Once the metallic material under deformation lead to the nucleation, growth and coalescence of voids that it is root of ductile damage.. 2 Figure 2.1 Ductile fracture mechanism of metall
INTRODUCTION
Nowadays, the sheet metals and their alloys have been widely applied in civil, automotive and aerospace industries thanks to their light weight and excellent strength characteristics These sheet metals are usually used to manufacture products which formed by forming process under plastic deformation Nevertheless, the phenomenon of ductile fracture is usually appearing in such metallic forming process Today, although high quality design and manufacturing processes can result in robust, strong products, the causation of fail cannot be avoided in some cases Figure 1.1(a) shows the ductile fracture of Ti6Al4V alloy which commonly used in aerospace industry under hydromechanical deep drawing [1] and Figure 1.1(b) displays ductile fracture in single point incremental forming of pure titanium [2]
(b) Figure 1.1 The ductile fracture under forming process of plastic deformation [1, 2]
2 The ductile fracture phenomenon will not be an issue if processes such as damage causation can be predicted or defective healing [3] is 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 For instance, the fractured position and moment can be exactly predicted by using the reliable damage models as shown in Figure 1.2 [2, 4]
Damage modeling and predicting of metal and nonmetal materials 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 (FEM) has mostly been used for numerical modeling of a material damage Two phenomenological approaches are usually used
Figure 1.2 Exact prediction of ductile fracture by numerical simulation [2, 4]
The first approach is based on theory of continuum damage mechanics (CDM) In this method, damage variable is modeled by either scalar variable integrated with an isotropic yield function for isotropic material [5, 6] or scalar variable integrated with an anisotropic yield function for anisotropic material [7] The main advantage of CDM theory-based model is that there are fewer material parameters compared with the
3 porous ductile material model that will be described in this work However, one of the main drawbacks of the CDM theory-based models is their limited validity in the region of lower and negative stress triaxiality when applied to multiaxial stress states [8] The several CDM theory-based typical damage models have been proposed by many researchers that could be found from literature [9-13] In this dissertation, a ductile fracture criterion based on the micro-void growth, N L Dung model [12], will be integrated with CDM theory to investigate the ductile fracture of anisotropic sheet metal
By observing the fracture surface measurement of high strength structural steel FeE690 using scanning electron microscope (SEM), Schiffman et al [14] revealed that the N L
Dung void growth model can be used to predict void coalescence Hoa et al [15] employed N L Dung model for seeking a maximum accumulated damage of the upsetting problem, the results displayed a coincidence with classical Gurson – Tvergaard – Needleman (GTN) model Trung et al [16] attempted to modify the N L
Dung model for only using stress triaxiality and Lode angle space, the modified model then shown excellently failure predictability of Mg alloy AZ31B [17]
The second approach is based on assumption that the original metallic material containing the second phase particles and/or the inclusions Once the matrix material under plastic deformation, the voids will be borne by separating between hard particles and matrix material or cracking of soft inclusions If the deformation of matrix material is continuous, the voids will be grown and coalesced together that is root of micro-crack [9, 10] With increased concentration of micro damage in a material, the spread and coalescence of which will cause the final macroscopic fracture of the material as critical conditions are reached The advantage of these damage models is that they can be used to describe micromechanical behavior of materials and the physical meaning of the damage parameters as porosity Hence these damage models are also known as the porous ductile material models A classical model that constituted based on this approach is that GTN [18-20] model The original GTN model is proposed based on the assumptions of isotropic and plastic-rigid material These assumptions have attracted attention of researchers, and many kinds of modifications to original GTN model have been proposed In order to consider effect of matrix material hardening on void evolution, the explicitly hardening exponent has supplied by N L Dung [21] into the
4 porous ductile yield function Grange et al [22] extended the GTN model by taking plastic anisotropy and viscoplasticity to represent crack propagation of hydrided Zircaloy-4 sheets The original GTN model is also cannot be used to predict the fracture under pure and simple shear stress states [23, 24] To improve the prediction accuracy in shear stress state, Xue [25] and Nahshon and Hutchinson [24] extended the GTN model through introducing the third invariant of stress tensor to consider the shearing mechanism The GTN model has been widely applied to predict ductile fracture of metallic materials [26-28]
Although there are many damage models can be found from literature and they are also having accurate predictability of ductile fracture for the various materials Recently, advanced high strength steel sheet and aluminum alloy sheet were extensively employed in automobile industry to satisfy the increasing requirement for high fuel efficiency and improved safety, the suitable material models are required for more accurately describing the plastic behavior in sheet metal Therefore, improving accuracy of ductile fracture prediction for various sheet metals by using various damage models is still needed to continue This research will therefore focus on developing the realistic damage models including CDM theory – based ductile fracture model and micromechanics theory - based porous ductile material model The N L Dung models [12, 21] will be enhanced into account the anisotropy and shear damage in sheet metals
An investigation of ductile fracture of sheet metallic material and their alloy using the modified GTN model and modified McClintock model, the N L Dung models [12, 21], will be performed in this dissertation Therefore, the following specific objectives had to be achieved in this work:
Firstly, Understanding the mechanism of microscopic ductile fracture of metallic materials and their alloys
Secondly, Improving the original damage models for predicting ductile fracture of anisotropic sheet metals and enhancing them for the case of shear damage prediction
Thirdly, Developing the user-defined material subroutine (VUMAT) for both porous ductile material model and CDM theories-based model
5 Fourthly, Conducting the experiments to determine the mechanical behavior of material and calibrate the material parameters for the constitutive models
Fifthly, Applying the damage models to predict the ductile fracture of metallic material and their alloys
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 [12, 21] are firstly enhanced to anisotropic sheet metal and they are also 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 FEM package using the numerical algorithms [29, 30] 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 verified via predicting ductile fracture of practical application (tensile test, deep drawing…).
A modified approach to the original damage model of isotropic materials was proposed by integrating the ductile damage criterion and yield function of porous ductile material with the quadratic yield criterion Hill48 The aim of this enhancement is to improve the predicted accuracy of ductile facture in the sheet metals This approach can be applied to the more sophisticated yield criteria
A modification of N L Dung models [12, 21] for shear damage case was also performed in this dissertation It is noted that the original N L Dung models [12, 21] are lack of the ductile fracture predictability under pure and simple shear loading conditions
Based on the numerical results, the analytical damage criteria that constructed by relation between the equivalent ductile fracture strain and the stress triaxialities have been proposed These criteria facilitate the implementing become more easier and they are helping to save computed time by using CDM theory instead of using sophisticated porous ductile material model
6 The forming limit diagram (FLD) of sheet aluminum alloy AA6061-T6 was also predicted by employing the proposed damage models together with the ISO 12004- 2:2008 standard
In this work, the user material subroutines incorporating CDM plasticity and micromechanical plasticity into the ABAQUS/Explicit finite element program have been successfully developed They can be used as source codes for implementing the various material models
The present thesis consists of seven chapters, the appendixes and a bibliography of cited references Below is a list of contents in each chapter
Chapter 1 gives motivation, defines objectives and presents an outline of the dissertation
DUCTILE FRACTURE OF METALLIC MATERIAL
Experimental investigations
Void nucleation can occur at particles by decohesion of the particle-matrix interface or particle fracture The void nucleation under plastic deformation of matrix material is categorized into two types of homogeneous and heterogeneous nucleation by Good and Brown [33] The homogeneous nucleation of voids within grains at lower temperature where grain boundary sliding does not make a significant contribution to the over-all deformation While void nucleation sites within grains are generally conjugated with hard second phase particles of non-metallic inclusions The heterogeneous nucleation of voids occurs at higher temperature and at lower strain rates Under those conditions the void appears more readily due to bulk deformation to grain boundary sliding Figure 2.2 shows typical void nucleation that observed by Kanetake et al [34]
Figure 2.2 Micro-void nucleation inside AA6061 aluminum alloy specimens: (a) interface deboning and (b) particle cracking [34]
9 The inclusions and the particles are cracked by heterogeneities of load carrying between clusters and the matrix, direction of microscopic crack is usually normal to tensile direction [35-37] Nevertheless, it found by many reseachers that, crack direction of inclusions was not necessarily perpendicular to the applied load, it was also in many different directions [38, 39] In addition, the voids are nucleated by interface deboning between the particles and matrix Based on observations by using electron microscope, Abbassi et al [40] have been found the second phase particles and non-metallic inclusions in the matrix material under uniaxial tensile load at room temperature (Figure 2.3)
Figure 2.3 Second phase and non-metallic particles in alloy steel [40]
A recent research based on X-ray microscopic photography of Landron et al [41] has been obtained three dimensional (3D) images describing process of void nucleation of double phase steel under deformation (Figure 2.4) According to Landron, in the double phase steel with small fraction of martensite, where the ferritic matrix is considered soft, and the inclusions of martensite are hard particles, voids mainly borne by decohesion at the interface between ferrite and martensite phases
10 Figure 2.4 Void nucleation in double phase steel: (a) 2D view, (b) 3D view [41]
Void nucleation models
The study of void nucleation using micromechanical modeling is much more limited than the study of void growth This is due to intrinsic difficulties of the topic which requires to introduce specific material properties for the inclusions and in some cases for the inclusion-matrix interfaces Depending on the particle size, modeling must either consider strain and stresses at the dislocation scale (small particles) or at the continuum mechanics scale (large particles) [42] The void nucleation is often quantified by a nucleation strain, the nucleation strain does not appear to be an absolute quantity for a given type of second phase particle Factors which influence the nucleation strain appear to be the strength of the particle-matrix interface, the stress state, and the flow strength of the matrix
Argon et al [43] proposed closed-form criterion that based on critical stress Once the stress of matrix material at interface surface reaches a critical value, the void will initialize at interface position between the particle and matrix material
Here, crit Argon is critical stress; e is equivalent stress of matrix material, 1
is mean stress; ij is delta Kronecker
Beremin et al [44] introduced void initiation criterion of ASTM A508 steel This criterion said that once stress inside second phase particle reaches a critical value, the
11 interface deboning between hard particle and matrix material or cracking inclusion will initialize
Where, i max is maximum principal stress in i - direction; 0 is initial yield stress of matrix material; K B is a function of particle shape
Based on the work of Chu and Needleman [45], the void nucleation is taken to obey a normal distribution Needleman and Tvergaard [46] proposed two void nucleation criterions that are based on a plastic strain-controlled nucleation and stress-controlled nucleation as shown in the Eq (2.3) and Eq (2.4), respectively:
Where, A N , B N are number of nucleated voids during the deformation of material; p is equivalent plastic strain of matrix material; N is average nucleated strain; s N is standard deviator; f is the flow stress of matrix material; N is average nucleated stress The Eq (2.3) is widely used because of its simplicity to determine material parameters via the experiment and calibration
The improvement of observed technique available today to observe microstructure so the precise quantitative measurements of void initiation can be achieved Based on the synchrotron X-ray tomography method, Bouaziz et al [47] and Maire et al [48] improved Needleman and Tvergaard nucleated model and precisely identified the stage of void initiation This criterion states that the number of nucleated voids N BM per a
12 volume unit is related to stress triaxiality and equivalent plastic strain of matrix material p The Bouaziz and Maire model is written as Eq (2.5) exp p p
Where, K BM is material constant; N crit N shear exp is the critical strain that the voids will be nucleated; N shear is critical shear strain that the voids will be initiated;
After nucleation stage the voids will continue to grow to a volume and shape determined by material properties and loading conditions.
Experimental investigations
By observing of micrographic slices obtaining by X-ray technique, it is readily to recognize the void size and shape change during matrix material under deformation [49- 52] These investigations mainly found the process of void growth in the necked area of the specimens, the results revealed that the void growth process is dominated by plastic strain and stress states in the specimens The early investigations are properly providing the volume, shape and two-dimensional distribution of the voids
Figure 2.5 Microscopic graph of AA6061 alloy [53]: (a) microscopic structure in an unetched condition; (b) void growth in notched specimen under uniaxial tension
13 For instance, the Figure 2.5 displays the observation of Agawal et al [53] for the AA6061 aluminum alloy specimen under uniaxial tension
Recently, using the advanced fractography devices, the researchers exactly shown three dimensional quantitative measures of void volume fraction (VVF) and density of the materials [54-57] Weck et al [55] used the series of micrographic slices obtaining from X-ray technique to present three dimensional void growth of the composite material (aluminum matrix contains zirconia/silica inclusions) The results exhibited that the void shape is cylinder when growing before stage of localized deformation (Figure 2.6a) while the shape of the void after this stage is diamond-like (Figure 2.6b)
Figure 2.6 The void nucleation (red color) by Zirconia inclusions (light blue color): (a) homogeneous deformation; (b) localized deformation [55]
Void growth model
By observing copper specimens, McClintock [9] considered the growth of cylindrical holes in a plastic deforming material (Figure 2.7 (a)) To facilitate the calculation, McClintock has only considered the growth of cylindrical single hole with elliptical section in cylindrical unit cell (Figure 2.7 (b)) According to McClintock the matrix material is also supposed to obey incompressible condition, i.e the volume of matrix material is always conserved before and after the matrix material under deformation
The evolution of void radius is calculated as Eq (2.6)
14 Here, r 0 is initial void radius and r is the instant value; r , r are the strain and stress in radial direction, respectively; z is axial stress The superscript indicates remote strain and stress fields
Figure 2.7 McClintock’s void growth model (a) solid contains the cylindrical voids;
In the case of cylindrical void surrounded by a cylindrical plasticity medium with hardening matrix under axisymmetric applied load, McClintock found the calculated equation for void radius evolution as follow,
Where, n is hardening exponent of matrix material; σ 1 and σ 2 are the principal stresses in 1 and 2 directions, respectively; ε 1 and ε 2 are the principal strains in 1 and 2 directions, respectively
Rice and Tracey [10] investigated evolution of an isolated spherical void in perfectly elasto-plastic cube A spherical void in an infinite non-hardening medium under remote simple tension stress field as shown in Figure 2.8, Rice and Tracey showed that the change of the void radius depends on the remote stress triaxiality
The rule of void radius evolution is proposed by Rice and Tracey as follow: σ 1 σ 2 σ 3
Figure 2.8 Rice and Tracey void growth model [10]
Where, m 1 2 3 / 3 is mean remote stress, 0 is initial yield stress of matrix material
2.3.2.3 Gurson-Tvergaard-Needleman (GTN) model
Based on an approximate calculation of plastic potential for any applied system of combined stresses using upper and lower bounds which obtained by Bishop and Hill [58], Gurson [18] employed a porous medium with its total volume of V total to obtain a close-form solution Two void geometries are considered: the long circular cylinder and the sphere The outer cell wall is idealized as geometrically similar to and centered around the void The matrix material is homogeneous, incompressible, perfectively elasto-plastic, and von Mises material is supposed The total volume of porous solid total
V is decomposed into two parts: matrix volume V matrix and void volume V void : total matrix void
Here, V matrix is matrix material volume, V void is total volume of the voids
Gurson employed the approximated velocity fields with upper bound calculations and proposed the two yield functions for porous ductile material corresponding to two kinds of void in matrix material The Gurson yield function has a form of truncated cylinder
16 as depending on hydrostatic pressure This yield function is also containing a damage variable (f) which describes the softening phenomenon of matrix material under plastic deformation void total f V
Figure 2.9 The Gurson void growth model: a) arbitrary voids in cubic solid, b) a spherical void in spherical solid [18]
For the spherical void (Figure 2.9b), the initial VVF of f 0 r a /r b 3 , current VVF
/ b 3 f r r Here, r a , r, r b are initial void radius, current void radius and spherical solid radius, respectively
The Gurson yield function in case of spherical void (Figure 2.9) is expressed as follow:
For the circular cylindrical void (Figure 2.10) with the initial VVF of f 0 r a /r b 2 and the current VVF f r r / b 2 , the yield function has form:
17 Figure 2.10 Circular cylindrical void in the cylindrical solid [18]
It is noted that the Gurson yield function become von Mises yield criterion once VVF equal to zero
Because Gurson model has not considered for hardening of matrix material and dropped interaction of adjacent voids Therefore, Yamamoto [59] and Tvergaard [19] recognized that the significant difference between their investigations and the numerical calculations that obtaining from original Gurson model From FEM analysis, Tvergaard [19] concluded that Gurson model underestimates the effect of porosity and stress triaxiality on the plastic response of voided solids He then incorporated three independent material parameters q 1, q 2, q 3 with the Gurson yield function The Gurson- Tvergaard model is given as follow,
In attempt to continue the McClintock’s work [9], N L Dung [60] investigated the evolution of the cylindrical and spherical voids in a cubic solid (see Figure 2.11) A remarkable aspect of the McClintock model is that consideration of void anisotropy during matrix material under plastic deformation Furthermore, McClintock’s investigation is also using an assumption of hardening matrix material The initial void shape is assumed by a circular cross-section and its shape will be becoming elliptically h 1
18 once matrix material is inelasticity In the N L Dung model, matrix material is assumed to obey exponent hardening rule To consider the void-void interaction, N L Dung employed the assumption of initial void shape is neither the cylinder nor the sphere and periodic distribution in the hardening material solid as illustrated on Figure 2.11c
Accordingly, the evolutive rule of void radius must be calculated for each void shape and for each various direction For the cylindrical void, the evolution of void radii is calculated in the two directions that normal to cylindrical axis while that the evolution of three void radii correspond to three various directions is calculated for ellipsoidal void The closed-form expression for void growth is then obtained by N L Dung as Eq.(2.15) and Eq (2.17) for cylindrical and ellipsoidal voids, respectively
Figure 2.11 N L Dung void growth model: a) cylindrical void; b) ellipsoidal void; c) void distribution in matrix material [60]
Two cylindrical void growth factors in i - direction are calculated as follow:
The rate of cylindrical void growth in i - direction is given by:
19 For the spherical void, three void growth factors in i - direction (i = 1, 2, 3) are calculated by Eq (2.16):
The rate of spherical void growth in i - direction is also proposed by N L Dung [12]:
Based on the analysis results of the void growth, a yield function for porous plastic material is introduced by N L Dung [21] as follow,
The Eq (2.18) has a form similar to the Gurson - Tvergaard [61] yield function
However, as a difference from Gurson - Tvergaard model because it has taken into account the influence of matrix hardening on VVF evolution by adding an explicit hardening exponent (n) into the second term of yield function For matrix material obeys a power rule hardening, N L Dung [21] revealed that the yield stress is decreased with the increasing of hardening exponent Several previous researches have indicated the dependence of matrix material on hardening exponent Koplik and Needleman [62] investigated the effect of initial VVF (f 0 = 0.0013 and f 0 = 0.0104) and that of the strain hardening exponent of matrix n (0, 0.1 and 0.2) on the evolution of VVF They concluded that the void growth is affected by f 0 and n The void evolution is decreased with decreasing f 0 or with increasing n, therefore the strain value to micro-crack occurrence (coalescence of voids) will be increased Corigliano et al [63] found that for high values of second factor in GTN model the strain hardening properties of the matrix
20 material are almost eliminated by softening due to intensified void growth, which strongly reduce the yield limit of the matrix material
Void coalescence leads to microscopic crack
The final step in the three idealized stages of ductile fracture process is neighbor void coalescence Once the size of voids reaches to a critical dimension the vicinity voids are coalesced together, microscopic crack is initiated, and matrix material is ruptured This mechanism occurs in very short time interval and is extremely difficult to recognize.
Experimental investigations
The stage of void coalescence occurs quite quickly Therefore, the researchers encountered the difficulties to exactly identify the time of void coalescence The advent of scanning electron microscopy (SEM) helped us deeply understanding about ductile fracture mechanism of metallic materials and their alloys Because ductile fracture mainly involves the growth and coalescence of microscopic voids that nucleated by the particles and inclusions, thus leading to the other modes of void coalescence Weck [64] and Benzerga [65] employed micrography technique to observe the coalescence of voids and they concluded that the void coalescence takes place when the deformation is localized within the ligament of the matrix material between the voids that located in the extreme region of the sample There are three void coalescence modes have recorded by the scientists are that void coalescence occurring in direction perpendicular to applied load (red arrows) is observed by Thompson [66], Benzerga et al [65] and Weck [64]
The first mode of coalescence, exhibited in the sequence (a–c) on Figure 2.12, is the
“internal necking” mode of coalescence, where the ligament between the two voids shrinks with a shape typical of a necking process
The second mode of void coalescence, called “necklace coalescence” is less common
The “necklace coalescence” mode has been observed to take place in columns of neighbor voids It comprises a localization deformation in a direction parallel to the applied loading axis [67] (see Figure 2.13)
The third mode of void coalescence shown in the sequence Figure 2.14 consists in a
“shear localization” between larger voids observed when the initial voids are distributed
21 Figure 2.12 The first void coalescence mode: a) Benzerga [65] and b, c ) Weck [64]
Figure 2.13 Second mode of void coalescence a) and b) Benzerga [67] and c) Pardoen et al [68])
Figure 2.14 Third void coalescence mode a) and b) Weck [64]; c) Benzerga [67] a) b) c) a) b) c) a) b) c)
22 along lines that lie in incline angle from the applied loading direction This mode of coalescence is frequently observed in high strength metal with low or moderate strain hardening capacity [67] By using technology of laser drill, Weck [64] found coalescence in a micro-shear band of AA5052 aluminum alloy sheet (see Figure 2.14a, b).
Void coalescence models
McClintock [9] proposed a ductile fracture criterion based on the analysis of cylindrical void growth This criterion is verified by McClintock for the rigid-plastic material (hardening exponent n0) and linear hardening material (n1) The damaged rate based on the cylindrical void growth in 2-direction (cylindrical axis of 3 direction) as follow:
Once equivalent plastic strain reaches to a value of fracture strain then coalescence of neighbor voids will be taken place The equivalent plastic strain at fracture is calculated as follow,
Where f p is equivalent plastic fracture strain, l 2 0 is initial space of two neighbor cylindrical voids in 2-direction and r 0 is initial radius of a cylindrical void
Brown and Embury [69] stated that the void coalescence takes place when the space of two neighbor voids, L m nearly equal to the size of void At low level of stress triaxiality, the change of neighbor void space is not significant because the void size mainly increases in tensile direction However, at medium level of stress triaxiality (biaxial
23 tension), the void size was simultaneously extended in the tensile direction and perpendicular direction to tensile direction so that the coalescence of voids occurs more quickly The Brown and Embury employed the assumption of homogeneous void size and periodic void distribution, they proposed a formula for the mean planar spacing of spherical voids as Eq (2.21)
Where, L m is mean space of two neighbor voids, f is VVF of void and r m denotes the mean void radius
Tvergaard and Needleman [20] proposed a void coalescence criterion that based on the critical VVF (f c ) This criterion is regardless to initial void shape The void distribution in matrix material is assumed to obey standard distribution rule The evolution rate of VVF is expressed as follow, g N df df df (2.22)
Where, df g is the rate of the growth of existing voids in matrix material:
1 : g kk ij df f d (2.23) df N denotes the rate of evolution of the VVF due to nucleation of voids during matrix material under deformation p
Here, A N is number of nucleated voids during matrix material under deformation and given in Eq (2.3)
The function of VVF to account for the effects of rapid void coalescence from the moment of micro-crack initiation to the fractured occurrence:
Where, f F is VVF at fracture (macro-crack initiation), f u 1/q 1 is VVF at complete loss of load capacity of matrix material
Using the evolutive rule and coalesced law of voids due to Tvergaard and Needleman proposed above, the GT model (Eq (2.13)) is now called GTN model and it is rewritten as follow,
2.4.2.4 Void coalescence model due to shear mechanism
The original GTN model has some limitations when it is employed to predict ductile fracture under the range of very low and negative stress triaxialities [24, 25, 70] To improve the shear damage predictability (stress triaxiality nearly equal to zero) of the original GTN model, Xue [25] and Nahshon and Hutchinson [24] have proposed the void coalescence models due to the relatively shearing voids during process of plastic deformation
The VVF evolution rate proposed by Xue [25] was derived from circular void growth under the simple shear strain as follow,
Where q 4 and q 5 are geometrical parameters, q 4 1.69 and q 5 1/ 2 for 2D case,
4 1.86 q and q 5 1/ 3 for 3D problem g L represents a function of Lode angle,
The Lode angle is defined as below,
Where s 1 , s 2 and s 3 are the components of the deviatoric stress tensor on principal plane
Nahshon and Hutchinson [24] have also introduced a VVF evolution rate that it is able to describe the softening in the material under pure shear load The basis difference between Xue and Nahshon and Hutchinson models is that the Nahshon and Hutchinson model allow to implement the GTN model in a way that does not change model for both tensile and axisymmetric loading conditions, i.e., stress states are dominated by stress function ij as follow
In the above expression, k is a numerical constant, s ij is deviatoric stress tensor,
is a function of stress state,
For all axisymmetric (equi-biaxial) stress states, ij is equal to zero while for all states consisting of pure or simple shear plus a hydrostatic stress, ij is equal to unity; J 3 1/ 3 s s s ij jk ki is third invariant of deviatoric stress tensor
N L Dung [60] introduced a criterion of void coalescence that based on critical accumulated damage variable
For the cylindrical void in hardening material solid (cylindrical axis in 3 direction), there are two critical accumulated damage variables in 1 and 2 directions:
For the ellipsoidal void in hardening material solid, there are three critical accumulated damage variables in 1, 2 and 3 directions:
Where, the subscripts i = 1, 2, 3; j = 2, 3, 1; k = 3, 1, 2; respectively, f p is ductile fracture equivalent strain of matrix material
DUCTILE FRACTURE MODELLING
The constitutive equations of void growth based CDM model
The CDM theory has been widely used to investigate process of ductile fracture in metallic forming [71-74] Because of its advantage is easy to implement and requires least the calibrated material parameters The CDM models are based on the concept of Kachanov [75], who was the first to introduce a scalar damage variable, which may be interpreted as the effective surface density per unit volume Kachanov pioneered the topic of CDM by introducing the concept of effective stress This concept is based on comparing between a fictitious undamaged configuration and the actual damage configuration Kachanov’s work is continued by researchers in different fields applied CDM to their field e.g., Lemaitre [76] for ductile materials
The yield criterion in stress space is expressed in the following form:
(3.1) where is softening exponent and e denotes equivalent stress, For isotropic matrix material, von Mises equivalent stress is adopted as
e (3.3) For anisotropic matrix material, Hill48 equivalent stress is used as follow
28 The constant tensor H ijkl for Hill48 material is represented by 6x6 matrix as in the material principal coordinate system:
The damage variable D is considered as internal variable corresponding to a material degradation 0 D 1, where D0 corresponds to a virgin material while D1 corresponds to fully damaged material
D crit is critical value of damage variable due to void growth, dD i g denotes the evolution rate of damage variable For the ellipsoidal void, N L Dung [21] has proposed three critical accumulated damage variables in i (i = 1, 2, 3) directions i.e.,
The damage accumulated criterion of ellipsoidal void:
29 The subscripts i = 1, 2, 3; j = 2, 3, 1; k = 3, 1, 2; respectively, p and f p is equivalent ductile strain and plastic strain at fracture of matrix material, respectively i j k , , denote the principal stress components
For recent studies, beside the damage evolution as function of stress triaxiality, the Lode angle parameter is usually used to describe the dependence of damage envelope on the third stress invariant [77, 78] In attempt to consider the simultaneous effect of stress triaxiality and Lode angle on damage evolution, Trung et al [16] has modified the N L
Dung model by transforming the principal stress components from the principal stress space to a space represented by stress triaxiality and Lode angle in the cylindrical coordinates Accordingly, three damage variables for von Mises material are written in explicit form as follows,
However, for the past studies, the Lode angle parameter is usually used as a way to overcome the limit of void growth based damage model under shear damage [25, 79]
Therefore, in this study, the original N L Dung damage criterion is conjugated with shear damage component containing Lode angle parameter as presented below
Noting that the original N L Dung damage criterion has developed based on an assumption of von Mises matrix material so that applying this criterion to Hill48 matrix
30 material just only is a heuristic enhancement The difference of damage evolution and loading response when using Hill48 yield criterion compared to those of von Mises criterion is adjusted by calibrating the material parameters through uniaxial tension tests in ASTM-E8 standard
The Hill48 yield criterion widely used to describe the anisotropy of metallic material
Because it has the advantage that its basic assumptions are easy to understand; the parameters included in the yield functions have a direct physical meaning In addition, the model has a simple formulation for the 3D case The criterion needs a small number of mechanical parameters for determining the yield function
Employing the associated flow law, the rate of plastic strains of Hill48 yield criterion can be found as
Where d plastic multiplier of Hill48 yield surface
From plastic strain rate components, the Hill48 equivalent plastic strain rate is defined as:
48 2 48 : 1 : 48 p Hill p Hill p Hill ij ijkl kl d d H d (3.19)
Where H ijkl 1 is the pseudo-inverse matrix of Hill48 material
Using above pseudo-inverse matrix, the Eq (3.19) can be written explicitly as follow
2 2 2 p Hill p Hill p Hill p Hill p Hill p Hill p Hill
The relations between the Lankford’s coefficients ( R R 0 , 45 , R 90 ) and anisotropic coefficients (F G H L M N, , , , , ) can be easily obtained from the associated flow rule to Hill48 plastic potential function
The Lankford’s coefficient in rolling direction (1-direction) as
The Lankford’s coefficient in transverse direction (90 o direction to rolling direction):
To calculate Lankford’s coefficient in 45 o direction to rolling direction, counter- clockwise rotation of the 1-2-3 coordinates system about its 3-axis an angle 45 o A 1’- 2’-3’ new coordinates system will be obtained as the same time Assuming uniaxial tensile load, , acting on 1’- direction the stress components can be calculated as follow
Using a rotated matrix about 3-axis with an angle of as
The stress components are written as
RT T is transposed matrix of rotated matrix RT 3
Figure 3.1 Illustration of rotation of coordinates system about 3-axis
33 Where f is uniaxial tensile stress (see Figure 3.1)
The rates of the plastic strain are calculated by 1’-2’-3’ coordinates system as
The Lankford’s coefficient in 45 o -direction to rolling direction as
To aid the calculations in the next chapters, the anisotropic coefficients of Hill48 yield criterion (F G H L M N, , , , , ) are calculated herein Assuming that the direction of flow stress that of uniaxial tensile test coincide with the rolling direction and the Hill48 yield surface is written in form,
Therefore, from Eq (3.6) and Eq (3.32) we can be obtained Eq (3.33)
By using the Eq (3.24), (3.25), (3.31) and (3.33), the anisotropic coefficients can be determined as
Two remaining anisotropic coefficients are chosen as L M 3 / 2
It is noted that, with the above selection of anisotropic coefficients then the Hill48 equivalent stress (Eq (3.4)) and plastic strain (Eq (3.23)) become those of von Mises material once the Lankford’s coefficients R 0 R 45 R 90 1.
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 [80] 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
2 damage due to relatively shearing and rotational voids damage due to void growth ln ln 1 sinh 1
In the Eq.(3.38), the first term on the right-hand side is damage evolution due to relative shear voids and the second term is due to void growth
Xue [25] has modified McClintock’s solution for shear damage by introducing a void shear configuration as shown in Figure 3.2
The distance between void surface and cell boundaries is calculated as a 2 r (3.39)
The shear strain component is determined via the deformation angle , i.e.,
35 Figure 3.2 Illustration of void shear mechanism [25]
The minimum distance after configuration is deformed
The artificial strain is defined by Xue [25] to consider for the reducing of the ligament as ln ln 1 2
Relation between engineering shear strain and equivalent plastic strain is expressed
Assuming that 1 2 3 the damage variable of N L Dung criterion (Eq (3.7)) can be written as follow
Where dD g is rate of damage evolution due to void growth
36 Where D crit g is the critical value of damage variable due to void growth
According to McClintock [80], the shear strain due to void shear is calculated
Xue [25] defined the shear damage variable associated with the shearing of the void as follow, art shear
From Eq.(3.42) and (3.47), the shear damage variable is calculated
The rate of shear damage evolution is
Ratio of 0 / 2r 0 calculated through initial VVF i.e.,
For N = 3/2 the Eq (3.50) is applied to von Mises material
37 Beside the ductile fracture depend on stress triaxiality, the Lode angle is frequently used to determine the fracture strain for each loading case The Lode angle is defined by the smallest angle between the line of pure shear and projection of the stress tensor on the deviatoric plane It is related with the third invariant of deviatoric stress tensor and is expressed as
The Lode angle parameter L is defined here in to normalized Lode angle, i.e.,
To accounts for the effect of different stress states on damage evolution, Trung et al
[16] introduced a weighting function depend on Lode angle as follows
(3.54) where ϛ is a constant to be determined from the experimental–numerical correlation analysis
Finally, the damage criterion for general loading case is written
Complete damage occurs once damage variable reaches to unity i.e., D1
Beside CDM theory based damage models, the models for coupling between plasticity and damage can be achieved by using damage micromechanics theory – micromechanical damage models, e.g., Gurson type models [18, 81, 82] These models are usually using a yield function integrated with a porosity (f) to describe the softening phenomenon of matrix material under plastic deformation Beside the capability of predicting macro-crack initiation by VVF at fracture (f F ), the porous ductile model can
38 be helped to estimate moment of micro-crack, it is interested in forming process to avoid waste, through the critical VVF (f c )
Anisotropic metal plasticity has been a hot topic for many decades and numerical models have been proposed, including both phenomenological models at the microscopic level and crystal plasticity models at the microstructure level [83-88] The sheet metals for stamping applications usually display certain extent of plastic anisotropy due to cold or hot rolling processes
There are two key limitations of the GTN type model, namely the lack of capability in modeling shear damage and anisotropic materials First, shear damage taking place under zero hydrostatic stress conditions cannot be described by the void evolution process In order to model shear failures using the GTN model, Xue [25] and Nahshon and Hutchinson [24] introduced an additional shear damage variable into the void evolution This shear damage term is superposed ad hoc in the GTN model Second, applications of the GTN model are limited to isotropic materials obeying the von Mises yield criterion Several efforts have been introduced to incorporate the anisotropy of matrix materials, e.g [26, 82, 89-91] In these studies the Hill48 quadratic yield criterion of orthotropy [92] was applied to consider the anisotropy of matrix materials With these enhancements, the GTN becomes capable to predict failures in sheet metal forming applications of anisotropic materials
Hardening of the matrix material is another important aspect, yet received less attention [93] being related to a key assumption of the original Gurson yield function The original Gurson model [18, 94] assumed an isotropic elastic-perfectly plastic matrix material
Influence of the strain hardening of matrix materials in modified versions of the model is commonly accounted for by adjusting the model factors in a heuristic manner Prior studies reported that the evolution of void growth decreases with increasing strain hardening exponent, thereby leading to an increase in the strain to micro-crack (coalescence of voids) [62] Also, by setting high values of the second factor the strain hardening properties of the matrix material are almost eliminated by softening due to intensified void growth, which strongly reduce the yield limit of the matrix material [63] N L Dung [21] argued that the hardening effect of matrix materials should be
39 included in the yield function The author then introduced a modification to the Gurson model by having an explicit strain hardening term in the place of the second factor of the GTN yield function The N L Dung model is, however, limited to isotropic matrix materials [21] and not equipped with a porosity evolution law accounting for the shear damage The focus of this work is to introduce an extension of the N L Dung model to incorporate both the shear damage effect and material anisotropy Material anisotropy is accounted for by assuming that the matrix material obeys the Hill48 quadratic yield criterion
Now the original N L Dung model [21] will be integrated with Hill48 quadratic yield criterion and rewritten as follow:
The Figure 3.3 shows the yield surface of the Dung-Hill48 model with a closing shape as the dependence of yield function on hydrostatic stress
Figure 3.3 The yield surface presentation of the Dung-Hill48 and pure Hill48 models in normalized principal stress space
In the case of Lankford’s coefficients, R 0 R 45 R 90 1, Hill48 equivalent stress e Hill 48 become von Mises stress e Mises and the modified N L Dung model, Eq.(3.56) become original N L Dung model
40 Three special cases should be noted of the Dung-Hill48 yield function as follows:
First, for the case of VVF, f 0, the yield surface (3.56) become pure Hill48 yield surface (3.32)
Secondly, once the hydrostatic stress is zero, 1
, value q 2 q 1 2 can be also chosen, the yield function (3.56) will be reduced as Eq (3.57) and the strength of porous ductile material is decreased by the factor (1 – q 1 f) The ductile damage without occurrence in this case, therefore, there is a drawback of ductile model dependent on hydrostatic pressure for predicting ductile fracture under pure shear stress state
Thirdly, in the case of equivalent and hydrostatic stress components equal to zero, i.e
Since the corrected parameters in the Dung-Hill48 yield function can be selected as
The Eq (3.59) and (3.60) shows the VVF value at which matrix material complete lose carrying load
The plastic strain rate tensor can be calculated by the associated flow rule to the plastic potential function Dung Hill 48 as below:
is given as the Eq from (3.13) to (3.18)
The rate of volume plastic strain is calculated as
Based on incompressible condition, the rate of the evolution of VVF is calculated in the rate of volume plastic strain The VVF is given as void void total void matrix
Taking the derivative of f with respect to time and using incompressible condition, i.e., dV matrix 0, and d kk p dV void / V void V matrix , the rate of the VVF evolution without the void nucleation can be obtained as
By using the Eq (3.63) and Eq (3.65) the rate of VVF evolution can be calculated for the porous ductile material model
NUMERICAL IMPLEMENTATION OF THE DUCTILE DAMAGE MODELS
Constitutive equations
A numerical algorithm for pressure dependent elasto-plastic model of Aravas [30] and Zhang [97] is employed in this dissertation The flowchart of stress integration algorithm of porous ductile model is given in Appendix II
The total strain rate tensor is divided into two parts: elastic and plastic strain rates e p ij ij ij d d d (4.18)
Where the subscripts e and p imply elastic and plastic states, respectively
The Cauchy stress tensor is calculated based on assumption of linear elasticity, thus
: e : p ij C ijkl kl C ijkl kl kl
2 2 ijkl ik jl 3 ij kl
The shear and bulk moduli are
The trial stress state is calculated as
The Hill48 equivalent and hydrostatic stresses are given below:
The Cauchy stress tensor can be written as
Here n ij is unit normality vector in deviatoric stress space
With s ij is deviatoric stress tensor
Since the porous ductile material model based on the evolution of VVF f , once f reaches to a critical value the micro-crack is initialized Supposing that the matrix material obeys isotropic hardening rule f p , here e p is the equivalent plastic strain of matrix material The flow stress of matrix material is usually exponent law as
48 The evolution of equivalent plastic strain p is found by condition that the plastic work of macroscopic stresses and strains equal to the energy dissipated in plastic deformation at the microscopic level
1 f f d p ij : d ij p (4.30) Which equivalent plastic strain rate can be obtained as
The Dung-Hill48 model is rewritten to assist analysis of the parameters and the variables in the yield function
Two parameters of q 1 and q 2 are proposed by Tvergaard [19, 61] for original Gurson model [18], to correct numerical results with original Gurson model These parameters are also recommended by N L Dung [21] for his original yield function by choosing q 1
= 1.5 and q 2 = 2.25 For this study, these parameters are identified by global optimization using Genetic Algorithm in chapter 5
The function of evolution of VVF f * f is introduced by Tvergaard and Needleman
[20] to account for the stage of void coalescence (from critical VVF f c to VVF at macro- crack f F ) take place more quick than before void fraction growth stage
The evolution of VVF includes three parts: a part of presence voids, a part of nucleated voids during matrix material under deformation and evolution of VVF due to shear load
Since matrix material is assumed incompressible, the evolution of VVF due to volume expansion as follow
The nucleated voids during the matrix material under deformation are assumed that the random and standard distribution p
The function of normal distribution proposed by Chu and Needleman [45] containing the average nucleation strain N and standard deviator s N as
Figure 4.2 Presentation of normal distribution function of void nucleation respect to equivalent plastic strain
The evolution of VVF due to shear dominated loads is recently proposed by Nahshon and Hutchinson [24] for the GTN model As describing in previous chapter, once hydrostatic stress equal to zero, the Dung-Hill48 model could not be used to predict the ε N
50 ductile damage under pure shear load conditions Therefore, the modification of original GTN model for shear damage is also applied to this study
48 : ij p s Hill ij ij e df k f s d
Where k a numerical constant considering effect of pure shear load, ij a function of stress state
The third invariant of the deviatoric tress tensor J 3 det s ij
By using the associated flow rule, the plastic strain rate tensor is calculated
Dung Hill Dung Hill Dung Hill p ij ij Hill ij ij m e d d d n
Substituting the Eq (4.41) and (4.42) into equation (4.40) yields
Elimination of d from (4.41) and (4.42), we obtain
Dung Hill Dung Hill p Hill q e m
51 Inserting the Eq (4.43) into (4.19) and integrating from t k to t k+1 gives
The hydrostatic and equivalent stresses in the Eq (4.45) are e m m K p
Aravas [30] was found that the stress components in the Eq (4.45) return yield surface along the unit vector n ij k 1 , thus n ij k 1 can be calculated from trial stress state
Implemented procedure
Given the stress tensor ij k at time t k , the size of time step t and plastic strain rate tensor d ij p The main objective is to find ij k 1 at time t k 1 In section 4.3.1 was found that set of nonlinear equations at time t k 1 contains two principal unknowns p and
Set of the nonlinear equations as below:
- The Dung-Hill48 yield function (4.32) - The nonlinear equation (4.44)
- The equation of stresses state (4.45) - The equations of evolution of two state variables (4.31) and (4.34) Two internal variables of equivalent plastic strain p and VVF f are considered as two state variables and denoted by vector H j (j = 1, 2)
The derivatives of two state variables respect to time are calculated as the Eq (4.31) and (4.34)
The integration in time of two internal variables are given
In order to find p and q at time t k 1 which simultaneous satisfy the both of hardening rule (4.29) and yield function (4.32) The two Eq (4.32) and (4.44) would be solved by an iterative method of Newton-Raphson The two equations are re-written as
Dung Hill Dung Hill p q Hill p q e m
To simple, the superscript k 1 will be aborted The roots of p and q are found by using the plastic-corrected iteration algorithm
Where i denotes an iterative counter Two plastic corrections of p and q are found by first order Taylor expansion of the functions of E 1 and E 2 as below
Updating the stress state and solution dependent variables
In this study, stress integrating algorithm “Euler backward algorithm” of Aravas [30] is employed to update stress state at time steps The algorithm applied to Dung-Hill48 model as follow
The set of nonlinear Eq (4.58) and (4.59) are written in matrix form as
Where the factors K ij are calculated as below
Dung Hill Dung Hill p Hill Hill e m e p
Dung Hill Dung Hill p Hill Hill e e q
Dung Hill Dung Hill q Hill m e m q
Two equations of E 1 and E 2 can be written as
Once trial stress state is calculated and plastic condition is satisfied, i.e., Dung Hill 48 0 an iterative program that based on Newton-Raphson method will be employed through the steps as follow
- Step 1: Set the iterative counter i = 0
- Step 2: Initialize the values at time step t k If t k t 0 0 then p q 0
- Step 3: Calculate the factors K ij
- Step 4: Calculate the equations of E 1 and E 2
- Step 6: Check convergence, if E 1 10 7 and E 2 10 7 then update new stress state
If not convergence, i.e., E 1 10 7 and E 2 10 7 then go to step 3
The derivatives of Dung-Hill48 model
To solve the set of nonlinear equations in section 4.3.2 the derivatives of Dung-Hill48 model is required
denotes the slope of equivalent plastic strain versus flow stress curve that fitted from uniaxial tensile data
57 In order to calculate the derivatives of H / p and of H / q First, the functions of H 1 and H 2 are written in a discrete-time form
Supposing that the derivatives of function H 2 respect to time are constant due to the time step t t k 1 t k can be chosen as tiny number
H t t t H H th H (4.93) Taking the derivative of Eq (4.93) with respect to p gives
so the Eq (4.95) and (4.96) are written in matrix form as
Similar to above, the derivative of Eq (4.93) with respect to q and
We can obtain the set of nonlinear equations with unknown q as
Solving two set of Eq (4.96) and (4.97), the variables 1 , 2 , 1 , 2 p p q q
will be obtained Substituting all values of derivatives in this section into the factors K ij in section 4.3.3, the plastic corrections p and q will be found
Verification of user-defined material subroutine
Verification by the unit elements
After successfully developing any user-defined material subroutine and compiling it, a verification of VUMAT subroutine is strongly recommended by the VUMAT experts to verify the accuracy of the implementation against known values and element distortion
The VUMAT subroutine will be verified by the unit element under uniaxial tension and under simple shear load Element type of linear brick element, with reduced integration element (C3D8R) is used for verifying
Initial size of each element edge is 1 mm Geometries and boundary conditions and displacement control load of 2.5 mm are given as Figure 4.3
(a) (b) Figure 4.3 Unit element (a) uniaxial tension and (b) simple shear y x z y x z
The matrix material is assumed to obey a Swift isotropic hardening rule with exponent form as follow
Where K and ε 0 are material constants, K 489.74 MPa and 0 0.02 Three hardening exponents of n0.1; 0.134; 0.179 are used The true stress-strain curve that used as input of numerical simulations is given in Figure 4.4
Figure 4.4 True stress-strain curve
Young’s modulus E74600 MPa, Poisson’s ratio 0.3 For anisotropic material, the Lankford’s coefficients r 0 0.55; r 45 0.52; r 90 0.53 are used
The material parameters of the CDM model: D crit g 0.75; 1.0; 1.25, 1.0; 2.0; 3.0 The material parameters of the porous plastic model: q 1 1.5 and q 2 2.25, initial VVF
0 0.04 f , N 0.3, s N 0.1, f N 0.04 The VVF growth acceleration stage from f c to f F is not considered herein i.e., a value of f c = f F = 1 is set
4.4.1.3 The results using CDM model
Figure 4.5 plots the evolution of damage variable as function of equivalent plastic strain
With the higher softening exponent, damage quicker accumulates which reflects matrix material will be failed earlier
Figure 4.5 Effect of softening exponent on evolution of damage variable D crit g 1
Figure 4.6 Effect of softening exponent on equivalent stress D crit g 1
The relation between equivalent stress and equivalent plastic strain corresponding to various softening exponents (β) is shown in Figure 4.6 For the lower softening exponent, the maximum value of equivalent stress is lower than that of higher softening exponent However, the material fails faster because of material degradation induced by damage variable evolution
Effect of critical damage parameter D crit g
The damage evolution versus equivalent plastic strain relationship is given in Figure 4.7, whereas Figure 4.8 shows the element equivalent stress corresponding to equivalent plastic strain For both the uniaxial and shear loading conditions, it is showed that the smaller value of critical damage variable prompting the damage evolution and decline of element stress which reflects the material damage occurs earlier comparing to higher value of critical damage variable
Figure 4.7 Effect of critical damage parameter on the evolution of damage variable of unit element under uniaxial tension 3
Figure 4.8 Effect of critical damage parameter on the equivalent stress of unit element under uniaxial tension 3
4.4.1.4 The results using porous ductile model
Figure 4.9 Effect of hardening exponent on VVF evolution
Figure 4.10 Effect of hardening exponent on equivalent stress To have investigation about effect of hardening exponent on evolution of VVF and element equivalent stress - strain response of Dung-Mises model, the several values of
0.1; 0.134; 0.179 n are selected Figure 4.9 shows the VVF evolution corresponding to equivalent plastic strain, the high hardening exponent delays the VVF evolution which leads to local damage happening more slowly For the hardening exponent
0.134 n the VVF evolution of Dung-Mises model is coincident with that of GTN model which reflects difference between Dung-Mises model and GTN model can be adjusted by the second term in each yield function Nevertheless, it is noted that the hardening exponent in N L Dung model is always conjugated with a hardening rule whereas the second factor in GTN model is independent parameter Here, the second
63 factors of n = 0.1 and n = 0.134 in N L Dung model is considered as an independent parameter to convenience in comparison between results by N L Dung model and those of GTN model
Figure 4.10 shows relation between the equivalent stress and equivalent plastic strain of unit element under uniaxial tension For the various hardening exponents, at beginning the curves are coincident But after equivalent plastic strain reaches about 0.4, the equivalent stress corresponding to smaller exponent decreases rapidly than that of higher exponent because material properties are deteriorated due to the VVF evolution
The VVF evolution is presented as a function of equivalent plastic strain and shown Figure 4.11 while Figure 4.12 displays element equivalent stress - strain response when using user-developed model (N L Dung model) and porous plasticity model (GTN model) in ABAQUS/Explicit constitutive model libraries It shows that the results by Dung-Mises model and Dung-Hill48 (R 0 = R 45 = R 90 = 1) model are coincident with those of GTN model These results reflect the correctness of user-developed subroutine (VUMAT) for N L Dung model For anisotropic material, the evolutional rate of VVF and equivalent stress by Dung-Hill48 (R 0 = 0.55, R 45 = 0.52, R 90 = 0.53) corresponding to equivalent plastic strain has slightly influence on the results Particularly, the equivalent stress by Dung-Hill48 model is smaller and the damage evolution is slower
64 Figure 4.11 Effect of Lankford’s coefficients on VVF evolution versus equivalent plastic strain
Figure 4.12 Effect of Lankford’s coefficients on the equivalent stress correspond to equivalent plastic strain
Figure 4.13 Effect of shear coefficient ( k ) on VVF evolution
65 Figure 4.14 Effect of shear coefficient ( k ) on the element equivalent stress
For the case of the single element under simple shear condition
xx yy , zz 0; and d xx d yy 0, d kk 0 , the evolution of VVF depend on the shear damage variable f s with a numerical constant k For k 0, no damage occurrences because of the VVF of existing voids f g 1 f d kk p : ij 0 and VVF due to the shear voids f s 0, the evolution of VVF is only dominated by initial VVF f 0 and nucleated void growth f N A d N p with maximum value of f f 0 f N For the other values of k 0, k 1.0, 2.0, the VVF includes two components of f N and f s The evolution of VVF is increasing correspond to with increasing of value of k (see Figure 4.13) Figure 4.14 shows equivalent stress versus equivalent plastic strain curve of single element under pure shear load for three cases of k 0, 1, 2 using Dung-Hill48 and Dung-Mises models It is showed that the equivalent stress by Dung-Hill48 model is bigger than that of Dung-Mises model This is due to Lankford’s coefficients R 0 = 0.55,
R 45 = 0.52, R 90 = 0.53 have effect on the material behavior, i.e anisotropic material has better shear damage subjected capability than the isotropic material For case of k 0 the equivalent stress of element does not decrease because of VVF evolution does not exceed the VVF value of f f 0 f N , it means that no damage occurrences in matrix material
Verification by tensile and deep drawing tests of AA6016-T4 aluminum alloy
In this section, the verification of VUMAT subroutine would be conducted for uniaxial tensile and deep drawing tests Sheet aluminum alloy material AA6016-T4 with its thickness of 1 mm, after Kami et al [91], is used Young’s modulus and Poisson ratio of 70 GPa and 0.33, respectively The Swift hardening law with the coefficients: K 525.77 MPa, 0 0.011252, n = 0.2704 The Lankford’s coefficients: R 0 = 0.5529, R 45
= 0.4091, R 90 = 0.5497 The material parameters for Dung-Hill48 model are taken following suggestion of Kami et al [91] and given in Table 4.1, whereas the material parameters for CDM-Hill48 model are fitted by displacement-force curve of tensile test and are given in Table 4.2
Table 4.1 The material parameter for Dung-Hill48 model
Table 4.2 The material parameter for CDM-Hill48 model
To save computational time, the numerical simulation of tensile test is only performed by the gauge section (length x width x thickness: 80 mm x 20 mm x 1 mm) of the whole experimental specimen (length x width x thickness: 200 mm x 20 mm x 1 mm) Mesh, boundary conditions and dimensions of finite element model are shown in Figure 4.15
Element type of 3D stress, 8 nodes, reduced integration (C3D8R) and the initial size of each element edge of 0.5 mm have been used to mesh model The displacement load of 25 mm is applied to the right edge of model
67 Figure 4.15 Geometry, mesh and boundary conditions of tensile test
Figure 4.16 Comparison of the crack path (a) experiment [91], (b) CDM-Hill48 model, (c) Dung-Hill48 model
Figure 4.16 displays the identical crack path obtaining by numerical simulations using the CDM-Hill48 model and Dung-Hill48 model and experiment It can be concluded that the damage models are suitable to predict the ductile fracture of aluminum alloy in the case of uniaxial tension
Figure 4.17 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
68 experiment This may be due to effect of hardening exponent in Dung-Hill48 model on the VVF evolution during plastic deformation process
Figure 4.17 Force – displacement curves of tensile test
This section presents the testing VUMAT for the case of more complicated load - the deep drawing of square cup Due to the symmetry, the square cup deep drawing test is modelled by one quarter of whole finite element model as shown in Figure 4.18 The circular blank with dimension of 85 mm and its thickness of 1 mm In order to avoid any effect of wrinkling and damage phenomenon of the blank, the holding load of 10 kN is applied holder Diagram of the tooling setup and finite element model are given in Figure 4.18
The circular blank with dimension of 85 mm and its thickness of 1 mm is used The initial mesh size of the blank is 0.5 mm x 0.5 mm and three element layers through sheet thickness The element type of 3D, 8-nodes, reduced integration (C3D8R) used for all analyses The punch, holder and blank are modeled by an assumption of the absolute rigid body with the 3D discrete rigid element type (R3D4) The contact interaction is assumed to obey Coulomb law, i.e., friction coefficient of 0.05 is applied to all contact surfaces
Figure 4.18 Diagram of the tooling setup in square cup drawing (a) dimensions (unit: mm) and (b) finite element model
Figure 4.19 Comparison of fracture path between experiment and numerical simulations (a) experiment, (b) CDM-Hill48 model, (c) Dung-Hill48 model
The simulation process takes place in 2 steps In the first step, die and punch are fixed whereas blank is clamped by moving holder in vertical direction and kept by a holding force of 10 kN during this step In the second step, while the boundary conditions of the
70 first step are maintained, the blank is stretched by moving the punch until its fracture occurs
The predicted fracture path by numerical simulations have identical shape and location with experiment is shown Figure 4.19 The forming force versus punch stroke curve is presented by Figure 4.20 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 4.20 Comparison of forming force curve between experiment and the numerical simulations
IDENTIFICATION OF MATERIAL PARAMETERS
The calibrated approach and procedure
In the previous studies, the calibration of damage model parameters was usually carried out by combining the results from the experiments and the numerical simulation It was also known as inverse numerical technique [101, 102] Accordingly, a few material parameters can be fixed based on its physical significant and the rest are used as the fitting parameters to fit the global load-displacement curves between experiment and numerical analysis [103, 104] For this work, a program was developed to optimize the material parameters by finding the values that minimize the difference in a least squares way between the load-displacement curve predicted by the numerical simulation and that of the experimental data Accordingly, the python object-orient programing language (https://www.python.org/) is employed to create FEM model (pre-processor) and access the ABAQUS results from the output database (.odb) file In the next step, the difference between the load-displacement curve by FEM simulation and that of experiment is calculated in MATLAB The Genetic Algorithm (GA) global optimization approach, which is a built-in optimization function in MATLAB, is then employed to optimize the necessary parameters The details of the optimization procedure can be explained as follow
First, the FEM simulation takes the material parameters p i as an input to predict the load-displacement curve The objective is to search set of optimum parameters p obj in such a manner that the difference between the two load-displacement curve is minimum
This would be archived by minimizing an objective function F obj p , defined as follows
(5.14) where p ( , p p 1 2 , , , p p i k ), in which p i denotes the material parameters, k is the total number of data points u F i , exp i resulted from segmentation of the force-
79 displacement experimental curves, F exp u Values of F sim i are evaluated in simulations at displacement point u i
Next, a suitable optimization approach is selected to find optimum parameters There are the optimization approaches can be found in the literature They are classified into gradient based and direct search approaches depending on optimized procedure The gradient based methods are not useful because they require both the objective function and the gradient of this function meanwhile the objective function in this work does not have an explicit analytical definition In addition, the gradient based methods are also known as the optimization methods converge to a local minimum rather that a global minimum Two direct search based global optimization methods are often used to optimize the material parameters are that Neural Network and GA methods [32, 105, 106] For this work, GA method that it built in MatLab would be applied to search optimum set of material parameters The flowchart of this process is given Figure 5.8
Figure 5.8 Flowchart of optimized process
The ASTM specimen is used to calbrate damage model The initial mesh size of 0.5 mm x 0.5 mm at critical zone is used to mesh for dog-bone specimen The displacement- controlled load is applied to right edge of specimen The element type of 3D, reduced
Experimental data Optimization Yes End
No Update new material parameters
80 integration, 8-nodes (C3D8R) and three element layers though thickness are applied to dog-bone specimen Mesh and boundary conditions are given in Figure 5.9
Figure 5.9 FEM mesh and boundary condition of dog-bone specimen
CDM model
For CDM model, assuming that initial VVF f 0 0.0016, ratio of 0 / 2 r 0 6.891 is obtained by Eq (3.51) The constant of 1 in weight function is selected Three material parameters of critical damage parameter due to void growth D crit g , softening exponent would be calibrated The Table 5.6 shows boundary condition and initial values of D crit g and Using Genetic Algorithm (GA) optimization method, the best-fit material parameters that archived after optimization process are given in Table 5.7 and the optimum force-displacement curves are shown in Figure 5.10 Now CDM model is ready to use for predicting the ductile fracture
Table 5.6 Initial guess values and constrains for optimization process
Table 5.7 The best-fit material parameters for CDM model
81 Figure 5.10 The best-fit force-displacement curve using CDM model
Porous ductile model
To apply the porous plastic material model to prediction of ductile fracture, 9 parameters
n q q f , , 1 2 , F , f f c , 0 , N , s N , f N must be identified The hardening exponent n was fitted by Swift model that described in section 5.1 and fixed in this study to have respect for recommendation of original N L Dung [21] model Therefore, eight remaining parameters q q f 1, 2, F ,f c ,f 0, N ,s N ,f N would be determined in this study In general, any identification procedure that used to identify all these parameters would be still challenge task because of the time cost for experimental data calibration In addition, for each material type, may be have more one set of material parameter (non-uniqueness of the solution) [104, 107, 108] Abendroth and Kuna [109] was used to Neural Network method for deification of two parameters f c and f N of the materials 18Ch2MFA and StE-690 Abbassi et al [32] were also succeed in identification of five material parameters f F , f c , N , s N , f N of GTN model For Genetic Algorithm method, Abbasi et al [110] were 4 optimal parameters f F , f c , f 0,f N for IF-steel Muủoz-Rojas et al
[105] were also employed this algorithm for identifying 4 parameters f c , N , s N , f N of material Aluminum at 400 o C This approach requires computational expensive time and difficult to obtain a convergence solution due to non-uniqueness of the solution
In an attempt to find the little physical meaning of porous ductile model parameters, Benseddiq and Imad [111], Kiran and Khandelwal [104] used the semi-experienced approach that based on the global analysis of large data from literature This method is
82 necessary to review the literature of GTN model parameters for many material types
However, it allows to reduce numerical simulations, so it helps to save the computational time
In this study, the GA algorithm-based optimization approach would be used to identify the N L Dung porous ductile model parameters Due to non-uniqueness of material parameters of the porous ductile model [104, 107, 108] so that GA optimization method based on inverse engineering process can be applied to find the set of material parameters for N L Dung model Although Kiran et al [104] tried to add little physical meaning to the nucleated strain parameter N but they also pointed out thatit is unclear whether a best choice for this parameter Therefore, the range limits and initial guess values are relatively defined as Table 5.8 The calibration is similar to that of CDM model using calibrated procedure in section 5.2.1
Figure 5.11 Force – displacement curve using set of optimum material parameters Table 5.8 Initial guess values and constrains for optimization process q 1 q 2 f F f c f 0 ε N s N f N
Upper limit 2.0 4.0 0.25 0.1 0.0020 0.65 0.15 0.07 Initial guess 1.5 2.25 0.15 0.06 0.0018 0.3 0.1 0.05 Lower limit 1.0 1.5 0.101 0.02 0.0014 0.085 0.05 0.03
83 Table 5.9 The optimal values for Dung-Hill48 model
Noting that the shear damage mechanism of Nahshon and Hutchinson [24] is strongly dependent on a numerical constant k ω Reis et al [112] revealed that is not possible to adopt a constant value of k ω for a wide range of stress triaxiality It only should be adopted for the low and very low value of stress triaxiality to overcome the limit of the traditional GTN model Therefore, the Nahshon-Hutchinson shear effects of dog-bone and R-notched specimens (high stress triaxiality) in this work was ignored and a value of k ω = 0 has used for calibrating the general material parameters The pure shear damage (very low stress triaxiality) is considered as a specific failure case and the value of k ω is separately calibrated for shear damage specimen In addition, because of traditional GTN modification for shear damage assumed by von Mises matrix material, therefore, it should be taken a careful consideration of Nashshon and Hutchinson criterion before applying it to Dung-Hill48 model It has been reported by Nashshon and Hutchinson [24] the evolution of void volume fraction f s in Eq.(4.34) is linear in f under an assumption that the effective void volume fraction (VVF) is small (simple or pure shear state) This linear dependence has been used in the absence of further mechanistic calibration Therefore, the evolution of VVF under pure shear load is just a linear accumulation of the shear VVF (f s ) For this work, before the Nashshon and Hutchinson criterion is used with an anisotropic material, the difference of VVF evolution between the anisotropic material (Dung-Hill48) and that of isotropic material (Dung-Mises) is verified via a single element under the pure shear load (see Figure 4.13) Its result curves are coincident between the two above-mentioned models, which validates the correctness of the Nashshon and Hutchinson criterion application to Dung-Hill48 model
DUCTILE FRACTURE PREDICTION OF AA6061-T6 ALUMINUM ALLOY
Geometries, mesh and boundary conditions
(d) Figure 6.1 Dimensions and geometries (a) R6 specimen, (b) R3 specimen, (c) R1.5 specimen and (d) shear specimen
85 The R-notched specimens are cut from thin sheet that its nominal thickness of 2 mm
The geometries and dimensions are shown in Figure 6.1 The initial mesh size at analysis zone is 0.5 mm x 0.5 mm The eight-node brick element type with reduced integration and hourglass control (C3D8R) has been used
Figure 6.2 Mesh and boundary condition (a) R6 specimen, (b) R3 specimen, (c) R1.5 specimen and (d) shear specimen
Ductility prediction
Ductility can be understood as the ability of a material to accept an amount of deformation without fracture Ductility is defined as (L f - L 0 )/L 0 ×100, where L 0 is the
86 initial gage length and L f is the gage length at fracture (the point where the displacement- load curve drops suddenly)
The predicted values of ductility and corresponding errors are presented in Table 6.1
The relative error is calculated by Eq (6.1) as below,
Where the subscripts exp and sim imply the experimental and predicted ductility, respectively
Table 6.1 The ductility predictions of the R-notched specimen
The maximum error values of predicted ductility by CDM-Hill48 model is 16.88 % and by Dung-Hill48 model is 7.46 % The average error values of 5.65 % and of 2.33 %are obtained by CDM-Hill48 model and Dung-Hill48 model, respectively The detail results are given in Table 6.1
Figure 6.3 shows the comparison experimental load-displacement curves and those of the numerical simulation corresponding to the various specimens The results indicated that using the sets of calibrated parameters, load-displacement behavior of all the R- notched specimens can be predicted by the anisotropic material models (CDM-Hill48 model and Dung-Hill48 model) The load-displacement response when using GTN model and CDM-Mises model is always higher than that of experiment This matter shown that von Mises material-based damage model is not suitable for predicting ductile fracture for sheet aluminum alloy AA6061-T6 The moment of fracture initiation depends on which type of damage model is used
87 Continuum damage mechanics model Porous ductile model
Figure 6.3 Force- displacement response of tensile tests
Figure 6.3 Force- displacement response of tensile tests (cont.)
The Figure 6.4 shows contour of damage variable by CDM-Hill48 model without shear damage variable and contour of VVF by original porous ductile material model (GTN model in ABAQUS) at displacement value of 0.8 mm The damage accumulation of CDM-Hill48 model and VVF evolution of GTN model are so small that force- displacement curve cannot describe softening phenomenon of matrix material (see Figure 6.3 for shear specimen) It revealed that for zero or very low stress triaxialities the previous void growth-based damage models alone are insufficient to predict ductile fractured features The limitation of GTN-like model for predicting shear-dominated damage has become a studied topic in recent years [24, 25] For Dung-Hill48 model, a discussion about damage predictability under pure shear stress state is addressed in
89 section 4.4.1 With revelation above, an extension of original N L Dung models for predicting shear-dominated fracture is necessary in this dissertation a) b)
Figure 6.4 No damage occurrence when using a) CDM-Hill48 and b) GTN models without shear damage variable
Crack initiation and propagation prediction
Beside the ductility prediction, the initiated fracture location is also identified using the CDM-Hill48 and Dung-Hill48 models The advantage of the porous model can be used to predict microscopic crack initiation that recognized by physical meaning of the critical VVF (f c ) Once the VVF reaches to this value, the coalescence of micro-voids in matrix material is assumed to occur The Figure 6.5 shows the contour of state variables at the moment of micro-crack initiation of the specimens under uniaxial tension The micro-crack initiation locations are determined by extracting VVF along minimum section of the specimens and are given in Figure 6.6 It predicted that in the case of dog- bone and R6 notched specimens micro-crack initiated at center, whereas in the case of R3, R1.5 and shear specimens micro-crack initiates near periphery
Figure 6.5 The contour of state variables at micro-crack initiation when using Dung- Hill48 model: (a) dog-bone specimen, (b) R6 specimen, (c) R3 specimen, (d) R1.5 specimen and (e) shear specimen
91 Figure 6.6 Micro-crack location of R-notched specimen
The evolution damage variable corresponds to the equivalent plastic strain is given in Figure 6.9 The various values of equivalent plastic strain to fracture are obtained corresponding to various specimens The Figure 6.8 shows contour of the state variables at moment that just before fracture occurrences when using CDM-Hill48 model The ductile fracture initiation locations can be predicted by stress triaxiality For the dog- bone and R6 notched specimens, the maximum value of stress triaxiality is obtained at center element whereas the maximum value of stress triaxiality at peripheral element is archived by R3 and R1.5 notched specimens The effect of the Lode angle on the ductile fracture process in each specimen is presented on Figure 6.10 Noting that to consider effect of loading condition on plastic deformation, the average value (dash line in Figure 6.9) of Lode angle parameter is expressed by Eq.(6.2) Accordingly, average values of Lode angle parameter corresponding to specimens are calculated and shown in Table 6.2
From Table 6.2 it can be recognized that the shear damage strongly governed by the Lode angle weight function in Eq (3.54), whereas the dominance of shear damage component in dog-bone specimen is niggling
(c) Figure 6.7 Contour of state variables at moment just before fracture occurrence when using CDM-Hill48 model: (a) dog-bone specimen and (b) R6-specimen
Figure 6.8 Contour of state variables at moment just before fracture occurrence when using CDM-Hill48 model: (c) R3 specimen, (d) R1.5 specimen and (e) shear specimen
(cont.) Table 6.2 Average Lode angle parameter values
94 Figure 6.9 The damage evolution corresponds to equivalent plastic strain
Figure 6.10 The variation of Lode angle parameter in equivalent plastic strain
The predicted fracture initiation locations using CDM-Hill48 model are identical to those of Dung-Hill48 model For all R-notched and shear specimens, the fracture propagates along minimum section The fracture initiates at center of the dog-bone and R6 specimens, whereas fracture initiates at periphery of R1.5 and shear specimens For the R3 specimen, there is small deference between predicted result by CDM-Hill48 model and Dung-Hill48 model Fracture occurs at periphery before when predicted by CDM-Hill48 while that the fracture initiates at very close to periphery when using Dung-Hill48 model The summary of predicted result of fracture initiation locations is given in Table 6.3
Fracture initiation Dog-bone specimen
Figure 6.11 Predicted fracture path by CDM-Hill48 and Dung-Hill48 models
Figure 6.12 Predicted fracture path by CDM-Hill48 and Dung-Hill48 models (cont.)
Figure 6.13 Predicted fracture path by CDM-Hill48 and Dung-Hill48 models (cont.)
Table 6.3 Summary of fracture initiation location prediction
CDM-Hill48 Center Center Periphery Periphery Periphery
Dung-Hill48 Center Center Near periphery Periphery Periphery
Ductile fracture strain prediction
Figure 6.14 The variation of stress triaxiality in equivalent plastic strain using Dung-
A dependence of the equivalent plastic strain at ductile fracture on stress triaxiality is of practical interest [10, 53] By adopting the work by Bai et al [77] a ductile fracture strain is defined as an exponential function of stress triaxiality:
(6.3) with C 1 and C 2 are material parameters, the average stress triaxiality defined by
, (6.4) with f p is the equivalent plastic strain at crack initiation
Using the form of Eq.(6.3) , the predicted micro-crack and fracture plastic strains of the AA6061-T6 aluminum alloy sheet are described, Figure 6.15, as follows
For micro-crack initiation by Dung-Hill48 model:
For fracture initiation by Dung-Hill48 model:
99 From the Eq (6.5) and Eq (6.6), one may notice that the porous plasticity model is employed to establish a ductile fracture criterion of the material via the construction of a plastic strain – triaxiality relation Such relation can later be used as the alternative micro-crack and fracture criterions
Figure 6.15 Equivalent plastic fracture strain as a function of average stress triaxiality
Forming limit diagram (FLD) prediction
The forming limit diagram (FLD) of sheet metal are always of interest in forming process under plastic deformation to avoid waste The AA6061-T6 aluminum alloy sheet is known as industrial material with its light weight and high strength Therefore, finding a suitable model to accurately predict the its FLD curve is still need to continuous
Theory of FLD is developed over the decades The pioneers in this field can be mentioned as Keeler [113] and Goodwin [114] Marciniak-Kuczynski (M-K) [115] and Marciniak et al., [116] continued it and proposed a model based on an inconsistency in sheet metal that is able to predict localized necking Today, it is known as M-K theory model and commonly used to estimate FLD of sheet metal [117, 118] Beside the FLD theory prediction, the Nakajima test model is also applied widely in experiment and numerical simulation to determine the forming limit curve According to the ISO 12004-2:2008 standard, the Nakajima test is usually conducted for the several specimens to
100 present the strain paths of sheet from uniaxial to biaxial stretched loading state In the Nakajima test, a hemispherical punch with relatively large diameter (approximation of 100 mm) is used to deform the notched specimen until occurred failure The limit strains are determined by alternative time-dependent or cross section methodology
In this work, the Nakajima’s type deep drawing is conducted for the seven specimens with waist width w = 30, 55, 70, 90, 120, 145 and the circular shape as Figure 6.16a
The setup of deep drawing is presented in Figure 6.16b, after Kami et al [26] The blank used mesh type of 3D, 8-nodes, reduced integration (C3D8R) whereas the punch, holder and die are assumed absolute hard body with analytical rigid type The initial mesh size at analysis zone is 0.5 mm x 0.5 mm Three element layers through the thickness of blank are used The blank holding force F hold 450 kN is used to avoid any wrinkling phenomenon and early damage at the blank holding region The friction coefficient between the blank and punch surfaces is 0.03 whereas the friction coefficient value of 0.1 on all remain contact surfaces is adopted The finite element mesh and model are given in the Figure 6.17
(a) (b) Figure 6.16 (a) blank and (b) deep drawing setup (unit: mm)
Figure 6.18 shows an illustration of a procedure to obtain a forming limit diagram of the material following the ISO 12004-2:2008 standard [119] Accordingly, the limit strains
1 and 2 are determined using an inverse parabola that fits major strain data at the
101 instant just prior to macro-crack occurrences in the specimens Three paths of the principal strains are extracted along each cross section of the necking region in each specimen Averaged value of the three paths is used to establish a fitting window Inner boundaries of the fit window (purple dash-dot lines) are determined by the highest peak of the second derivative respect to arc length of three consecutive major strain points within a range of 6 mm of both crack sides Outer boundaries of the fit window (green solid lines) are calculated by
W W W (6.7) where W L is the left fit window width, W R the right fit window width, and the window width
W , (6.8) with 2 1 22, BL 2, BR and 11 21, BL 1, BR Subscripts BL and BR indicate left and right inner boundaries, respectively Within the defined fit window for each specimen, an inverse best-fit parabola of all data points approximates the location of the forming limit state which is located at the parabolic peak, Figure 6.18 This technique is used repetitively for remaining specimens to obtain locations of limit values 2, 1 , Figure 6.19
Forming limit points identified from earliest failure elements in each specimen using the combined Dung-Hill48 model are plotted in Figure 6.19 The same procedure is applied to simulation results with the GTN model using the same parameter set to determine limiting strains (contour plots of this case are not shown) Based on these points forming limit diagrams of the material are obtained for the two models The two curves are just a small distance apart resulting from a mild anisotropy of the material In particular, the GTN isotropic material model underestimates the forming limits in most cases except for in proximity of the equi-biaxial stretching mode Equivalent plastic strains at the forming limits are computed and plotted against corresponding average triaxiality values in Figure 6.20 Representation of these states is located in the zone between micro-crack
102 and macro-crack loci Hence, the micro-crack condition ( f f c ) can be applied as a conservative measure of the formability of the material Also, the micro-crack condition ( f f F ) serves as an upper bound
Figure 6.17 Finite element mesh and model: (a) W30 specimen, (b) W55 specimen, (c) W70 specimen, (d) W90 specimen, (e) W120 specimen, (f) W145 specimen, (g) circular specimen, (h) finite element model
(b) Figure 6.18 Illustration of the method to determine limit strains (a) W30 and (b)
104 Figure 6.19 The forming limit diagram of AA6061-T6 aluminum alloy
Figure 6.20 The equivalent plastic fracture strain of AA6061-T6 aluminum alloy
CONCLUSIONS AND FUTURE WORK
This research has focused on improving and applying the ductile damage models that can predict the plastic failures of the anisotropic metallic sheet This has required a combination of high-quality experimental data obtaining by tensile tests and the material parameter calibration of the damage models The principal conclusions have been summarized in detail below:
The original ductile damage models were enhanced to be applied to anisotropic matrix material The applicability of both continuum damage mechanics theory-based model (CDM-Hill48 model) and porous ductile theory-based model (Dung-Hill48 model) were examined The user-defined material subroutines (VUMAT) in ABAQUS/Explicit were successfully developed for the numerical computation
The finite element simulations were performed on uniaxial tension, deep drawing and Nakajima tests The numerical results show that the response of force-displacement curve obtaining by Hill48 matrix material-based damage models is more suited to experimental data than those of the von Mises matrix material assumption This reflects the correctness of the enhanced models comparing with original damage models
The plasticity models are calibrated using macroscopic scale quantities using an inverse engineering technique which employs genetic algorithm to find an optimal set of parameters minimizing an objective function The function is defined as the error between simulated and experimental results of force-displacement relations with the uniaxial tension tests on dog-bone specimens following ASTM-E8 standard
The dependency of equivalent plastic strain on stress triaxiality was studied and the loci of equivalent plastic strain at micro-crack and fracture initiation were obtained from tensile tests by the porous ductile material model (Dung-Hill48 model) Accordingly, using critical values of the void volume fraction, the predictions of micro- and fracture initiation of the material are performed to obtain the equivalent plastic strains at fracture
In that sense, the porous plasticity model is employed to establish a ductile fracture
106 criterion of the material via the construction of a plastic strain – triaxiality relation Such relation can later be used as an alternative fracture criterion based on CDM theory
The FLD of AA6061-T6 sheet is predicted using the Dung-Hill48 model The predicted result sits above the one obtained with GTN model for most deformation mode except for the neighborhood of the equi-biaxial stretching state, the different apart reflects anisotropic assumption of matrix material in Dung-Hill48 model The prediction of forming limits are consistent with the established fracture plastic strain - triaxiality relation that obtained by tension and Nakajima test when using both damage models (CDM-Hill48 and Dung-Hill48)
The recommends for future work
In this work, to save computational time, a consistent mesh size is used for calibrating and validating the ductile fracture analyses However, in finite element method, the mesh dependence is well known problem for damage analyses due to softening behavior of matrix material Therefore, it would be adequate evaluation if a particular mesh size which gives accurate results is investigated
The porous ductile material model mainly deals with the defects and dislocation in the crystalline structure of material In addition, the aluminum alloys are known as the crystalline structure material so that a further damage investigation based on grain structure analyses at microscopic level is important
This work marks the first attempt to incorporate sheet metal anisotropy into a porous plasticity model to predict ductile fracture and forming limits of the aluminum alloy As such, a simplest extension of the J2 flow theory, the Hill’s quadratic plasticity model, is chosen Nevertheless, it is rather well-known that the aluminum alloy considered obeys non-quadratic yield function Therefore, the present approach should be applied to more sophisticated yield functions
The effect of strain rate and temperature in warm and hot forming process should be studied The tests should be carried out on specimens under lager range of stress triaxialities to archive full locus of fracture strain for AA6061-T6 aluminum alloy
APPENDIX I: Flow chart of stress integrated algorithm of CDM-Hill48 model k + 1
APPENDIX II: Flow chart of stress integrated algorithm of Dung-Hill48 model
APPENDIX III-1: The contour of major and minor principal strains obtaining by using CDM-Hill48 model
APPENDIX III-1: The contour of major and minor principal strains obtaining by using CDM-Hill48 model (cont.)
APPENDIX III-2: The contour of equivalent plastic strain and stress triaxiality obtaining by using CDM-Hill48 model
APPENDIX III-2: The contour of equivalent plastic strain and stress triaxiality obtaining by using CDM-Hill48 model (cont.)
APPENDIX III-3: The contour of Lode angle parameter and damage variable obtaining by using CDM-Hill48 model
APPENDIX III-3: The contour of Lode angle parameter and damage variable obtaining by using CDM-Hill48 model (cont.)
APPENDIX IV-1: The contour of major and minor principal strains obtaining by using Dung-Hill48 model
APPENDIX IV-1: The contour of major and minor principal strains obtaining by using Dung-Hill48 model (cont.)
APPENDIX IV-2: The contour of equivalent plastic strain and void volume fraction obtaining by using Dung-Hill48 model
APPENDIX IV-2: The contour of equivalent plastic strain and void volume fraction obtaining by using Dung-Hill48 model (cont.)
APPENDIX V: The information about experimental data V.1 Experimental place:
Construction material laboratory, SGS Vietnam LTD company Address: Lot III/21, 19/5A St., Industrial group III, Tan Binh IZ, Tan Phu Dist., Hochiminh city, Vietnam
V.2 Analysis of chemical element content
Test method: ASTM E1251 – 11 Standard Test Method for Analysis of Aluminum and Aluminum Alloys by Spark Atomic Emission Spectrometry
The test was accredited ISO 17025:2005 by VILAS (VILAS is the full member and signatory of Mutual Recognition Arrangement with International Laboratory Accreditation)
Table V.1 Test result of chemical composition
Unit Test result Chemical composition
The AA6061-T6 aluminum alloy sheet (length x width x thickness: 2500 mm x 1250 mm x 2 mm) is imported from South Korea and is provided by Phuong Nam company (Address: AH1-Hight way, Thanh Loc Ward, District 12, Hochiminh city) The specimens are completed by precision CNC machining services
V.4.1 Test condition: Temperature: 23 ± 2 o C; humidity: 50 ± 5 %; test speed: 1 mm/min; gauge length: 50 mm
Table V.2 Test result of dog-bone specimen in rolling direction (3 specimens)
Tensile strength (stress at peak) MPa 288.3 290.8 290.1
121 Table V.3 Test result of dog-bone specimen in diagonal direction (3 specimens)
Tensile strength (stress at peak) MPa 289.4 295.0 290.9
122 Table V.4 Test result of dog-bone specimen in transverse direction (3 specimens)
Tensile strength (stress at peak) MPa 297.7 296.6 296.1
123 Table V.5 Test result of R6 specimen (3 specimens)
Tensile strength (stress at peak) MPa 318 328 310
124 Table V.6 Test result of R3 specimen (3 specimens)
Tensile strength (stress at peak) MPa 320 314 304
125 Table V.7 Test result of R1.5 specimen (3 specimens)
Tensile strength (stress at peak) MPa 315 295 292
126 Table V.8 Test result of shear specimen (3 specimens)
Specimen name Sh-1 Sh-2 Sh-3
Tensile strength (stress at peak) MPa 201 210 195
[1] H H Nguyen, T N Nguyen, and H C Vu “Ductile fracture prediction and forming assessment of AA6061-T6 aluminum alloy sheets” International Journal of Fracture, 2018, 209.1-2: 143-162
[2] H H Nguyen, T N Nguyen, and H C Vu, "Forming limit curve determination of AA6061-T6 aluminum alloy sheet,"Journal of Science and Technology
[3] H H Nguyen, T N Nguyen, and H C Vu “Forming Limit Diagram Prediction of AA6061-T6 Sheet Using a Microscopic Void Growth Model” In: in in AETA
2017: Recent Advances in Electrical Engineering and Related Sciences, Part of the Lecture Notes in Electrical Engineering book series, Springer, Cham, 2017 p 1026-1036
[4] H H Nguyen, T N Nguyen, and H C Vu "Ductile Fracture Prediction and
Formability Assessment of AA6061-T6 Sheets using a Porous Plasticity Model" in Proceedings of the 11th Southeast ASEAN Technical University Consortium
Symposium – SEATUC-11, March 13 and 14, 2017 ed 268, Ly Thuong Kiet, TP
Ho Chi Minh, Viet Nam: Conference Proceedings by CD-ROM with International Standard Serial Number, ISSN: 2186-7631
[5] H H Nguyen, T N Nguyen, and H C Vu, "Implementation and Application of Dung Model to Analyze Ductile Fracture of Metallic Material" in in AETA 2015: Recent Advances in Electrical Engineering and Related Sciences, Part of the Lecture Notes in Electrical Engineering book series, Springer, Cham, 2016, p 903-913
[6] H H Nguyen, T N Nguyen, and H C Vu, "Application of the Dung
Microscopic Damage Model to Predict Ductile Fracture of the Deep Drawn Aluminum Alloy Sheets" in AETA 2015: Recent Advances in Electrical Engineering and Related Sciences, Part of the Lecture Notes in Electrical Engineering book series, Springer, Cham, 2016, p 891-901
[7] H H Nguyen, T N Nguyen, and H C Vu, "Ductile Fracture Prediction of
Docol DP600 Steel by the Numerical Simulations," in Proceedings of the 4rd
International Conference on Engineering Mechanics and Automation – ICEMA4,
August 25 and 26, 2016 ed 18, Hoang Quoc Viet, Ha Noi, Viet Nam: Publishing House for Science and Technology, 2016, p 318-325
[8] H H Nguyen, T N Nguyen, and H C Vu, "Phân tích động lực học sự tăng trưởng lỗ hổng vi mô trong vật liệu đàn dẻo," in Tuyển tập Hội nghị Cơ học toàn quốc, Hà Nội, Việt Nam, 2014, p 175-180
[9] H H Nguyen, T N Nguyen, and H C Vu, "The Dung Void Growth Model for
Shear Failure," Journal of Science and Technology, vol 52(2C), p 214-255, 2014
[10] H H Nguyen, T N Nguyen, and H C Vu, "Ductile Fracture Analysis of API
X65 Steel by Modified Gurson Model in ABAQUS," in Proceedings of the 3rd
International Conference on Engineering Mechanics and Automation - ICEMA3,
October 15, 2014 ed 18, Hoang Quoc Viet, Ha Noi, Viet Nam: Publishing House for Science and Technology, 2014, p 427-432
[1] H N Esfahlan, S A Dizaji, and F Djavanroodi, "Experimental and Numerical
Analysis for Hydroforming of Ti6Al4V Alloy Used in Aerospace, Assisted by Floating Disk," Journal of Applied Sciences, vol 9, no 16, pp 2925-2932, 2009
[2] S Gatea, H Ou, B Lu, and G McCartney, "Modelling of ductile fracture in single point incremental forming using a modified GTN model," Engineering Fracture Mechanics, vol 186, pp 59-79, 2017
[3] S Smirnov, "The healing of damage after the plastic deformation of metals,"
Frattura ed Integrità Strutturale, no 24, p 7, 2013
[4] M Gorji, B Berisha, N Manopulo, and P Hora, "Effect of through thickness strain distribution on shear fracture hazard and its mitigation by using multilayer aluminum sheets," Journal of Materials Processing Technology, vol 232, pp 19- 33, 2016
[5] J Lemaitre, "A continuous damage mechanics model for ductile fracture,"
Journal of engineering materials and technology, vol 107, no 1, pp 83-89, 1985
[6] J Lemaitre, "Coupled elasto-plasticity and damage constitutive equations,"
Computer methods in applied mechanics and engineering, vol 51, no 1-3, pp
[7] Z Yue, H Badreddine, and K Saanouni, "A new model describing plastic distortion fully coupled with ductile damage," Procedia Engineering, vol 81, pp
[8] M Brünig, O Chyra, D Albrecht, L Driemeier, and M Alves, "A ductile damage criterion at various stress triaxialities," International Journal of Plasticity, vol 24, no 10, pp 1731-1755, 2008
[9] F A McClintock, "A Criterion for Ductile Fracture by the Growth of Holes,"
Journal of Applied Mechanics, vol 35, no 2, pp 363-371, 1968
[10] J R Rice and D M Tracey, "On the ductile enlargement of voids in triaxial stress fields," Journal of the Mechanics and Physics of Solids, vol 17, no 3, pp
130 [11] M Cockcroft and D Latham, "Ductility and the workability of metals," J Inst
[12] N L Dung, "Plasticity theory of ductile fracture by void growth and coalescence," Forschung im Ingenieurwesen, vol 58, no 5, pp 135-140, 1992
[13] Y Bai and T Wierzbicki, "Application of extended Mohr–Coulomb criterion to ductile fracture," International Journal of Fracture, vol 161, no 1, p 1, 2010
[14] R Schiffmann, W Dahl, and W Bleck, "Different CDM-models and their ability to describe the damage development at ductile fracture of steel," in ECF13, San
[15] V C Hoa, D W Seo, and J K Lim, "Site of ductile fracture initiation in cold forging: A finite element model," Theoretical and Applied Fracture Mechanics, vol 44, no 1, pp 58-69, 9// 2005
[16] N T Nguyen, D Y Kim, and H Y Kim, "A continuous damage fracture model to predict formability of sheet metal," Fatigue & Fracture of Engineering Materials & Structures, vol 36, no 3, pp 202-216, 2013