1

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: zhusheng.shi@imperial.ac.uk

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

2

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

3

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

4

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

5

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].

6

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

7

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

8

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].

9

(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.

10

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

11

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:

12

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.

13

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.

14

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

9

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

15

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.

References

1. Bai, Q., et al., A study of direct forging process for powder superalloys. Materials Science and Engineering: A, 2015. 621: p. 68-75.

33

2. Wang, S., et al., Direct powder forging of PM nickel-based superalloy: densification and recrystallisation. The International Journal of Advanced Manufacturing Technology, 2016: p. 1-10.

3. Ning, Y., et al., Investigation on hot deformation behavior of P/M Ni-base superalloy FGH96 by using processing maps. Materials Science and Engineering: A, 2010. 527(26): p. 6794-6799.

4. Zhang, H., et al., Hot deformation behavior and processing map of a γ′-hardened nickel-based superalloy. Materials Science and Engineering: A, 2014. 604: p. 1-8.

5. Lewis, R.W. and A.R. Khoei, Numerical modelling of large deformation in metal powder forming. Computer Methods in Applied Mechanics and Engineering, 1998. 159(3-4): p. 291-328.

6. Khoei, A. and S. Iranfar, 3D numerical simulation of elasto-plastic behaviour in powder compaction process using a quasi-nonlinear technique. Journal of materials processing technology, 2003. 143: p. 886-890.

7. Khoei, A., M. Mofid, and A. Bakhshiani, Modelling of powder compaction process using an endochronic plasticity model. Journal of materials processing technology, 2002. 130: p. 175-180.

8. Khoei, A.R. and S. Azizi, Numerical simulation of 3D powder compaction processes using cone-cap plasticity theory. Materials & design, 2005. 26(2): p. 137-147.

9. Gurson, A.L., Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media. Journal of engineering materials and technology, 1977. 99(1): p. 2-15.

10. Kuhn, H. and C. Downey, Deformation characteristics and plasticity theory of sintered powder materials. International Journal of Powder Metallurgy, 1971. 7: p. 15-25.

11. Kuhn, L. and R. McMeeking, Power-law creep of powder bonded by isolated contacts. International journal of mechanical sciences, 1992. 34(7): p. 563-573.

12. Green, R., A plasticity theory for porous solids. International Journal of Mechanical Sciences, 1972. 14(4): p. 215-224.

13. Wilkinson, D.S. and M. Ashby, Pressure sintering by power law creep. Acta Metallurgica, 1975. 23(11): p. 1277-1285.

14. Shima, S. and M. Oyane, Plasticity theory for porous metals. International Journal of Mechanical Sciences, 1976. 18(6): p. 285-291.

15. Tvergaard, V. and A. Needleman, Analysis of the cup-cone fracture in a round tensile bar. Acta metallurgica, 1984. 32(1): p. 157-169.

16. Cocks, A., Inelastic deformation of porous materials. Journal of the Mechanics and Physics of Solids, 1989. 37(6): p. 693-715.

17. Doraivelu, S., et al., A new yield function for compressible PM materials. International Journal of Mechanical Sciences, 1984. 26(9-10): p. 527-535.

18. Park, J.J., Constitutive relations to predict plastic deformations of porous metals in compaction. International journal of mechanical sciences, 1995. 37(7): p. 709-719.

19. Duva, J. and P. Crow, The densification of powders by power-law creep during hot isostatic pressing. Acta Metallurgica et Materialia, 1992. 40(1): p. 31-35.

20. Duva, J. and P. Crow, Analysis of consolidation of reinforced materials by power-law creep. Mechanics of materials, 1994. 17(1): p. 25-32.

21. Castañeda, P.P., The effective mechanical properties of nonlinear isotropic composites. Journal of the Mechanics and Physics of Solids, 1991. 39(1): p. 45-71.

22. Sofronis, P. and R. McMeeking, Creep of power-law material containing spherical voids. Journal of Applied Mechanics, 1992. 59(2S): p. S88-S95.

23. Abouaf, M., et al., Finite element simulation of hot isostatic pressing of metal powders. International journal for numerical methods in engineering, 1988. 25(1): p. 191-212.

34

24. Abouaf, M. and J. Chenot, A NUMERICAL-MODEL FOR HOT DEFORMATION OF METAL POWDERS. Journal De Mecanique Theorique Et Appliquee, 1986. 5(1): p. 121-140.

25. ElRakayby, H., et al., An investigation of densification behavior of nickel alloy powder during hot isostatic pressing. Advanced Powder Technology, 2015. 26(5): p. 1314-1318.

26. ElRakayby, H. and K. Kim, Deformation and densification behaviours of nickel-based superalloy during hot isostatic pressing. Powder Metallurgy, 2017: p. 1-8.

27. Nicolle, C., Mise en forme de poudres de bore par Compression Isostatique à Chaud: Détermination des propriétés rhéologiques et simulation numérique du procédé. 1999.

28. Geindreau, C., D. Bouvard, and P. Doremus, Constitutive behaviour of metal powder during hot forming. Part I: Experimental investigation with lead powder as a simulation material. European Journal of Mechanics-A/Solids, 1999. 18(4): p. 581-596.

29. Lin, J. and Y. Liu, A set of unified constitutive equations for modelling microstructure evolution in hot deformation. Journal of materials processing technology, 2003. 143: p. 281-285.

30. Lin, J., et al., The development of continuum damage mechanics-based theories for predicting forming limit diagrams for hot stamping applications. International Journal of Damage Mechanics, 2014. 23(5): p. 684-701.

31. Lin, J., et al., Development of dislocation-based unified material model for simulating microstructure evolution in multipass hot rolling. Philosophical magazine, 2005. 85(18): p. 1967-1987.

32. Manière, C., et al., Identification of the Norton‐Green Compaction Model for the Prediction of the Ti–6Al–4V Densification During the Spark Plasma Sintering Process. Advanced Engineering Materials, 2016. 18(10): p. 1720-1727.

33. Lin, J., Fundamentals of materials modelling for metals processing technologies: theories and applications. 2015: World Scientific Publishing Co Inc.

34. Chaboche, J., A review of some plasticity and viscoplasticity constitutive theories. International Journal of Plasticity, 2008. 24(10): p. 1642-1693.

35. Stupkiewicz, S., A. Piccolroaz, and D. Bigoni, Elastoplastic coupling to model cold ceramic powder compaction. Journal of the European Ceramic Society, 2014. 34(11): p. 2839-2848.

36. Lin, J. and T. Dean, Modelling of microstructure evolution in hot forming using unified constitutive equations. Journal of materials processing technology, 2005. 167(2): p. 354-362.

37. Li, Y., et al., A unified constitutive model for asymmetric tension and compression creep-ageing behaviour of naturally aged Al-Cu-Li alloy. International Journal of Plasticity, 2016.

38. McMeeking, R., Creep of power-law material containing spherical voids. Stress, 1992. 1(1): p. 1.

39. Ramakrishnan, N. and V. Arunachalam, Effective elastic moduli of porous ceramic materials. Journal of the American Ceramic Society, 1993. 76(11): p. 2745-2752.

40. Ramakrishnan, N. and V. Arunachalam, Effective elastic moduli of porous solids. Journal of materials science, 1990. 25(9): p. 3930-3937.

41. Ji, H., et al., Microstructure evolution and constitutive equations for the high-temperature deformation of 5Cr21Mn9Ni4N heat-resistant steel. Journal of Alloys and Compounds, 2017. 693: p. 674-687.

42. Kim, K. and H. Yang, Densification behavior of titanium alloy powder during hot pressing. Materials Science and Engineering: A, 2001. 313(1): p. 46-52.

43. Chavoshi, S.Z., A study on the influences of Coulomb–Amonton’s and Prandtl’s constant friction laws on the hot closed-die forging process of AA7075. International Journal on Interactive Design and Manufacturing (IJIDeM), 2014: p. 1-8.

44. Zare Chavoshi, S., M. Tajdari, and X. Luo, An investigation into the effect of variable flash thickness on the strength of an AA7075 part obtained by hot closed-die forging. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2015. 229(5): p. 916-925.

35

45. Wells, W., Heat transfer aspects of nonisothermal axisymmetric upset forging. Journal of Engineering for Industry, 1984. 106: p. 187.

46. Bogaard, R., et al., Thermophysical properties of stainless steels. Thermochimica Acta, 1993. 218: p. 373-393.

47. Cho, J. and K. Kim, Densification of mixed metal powder at high temperature. International journal of mechanical sciences, 2001. 43(4): p. 921-933.

48. Yang, H., J. Lee, and K. Kim, Rubber isostatic pressing of metal powder under warm temperatures. Powder technology, 2004. 139(3): p. 240-251.

49. Wu, Y. and G.-Y. Kim, Compaction behavior of Al6061 powder in the semi-solid state. Powder technology, 2011. 214(2): p. 252-258.

50. Kang, C. and H. Jung, Finite element analysis with deformation behavior modeling of globular microstructure in forming process of semi-solid materials. International Journal of Mechanical Sciences, 1999. 41(12): p. 1423-1445.

51. Kim, H.S. and D.N. Lee, Power-law creep model for densification of powder compacts. Materials Science and Engineering: A, 1999. 271(1): p. 424-429.

52. Kang, C.S., et al., Densification behavior of iron powder during cold stepped compaction. Materials Science and Engineering: A, 2007. 452: p. 359-366.

53. Wolff, C., et al., Modeling of conventional hot compaction and spark plasma sintering based on modified micromechanical models of porous materials. Mechanics of Materials, 2012. 49: p. 72-91.

54. Rice, J.R. and D.M. Tracey, On the ductile enlargement of voids in triaxial stress fields∗. Journal of the Mechanics and Physics of Solids, 1969. 17(3): p. 201-217.

55. Budiansky, B., J. Hutchinson, and S. Slutsky, Void growth and collapse in viscous solids. 1982: Pergamon Press, London.

56. Duva, J. and J. Hutchinson, Constitutive potentials for dilutely voided nonlinear materials. Mechanics of materials, 1984. 3(1): p. 41-54.

57. Zhang, X.-x. and Z.-s. Cui, Theoretical study of void closure in nonlinear plastic materials. Applied Mathematics and Mechanics, 2009. 30(5): p. 631-642.

58. Zhang, X.-X., et al., A criterion for void closure in large ingots during hot forging. Journal of Materials Processing Technology, 2009. 209(4): p. 1950-1959.

59. Park, C. and D. Yang, A study of void crushing in large forgings I: Bonding mechanism and estimation model for bonding efficiency. Journal of materials processing technology, 1996. 57(1-2): p. 129-140.

60. Saby, M., Understanding and modeling of void closure mechanisms in hot metal forming processes. 2013, Ecole Nationale Supérieure des Mines de Paris.