simulation of hot isostatic pressing of a powder metal component with an internal core

ELSEVIER Comput. Methods Appl. Mech. Engrg. 148 (1997) 299-314

Computer methods in applied

mechanics and engineering

Simulation of Hot Isostatic Pressing of a powder metal component with an internal core

,41es Svoboda" , Hans-Ake Hiiggblad, Lennart Karlsson Department of Mechanical Engineering, Luled University of Technology, S-971 87 Luled, Sweden

Received 3 April 1996; revised 21 August 1996; accepted 26 August 1996

Abstract

This paper presents a finite element simulation of the thermomechanical phenomena occurring during Hot Isostatic Pressing (HIP) of a powder metal component which includes a graphite core. The thermomechanical coupling is achieved in a staggered step manner. The staggered step approach considers the coupled thermomechanical response of solids, including nonlinear effects in both the thermal and mechanical analyses. The creep behaviour of the powder material during densification is modelled using the constitutive equations of thermal elasto-viscoplastic type with compressibility. The various mechanical material properties are assumed to be functions of temperature and relative density. The mechanical solution also includes large deformation and strains. The thermal problem includes temperature and relative density dependent specific heat and thermal conductivity. The constitutive equations and relations for thermal characteristics are implemented into the implicit nonlinear finite element code, PALM2D.

The simulation of the HIP process of a component with internal core is chosen as an application example. The component, injection molding tool, is produced of a hot isostatically pressed stainless tool steel with an internal cavity which is achieved by inserting a graphite colre into the HIP container. To verify the result of the simulation, the geometry of the capsule and the coated core are measured both before and after pressing using a computer controlled measurement machine (CMM). The measured geometry is compared with the simulated final shapes of the container and internal core. A computer-aided concurrent engineering system (CACE) is used for the complete manufacturing process from the design of the component and finite element simulation to the inspection of the final geometry.

1. Introduction

The powder metallurgy (P/M) f orming processes promote savings in energy and material in addition to the quality improvement. Advanced P/M technology is highly competitive with conventional manufacturing processes such as die casting, forging or machining. One of these P/M processes, the Hot Isostatic Pressing (HIP), is a technique used for manufacturing of components with complex geometry to near net shape (NNS). In the HIP process, the metal powder mass is packed into metallic container of the desired shape. The out-gassed and sealed container is submitted to the simultaneous application of isostatic pressure through an argon gas medium (about 100 MPa) and an elevated temperature usually higher than 0.6T,,, for several hours of duration, T,,, being the melting temperature of the metal comlposing the powder. Due to the simultaneous application of heat and pressure, the porosity in the material is eliminated and the powder is compacted into a fully dense metal. A porous material is define’d as a powder packing consisting of voids and solid material. The physical properties of porous material1 are assumed to be dependent on relative density which is defined as the ratio of the density of metal .matrix including voids to the density of the void-free matrix material.

* Corresponding author. Phone: +46 920 91824; Fax: +46 920 97288; e-mail: [email protected]

0045-7825/97/$17.00 @ 1997 Published by Elsevier Science S.A. All rights reserved PII SOO45-7825( 96)01180-2

300 A. Svoboda et al. I Comput. Methods Appl. Mech. Etlgrg. 11X (1997) -799-313

During the densification process the volume of the component decreases by about 30-40%. Due to effects of container rigidity and non-uniform distribution of the temperature and relative density, the final shape and size of the component can differ from the required shape and size. In many cases of HIP applications the final shape of the product is mainly determined by the geometry of a container. In our application example, an injection molding tool, the final shape is also influenced by the graphite core. It is important to obtain HIP products with NNS geometry, especially in the case of less machinable materials, in order to reduce costs for extra machining.

It is a hard problem for the designers to predict the size and shape of a container in order to reach the required geometry of a component. Recent advances in computer-aided concurrent engineering technology facilitate the development of special-purpose CACE systems. In these systems important tools such as computer aided design (CAD), finite element analysis (FEA) and computer aided manufacturing (CAM) are integrated (see [1,2]). Using the special-purpose CACE system makes it easier for the designer to choose effectively the initial form of the container for HIP. The numerical simulation of HIP process using the finite element method (FEM) aims at the prediction of the final geometry. An intense scientific activity in the field of finite element modelling of the HIP process has resulted in many papers presenting different simulation models. One of the earliest contributions in this field is the work by Cassenti [3]. Recent developments include the papers of Nohara et al. [4] and Abouaf et al. [5,6] for superalloys and Besson and Abouaf [7] for ceramics. A simulation of HIP based on the framework of a mixed formulation was presented by Jinka and Lewis [8].

Our approach to model the densification of the powder is similar to the one presented in [S]. We use the rate form constitutive equations of thermal-elastoviscoplastic type with compressibility. Our model uses modified power-law breakdown deformation mechanism to predict the deformation rates during the HIP process. For the successful simulation of HIP manufacturing to NNS geometry by means of FEM, the constitutive equations have to be based on proper experimental results. The set of material parameters for a given material in the numerical simulation of metal powder HIP process is extensive. In order to model the behaviour of the powder, a fitting of the parameters to experimental data has to be made. Incompressible constitutive equations are used for the container material and for the graphite core. The constitutive model considered for the container is thermal-elastoplastic and for the core thermoelastic. The proper coupling between the mechanical deformation and heat transfer is achieved in a staggered step manner.

An important stage of our approach to manufacturing simulation is the verification of the simulated geometry in comparison with the final geometry of HIP product which is extracted using a coordinate measurement machine (CMM). The simulation of manufacturing provides then important information about the final form of the product as well as the intermediate stages of HIP process. Moreover, it contributes to a better understanding of the HIP process and to optimization of process parameters, which is essential for successful industrial application of NNS technology.

1.1. Present study

The area of applications of the HIP process is widened to many industrial materials including tool steel, superalloys, metal matrix composites and toughened ceramics. This innovative technique is not only used by manufacturers of high-technology products as aeronautical engines. The hot isostatically pressed injection molding tools can be applied for example in mass production of polymer components. The manufacturing simulation of such a tool with internal cavity is the aim of the present study. The use of a graphite core is suitable in order to achieve an internal cavity with relatively large volume. The core material requires a surface coating to enhance release of the core and prevent carbon diffusion between the core and surrounding powder material during pressing. In our application the graphite core was coated with 50-100 pm sol-gel layer. The sol-gel was spray-painted on the surface of the core and subsequently dried in air and sintered in an inert atmosphere at 900°C. Details about the constituency of the sol-gel mix and the coating technology are presented in [9]. The coated core was mounted inside the HIP container which was then filled with APM 2390 which is a hot-working stainless steel powder material. The container was subsequently outgassed, sealed and submitted for the pressure 100 MPa

A. Svoboda et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 299-314 301

and temperature 1150°C for 1.1 h of dwell time. After HIP the compacted component was sectioned and relevant dimensions of the product was measured using CMM.

2. Formulation

Simulation of the HIP process requires a coupled thermomechanical analysis. Strong thermomechanical coupling arise s in problems where the mechanical response affects the solution of the thermal problem. During tlhe HIP process two sources are contributing to the thermomechanical coupling. The mechanical deformations governed by creep mechanisms are strongly dependent on temperature in addition to relative density. The heat generation due to plastic work dissipation is the other contributing factor to the thermomechanical coupling. The thermomechanical contact effects are also included. Following these aspects the coupling is achieved in a staggered manner. At each time step increment the staggered step formulation considers heat generation due to mechanical plastic deformation incorporated into the energy equation and energy equation resolved on the current deformed geometry, see [lo].

Standard notation is used throughout. Dots denote time differentiation and indices denote Cartesian components and range over the number of spatial dimensions. The convention of summation of repeated indices is used.

2.1. The energy equation

The heat transfer through a loose powder packing is slower than that through a solid body. The thermal characteristics of the powder material, e.g. thermal conductivity and heat capacity are dependent on relative density as well as temperature. In powder materials, heat transfer takes place by conduction through the base material and by convection and radiation through the pores. Radiation can be neglected by low temperature. Convection is also negligible when the size of the pores is sufficiently small (see [ll]). E ven though the radiation and convection can be important, the conduction through the solid phase plays the dominant role in the heat transfer in porous powder material. The effective conductivity used in the energy equation should take into account all phenomena occurring, that is pure conduction throug‘h the solid phase, conduction and convection through the voids and radiation. We use the following analytical continuous relation for the effective conductivity proposed by Argento and Bouvard [12]. This relation which establishes the dependence of the effective conductivity upon relative density is relevant either at low density or at high density. Thus

3’*(1-po) :ff _

s ( > ; I$

'0 (1)

where p. is the initial relative density and p the intermediate relative density of the powder packing, k, is the thermal conductivity of the solid and keff the effective thermal conductivity of porous material.

The energy equ.ation on a continuum domain R is written for an anisotropic solid as

p,cp = (kijT.j)i + Q (2)

with

k, = k,,,S,,

where pm is the material density, c is the heat capacity, T is the temperature, k, is the effective thermal conductivity tensor, Q is the volumetric rate of heat generation and Sij Kronecker delta. The heat generation source Q in Eq. (2) has several components. In the present study the components are: The contribution of the heat generation due to plastic work during the mechanical deformation; the contribution of the: heat conduction between powder packing and graphite core through their interfaces governed by the contact heat transfer coefficient; the contribution of convective heat transfer due to convective heating, via argon gas medium external to the container.

302 A. Svoboda et al. I Comput. Methods Appl.

The source of heat generation due to the dissipation

ci, = KC+;

Mech. Engrg. I-L? (1997) 299-314

of plastic work can be expressed as

(3)

where ulj represents the Cauchy stress in the powder material, E :;” is the viscoplastic strain rate and ti is an internal power source. The heat generation efficiency K represents the fraction of plastic work converted into heat. It is usually assumed to be 0.9 (see [ll]). The fraction of the remainder of the plastic deformation energy (1 - K) is expended in changes of dislocation density and grain boundaries.

Boundary conditions for the boundary r, of the deforming solid 0 are defined below: Boundary l& with specified temperature

T= T,(x, t) (4)

where T, is a given function, x is a position vector in 0 and t is the time; boundary r,, where conductive flux between powder packing and core domain L$ is prescribed

4ini = W, W, - T,) (5)

where T, and T, represent the surface temperatures of powder mass and core body, h, is a pressure dependent contact thermal resistance of the contact interface, qi are the components of heat flux vector and n, are the components of the outward surface normal vector; boundary r,, where convective heating is prescribed

q,ni = h,(x, t)(Te - Tp) (6)

where h, is the convective heat transfer coefficient, T, is the temperature of gaseous environment and T, is the temperature of the powder compact.

For transient problems the initial conditions of the form

T(x,O)= T,(x) VxELl

must be specified.

(7)

From the weak form of Eq. (2) and the boundary conditions in Eqs. (4)-(6) we get, using a finite element spatial discretization, a coupled set of first-order ordinary differential equations expressed in matrix form as

Cf + G’“‘(T) = R(T, Q, t) (8)

where C is a capacitance matrix, T is a vector of nodal temperatures, G i”t is a vector of internal nodal fluxes, R is a vector of external nodal fluxes (includes also contribution of internal heat generation), Q is a vector of element thermal and mechanical heat generation rates and t is the time. Eq. (8) represents an initial boundary-value nonlinear transient problem. Using the midpoint rule for time discretization yields a nonlinear algebraic system of equations to be solved at each time step of the analysis.

2.2. The momentum equation

The conservation of momentum equation and the conservation of mass equation are the equations governing the mechanical behaviour of the powder material during densification. In the HIP simulation the inertial effects are eliminated and the conservation of momentum equation on a domain 0 has the form

uiji. j + bi = 0

where aij is the Cauchy stress tensor, bj is the body force The conservation of mass equation can be expressed as

per unit volume.

(9)

$ p t; Iii,) = 0 (10)

where tii represents the velocity.


A set of boundary conditions must be defined for the boundary Z, which is divided into a boundary Z,, where displacements ui are prescribed and Z,, where the pressure, in the form of equivalent forces, is specified for those nodes of the container which are in contact with the argon gas.

Accordingly, the boundary conditions can be written in the following form

ui = &(x, t) on Z,,

where Ui is a given function and

(11)

uijn, = pi@, t) on Z,, (12)

where pi is a prescribed function and nj denotes outward normal vector. From the weak form of Eqs. (9), (ll), (12) we get, using spatial discretization, a coupled system of

second-order differential equations. For a quasi-static analysis the finite element equations have the form

Z+‘(U, li, T) := P(U, b, t, T) (13)

where F i”t ’ 1s an internal nodal force vector, P is an external nodal force vector containing contributions from mechanical boundary conditions and contact pressure, II is a vector of nodal displacements, T is a vector of nodal temperatures. Viscous effects from a rate-dependent constitutive equation are incorporated into F int. The nonlinear algebraic system in Eq. (13) must be solved at each time step.

2.3. Constitutive relations for hot powder compaction

During the HIP process the initial porous powder packing transforms into fully dense material. The densification of the powder material is governed by different deformation mechanisms. Creep deformation was found to be the main contributing factor to the densification of powder material (see [6,13]). In the present study the densification behaviour of the metal powder is modelled using constitutive equations of thermal elasto-viscoplastic type. The use of the elasto-viscoplastic material model enable us to calculate the residual stress field. The residual stresses are important for the behaviour and function of the component. The total deformation rate D is defined as the sum of an elastic part and a viscoplastic part, i.e.

D=D”+D”” (14)

The powder is considered as a continuum with relative density p as an internal variable. It is known that the plastic yielding of the porous material is, in contrast to the dense metallic material, a function of both deviatoric and hydrostatic stress state. Taking into account these aspects we use constitutive equations with pressure sensitivity which are extensions of von Mises theory to porous materials. This concept was proposed by Shima and Oyane [14] among others. Accordingly, the natural way to describe the stress state, sensitive to both shear and volumetric deformation, is to use the equivalent stress u_, defined as

G, = 3pJ, + yIf (15)

with

J,=+S:S I, = tr(cr)

where u is the Cauchy stress tensor, S is the deviator of the stress tensor, 1 is the second-order unitary tensor and tr(.) designates the trace operator. The functions of relative density p(p) and y(p) are material parameters which represent localization of stress state generated by changes of porosity. The first invariant I, of the stress tensor represents the pressure, and the second invariant J2 of the deviatoric stress tensor represents the shear applied to the material. At full density, corresponding the value of p = 1, the functions p(p) and y(p) must reach the values /3 = 1 and y = 0. Simple analytical equations of the form

304 A. Svoboda et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 299-314

p=1+p,- 1-P P-Pz’ Y’Ylb_r

2 (16)

are used for the description of functions /3(p) and y(p). Further, we assume the normality rule (see e.g. [1.5]), between the viscoplastic strain rate tensor D” and viscoplastic potential @

@ = @(a, P, T) (17)

where the relative density p and the temperature T are incorporated as history-dependent variables. The viscoplastic strain rate tensor can be derived from the viscoplastic potential @ as

(18)

The equivalent viscoplastic strain rate can be defined (see [14]), by assuming that the macroscopic plastic work rate ti must be equal to the average plastic work done by powder particles per unit volume of the porous body with relative density p, which gives the relation

ti = u : Dvp = pa,,Drfl (19)

Using Eqs. (15), (18), (19) we obtain the expression for the viscoplastic strain rate tensor in the form

where a,, is given by Eq. (15). From the mass conservation, Eq. (lo), we get variations of relative density related to the volumetric component of the strain rate as

p = -p tr(DvP) (21)

Densification of the powder occurs due to the coupled deformation mechanisms as plastic flow, creep and diffusion in the metal particles (see [16]). D uring the densification process strain rate is influenced by stress as well as temperature. The relation between equivalent strain rate 0:: and equivalent stress oeeq is determined using the rate equation for power-law breakdown. This mechanism is identified as the main contributing deformation mechanism. We consider a modified form of hyperbolic sine equation proposed by Raboin [17] for the hot deformation behaviour of a dense metal. The modified equation which is also used for a fitting of the creep parameters to experimental data (see [18]), has the form

VP_ a@ A pD-4 - - = 7 (sinh(apeq))” exp(-Q/(RT)) aa

eq (22)

where A, CY, m, Q are material parameters. The parameter A is a pre-exponential temperature factor, (Y determines power-law break down stress level, m is strain rate sensitivity, Q is activation energy, R is the universal gas constant and T is the absolute temperature.

The constitutive relations for simulation of hot powder compaction are developed within the framework of finite-deformation theory to be able to represent correctly the large deformations and rotations which can occur during the HIP process. To achieve second order accuracy, the algorithm is provided with midpoint strain increment, see [19], which is combined with the Green-Naghdi stress rate (see [20]). Details about the implementation of the constitutive model into the finite element code can be found in [18]. To avoid locking due to incompressibility constraints, a mean dilatational element is considered according to Engelmann and Hallquist [21]. Four nodes isoparametric elements are used for spatial discretization with 2 x 2 point Gauss quadrature rule. The relative density is calculated at each Gauss point. The constitutive relations and analytical form for effective conductivity, Eq. (l), are implemented into the implicit nonlinear finite element code PALM2D [22].


2.4. Thermomechanical coupling and solution algorithms

The thermomechanical coupling stems from the contribution of the following sources. The mechanical behaviour is dependent on temperature in addition to the relative density. The energy equation which is solved on the deformed geometry incorporates the heat source term due to plastic work dissipation in addition to dependence of the thermal parameters on temperature and relative density. In a staggered step approach, the thermal calculations are performed on the geometry at the end of the previous step. The resulting temperature field is used in the mechanical calculations to update geometry and stress field in the current step. The thermal problem and the mechanical problem of the form of nonlinear algebraic systems are individually solved using a linearization and iteration procedure.

In the staggered step approach different time step sizes can be used in the solution of the thermal problem and the mechanical problem. Within the thermal step or the mechanical step, the substepping strategy can also be used. It is of advantage to have an adaptive solution strategy which facilitates automatic adjustment of thermal and mechanical step size based on given criteria. The driver ISLAND by Engelmann and Whirley [23] proved to be an efficient tool together with PALM2D (both public codes from Lawrence Livermore National Laboratory). ISLAND is a solution control language which allows temporal and algorithmic adaptivity. This adaptivity may be based on different solution control parameters. In the simulation case presented here we use displacement norms, residual norm and effective plastic strain rate of powder mass as criteria for convergence checks. The time step size for either the thermal or mechanical problem can be cut down when convergence problems occur during the solution. In the case of nonconvergence a backstep followed by a restart from previous converged step with altered solution control parameters may be used. In the case of stable convergence the time step length may be automatically increased.

Several numerical procedures for the equilibrium solution are available. We have tested the following Quasi-Newton methods: BFGS, Broyden, Davidon-Fletcher-Powell (DFP). The Davidon-Fletcher- Powell (DFP) method is chosen which offers the most cost-effective solution of our problem. The details about Qua.si-Newton methods can be found in [24] and [25].

Details about the staggered step formulation in PALM2D can be found in [lo]. In the following subscripts denote time step number. Index for substepping is dropped to make notations more readable. Assuming thermal quantities and the relative density p,, to be known at time t, we compute the current mechanical volumetric rate of heat generation tin and new temperature field T,,, on current geometry x,. Thle nonlinear algebraic system for the thermal problem, Eq. (8), is solved by an incremental iterative Newton-Raphson method. The mechanical contact conditions are held constant during the thermal step. Assuming that all mechanical quantities at time t, are known the nonlinear algebraic system for the mechanical problem, Eq. (13) is solved’ by linearization and iteration procedure to find displacement vector u, + *. During the solution of the mechanical problem the current nodal temperatures T, + 1 and the relative density P,,+~ are used to evaluate terms in Eq. (13) which are dependent on temperature and relative density. When the solution at time step tn+l has converged and u n+l is found the mechanical volumetric rate tiitn+l can be calculated from the constitutive relation.

3. Application

3.1. Manufacturing simulation of injection molding tool

The objective of the manufacturing simulation is to predict final geometry of the tool and residual stresses distribution at the end of HIP cycle. The distribution of relative density, pressure and temperature gradients in the intermediate stages are also studied. The HIP cycle used during the simulation study is shown in Fig. 1. The container was filled with the APM 2390 powder material having the initial relative density of 0.756 and the mass density of full dense material p, = 7700 kg/km3. The geometry of the outgassed and sealed container was measured by means of CMM. The coordinate measurement machine used in our system was Johansson 12 with measurement accuracy 10 pm. The result of the measurement in the form of a set of coordinates, describing the profile of the container,


. , . _ - _ - _ _ _ - _ - _ _ #’

1

i \

i Tempera&m ---

,' F?e-ssurc. -

Fig. 1. The HIP cycle-temperature and pressure load.

was used in I-DEAS system [26] for the definition of the computational model. By taking advantage of the axial symmetry in geometry and loading, the finite element model is created using 499 four-node isoparametric elements with 560 nodes. The complete set of boundary conditions including pressure and temperature loads and displacement restraint is also created in I-DEAS. The right half of the geometry is used for the model as shown in Fig. 3. The pressure load is applied on the element free edges and the temperature load is imposed to the nodes on the free surfaces. We assume thermal and mechanical contact between the graphite domain and powder material. The penalty method is used for the resolution of the mechanical contact problem with penalty factor p = 10. For the interface between graphite core and powder mass we use a contact thermal resistance h, = 3 x lo4 W/m2 K (see [27]). In the contact between the steel container and the powder no sliding is allowed. Using this information the topology of finite elements and boundary conditions were generated in I-DEAS.

Three different types of material description are assumed in our computational model: The graphite core is modelled as a pure thermoelastic body. The thermal and mechanical properties data for graphite are given in [28-301. We chose ,Poisson ratio as 0.07 and thermal expansion coefficient as 2.36 X 10m6 K-‘. Young’s modulus was chosen as an average value from literature [28-301 to 4900 MPa. Other values (in the range 3000-10OOOMPa) were tested. The chosen value gave the best comparison between calculated and measured values of final size and shape of the component.

The material model for the steel container is thermal-elastoplastic with temperature dependent mechanical and thermal properties [l&31]. Material used for the manufacturing of the container is stainless steel AISI 304. The materials of the core and container are both assumed to be incompressible.

The densification behaviour of the powder material APM 2390 is modelled using power-law breakdown mechanism as mentioned above. The composition of the powder is given in [32]. The numerical values of the various parameters in Eqs. (16) and (22) are taken from Svoboda et al. [33]:

A = 8.308 x 10” K/s /?i = 2.54 x 10’

(Y = 1.092 x lo-* Pa-’ & = 6.0 x 10-l

m = 2.45 y1 = 2.89

Q/R = 6.774 x lo4 K y2 = 6.25 x 10-l

.A. Svoboda et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 299-314 307

0.8

0.75 0.8 0.85 0.9 0.95 1

Norm&ad effective conductivity bff/ks

Fig. 2. Effective thermal conductivity as a function of relative density. Comparison between relation (Eq. 1) and experimental results, [32].

Optimization methods were used to fit the parameters of the constitutive equations to experimental data, see [33] for the optimization procedures used. Details about experimental procedures can be found in [18] and [33]. The relation for dependence of elasticity modulus E on the relative density (see [IS]), is implemented into the code as

Eh~)=EUJ)/ 20(1 - p)’

. > p_o7 (23)

where E( 1, T) is temperature dependent Young’s modulus. A continuous relation for the effective thermal conductivity of a densifying powder mass, Eq. (1) is used for fitting to experimental data, see Fig. 2. Information about experimental measurements of thermal conductivity of full dense and porous APM 2390 powder material may be found in [32]. Here results of measurements of specific heat are also presented.

The set of information about the finite element model was saved in the form of I-DEAS universal file and transferred to the product database. The product database was used for generation of the input datafiles for the coupled thermomechanical analysis with PALM’LD. After the HIP process the same profile of fhe container geometry was measured again using CMM. The container was also cut to measure geometry of the deformed internal shape. The result of measurements by means of CMM and result of the finite element simulation were transferred into I-DEAS system via the product database. I-DEAS was used for the comparison of the final shape resulting from measurements with that obtained from computations as illustrated in Fig. 3. In this figure the good agreement between simulated and measured geometry can be seen. For the illustration of the volume decrease some significant dimensions are chosen. The dliameter of the container on the initial geometry was d = 125.26 mm and the axial dimension in the middle of the container was h = 96.15 mm. Changes of the same dimensions during HIP process are following:

Deformed geometry-simulated: diameter: d = l17.56 mm height: h = 86.55 mm

Deformed geometry-measured: diameter: d = 117.62 mm height: h = 87.58 mm


Ebzment No.106,

‘1 ,a

I- I-

Elcnrnt No. 433

Fig. 3. Comparison between initial undeformed geometry (left side of the figure), resulting geometry from the simulation (right side) and from the inspection. The result of the simulation shown by FEM mesh. Bold lines show initial shape of the container and the core, dashed lines show result from the inspection obtained using CMM.

The initial diameter of the core was d = 97.06 mm and the axial dimension in the middle of the core was h = 27.92 mm. The dimensions of the core after HIP were:

Deformed geometry-simulated: Deformed geometry-measured: diameter: d = 93.06 mm diameter: d = 92.87 mm height: h = 27.27 mm height: h = 27.69 mm

It can be concluded from the measurements that the graphite core was almost incompressible during the HIP process. Details about the CACE system used for the definition of simulation model and comparison of results are presented in [l] and [2].

The densification behaviour of the powder during the HIP process is illustrated in the following sequence of figures. Two intermediate stages of the densification process at time t = 7000 s and 9000 s of the HIP cycle, see Fig. 1, are chosen. Figs. 4-6 illustrate the isotherm, isodensity and isobar maps in the tool at time t = 7000 s. Heat, which is applied on the exterior of the part in the form of temperature load, diffuses into the powder. The hotter outer surface is compacted faster than the interior. Heat is then conducted quicker through this sintered layer which adds to the temperature differences between interior and exterior. The temperature varies, see Fig. 4, from the maximum value of T = 1083 K at the surface to the minimum value of T = 1051 K at the centre of the core. It can be seen from isodensity map, Fig. 5, that the relative density varies from p = 0.757 to p = 0.772. At this time the influence of thermal strain on the relative density is still evident. The external applied pressure is 81.1 MPa. The distribution of pressure in the tool is shown in Fig. 6. It can be seen that the relative density and temperature create fronts moving through the part from the outside to the inside. The large differences between the thermal conductivity and the specific heat capacity in the powder mass and in the graphite core are also contributing to the non-uniform distribution of temperature in the powder. The phenomenon of non-uniform sintering affects directly the pressure distribution in the powder. It is illustrated in Figs. 6 and 9 that the pressure experienced in the powder mass is far from being hydrostatic, although the pressure applied on the exterior of the part is hydrostatic. This behaviour is enhanced by the stiffness of the container taking up the load which should be otherwise transmitted into the powder mass. In addition, the pressure distribution is also influenced by the stiffness of the thermoelastic graphite core. At time t = 9000 s the relative density varies from p = 0.954 close to the

.A. Svoboda et al. I Comput. Methods Appi. Mech. Engrg. 148 (1997) 299-314 309

Fig. 4. The isotherm map at time t = 7000 s.

fringe levels

[ “Kl

fringe levels

[kg/n?

5.839E+3

5.847E+3

5*856E+3

5,86!B+3

5.8743+3

5.882E+3

5.891E+3

5.9mE+3

5.908E+3

,5*917Ei+3 I ,5.926E+3

5.934E+3

milml = 5.83mk3 maxval = 5.9433+3

310 A. Svoboda et al. / Comput. Methods Appl. Mech. Engrg. 148 (1997) 299-314

fringe levels

IMPal

_ 6.439E+7

6.72 lE+7

7.002E+7

7.283E+7

7.565E+7

7.846E+7

- 8.127E+7

minval = 5.314E+7 maxval = 8.971E+7

Fig. 6. The isobar map at time t = 7000 s.

fringe levels

[ Wm-?

n n -7mE+3

-7215E+3

D.hval= 7.19oE+3 mauvai = 7.353Eb3

u Fig. 7. The isodensity map at time t = 9000 s.


Fig. 8. The isotherm map at time t = 9000 s.

fringe levels

[OKI

- 1385

- 1386’

- 1387’

- 1388’ m *

- 138g*

- 1390-

I 13g4-

- 13g5.

+g

- -

minval = 1383. maxval = 1398.

fringe levels

IMPal

- 3_848E+7

6.67OE+7

&ONE+7

9.492E+7

- l.O9OE+8

- 1.2903+8

- 1.37OlS8

I, m

151OE+8

1.65OE+8

- 1.800&8

1 -1.94OE+8

minval = 2.437Eh7 muxval = 2.078lZ+8

Fig. 9. The isobar map at time t = 9000 s.


container to p = 0.933 close to the core (see Fig. 7). Temperature varies from a maximum of T = 1398 K to a minimum of T = 1383 K (see Fig. 8). The maximum value of pressure predicted by the simulation is p = 207.8 MPa found in the core and the minimum value found in the powder mass is p = 24.4 MPa,. whereas the external applied pressure load at this time is pext = 97.1 MPa (see Fig. 9). The difference between predicted and applied values of pressure can be explained by the presence of thermal stresses induced by temperature gradients. These thermal stresses are enhanced due to elastic energy accumulated in the incompressible core. The energy accumulated in the graphite core is also contributing to the residual stresses after HIP process. It can be concluded from Fig. 10 showing large residual stresses at the end of HIP cycle at time t = 21000 s. The knowledge of the stress field distribution including residual stresses is important for the subsequent heat treatment of the resulting martensitic microstructure.

Two elements, No. 433 in the interior of the tool closed to the core and No. 106 in the exterior layer closed to the can, as illustrated in Fig. 11, are chosen to show evolution of the relative density during the densification. It can be concluded from the figure that the large powder domain becomes almost full dense close to the time when the maximum of pressure and temperature load is reached, compare with Fig. 1. The influence of the thermoelastic strain upon the macroscopic deformation becomes evident at the end of the dwell time when the temperature-load begins to diminish. At the end of HIP cycle the relative density for element No. 106 is p = 0.998’ and for element No. 433 p = 0.997.

fringe levels

IMPal

7641E+7

- 1.13OE+*

- lSooE+8

- 1.870%8

2.24OE+8

-2.WEZ+8

-2.97(X+8

-3.340%8

- 3.71OE+8

-4.08oE+8

-4.45oE+8

minval = 2.794E+6 maxval = 4.813E+8

Fig. 10. The residual stresses at the end of HIP cycle, time r = 21000 s.

.4. Svoboda et al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 299-314 313

Fig. 11. The evolution of relative density p for elements No. 106 and No. 433.

4. Conclusions

The manufacturing simulation of the Hot Isostatic Pressing can be successfully modelled by a continuum mechanics approach. The elasto-viscoplastic constitutive relation used in the material description proved to predict successfully the densification behaviour of the powder material in comparison with experimental measurements. It is demonstrated that the simulation techniques can be applied with satisfactory accuracy and effectiveness to the simulation of complicated parts consisting of different types of material models. The mean dilatational formulation is shown to represent consistently the incompressibility in the analysis of an axisymmetric approximation. The thermomechanical effects of the hot densification behaviour of metal powder are resolved in the frame of the staggered step formulation for fully coupled thermomechanical problems.

The computer-aided concurrent engineering system (CACE) used for the simulation of HIP process offers powerful tool for cost-effective way to fabricate parts with complicated geometry. The direct flow of information between solid modeller of the computer aided design (CAD) system, codes for the finite element analysis (FEA) and the coordinate measurement machine (CMM), based on the product database, makes the process of product development more flexible.

Acknowledgements

Support for this work was provided by the Swedish Board for Industrial and Technical Development (NUTEK).

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