Enhancement of void growth model for the anisotropic ductile metal (tt)

ABSTRACT The aim of work presented in this dissertation was to perform the improvement of the existing void growth-based damage models used for the ductile fracture analysis and predicti

ABSTRACT

The aim of work presented in this dissertation was to perform the improvement

of the existing void growth-based damage models used for the ductile fracture analysis and prediction of sheet metals, which are subjected plastic deformation The original metal material is usually containing the second phase particles or/and inclusions Once the metallic material under deformation lead

to the nucleation, growth and coalescence of micro-voids that it is root of ductile damage in industrial and civil products

The first objective of this work was enhancement of N L Dung micro-void growth model to predict ductile fracture behavior of sheet aluminum alloys, typical for civil structures with anisotropic properties and their implementation

in user-defined material subroutine (VUMAT) The explicit finite element code has been chosen for implementation of new material models Constitutive model with anisotropic yield criterion, damage growth and failure mechanism has been developed and implemented into ABAQUS/Explicit software

The second important aspect of this dissertation was performance of tensile experiments in three different orientations of materials for identification of mechanical behavior of high strength sheet aluminum alloys AA6061-T6 The results from these tests allowed derivation of material constants for constitutive models and help to a better understanding of anisotropic material behavior The tensile tests were also used to validate the accuracy and applied capability of enhanced constitutive material models

The constitutive models were developed within the general framework of ductile damage mechanics Coupling of the quadratic yield function Hill48 with damage model based on micro-mechanical and continuum damage mechanics (CDM) theories has been chosen to suit the anisotropic behavior of sheet material The validation of the constitutive models has been performed by numerical simulations of tensile and Nakajima tests The micro-crack and

fracture initiation, crack path, damage criteria and forming limit diagram (FLD)

of aluminum alloy AA6061-T6 are predicted using these constitutive models

INTRODUCTION

1.1 The research motivation

The fracture phenomenon can be observed any everywhere in our daily lives The phenomenon of ductile fracture is usually happening in metallic forming process under plastic deformation Today, although high quality design and manufacturing processes can result in robust, strong products, the causation of fail cannot be avoided in some cases However, this will not be an issue if processes such as damage elimination or healing are included in the application Therefore, the prediction and characterization of micro-crack initiation, fracture propagation and the final failure of the material is of such importance that it has become a special field in materials science

Recently, damage modeling and predicting of metal material and their alloys are becoming more and more important as an object of research in recent years Until now, phenomenological continuum damage mechanics combined with finite element method has mostly been used for numerical modeling of a material damage Two phenomenological approaches are usually used

The first approach is based on theory of continuum damage mechanics (CDM)

In this method, damage variable is modeled by a scalar variable integrated with

a suitable yield function [1, 2] One of the disadvantages of classical continuum mechanics is the impossibility of predicting the micro-crack initiation

The second approach is based on the micro-mechanical theory This method using a yield function containg a porousity (void volume fraction) that descibed softening phenomenon of matrix material Therefore, this theory-based damage model is also known as porous ductile material model [3-5] The primary advantage of this approach is its micro-crack and fracture predictability via porousity of matrix material

Recently with the rapid development of advanced high strength steel and aluminum alloy sheets, suitable material models are required for accurately describing their anisotropic behavior Therefore, improving the accuracy of ductile fracture prediction for various metals by using various fracture predicted models is still needed to continue

1.2 The research objectives

An investigation of ductile fracture of sheet metallic material and their alloy using N L Dung micro-void growth models [5, 6] was performed in this dissertation The following specific objectives had to be achieved in this work: Understanding the mechanism of microscopic ductile fracture of metallic materials and their alloys

Improving the original damage models for predicting ductile fracture and shear damage of anisotropic sheet metals

Developing the user-defined material subroutine (VUMAT) for both porous ductile material and continuum damage mechanics (CDM) theory-based models

Conducting the experiments to determine the mechanical behavior of material and calibrate the material parameters for the constitutive models

Applying the damage models to predict the ductile fracture of isotropic and anisotropic metals

1.3 Research methodology

An approach based on theoretical framework of ductile fracture together with experimental observation is applied to this dissertation The ductile damage models of N L Dung [5, 6] are first enhanced to anisotropic sheet metal and modified for shear damage by using the classical CDM and micromechanical ductile damage theories After this enhancement the damage models are written

in Fortran program language as the user material subroutines (VUMAT) for the Abaqus/Explicit finite element package using the numerical algorithms [7, 8]

During this process the code is frequently verified from various aspects in such

a way that it works along with the existing capability of Abaqus/Explicit software without any errors Once the VUMAT subroutines are successfully developed, the damage models would be verifed via predicting ductile fracture

of practical application (tensile test, deep drawing…)

1.4 The contributions of dissertation

An enhancement of N L Dung void growth model [5, 6] for anisotropic metal using quadratic yield criterion Hill48 is proposed in this work This approach can be applied to the various yield criteria

This work is also modified the N L Dung model [5, 6] for ductile fracture prediction under pure and simple shear loading states

The micro-crack and fracture criteria are formed by relationship between equivalent plastic fracture strain and stress triaxiality They help to reduce computational time once they associated with CDM theory

The forming limit diagram (FLD) of aluminum alloy sheet AA6061-T6 is suggested in this dissertation

The VUMAT subroutines can be used as the sourced codes for implementing a new material model and developing current work in future

1.5 Dissertation outline

The outline of this dissertation is as follows

An introduction to ductile fracture mechanism of metallic material that occurs due to the nucleation, growth and coalescence of voids is presented in chapter

2 A literature review of the existing porous ductile material models is also shown in this chapter

Chapter 3 details the enhancing ductile fracture criterion and porous ductile material model (N L Dung models) for anisotropic material

The numerically implemented procedure using the stress integrating algorithms

to seek element stress and state variables of damage models are represented in chapter 4

The experiments and an optimized tool are conducted to identify the input material parameters of damage models would be outlined in chapter 5

In chapter 6, the finite element calculations for tensile tests and deep drawing process are performed The numerical simulations of prediction of ductile fracture in anisotropic metals are also compared to experimental results A ductile fracture criterion of the aluminum alloy is also proposed based on finite element analysis

Finally, in chapter 7 conclusions are made based on the results gained in previous chapters and some discussions are presented for future work

In addition, the appendixes and cited documents are also included in this dissertation

DUCTILE FRACTURE OF METALLIC MATERIAL

Ductile fracture of metallic material is due to heterogeneous microscopic structures that decline the mechanical properties of the material The metallic material is usually containing the resource of microscopic damage such as distributed micro-voids which might be process induced during loading are more tending to crack or failure The Figure 2.1 shows the formation process of ductile fracture due to micro-void nucleation, growth and coalescence in the metal sheet under uniaxial tension [9]

In the recent time of several decades, the experimental studies and analytical models of void nucleation, growth and coalescence have been conducted by many researchers [3, 10-16]

The original void growth models are not feasible for ductile fracture prediction under the range of low and negative stress triaxialities [17-19], i.e., they cannot

be used to predict ductile fracture in the cases of compressed and pure shearing loads To improve the predictive ability of the GTN-like model, Xue [18], Nahshon and Hutchinson [19] have proposed the void coalescence models due

to the relatively shearing and rotational voids during plastic deformation In this work, the N L Dung porous ductile model [5] would be associated with Nahshon and Hutchinson [19] shear damage criterion In addition, a modification of the N L Dung ductile fracture criterion [6] for shear damage using CDM theory is also performed in this work

Figure 2.1 Ductile fracture mechanism of metallic material: a) specimen, b) the process of void nucleation, growth and coalescence versus plastic strain evolution [9]

DUCTILE FRACTURE MODELLING

3.1 The continuum damage mechanics (CDM) model

3.1.1 The constitutive equations

The yield criterion in stress space is expressed in the following form:

b)

where  is softening exponent and edenotes equivalent stress,

For isotropic matrix material

3 : 2

0

1

1

p f

3.1.2 An extension of the void growth model for shear damage

As a drawback of void growth-based damage model under simple or pure shear load, the past studies indicated that there is no void growth under shear loading states because of zero stress triaxiality but the voids are still rotated under this condition In an earlier study by McClintock [20] for void growth in shear bands, the fracture due to void growth in the longitudinal direction of the shear bands and the void shear is given by

ln 1 2

1 ln

2

p f

p s

g g

crit

N D

Ratio of 0 / 2 r0 calculated through initial VVF

The damage criterion for general loading case is written

3.2 The porous ductile model

In order to consider anisotropic aspect of sheet material, the original Dung model [5] will be associated with ewuivalent stress Hill48 as follow,

Figure 3.1 The yield surface presentation of the Dung-Hill48 model in normalized principal stress space

NUMERICAL IMPLEMENTATION OF THE DUCTILE DAMAGE MODELS

This chapter describes the implementation of the constitutive models in chapter

3 into FEM code of ABAQUS/Explicit software via the VUMAT subroutines For this dissertation, the VUMAT subroutines are written by program language Fortran 90 and integrated with FEM code of ABAQUS/Explicit software package version 6.14-3

4.1 Numerical implementation of CDM model

The Hill48 yield criterion would be implemented by using “cutting-plane” algorithm [7] The flowchart of stress integration algorithm of CDM model is given in appendix 1

4.2 Numerical implementation of the porous ductile model

A numerical algorithm for pressure-dependent plasticity models [8] is applied

to this dissertation

The flowchart of stress integration algorithm of porous ductile model is shown

in appendix 2

4.3 Verification of user-defined material subroutine (VUMAT)

To verify the accuracy of the implementation against known values and element distortion, the VUMAT subroutines would be verified via single element, tensile specimen and deep drawing process

Results of tensile test: Figure 4.1 shows an identical crack path obtaining by

experiment and numerical simulation of uniaxial tensile specimen Figure 4.2 shows a comparison of force-displacement curve between the experiment and numerical simulations The numerical results show a good agreement with experimental data when using the CDM-Hill48 model with fitted parameter set, whereas a larger displacement amount is archived by Dung-Hill48 model comparing with that of experiment This may be due to effect of hardening exponent in Dung-Hill48 model during plastic deformation process

a) b) c) Figure 4.1 Comparison of the crack path (a) experiment [21], (b) CDM-Hill48

model, (c) Dung-Hill48 model

Figure 4.2 Force – displacement curves of tensile test

(a) (b)

(c)

Figure 4.3 Comparison of fracture path between experiment and numerical simulations (a) experiment, (b) CDM-Hill48 model, (c) Dung-Hill48 model

The result of deep drawing test: The predicted fracture path by numerical

simulation having identical shape and location with experiment is shown Figure 4.3 The forming force versus punch stroke curve is presented by Figure 4.4 The predicted curves underestimate the experimental data and punch depth at moment of fracture occurrence of 16.1 mm, 19.3 mm and 18.7 mm are archived

by CDM-Hill48, Dung-Hill48 model and experiment, respectively

Figure 4.4 Comparison of forming force curve between experiment and

Table 5.1 Chemical composition of AA6061-T6 aluminum alloy

The isotropic hardening model is assumed to obey Swift [24] law The

unknowns (K, ε0, n) that should be determined by the equation (5.1) following

the least square root method

n p

Where K and 0 are material constants, n hardening exponent The optimum curve is shown in Figure 5.1

Figure 5.1 The best fit hardening curve The value of anisotropic coefficients is given as Table 5.2 The best fit parameters are given in Table 5.3

Table 5.2 The anisotropic coefficients of Hill48 equivalent stress function

5.2 Calibration of the material parameters for the damage models

5.2.1 The calibrated approach

The Genetic Algorithm (GA) global optimization approach, which is a built-in optimization function in MATLAB, is employed to optimize the necessary parameters Parameters of the model are obtained from an inverse engineering analysis which aims at minimizing the discrepancy of force-displacement curves provided by simulation and tensile test of a dog-bone specimen

5.2.2 CDM model

Assuming that initial VVF of f =0 0.0016 The constant of  = 1 in weight function is selected

The critical damage parameter of due to void growth ( g )

crit

D and softening exponent ( ) would be calibrated

Table 5.4 Initial guess values and constrains for optimization process

in Table 5.5 and the optimal force-displacement curves are shown in Figure 5.2 Table 5.5 The best-fit material parameters for CDM model

Figure 5.2 The best-fit force-displacement curve using CDM model

5.2.3 Porous ductile model

The Table 5.6 shows boundary condition and initial values of input pparameters for the porous ductile model The best-fit material parameters that archived