31.

� � �� � �� ����

B. The stress is evaluated by means of hyperelastic potential in terms of the elastic strain
and expressed as:

��� � ��뼘
�� ���� � ��t�

C. For plastic flow, the plastic flow rule which determines the plastic flow rate ���� � is
given as:

��� � �� �� ��� �� � ���� ����t�� ����

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Here the internal variable is defined in the intermediate domain ��� as shown in figure 5.

D. The yield condition in terms of the effective second Piola-Kirchhoff stress ���� is
expressed as:

�� ����t�� � h ��t�

E. The rate of evolution of internal variables given in equation 35 can be derived from the
hardening (softening) law as:

t�� � �� ������t�� ����

F. While the effective second Piola-Kirchhoff stress rate is given as in equation 36.

��� � ��뼘
�� ���� � ��뼘

�� ���� � ���� ����

Where:
��� � is the plastic part of the spatial velocity gradient.
��뼘
�� - Is the anisotropic elasticity tensor
��� and ��� are the elastic and plastic parts of the rate of deformation.
�� � is the plastic flow direction
�� � is the plastic moduli
� � Position vectors in the current configuration
�� Position vectors in the reference configuration (see figure 6 below) (Belytschko et al.,
2014; Hashiguchi 2017; Hashiguchi 2019).

For large deformations, it will no longer be viable to additively decompose the strains into
elastic and plastic parts, rather the deformation gradient (�) is multiplicatively decomposed into
elastic and plastic parts (Kim, 2018). This multiplicative decomposition of the deformation
gradient into elastic and plastic components as given in equation (31) sets up the configuration
as shown in Figure 6 (Belytschko et al., 2014). This configuration shows the mapping of the
point X in the reference domain �Ωh� to �� in the intermediate domain �Ω�� by the plastic part
of the deformation gradient ��. It also shows the mapping of �� in the intermediate domain ���
to x in the current domain (Ω) by the elastic part (��) of the deformation gradient (Belytschko
et al., 2014; Kim, 2015).

The formulation of the above constitutive equations is done on the virtual configuration as
shown in Figure 6.

4.3 Structural Model
The rate dependent elasto-viscoplastic material model has been used in the transient finite
element analysis of a steel beam (UB ���xt�hxtt with a thickness of �h.�mm ) subject to

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impact loading tested by Aliyu, 2019. This is because mild steels are generally strain rate
sensitive with its stress strain relationship highly dependent on the rate of loading. The mild
steel beam was subjected to impact loads at both the mid span and quarter span respectively
where the loading was applied incrementally. According to Aliyu, 2019 loading of this nature
tends to generate strain rates in the range of �ht /sec. Hence, the material model specification
adopted from the Ansys material library used in the transient nonlinear finite element analysis
of the beam to determine the plastic deformation capacity is as given below in the material
model parameter section.

4.4 Material Model Parameters
For this particular finite element (FE) simulation, the ANSYS nonlinear finite element code
adapted for rough nonlinear elasto-plasticity problems was used; with the following material
model specification: Non-linear > Inelastic > Rate Dependent > Visco-Plasticity > Isotropic
Hardening Plasticity > Mises Plasticity > Bilinear Isotropic This was to ensure that the beam is
properly modelled in both the elastic and the plastic range.

The end condition of the beam was fixed. The mesh discretion was done using the Beam4 3-D
finite element (Figure 7) which has two nodes per element and six degrees of freedom per
node. This particular element had tension, compression, torsion and bending capabilities as well
as special features of stress stiffening and large deformation capabilities.

Figure 7: Details of the Beam 4 3D geometry adapted from the ANSYS material library

Figure 8 shows the Idealized elasto-plastic relationship assumed in Aliyu, 2019; the values of the
material properties adopted were:
Modulus of elasticity 2.1 x �h�� �

�t ; Passion ratio 0.3; Yield stress 4.10

x �h� � �t Tangent modulus of elasticity 0 (Aliyu, 2019)

ET

Et
Fy

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where:

�� = tangent modulus

�� = elastic modulus

For the adopted idealized elasto-plastic stress strain curve in Figure 8, the initial behaviour was
assumed elastic until yield stress fy was reached after which the plastic phase was assumed to
have begun. The results from the nonlinear transient finite element simulation using the above
elasto-viscoplastic material model showed good agreement with the results obtained from the
analytical model. Figures 9 and 10 compares the resulting displacements (from both the
simulation and analytical model) of the steel beam after being struck at mid-spans and quarter-
spans respectively by varying impactors from height ranges of 15m, 20m, and 25m respectively.
The resulting displacements obtained from the mid span appeared to match exactly both the
allowable and actual deflections predicted from the analytical calculations as given in Aliyu,
(2019) which was arrived at by using the energy momentum balance technique suggested by
Mughal et al. (1994). However, a very minor variation was observed with the actual quarter
span deflections.

Figure 9: A plot of actual versus allowable displacement considering the effects of other loads
for mid span deflection adapted from Aliyu, (2019)

Figure 10: A graph of actual versus allowable displacement considering the effects of other
loads for quarter span deflection adapted from Aliyu, (2019)

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5 Conclusion
It can be concluded that, the choice of a suitable constitutive material model is considered key
in any analysis simulation in order to properly model the exact material behaviour under
loading conditions. This recommendation has been verified in the study carried out by Aliyu,
2019 where the rate dependent elasto-viscoplastic material model was selected from the
material library of the Ansys standard finite element code as the appropriate model for use in
the deformation regime where the mild steel beam behaves ductile for low speed (high period)
loading and becomes brittle for high speed (low periods) loading. The model predicted with
reasonable accuracy the behaviour of mild steel beams under impact loading when compared to
comparable results from an analytical model reported by Aliyu, 2019. An improvement in the
simulation results for the quarter span deflection may be achieved by using higher order finite
elements such as SOLID 186 which is an eight noded finite element with six degrees of
freedom per node. This particular element, supports plasticity, hyperelasticity, creep, stress
stiffening, large deflection, and has large strain capabilities. The use of this higher order finite
element may result in an increased simulation time and cost.

Generally, the elastoplastic constitutive equations discussed can be used in modelling the
behaviour of geomaterials (such as: soils, concrete and so on) where permanent/irreversible
strains are developed upon unloading.

Similarly, the stress strain relationships discussed in the text, postulates that plastic strain rate
develops when stress approaches yield surface. Hence, models developed from this principle
can be used to describe rate-independent/dependent elasto-plastic deformations of materials
under monotonic/cyclic loading and proportional/non-proportional loading.

Furthermore, elasto-plastic constitutive equations can also be employed in describing friction
between solids; formulating damage and plastic models used in fracture and damage analysis of
composite material subject to dynamic loadings.

The above applications of elasto-plastic constitutive equations are however, not exhaustive but
only to mention a few.

6 Future trends
Elasto plastic constitutive equations in formulating damage and plastic models for simulating the
behaviour of geomaterial have also been proposed in literature but this is still a long standing
challenge as more studies are required particularly in understanding the influence of combined
damage and plasticity on analysing failure patterns in these material under monotonic/cyclic
loading and proportional/non-proportional loading. Improvement in this area would be highly
invaluable in automobile, aerospace industries and so on.

Glossary
� � Elastic (Young’s) modulus (Pa or kPa�
� � Plastic switch parameter with a value of 1(� � �) corresponding to plastic loading
` and (� � h) corresponding to pure elastic loading
� � Back stress 1Pa
���� � Slope of a line tangent to the stress-stain cure at a point of interest it is expressed as
a percentage of the Young’s modulus E
d� Plastic modulus �����
� � Overstress which is the stress above the yield surface which describes viscoplastic `
` deformation

� � Viscosity with S.I unit of pascal-second (Pa s)
�� � Super imposed dot denotes the time derivative of the backstress

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��� � Effective plastic strain rate
t � Kinematic hardening parameter
σ� � Difference between the stress on the yield surface and the backstress
�� Directional vector
��� �Viscoplastic rate
μ� � Viscoplastic coefficient
f�σ� � Yield condition
��� �� � Elastic plastic parts of the deformation gradient � � �� � ��

� � Kirchhoff stress tensor
� �Cauchy stress tensor

��� � Effective second Piola-Kirchhoff stress rate

��� � Effective Green strain rate
�� � internal variable

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