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Density-based constitutive modelling of P/M FGH96 for ...
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Density-based constitutive modelling of P/M FGH96 for powder forging
Saeed Zare Chavoshi1, Jiaying Jiang1, Yi Wang1, Shuang Fang2, Shuyun Wang2, Zhusheng
Shi*1, Jianguo Lin1
1Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK
2Beijing Institute of Aeronautical Materials, Beijing 100095, China
*Corresponding author: [email protected]
Abstract
A set of viscoplastic constitutive equations is presented in this study to predict hot
compressive deformation behaviour and densification levels of powder metallurgy (P/M)
FGH96 nickel-base superalloy during direct powder forging (DPF) process. The constitutive
equations make use of the elliptic equivalent stress proposed in porous material models, and
unify the evolution of relative density, normalised dislocation density, isotropic hardening
and flow softening of the powder compact. A gradient-based optimisation technique is
adopted to determine the material constants using the experimental data obtained from
Gleeble isothermal uniaxial compression tests of HIPed FGH96 at different temperatures and
strain rates. The developed constitutive equations are incorporated into finite element code
DEFORM via user-defined subroutine for coupled thermo-mechanical DPF process
modelling. The constitutive equations benefiting from the viscoplastic densification model of
the calibrated Abouaf, among the six studied porous material models, compare favourably
with the experimental data, while the equations integrating the porous material model of
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Shima & Oyane provide excellent agreement with experiments in the low density outer
region of the powder compact.
Keywords: Direct powder forging; FGH96; Densification; Constitutive equations; Finite
element modelling
Nomenclature
𝑐(𝐷) Relative density related term
𝐶𝐶 Correlation coefficient
𝐷 Relative powder density
𝑓(𝐷) Relative density related term
µ, 𝜆 Lamé parameters (MPa)
𝑁 Number of data
𝑅 Isotropic hardening (MPa)
𝑋𝑖 Experimental flow stress (MPa)
�� Mean value of 𝑋𝑖
𝑌𝑖 Computed flow stress (MPa)
�� Mean value of 𝑌𝑖
𝜌 Dislocation density
𝜌 Normalised dislocation density
𝜌𝑖 Initial dislocation density
𝜌𝑠 Saturated dislocation density
휀𝑖𝑗 Strain tensor
휀𝑖𝑗𝑣𝑝 Viscoplastic strain tensor
휀𝑒 Effective strain
휀𝑒𝑣𝑝 Effective viscoplastic strain
휀𝑒𝑝𝑣𝑝 Effective viscoplastic strain of powder compact
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𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐 Elliptic equivalent stress (MPa)
𝜎𝑖𝑗 Stress tensor (MPa)
𝜎𝑚 Hydrostatic stress (MPa)
𝜎𝑒 Effective stress (MPa)
𝑆𝑖𝑗 Stress deviator tensor (MPa)
𝑇𝑋 Stress triaxiality factor
∅ Strain rate potential
𝛿𝑖𝑗 Kronecker delta
𝜔 Damage (softening) parameter
𝑘, 𝐾, 𝐶, 𝐵, 𝐸 Temperature-dependent material constants
𝑛, 𝜎0, 휀0, 𝛾, 𝐴,
𝛿1, 𝛿2, 𝛽, 𝜑,
𝑘0, 𝐾0, 𝐶0, 𝐵0,
𝐸0, 𝑄1, 𝑄2, 𝑄3,
𝑄4, 𝑄5
Temperature-independent material constants
1. Introduction
Direct powder forging (DPF) is a new powder forming process which can be used for low-
cost manufacturing of components with superior mechanical properties. The DPF process
comprises compaction of encapsulated, vacuumed and heated powder particles under high
forming loads within a short time, and holding for a given period of time to produce
sufficient bonding between powder particles. The primary advantages of this novel process
are low quantities of prior particle boundary (PPB) networks which are inherently brittle and
provide an easy fracture path in the component, controllability of microstructure at different
locations of the component, near net shape manufacturing, low energy consumption and the
possibility to apply conventional hot forging presses [1, 2]. In DPF process, both powder
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consolidation and deformation occur concurrently and interact with each other, which plays a
crucial role in determining the mechanical properties of the DPFed components.
Powder metallurgy (P/M) nickel-base superalloys are commonly used in aerospace industry
for manufacturing of combustion systems, advanced gas turbines and other related high-
temperature applications owing to their outstanding high-temperature strength,
microstructural stability, creep and corrosion resistance. FGH96 is a relatively new damage-
tolerant P/M superalloy which is widely used in fabrication of turbine engine disks and other
bearing components. This superalloy exhibits poor workability and high strain rate sensitivity
index, and requires narrow forging temperature range [3, 4]. The DPF process could be
adopted to improve the workability and control the deformation of P/M FGH96. However,
the constitutive behaviour of P/M FGH96 under the DPF conditions has to be fully
understood in order to design an optimum process. Constitutive modelling of powder
densification, evolution of dislocation density, hardening and softening mechanisms are
obviously keystones of successful quantitative solutions, which can be used for the numerical
analysis.
Numerical simulation is an essential step for obtaining a clear and unequivocal understanding
of the mechanics of powder behaviour during any metal powder forming process. Such
numerical analysis can assist the controlling of the densification and microstructure evolution
of powder, which in turn would culminate in attaining exceptional mechanical and physical
properties. Numerical modelling of consolidation processes is primarily divided into two
categories, i.e., micromechanical and macromechanical approaches. The micromechanical
method deals with the inter-particle behaviour of metal powders whereas macromechanical,
or continuum, method considers the overall behaviour of powder mass by idealising powder
mass as an equivalent continuum material or a solid continuum owing to the characteristic of
dilatancy [1, 5]. From an industrial perspective, the macromechanical method has an
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absolute preference over the micromechanical approach due to its capability in predicting
macroscopic behaviour of the powder, viz. density distribution, equivalent stress state and
final shape of the component. Specifically, achieving a homogeneous density distribution
within the component is of great interest because of its significant influence on the final
performance of the engineered part. Accordingly, an appropriate constitutive modelling to
predict powder material behaviour, consolidation levels and density distribution under
complicated loading conditions could be very beneficial for a successful manufacturing [6-8].
Numerous macromechanical porous material models such as Gurson [9], Kuhn & Downey
[10], Kuhn & McMeeking [11], Green [12], Wilkinson & Ashby [13], Shima & Oyane [14],
Gurson-Tvergaard-Needleman (GTN) [15], Cocks [16], Doaraivelu et al. [17], Park [18],
Duva & Crow [19, 20], Ponte-Castaneda [21], Sofronis & McMeeking [22], Abouaf [23, 24],
calibrated Abouaf [25-28], etc. have been proposed for the prediction of stresses and
deformations in porous media. On the other hand, Lin et al. [29-31] have developed various
sets of constitutive equations to model the evolution of dislocation density, hardening,
softening, recrystallisation and damage in warm/hot metal forming processes. For the DPF
process, however, reliable unified density-based constitutive models to accurately predict
both the powder densification and deformation mechanisms including dislocation density,
hardening and softening are lacking in the literature [1, 2]. It should be noted that the powder
compaction parameters for the Abouaf model [23, 24] are conventionally determined using
hot isostatic pressing (HIP) cycles which needs several hours of handling and therefore limits
the identification of such parameters to coarse microstructures. An efficient methodology has
recently been proposed which uses spark plasma sintering (SPS), instead of HIPing, thus
short processing time associated with SPS allows an identification of the parameters on
controlled microstructures [32].
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In order to precisely simulate the powder densification and deformation mechanisms during
the DPF process, appropriate physically-based constitutive models must be developed. In
tandem with this goal, the current study presents a set of unified density-based viscoplastic
constitutive equations incorporating porous material models of Shima & Oyane [14], Cocks
[16], Duva & Crow [20], Ponte-Castaneda [21], Sofronis & McMeeking [22] and calibrated
Abouaf [26] to describe powder densification as well as evolution of internal variables such
as dislocation density, isotropic hardening and flow softening during the DPF process. To
calibrate the equations, hot compression tests of P/M FGH96 nickel-base superalloy are first
performed on a Gleeble thermo-mechanical simulator at different temperatures and strain
rates. Similarly, hot compression experiments are also conducted isothermally on stainless
steel AISI 304 which is used as the container material, to characterise its flow behaviour. A
gradient-based optimisation technique is employed to determine the material constants arising
in the constitutive equations, and unbiased statistical parameters are calculated to evaluate the
prediction accuracy of the constitutive models. The developed unified material models are
then implemented into finite element (FE) solver DEFORM via user-defined subroutine to
model the DPF process of P/M FGH96. The reliability and accuracy of the unified density-
based constitutive equations and porous material models are assessed through making
comparison with the DPF experiments. The developed FE model is also used to analyse the
evolution of internal state variables such as relative density, normalised dislocation density,
isotropic hardening, stress triaxiality factor, etc. during the DPF process.
2. Experimental programme
The chemical composition of FGH96 superalloy used in this study is summarised in Table 1.
To characterise the flow behaviour of HIPed FGH96 powder compact, isothermal uniaxial
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hot compression tests are conducted under vacuum condition on cylindrical specimens of
12.0 mm height and 8.0 mm diameter using the Gleeble 3800 thermo-mechanical testing
system. A graphite disc is used on each side of the specimens and high temperature lubricant
paste is applied to the interfaces to decrease friction and to reduce non-uniform deformation
during compression. Figure 1 presents the schematic illustration of the heating cycle of the
hot compression tests. The fully dense HIPed FGH96 samples are resistance-heated to
deformation temperature (1000, 1050 and 1100 °C) and are held for 5 min at the temperature
before performing the hot compression tests. The compression tests are performed at constant
strain rates of 0.1, 1 and 10 s-1. Likewise, compression tests on the stainless steel AISI 304
are carried out at temperatures of 900, 1000 and 1100 °C, and strain rates of 0.1, 1 and 10 s-1.
Stainless steel AISI 304 is selected as the container material since it possesses good
weldability, high stiffness and strength at room temperature as well as high ductility at
elevated temperatures, which makes it an ideal choice for container material. The temperature
and strain rate ranges are carefully selected to mimic the most suitable conditions commonly
used in industry for manufacturing FGH96 components. The collected stress-strain data are
subsequently utilised to calibrate the constitutive models.
Table 1 Nominal composition of FGH96 nickel-base superalloy.
Element Cr Co Mo W Ti Al Nb Zr C B Ni
wt% 16.0 13.0 4.0 4.0 3.7 2.2 0.8 0.036 0.03 0.011 Balance
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Figure 1 Schematic illustration of the heating cycle of the hot compression tests.
Before DPF experiments, argon atomised FGH96 powder with an average particle size of 35
μm is encapsulated in the stainless steel AISI 304 cylindrical container, vacuumed to
1.0×10−5 Pa to alleviate the oxidation of the powder at elevated temperatures, and then
sealed. Figure 2 demonstrates dimensions of the cylindrical container, experimental setup,
preforms before and after the DPF process, morphology of FGH96 powder particles and
microstructure of the DPFed compact. The process conditions used for the DPF experiments
are summarised in Table 2. The powder compact with an initial relative density of ~0.7 is
compressed using a 250 kN high rate servo-hydraulic machine which is connected to an
oscilloscope to record the stroke-load data during DPF process. Image analysis is adopted to
measure the average porosity of the powder compact at different locations, by counting the
pixels for the porosity to obtain the percentage area. Accordingly, the relative density can be
calculated as “1.0 minus porosity”. More details can be found in [1].
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(a) Geometry of container
(b) Test set-up (c) Samples before and after forging
(d) FGH96 powder particles (e) Microstructure of the DPFed material
Figure 2 (a) The container geometry, (b) test set-up, (c) samples before and after the process,
(d) powder morphology and (e) microstructure of the DPFed compact with an initial relative
density of ~0.7.
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Table 2 Process conditions adopted for the DPF experiments.
Equipment Ram speed
(mm/s)
Die
material
Container
material
Forging
temp. (°C)
Soaking
time (min) Lubricant
250 kN
servo-
hydraulic
machine
200 H13 AISI 304 1150 20 Glass
powder
3. Unified viscoplastic constitutive equations
Multiaxial flow for viscoplastic materials, represented by a power-law, can be obtained by
defining a viscoplastic strain rate potential and by assuming von Mises behaviour for perfect
viscoplasticity (secondary creep), i.e. without the consideration of strain hardening. The
viscoplastic strain rate potential for a fully dense, incompressible, power-law creeping
material can be expressed in the form of [33]:
∅ = 0𝜎0
𝑛+1(
𝜎𝑒
𝜎0)𝑛+1 (1)
𝜎𝑒 = (3𝑆𝑖𝑗𝑆𝑖𝑗
2)1/2 (2)
𝑆𝑖𝑗 = 𝜎𝑖𝑗 − 𝛿𝑖𝑗𝜎𝑚 , 𝜎𝑚 =1
3𝜎𝑘𝑘 (3)
where ∅ is the viscoplastic strain rate potential, 휀0, 𝜎0 and n are material parameters, 𝜎𝑒
and 𝜎𝑚 stand for the effective and hydrostatic stresses, 𝑆𝑖𝑗 and 𝛿𝑖𝑗 represent the deviatoric
part of the Cauchy stress tensor and Kronecker delta, respectively. Eq. (1) corresponds to
Norton’s equation (or Odqvist’s law in three-dimensional context) for the secondary creep
without considering the hardening effect, and it ignores the elastic domain, thus representing
rigid perfect viscoplasticity [34]. Accordingly, such proposed model is not capable of
describing the increase of elastic properties with plastic deformation, a phenomenon known
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as elastoplastic coupling [35]. The strain rate tensor 휀��𝑗 can be obtained by differentiating Eq.
(1) in terms of deviatoric stresses:
휀��𝑗 =𝜕∅
𝜕𝑆𝑖𝑗=
3
2
0
𝜎0(
𝜎𝑒
𝜎0)𝑛−1𝑆𝑖𝑗 (4)
Considering the porous (or powder) material as a continuum medium, the tensor form of the
constitutive equation of a porous material can be written in the same form as that of the dense
material, using the same strain rate potential as in the dense state, yet with a modified
expression of the generalised stress. The classical J2 theory of metal plasticity assumes that
the effect of hydrostatic pressure on plastic flow is negligible. However, for porous materials,
both the deviatoric and hydrostatic components of stress cause yielding. Accordingly, the
yield criterion for an isotropic porous media must depend on both the second invariant of the
deviatoric stress tensor and the first invariant of the stress tensor. Consequently, the
viscoplastic strain rate potential for an isotropic, macroscopically homogeneous, power-law
creeping powder material (∅𝑝𝑜𝑤𝑑𝑒𝑟) can be formulated as a function of elliptic equivalent
stress (𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐) [12, 19, 20]:
∅𝑝𝑜𝑤𝑑𝑒𝑟 = 0𝜎0
𝑛+1(
𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐
𝜎0)𝑛+1 (5)
𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐2 = 𝑐(𝐷)𝜎𝑒
2 + 𝑓(𝐷)𝜎𝑚2 (6)
where 𝑐(𝐷) and 𝑓(𝐷) are decreasing functions of the relative density and represent the
localisation of stress state generated by changes of porosity. Eq. (5) is in fact an extension of
the Odqvist’s law to porous material, which considers not only the effect of hydrostatic
components of stress state, but also the influence of deviatoric stress. By differentiating Eq.
(5) in terms of deviatoric stress, the strain rate tensor for powder compact (휀��𝑗,𝑝𝑜𝑤𝑑𝑒𝑟) can be
written as:
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휀��𝑗,𝑝𝑜𝑤𝑑𝑒𝑟 =𝜕∅𝑝𝑜𝑤𝑑𝑒𝑟
𝜕𝑆𝑖𝑗= 0
𝜎0(
𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐
𝜎0)
𝑛−1
(3
2𝑐(𝐷)𝑆𝑖𝑗 +
1
3𝑓(𝐷)𝛿𝑖𝑗𝜎𝑚) (7)
In the aforementioned power-law viscoplastic equations, the initial yield stress (threshold
stress, k), isotopic hardening (R), and damage (softening) parameter (ω) can be introduced to
the perfect viscoplastic model, and the equations can be written as:
∅ =𝐾
𝑛+1(
𝜎𝑒−𝑅−𝑘
𝐾)𝑛+1 1
(1−𝜔)𝛾 (8)
휀��𝑗𝑣𝑝 =
3
2𝜎𝑒(
𝜎𝑒−𝑅−𝑘
𝐾)𝑛 1
(1−𝜔)𝛾 𝑆𝑖𝑗 (9)
where 휀��𝑗𝑣𝑝
is the viscoplastic strain rate tensor. For the powder compact material:
∅𝑝𝑜𝑤𝑑𝑒𝑟 =𝐾
𝑛+1(
𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐−𝑅−𝑘
𝐾)
𝑛+1 1
(1−𝜔)𝛾 (10)
휀��𝑗,𝑝𝑜𝑤𝑑𝑒𝑟𝑣𝑝 =
3
2𝜎𝑒(
𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐−𝑅−𝑘
𝐾)
𝑛 1
(1−𝜔)𝛾 (3
2𝑐(𝐷)𝑠𝑖𝑗 +
1
3𝑓(𝐷)𝛿𝑖𝑗𝜎𝑚) (11)
where K and γ are material constants. In this representation, it is assumed that below the
threshold stress k, no plastic flow would occur. When strain increases, dislocation density
increases and dislocation trapping and tangling happens, leading to the hardening of the
material. The stress has to overcome R to continue the viscoplastic flow [31]. As can be seen
from the experimental data of fully dense HIPed FGH96 shown in Figure 3(a), there is a drop
in flow stress during the hot compression tests. This is attributed to a series of complex
interaction and cooperation of recovery and recrystallisation processes, the two main
softening processes that occur in hot deformation and contribute to the reduction of
dislocation density. To account for this, a typical damage model which was proposed to
approximate the flow softening behaviour of material during hot deformation [33] is used as a
simplified softening model. The damage parameter (ω), referred to as softening parameter in
this paper, would be included in the viscoplastic equations.
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Accordingly, the full set of the physical-based constitutive equations for a fully dense
material can be summarised as follows:
휀��𝑣𝑝 = (
𝜎𝑒−𝑅−𝑘
𝐾)𝑛 1
(1−𝜔)𝛾 (12)
�� = 𝐴(1 − 𝜌)|휀��𝑣𝑝|
𝛿1− 𝐶𝜌
𝛿2 (13)
�� = 0.5𝐵𝜌−0.5
�� (14)
�� = 𝛽(1 − 𝜔)𝜑휀��𝑣𝑝
(15)
휀��𝑗𝑒 = (휀��𝑗
𝑇 − 휀��𝑗𝑣𝑝) (16)
��𝑖𝑗 = 2µ휀��𝑗𝑒 + 𝜆휀��𝑘
𝑒 (17)
where 휀��𝑣𝑝
, ��, ��, ��, ��𝑖𝑗, 휀��𝑗𝑒 and 휀��𝑗
𝑇 are the rates of effective viscoplastic strain, normalised
dislocation density, isotropic hardening, softening parameter, stress tensor, elastic strain
tensor and total strain tensor, respectively. µ = 𝐸/[2(1 + 𝑣)] and 𝜆 = 𝐸𝑣/[(1 + 𝑣)(1 −
2𝑣)] are the Lamé parameters, in which 𝐸 and 𝑣 are respectively Young’s modulus and
Poisson’s ratio of the material. 𝛾, 𝐴, 𝛿1, 𝛿2, 𝛽, 𝜑, C, B are material constants.
As can be seen, all the equations are developed in their rate forms so as to represent the
evolutionary nature of the associated material properties during hot compressive loading
processes [31, 36]. It should be mentioned that a normalised dislocation density 𝜌 [31, 37] is
adopted here since the measurement of the absolute dislocation density of a material is too
intricate. The normalised dislocation density is defined as follows:
𝜌 =𝜌−𝜌𝑖
𝜌𝑠 (18)
𝜌𝑖 and 𝜌𝑠 are respectively the dislocation density of the material in its initial state and the
saturated state. Normally 𝜌𝑖≪𝜌𝑠, thus 𝜌 changes from 0 (initial) to 1 (saturated) during the
hot compressive loading.
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To formulate the unified density-based constitutive equations for a powder compact material,
𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐 replaces 𝜎𝑒 and is incorporated into the effective viscoplastic strain rate of powder
compact (휀��𝑝𝑣𝑝). In essence, 𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐 can reflect the impact of powder density 𝐷 on the
effective viscoplastic strain rate during any hot compressive loading problem. Therefore,
휀��𝑝𝑣𝑝 = (
𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐−𝑅−𝑘
𝐾)𝑛 1
(1−𝜔)𝛾 (19)
𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐 = (𝑐(𝐷)𝜎𝑒2 + 𝑓(𝐷)𝜎𝑚
2 )1/2 (20)
The relative density related terms 𝑐(𝐷) and 𝑓(𝐷) have various mathematical expressions
owing to different approaches by different researchers from the literature, as listed in Table 3.
𝐷0 in the calibrated Abouaf model is the initial density of the powder compact. Considering
the incompressibility of the matrix phase, the change in volume of the porous material equals
the change in volume of the voids, and the evolution law for the density, i.e. the densification
rate (��), takes the form [9]:
�� = −𝐷휀��𝑘𝑣𝑝
(21)
Table 3 Expressions of the relative density dependent terms c and f.
Model 𝒄(𝑫) 𝒇(𝑫)
Cocks [16] 1 +2
3(1 − 𝐷)𝐷−
2𝑛𝑛+1
9𝑛(1 − 𝐷)
2(𝑛 + 1)(2 − 𝐷)𝐷−
2𝑛𝑛+1
Duva & Crow [20] 1 +2
3(1 − 𝐷)𝐷−
2𝑛𝑛+1 (
𝑛(1 − 𝐷)
(1−(1 − 𝐷)1𝑛)𝑛
)
2𝑛+1
(3
2𝑛)2
Ponte-Castaneda [21] 1 +2
3(1 − 𝐷)𝐷−
2𝑛𝑛+1
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4(1 − 𝐷)𝐷−
2𝑛𝑛+1
Sofronis & McMeeking [38] (2
𝐷− 1)
2𝑛𝑛+1
(𝑛(1 − 𝐷)
(1−(1 − 𝐷)1𝑛)𝑛
)
2𝑛+1
(3
2𝑛)2
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Calibrated Abouaf [26] 1 + 3.1(1 − 𝐷
𝐷 − 𝐷0) 0.72(
1 − 𝐷
𝐷 − 𝐷0)0.05
Shima & Oyane* [14] 4.32
3(2.44 − 𝐷)
9(1 − 𝐷)
2.44 − 𝐷
*In the yield function 𝛿 =1.44𝐷5
2.44−𝐷
It should be mentioned here that the porous material model of Shima & Oyane [14] was
developed based on determining the parameters in the yield function using uniaxial
compression tests of sintered porous copper and iron. This model inherently incorporates
isotropic hardening effects in the formulation, where the geometrical hardening and
dislocation hardening are not dissociated. In other words, the hardening effects of relative
density and dislocations are integrated in the parameters c(D) and f(D), and thus 𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐.
Accordingly, when using porous material model of Shima & Oyane, (𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐 − 𝑅) is
replaced with 𝜎𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐 which considers the combined hardening effects in the constitutive
equations presented in this study.
The normalised dislocation density rate (��), hardening rate (��), softening rate (��) and stress
rate (��) for the powder material take similar form to that for the fully dense material,
however, the effective viscoplastic strain rate of powder compact (휀��𝑝𝑣𝑝) is integrated in the
equations to consider the influence of relative density of the powder compact.
�� = 𝐴(1 − 𝜌)|휀��𝑝𝑣𝑝|
𝛿1− 𝐶𝜌
𝛿2 (22)
�� = 0.5𝐵𝜌−0.5
�� (23)
�� = 𝛽(1 − 𝜔)𝜑휀��𝑝𝑣𝑝
(24)
휀��𝑗,𝑝𝑜𝑤𝑑𝑒𝑟𝑒 = (휀��𝑗
𝑇 − 휀��𝑗,𝑝𝑜𝑤𝑑𝑒𝑟𝑣𝑝 ) (25)
��𝑖𝑗,𝑝𝑜𝑤𝑑𝑒𝑟 = 2µ휀��𝑗,𝑝𝑜𝑤𝑑𝑒𝑟𝑒 + 𝜆휀��𝑘,𝑝𝑜𝑤𝑑𝑒𝑟
𝑒 (26)
16
The Lamé parameters (µ and 𝜆) are affected by density. In fact, 𝐸 and 𝑣 for the porous
material could dynamically change during the powder forging process, as they depend on the
material’s relative density, which could change from 0.7 to 1.0 in this study. It is expected
that 𝑣 does not considerably vary with the relative density and therefore a constant value, 𝑣 =
0.319, is adopted in this work for the studied nickel-base superalloy. To consider the effect of
density on the Young’s modulus of powder compact 𝐸𝑝𝑜𝑤𝑑𝑒𝑟, the theoretical model of
Ramakrishnan and Arunachala (RA model) [39, 40] is employed:
𝐸𝑝𝑜𝑤𝑑𝑒𝑟 = 𝐸 [𝐷2
1+𝑘𝐸(1−𝐷)] (27)
𝑘𝐸 = 2 − 3𝑣 (28)
where 𝐸 and 𝑣 are respectively the Young’s modulus and Poisson’s ratio at fully dense
condition.
Within the above equations, the constants 𝑘, 𝐾, 𝐶, 𝐵 and 𝐸 are temperature-dependent
material constants, and 𝑛, 𝛾, 𝐴, 𝛿1, 𝛿2, 𝛽, 𝜑 are temperature-independent ones. The
temperature-dependent parameters are defined using the following expressions:
𝑘 = 𝑘0exp (𝑄1
𝑅𝑔𝑇) (29)
𝐾 = 𝐾0exp (𝑄2
𝑅𝑔𝑇) (30)
𝐶 = 𝐶0exp (−𝑄3
𝑅𝑔𝑇) (31)
𝐵 = 𝐵0exp (𝑄4
𝑅𝑔𝑇) (32)
𝐸 = 𝐸0exp (𝑄5
𝑅𝑔𝑇) (33)
where Q is the activation energy, 𝑅𝑔 = 8.31 J/mol K is the universal gas constant, and T is
the absolute temperature in Kelvin.
17
4. Calibration of the constitutive equations
The developed unified constitutive equations for powder and fully dense material outlined in
the foregoing section comprise a set of non-linear ordinary differential equations (ODE) for
initial value problem, which can be solved using any numerical integration method such as
Euler, Runge Kutta, etc. by giving initial values of the variables. To calibrate the unified
constitutive equations, a gradient-based scheme is adopted and the material constants are
determined by minimising the residuals of the computed and experimental data of the stress-
strain curves through a least square objective function. The determined values of the material
constants for FGH96 nickel-base superalloy and stainless steel AISI 304 are listed in Table 4
and Table 5 respectively. Figure 3 presents the computed and experimental strain-stress
results for both materials at different strain rates and deformation temperatures. As can be
seen form the figure, the computed and experimental stress data give a satisfactory match. In
order to quantify the prediction accuracy of the determined unified viscoplastic constitutive
models, the unbiased statistical parameters, correlation coefficient (CC) and average absolute
relative error (AARE), are calculated. These parameters are formulated as follows [41]:
𝐶𝐶 =∑ (𝑋𝑖−��)(𝑌𝑖−��)𝑁
𝑖=1
√∑ (𝑋𝑖−��)2𝑁𝑖=1 √∑ (𝑌𝑖−��)2𝑁
𝑖=1
(34)
𝐴𝐴𝑅𝐸 =1
𝑁∑ |
𝑌𝑖−𝑋𝑖
𝑋𝑖|𝑁
𝑖=1 × 100% (35)
where 𝑋𝑖 and 𝑌𝑖 designate the experimental and computed flow stresses respectively. �� and ��
represent the mean values of 𝑋𝑖 and 𝑌𝑖, and 𝑁 is the number of data points. For all the test
conditions, the calculated CC and AARE are 0.9526 and 8.57% for FGH96 and 0.9606 and
3.52% for AISI 304 respectively, signifying that the determined unified viscoplastic
18
constitutive equations can satisfactorily model the viscoplastic flow and physical behaviours
of FGH96 and AISI 304 in thermo-mechanical processing conditions.
Table 4 Determined material constants for FGH96 superalloy.
𝐾0(𝑀𝑃𝑎) 𝑄1(𝐽/𝑚𝑜𝑙) 𝑘0(𝑀𝑃𝑎) 𝑄2(𝐽/𝑚𝑜𝑙) 𝐶0(−)
1.711 × 10−6 2.008 × 105 4.786 × 10−8 2.247 × 105 1.888 × 103
𝑄3(𝐽/𝑚𝑜𝑙) 𝐵0(𝑀𝑃𝑎) 𝑄4(𝐽/𝑚𝑜𝑙) 𝐸0(−) 𝑄5(𝐽/𝑚𝑜𝑙)
1.291 × 104 1.29 × 102 8.367 × 103 4.748 × 104 8.018 × 103
𝛿1(−) 𝛿2(−) 𝑛(−) 𝐴(−) 𝛽(−)
0.618 1.995 9.575 8 0.757
𝜑(−) 𝛾(−)
0.806 11.303
Table 5 Determined material constants for stainless steel AISI 304.
𝐾0(𝑀𝑃𝑎) 𝑄1(𝐽/𝑚𝑜𝑙) 𝑘0(𝑀𝑃𝑎) 𝑄2(𝐽/𝑚𝑜𝑙) 𝐶0(−)
7.274 × 10−1 4.622 × 104 1.559 × 10−1 5.757 × 104 1.058 × 104
𝑄3(𝐽/𝑚𝑜𝑙) 𝐵0(𝑀𝑃𝑎) 𝑄4(𝐽/𝑚𝑜𝑙) 𝐸0(−) 𝑄5(𝐽/𝑚𝑜𝑙)
6.557 × 104 1.443 4.311 × 104 4.86 × 103 9.322 × 103
𝛿1(−) 𝛿2(−) 𝑛(−) 𝐴(−) 𝛽(−)
1.274 9.896 4.562 5.068 6.277 × 10−4
𝜑(−) 𝛾(−)
6.79 × 10−1 2.817 × 10−1
19
(a) Fully dense HIPed FGH96 superalloy
(b) Stainless steel AISI 304
Figure 3 Comparison of the computed (solid curves) and experimental (symbols) stress-strain
relationships at different temperatures and strain rates.
5. FE modelling of DPF process
3D coupled thermo-mechanical FE simulations of the DPF process are performed in order to
assess the reliability and efficacy of the developed unified constitutive equations by making
comparisons between experimental and simulation results of relative density distribution
20
across the powder compact as well as forming loads. The suitability of those six porous
material models for the DPF process of FGH96 is also discussed in this section.
5.1. Development of FE model
The developed FE model of the DPF process is demonstrated in Figure 4. Due to the
symmetrical feature of the model, only 1/8 of the full geometry is considered so as to reduce
the overall computing time. The geometry and dimensions of the container are identical to
those shown in Figure 2. The powder tube is omitted in the FE model for the sake of
simplicity, and its effect on the DPF process is expected to be trivial. The powder compact
and container are modelled as 3D deformable continuum, which are meshed with tetrahedral
elements. The viscoplastic constitutive Eqs. (1-33) with the determined material constants
listed in Table 4 and Table 5 are implemented into the commercial FE code DEFORM via
user-defined subroutine to simulate the DPF process. The geometrical nonlinearities
associated with large deformations are taken into account in the FE formulation to satisfy the
force equilibrium after incremental deformation, however, nonlinear elastic behaviour and
elastoplastic coupling are not considered. The initial relative density of FGH96 powder
compact is considered as 0.7. The friction between the powder compact and the container
wall is modelled as a friction surface interaction with a classical isotropic shear friction
model by using contact with conforming coupling. The friction between the outer surface of
the container and the H13 die is modelled using contact with penalty. The value of friction
coefficient is set as 0.3 [42-44]. The Lagrangian description is generally employed for non-
steady metal forming processes and is adopted here, with dependency on material history.
Mesh convergence studies are performed to obtain the appropriate mesh size in order to
precisely capture the non-linear material behaviour of the deformable bodies. As only the
21
heat transfer characteristics of the die are of interest, its deformation is neglected in the
simulation. Thus, the die is considered as a rigid body and a coarser mesh is selected to
decrease the simulation time. Heat transfer coefficient between the H13 die and AISI 304
container is assumed to be 20 kW/m2 K [45]. Besides, the thermal properties for AISI 304
published by Bogaard et al. [46] are adopted; for FGH96 superalloy, the thermal properties of
a similar nickel-base superalloy, IN718, are utilised. Effective thermal conductivity for the
superalloy powder compact and stainless steel container at a temperature of 1150 °C is
considered as 28.75 and 22 W/m K, respectively.
Figure 4 Geometric model for FE simulation of the DPF process.
5.2. Model validation and discussion
To evaluate the accuracy and reliability of the developed unified constitutive equations
incorporating porous material models, the relative density distribution on a longitudinal
22
cross-section of the powder compact is measured after the DPF process of P/M FGH96.
Figure 5(a) shows the comparison between the experimental data and FE simulation for the
relative density along the horizontal centre line at seven locations. As can be seen in Figure
5(a), the developed constitutive model making use of the porous material model of the
calibrated Abouaf is qualitatively consistent with the experimental relative density data,
although it overestimates the density evolution toward the outer edge of the part. The
constitutive model incorporating the porous material model of Shima & Oyane
underestimates the densification levels near the centre of the powder compact. However, its
prediction agrees excellently with experiments in the outer region of the DPFed part. The
other four porous material models of Cocks, Sofronis & McMeeking, Duva & Crow, and
Ponte-Castaneda all significantly overestimate the achieved density. Figure 5(b) depicts the
load variation with stroke during DPF process, obtained from the oscilloscope record and FE
simulations. The maximum forming load predicted by the density-based viscoplastic
constitutive equations benefiting from the calibrated Abouaf and Shima & Oyane exhibits
roughly identical match with the experimental results. Table 6 summarises the relative error
(RE) for the relative density and the forming load calculated for different porous material
models. As can be seen from Table 6, the porous material models of the calibrated Abouaf
and Shima & Oyane achieve lower relative errors for both the relative density and the
forming load, as opposed to the other studied models.
23
Figure 5 Comparison between experimental data and FE simulation for: (a) Relative density
distribution along the horizontal distance from the centre; and (b) Forming load variation
(a)
(b)
24
with stroke. The insert in 5(a) shows the final distribution of the relative density from the
calibrated Abouaf model.
Table 6 Relative error (RE) for the relative density and forming load.
Model Average RE for relative
density (%)
RE for maximum forming
load (%)
Cocks [16] 11.8 17.1
Duva & Crow [20] 11.3 16.3
Ponte-Castaneda [21] 10.6 16
Sofronis & McMeeking [38] 10.7 16.5
Shima & Oyane [14] 3.2 3.7
Calibrated Abouaf [26] 3.4 3.1
It is instructive to mention that densification of metal powders is generally considered by two
stage models, i.e., initial stage (D<0.9) and final stage (0.9<D<1). The densification at the
initial stage is modelled by the growth of necks where the particles are in contact. The
porosity which is initially interconnected is reduced progressively. At the end of stage 1,
interconnected porosity is eliminated. During final stage, the pores are pinched off, forming
an array of isolated pores, and densification is due to the shrinkage of closed pores [47]. As
mentioned earlier, the Shima & Oyane porous material model [14] is based on compaction
studies for sintered copper and iron at various apparent densities. The use of the relative
density related terms 𝑐(𝐷) and 𝑓(𝐷) in the current work may result in some errors, although
it has been successfully used for the prediction of densification of other materials, e.g.,
aluminium alloy Al6061 [48, 49], A356 [50] and stainless steel 316L [51]. This model is built
into the DEFORM and is known to be appropriate for powder compaction with D≥0.7 at
25
room and high temperatures [49, 50, 52]. Porous material models such as Cocks [16], Ponte-
Castaneda [21], Duva & Crow [20] and Sofronis & McMeeking [38] are originally valid for
the final stage of the compaction process where voids are isolated in the matrix. A central
assumption of these models is the spherical shape of the pores. For these models, the unit cell
is a void (spherical) embedded into a matrix. Accordingly, the porosity in the material is not
connected. Consequently, these models are not appropriate for the early stage of compaction
where the dominant densification mechanism is rearrangement and relative motion/rotation
between powders as well as growth of inter-particle contacts. However, heuristic
modification can be made in these models through introducing a critical porosity 𝜃0 in the
formulation to describe the loss of stress carrying capacity of the powder material when the
porosity approaches 𝜃0 [15, 53]. It is of note that the viscoplastic model of Abouaf [23, 24]
owns by construction such critical porosity. For porosity level close to the porosity of the
close packed powder material, in the porous material or aggregate of powders, the contact
zone between powder particles is limited, leading to a significant contact pressure under
limited applied external loading and also large plastic deformation in the contact area. This
effect is well described by the model of Abouaf and not by many other porous material
models [53]. The observed discrepancy between experimental and simulated data of the
calibrated Abouaf and Shima & Oyane models in this study could result from a slightly
different chemical composition of the superalloy used for the calibration of Abouaf model,
copper and iron instead of nickel alloy used for the calibration of Shima & Oyane model,
error arising from the simplified softening model, rough values of friction coefficients
defined in the FE model, etc. In addition, it can be postulated that some available porous
material models may not be quite accurate for the DPF of metal powders in which the
contribution of deviatoric stress to creep rates increases, unlike HIPing which is a
26
hydrostatic-dominative process. Furthermore, the current experimental results were obtained
at much higher strain rate than HIPing which may also lead to discrepancies in the results.
In summary, it is established that the developed unified density-based constitutive equations
integrating the calibrated Abouaf model provide the best approximation, among the six
porous material models employed in this study, to the experimental results in terms of density
distribution and forming load variation during DPF process. It overestimates the relative
density toward the outer edge of the DPFed part. On the other hand, the constitutive
equations integrating Shima & Oyane results in excellent agreement with experiments in the
outer region of the forge. Accordingly, it remains a topic of ongoing research to further refine
the models so as to attain highly accurate predictions of relative density for the DPF process.
In the following section, FE simulation results based on the calibrated Abouaf model are
reported to demonstrate the evolution of in-process variables during the DPF of P/M FGH96.
5.3.Evaluation of internal state variables
In addition to the macro deformation behaviour and powder consolidation levels, evolution of
internal state variables during DPF process can be predicted using the unified viscoplastic
constitutive equations, which are embedded into the FE model. Figure 6 demonstrates the
distribution of effective viscoplastic strain, normalised dislocation density, hydrostatic stress
and stress triaxiality factor of the powder compact after DPF process of P/M FGH96. It can
be found that the field pattern of the normalised dislocation density is somewhat similar to
that of the effective viscoplastic strain, since its increment is associated to the effective
viscoplastic strain rate, as described in Eq. (22). Further analysis reveals that, due to high
levels of stress, the centre of the powder compact undergoes larger plastic deformations, thus,
the magnitude of normalised dislocation density in this zone is higher than those of other
27
areas. The evolution pace and values of normalised dislocation density decreases from the
centre region to the outer edge of the powder compact. It is believed that when the
hydrostatic stress is compressive (negative), the porosity closure could be promoted. As seen
in Figure 6(c), the hydrostatic stress is tensile (positive) at the outer edge of the powder
compact, which may lead to low levels of densification as well as the initiation of cracks in
this region. The influence of hydrostatic stress on the void (pore) closure is usually expressed
in terms of stress triaxiality factor, 𝑇𝑋 = 𝜎𝑚/𝜎𝑒. Based on micro-analytical approaches of Rice
and Tracey [54], Budiansky et al. [55], Duva & Hutchinson [56] and Zhang et al. [57, 58],
the void volume evolution at each increment step is assumed proportional to the product of
the stress triaxiality factor and the effective viscoplastic strain (or effective strain). Thus,
stress triaxiality factor and effective viscoplastic strain have profound influence on
mechanical void closure (reducing the void volume to zero). The internal void crushing
comprises two stages: the mechanical void closure, and the final metallic bonding of internal
surfaces which provides complete healing and a dense material [59]. It is observed from
Figure 6(d) that the stress triaxiality factor is positive at the outer edge of the powder
compact, implying that tensile deformation takes place in this zone, which contributes to void
growth. Furthermore, low viscoplastic strain (~0.5) is witnessed in this area and thus high
level of powder densification cannot be obtained in this region. On the other hand, at the
central regions, high negative stress triaxiality factor and large effective viscoplastic strains
occur, indicative of high levels of void crushing and densification. In the following
paragraphs, variation of the internal state variables with time is discussed in greater depth at
four locations depicted in Figure 6(a).
28
Figure 6 Distribution of internal state variables at the end of DPF process of P/M FGH96
from the constitutive equations incorporating the calibrated Abouaf model: (a) Effective
viscoplastic strain; (b) Normalised dislocation density; (c) Hydrostatic stress; and (d) Stress
triaxiality factor.
Figure 7 displays the evolution of the internal state variables, recorded at four points
indicated Figure 7(a), during the DPF process. As seen in Figure 7(a), the relative density
rises rapidly at the location of P1 primarily due to high levels of viscoplastic strain and
negative stress triaxiality factor. However, at the locations of P2, P3 and P4, the relative
density increases gradually with the DPF time, yet it increases sharply at the final stage
(𝑡/𝑡𝑓 > 0.8) of the process. The powder compact is fully densified at the location of P1 and
P2. Further to the left of P1 towards the centre, the relative density increases faster and
reaches full density earlier than P1. P2 is actually a critical location where it just reaches full
density at the end of the DPF process. Any locations right of P2 will not reach full density in
the current conditions. From Figure 7(a) and Figure 7(b), it is recognised that a large amount
29
of viscoplastic strain (> 1.0) is needed to attain full density. Thus, sufficient effective
viscoplastic strain could accelerate densification of the powder compact during the DPF
process. The magnitudes of normalised dislocation density and isotropic hardening reach
~0.09 and ~80 MPa, respectively, as soon as the corresponding location of the powder
compact is fully densified. These values reach ~0.156 and ~97 MPa, respectively, at the
centre of the powder compact at the end of the DPF process. Notice that the magnitude of
these state variables increases continuously with the DPF time and they evolve faster towards
the centre of the powder compact. Figure 7(e) and Figure 7(f) suggest that hydrostatic stress
and stress triaxiality factor share the similar evolution pattern i.e. at the locations of P1 and
P2, hydrostatic stress and stress triaxiality factor decrease constantly with the DPF time
whereas at the locations of P3 and P4 they experience a decrease followed by a sharp increase
after 𝑡/𝑡𝑓 > 0.9. In general, the stress triaxiality factor decreases to 𝑇𝑋 ≈ −1.1 when full
density is attained. However, the stress triaxiality factor reaches 𝑇𝑋 ≈ −1.8 at the centre of the
powder compact at the end of the process. The magnitudes of stress triaxiality factor are in
the range of values obtained in the industrial processes such as hot forging and hot rolling
[60]. The values of hydrostatic stress and stress triaxiality factor at the locations of P1, P2 and
P3 remain negative during the DPF process, so improving the porosity elimination with a
higher rate. The variation of stress triaxiality factor at the location of P3 indicates that at the
final stage of the process the value of stress triaxiality factor is approximately zero. On the
other hand, the magnitude of effective viscoplastic strain is around 0.8. Therefore, full
density is not obtained at this location. Restricting the material flow using shaped-dies at the
outer edge of the powder compact would aid to achieve more compressive stress triaxiality
state at these regions and thus lead to high levels of densification throughout the powder
compact.
30
(a) (b)
(c) (d)
31
Figure 7 Evolution of the internal state variables for material model incorporating the
calibrated Abouaf: (a) Relative density; (b) Effective viscoplastic strain; (c) Normalised
dislocation density; (d) Isotropic hardening; (e) Hydrostatic stress; and (f) Stress triaxiality
factor. The insert in 7(a) shows the distribution of the final relative density and the four
positions selected for the analysis.
6. Conclusions
A set of unified density-based viscoplastic constitutive equations incorporating porous
material models has been formulated in this study to describe the complex powder
consolidation behaviour, dislocation density, isotropic hardening and flow softening of P/M
FGH96 nickel-base superalloy during the DPF process. The FE simulation results of DPF
process reveal that the developed unified constitutive equations embedding the porous
material model of the calibrated Abouaf provide the best approximation to the experimental
results among the six porous material models studied, in terms of relative density distribution
and forming load variation, while the equations integrating the porous material model of
(e) (f)
32
Shima & Oyane result in excellent agreement with experiments in the outer region of the
DPFed part. Further analysis of the FE results demonstrated that a large amount of
viscoplastic strain (> 1.0) and high levels of negative stress triaxiality factor is required to
attain full density. The stress triaxiality factor decreases to 𝑇𝑋 ≈ −1.1, when corresponding
locations of the powder compact is fully densified. The evolution pace and values of the state
variables i.e. relative density, normalised dislocation density and isotropic hardening were
found to decrease from the centre region to the outer edge of the powder compact.
The preliminary study discussed in this paper shows that unified density-based constitutive
models can be used for process optimisation and die design through exploring the
concurrently occurring phenomena i.e. consolidation and deformation mechanisms in the
DPF process. It sets a framework for integrating other fundamental deformation mechanisms
to capture the full powder compact response under hot deformation conditions. Further work
is underway to develop new sets of unified density-based viscoplastic constitutive equations
with higher accuracy which are able to model the kinetics of grain size and dynamic/static
recovery and recrystallisation of powder compact during hot plastic deformation processes.
Acknowledgement
Much appreciated is the strong support received from Beijing Institute of Aeronautical
Materials (BIAM). The research was performed at the BIAM-Imperial Centre for Materials
Characterisation, Processing and Modelling at Imperial College London.
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