Elastoplastic–viscoplastic modelling of metal powder compaction: Application to hot isostatic pressing

Abstract

A constitutive model is proposed for the densification of metal powder during hot isostatic pressing. The model considers an inelastic deformation resulting from time dependent (viscoplastic) and time independent (plastic) mechanisms during loading. With employing the Abouaf's formalism for viscoplastic part, the paper is focused mainly on the plasticity contribution including the hardening effects of both relative density and the equivalent plastic strain. The proposed plastic–viscoplastic model is summarised to an equation expressing the densification rate under hydrostatic loading. The gas atomised 316LN stainless steel powder was used in this study. Model parameters necessary for simulation of HIP process during a given pressure ramp at constant temperature are identified using previously published data. The identified model is then verified using experimental data obtained from HIP trials performed at another fixed temperature but with varied pressure ramp rates. A good agreement was found between the model prediction and the experimental results.

Keywords

HIP Modelling Plastic Viscoplastic Densification Powder

Introduction

Because of its advantages such as energy and material savings, modelling thermomechanical behaviour of materials during fabrication process has increased during recent decades. There has been considerable interest to model the process of hot isostatic pressing (HIP) in the field of powder metallurgy. However, complex HIP cycles with different temperatures and pressure ramp rates are common within industrial practices. These facts as well as the presence of can resistance preventing a perfect and uniform transfer of applied pressure and temperature to the powder make it difficult to obtain a simple analytical relationship between HIP parameters such as pressure and temperature and the results of material deformation such as compact density or dimensional changes. This is why a modelling work on material behaviour consists usually of constitutive equations for both powder and the can surrounding powder. The constitutive equations are then integrated in a numerical procedure such as finite element analysis in order to predict the material deformation during the HIP cycle. However, it is evident that some valid constitutive models must be used in the numerical procedure in order to obtain reliable results.

There are two approaches that are commonly used to model the compaction of metal powder. The first one is a rate independent plastic modelling that is appropriate for cold compaction processes such as metal powder compaction in a closed die at ambient temperature and the second one is the rate dependent viscoplastic modelling that is applicable for powder compaction processes at relatively high temperatures such as the holding stage in a HIP process. In other words, the inelastic deformation mechanisms are considered to have a rate independent plastic nature in the former and a rate dependent viscoplastic nature in the latter. The work reported for modelling of the compaction of metal powder is essentially classified in these two categories.

For modelling the plastic behaviour of metal powders, it is possible to consider a yield function as an extension of von Mises yield criterion. The extension is made by taking into account the effect of hydrostatic pressure on materials’ inelastic deformation in addition to the second invariant of stress deviator. In this category called as porous material model, several models have been introduced.^1–5 This type of modelling may not be useful for the initial stage of compaction where the particle rearrangement and not the particle plastic deformation is the main mechanism of densification.⁶ In fact, for the initial stage of compaction, we deal with a granular material and we should thus use a granular material model. Brown and Abou-chedid⁷ suggested the usage of two-mechanism models to cover the whole range of powder compaction. An improvement in the modelling works is the consideration of strain hardening of matrix material into account.^8,9 It is interesting to note that in the work of Sun et al.⁸ the model of Shima and Oyane has been used with considering the hardening effects of matrix material although the original version of Shima and Oyane's does not consider these effects. As the model of Gurson and that of Fleck et al.⁹ are appropriate for low and high porosity ranges respectively, Redanz¹⁰ has used a combination of two models to simulate the whole cycle of compaction from high to low porosities with considering a power law behaviour for matrix material. It was seen that hardening of matrix material could have a significant influence on the evolution of applied load as well as the stress in the piece.

For modelling rate dependent (viscoplastic) deformation of metal powder, many works have been previously reported. Following Cassenti,¹¹ Abouaf et al.¹² presented a constitutive modelling applicable to hot isostatic pressing of metal powders. In fact, the Norton power law creep of matrix material¹³ was extended to the case of porous media in the work of Abouaf et al.¹² By considering a modified form of hyperbolic sine equation proposed by Raboin¹⁴ for the matrix material and using the model suggested by Abouaf et al.,¹² Svoboda et al.¹⁵ simulated HIP of metal powders. Bouvard¹⁶ and Bouvard and Ouedraogo¹⁷ have modelled the HIP process with considering power law creep for the matrix material. The behaviour of metal powders has also been considered based on a diffusional creep mechanism occurring on the interparticle contacts.¹⁸ Storakers et al.¹⁹ consider the early stage of cold and hot compaction by a viscoplastic modelling. They consider a viscosity hardening equation for uniaxial deformation of matrix material²⁰ making it possible to be used in a large range of temperatures. Govindarajan and Aravas,²¹ Aryanpour,²² Wikman et al.²³ have also applied plastic–viscoplastic modelling to describe materials’ inelastic deformation in HIP process. They assume in fact an inelastic deformation due to simultaneous effects of plasticity and viscoplasticity.

The present work describes a phenomenological constitutive modelling in which the inelastic deformation is proposed to consist of rate independent plastic and rate dependent viscoplastic parts. It is thus classified in the plastic–viscoplastic category. After presenting the general constitutive equations, the expression applicable to the pressure ramp of HIP process at a constant temperature is derived. Integration of this expression results in density as a function of time during the pressure ramp. The parameters in this equation are identified using data from prior HIP experiments carried out at the pressure ramp stage on 316LN stainless steel metal powder. Then, the model predictability is verified against new HIP experiments carried out at different pressure ramp rates other than the one utilised to identify the model parameters.

Modelling

General concept

As stated previously, an elastoplastic–viscoplastic modelling is introduced in this work. The isotropic Hook elasticity is assumed for elastic behaviour of the material. However, the inelastic deformation is decomposed into plastic and viscoplastic parts. From a physical point of view, one can consider that there would be permanent deformation mechanisms whose characteristic time is much less than the loading time and in this case the time independent plastic deformation occurs. On the other hand, it could be considered that there are some other permanent deformation mechanisms with characteristic time comparable with the loading time.²⁴ Presence of these mechanisms leads to the time dependent deformation. The coexistence of time dependent and time independent mechanisms could be apparent when we think of dislocation slip and disclocation creep as two phenomena being active simultaneously in material deformation in the loading condition at relatively high temperatures. Dislocation slip and dislocation creep are considered as two sources of plastic and viscoplastic behaviours respectively. After this assumption, two general points are remarked:

(i)

a plastic problem can be usually treated by a classical approach including yield criterion. However, care must be taken when dealing with plastic deformation simultaneously with viscoplastic deformation. This subject will be further discussed in the subsequent sections

(ii)

different assumptions can be made for viscoplastic behaviour.^25,26 However, in this work, we consider secondary creep as the most important contribution of time dependent inelastic deformation in comparison with primary and tertiary creeps.

In the following sections, first the viscoplastic and plastic modelling approaches are described separately and then the plastic–viscoplastic modelling that is the main topics of this work will be presented.

Viscoplastic modelling

Depending on the material, temperature and loading conditions, it can be considered that all of the inelastic deformation in material occurs due to viscoplastic behaviour. For example, during the holding stage in a HIP process, a viscoplastic model such as Abouaf's model is adequately appropriate to describe material deformation. In the following a brief review will be done on this model. In fact, Abouaf's model is an extension of Norton power law creep model for the porous material. The Norton power law used to describe secondary creep of dense material in a uniaxial test is given by the following equation 1 where d^vp is the axial deformation rate, and σ is the axial stress. A and n are two material parameters that are dependent on temperature. Based on Norton power law for uniaxial behaviour, Odqvist introduced a viscoplastic dissipation potential that is a generalisation of Norton law to three-dimensional loading 2 where Ω is the viscoplastic dissipation potential, and σ_eq is an equivalent stress. To generalise the Odqvist's work to porous material, Abouaf et al.¹² considered the following equivalent stress 3 Therefore, the viscoplastic dissipation potential for porous material is given as the following equation 4 where S₁ is the first invariant of the stress tensor applied on the material point and is an invariant of the stress deviator tensor. If is the Cauchy stress tensor, we have 5 6 where is the stress deviator tensor given by 7 is the Kronecker delta, and f and c are two functions of relative density of the material and are defined as the following equations 8 9 where a, b, a′ and b′ are constants. Relative density ρ is the ratio of porous material density to the density of completely dense material and ρ₀ is the relative density of the loose powder including point-like contacts between powder particles. The viscoplastic deformation rate tensor can be obtained by the following equation 10 Applying equation (4) in equation (10), the following expression can be obtained 11

Plastic modelling

Plasticity of non-porous material in uniaxial test

Consider the elastoplastic behaviour in uniaxial test. A common formalism to describe material behaviour is the Ramberg–Osgood's equation 12 where 13 and 14 ϵ^e and ϵ^p are the axial elastic and plastic strains respectively, and k and m are two material parameters defining plastic behaviour and depending on temperature. It is important to note that no elastic limit is considered for the material in the above treatment. In other words, it is considered that plastic behaviour occurs from the beginning of loading. This assumption is based on the observation that at certain temperature and strain rate, the elastic limit cannot be clearly defined from the uniaxial stress–strain curve. The reasoning behind this observation is not only due to the precision of measuring system but also the fact that microplasticity occurs at microscales just after onset of loading and before any measurable macroscopic plasticity is produced. In this work, this assumption of onset of plastic deformation from the beginning of the loading is employed to treat the plasticity of porous material in subsequent sections.

Plasticity of porous material

If we consider deformation of the material at relatively low temperatures, the rate independent mechanisms are basically responsible of material's inelastic deformation. In such cases, with maintaining loading condition, the classical plasticity theory is applied so that the following concepts are introduced to calculate plastic deformation rate tensor:

(i)

yield criterion or yield function that defines a surface in the stress space. If the state of stress is found on the yield surface then the plastic deformation will occur provided that the loading condition is conserved. The yield criterion is shown by 15 where F is a function of stress tensor and hardening parameter

(ii)

the consistency condition states that the variation rate of F is zero during plastic deformation 16

(iii)

an associated flow rule is usually applied to calculate the plastic deformation rate tensor of porous metal 17 where is the plastic multiplier.

The main task among the three steps stated above is to introduce a yield function (F = 0). Based on physical phenomenon occurring during compaction of the metal powder, the plastic deformation of metal particles could result in the flattening of interparticle contacts leading to the material densification. This point is schematically shown for deformation of two spherical particles in Fig. 1.

Flattening of particle contact due to plastic deformation

Moreover, based on the materials engineering concepts such as dislocation glide and thus dislocation generation during plastic deformation, metal particles will be strain hardened during compaction. Hence, work or strain hardening of particles and increasing material density are two simultaneous phenomena when plastic deformation of metal particles is considered. Therefore, material relative density ρ and the equivalent plastic strain could be considered as two isotropic hardening parameters for plastic deformation. In other words, porous material strengthening (work hardening) is influenced by both relative density and equivalent plastic strain . Therefore, the simultaneous application of Green's porous metal plasticity formalism and Ramberg–Osgood's power law formalism in the yield function enables us to take into account both effects of relative density and equivalent plastic strain respectively in material hardening. The following yield function is thus introduced 18 where H and G are again two functions of relative density. We consider H and G with mathematical forms similar to f and c respectively, but with different constants 19 20 It is seen that for some given values of and ρ, the yield function defines an ellipse in the plane of versus S₁. Aryanpour and Farzaneh²⁷ have studied porous material plasticity based on the equation (18). They have considered different situations for the initial material strain hardening state and relative density. As example, one can consider the situation that there are point-like contacts between particles and the initial stain hardening of particles is null. This situation means that the initial relative density is equal to ρ₀ and the initial equivalent plastic strain is equal to zero. In this case, the yield function (equation (18)) is obviously valid from the beginning of loading. With considering constant temperature, applying equation (16) results in 21 where H′ and G′ are dH/dρ and dG/dρ respectively. and are the rates of S₁ and respectively resulted from the Jaumann rate of Cauchy stress and is the variation rate of relative density due to plastic deformation. From equation (17), we can write 22 can be obtained from the principle of mass conservation 23 Equations (22) and (23) give therefore 24 in equation (21) is the variation rate of and can be written as 25 where the operator ‘:’ means the scalar product of two tensors. Applying equation (22) in equation (25) gives thus 26 Applying equations (24) and (26) in equation (21) results in 27 Applying equation (27) in equation (22) yields therefore 28 Equation (28) is the explicit presentation of deformation rate tensor applicable to a stress driven deformation process.

If elastic deformation occurs along with the plastic one, we can write 29 where and are the elastic and elastoplastic deformation rate tensors respectively. We can also write as the following equation 30 The consistency condition ( ) can be formulated in another way 31 From equations (18) and (30), we can write 32 From equations (18) and (24), we can write 33 From equations (18) and (26), we can write 34 Three terms in the equation (31) are replaced with equations (32)–(34). The result is 35 From equation (29), we have 36 or 37 We can also write 38 or 39 If equations (37) and (39) are applied in equation (35), an expression is obtained from which the plastic multiplier is calculated as follows 40 From equation (22), we have 41 Equation (41) is another expression of deformation rate tensor applicable for deformation driven processes.

The elliptical curve characterised by some values of ρ and shows the position of yield surface in the plane (Fig. 2). It is considered that the yield surface during an elastoplastic deformation process is expanded due to the increase in relative density and material equivalent plastic strain that are considered as the isotropic hardening parameters.

Schematic illustration of elliptical yield surface and plastic deformation rate

Plastic–viscoplastic modelling

A typical HIP cycle may often involve some large variation of temperature and pressure. For example, a HIP process including ramp stages starting from low values of pressure and temperature and ending to some high values of these parameters is considered. In such situations, material's inelastic deformation may be described by a plastic–viscoplastic approach. According to physical phenomena occurring during deformation, it is comprehensible that certain deformation mechanisms may be so rapid (i.e. plastic mechanisms) in comparison with loading times applied in practice and other deformation mechanisms may be active in slower rates (i.e. viscoplastic mechanisms). Therefore, we adopt a plastic–viscoplastic approach in this work. Aryanpour²² has previously applied such a modelling in which Abouaf's model and Green's model were selected to describe viscoplastic and plastic parts of deformation respectively in a way that material relative density is the unique scalar hardening parameter. However, in this work the plastic model is modified with inclusion of equivalent plastic strain as an additional hardening parameter. The general expression of deformation rate applicable to any loading path will be presented in the following. To obtain the necessary formulation, the deformation rate tensor at a material point is assumed to be the sum of two functions 42 such that 43 44 and are supposed to be two homogeneous tensorial functions such that is of order null with respect to time and is of order n with respect to . Therefore, and can be considered as two functions describing rate independent elastoplastic and rate dependent viscoplastic deformation rate components respectively 45 46 Some important points to consider in the definition of functions and are as follows:

(i)

parameter ρ in both and is the relative density resulting from both plastic and viscoplastic deformations

(ii)

since Abouaf's model as an extension of secondary creep is used to model viscoplastic deformation, it is thus considered that the strain resulting from viscoplastic deformation does not contribute towards material's strain hardening in plastic behaviour. In other words, only the plastic contribution of equivalent inelastic strain, i.e. is considered in addition to the material relative density to describe plastic behaviour (see equation (43)).

From above definitions of function and , the Abouaf's model explained in the section on ‘Viscoplastic modelling’ and the plastic model explained in the section on ‘Plastic modelling’ are selected to formulate the elastoplastic–viscoplastic modelling. We can thus simply use equation (11) in order to calculate the viscoplastic part of deformation rate. However, for the plastic part of deformation rate, some issues must be resolved in order to obtain the explicit equation of plastic deformation rate. In fact, an important consequence of considering simultaneous effects of plasticity and viscoplasticity is to obtain some assumed values of relative density and equivalent plastic strain at a stress state that is located within the interior of the yield surface characterised by the assumed values of relative density and equivalent plastic strain. Figure 3 illustrates this concept where the values of relative density and equivalent plastic strain are obtained at a stress state shown by point B within the interior of the elliptical yield surface defined by equation (18). Concept of onset of the plastic deformation from the beginning of material loading instead of an elastic limit analysis for dealing with plastic behaviour has been previously suggested.^22,25,28 We follow the same concept in this work and make this hypothesis that even in the condition of point B located within the interior of yield surface (Fig. 3), a plastic deformation occurs at point B in the condition that the loading condition is satisfied. In the following, the expression of deformation rate tensor at a point such as B will be presented. For this aim, consider the following points that are also illustrated in Fig. 3:

Schematic illustration showing plastic deformation rate at point B located within interior of yield surface with considering elastoplastic–viscoplastic behaviour for material

(i)

the point B* shows a stress state located on the yield surface and obtained from radial projection of point B on the surface

(ii)

as the position located on the yield surface is shown by an asterisk, equation (38) can be rewritten as follows where an asterisk is used to define the parameters at point B* 47 At this step, we should determine the expression of plastic deformation rate at point B that is denoted as . As the same ratio is conserved between the stress components at B and B*, one may make the hypothesis that is proportional to by a proportionality factor ψ such that is defined by equation (47). Therefore 48 or 49 It is considered in fact that with assuming a deformation driven process, the plastic deformation rate at point B is proportional to the corresponding value at a stress point located on the yield surface obtained by radial projection (point B*). It should be pointed out that equation (49) is written with considering viscoplastic deformation besides elastoplastic deformation. Therefore, the term in equation (49) could be replaced with the following equation 50 where is the total deformation rate tensor.

Application of plastic–viscoplastic modelling

The HIP process is used for compaction of metal powders by simultaneous application of temperature and isostatic pressure on the parts. Typically, pressure and temperature are applied at a certain ramp rate until desired values are reached. The densification of metal powder during this ramp stage may be important and should thus be accounted in the approach being used to model densification during HIP. In this study, the HIP cycles were carried out such that the pressure is ramped with certain ramp rate at a constant high temperature. This allows the decoupling of the simultaneous effect of pressure and temperature by keeping the temperature constant while increasing the pressure as illustrated schematically in Fig. 4. For the ramp stage presented in Fig. 4, it is assumed that during each time increment, a time independent plastic deformation occurs due to increase in pressure (pressure increment), whereas a time dependent viscoplastic deformation occurs due to the applied hydrostatic pressure, simultaneously at the constant high temperature.

Pressure ramp applied at relatively high temperature may be part of different HIP cycles

If the container used for powder encapsulation is very thin and the applied temperature is relatively high, one may assume a perfect transfer of hydrostatic pressure to the powder by neglecting the container resistance. Hence we can apply the plastic–viscoplastic model to describe metal powder inelastic deformation under an increasing hydrostatic pressure at a constant high temperature (Fig. 4). To apply the general equation, i.e. equation (49), for the particular case of pressure ramp in HIP process, we transform first equation (49) to its equivalent expression applicable for stress driven processes. To realise this task, we follow a mathematical procedure inversing to what was explained from equation (28) to equation (41). The final result is 51 For hydrostatic loading where is null, equation (51) changes to the following equation 52 As seen in this equation, the plastic deformation rate is given as a function of the rate in S₁ ( ) where ψ could be assumed as below 53 Moreover, equation (18) is summarised to the following equation since is null 54 If the normal component of plastic deformation rate is designated as in hydrostatic loading, one can write 55 and 56 Using equation (52) to calculate and then applying equation (55) result in 57 From equation (11) the viscoplastic deformation rate tensor is calculated and we have 58 If the inelastic deformation rate tensor is indicated as , one can write 59 so that 60 where is the rate of change of relative density. In hydrostatic loading we have 61 62 where P and are the pressure and pressurisation rate respectively. Using the equations (57)–(60), we can thus write 63 Equation (63) states the densification rate as a function of P and . This is the useful expression resulted from the suggested plastic–viscoplastic modelling that can be applied to describe the powder densification during a pressure ramp at a given temperature. This equation is solved numerically to obtain the evolution of density as a function of time at given temperature and pressure ramp rate.

Model identification

Assuming that 316L and 316LN stainless steel have similar themomechanical behaviour, the comprehensive experimental data published in the literature^22,29,30,31 for these two materials can be used to identify following parameters.

Young's modulus E and Poisson's ratio ν of porous material

The Poissons’ ratio is considered to be equal to 0·33 in this work. However, the following expression is considered for E as a function of realtive density 64 where E_dense is the Young's modulus of dense material³¹ 65 where T is the temeprature in Kelvin.

k and m

Uniaxial tests on 316L stainless steel between 450 and 600°C³⁰ show that the material behaviour in this temperature range is essentially elastoplastic. In fact, no dependence on deformation rate was seen in the material uniaxial test in this temperature range. It is thus possible to apply the Ramberg–Osgood's equation to describle elastoplastic uniaxial behaviour and thus the following parametric equations are obtained 66 67 where T is the temperature in Kelvin. In the following, we use these parametric equations to approximate the values of k and m at other temperatures.

A and n

From uniaxial tests on 316LN stainless steel, the parameters A and n were found for this material for temperatures above 980°C.³¹ The following parametric equations on the experimental values of A and n enable us to estimate the values of these parameters at other temperatures²² 68 69 where T is the temperature in Kelvin.

Function f

From densification experiments on 316LN stainless steel powder interrupted at different times of the holding stage in HIP tests, function f is estimated as^22,31 70 With respect to equation (63), all model parameters are identified except the function H. In the following section, the identification of this function is explained.

Function H

Experimental points obtained from HIP experiments on 316LN stainless steel in a pressure ramp of 2·8 MPa min⁻¹ at 1125°C are shown by points in Fig. 5.²² As mentionned before, we can neglect the effect of container and thus equation (63) can be used to find the values of H. Considering ρ₀ equal to 0·64, function H was basically estimated by a trial end error approach as 71 It is seen in Fig. 5 (with positive sign for pressure) that the results predicted by equation (63) are close to the experimental points. Data published in the literature from HIP experiments on 316LN stainless steel powder under 0·5 and 1 MPa min⁻¹ at 1125°C (Ref. 31) are then used to verify the identified model parameters as shown in Fig. 6. This figure shows that the model prediction using equation (63) is in good agreement with the experimental data.

Comparison of experimental results²² from HIP trials on 316LN stainless steel powder at 1125°C with pressurisation rate of 2·8 MPa min⁻¹ (points) and simulation result (curve)

Verification of identified model parameters against experimental results from HIP trials³¹ on 316LN stainless steel powder at 1125°C with pressurisation rates of a 0·5 MPa min⁻¹ and b 1 MPa min⁻¹

The validity of the proposed model is reinforced using experimental data obtained from HIP trials on 316L stainless steel powder at various pressure ramp rates and at temperature of 1115°C, as outlined in Table 1. For these HIP trials, the powder was encapsulated in a cylindrical, low carbon steel container with an internal height of 76·2 mm and an inner diameter of 17·3 mm. The container thickness was 0·89 mm. While using two or three samples for each trial, a low pressure about 0·2 MPa of argon was applied during the time that samples were being heated to 1115°C. Then, at 1115°C, a pressure ramp was applied in order to achieve a final pressure. The parameters used for each trial are reported in Table 1. As seen in this table, the pressurisation rate had a different value in each trial. After the final pressure was obtained, the furnace power was cut and the chamber pressure was released as quickly as possible. It is thus assumed that no significant densification occurs in the cooling stage. Ignoring the effects of the container, these HIP trials were also simulated by applying equation (63). It should be pointed out that the following values were selected for the temperature of 1115°C according to equations (65)–(69). 72 As seen in Fig. 7, the simulation and experimental results are in good agreement. It should be pointed out that parameters α₁ and β₁ in function H might depend on ρ₀;²⁷ however, for the powders used in this work, one with ρ₀ = 0·64 ( 5 Figs. 5 and 6) and the other with ρ₀ = 0·6684 (Fig. 7), the values of α₁ and β₁ were approximated constant equal to 20 and 1·3 respectively.

Relative density attained during five different pressure ramps Initial point (ρ = 0·71) identifies the densification due to sintering during the heating portion of the experiments. In this work, ρ₀ is equal to 0·6684.

Table 1

Results of HIP during pressure ramp at 1115°C

Pressurisation rate/MPa min⁻¹	Maximum pressure/MPa	Relative density (measured)	Relative density (calculated)	Percent difference (measured versus calculated)
1·8	2	0·728	0·720	1·1
3·3	7	0·805	0·778	3·3
1·0	18	0·916	0·893	2·5
0·9	24	0·942	0·931	1·2
1·3	30	0·949	0·944	0·5

At the end of this discussion, it should be pointed out that although from a mathematical point of view, a relatively large number of parameters were needed to obtain simulation results adequately close to the experimental values, knowing their physical significance in the model allowed us to identify these parameters using well known experimental routes available in the literature.

Conclusions

Considering that the inelastic deformation behaviour of powder materials is governed by plastic and viscoplastic mechanisms, an elastoplastic–viscoplastic model was proposed for densification of metal powders during compaction process such as hot isostatic pressing of metal powders. In this model, relative density and equivalent plastic strain are considered as two isotropic hardening parameters influencing the plastic component of the model while the viscoplastic component is uniquely influenced by relative density. After the derivation of a suitable expression for powder material's densification under hydrostatic loading, the model parameters were identified using previously published data and also from HIP trials on 316LN stainless steel at 1125°C with a pressure ramp of 2·8 MPa min⁻¹. The model predictions are then compared with previously published results from two other series of HIP trials at pressure ramps of 0·5 and 1 MPa min⁻¹. Finally, a test programme of HIP trials was outlined and carried out on 316L stainless steel powder at 1115°C with different pressure ramps and it was found that the simulation results were in good agreement with the experimental data. The model must be applied for more complicated loading paths including some deviator components. Some experimental routes should be outlined to identify other model parameters important in deviatoric loading.

Footnotes

Acknowledgements

The authors gratefully acknowledge collaboration, support, and in kind contributions from Bodycote HIP, Inc., Andover, MA, USA and the research group REGAL at Université Laval, PQ, Canada.

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