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Analytical and numerical modelling of open-die forging process for elliptical cross-section of billet
Accepted Manuscript Analytical and Numerical Modelling of Open-Die Forging Process for Elliptical Cross-Section of Billet R. Hari Krishna, D.P. Jena PII: DOI: Reference:
S0263-2241(18)31170-9 https://doi.org/10.1016/j.measurement.2018.12.023 MEASUR 6157
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
13 June 2017 9 October 2018 5 December 2018
Please cite this article as: R.H. Krishna, D.P. Jena, Analytical and Numerical Modelling of Open-Die Forging Process for Elliptical Cross-Section of Billet, Measurement (2018), doi: https://doi.org/10.1016/j.measurement.2018.12.023
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Analytical and Numerical Modelling of Open-Die Forging Process for Elliptical Cross-Section of Billet R. Hari Krishna, D.P. Jena∗
4
Industrial Acoustics Laboratory, Department of Industrial Design, National Institute of Technology Rourkela, India-769008
5
Abstract
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The present work proposes a novel analytical solution for open-die forging for the billet
7
of elliptical cross-section. From design and evaluation, moreover, from industrial prospec-
8
tive, in order to overcome the limitations and complexities of mathematical modelling for
9
complex shape billets, the numerical simulation technique has also been demonstrated to
10
establish and exploit the potential of commercially available finite element analysis (FEA)
11
software. To start with, testing of fundamental material properties such as tensile and com-
12
pressive tests have been simulated and validated against experimental data of aluminum
13
alloy such as AA6063-T7 and AA6061, respectively. In a similar manner, the FEA sim-
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ulation for a closed-die forging process has been carried out and validated against exper-
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imental result. Subsequently, a mathematical model for open-die forging having elliptical
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cross-section of billet has been derived with certain predefined assumptions. The potentials
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of FEA simulation for different materials such as aluminum alloy (AA6063), mild steel and
18
brass, have also been investigated to justify the proposed analytical solution, which has been
19
noticed to have less than 5% error. The intricacies involved in the simulations along with
20
mathematical modelling have been discussed in detail in the present article.
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Keywords: Analytical modelling, Open-Die Forging Process, Finite Element Analysis
22
(FEA), Tensile Test, Compression Test.
3
Email address: [email protected], +91-9938084602 (D.P. Jena∗ ) Preprint submitted to Measurement
October 5, 2018
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39 40 41 42 43 44 45 46 47 48 49 50 51
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1. Introduction Principally, every industry wants to design, develop and manufacture a finished or semifinished component with low cost, however, having adequate quality, reliability and ecofriendly in nature. In order to attain those requirements, one needs deterministic approach such as from a selection of raw material to the final product including intermediate manufacturing processes, where each and every step is important. During making of any component or product, the possibility of forming processes has been assessed at first sight because of its potential in diminishing the waste of raw material. Out of many forming processes, the forging process is the one, where a stronger product having better fatigue resistance can be obtained [1]. Present research has been provoked by considering the problems faced by designers and manufacturers while designing a new cross-section of billet for different materials and estimating the desired forging force to forge the billet. A robust methodology needs to be established to address aforementioned lacuna and enables designer in designing a new cross-section of billet for different materials with varying dimension using analytical solutions or commercially available finite element software from industrial design prospective. It is worth to remember that, the smith forging of bronze and wrought iron had begun in earlier 4000 B.C, for producing hand tools and weapons of war. Later in 1856, with the invention of Bessemer steel making process, a sudden break through occurred in the ferrous forging industry [2]. Successively, in 1862, for the first time, cavity steel forging using closed-die had commenced in Australia for the production of colt revolver. Further, with introduction of motor vehicles, demand for forging increased suddenly. Therefore, researchers tried to identify new methods for producing forged parts. Recently, with advancement in technology, computer controlled hammers have been used for forging the billets with different materials having complex shapes. Moreover, in recent times, forging has been used to evaluate the properties of concrete [3, 4]. Nevertheless, the 3D printed rapid prototyping structures also used to be evaluated using forging technique to estimate the compressive strength [5, 6]. However, a brief literature review had been carried out in the contest of the demonstrated research and briefed below. One can refer any test book on manufacturing process to get familiar with open-die forging process for a circular cross-section shape of billet [1, 7]. In earlier days, Y. Choi and J.C. Choi (1998) have suggested an analytical approach for the helical gear (guiding and clamping forging) using upper bound analysis [8] and compared with numerical simulations. In recent years, Parveen et al. (2011) have also proposed a new technique in forging the preform sintered billet through open-die [9] and have investigated with different coefficient of frictions on upper die and lower die to estimate the flow of billet and die load. Same authors have observed that die load increases with increase in the reduction of height and coefficient 2
60 61 62 63
64 65 66 67 68 69 70 71 72 73 74 75
of friction. In a similar manner, Phogat et al. (2012) have also demonstrated the elliptical disc open-die forging using upper bound analysis and have examined forging load involved in the process analytically for different reduction in heights and increase in length of major and minor axis [10]. It is an expensive and time consuming practice, however, trustworthy, that the researchers optimize the forging performance and corresponding parameters experimentally. In order to overcome such expensive and time consuming necessities, Xu and Rao (1977) have investigated the process parameters effect on spike forging through finite element method (FEM) and experiments [11]. Similarly, in detail, Fereshteh and Jaafari (2002) have also carried out analytical, numerical and experimental investigations for closed-die forging process of lead [12]. Likewise, Kamble and Nandedkar (2011) have also demonstrated the modified slab method for calculating the forging of hot forming process [13] and substantiated with experimental investigations. With growing industrial requirements, the necessities of cost effective solution provoke researchers to exploit the potential of finite element analysis (FEA) extensively [14, 15]. One can refer few recent developments to get an overview about the intricacy involved in FEA simulations and experiments [16, 17].
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Summarizing, it can be noticed that the analytical solution of open-die forging for elliptical cross-section of billet has not been addressed adequately. The correlation of analytical model with numerical analysis followed by experimental validation need a next level of investigation. In order to address the lacuna, the proposed research aims to model the open-die forging process for elliptical cross-section of billet analytically. In order to validate the mathematical model, the finite element method (FEM) based simulation has been established and validated against reported experimental investigations. The potential of analytical modelling and simulations have been demonstrated in detail with varying various forging parameters such as materials and coefficient of friction etc., discussed in detail in consecutive sections.
85
2. Underlying Theory
76 77 78 79 80 81 82 83
86 87 88 89 90 91 92 93 94 95
In present work, the finite element analysis (FEA) based simulation has been carried out in R ANSYS⃝ structural analysis platform. Various tasks involved for an appropriate simulation may be noted as precise modelling, adequate meshing, proper physical contact modelling, material behavior curve fitting, relevant element type selection, and apposite solver configuration. One can refer the Appendix-I to understand the fundamental background of finite element modelling, used in the present simulation. As the simulation of aforementioned manufacturing process involve extremely large deformation, the dynamic meshing or remeshing using predefined displacement/distortion criteria are indispensable and should be considered in FEA simulation. The details of displacement criteria for large deformation and re-meshing criteria have been discussed in Appendix-II and III, respectively. However, 3
96 97 98
99 100 101 102 103 104 105
the selection of appropriate numerical solver plays a vital role in generating converged solutions and has been discussed in Appendix-IV. In current work a non-linear solver has been used. To start with, first, the aluminium alloy materials have been used to simulate and validate the tensile test (AA6063-T7), compression test (AA6061) followed by benchmarking of forging process using aluminium 5154 material. The corresponding material properties such as experimental stress verses strain data [18, 19] have been used to model the accurate material behaviour using Chaboche nonlinear kinematic hardening technique [20]. The mathematical formulations to calculate the back stress tensor in such situations can be stated as: n
.
α = ∑ αi and αi = 23 Ci ε˙ pl − γi˙¯ε pl α
(1)
i=1
.
106 107 108 109 110
where, n is the number of back stress components, ε pl is the magnitude of plastic strain rate, . α is the back stress, αi is the ith component of back stress rate, and ˙¯ε pl is the plastic strain rate. The Ci and γi are material properties, where Ci represents the plastic modulus and γi indicates the parameters for previous data history. Such curve fittings using material datas R for the simulations have been carried out in ANSYS⃝ engineering data module.
Figure 1: Tensile and compression tests curve fitting
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112 113 114 115 116 117
3. Benchmarking Simulations and Validations The present work attempts to derive an analytical solution for open die forging using elliptical billets. Moreover, in order to attain a robust design, from industrial prospective, the numerical simulation using commercially available FEA software have been observed to be indispensable. The specifications of any design can only be achieved by choosing appropriate material and corresponding manufacturing process. To achieve so, the material properties which have been obtained from experiments needs to be used for corresponding 4
123
FEA simulations. In line with the objective, first, the tensile and compression tests have been simulated using finite element method. Moreover, these basic tests also help in undermining the corresponding material behaviour on different type of loading. In present simulations, the aluminum alloy AA6063-T7 and aluminum alloy AA6061 have been taken as material for tensile and compressive tests, respectively. The corresponding experimental measurements have been collected from literature [18, 19, 21].
124
3.1. Simulations of Tensile and Compression Tests
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125 126 127 128 129 130 131 132 133 134 135
To start with, the sample specimen (following ASTM E8/E8M-16a, ASTM E9-09) have R been modelled in ANSYS DesignModeller⃝ followed by meshing using surface-156 (quadrilateral and triangular elements) for tensile test (904 nodes and 790 elements) and solid-285 element for compression test (634 nodes and 2104 elements). The aforementioned element types support re-meshing criteria for desired non-linear simulation by defining nonlinear adaptive zone in the body. The boundary conditions have been applied to imitate the realtime experiment such as one end was fixed (by making degree of freedom=0). The desired loads have been applied on other end. As mentioned in the earlier section, the static analysis solver has been used using 100 steps for estimating the Von-Mises stress and plastic strain in the body, correspondingly. The estimated results for tensile and compressive tests have been shown in Fig.2 (a) and (b), correspondingly.
144
It can be noticed from Fig.2 that the estimated results from proposed numerical solution agree to reported experimental results [18, 19] satisfactorily. In sequence, the simulation technique has been established for consequent investigations. Moreover, a grid testing has also been carried out to quantify the desired meshing criteria. As the proposed simulation uses re-meshing technique to simulate the large distortion, rather than the element quality (aspect ratio of element should be less than 3-4), the re-meshing distortion criteria value is more important, since, the number of element changes dynamically in simulation. Moreover, it has been noticed that the computational requirement to simulate such problem is not significant (≈ 10 minutes using HP-Elite, 16 GB RAM and having 3.40 GHz i7 processor).
145
3.2. Simulating Closed-Die Forging Process
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Subsequently, the closed die forging simulation has been carried out. The sample specimen has been modelled using ANSYS Parametric Design Language (APDL) platform. The finite element meshing has been carried out by using Plane-182 element (generating 16928 nodes and 8802 elements) to simulate the closed-die forging process [12]. The aforementioned element type supports desired re-meshing criteria required for non-linear simulation by defining nonlinear adaptive zone in the body to simulate extremely large deformation. The boundary conditions have been applied to imitate the real-time experiment such as die 5
Figure 2: Validation of simulation results with experimental results
Experimental Force (N) 368.7
Simulation Force (N) 403
Percentage (%) 9.3
Table 1: Comparison of simulation result with experimental result
153 154 155 156 157
has been fixed (by making degree of freedom=0). The desired load has been applied on the one end of the upper-die surface. As mentioned in the earlier section, the static analysis solver has been used using 3000 steps for estimating the forging force and final deformation on the body. The estimated results for closed-die forging of lead [12] has been tabulated in Table 1.
168
It can be noticed from the Table-1 that the estimated results from proposed numerical solution agree to experimental results [12] adequately. In recent times, similar numerical R solutions have also been reported by researchers [22] using DEFORM-3D⃝ platform. However, in those environment, a grid testing has been carried out to quantify the desired meshing and trial and error method has been used for finding the optimum billet dimensions. As the proposed simulation uses re-meshing criteria to simulate the large distortion, rather than the element quality, the re-meshing distortion criteria has been implemented. Since, the number of element changes dynamically in simulation, the re-meshing based simulation is more appropriate one. Moreover, it has been noticed that the computational requirement to simulate such problem is not significant (≈ 7 minutes using HP-Elite, 16 GB RAM and 3.40 GHz i7 processor).
169
4. Analytical Modelling of Open-Die Forging Process
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170 171
The analytical solution is very much essential during designing the open-die forging process for small and medium scale industry who really could not effort for a commercially available 6
174
FEA software. In line with the objective, the analytical modelling for elliptical cross-section has been derived by taking analytical derivation for circular cross-section as a reference, explained in subsequent section [7].
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4.1. Elliptical Cross Section
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182 183 184 185 186
Assumptions: Considering the forging force F will attain its maximum value at the end of the operation and the coefficient of friction between the work-piece and the dies (upper and lower die) will remain constant. The variation of the stress field along the y-direction is negligible and is not considered. Because, the thickness of the work-piece is assumed to be small as compared with its other dimensions, and during the forging process, the entire work-piece will be in the plastic state. Fig.3 (a) shows a typical open die forging of an elliptical disk at the end of the operation. The r is radial direction and h is final thickness of the disc at the end of operation. The maximum and minimum radius of the disk are a and b, respectively. The origin of cylindrical co-ordinate system has been taken at the center of disk. The corresponding stresses acting on radial direction have been shown in Fig.3 (b).
Figure 3: (a) Forging of elliptical disk, (b) Stresses acting on cylindrical-symmetry of Elliptical cross- section.
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Considering the cylindrical symmetry, it has been assumed that the σθ = σr and both the σθ and σr are independent of θ . Now, considering the radial equilibrium, the stresses acted of the element can be written as: √ √ (σr + d σr )h( (a + da)2 d θ 2 + (b + db)2 d θ 2 ) − σr h( a2 d θ 2 + b2 d θ 2 ) − σθ hda sin( d2θ ) −σθ hdb sin( d2θ ) − τ ad θ da − τ bd θ db = 0
(2) 190
On simplifying, neglecting higher order terms, the above equation can be reduced to:
7
(σr +d σr )h((a+da)+(b+db))d θ − σr h(a+b)d θ − σθ hda
191
dθ dθ − σθ hdb − τ ad θ da− τ bd θ db = 0. 2 2 (3)
Further on simplification, the above equation can be stated as:
σr h (da + db) + d σr h((a + da) + (b + db)) − τ (ada + bdb) = 0. 2 192 193
Again to simplify the analysis, take σr , σθ and p as the principal stresses. Using below equations (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 = C (cons tant)
194 195
and C = 6K 2 with σ1 = σr , σ2 = σθ (= σr ), has been obtained.
σr + p = 196
√ 3K
(6)
or
(7)
197
(since the shear stress K is constant)
198
Substituting d σr in the equation (4), it can be re-written as:
σr h σr h da + db − d pha − d phb − τ ada − τ bdb = 0. 2 2
201
(8)
Now, substitute the value of σr in the above equation, the following equation is obtained. √ √ 3K − p 3K − p hda + hdb − d pha − d phb − τ ada − τ bdb = 0 2 2
200
(5)
and σ3 = −p , the following conditions
d σr = −d p
199
(4)
(9)
In this process, beyond certain radii, say, as and bs , a sliding will take place at the interface which allow the radial expansion of the work-piece. With an assumption: (
τ = µp
as ≤ a ≤ a f bs ≤ b ≤ b f 8
) (10)
(
τ =K
202 203
0 ≤ a ≤ as 0 ≤ b ≤ bs
) (11)
Thus, in these two zones, the above equation (10) and (11) takes the below mentioned mathematical form. √ ( 3K−p) (da+db) 2 p
( − d p(a+b) − µh (ada + bdb) = 0 p
√ ( 3K−p) (da + db) − d p(a + b) − µhK (ada + bdb) = 0 2
204
p
p=
206
= C1 exp
√1 3
−C2 −
µ 2h
209
) (13)
a2 +b2 a+b
(
)
( + 12
a2 +b2
as ≤ a ≤ a f bs ≤ b ≤ b f
(
)
a+b
0 ≤ a ≤ as 0 ≤ b ≤ bs
) (14) ) (15)
√ 3K (at a = a f and b = b f )
(16)
Using equation (16) in equation (14), the following equation has been obtained.
exp
µ − 2h
(
a f 2 +b f 2 a f +b f
)
√ 3K
(17)
At a = as and b = bs , equating the right−hand sides of equations (10) and (11) to the value of p (p = Kµ ), the equations (14) and (17) can be written as: √ 3µ exp 2 K µ
210
(
µK 3h
C1 = 208
0 ≤ a ≤ as 0 ≤ b ≤ bs
(12)
As the periphery of the disc is free, means a = a f , b = b f , and σr = 0, the equation (6) can be re-written as:
p= 207
(
)
Integrating the above two equations, the following equation is obtained. √ 3K exp 2p
205
as ≤ a ≤ a f bs ≤ b ≤ b f
µ − 2h
=
exp
(
a f 2 +b f 2 a 2 +b 2 s s a f +b f − as +bs
√ 3K
Or
9
) −1 2
(
as ≤ a ≤ a f bs ≤ b ≤ b f
) (18)
[ ] √ √ as 2 + bs 2 a f 2 + b f 2 2h 3 1 = + ln ln 3µ + µ+ as + bs af +bf µ 2 2
(19)
a
211 212 213
For simplification, ab = b ff = abss = C, where the value of constant, C, used to be determined by the designer. Now, substituting, a = as and b = bs in equation (15), the following equation can be obtained. (
1 K µK C2 = √ − + 3h 3 µ 214
µ − 2h
= exp
p
(
a f 2 +b f 2 a2 +b2 a f +b f − a+b
as 2 +bs 2 as +bs
−
a2 +b2
)
a+b
(
) − 12
( + Kµ
)
as ≤ a ≤ a f bs ≤ b ≤ b f
(21)
)
0 ≤ a ≤ as 0 ≤ b ≤ bs
(22)
[∫
∫ bs
as
p2 ada +
0
∫ af
p2 bdb+
0
p1 ada+
as
]
∫ af bs
p1 bdb
(23)
where, p1 and p2 are pressures given by equations (20) and (21), respectively. The first to fourth terms have been defined by equations 24-27. First term
[
Second term
218
(20)
The total forging force can be written as: F = 2π
217
(
µK 3h
p=
216
)
Following, the pressures for the non-sticking and sticking zone can be written as: √ 3µ exp 2
215
as 2 + bs 2 as + bs
µK 3h
[
µK 3h
((
as 2 + bs 2 as + bs
((
)(
as 2 + bs 2 as + bs
a2 2
)(
)
b2 2
(
a3 − 0.934 3
)
(
b3 − 1.09 3
))
))
K a2 + µ 2
K b2 + µ 2
]as (24) 0
]bs (25) 0
Third term
[√
exp
3µ µ K 2 + 2h
((
a f 2 +b f 2 a f +b f
)
) ] −0.934a + 12
[√
+
2.14exp
3µ µ K 2 + 2h
((
a f 2 +b f 2 a f +b f
a2
)
) ] a f −0.934a + 12
as
10
(26)
3.5
×10 9
3.5
×10 8 Mild Steel Brass
3
2.5
2.5
Maximum Force (N)
Maximum Force (N)
Aluminum Alloy
3
2
1.5
1
0.5
2
1.5
1
0.5
0
0 0
50
100
150
200
250
300
0
Major radius (in mm), E.R=0.5, a/b=1.5
50
100
150
200
250
300
Major radius (in mm), E.R=0.5, a/b=1.5
Figure 4: Analytical Results for constant values of coefficient of friction for elliptical cross-section with different materials
Fourth term
[√
exp
3µ µ K 2 + 2h
((
a f 2 +b f 2 a f +b f
)
) ] −1.09b + 12
[√
+
1.83exp
3µ µ K 2 + 2h
((
a f 2 +b f 2 a f +b f
b2
)
) ] b f −1.09b + 12
(27)
bs
219
Now, the final forging equation for an elliptical cross-section billet can be written as: [ µ K (( F = 2π
( 3 )) ]a a K a2 s − 0.934 + 3h 3 µ 2 0 + [ (( 2 2 ) ( 2 ) ( 3 )) ]b µK as +bs b b K b2 s − 1.09 + 3 µ 2 0 + 3h [ as +bs (( 2 [√ ) ) ] as 2 +bs 2 as +bs
)( 2)
a 2
(( ) ) ] a f a f 2 +b f 2 √ 3µ µ K 2 +b 2 1 a −0.934a + 2 3µ µ K f f 1 2 + 2h a f +b f −0.934a + + 2 2h a f +b f 2 2.14exp + + exp 2 a as [ [√ (( ) ) ] b 2 2 f (( 2 ) ) ] a f +b f √ 3µ µ K 2 1 a +b + −1.09b + 3µ µ K f f 1 2 2 2h a f +b f + −1.09b + 2 2 2h a f +b f 1.83exp exp + b2 bs
220 221 222 223
(28)
Using the above expression, the analytical results for a constant value of coefficient of friction, such as µ =0.3, for an elliptical cross-section billet with different materials has been estimated and have been shown in Fig.4 (a-c). The required maximum force for forging the billet has been calculated using above final expression (equation 28). 11
Materials Density (Kg*/m()3 ) Youngs modulus (Mpa) Poissons ratio
Aluminium Alloy (AA6063) 2700 61300 0.33
Mild Steel 7860 1.9E5 0.28
Brass 8450 1E5 0.331
Table 2: Material Properties
224
225 226 227 228 229
230 231 232 233 234 235 236 237 238 239 240 241
5. FEM Simulation Next, the finite element method based simulation has been carried out to verify the performance of the derived analytical solution. In line with this, the numerical investigations R have been carried out in ANSYS⃝ workbench which is based on finite element analysis (FEA). The fundamental of steady state structural solver is used in the present work has been discussed in Appendices. In order to verify the analytical solution (equation 28), further, the open-die forging process has been simulated for elliptical cross-section, as discussed in section 3.2. The aluminium alloy (AA6063), mild steel and brass materials have been used in the present simulations. The corresponding material properties are tabulated in Table 1. As discussed in the earlier section, the contact has been created between the billet (contact, first body) and die (target, second body) with friction type (having constant friction coefficient µ ). The boundary conditions used in the simulation are same as discussed in the earlier section, i.e., the lower die is fixed by assigning degree of freedom zero. Displacement has been applied to the upper die by allowing the billet to displace in lateral direction rather than in axial direction. The nonlinear adaptive region to the billet has been applied to simulate extremely large deformation. The Solid-285 element type has been used for meshing the billet and has been shown in Fig. 5. The solution has been achieved by using steady state structural solver with 100 steps.
Figure 5: Meshing of components: (a) 3D-model view, (b) front view, and (c) isometric view (without dies)
242
243 244 245
6. Results and Discussion The steady state structural solutions have been achieved using aforementioned simulation technique for elliptical cross-section of billet with different materials (aluminium alloy (AA6063), mild steel and brass). From steady state solution, the contour plot of directional 12
246 247 248 249 250 251 252
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271
272 273 274 275 276 277 278 279 280 281 282 283
deformation, maximum principle stress and equivalent total strain have been extracted for circular cross-section with different materials have been shown in Fig.6 (a-c (aluminium alloy), g-i (mild steel), m-o (brass)). In a similar nature, the billet with elliptical cross-section having aforementioned materials have been analysed with constant coefficient of friction (µ =0.3). The contour plot of parameters such as directional deformation, maximum principal stress, and equivalent total strain have been extracted for the above billet of different materials and have been shown in Fig. 6 (d-f (aluminium alloy), j-l (mild steel), p-r (brass)). Subsequently, the maximum force observed at steady state structural simulation solutions have been extracted for different diameter of circular cross-section with different materials and examined against analytical solutions reported in literature [7], as shown in Fig. 7 (a-c). From figure, it can be inferred that the numerical results agree to well-established analytical solution. From these, it can be inferred that the proposed numerical simulations agree to analytical solutions having error ≤ 5%. The error has been calculated by ratio of difference between analytical and simulated results to analytical result. In a similar way, at steady state structural simulation solutions for different elliptical cross-section, (a/b = 1.5), of billet with different materials have been estimated and have been shown in Fig. 7 (d-f), respectively. One can notice that the numerical results agree to analytical results adequately. The observed error percentage in simulations has been tabulated in Table 4. The observed error is ≤ 5%, which is acceptable in such numerical solution, where extremely large deformation take place. Moreover, the error analysis carried out in detail to understand the performance of the simulation. The observed error in simulation has been shown in Fig. 8. From figure, it can be inferred that the numerical error increases with increase of major length (a/b =1.5), however, lie in acceptable range. Later, similar analyses have been carried out for different contact conditions by varying coefficient of frictions (µ =0.05 to 0.3 in six steps). The simulated results again agree to analytical results with error ≤ 5%, as shown in Fig. 9. Summarizing, the above analyses are to overcome the imitation of a real-time open-die forging process. So, every constraint is imperative and should be incurred in the simulation process to achieve desired result. In the open-die forging process, the shape of billet and length to diameter ratio are important in designing the geometry. The selection of appropriate material is also important to get a quality product. The proposed simulation process may be considered as the one which enables designer to address the solution to the selection of the material. It provides chance to interpret the material and can detect the suitability. Moreover, the proposed methodology can also assist in optimizing different process parameters. It finally tells which parameters are best suitable at given loading condition. As the proposed simulation based on finite element method does not involve large computational resource requirement, it can be reckoned to investigate a real world open-die forging process, eventually, reduces the manufacturing cost. 13
Figure 6: Directional deformation, maximum principal stress and equivalent total strain for circular, elliptical, square and rectangular cross-sections
284
285 286 287 288
7. Conclusion The proposed research exploits the potential of analytical modelling to estimate the desired forging force for geometrical shapes such as circular and elliptical with different materials. However, from industrial requirement point of view, the proposed numerical method based on FEA using a commercially available platform has been observed apposite and agrees 14
9
8
(a)
x 10
4 3 2 1
0
100
200
Mild Steel (Analytically) Mild Steel (Simulaiton)
10 8 6 4 2 0
300
8
0
100
Diameter (in mm) 9
200
6 5 4 3 2 1 0
300
Brass (Analytically) Brass (Simulaiton)
7
0
100
Diameter (in mm) 8
(d)
x 10
Maximum Force (N)
Maximum Force (N)
Maximum Force (N)
Aluminum Alloy (Analytically) Aluminum Alloy (Simulaiton)
(c)
x 10
12
5
0
8
(b)
x 10
6
8
(e)
x 10
200
300
Diameter (in mm) (f)
x 10 4.5
6
5 4 3 2 1 0
0
100
200
300
Major radius (in mm), a/b=1.5
Mild Steel (Analytically) Mild Steel (Simulaiton)
Maximum Force(N)
Aluminum Alloy (Analytically) Aluminium Alloy (Simulaiton)
Maximum Force(N)
Maximum Force(N)
6
5 4 3 2 1 0
Brass (Analytically) Brass (Simulaiton)
4 3.5 3 2.5 2 1.5 1 0.5
0
100
200
300
Major radius (in mm), a/b=1.5
0
0
100
200
300
Major radius (in mm), a/b=1.5
Figure 7: Validation of simulation results with theoretical results for circular and elliptical cross-section with different materials, a/b= major dimension/minor dimension=1.5
289 290 291 292 293 294 295 296 297 298
adequately with experimental result. Moreover, the analytical results have been examined against the FEA result and the observed error is less than 5%. Subsequently, the potential of proposed analytical model and FEA technique have been investigated for different coefficient of frictions to demonstrate the appropriateness. It has been observed from above investigations that the forging force required to forge the billet is more for circular cross-section than the elliptical cross-section, for the different materials (aluminium alloy (AA6063), mild steel and brass) with varying cross-section dimensions of the billet. Similarly, for the different coefficient of friction, the forging force is more for circular cross-section billet than the elliptical cross-section billet. It can be inferred from above analysis that the simulation results are well below the analytical results with ≤ 5% error. The established fact such as the Cross-section Circular -Aluminum Alloy Elliptical- Aluminum Alloy Circular -Mild Steel Elliptical-Mild Steel Circular - Brass Elliptical- Brass
Percentage variation 2.93 to 5.2 2.73 to 5.61 2.75 to 5.02 2.63 to 5.69 2.36 to 5.28 2.81 to 5.73
Table 3: Percentage error variation for circular and elliptical cross-sections for various dimensions (constant coefficient of friction)
15
(a)
(b) 0.06
ALuminum Alloy Mild Steel Brass
0.05 0.045
Percentage of error (%)
Percentage of error (%)
0.055
0.04 0.035 0.03 0.025 0.02
0
2
4
6
8
0.05 0.045 0.04 0.035 0.03 0.025
10
ALuminum Alloy Mild Steel Brass
0.055
0
2
case no: x (x=1 to 10)
4
6
(c) 0.1 ALuminum Alloy Mild Steel Brass
0.08
Percentage of error (%)
Percentage of error (%)
10
(d)
0.1
0.06 0.04 0.02 0
8
case no: x (x=1 to 10)
0
1
2
3
4
5
6
0.06 0.04 0.02 0
7
case no: x (x=1 to 6)
ALuminum Alloy Mild Steel Brass
0.08
0
1
2
3
4
5
6
7
case no: x (x=1 to 6)
Figure 8: percentage error for cross-sections from its analytical results
299 300 301 302 303
forging force increases with increase in the volume of the billet has also been observed. The limitation of the proposed FEA may be identified to simulate the billet up to reduction of height ratio (0.3) to attain reliable estimation. Summarizing, the demonstrated methodology can be reckoned to estimate the desired forging force, detect the suitability of given material for corresponding open-die forging process in real- time, from industrial prospective. Cross-section Circular -Aluminum Alloy Elliptical- Aluminum Alloy Circular -Mild Steel Elliptical-Mild Steel Circular - Brass Elliptical- Brass
Percentage variation 3.35 to 5.76 3.65 to 5.73 3.24 to 5.84 3.52 to 5.41 3.02 to 5.91 3.83 to 5.97
Table 4: Percentage error variation for circular and elliptical cross-sections (varying coefficient of friction)
16
9
8
(a)
x 10
1.5 1 0.5 0
0.1
0.2
Mild Steel (Analytically) Mild Steel (Simulaiton)
4 3 2 1 0
0.3
5
Maximum Force (N)
2
Maximum Force (N)
Maximum Force (N)
Aluminum Alloy (Analytically) Aluminum Alloy (Simulaiton)
(c)
x 10
5
2.5
0
8
(b)
x 10
3
0
0.1
0.2
4 3 2 1 0
0.3
Brass (Analytically) Brass (Simulaiton)
0
0.1
0.2
0.3
coefficient of friction, (radius=150mm) coefficient of friction, (radius=150mm) coefficient of friction, (radius=150mm) 8
8
(d)
x 10
8 6 4 2 0
0.1
0.2
0.3
coefficient of friction, (M.R=150mm)
2
Maximum Force(N)
10
Maximum Force(N)
Maximum Force(N)
Aluminum Alloy (Analytically) Aluminium Alloy (Simulaiton)
Mild Steel (Analytically) Mild Steel (Simulaiton) 1.5
1
0.5
0
(f)
x 10
2
12
0
8
(e)
x 10
14
0
0.1
0.2
0.3
coefficient of friction, (M.R=150mm)
Brass (Analytically) Brass (Simulaiton) 1.5
1
0.5
0
0
0.1
0.2
0.3
coefficient of friction, (M.R=150mm)
Figure 9: Validation of Analytical results with Simulation results for different coefficient of friction for circular and elliptical cross-sections with different materials, a/b=1.5
304
305 306 307
Acknowledgement One of the authors, R. Hari Krishna, acknowledges the generous funding received from the Ministry of Human Resource Development (MHRD), Government of India, and Industrial Acoustics Laboratory for carrying out this work at NIT Rourkela.
17
308
APPENDICES
309
310
311 312
Appendix-I (Finite Element Modelling) Mathematically, in finite element method, the element formulations are based on the principle of virtual work, which can be stated as: ∫ v
313 314 315 316 317 318
σi j δ ei j dV =
v
fiB δ ui dV
∫
+ s
fis δ ui ds
. j
. j
where, σ
ij
.
.
.
ij
jk
ik
= σ − σik ω − σ jk ω .
ij
is the Jaumann rate of Cauchy stress, ω = ij
1 2
(
(A.2) ∂ vi ∂xj
320
the time rate of Cauchy stress.
321
The stress change due to applied strain can be expressed as: . j
σ 322 323
(A.1)
( ) ∂u where, σi j is the Cauchy stress component, ei j = 12 ∂∂ xuij + ∂ xij is the deformation tensor [23]. The ui is the displacement, xi is the current coordinate, fiB is the component of body force, fis is the component of surface traction, V is the volume of deformed body, and s is the surface of deformed body on which tractions are applied [24, 25]. The desired element formulation can be obtained by differentiating the aforementioned virtual work. The corresponding Cauchy stress, σi j , can be estimated using below relation [26].
σ
319
∫
ij
= ci jkl dkl
where, ci jkl is the material constitutive tensor, di j = tensor, vi is velocity.
∂v
− ∂ xij
)
.
is the spin tensor, σ is ij
(A.3) 1 2
(
∂ vi ∂xj
) ∂v + ∂ xij is the rate of deformation
324
325
326 327 328
Appendix-II (Displacement Criteria) The displacement of element is essential in simulating large deformation processes which depends upon element formulation. In present simulation, the mixed u − p formulation has been found suitable due to ease of calculating the hydrostatic pressure or volume change 18
329 330
rate [23]. The finite element formulation to calculate the stiffness of the element can be expressed as: (
Kuu Kup K pu K pp
) {
∆u ∆ p˙
}
{ =
∆F 0
} (A.4)
.
331
where, ∆u, ∆ P are the increment in displacement and hydrostatic pressure, correspondingly.
332
333
334 335 336 337 338 339
Appendix-III (Re-Meshing Criteria) When a material undergoes maximum distortion, the re-meshing is required to achieve the desired numerical stability. However, the most critical is to identify the re-meshing criteria values to initiate re-meshing during simulation. One can refer literature [27] to find the detail on finite element modelling based on re-meshing. The desired parameters corresponding to present simulation such as equilibrium, pressure constitutive and weighted residual form can be written as: ∫ Ω
δ εi j (si j δi j p)dΩ −
∫ Ω
δ ui bi dΩ −
∫ Γt
δ uiti dΓ = 0
( ) ) ∫ nd P ∂q q − εv dΩ + ∑ τiri dΩ = 0 K Ω Ω i=1 ∂ xi
∫
(
∫ Ω 340 341 342 343 344 345 346 347
(A.5)
(
ωi τi
) ∂p + πi dΩ = 0 ∂ xi
(A.6)
(A.7)
where, ωi is the appropriate weighting function (ωi = δ πi , πi is the negative of projected pressure gradient. The τi is the term traduced in order to ensure symmetry in final system of equations. The P is pressure, q is the arbitrary test function representing virtual pressure field, K is the bulk modulus, εv is the volumetric strain, nd is the number of space dimensions, τi is the intrinsic time parameter, and rt is the standard form of ith differential equation. When ui , p, and πi are substituted in aforementioned equations in its interpolation form, it leads to the Galerkin discretized equations (for δ ui =q=ωi =Ni) as [28]: A G 0 u¯ T p¯ G −(C + L) −Q 0 −QT −C˙ π¯
19
f = 0 0
(A.8)
348
where, ∫
Ai j =
Bi T Dd B j dΩ
(A.9)
(∇Ni )N j dΩ
(A.10)
∇T Ni [τ ]∇N j dΩ
(A.11)
Ωe
∫
Gi j =
Ωe
∫
Li j =
Ωe
∫
1 Ni N j dΩ Ωe K
Ci j =
(A.12)
C¯ 1 0 0 C¯ = 0 C¯ 2 0 0 0 C¯ 3 ∫
C¯ikj = [ Q=
Ωe
Q
ij
=
Ωe
∫
fi = In the above
Ni bdΩ + ]T
b1 b2 b3
∇=
349
Ωe
Q2 ,
τk
Q3
(A.14) ] (A.15)
∂ Ni N j dΩ ∂ xk
(A.16)
Nit¯dΓ, i, j = 1, nd
(A.17)
∫
[ b=
τk Ni N j dΩ
Q1 , ∫
k
(A.13)
∂ ∂ x1 ∂ ∂ x2 ∂ ∂ x3
Γ
and t¯ = , [τ ] =
[ t¯1 t¯2 t¯3
τ1
0
τ2 0
]T
,
(A.18)
τ3
where B is the standard infinitesimal strain matrix, Dd is the deviatoric constitutive matrix [27].
20
350
351
Appendix-IV (Numerical Solver)
352
Finite element analysis platform used below stated equations to estimate the stiffness and the force [29]. [K]{u} = {F a } + {F r }
(A.19)
N
where, [K] = ∑ [ke ] is the total stiffness matrix, {u} is the nodal displacement vector, N is m=1
353
the number of elements, [Ke ] is the element stiffness matrix (may include the element stress stiffness matrix), {F r } is the reaction load vector, {F a }, the total applied load vector, is given by:
{F a } = {F nd } + {F ac } +
N
∑ ({Feth} + {Fepr })
(A.20)
m=1
where, {F nd } is the applied nodal load vector, {F ac }= -[M]{ac }is the acceleration load N
vector, [M]= ∑ [Me ]is the total mass matrix, [Me ] is the element mass matrix, {ac } is the m=1
354
total acceleration vector, {Feth } is the element thermal load vector, {Fepr } is the element pressure load vector.
21
355
References
356
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[8] Y. choi, J. choi, Forging of helical gears: Upper bound analyses and experiments, Materials and Materials 4(4) (1998) 747–754.
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[9] R. K. R. Parveen Kumar, R. Kumar, Mechanics of deformation during open die forging of sintered preform: comparative study by equilibrium and upper bound methods, ARPN Journal of Engineering and Applied Sciences 6(6) (2011) 83–93.
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[11] Xu, Rao, Analysis of the deformation characteristics of spike-forging process through fe simulations and experiments, Journal of Material Processing Technology 70 (1977) 122–128.
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[12] Saniee, jaafari, Analytical, numerical and experimental analysis of closed-die forging, Journal of Material Processing Technology 125-126 (2002) 334–340.
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[16] X. Z. Wen Chen, Z. Cui, Numerical simulations of open-die forging process for manipulator design, National Die and Mold CAD Engineering Research Center, Shanghai Jiao Tong University (2008) 650 – 658.
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[17] Y. C. Cenk Misirli, Isik Cetintav, Experimental and fem study of open die forging for bimetallic cylindrical parts produced using different materials, International Journal of Modern Manufacturing Technologies 8(1) (2016) 69–74.
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[18] L. X. George E. Totten, Kiyoshi Funatani, Handbook of metallurgical process design, CRC Press.
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[19] N. Khlystov, D. Lizardo, K. Matsushita, J. Zheng, Uniaxial tension and compression testing of materials.
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[23] K. J. Bathe, Finite element procedures, Prentice-Hall. Englewood Cliffs.
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[24] J. Bonet, R. D. Wood., Nonlinear continuum mechanics for finite element analysis, Cambridge University Press.
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[25] M. Gadala, J. Wang, Simulation of metal forming processes with finite element methods, International Journal for Numerical Methods in Engineering 44 (1999) 1397 – 1428. 23
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[26] R.M.McMeeking, J.R.Rice, Finite element formulations for problems of large elasticplastic deformation, International Journal of Solids and Structures 121 (1975) 601 – 616.
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[27] E. Onate, J. Rojek, R. L. Taylor, O. C. Zienkiewicz, Finite calculus formulation for incompressible solids using linear triangles and tetrahedra, International Journal for Numerical Methods in Engineering 59 (2004) 1473 – 1500.
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S0263-2241(18)31170-9 https://doi.org/10.1016/j.measurement.2018.12.023 MEASUR 6157
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
13 June 2017 9 October 2018 5 December 2018
Please cite this article as: R.H. Krishna, D.P. Jena, Analytical and Numerical Modelling of Open-Die Forging Process for Elliptical Cross-Section of Billet, Measurement (2018), doi: https://doi.org/10.1016/j.measurement.2018.12.023
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1
2
Analytical and Numerical Modelling of Open-Die Forging Process for Elliptical Cross-Section of Billet R. Hari Krishna, D.P. Jena∗
4
Industrial Acoustics Laboratory, Department of Industrial Design, National Institute of Technology Rourkela, India-769008
5
Abstract
6
The present work proposes a novel analytical solution for open-die forging for the billet
7
of elliptical cross-section. From design and evaluation, moreover, from industrial prospec-
8
tive, in order to overcome the limitations and complexities of mathematical modelling for
9
complex shape billets, the numerical simulation technique has also been demonstrated to
10
establish and exploit the potential of commercially available finite element analysis (FEA)
11
software. To start with, testing of fundamental material properties such as tensile and com-
12
pressive tests have been simulated and validated against experimental data of aluminum
13
alloy such as AA6063-T7 and AA6061, respectively. In a similar manner, the FEA sim-
14
ulation for a closed-die forging process has been carried out and validated against exper-
15
imental result. Subsequently, a mathematical model for open-die forging having elliptical
16
cross-section of billet has been derived with certain predefined assumptions. The potentials
17
of FEA simulation for different materials such as aluminum alloy (AA6063), mild steel and
18
brass, have also been investigated to justify the proposed analytical solution, which has been
19
noticed to have less than 5% error. The intricacies involved in the simulations along with
20
mathematical modelling have been discussed in detail in the present article.
21
Keywords: Analytical modelling, Open-Die Forging Process, Finite Element Analysis
22
(FEA), Tensile Test, Compression Test.
3
Email address: [email protected], +91-9938084602 (D.P. Jena∗ ) Preprint submitted to Measurement
October 5, 2018
23
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
39 40 41 42 43 44 45 46 47 48 49 50 51
52 53 54 55 56 57 58 59
1. Introduction Principally, every industry wants to design, develop and manufacture a finished or semifinished component with low cost, however, having adequate quality, reliability and ecofriendly in nature. In order to attain those requirements, one needs deterministic approach such as from a selection of raw material to the final product including intermediate manufacturing processes, where each and every step is important. During making of any component or product, the possibility of forming processes has been assessed at first sight because of its potential in diminishing the waste of raw material. Out of many forming processes, the forging process is the one, where a stronger product having better fatigue resistance can be obtained [1]. Present research has been provoked by considering the problems faced by designers and manufacturers while designing a new cross-section of billet for different materials and estimating the desired forging force to forge the billet. A robust methodology needs to be established to address aforementioned lacuna and enables designer in designing a new cross-section of billet for different materials with varying dimension using analytical solutions or commercially available finite element software from industrial design prospective. It is worth to remember that, the smith forging of bronze and wrought iron had begun in earlier 4000 B.C, for producing hand tools and weapons of war. Later in 1856, with the invention of Bessemer steel making process, a sudden break through occurred in the ferrous forging industry [2]. Successively, in 1862, for the first time, cavity steel forging using closed-die had commenced in Australia for the production of colt revolver. Further, with introduction of motor vehicles, demand for forging increased suddenly. Therefore, researchers tried to identify new methods for producing forged parts. Recently, with advancement in technology, computer controlled hammers have been used for forging the billets with different materials having complex shapes. Moreover, in recent times, forging has been used to evaluate the properties of concrete [3, 4]. Nevertheless, the 3D printed rapid prototyping structures also used to be evaluated using forging technique to estimate the compressive strength [5, 6]. However, a brief literature review had been carried out in the contest of the demonstrated research and briefed below. One can refer any test book on manufacturing process to get familiar with open-die forging process for a circular cross-section shape of billet [1, 7]. In earlier days, Y. Choi and J.C. Choi (1998) have suggested an analytical approach for the helical gear (guiding and clamping forging) using upper bound analysis [8] and compared with numerical simulations. In recent years, Parveen et al. (2011) have also proposed a new technique in forging the preform sintered billet through open-die [9] and have investigated with different coefficient of frictions on upper die and lower die to estimate the flow of billet and die load. Same authors have observed that die load increases with increase in the reduction of height and coefficient 2
60 61 62 63
64 65 66 67 68 69 70 71 72 73 74 75
of friction. In a similar manner, Phogat et al. (2012) have also demonstrated the elliptical disc open-die forging using upper bound analysis and have examined forging load involved in the process analytically for different reduction in heights and increase in length of major and minor axis [10]. It is an expensive and time consuming practice, however, trustworthy, that the researchers optimize the forging performance and corresponding parameters experimentally. In order to overcome such expensive and time consuming necessities, Xu and Rao (1977) have investigated the process parameters effect on spike forging through finite element method (FEM) and experiments [11]. Similarly, in detail, Fereshteh and Jaafari (2002) have also carried out analytical, numerical and experimental investigations for closed-die forging process of lead [12]. Likewise, Kamble and Nandedkar (2011) have also demonstrated the modified slab method for calculating the forging of hot forming process [13] and substantiated with experimental investigations. With growing industrial requirements, the necessities of cost effective solution provoke researchers to exploit the potential of finite element analysis (FEA) extensively [14, 15]. One can refer few recent developments to get an overview about the intricacy involved in FEA simulations and experiments [16, 17].
84
Summarizing, it can be noticed that the analytical solution of open-die forging for elliptical cross-section of billet has not been addressed adequately. The correlation of analytical model with numerical analysis followed by experimental validation need a next level of investigation. In order to address the lacuna, the proposed research aims to model the open-die forging process for elliptical cross-section of billet analytically. In order to validate the mathematical model, the finite element method (FEM) based simulation has been established and validated against reported experimental investigations. The potential of analytical modelling and simulations have been demonstrated in detail with varying various forging parameters such as materials and coefficient of friction etc., discussed in detail in consecutive sections.
85
2. Underlying Theory
76 77 78 79 80 81 82 83
86 87 88 89 90 91 92 93 94 95
In present work, the finite element analysis (FEA) based simulation has been carried out in R ANSYS⃝ structural analysis platform. Various tasks involved for an appropriate simulation may be noted as precise modelling, adequate meshing, proper physical contact modelling, material behavior curve fitting, relevant element type selection, and apposite solver configuration. One can refer the Appendix-I to understand the fundamental background of finite element modelling, used in the present simulation. As the simulation of aforementioned manufacturing process involve extremely large deformation, the dynamic meshing or remeshing using predefined displacement/distortion criteria are indispensable and should be considered in FEA simulation. The details of displacement criteria for large deformation and re-meshing criteria have been discussed in Appendix-II and III, respectively. However, 3
96 97 98
99 100 101 102 103 104 105
the selection of appropriate numerical solver plays a vital role in generating converged solutions and has been discussed in Appendix-IV. In current work a non-linear solver has been used. To start with, first, the aluminium alloy materials have been used to simulate and validate the tensile test (AA6063-T7), compression test (AA6061) followed by benchmarking of forging process using aluminium 5154 material. The corresponding material properties such as experimental stress verses strain data [18, 19] have been used to model the accurate material behaviour using Chaboche nonlinear kinematic hardening technique [20]. The mathematical formulations to calculate the back stress tensor in such situations can be stated as: n
.
α = ∑ αi and αi = 23 Ci ε˙ pl − γi˙¯ε pl α
(1)
i=1
.
106 107 108 109 110
where, n is the number of back stress components, ε pl is the magnitude of plastic strain rate, . α is the back stress, αi is the ith component of back stress rate, and ˙¯ε pl is the plastic strain rate. The Ci and γi are material properties, where Ci represents the plastic modulus and γi indicates the parameters for previous data history. Such curve fittings using material datas R for the simulations have been carried out in ANSYS⃝ engineering data module.
Figure 1: Tensile and compression tests curve fitting
111
112 113 114 115 116 117
3. Benchmarking Simulations and Validations The present work attempts to derive an analytical solution for open die forging using elliptical billets. Moreover, in order to attain a robust design, from industrial prospective, the numerical simulation using commercially available FEA software have been observed to be indispensable. The specifications of any design can only be achieved by choosing appropriate material and corresponding manufacturing process. To achieve so, the material properties which have been obtained from experiments needs to be used for corresponding 4
123
FEA simulations. In line with the objective, first, the tensile and compression tests have been simulated using finite element method. Moreover, these basic tests also help in undermining the corresponding material behaviour on different type of loading. In present simulations, the aluminum alloy AA6063-T7 and aluminum alloy AA6061 have been taken as material for tensile and compressive tests, respectively. The corresponding experimental measurements have been collected from literature [18, 19, 21].
124
3.1. Simulations of Tensile and Compression Tests
118 119 120 121 122
125 126 127 128 129 130 131 132 133 134 135
To start with, the sample specimen (following ASTM E8/E8M-16a, ASTM E9-09) have R been modelled in ANSYS DesignModeller⃝ followed by meshing using surface-156 (quadrilateral and triangular elements) for tensile test (904 nodes and 790 elements) and solid-285 element for compression test (634 nodes and 2104 elements). The aforementioned element types support re-meshing criteria for desired non-linear simulation by defining nonlinear adaptive zone in the body. The boundary conditions have been applied to imitate the realtime experiment such as one end was fixed (by making degree of freedom=0). The desired loads have been applied on other end. As mentioned in the earlier section, the static analysis solver has been used using 100 steps for estimating the Von-Mises stress and plastic strain in the body, correspondingly. The estimated results for tensile and compressive tests have been shown in Fig.2 (a) and (b), correspondingly.
144
It can be noticed from Fig.2 that the estimated results from proposed numerical solution agree to reported experimental results [18, 19] satisfactorily. In sequence, the simulation technique has been established for consequent investigations. Moreover, a grid testing has also been carried out to quantify the desired meshing criteria. As the proposed simulation uses re-meshing technique to simulate the large distortion, rather than the element quality (aspect ratio of element should be less than 3-4), the re-meshing distortion criteria value is more important, since, the number of element changes dynamically in simulation. Moreover, it has been noticed that the computational requirement to simulate such problem is not significant (≈ 10 minutes using HP-Elite, 16 GB RAM and having 3.40 GHz i7 processor).
145
3.2. Simulating Closed-Die Forging Process
136 137 138 139 140 141 142 143
146 147 148 149 150 151 152
Subsequently, the closed die forging simulation has been carried out. The sample specimen has been modelled using ANSYS Parametric Design Language (APDL) platform. The finite element meshing has been carried out by using Plane-182 element (generating 16928 nodes and 8802 elements) to simulate the closed-die forging process [12]. The aforementioned element type supports desired re-meshing criteria required for non-linear simulation by defining nonlinear adaptive zone in the body to simulate extremely large deformation. The boundary conditions have been applied to imitate the real-time experiment such as die 5
Figure 2: Validation of simulation results with experimental results
Experimental Force (N) 368.7
Simulation Force (N) 403
Percentage (%) 9.3
Table 1: Comparison of simulation result with experimental result
153 154 155 156 157
has been fixed (by making degree of freedom=0). The desired load has been applied on the one end of the upper-die surface. As mentioned in the earlier section, the static analysis solver has been used using 3000 steps for estimating the forging force and final deformation on the body. The estimated results for closed-die forging of lead [12] has been tabulated in Table 1.
168
It can be noticed from the Table-1 that the estimated results from proposed numerical solution agree to experimental results [12] adequately. In recent times, similar numerical R solutions have also been reported by researchers [22] using DEFORM-3D⃝ platform. However, in those environment, a grid testing has been carried out to quantify the desired meshing and trial and error method has been used for finding the optimum billet dimensions. As the proposed simulation uses re-meshing criteria to simulate the large distortion, rather than the element quality, the re-meshing distortion criteria has been implemented. Since, the number of element changes dynamically in simulation, the re-meshing based simulation is more appropriate one. Moreover, it has been noticed that the computational requirement to simulate such problem is not significant (≈ 7 minutes using HP-Elite, 16 GB RAM and 3.40 GHz i7 processor).
169
4. Analytical Modelling of Open-Die Forging Process
158 159 160 161 162 163 164 165 166 167
170 171
The analytical solution is very much essential during designing the open-die forging process for small and medium scale industry who really could not effort for a commercially available 6
174
FEA software. In line with the objective, the analytical modelling for elliptical cross-section has been derived by taking analytical derivation for circular cross-section as a reference, explained in subsequent section [7].
175
4.1. Elliptical Cross Section
172 173
176 177 178 179 180 181
182 183 184 185 186
Assumptions: Considering the forging force F will attain its maximum value at the end of the operation and the coefficient of friction between the work-piece and the dies (upper and lower die) will remain constant. The variation of the stress field along the y-direction is negligible and is not considered. Because, the thickness of the work-piece is assumed to be small as compared with its other dimensions, and during the forging process, the entire work-piece will be in the plastic state. Fig.3 (a) shows a typical open die forging of an elliptical disk at the end of the operation. The r is radial direction and h is final thickness of the disc at the end of operation. The maximum and minimum radius of the disk are a and b, respectively. The origin of cylindrical co-ordinate system has been taken at the center of disk. The corresponding stresses acting on radial direction have been shown in Fig.3 (b).
Figure 3: (a) Forging of elliptical disk, (b) Stresses acting on cylindrical-symmetry of Elliptical cross- section.
187 188 189
Considering the cylindrical symmetry, it has been assumed that the σθ = σr and both the σθ and σr are independent of θ . Now, considering the radial equilibrium, the stresses acted of the element can be written as: √ √ (σr + d σr )h( (a + da)2 d θ 2 + (b + db)2 d θ 2 ) − σr h( a2 d θ 2 + b2 d θ 2 ) − σθ hda sin( d2θ ) −σθ hdb sin( d2θ ) − τ ad θ da − τ bd θ db = 0
(2) 190
On simplifying, neglecting higher order terms, the above equation can be reduced to:
7
(σr +d σr )h((a+da)+(b+db))d θ − σr h(a+b)d θ − σθ hda
191
dθ dθ − σθ hdb − τ ad θ da− τ bd θ db = 0. 2 2 (3)
Further on simplification, the above equation can be stated as:
σr h (da + db) + d σr h((a + da) + (b + db)) − τ (ada + bdb) = 0. 2 192 193
Again to simplify the analysis, take σr , σθ and p as the principal stresses. Using below equations (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 = C (cons tant)
194 195
and C = 6K 2 with σ1 = σr , σ2 = σθ (= σr ), has been obtained.
σr + p = 196
√ 3K
(6)
or
(7)
197
(since the shear stress K is constant)
198
Substituting d σr in the equation (4), it can be re-written as:
σr h σr h da + db − d pha − d phb − τ ada − τ bdb = 0. 2 2
201
(8)
Now, substitute the value of σr in the above equation, the following equation is obtained. √ √ 3K − p 3K − p hda + hdb − d pha − d phb − τ ada − τ bdb = 0 2 2
200
(5)
and σ3 = −p , the following conditions
d σr = −d p
199
(4)
(9)
In this process, beyond certain radii, say, as and bs , a sliding will take place at the interface which allow the radial expansion of the work-piece. With an assumption: (
τ = µp
as ≤ a ≤ a f bs ≤ b ≤ b f 8
) (10)
(
τ =K
202 203
0 ≤ a ≤ as 0 ≤ b ≤ bs
) (11)
Thus, in these two zones, the above equation (10) and (11) takes the below mentioned mathematical form. √ ( 3K−p) (da+db) 2 p
( − d p(a+b) − µh (ada + bdb) = 0 p
√ ( 3K−p) (da + db) − d p(a + b) − µhK (ada + bdb) = 0 2
204
p
p=
206
= C1 exp
√1 3
−C2 −
µ 2h
209
) (13)
a2 +b2 a+b
(
)
( + 12
a2 +b2
as ≤ a ≤ a f bs ≤ b ≤ b f
(
)
a+b
0 ≤ a ≤ as 0 ≤ b ≤ bs
) (14) ) (15)
√ 3K (at a = a f and b = b f )
(16)
Using equation (16) in equation (14), the following equation has been obtained.
exp
µ − 2h
(
a f 2 +b f 2 a f +b f
)
√ 3K
(17)
At a = as and b = bs , equating the right−hand sides of equations (10) and (11) to the value of p (p = Kµ ), the equations (14) and (17) can be written as: √ 3µ exp 2 K µ
210
(
µK 3h
C1 = 208
0 ≤ a ≤ as 0 ≤ b ≤ bs
(12)
As the periphery of the disc is free, means a = a f , b = b f , and σr = 0, the equation (6) can be re-written as:
p= 207
(
)
Integrating the above two equations, the following equation is obtained. √ 3K exp 2p
205
as ≤ a ≤ a f bs ≤ b ≤ b f
µ − 2h
=
exp
(
a f 2 +b f 2 a 2 +b 2 s s a f +b f − as +bs
√ 3K
Or
9
) −1 2
(
as ≤ a ≤ a f bs ≤ b ≤ b f
) (18)
[ ] √ √ as 2 + bs 2 a f 2 + b f 2 2h 3 1 = + ln ln 3µ + µ+ as + bs af +bf µ 2 2
(19)
a
211 212 213
For simplification, ab = b ff = abss = C, where the value of constant, C, used to be determined by the designer. Now, substituting, a = as and b = bs in equation (15), the following equation can be obtained. (
1 K µK C2 = √ − + 3h 3 µ 214
µ − 2h
= exp
p
(
a f 2 +b f 2 a2 +b2 a f +b f − a+b
as 2 +bs 2 as +bs
−
a2 +b2
)
a+b
(
) − 12
( + Kµ
)
as ≤ a ≤ a f bs ≤ b ≤ b f
(21)
)
0 ≤ a ≤ as 0 ≤ b ≤ bs
(22)
[∫
∫ bs
as
p2 ada +
0
∫ af
p2 bdb+
0
p1 ada+
as
]
∫ af bs
p1 bdb
(23)
where, p1 and p2 are pressures given by equations (20) and (21), respectively. The first to fourth terms have been defined by equations 24-27. First term
[
Second term
218
(20)
The total forging force can be written as: F = 2π
217
(
µK 3h
p=
216
)
Following, the pressures for the non-sticking and sticking zone can be written as: √ 3µ exp 2
215
as 2 + bs 2 as + bs
µK 3h
[
µK 3h
((
as 2 + bs 2 as + bs
((
)(
as 2 + bs 2 as + bs
a2 2
)(
)
b2 2
(
a3 − 0.934 3
)
(
b3 − 1.09 3
))
))
K a2 + µ 2
K b2 + µ 2
]as (24) 0
]bs (25) 0
Third term
[√
exp
3µ µ K 2 + 2h
((
a f 2 +b f 2 a f +b f
)
) ] −0.934a + 12
[√
+
2.14exp
3µ µ K 2 + 2h
((
a f 2 +b f 2 a f +b f
a2
)
) ] a f −0.934a + 12
as
10
(26)
3.5
×10 9
3.5
×10 8 Mild Steel Brass
3
2.5
2.5
Maximum Force (N)
Maximum Force (N)
Aluminum Alloy
3
2
1.5
1
0.5
2
1.5
1
0.5
0
0 0
50
100
150
200
250
300
0
Major radius (in mm), E.R=0.5, a/b=1.5
50
100
150
200
250
300
Major radius (in mm), E.R=0.5, a/b=1.5
Figure 4: Analytical Results for constant values of coefficient of friction for elliptical cross-section with different materials
Fourth term
[√
exp
3µ µ K 2 + 2h
((
a f 2 +b f 2 a f +b f
)
) ] −1.09b + 12
[√
+
1.83exp
3µ µ K 2 + 2h
((
a f 2 +b f 2 a f +b f
b2
)
) ] b f −1.09b + 12
(27)
bs
219
Now, the final forging equation for an elliptical cross-section billet can be written as: [ µ K (( F = 2π
( 3 )) ]a a K a2 s − 0.934 + 3h 3 µ 2 0 + [ (( 2 2 ) ( 2 ) ( 3 )) ]b µK as +bs b b K b2 s − 1.09 + 3 µ 2 0 + 3h [ as +bs (( 2 [√ ) ) ] as 2 +bs 2 as +bs
)( 2)
a 2
(( ) ) ] a f a f 2 +b f 2 √ 3µ µ K 2 +b 2 1 a −0.934a + 2 3µ µ K f f 1 2 + 2h a f +b f −0.934a + + 2 2h a f +b f 2 2.14exp + + exp 2 a as [ [√ (( ) ) ] b 2 2 f (( 2 ) ) ] a f +b f √ 3µ µ K 2 1 a +b + −1.09b + 3µ µ K f f 1 2 2 2h a f +b f + −1.09b + 2 2 2h a f +b f 1.83exp exp + b2 bs
220 221 222 223
(28)
Using the above expression, the analytical results for a constant value of coefficient of friction, such as µ =0.3, for an elliptical cross-section billet with different materials has been estimated and have been shown in Fig.4 (a-c). The required maximum force for forging the billet has been calculated using above final expression (equation 28). 11
Materials Density (Kg*/m()3 ) Youngs modulus (Mpa) Poissons ratio
Aluminium Alloy (AA6063) 2700 61300 0.33
Mild Steel 7860 1.9E5 0.28
Brass 8450 1E5 0.331
Table 2: Material Properties
224
225 226 227 228 229
230 231 232 233 234 235 236 237 238 239 240 241
5. FEM Simulation Next, the finite element method based simulation has been carried out to verify the performance of the derived analytical solution. In line with this, the numerical investigations R have been carried out in ANSYS⃝ workbench which is based on finite element analysis (FEA). The fundamental of steady state structural solver is used in the present work has been discussed in Appendices. In order to verify the analytical solution (equation 28), further, the open-die forging process has been simulated for elliptical cross-section, as discussed in section 3.2. The aluminium alloy (AA6063), mild steel and brass materials have been used in the present simulations. The corresponding material properties are tabulated in Table 1. As discussed in the earlier section, the contact has been created between the billet (contact, first body) and die (target, second body) with friction type (having constant friction coefficient µ ). The boundary conditions used in the simulation are same as discussed in the earlier section, i.e., the lower die is fixed by assigning degree of freedom zero. Displacement has been applied to the upper die by allowing the billet to displace in lateral direction rather than in axial direction. The nonlinear adaptive region to the billet has been applied to simulate extremely large deformation. The Solid-285 element type has been used for meshing the billet and has been shown in Fig. 5. The solution has been achieved by using steady state structural solver with 100 steps.
Figure 5: Meshing of components: (a) 3D-model view, (b) front view, and (c) isometric view (without dies)
242
243 244 245
6. Results and Discussion The steady state structural solutions have been achieved using aforementioned simulation technique for elliptical cross-section of billet with different materials (aluminium alloy (AA6063), mild steel and brass). From steady state solution, the contour plot of directional 12
246 247 248 249 250 251 252
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271
272 273 274 275 276 277 278 279 280 281 282 283
deformation, maximum principle stress and equivalent total strain have been extracted for circular cross-section with different materials have been shown in Fig.6 (a-c (aluminium alloy), g-i (mild steel), m-o (brass)). In a similar nature, the billet with elliptical cross-section having aforementioned materials have been analysed with constant coefficient of friction (µ =0.3). The contour plot of parameters such as directional deformation, maximum principal stress, and equivalent total strain have been extracted for the above billet of different materials and have been shown in Fig. 6 (d-f (aluminium alloy), j-l (mild steel), p-r (brass)). Subsequently, the maximum force observed at steady state structural simulation solutions have been extracted for different diameter of circular cross-section with different materials and examined against analytical solutions reported in literature [7], as shown in Fig. 7 (a-c). From figure, it can be inferred that the numerical results agree to well-established analytical solution. From these, it can be inferred that the proposed numerical simulations agree to analytical solutions having error ≤ 5%. The error has been calculated by ratio of difference between analytical and simulated results to analytical result. In a similar way, at steady state structural simulation solutions for different elliptical cross-section, (a/b = 1.5), of billet with different materials have been estimated and have been shown in Fig. 7 (d-f), respectively. One can notice that the numerical results agree to analytical results adequately. The observed error percentage in simulations has been tabulated in Table 4. The observed error is ≤ 5%, which is acceptable in such numerical solution, where extremely large deformation take place. Moreover, the error analysis carried out in detail to understand the performance of the simulation. The observed error in simulation has been shown in Fig. 8. From figure, it can be inferred that the numerical error increases with increase of major length (a/b =1.5), however, lie in acceptable range. Later, similar analyses have been carried out for different contact conditions by varying coefficient of frictions (µ =0.05 to 0.3 in six steps). The simulated results again agree to analytical results with error ≤ 5%, as shown in Fig. 9. Summarizing, the above analyses are to overcome the imitation of a real-time open-die forging process. So, every constraint is imperative and should be incurred in the simulation process to achieve desired result. In the open-die forging process, the shape of billet and length to diameter ratio are important in designing the geometry. The selection of appropriate material is also important to get a quality product. The proposed simulation process may be considered as the one which enables designer to address the solution to the selection of the material. It provides chance to interpret the material and can detect the suitability. Moreover, the proposed methodology can also assist in optimizing different process parameters. It finally tells which parameters are best suitable at given loading condition. As the proposed simulation based on finite element method does not involve large computational resource requirement, it can be reckoned to investigate a real world open-die forging process, eventually, reduces the manufacturing cost. 13
Figure 6: Directional deformation, maximum principal stress and equivalent total strain for circular, elliptical, square and rectangular cross-sections
284
285 286 287 288
7. Conclusion The proposed research exploits the potential of analytical modelling to estimate the desired forging force for geometrical shapes such as circular and elliptical with different materials. However, from industrial requirement point of view, the proposed numerical method based on FEA using a commercially available platform has been observed apposite and agrees 14
9
8
(a)
x 10
4 3 2 1
0
100
200
Mild Steel (Analytically) Mild Steel (Simulaiton)
10 8 6 4 2 0
300
8
0
100
Diameter (in mm) 9
200
6 5 4 3 2 1 0
300
Brass (Analytically) Brass (Simulaiton)
7
0
100
Diameter (in mm) 8
(d)
x 10
Maximum Force (N)
Maximum Force (N)
Maximum Force (N)
Aluminum Alloy (Analytically) Aluminum Alloy (Simulaiton)
(c)
x 10
12
5
0
8
(b)
x 10
6
8
(e)
x 10
200
300
Diameter (in mm) (f)
x 10 4.5
6
5 4 3 2 1 0
0
100
200
300
Major radius (in mm), a/b=1.5
Mild Steel (Analytically) Mild Steel (Simulaiton)
Maximum Force(N)
Aluminum Alloy (Analytically) Aluminium Alloy (Simulaiton)
Maximum Force(N)
Maximum Force(N)
6
5 4 3 2 1 0
Brass (Analytically) Brass (Simulaiton)
4 3.5 3 2.5 2 1.5 1 0.5
0
100
200
300
Major radius (in mm), a/b=1.5
0
0
100
200
300
Major radius (in mm), a/b=1.5
Figure 7: Validation of simulation results with theoretical results for circular and elliptical cross-section with different materials, a/b= major dimension/minor dimension=1.5
289 290 291 292 293 294 295 296 297 298
adequately with experimental result. Moreover, the analytical results have been examined against the FEA result and the observed error is less than 5%. Subsequently, the potential of proposed analytical model and FEA technique have been investigated for different coefficient of frictions to demonstrate the appropriateness. It has been observed from above investigations that the forging force required to forge the billet is more for circular cross-section than the elliptical cross-section, for the different materials (aluminium alloy (AA6063), mild steel and brass) with varying cross-section dimensions of the billet. Similarly, for the different coefficient of friction, the forging force is more for circular cross-section billet than the elliptical cross-section billet. It can be inferred from above analysis that the simulation results are well below the analytical results with ≤ 5% error. The established fact such as the Cross-section Circular -Aluminum Alloy Elliptical- Aluminum Alloy Circular -Mild Steel Elliptical-Mild Steel Circular - Brass Elliptical- Brass
Percentage variation 2.93 to 5.2 2.73 to 5.61 2.75 to 5.02 2.63 to 5.69 2.36 to 5.28 2.81 to 5.73
Table 3: Percentage error variation for circular and elliptical cross-sections for various dimensions (constant coefficient of friction)
15
(a)
(b) 0.06
ALuminum Alloy Mild Steel Brass
0.05 0.045
Percentage of error (%)
Percentage of error (%)
0.055
0.04 0.035 0.03 0.025 0.02
0
2
4
6
8
0.05 0.045 0.04 0.035 0.03 0.025
10
ALuminum Alloy Mild Steel Brass
0.055
0
2
case no: x (x=1 to 10)
4
6
(c) 0.1 ALuminum Alloy Mild Steel Brass
0.08
Percentage of error (%)
Percentage of error (%)
10
(d)
0.1
0.06 0.04 0.02 0
8
case no: x (x=1 to 10)
0
1
2
3
4
5
6
0.06 0.04 0.02 0
7
case no: x (x=1 to 6)
ALuminum Alloy Mild Steel Brass
0.08
0
1
2
3
4
5
6
7
case no: x (x=1 to 6)
Figure 8: percentage error for cross-sections from its analytical results
299 300 301 302 303
forging force increases with increase in the volume of the billet has also been observed. The limitation of the proposed FEA may be identified to simulate the billet up to reduction of height ratio (0.3) to attain reliable estimation. Summarizing, the demonstrated methodology can be reckoned to estimate the desired forging force, detect the suitability of given material for corresponding open-die forging process in real- time, from industrial prospective. Cross-section Circular -Aluminum Alloy Elliptical- Aluminum Alloy Circular -Mild Steel Elliptical-Mild Steel Circular - Brass Elliptical- Brass
Percentage variation 3.35 to 5.76 3.65 to 5.73 3.24 to 5.84 3.52 to 5.41 3.02 to 5.91 3.83 to 5.97
Table 4: Percentage error variation for circular and elliptical cross-sections (varying coefficient of friction)
16
9
8
(a)
x 10
1.5 1 0.5 0
0.1
0.2
Mild Steel (Analytically) Mild Steel (Simulaiton)
4 3 2 1 0
0.3
5
Maximum Force (N)
2
Maximum Force (N)
Maximum Force (N)
Aluminum Alloy (Analytically) Aluminum Alloy (Simulaiton)
(c)
x 10
5
2.5
0
8
(b)
x 10
3
0
0.1
0.2
4 3 2 1 0
0.3
Brass (Analytically) Brass (Simulaiton)
0
0.1
0.2
0.3
coefficient of friction, (radius=150mm) coefficient of friction, (radius=150mm) coefficient of friction, (radius=150mm) 8
8
(d)
x 10
8 6 4 2 0
0.1
0.2
0.3
coefficient of friction, (M.R=150mm)
2
Maximum Force(N)
10
Maximum Force(N)
Maximum Force(N)
Aluminum Alloy (Analytically) Aluminium Alloy (Simulaiton)
Mild Steel (Analytically) Mild Steel (Simulaiton) 1.5
1
0.5
0
(f)
x 10
2
12
0
8
(e)
x 10
14
0
0.1
0.2
0.3
coefficient of friction, (M.R=150mm)
Brass (Analytically) Brass (Simulaiton) 1.5
1
0.5
0
0
0.1
0.2
0.3
coefficient of friction, (M.R=150mm)
Figure 9: Validation of Analytical results with Simulation results for different coefficient of friction for circular and elliptical cross-sections with different materials, a/b=1.5
304
305 306 307
Acknowledgement One of the authors, R. Hari Krishna, acknowledges the generous funding received from the Ministry of Human Resource Development (MHRD), Government of India, and Industrial Acoustics Laboratory for carrying out this work at NIT Rourkela.
17
308
APPENDICES
309
310
311 312
Appendix-I (Finite Element Modelling) Mathematically, in finite element method, the element formulations are based on the principle of virtual work, which can be stated as: ∫ v
313 314 315 316 317 318
σi j δ ei j dV =
v
fiB δ ui dV
∫
+ s
fis δ ui ds
. j
. j
where, σ
ij
.
.
.
ij
jk
ik
= σ − σik ω − σ jk ω .
ij
is the Jaumann rate of Cauchy stress, ω = ij
1 2
(
(A.2) ∂ vi ∂xj
320
the time rate of Cauchy stress.
321
The stress change due to applied strain can be expressed as: . j
σ 322 323
(A.1)
( ) ∂u where, σi j is the Cauchy stress component, ei j = 12 ∂∂ xuij + ∂ xij is the deformation tensor [23]. The ui is the displacement, xi is the current coordinate, fiB is the component of body force, fis is the component of surface traction, V is the volume of deformed body, and s is the surface of deformed body on which tractions are applied [24, 25]. The desired element formulation can be obtained by differentiating the aforementioned virtual work. The corresponding Cauchy stress, σi j , can be estimated using below relation [26].
σ
319
∫
ij
= ci jkl dkl
where, ci jkl is the material constitutive tensor, di j = tensor, vi is velocity.
∂v
− ∂ xij
)
.
is the spin tensor, σ is ij
(A.3) 1 2
(
∂ vi ∂xj
) ∂v + ∂ xij is the rate of deformation
324
325
326 327 328
Appendix-II (Displacement Criteria) The displacement of element is essential in simulating large deformation processes which depends upon element formulation. In present simulation, the mixed u − p formulation has been found suitable due to ease of calculating the hydrostatic pressure or volume change 18
329 330
rate [23]. The finite element formulation to calculate the stiffness of the element can be expressed as: (
Kuu Kup K pu K pp
) {
∆u ∆ p˙
}
{ =
∆F 0
} (A.4)
.
331
where, ∆u, ∆ P are the increment in displacement and hydrostatic pressure, correspondingly.
332
333
334 335 336 337 338 339
Appendix-III (Re-Meshing Criteria) When a material undergoes maximum distortion, the re-meshing is required to achieve the desired numerical stability. However, the most critical is to identify the re-meshing criteria values to initiate re-meshing during simulation. One can refer literature [27] to find the detail on finite element modelling based on re-meshing. The desired parameters corresponding to present simulation such as equilibrium, pressure constitutive and weighted residual form can be written as: ∫ Ω
δ εi j (si j δi j p)dΩ −
∫ Ω
δ ui bi dΩ −
∫ Γt
δ uiti dΓ = 0
( ) ) ∫ nd P ∂q q − εv dΩ + ∑ τiri dΩ = 0 K Ω Ω i=1 ∂ xi
∫
(
∫ Ω 340 341 342 343 344 345 346 347
(A.5)
(
ωi τi
) ∂p + πi dΩ = 0 ∂ xi
(A.6)
(A.7)
where, ωi is the appropriate weighting function (ωi = δ πi , πi is the negative of projected pressure gradient. The τi is the term traduced in order to ensure symmetry in final system of equations. The P is pressure, q is the arbitrary test function representing virtual pressure field, K is the bulk modulus, εv is the volumetric strain, nd is the number of space dimensions, τi is the intrinsic time parameter, and rt is the standard form of ith differential equation. When ui , p, and πi are substituted in aforementioned equations in its interpolation form, it leads to the Galerkin discretized equations (for δ ui =q=ωi =Ni) as [28]: A G 0 u¯ T p¯ G −(C + L) −Q 0 −QT −C˙ π¯
19
f = 0 0
(A.8)
348
where, ∫
Ai j =
Bi T Dd B j dΩ
(A.9)
(∇Ni )N j dΩ
(A.10)
∇T Ni [τ ]∇N j dΩ
(A.11)
Ωe
∫
Gi j =
Ωe
∫
Li j =
Ωe
∫
1 Ni N j dΩ Ωe K
Ci j =
(A.12)
C¯ 1 0 0 C¯ = 0 C¯ 2 0 0 0 C¯ 3 ∫
C¯ikj = [ Q=
Ωe
Q
ij
=
Ωe
∫
fi = In the above
Ni bdΩ + ]T
b1 b2 b3
∇=
349
Ωe
Q2 ,
τk
Q3
(A.14) ] (A.15)
∂ Ni N j dΩ ∂ xk
(A.16)
Nit¯dΓ, i, j = 1, nd
(A.17)
∫
[ b=
τk Ni N j dΩ
Q1 , ∫
k
(A.13)
∂ ∂ x1 ∂ ∂ x2 ∂ ∂ x3
Γ
and t¯ = , [τ ] =
[ t¯1 t¯2 t¯3
τ1
0
τ2 0
]T
,
(A.18)
τ3
where B is the standard infinitesimal strain matrix, Dd is the deviatoric constitutive matrix [27].
20
350
351
Appendix-IV (Numerical Solver)
352
Finite element analysis platform used below stated equations to estimate the stiffness and the force [29]. [K]{u} = {F a } + {F r }
(A.19)
N
where, [K] = ∑ [ke ] is the total stiffness matrix, {u} is the nodal displacement vector, N is m=1
353
the number of elements, [Ke ] is the element stiffness matrix (may include the element stress stiffness matrix), {F r } is the reaction load vector, {F a }, the total applied load vector, is given by:
{F a } = {F nd } + {F ac } +
N
∑ ({Feth} + {Fepr })
(A.20)
m=1
where, {F nd } is the applied nodal load vector, {F ac }= -[M]{ac }is the acceleration load N
vector, [M]= ∑ [Me ]is the total mass matrix, [Me ] is the element mass matrix, {ac } is the m=1
354
total acceleration vector, {Feth } is the element thermal load vector, {Fepr } is the element pressure load vector.
21
355
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