A VISCOPLASTIC CONSTITUTIVE THEORY
FOR MONOLITHIC CERAMICS - II
Lesley A. Janosik
NASA Lewis Research Center
Cleveland, OH 44135
Stephen F. Duffy, PhD, PE
Cleveland State University
Cleveland, OH 44115
ABSTRACT
This paper, which is the second of two in a series,
exercises the viscoplastic constitutive model developed by
the authors in the previous article (Janosik and Duffy,
1998).
The model accounts for time-dependent
phenomena (e.g., creep, rate sensitivity, and stress
relaxation) in monolithic ceramics. Additionally, the
formulation exhibits a sensitivity to hydrostatic stress, and
allows different behavior in tension and compression.
Here, the constitutive equations formulated for the flow law
(i.e., the strain rate) and the evolutionary law have been
incorporated into computer algorithms for predicting the
multiaxial inelastic (creep) response of a given
homogeneous state of stress. Numerically simulated
examples illustrate the model’s ability to capture the
time-dependent phenomena suggested above. For each
imposed service (load) history considered, creep curves and
viscoplastic flow surfaces are examined to demonstrate the
model’s ability to capture the inelastic creep deformation
response. No attempt is made here to assess the accuracy
of the model in comparison to experiment. A quantitative
assessment is reserved for a later date, after the material
constants have been suitably characterized for a specific
ceramic material.
INTRODUCTION
This paper, which is the second of two in a series,
exercises the theory developed in the previous manuscript
by examining specific time-dependent stress-strain
behavior that can be modeled with the viscoplastic
constitutive relationship. Here, the constitutive equations
formulated for the flow law and the evolutionary law have
been incorporated into computer algorithms for predicting
the multiaxial inelastic (creep) response of homogeneously
stressed elements. Examples are presented to illustrate
certain aspects of the constitutive model. Creep curves
and viscoplastic flow surfaces are examined to demonstrate
the model’s ability to capture the inelastic deformation
response in advanced monolithic ceramic materials.
In monolithic ceramics, creep deformation and creep
rupture are generally controlled by the physical and
chemical properties of the grain and grain boundaries.
These microstructural phenomena, including the
nucleation, growth, and coalescence of cavities, lead
eventually to macrocracks which propagate and ultimately
lead to component fracture. Much work continues in
determining the exact mechanisms responsible for the
observed creep deformation and rupture behavior of
ceramics. An in-depth discussion of the micromechanical
aspects of creep of ceramics is beyond the scope of this
work. Here attention will focus on the macromechanical
phenomena (specifically, creep deformation) that can be
accounted for in the proposed model. Creep rupture is not
considered in the model, although incorporating damage
mechanics concepts into the present theory could yield a
workable creep rupture model. This task is reserved for a
future enhancement.
Under isothermal conditions, a typical creep test
entails abruptly applying a constant stress (or load) to a test
specimen. The material response in this type of test is
conveniently described in terms of strain vs. time. In a
typical creep strain vs. time plot, the material response
usually (but not always) exhibits three distinct regions of
behavior. The first, denoted primary creep, is identified as
the initial period during which the strain rate diminishes
with time. Following the primary creep response, the
strain rate stabilizes and remains constant for a (relatively)
long period of time. This region is denoted as secondary,
or steady-state, creep. In the third region, the strain rate
accelerates, leading to specimen or component fracture
(failure) or rupture. The response in this region is denoted
tertiary creep.
Many types of monolithic ceramics
typically don’t exhibit tertiary creep behavior. Therefore,
many constitutive equations formulated for these material
systems appropriately describe only the primary and
secondary creep regions. This is the case with the
proposed model. However, the tertiary creep response
(including creep rupture) can be added to the current model
by incorporating continuum damage mechanics concepts.
This task is reserved for a future enhancement.
CONSTITUTIVE THEORY
The complete theory was derived in detail in the
previous manuscript by the authors (Janosik and Duffy,
1998). For brevity, the derivation of the constitutive
theory will only be summarized here. The theory is
derived from a scalar dissipative potential function
attributed to Robinson (1978), which is identified here as
.
Under isothermal conditions, this function is
dependent upon the applied Cauchy stress (ij) and internal
state variable (aij), i.e.,


    ,  
 ij ij 
 K
2
 1 n
 R m 
  2    F dF +  H   G dG 


F



1
c

~
 1  2 J2 
 r ~    5 


 i j
3
1/ 2
(2)
~
I
1


parameters , c, and r(  ) shown in equation (2) are
presented in detail in the previous article (Janosik and
Duffy, 1998) and will not be discussed herein.
With the functions F and G (and hence ) completely
defined, the flow law (i.e., the inelastic strain rate) and the
evolutionary law are derived from the potential function.
The flow law is derived from the potential function  by
taking the partial derivative with respect to the applied
stress, i.e.,
(1)
In this formulation, ij and ij represent second rank
Cartesian tensors. Indicial notation is used with the
convention that repeated indices imply summation. The
parameters K, , R, H, n, and m are material constants
related to viscoplasticity. The authors realize that several
of these quantitites identified as material constants in the
theory may be strongly temperature-dependent in a
non-isothermal environment. However, for simplicity, the
present work is restricted to isothermal conditions. A
paper by Robinson and Swindeman (1982) provides the
approach by which an extension can be made to
non-isothermal environments. This task is reserved for a
future enhancement to the current model.
Specific macroscopic behavior (e.g., different
behavior in tension and compression) is readily embedded
in the model through the use of invariant theory. A
three-parameter yield criterion originally proposed by
Willam and Warnke (1975) for concrete serves as the
threshold flow criterion, F, for the model
~ ~ ~
I1 , J 2 , J 3
advanced monolithic ceramics in practical engineering
applications. Specifically, the proposed formulation permits
the constitutive model to exhibit a sensitivity to the
hydrostatic component of stress, and allows different
behavior in tension and compression. The additional
1
c
The functional form of the scalar state function, G, has
similar mathematical formulation, following the framework
of previously proposed constitutive models based on
Robinson’s (1978) viscoplastic law.
For frame
indifference, the scalar functions F and G (and hence )
must be form invariant under all proper orthogonal
transformations. Combining the principal stress invariants
~ ~ ~
( I 1, J 2, J 3 for the scalar function F, and ^I 1, ^J2, ^J3 for the
scalar function G) in an intelligent fashion allows the
model to account for the viscoplastic response to general
multiaxial loading conditions encountered when utilizing

=
 i j
(3)
Here, the inelastic strain rate vector is the gradient (directed
outward normal) to level surfaces of , similar to the
structure encountered in classical plasticity.
The
evolutionary law is similarly derived from the flow
potential. The rate of change of the internal stress with
deformation history is expressed as
 i j

= -h
 i j
(4)
where h is a scalar hardening function of the inelastic state
variable (i.e., the internal stress) only.
The complete multiaxial statement of the viscoplastic
constitutive theory was presented in the previous
manuscript by the authors. The remainder of this paper
will focus on applying the theory to account for asymmetric
tensile and compressive behavior, as well as sensitivity to
hydrostatic stress exhibited by advanced monolithic
ceramics.
EXAMPLES
This section presents theoretical predictions of the
multiaxial creep response of homogeneously stressed
elements obtained using the viscoplastic constitutive
theory. The coupled system of differential equations for
the inelastic strain, ij, and the state variable, ij, as
expressed in the flow and evolutionary laws (equations (3)
and (4), respectively), have been incorporated into
computer algorithms. For a full multiaxial creep problem,
solution requires integration of twelve coupled differential
equations representing the constitutive law, i.e., six
equations from the flow law and six equations from the
evolutionary law.
Three numerical examples are presented below to
illustrate various aspects of the constitiutive model. In
each of the examples, it is assumed that the material is
initially in a virgin state, i.e., the centers of the flow
surfaces lie at the origin of stress space.
0.02
0
M
Pa
0.03
15
Example 1.
Uniaxial Tension vs. Uniaxial
Compression—Longitudinal Creep Curves.
This
example illustrates the model’s ability to qualitatively
capture the asymmetric behavior in tension and
compression exhibited experimentally by ceramic
materials.
The William-Warnke threshold stress
parameters are chosen as t = 10 MPa (1.450 ksi), c = 100
MPa (14.503 ksi), and bc = 116 MPa (16.824 ksi). Equal
magnitude uniaxial tensile and compressive stresses are
used to generate the creep curves (longitudinal strain, 11)
presented in Figure 1. Notice that the creep curves in this
figure exhibit the classical behavior described previously in
Example 2.
Uniaxial Tension—Multiaxial Creep
Curves and Inelastic Flow Surfaces. This example
examines both creep curves and viscoplastic flow surfaces
to illustrate the constitutive model’s ability to capture the
inelastic (creep) deformation response of homogeneously
stressed elements. For the uniaxial tensile case of 11 =
125 MPa (18.129 ksi), Figure 2 depicts the multiaxial
components of the creep strain tensor, ij, obtained using
the William-Warnke model with threshold stress
parameters t = 10 MPa (1.450 ksi), c = 100 MPa (14.503
ksi), and bc = 116 MPa (16.824 ksi). Notice that here the
entire inelastic creep strain tensor, ij, is represented; the
creep curves in the previous figure depicted only the
longitudinal strain 11. Here, it is observed that the largest
strain is in the direction of the applied uniaxial tensile
stress. The transverse strains, 22 = 33, are essentially
neglible in comparison, and the other remaining strain
components are equal to zero.
=
These quantities are consistent with stress having units of
ksi (1 ksi = 6.895 MPa) and time measured in hours.
While these viscoplasticity parameters remain constant
throughout each of the examples which follow, those
associated with the Willam-Warnke model, i.e., the
Willam-Warnke threshold stress parameters t, c, bc, will
vary within the limits imposed by the convexity
requirement (i.e., 0.5 < rt /rc = 1.0). The permissible
Willam-Warnke model parameters impart great flexibility
in capturing a wide range of multiaxial material behavior.
The examples that follow will demonstrate the ability of the
constitutive model to qualitatively capture the essential
features of the complex material deformation behavior.
No attempt is made here to assess the accuracy of the
model in comparison to experiment. It should be noted
that the examples presented are intended to show trends
only. A quantitative assessment is reserved for a later
date, after the material constants have been suitably
characterized for a specific ceramic material.
1
(5a)
(5b)
(5c)
(5d)
(5e)
(5f)
1 =
1
200 M
Pa
1
1 =
175
MP

a
1
m = 350,000
H = 10,000
R = 3.0x10-4
n=4
m=4
b = 0.75
that they display a primary creep period followed by a
region of steady state behavior. Notice also in this figure
that the tertiary region is not included in the response
generated by the current model, as indicated previously. It
is evident in these creep response curves that the duration
of primary creep varies inversely with the stress magnitude,
i.e., lower stresses produce a longer duration of primary
creep, while higher stresses result in a shorter region of
primary creep. The asymmetry between the tensile and
compressive (longitudinal) creep strain behaviors depicted
in Figure 1 is consistent with experimental results reported
in the literature. Luecke and Wiederhorn (1994) present
findings from several authors who report that at equivalent
stress, creep results for a particular ceramic in tension may
be up to a hundred or more times greater than those for the
same material in compression.
Longitudinal Strain, 11
In addition to the Willam-Warnke threshold stress
parameters discussed extensively in the development of the
model in the previous manuscript, the viscoplastic material
parameters m, m, n, b, R, and H, are needed to obtain the
inelastic response predictions. Since there is not yet a
complete data base for characterizing these material
constants for a particular monolithic ceramic material, the
following set of material parameters is adopted here:
25
=1
1
1
MP
a
0 MPa
 11 = 10
0.01
11 = 75 MPa
Time (h)
0
0
-0.01
50
100
150
200
250
300
11 = -150 MPa
11 = -175 MPa
-0.02
11 = -200 MPa
-0.03
Figure 1 Predicted longitudinal creep strain curves
(11) for uniaxial tensile and compressive states of
stress with t = 10 MPa, c = 100 MPa, and bc = 116
MPa (1 ksi = 6.895 MPa).
0.03
0.03
11
0.02
11
0.02
0.01
0
0
-0.01
50
100
150
200
250
300
22 = 33
Strain, ij
Strain, ij
0.01
Time (h)
Time (h)
0
0
50
100
12 = 13 =23 = 0
200
250
300
22 = 33
-0.01
-0.02
150
11 =222 = 233
-0.02
12 = 13 =23 = 0
-0.03
-0.03
Figure 2 Predicted multiaxial creep strain curves
(ij) for uniaxial tensile state of stress 11 = 125 MPa
with t = 10 MPa, c = 100 MPa, and bc = 116 MPa
(1 ksi = 6.895 MPa).
Both the Willam-Warnke threshold function, F, and
the constitutive equations (i.e., the flow and evolutionary
laws) degenerate to the “simpler” underlying models (i.e.,
Drucker-Prager and von Mises, or J2, models) when certain
limiting conditions are imposed. Here, specifically, it is
illustrative to notice that the constitutive equations reduce
to forms consistent with those of Robinson’s J2-dependent
(von Mises) model. This is equivalent to assuming
insensitivity to hydrostatic stress and identical behavior in
tension and compression.
Complete details can be
obtained in Janosik (1998).
Here, the results of
implementing these same limiting conditions are presented
as they apply to observations of the inelastic response of
homogeneously stressed elements. Figure 3 depicts the
multiaxial components of the creep strain tensor ij for the
same uniaxial tensile stress state, 11 = 125 MPa (18.129
ksi), presented above. However, in this figure, the
Willam-Warnke parameters are stipulated as t = c = bc =
10 MPa (1.450 ksi). This formulation is the equivalent of
a J2-dependent model. In figure 3, it is again observed that
the largest strain is in the direction of the applied uniaxial
tensile stress. However, in this case, the transverse strains,
22 = 33, are equal in magnitude to 0.5 times the
longitudinal strain 11. This appropriately reflects the
effect of the Poisson’s ratio which is approximately equal
to 0.5 in a ductile J2-dependent material.
Taking a slightly different perspective, the changes in
the inelastic state can be viewed with projections of the
scalar function F. Figure 4 depicts inelastic flow surfaces
projected onto the 11-22 stress space for the uniaxial
tensile case 11 = 125 MPa (18.129 ksi) presented above.
The load path is indicated in this figure. Here, again, the
Willam-Warnke parameters are chosen as t = c = bc = 10
MPa (1.450 ksi) to represent a J2-dependent model. Each
surface in this figure is one of a family representing distinct
inelastic states. The largest surface represents the virgin
state and corresponds to the initial stages of primary creep.
Figure 3 Predicted multiaxial creep strain curves
(ij) for uniaxial tensile state of stress 11 = 125 MPa
with t = c = bc = 10 MPa (1 ksi = 6.895 MPa).
Only the member of each family that passes through the
stress point 11 = 125 MPa (18.129 ksi) is shown.
Corresponding to each surface is an inelastic strain rate
vector. As the size of the flow surfaces diminish, the
magnitude of the inelastic strain rate vector decreases.
Eventually steady state creep is attained where the size of
the surface and the magnitude of the inelastic strain rate
vector remain unchanged. The elliptical shape of the
surfaces depicted in the 11-22 stress space is a result of
the limiting assumptions imposed to achieve identical
behavior in tension and compression. Figure 5 depicts the
inelastic flow surfaces for this same stress state projected
onto the -plane. Here, the member of each family that
~
passes through the point r(
) = 2J 2 along the positive
22 (ksi)
50
40
30
Load
Path,
20
11
10
11
(ksi)
0
-50
-40
-30
-20
-10
0
-10
10
20
30
40
50
-20
-30
-40
-50
Figure 4 Inelastic states (F = constant) depicted in
the 11-22 stress space for uniaxial stress state 11 =
125 MPa with t = c = bc = 10 MPa (1 ksi = 6.895
MPa).
1
1
2
2
3
Figure 5 Inelastic states (F = constant) depicted in
the  -plane for uniaxial stress state 11 = 125 MPa
with t = c = bc = 10 MPa (1 ksi = 6.895 MPa).
1 axis is shown. Recall that the -plane is a deviatoric
plane with no hydrostatic component of stress. The
circular surfaces produced in the -plane by this
formulation would appear as right circular cylinders in the
three-dimensional Haigh-Westergaard principal stress
space. As indicated in Figure 5, these surfaces would
intersect along a line parallel to the hydrostatic axis in this
stress space.
Figure 6 depicts inelastic flow surfaces projected onto
the 11-22 stress space for the uniaxial tensile case 11 =
125 MPa (18.129 ksi) with Willam-Warnke threshold stress
parameters t = 10 MPa (1.450 ksi), c = 100 MPa (14.503
ksi), and bc = 116 MPa (16.824 ksi). Again, the member
22 (ksi)
50
11
0
-300
-250
3
-200
-150
-100
-50
(ksi)
0
-50
50
Load
Path,
11
-100
-150
-200
-250
-300
Figure 6 Inelastic states (F = constant) depicted in
the 11-22 stress space for uniaxial stress state 11 =
125 MPa with t = 10 MPa, c = 100 MPa, and bc
= 116 MPa (1 ksi = 6.895 MPa).
Figure 7 Inelastic states (F = constant) depicted in
the  -plane for uniaxial stress state 11 = 125 MPa
with t = 10 MPa, c = 100 MPa, and bc = 116
MPa (1 ksi = 6.895 MPa).
of each family that passes through the stress point 11 =
125 MPa (18.129 ksi) is shown. Here, the shape of the
surfaces in the 11-22 stress space results from the
Willam-Warnke threshold stress parameters imposed to
achieve the asymmetric behavior in tension and
compression. Figure 7 presents the inelastic flow surfaces
for this same case in the -plane. These surfaces would
appear as right tetrahedrons in the three-dimensional
Haigh-Westergaard principal stress space.
Example 3. Thin-Wall Pressure Vessel—Multiaxial
Creep Curves and Inelastic Flow Surfaces. In this
example, the qualitative change in the inelastic deformation
response of a thin-wall pressure vessel resulting from
incorporating a sensitivity to hydrostatic stress and
asymmetric tensile and compressive behavior is presented.
With the postulated normality structure, the direction of the
inelastic strain rate vector for each inelastic state is directed
along the gradient to the respective state’s surface of F =
constant. One can easily view this in the predicted
structural response of a thin-wall pressure vessel.
Specifically, the stress path 11 = 222 = 125 MPa (18.129
ksi) is equivalent to the state of stress in a thin-wall
pressure vessel if 11 is taken as the circumferential stress,
22 the axial stress, and 33 the radial stress. Abruptly
applying an internal pressure such that F >> 0 and holding
the state of stress fixed at the circumferential and axial
stress values specified above allows the tube to deform
inelastically.
Figure 8 shows the multiaxial creep curves obtained
for an isotropic material with Willam-Warnke threshold
stress parameters t = c = bc = 10 MPa (1.450 ksi). This
formulation is the equivalent of a J2-dependent material; it
assumes insensitivity to hydrostatic stress and identical
behavior in tension and compression. In this figure it is
evident that ii  0. That is, 11 = -33 and 22 = 0,
indicating inelastic incompressibility. This results from
the assumptions of a J2-dependent theory.
22 (ksi)
50
0.02
-300
Strain, ij
0.01
-250
-200
-150
-100

11
0
-50 11
(ksi)
0
50
Load Path,
11 = 2 22
-50
Time (h)
0
-100
0
50
100
150
200
250
300
-150
-0.01
33
11 = -33
22 = 0
-0.02
-200
12 = 13 =23 = 0
-0.03
-250
-300
Figure 11
8 Predicted
multiaxial
creep strain
curves
ij)
Figure
Inelastic states
(F = constant)
depicted
in (the
for
thin-wall
pressure
vessel
(

=
2

=
125
MPa)
11
22


stress
space
for
thin-wall
pressure
vessel
(

11 22
11 =
t = MPa)
= 10 
MPa
(1 MPa,
ksi = 6.895
c = bc with
2with
22 =125
c = MPa).
100 MPa, and
t = 10
bc = 116 MPa (1 ksi = 6.895 MPa).
The changes in the inelastic state of the thin-wall
pressure vessel as it creeps can be illustrated with
projections of the scalar function F. Figure 9 depicts the
changes in inelastic state for a J2-dependent material. The
load path 11 = 222 = 125 MPa (18.129 ksi) is illustrated
in this figure. Each surface in Figure 9 is one of a family
representing distinct inelastic states. The largest surface
represents the virgin state and corresponds to the initial
stages of primary creep. Only the member of each family
that passes through the stress point 11 = 222 = 125 MPa
(18.129 ksi) is shown. Corresponding to each surface is
an inelastic strain rate vector. As the size of the flow
22 (ksi)
50
surfaces diminish, the magnitude of the inelastic strain rate
vector decreases. Eventually steady state creep is attained
where the size of the surface and the magnitude of the
inelastic strain rate vector remain unchanged. Note that
the inelastic strain rate vectors do not have  components.
In contrast to the previous two figures obtained by
applying the assumptions for an equivalent J2-dependent
formulation, the following two figures represent inelastic
deformation results for the same thin-wall pressure vessel
incorporating both a sensitivity to hydrostatic stress and
differing behavior in tension and compression. The
Willam-Warnke threshold stress parameters are stipulated
as t = 10 MPa (1.450 ksi), c = 100 MPa (14.503 ksi), and
bc = 116 MPa (16.824 ksi). The multiaxial creep strain
curves corresponding to this example are depicted in Figure
10. Here it is evident that ii  0. This is consistent with
the theory developed in this series of manuscipts, since
contraction of the flow law yields a nonzero result for the
stipulated loading condition.
Figure 11 depicts the inelastic flow surfaces projected
onto the 11-22 stress space for the pressure vessel
presented above. The load path 11 = 222 = 125 MPa
(18.129 ksi) is indicated in this figure. Each surface again
represents distinct inelastic states as the vessel deforms
under an identical internal pressure of 11 = 222 = 125
MPa (18.129 ksi). Whereas the 22 components vanished
in the J2-dependent formulation, it is evident that the
0.03
11
0.02
Time (h)
0
-0.01
40
30
Load
Path,
11 = 2 22
20
10
-40
-30
-20
-10
0
-10
20
30
40
50
100
150
200
250
300
33
12 = 13 =23 = 0
(ksi)
10
0
-0.02
11
0
-50
22
0.01
Strain, ij
0.03
50
-20
-30
-0.03
Figure 10 Predicted multiaxial creep strain curves (ij)
for thin-wall pressure vessel (11 = 222 = 125 MPa)
with t = 10 MPa, c = 100 MPa, and bc = 116 MPa (1
ksi = 6.895 MPa).
-40
-50
Figure 9 Inelastic states (F = constant) depicted in
the 11-22 stress space for thin-wall pressure vessel
(11 = 222 =125 MPa) with t = c = bc = 10 MPa (1
ksi = 6.895 MPa).
inelastic strain rate vectors would have relatively large 22
components in the current formulation. Thus, the mode of
inelastic deformation changes with sensitivity to
hydrostatic stress and asymmetric behavior. Eventually,
upon reaching steady state, the size of the surface and the
magnitude of the inelastic strain rate vectors remain fixed.
SUMMARY AND CONCLUSION
This paper has presented numerical examples to
illustrate the multiaxial viscoplastic constitutive theory
developed by the authors in the previous manuscript. The
first paper in this series focused on the formulation and
derivation of viscoplastic constitutive equations for the
flow law (strain rate) and the evolutionary law. The theory
was derived based on a threshold function that exhibits a
sensitivity to hydrostatic stress and asymmetric behavior in
tension and compression. This paper has presented the
results obtained after incorporating these constitutive
equations into computer algorithms for predicting the
multiaxial inelastic (creep) deformation response of
homogeneously stressed elements. Three numerically
simulated examples were presented with the intent to
qualitatively illustrate the model’s ability to capture and
predict the inelastic deformation response exhibited by
isotropic monolithic ceramics at elevated service
temperatures. A quantitative assessment is reserved for a
later date, after the material constants have been
characterized for a specific ceramic material.
Incorporating this model into a non-linear finite
element code would provide industry the means to
numerically simulate the inherently time-dependent and
hereditary phenomena exhibited by these brittle material
systems in service. This task is also reserved for a future
enhancement.
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