EXPERIMENTAL AND NUMERICAL STUDIES ON MODELING MECHANICAL PROPERTIES OF AUSFERRITIC DUCTILE IRON

EXPERIMENTAL AND NUMERICAL STUDIES ON MODELING MECHANICAL PROPERTIES OF AUSFERRITIC DUCTILE IRON

Cyprian Suchocki

^1a

, Dawid Myszka

^2b

1Department of Mechanics and Armament Technology, Warsaw University of Technology

2Department of Metal Forming and Casting, Warsaw University of Technology

ac.suchocki@imik.wip.pw.edu.pl, ^bd.myszka@wip.pw.edu.pl

Summary

In this study the experimental results obtained by testing different types of ausferritic ductile iron in compression are presented. In order to describe the material’s response a new constitutive equation of elasto-plasticity in large strains is utilized. The material parameter values determined for the examined materials are presented. The im- plementation of the model into finite element method (FEM) is discussed. The obtained results allow to conclude that the proposed constitutive model requires definition of a far lesser number of constants than the classical model of plastic flow theory with isotropic hardening. What is more, the numerical computation cost is lower.

Keywords: plasticity, ausferritic ductile iron, finite element method

EKSPERYMENTALNE I NUMERYCZNE BADANIA

NAD MODELOWANIEM MECHANICZNYCH WŁASNOŚCI ŻELIWA SFEROIDALNEGO AUSFERRYTYCZNEGO

Streszczenie

W pracy przedstawiono wyniki doświadczalne badań własności mechanicznych żeliwa sferoidalnego ausferrytycz- nego różnego typu w warunkach ściskania. Do opisu odpowiedzi materiału wykorzystano równanie konstytutywne sprężysto-plastyczności dla dużych odkształceń według własnej koncepcji. Dla badanych materiałów zestawiono wyznaczone wartości parametrów materiałowych. Omówiono implementację modelu w ramach metody elementów skończonych (MES). Uzyskane rezultaty pozwalają stwierdzić, że proponowany model konstytutywny wymaga podania wielokrotnie niższej liczby stałych materiałowych niż klasyczny model teorii plastycznego płynięcia z izo- tropowym wzmocnieniem. Ponadto koszt obliczeń numerycznych jest niższy.

Słowa kluczowe: plastyczność, żeliwo sferoidalne, metoda elementów skończonych

1. INTRODUCTION

Several different approaches can be distinguished in modeling the material elasto-plastic behavior. The most popular formulations of plasticity are the so-called flow theories. The constitutive equations of this type find their origin in mid. XIXth century and are together often referred to as the classical plasticity. The remain- ing elasto-plastic models which do not qualify to the aformentioned group are far less celebrated which is most probably caused by the fact that they are much younger. Some of those model formulations do not em-

ploy the flow rule and are called the deformation theo- ries. Other approaches do not even employ the yield criterion. Among the last subgroup the models which introduce the intrinsic time measures are called endochronic, cf Wu (2005). There also exist a number of hybrid models which combine the endochronic formula- tions and the yield criteria, sometimes along with the flow rules.

The numerical algorithms utilized to integrate the con- stitutive equations of classical plasticity in order to

implement them into the finite element method (FEM) are complicated. It is common that during the Newton- Raphson (N-R) procedure used to solve the global equi- librium equations at each iteration an additional N-R iterative process has to be launched in order to integrate the constitutive equation. All these algorithmic compli- cations are caused by the definition of the yield surface.

At each iteration it has to be checked whether a materi- al yielding occurs. If the plastic flow takes place the proper increment of the plastic strain must be calculated using the radial-return procedure, for instance. All these activities increase the computation cost of the constitu- tive equation integration algorithm. Thus, it is expected that a plasticity formulation which does not introduce neither the yield surface nor the flow rule could be char- acterized by a lesser computation cost.

In the present work the elasto-plastic properties of austempered ductile cast iron are investigated. For the purpose of simulating the mechanical behavior of cast iron subjected to monotonic loadings the classical flow theory of plasticity is usually used. The Huber-Mises- Hencky (HMH) yield surface with the isotropic or kine- matic hardening laws are most commonly utilized. Such models are often called J₂ plasticity. Apart from the J2

plasticity models, a group of specialized constitutive equations exist that are specifically dedicated to describe the behavior of cast iron. The aim of these model formu- lations is to capture the stress-strain curve’s asymmetry under tension and compression, particularly at elevated temperatures (e.g. Downing 1984, Augustins et al. 2016).

Probably the most popular model able to capture the asymmetry is the one developed by Hjelm (1994). This approach was later extended by the incorporation of kinematic hardening and nonassociated flow rules, cf Josefson et al. (1995). Other constitutive models based on the flow theory were proposed by Altenbach et al.

(2001) and Szmytka et al. (2010). All the aforemen- tioned models are aimed at modeling the material re- sponse in the case of infinitesimal strains.

In this study a new formulation of large strain elasto- plasticity (Suchocki 2015, Suchocki and Skoczylas 2016) is applied to capture the mechanical response of austempered ausferritic ductile cast iron in compression.

The considered model does not introduce neither the yield surface nor the yield criterion. A number of ductile cast irons are examined that were manufactured using different heat treatment procedures. The model con- stants are determined for all of those materials. An implementation into the finite element (FE) system ABAQUS is discussed and verified. Furthermore, the obtained results are compared to those generated when the classical J2 large strain plasticity is used. In particu- lar, the computation time required in both cases is com- pared.

2. MATERIALS AND METHODS

The performed compression experiments involved six types of ausferritic ductile iron. The materials were characterized by different austempering parameters utilized for their manufacturing. These aspects are de- scribed in the section below.

2.1 TECHNOLOGICAL PROCESS

An investment casting as a plate with geometry 11x83x137 mm was selected. The casting was made from ductile iron melted in a medium-frequency induction furnace of 500 kg capacity and treated with NiCuMg17 alloy in a Sandwich process. The metal was poured in a ceramic moulds cooled in the air after the pouring pro- cess. The casting of the chemical composition given in Table 1 was next heat treated in an FPM 450-01 device operating in two steps: austenitization and austempering. Austenitising was carried out for 2,5 hours at the temperature of 900 °C; austempering was carried out in a fluidized bed at different temperatures in order to obtain different mechanical properties. Re- sults of static tensile test and designation of samples are presented in Table 2. The abbreviations ADI_3201 and ADI_3202 correspond to the ductile iron austempered at the temperature of 320 °C for 30 min (variant 1) and 180 min (variant 2).

Table 1. Chemical composition of ductile iron [wt. %]

C Si Mn Mg Cu Mo

3.40 2.80 0.28 0.055 0.72 0.27

Table 2. Static tensile test results for different types of ausferritic ductile iron

Designation of samples

Yield Strength

[MPa]

Tensile Strength

[MPa]

Elongation [%]

ADI_3201 687 922 3.4

ADI_3202 887 1092 4

ADI_3701 607 761 2.3

ADI_3702 684 842 2.2

ADI_4001 541 833 5.7

ADI_4002 573 808 3.2

2.2 EXPERIMENTAL SETUP

The experimental arrangement and results are shown in Figs 1 and 2. The hydraulic universal testing machine 2D40 with the maximum force of 400 kN was used to perform the compression experiments. The cylindrical specimens used for the tests had the diameter Ø8 and height of 9 mm. In every case a wax was applied to the specimen’s upper and lower surfaces in order to elimi-

nate friction. The testing speed was set to 5 mm/min.

Each test was conducted for at least four times to e sure that the obtained results are repetitive. A video extensometer was utilized to measure the specimen’s axial and lateral displacements.

Fig. 1. Experimental setup

The assumed material and spatial coordinate systems, i.e. {Xk} and {xk} (k = 1, 2, 3), respectively, are illu trated in Fig. 1. The following kinematic relations hold:

, ,

where λ1 and λ are the axial and lateral stretch ratios, respectively. The deformation gradient tensor

Jacobian of the transformation J are given as

, det

Fig. 2. Compression test results: a) view of undeformed specimen stretch: experimental data and simulation, d)

3. CONSTITUTIVE MODELING

In order to describe the measured mechanical behavior of the cast iron a model of large strain elasto

without a yield surface is employed (cf Suchocki 2015, Suchocki and Skoczylas 2016).

eed was set to 5 mm/min.

Each test was conducted for at least four times to en- sure that the obtained results are repetitive. A video extensometer was utilized to measure the specimen’s

The assumed material and spatial coordinate systems, ), respectively, are illus- . The following kinematic relations hold:

,

⁽¹⁾

are the axial and lateral stretch ratios, respectively. The deformation gradient tensor F and the

are given as

.

(2)

For the considered case of uniaxial

application of material incompressibility condition yields:

1,

In contrast, following the definition of the Poisson’s ratio ν a different formula for the lateral stretch can be found, i.e.

1

The predictions of Eqs (3)2 and (4) were compared to the experimental measurements of

values of the Poisson’s ratio were considered, i.e.: 0.22 and 0.3, that are usually assumed

0.5 which corresponds to the case of ideal incompress bility as described by the infinitesimal strain theory. The experimental data and the theoretical values of displayed together in Fig. 2c. It can be seen that in compression the ductile cast iron acts like an ideally incompressible material which is well captured by Eq.

(3)2 .

In Figs 2a,b the cylindrical specimen prior to the defo mations and at the maximum shortening can be viewed, respectively. In Fig. 2d some exemplary t

stretch curves are presented.

iew of undeformed specimen, b) view of deformed specimen, c) lateral stretch versus axial d) measured stress versus axial stretch relations

CONSTITUTIVE MODELING

In order to describe the measured mechanical behavior of the cast iron a model of large strain elasto-plasticity without a yield surface is employed (cf Suchocki 2015,

3.1 BASIC NOTIONS

In terms of FEM it is profitable to decouple the material volumetric and isochoric responses as it simplifies the form of the material Jacobian (MJ). What is more, the material compressibility can be easily controlled this way by setting a proper value of the bulk modulus.

For the considered case of uniaxial compression the application of material incompressibility condition

⁄

.

(3) In contrast, following the definition of the Poisson’s ratio

a different formula for the lateral stretch can be found,

1

(4) and (4) were compared to the experimental measurements of λ. Three different values of the Poisson’s ratio were considered, i.e.: 0.22 and 0.3, that are usually assumed for the cast iron, and 0.5 which corresponds to the case of ideal incompressi-

ity as described by the infinitesimal strain theory. The experimental data and the theoretical values of λ are c. It can be seen that in n the ductile cast iron acts like an ideally incompressible material which is well captured by Eq.

a,b the cylindrical specimen prior to the defor- mations and at the maximum shortening can be viewed,

d some exemplary true stress -

ateral stretch versus axial

In terms of FEM it is profitable to decouple the material volumetric and isochoric responses as it simplifies the (MJ). What is more, the material compressibility can be easily controlled this way by setting a proper value of the bulk modulus.

The decoupling of the constitutive equation is facilitated by taking advantage of the multiplicative split of defor- mation gradient tensor (e.g. Simo and Hughes), i.e.

(5) where Fvol and are the volumetric and isochoric defor- mation gradients, respectively that are defined by the following relations:

^⁄

,

^⁄

.

(6) Consequently, the right Cauchy-Green (C-G) defor- mation tensor C along with its isochoric version can be defined as:

,

^⁄

.

⁽⁷⁾

The stored-energy function W is assumed in the decou- pled form, i.e.

! " + ! ,

(8) with U and ! being the volumetric and the isochoric components, respectively. The large strain elasto-plastic constitutive model is formulated by utilizing the formal- ism of the internal state variables. The total second Piola-Kirchhoff (P-K) stress S is assumed to be the sum of an elastic component S0 and N inelastic stress-like tensorial state variables $%& (k = 1, 2, … , N):

' '

₍

+ ∑

^*_&+

$%

_&

,

'

₍

'

₍

+ '

₍^{,- (9)}

with S0vol and S0iso being the volumetric and isochoric second P-K elastic stress tensors which are given as:

'

₍

. , '

₍^{,- ⁄}

/01['],

⁽¹⁰⁾

where

.

⁴₅

, ' 2

⁷

,

(11) and DEV [ ] is an operator extracting a deviatoric com- ponent of a second order tensor in the reference configu- ration, i.e.

/01[] [] [] ⋅ .

(12) The evolution of internal state variables is governed by the following differential equations:

$%9

_&

+

_:%

;< | 9 |

$%

_&

>

_&

'9

₍^,-

,

k = 1,2,…,N (13) with

? @| 9 |A ¯

^⁄

, ¯ tr 9 .

(14) This particular choice of M function ensures that the material response is rate-independent.

For the purpose of modeling the mechanical response of the ductile cast iron the isochoric stored-energy func-

tion ! is assumed as neo-Hooke, whereas the volumet- ric stored-energy U is chosen in the standard form, i.e.

! D

2 E ¯ 3 , " 1

/ 1 ,

where D1 is the inverse of bulk modulus. A single inter- nal state variable tensor was assumed in Eq. (9)1, i.e. N

= 1.

3.2 DISCRETIZED CONSTITUTIVE EQUATION

The system of Eqs (9-14) can be solved numerically using the central difference method in order to deter- mine the simulated material response for an arbitrary loading history in a one-dimensional process.

For the purpose of modeling three-dimensional process- es the developed elasto-plastic constitutive model was implemented into the FE system ABAQUS by taking advantage of the user subroutine UMAT (UserMA - Terial). The UMAT code is responsible for the calcula- tion of the Cauchy stress tensor and the fourth order MJ tensor. For the time increment n+1 Eq. (8)1 takes the form:

'

_GH

'

_{( GH}

+ ∑

^*_&+

$%

_{& GH} (15) By taking a directional derivative of Eq. (15) with respect to Cn+1 a linearized incremental form of the constitutive equation is found, i.e.

Δ'

_GH

ℂ

_GH^{K L}

⋅ Δ

_GH

,

(16) where ℂ^{K L} is the fourth order elasto-plastic stiffness tensor in the reference configuration:

ℂ

_GH^{K L}

2

^'MNO_MNO

.

(17) After pushing forward to the current configuration, Eq.

(16) can be written as

P

_GH^Q _GH

ℂ

_GH^<5

⋅ ΔR

_GH

,

(18) where

P

_GH^Q

ΔP

_GH

ΔS

_GH

P

_GH

P

_GH

ΔS

_GH

is the incremental Zaremba-Jaumann (Z-J) rate of the Kirchhoff stress τn+1 whereas Wn+1 and Dn+1 are the spin tensor and velocity gradient tensors, respectively, whose increments are given by the following formulas:

ΔS

_GH

1

2 [Δ

^{GH GH}

− (Δ

_{GH GH}

) ], ΔR

_GH

= 1

2 [Δ

^{GH GH}

+ (Δ

_{GH GH}

) ],

Δ

_GH

=

_{GH G}

.

The fourth order tensor J^<5 is the material Jacobian which has to be defined within the UMAT code. The perturbed deformation gradient method proposed by Sun et al. (2008) was utilized for calculating the a proximate values of the MJ components.

4. CURVE FITTING

The experimental data obtained from monot pression tests were used to evaluate the material p rameter values for each of the materials under study.

The pattern search algorithm described by Suchocki and Skoczylas (2016) was used. Durin

value optimization process the total weighted quadratic error is being minimized, i.e.

T UV[(W (X))V− (WY )V]

Z V+

→ \]^

where (W )V and (WY )V are the computed and mea ured true stresses, respectively, in the compression direction at the j-th collocation point (j

K is the number of collocation points, assigned to j-th collocation point and p matrix of optimized material parameters.

Fig. 3. Experimental data and model predictions of monotonic

ADI_3201, (b) ADI_3202, (c) ADI_3701, (d) ADI_3702, (e) ADI_4001, (f) ADI_4002 is the material Jacobian

which has to be defined within the UMAT code. The perturbed deformation gradient method proposed by Sun et al. (2008) was utilized for calculating the ap- proximate values of the MJ components.

The experimental data obtained from monotonic com- pression tests were used to evaluate the material pa- rameter values for each of the materials under study.

The pattern search algorithm described by Suchocki and Skoczylas (2016) was used. During the constant value optimization process the total weighted quadratic

\]^.,

are the computed and meas- ured true stresses, respectively, in the compression j = 1, 2, …, K), is the number of collocation points, wj is the weight p is the column matrix of optimized material parameters.

Based on the experimental measurements (

material near incompressibility was assumed and simulated by assigning a sufficiently small value to the inverse of the bulk modulus D1.

The comparison of the experimental monotonic co pression curves and model predictions can be seen in Fig. 3. The evaluated constitutive constants for the considered types of ductile cast iron have been gathered in Table 3.

Table 3. Determined material parameter values Material μ [MPa] γ1 [-]

ADI_3201 628.68 72.63 ADI_3202 660.54 83.47 ADI_3701 797.21 67.56 ADI_3702 728.21 65.0025 ADI_4001 668.703 57.14 ADI_4002 721.49 55.32

Fig. 3. Experimental data and model predictions of monotonic compression curves for the considered types of ductile cast iron: (a) ADI_3201, (b) ADI_3202, (c) ADI_3701, (d) ADI_3702, (e) ADI_4001, (f) ADI_4002

Based on the experimental measurements (Fig. 1c) the lity was assumed and mulated by assigning a sufficiently small value to the

The comparison of the experimental monotonic com- pression curves and model predictions can be seen in

The evaluated constitutive constants for the considered types of ductile cast iron have been gathered

Table 3. Determined material parameter values

/% [-] D1 [MPa^-1] 0.026 33E-9 0.023 33E-9 0.017 33E-9 0.019 33E-9 0.022 33E-9 0.0208 33E-9

compression curves for the considered types of ductile cast iron: (a)

5. FINITE ELEMENT SIMULATIONS

For each of the determined sets of material parameters a number of FE simulations was conducted in order to verify the implementation of the model into ABAQUS via the written user subroutine UMAT. The simulation involved uniaxial tension of a 15mm×15mm

iron block. In each of the simulations the excitat was kinematic, i.e. a ramp displacement of 1.5 mm was enforced on the block’s frontal face. After reaching the maximum displacement unloading was performed until the initial block’s configuration was reached. The defined boundary conditions ensured homogenous and uniaxial stress state with a three-dimensional strain state (Fig. 4).

Fig. 4. Homogeneous deformation of a single finite element: a) distribution of the displacement component in the te sion/compression direction; b)applied boundary

Fig. 5. Stress-stretch curves for cast iron block undergoing ramp loading

ADI_3201, (b)ADI_3202, (c) ADI_3701, (d) ADI_3702, (e) ADI_4001, (f) ADI_4002 For each of the determined sets of material parameters

umber of FE simulations was conducted in order to verify the implementation of the model into ABAQUS via the written user subroutine UMAT. The simulation 15mm×15mm cast iron block. In each of the simulations the excitation displacement of 1.5 mm was enforced on the block’s frontal face. After reaching the maximum displacement unloading was performed until the initial block’s configuration was reached. The homogenous and dimensional strain

Fig. 4. Homogeneous deformation of a single finite element: a) distribution of the displacement component in the ten- sion/compression direction; b)applied boundary conditions

In each case the block was initially meshed with a single FE. Subsequently, the simulation was repeated for the cube meshed with a larger number of elements (125 for cubic and 1032 for tetrahedral elements, respectively).

The following types of finite elements were tested:

• C3D8 (cubic, 8 nodes);

• C3D8H (cubic, 8 nodes, hybrid);

• C3D4 (tetrahedral, 4 nodes);

• C3D4H (tetrahedral, 4 nodes, hybrid).

In all cases linear shape functions were utilized. The same deformation histories were simulated with a s written using Scilab software. In this approach the set of algebraic and differential equations which describe the uniaxial process is integrated using the central difference method (cf Suchocki and Skoczylas 2016).

The results obtained using ABAQUS

compared in Fig. 5. The material response predicted using different methods is the same which allows one to conclude that the chosen methodology is correct.

cast iron block undergoing ramp loading-unloading for different material parameter sets: (a) ADI_3201, (b)ADI_3202, (c) ADI_3701, (d) ADI_3702, (e) ADI_4001, (f) ADI_4002

In each case the block was initially meshed with a single FE. Subsequently, the simulation was repeated for the cube meshed with a larger number of elements (125 for cubic and 1032 for tetrahedral elements, respectively).

finite elements were tested:

C3D8H (cubic, 8 nodes, hybrid);

C3D4 (tetrahedral, 4 nodes);

C3D4H (tetrahedral, 4 nodes, hybrid).

In all cases linear shape functions were utilized. The same deformation histories were simulated with a script written using Scilab software. In this approach the set of algebraic and differential equations which describe the uniaxial process is integrated using the central difference method (cf Suchocki and Skoczylas 2016).

The results obtained using ABAQUS and Scilab are compared in Fig. 5. The material response predicted using different methods is the same which allows one to conclude that the chosen methodology is correct.

unloading for different material parameter sets: (a)

6. COMPARISON TO J2 PLASTICITY

The material constants of the J2 flow plasticity with multilinear isotropic hardening were determined in order to compare the classical plasticity formulation to the proposed constitutive model. The J₂

constants for a selected type of ductile cast iron have been gathered in Table 4.

Table 4. Classical J2 elasto-plasticity constants for du tile cast iron (ADI_4001)

Elasticity constants E [MPa]

87150.75

Multilinear hardening parameters i σy(ε^p) [MPa]

1 653.38

2 803.042 5.64E

3 944.18 5.073E

4 1015.97 1.059E

5 1136.43 3.41E

6 1241.065 6.43E

7 1387.072 1.12E

Table 5. Comparison of computation time for different constitutive models and selected material parameter sets

Material CPU time [s] Wallclock time [s]

Present model

ADI_3201 14.5

ADI_3202 14.6

ADI_3701 15.1

ADI_3702 14.7

ADI_4001 14.1

ADI_4002 14.5

Large strain J2 plasticity (isotropic hardening)

ADI_4001 15.1

The experimental data used for constant determination along with the computed loading-unloading curve are shown in Fig. 6. The theoretical curve was obtained by performing a FE simulation in ABAQUS using the

2

flow plasticity with multilinear isotropic hardening were determined in order to compare the classical plasticity formulation to elasto-plasticity constants for a selected type of ductile cast iron have

plasticity constants for duc-

ν [-]

0.49 Multilinear hardening parameters

εp [-]

0 5.64E-4 5.073E-3 1.059E-2 3.41E-2 6.43E-2 1.12E-1

Table 5. Comparison of computation time for different constitutive models and selected material parameter

Wallclock time [s]

15 15 16 15 15 15 Large strain J2 plasticity (isotropic hardening)

16

The experimental data used for constant determination unloading curve are shown in Fig. 6. The theoretical curve was obtained by performing a FE simulation in ABAQUS using the

material parameters from Table 4 and the boundar conditions as depicted in Fig. 4. A single C3D8 element was utilized to mesh the block. It can be seen that no Bauschinger effect is predicted. After the plastic strain is reached that exceeds the maximum value defined in Table 4 the model acts like elastic

Fig. 6. Classical J2 plasticity: experimental data and model predictions for loading-unloading process

The computation time obtained for

plasticity was compared to that when the proposed elasto-plastic constitutive model is utilized.

have been gathered in Table 5.

7. CONCLUSIONS

In the present study a newly developed constitutive model of large strain elasto-plasticity without the yield surface was applied to describe the mechanical res of the ductile cast iron in compression. The material parameter values were evaluated for six d

of ductile cast iron. A good agreement between the experimental measurement and the model predictions was found (cf Fig. 3). The considered

equation was implemented into the FE system ABAQUS by utilizing the user subroutine UMAT sp cifically written for that purpose. A comparative study between the model presently discussed and the classical J₂ elasto-plasticity was conducted. It

the case of monotonic loadings both constitutive equ tions provide a good description of the stress curve (cf Figs 3 and 6). However, for the

this is achieved at the price of determining 16 material parameters (see Table 4) which is a rather lengthy and cumbersome procedure. The model proposed in this study requires a definition of 4 constants that can be evaluated within a few minute

regression methods. What is more, the performed comparative study reveals that the new elasto constitutive model is less costly in terms of the comp tation time. Finally, the proposed constitutive equations allows to simulate the Bauschinger effect which cannot be described by the J₂ elasto-plasticity with isotropic hardening.

material parameters from Table 4 and the boundary conditions as depicted in Fig. 4. A single C3D8 element was utilized to mesh the block. It can be seen that no Bauschinger effect is predicted. After the plastic strain is reached that exceeds the maximum value defined in

astic-ideally plastic.

plasticity: experimental data and model unloading process

The computation time obtained for the J₂ elasto- plasticity was compared to that when the proposed

plastic constitutive model is utilized. The results

In the present study a newly developed constitutive plasticity without the yield surface was applied to describe the mechanical response of the ductile cast iron in compression. The material parameter values were evaluated for six different types of ductile cast iron. A good agreement between the experimental measurement and the model predictions was found (cf Fig. 3). The considered constitutive equation was implemented into the FE system ABAQUS by utilizing the user subroutine UMAT spe- cifically written for that purpose. A comparative study between the model presently discussed and the classical plasticity was conducted. It is found that for the case of monotonic loadings both constitutive equa- tions provide a good description of the stress-stretch curve (cf Figs 3 and 6). However, for the J2 plasticity this is achieved at the price of determining 16 material parameters (see Table 4) which is a rather lengthy and cumbersome procedure. The model proposed in this study requires a definition of 4 constants that can be s using the nonlinear regression methods. What is more, the performed comparative study reveals that the new elasto-plastic constitutive model is less costly in terms of the compu- tation time. Finally, the proposed constitutive equations

the Bauschinger effect which cannot plasticity with isotropic

References

1. Altenbach H., Stoychev G. B., Tushtev K. N.: On elastoplastic deformation of grey cast iron. ,,International Journal of Plasticity”, 2001, Vol. 17, p. 719

2. Augustins L., Billardon R., Hild F.: Constitutive model for flake graphite cast iron automotive brake discs: from macroscopic multiscale models to a 1D rheological description. ,,Continuum Mechanics and The

2016, Vol. 28, p. 1009-1025.

3. Downing S. D.: Modeling cyclic deformation and fatigue behavior of cast iron under uniaxial loading. PhD th sis, University of Illinois at Urban-Champaign, 1984.

4. Hjelm H. E.: Yield surface for grey cast iron under nology”, 1994, 2, Vol. 116, p. 148-154.

5. Josefson B. L., Stigh U., Hjelm H. E.: A nonlinear kinematic hardening model for elastoplastic deformations in grey cast iron. ,,Journal of Engineering Mate

6. Pipkin A. C., Rivlin R. S.: Mechanics of rate

7. Simo J. C., Hughes T. J. R.: Computational inelasticity. New York: Springer Verlag Inc., 2000.

8. Suchocki C.: An internal-state-variable based viscoelastic and Applied Mechanics”, 2015, 3, Vol. 53, p. 593

9. Suchocki C., Skoczylas P.: Finite strain formulation of elasto

identification and applications. ,,Journal of Theoretical and Applied Mechanics”, 2016, 3, Vol. 54, p. 731 10. Sun W., Chaikof E. L., Levenston M. E.: Numerical approximation of tangent moduli for finite element impl

mentations of nonlinear hyperelastic material models. ,,Journal of Biomechanical Engineering”, 2008, Vol. 130, p. 061003-1 – 061003-7.

11. Szmytka F., Rémy L., Maitournam H., Kőster A., Bourgeois M.: New flow rules in elasto tive models for spheroidal graphite cast

12. Wu H. C.: Continuum mechanics and plasticity. New York: Chapman & Hall/CRC Press, 2005.

This work was supported by project PBS III/246715/NCBiR.

Artykuł dostępny na podstawie licencji Creative Commons Uznanie autorstwa 3.0 Polska.

http://creativecommons.org/licenses/by/3.0/pl

Altenbach H., Stoychev G. B., Tushtev K. N.: On elastoplastic deformation of grey cast iron. ,,International ournal of Plasticity”, 2001, Vol. 17, p. 719-736.

Augustins L., Billardon R., Hild F.: Constitutive model for flake graphite cast iron automotive brake discs: from macroscopic multiscale models to a 1D rheological description. ,,Continuum Mechanics and The

Downing S. D.: Modeling cyclic deformation and fatigue behavior of cast iron under uniaxial loading. PhD th Champaign, 1984.

Hjelm H. E.: Yield surface for grey cast iron under biaxial stress. ,,Journal of Engineering Materials and Tec 154.

Josefson B. L., Stigh U., Hjelm H. E.: A nonlinear kinematic hardening model for elastoplastic deformations in grey cast iron. ,,Journal of Engineering Materials and Technology”, 1995, Vol. 117, p. 145

Pipkin A. C., Rivlin R. S.: Mechanics of rate-independent materials. ,,ZAMP”, 1965, Vol. 16, p. 313 Simo J. C., Hughes T. J. R.: Computational inelasticity. New York: Springer Verlag Inc., 2000.

variable based viscoelastic-plastic model for polymers. ,,Journal of Theoretical and Applied Mechanics”, 2015, 3, Vol. 53, p. 593-604.

Suchocki C., Skoczylas P.: Finite strain formulation of elasto-plasticity without yield surface: theory, parameter identification and applications. ,,Journal of Theoretical and Applied Mechanics”, 2016, 3, Vol. 54, p. 731

Sun W., Chaikof E. L., Levenston M. E.: Numerical approximation of tangent moduli for finite element impl of nonlinear hyperelastic material models. ,,Journal of Biomechanical Engineering”, 2008, Vol. 130,

Szmytka F., Rémy L., Maitournam H., Kőster A., Bourgeois M.: New flow rules in elasto

al graphite cast-iron. International Journal of Plasticity, 2010, Vol. 26, p. 905 Wu H. C.: Continuum mechanics and plasticity. New York: Chapman & Hall/CRC Press, 2005.

This work was supported by project PBS III/246715/NCBiR.

podstawie licencji Creative Commons Uznanie autorstwa 3.0 Polska.

http://creativecommons.org/licenses/by/3.0/pl

Altenbach H., Stoychev G. B., Tushtev K. N.: On elastoplastic deformation of grey cast iron. ,,International

Augustins L., Billardon R., Hild F.: Constitutive model for flake graphite cast iron automotive brake discs: from macroscopic multiscale models to a 1D rheological description. ,,Continuum Mechanics and Thermodynamics”,

Downing S. D.: Modeling cyclic deformation and fatigue behavior of cast iron under uniaxial loading. PhD the-

biaxial stress. ,,Journal of Engineering Materials and Tech-

Josefson B. L., Stigh U., Hjelm H. E.: A nonlinear kinematic hardening model for elastoplastic deformations in rials and Technology”, 1995, Vol. 117, p. 145-150.

independent materials. ,,ZAMP”, 1965, Vol. 16, p. 313-327.

Simo J. C., Hughes T. J. R.: Computational inelasticity. New York: Springer Verlag Inc., 2000.

plastic model for polymers. ,,Journal of Theoretical

surface: theory, parameter identification and applications. ,,Journal of Theoretical and Applied Mechanics”, 2016, 3, Vol. 54, p. 731-742.

Sun W., Chaikof E. L., Levenston M. E.: Numerical approximation of tangent moduli for finite element imple- of nonlinear hyperelastic material models. ,,Journal of Biomechanical Engineering”, 2008, Vol. 130,

Szmytka F., Rémy L., Maitournam H., Kőster A., Bourgeois M.: New flow rules in elasto-viscoplastic constitu- iron. International Journal of Plasticity, 2010, Vol. 26, p. 905-924.

Wu H. C.: Continuum mechanics and plasticity. New York: Chapman & Hall/CRC Press, 2005.

podstawie licencji Creative Commons Uznanie autorstwa 3.0 Polska.