two dimensional elasto-viscoplastic analysis using … · two dimensional elasto-viscoplastic...
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Two Dimensional Elasto-Viscoplastic Analysis Using Finite Element Method
Dr. AHMED A. HUSSAIN and ABDULLA W. SHAKIR Mechanical Engineering Department , University of Technology, Baghdad, Iraq .
Abstract
The time-dependent behavior of metals is modeled in a simple manner by using the constitutive equations of elasto-viscoplastic behavior together with the linear strain-hardening. The finite element method (FEM) is used for solving the presented mathematical model by using the ANSYS software. The presented mathematical model is studied by using a rectangular metallic perforated strip with loading varying in its type and value. The study includes the relaxation process and the response of the model at the yielding and failure regions in the stress and strains curves and the effects of some factors on these behaviors like tangent modulus and fluidity parameter. It's found that the value of the applied tensional stress that causing failure in elasto-plastic behavior is (T=123 N/mm2) while it is (T=130 N/mm2) in elasto-viscoplastic behavior at ramp load. In another hand the time required for reaching steady state in relaxation process is (t ≈ 7 min) at (T=80 N/mm2) and it is not affected by changing the value of tangent modulus but it changes during the changing of fluidity parameter value.
NOTATIONS
Symbol Definition Unit
Et Tangent modulus
F Yield function [N/mm2]
J2 Second deviatoric stress invariant
k Strain-hardening parameter [N/mm2]
m Strain-hardening exponent
Q Plastic potential function
T Tensional stress [N/mm2]
t Time [min]
V Element volume
Y Current (strain-hardening) yield stress [N/mm2]
γ Fluidity parameter [min -1]
δ Reciprocal of strain-hardening exponent
ε Vector of total strain
εe Vector of elastic strain
εp Vector of plastic strain
εvp Vector of viscoplastic strain
εθ Vector of initial strain
θ Absolute temperature [k]
dλ Plastic multiplier
µ Viscosity of dashpot
ν Poisson's ratio
σ Total stress [N/mm2]
σd Friction slider stress [N/mm2]
σij Components of total stress tensor
σp Dashpot stress [N/mm2]
σy Uniaxial yield stress [N/mm2]
σyd Dynamic yield stress [N/mm2]
[B] Strain matrix
[D] Elasticity matrix
[Dvp] Equivalent elasticity matrix for elasto-viscoplastic behavior
[K] Stiffness matrix
[N] Global displacement shape functions
{R}b Body loads
{u} Displacement at any point
{V} Pseudo loads
{δ} Nodal displacement
{ε} Total strain
{εp}
Plastic strain rate
•
•
{εvp} Viscoplastic strain rate
{εθ}
Initial strain rate
{σ} Total stress
{σ} Arbitrary coordinate dependent stress distribution
Introduction The fundamental assumption of all theories of plasticity (that of time independence of the equation of state) makes simultaneous description of the plastic and rheologic properties of a material impossible. It is well known that in many practical problems the actual behavior of a material is governed by plastic as well as rheologic effects [1].
Both sciences (plasticity and rheology) are concerned with the description of very important mechanical properties of structural materials. Each of them has created its own methods of investigation and has developed within the framework of certain assumption which, unfortunately, cannot always be satisfied in reality. The results of rheology are confined to cases where plastic strain is of no decisive importance. On the other hand, the results of the theory of plasticity permit correct description of only such problems where the influence of rheologic effects may be considered unessential. Thus there is no need to point the advantages that can be gained simultaneous description of rheologic and plastic effects, and the general problem is that of viscoplasticity. The viscous properties of the material introduce a time-dependence of the states of stress and strain. These states are represented by state variables and constitutive equation and they are defining as follows:
• State variables : They are variables which represent the current state of a material (which is produced by the entire past history of deformation undergone by the material) to determine the instantaneous response of it. Such variables are called also (internal variables) [2].
• Constitutive equation : It is a suitable description of the material elastic and inelastic deformation behavior, this description is contained in stress analysis that yields the stresses and strains in structures or products under a given load history [3].
Cormeau [4] obtained a stability criterion for commonly used viscoplastic laws along with the finite element method, later he introduces [5] a three-dimensional elasto-plastic finite element solution for a thick shell structure via the stationary response of an elastic-viscoplastic model and devoted a special attention to the problem of numerical integration across the depth of the shell. Cristescu and Suliciu [6] , present a thermo mechanical theory which allows finite speed of propagation for both mechanical and thermal waves in viscoplastic media. Li and Weng's study [7] is concerned with the determination of the time-dependent, non-linear creep of such a two phase composite. Each phase is taken to be elastic-viscoplastic and capable of work-hardening. The main objective of the present study is to investigate the viscoplastic behavior in a perforated strip through several types of loading , also the effects of tangent modulus (Et) and fluidity parameter (γ) values on this behavior has been clarified .
Finite Element Formulation for Elasto-Viscoplastic Analysis The basic steps of finite element analysis here is the description of unknown displacement field {u} within each element in terms of nodal displacements {δ} associated generally with the values of the function at the nodal points, in the form [8]
{u} = [N] {δ} .............................. (1)
•
in which the shape functions matrix [N] depends on the spatial coordinates. With displacements known at all points within the element, the strain at any point can be determined by using a relationship of the form [8]
{ε} = [B] {δ} …....……………… (2)
the matrix [B] is the strain matrix which is generally composed of the derivatives of the shape functions
If body loads and boundary loads acting on an element are {R}b and assumed to be concentrated at the nodes for the present discussion, then for equilibrium it is required that
……..……….. (3)
which is called the total potential energy minimization principle , in which {σ} are the stresses associated with the strains {ε}. The constitutive equation for viscoplastic problem, Eqs. (4.22), along with Eq. (4.25) can be rewritten in matrix form as [8]
............. (4)
As the previous relation has been specified in a time rate, it is convenient to rewrite Eq. (3) as [8]
..…….(5)
Substituting from Eq. (2) in Eq. (5) results in
...…………… (6)
where the stiffness matrix [K] is ……..….. (7)
and represents the total loading rate obtained on addition of the strain components to be
...………. (8)
In matrix form,
……….... (9)
dVDBdVDBRR VPV
T
V
Tb }{][][}{][][}{}{••••
∫∫ ++= εε θ
0}{}{][ =−∫ b
V
T RdVB σ
[ ] θεσ
φσε }{}{
)(}{}{ 1 ••−
•+
∂∂
><+=QFD γ
0}{}{}{}{][ =−
••
∫ be
eV
T RdVddB εε
σ
0}{}{][ =−••RK δ
dVBDBKV
T ][][][][ ∫=}{
•R
θεεεε }{}{}{}{••••
++= VPe
………….… (10)
so in finite element form, Eqs. (9) & (10) above can be written as
……………. (11)
and
………..… (12)
Integration of Eq. (11) results in, for associated viscoplasticity [8]
……… (13)
Parabolic Isoparametric Element Fig. (1) shows an 8-noded parabolic isoparametric element. Greater accuracy is achieved by use of fewer complex elements in place of a larger number of simple elements, this fact is proved by the theoretical error analysis which provides a solution which is sufficiently smooth. In non-linear problems the benefits can be even more marked than for the linear case [9]. This element has a parabolic sides and a general node i is shown in fig. (1) with its displacement ui & vi along x & y axes respectively.
eD }{][}{••
= εσ
)}{}{}{]([][}{ θεεδσ••••
−−= VPBD
}{][}{ 1 σε −•
= VPVP D
}{)}{}{}{][(][}{ σεεδσ θ′+−−= VPBD
Figure 1. Typical two-dimensional parabolic isoparametric element [9]
Case Study The case is a perforated strip and because it is very thin; it can be assumed to be a plane stress problem. The main dimensions of this strip are shown in Fig. (2) . The material properties are listed in table (1) below . Due to symmetry along axes XX and YY, a quarter of the domain will be considered as shown in Fig. (3). 8-noded serendipity elements are used in the finite element mesh and as shown in Fig. (3), where 71 elements with total degrees of freedom equal to 504 is used. All the nodes along line CD are assumed to be fixed in x-direction and free in y-direction, while the nodes along line AB are assumed to be fixed in y-direction and free in x-direction.
Figure 2. Main dimensions of the perforated strip
u
vgeneral node
i
x
y
Parabolic shape and displacement
i i
Figure 3. Finite element mesh for the shaded quarter of strip
Table (1) Material properties for the strip . Material properties Value Unit
E 70000 N/mm2
ν 0.2 ______
σy 243 N/mm2
k 2250 N/mm2
m 1 ______
γ 0.01 min-1
The load, which is distributed tensional stress (T), is applied on the upper side of the strip with step-wise condition, where its value is increased through a 6 runs as listed in table (6.2)
Table (2) Tensional stress (T) values at each run [10]
Results and Discussion The results are shown in Figs. (4), (5) & (6). Where Fig. (4a) shows the variation of the axial stress along line AB at each run. Figs. (4a) & (4b) show the existing finite element results from the present model and the published finite element results [10] respectively. A good agreement between the two results is indicated.
Also, the variation of viscoplastic strain at each run along line AB is shown in Fig. (5a). It is clear that there is no viscoplastic strain at run no.1 and also the maximum value of the effective viscoplastic strain at each run is located at node B. The corresponding effective viscoplastic strain is shown in Fig. (5b), which is obtained from Ref. [10]. Again, a good agreement between the two results is concluded. Fig. (6a) shows the axial displacement of node C, at the top of the finite element mesh, which is plotted against the non -dimensional tensional stress increment (σeffective / σy) at each run. It can be observed that the tensional stress-displacement relation tends to be non -linear after run no.4 approximately. Fig.(6b) shows the corresponding axial displacement of node C from Ref. [10]. Again, a good agreement between the two results is concluded.
Run no. Tensional stress (T) values
[N/mm2]
1 51.638
2 65.367
3 79.947
4 97.2
5 112.388
6 128.8
a b
Figure 4. Axial stress distribution along line AB
(a) ANSYS software results.
(b) Published results [10].
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.0 1.0 2.0 3.0 4.0 5.0
Line A B [mm]
Effe
ctiv
e Vi
scop
last
ic S
trai
n [%
]
23
4
5
6
a b
Figure 5. Effective viscoplastic strain distribution along line AB
(a) ANSYS software results.
(b) Published results [10].
0
5
10
15
20
25
30
35
0 1 2 3 4 5
line A B [mm]
Axi
al s
tres
s [k
g/m
m2 ]
1
23
45
6
a b
Figure 6. Axial displacement at node C
(a) ANSYS software results.
(b) Published results [10].
Fig. (7) shows the relaxation process in the values of effective stress at each run. It can be seen that the time required for reaching the steady state increases with the increasing of the tensional stress (T) value as shown in Fig. (8), also the difference between the maximum value of effective stress, at the beginning of the relaxation process, and its steady state value, at the end of the relaxation process, increases along with the increasing of the tensional stress (T) value (as shown in Fig. (9).
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
axial displacement at C
2 * T / ? y
1 2
3 4
5 6
T = 4 kg/mm2
17
17.25
17.5
17.75
18
0 1 2 3 4 5 6 7 8 9 10 11
Time [min]
Effe
ctiv
e st
ress
[kg
/mm
2 ]
T= 5 kg/mm2
21.6
21.7
21.8
21.9
22
0 2 4 6 8 10 12
Time [min]
Effe
ctiv
e st
ress
[kg
/mm
2 ]
Figure 7. Stress relaxation at some values of tensional stress T = 40, 50, 80 & 90 N/mm2 , influence of loading type - gradual steady loading
T = 8 kg/mm2
22
24
26
28
30
32
34
0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0
Time [min]
Effe
ctiv
e st
ress
[kg
/mm
2 ]
T = 9 kg/mm2
24
26
28
30
32
34
36
0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0
Time [min]
Effe
ctiv
e st
ress
[k
g/m
m2 ]
0
20
40
60
80
100
120
140
160
180
0 2 4 6 8 10 12 14 16 18 20
Tensional stress (T) [kg/mm2]
Tim
e to
ste
ady
stat
e [m
in]
Figure 8. Time required for reaching steady state at each value of tensional stress (T) ,
influence of loading type - gradual steady loading
0
10
20
30
40
50
60
70
80
0 2 4 6 8 10 12 14 16 18 20
Tensional stress (T) [kg/mm2]
Effe
ctiv
e st
ress
[k
g/m
m2 ]
max
steady
Figure 9. Maximum and steady state values of effective stress against tensional stress (T) ,
influence of loading type - gradual steady loading
T = 123
Fig. (10) shows the region of yielding initiation of Fig. (9) with a range of tensional stress ( T = 60 to 65 N/mm2). It is clear that the yielding initiation happens at a tensional stress (T) value (between 61.5 and 62 N/mm2).
24.0
24.5
25.0
25.5
26.0
26.5
27.0
6.00 6.05 6.10 6.15 6.20 6.25 6.30 6.35 6.40 6.45 6.50
Tensional stress (T) [kg/mm2]
Effe
ctiv
e st
ress
[k
g/m
m2 ]
maxsteady
Figure 10. Maximum and steady state values of effective stress against tensional stress (T) , T
= 60 to 65 N/mm2 , influence of loading type - gradual steady loading
Fig. (11) shows the values of effective stress during the increasing of tensional stress (T) (from 0 to 180 N/mm2). The value of effective stress starts from zero then increases in a straight line until the value of tensional stress ( T = 61.63 N/mm2) which causes the yielding initiation and the related value of effective stress is (268.08 N/mm2). After that a rapid decrease in the value of effective stress happens until (243.21 N/mm2) which is at value of tensional stress ( T = 61.64 N/mm2). Then the effective stress increases again until a value of tensional stress ( T ≈ 130 N/mm2) where the effective stress increases rapidly after that because of the failure. The values of effective stress before yielding are the same as those of Fig. (9), while after yielding the values of effective stress in this figure, are neither equal to the maximum nor to steady state values of effective stress in Fig. (9).
Figure 11. Effective stress against tensional stress (T) , influence of loading type – ramp
loading
Fig. (12) shows the effective stress plotted against the effective viscoplastic strain. The yielding initiation can be seen here clearly which is happen when the effective stress reaches a value (268.08 N/mm2). After that a rapid decreasing in its value happens and then returns to increase during the increasing of the effective viscoplastic strain. The failure also can be seen when the values of both of effective stress and effective viscoplastic strain increase rapidly, while Fig. (13) shows the plot of effective stress against the effective elastic strain which is simply as a straight line with a slope equal to (70000 N/mm2), and it is the value of young's modulus that is used, in the entire range of the tensional stress (T) , i.e. (T = 0 to 180 N/mm2).
0
10
20
30
40
50
60
0 2 4 6 8 10 12 14 16 18 20
Tensional stress (T) [kg/mm2]
Effe
ctiv
e st
ress
[ k
g / m
m
2
]
0
10
20
30
40
50
60
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Effective viscoplastic strain [%]
Effe
ctiv
e st
ress
[k
g/m
m2 ]
Figure 12. Effective stress against Effective viscoplastic strain , influence of loading type –
ramp loading
0
10
20
30
40
50
60
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
Effective elastic strain [%]
Effe
ctiv
e st
ress
[kg
/mm
2 ]
Figure 13. Effective stress against Effective elastic strain , influence of loading type – ramp
loading
failure
slope = E = 7000 kg/mm2
Conclusions
The results obtained from the strip at the end of relaxation process of the elasto-viscoplastic behavior at steady loading are the same as those obtained from the same strip at the elasto-plastic behavior. The effective stress-effective plastic strain relation obtained at the elasto-plastic behavior is linear after yielding due to the linear strain-hardening that is used, while this relation is not linear after yielding when obtained at the elasto-viscoplastic behavior due to the dependence of the dynamic yield stress (σyd) on viscoplastic strain rate. The relaxation process time is increased with the increasing of tensional stress (T) value because the difference between the maximum and steady state values of effective stress increases.
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