Elastic Perfectly Plastic Behavior modeling
by a Meshless Method
Khadija KHAMMARI
Laboratory of Engineering, Industrial Management and Innovation
Hassan 1erUniversity, Faculty of Sciences and Technology
Hicham FIHRI FASSI
Laboratory of Engineering, Industrial Management and Innovation
Hassan 1erUniversity, Faculty of Sciences and Technology
Abstract—Meshless methods are computational techniques that do not require the use of any connectivity concept, such as those used in the finite element method (FEM). They offer the great advantages and promising potential in linear and nonlinear problems of structures. In this paper we present the incremental formulation of elasto-plastic problems based on the element-free Galerkin method (EFG) using the Moving Least Squares (MLS) approximation, the Lagrange multipliers method is applied for imposing the boundary conditions. The elastic perfectly plastic behavior of the material is considered and the implicit integration scheme is used. Numerical example is given to show the performance of the current approach.
Index Terms— Elastic perfectly plastic behavior, Element-free Galerkin method, Implicit integration scheme, Incremental formulation, Meshless methods.
I. INTRODUCTION
T
he finite element method (FEM) [1] is a powerful tool for modeling linear and non-linear problems in applied mechanics. In this method, the continuum is divided into a finite number of elements, the behavior of each element is specified by a finite parameters and the solution of the complete system is an assembly of this elements.Meshless methods (MM) have been introduced and developed for solid mechanics since 1990 [2], [3] as an attractive alternative numerical approach. The interpolation in meshless methods is entirely based on a set of scattered nodes for the problem discretization instead of meshes.
The element free Galerkin method (EFG) is one of the most widely used to model a variety of physics e.g. 2D linear elasticity [4], [5], static and dynamic fracture mechanics [6], [7], plate and shell analysis [8],vibration [9], electromagnetic [10], heat transfer [11], metal forming [12], biomechanics [13] and geomechanics [14].
Plasticity is the evolution problem which needs to use a fitting integration scheme. Applying the implicit time integration scheme proposed by Moreau [15], we obtain the incremental formulation which has given the best results for seems problems [16].
In this work, the plastic law leads us to minimize the energy of deformation based on the incremental formulation and the inf-convolution concept [17]. The EFG method and the mathematical programming [18] are used to perform numerical solutions. The paper is structured as follows. In section 2, the MLS approximation is briefly reviewed and the Lagrange multipliers method is presented for the purpose to impose the boundary conditions. The elastic perfectly plastic behavior is given in Section 3. Section 4 describes the elastoplastic analysis for the evolution problems and its application for a uniaxial load. The Section 5 presents the EFG discretization. The variational principles are discussed in Section 6. Numerical example follows in Section 7 to show the efficiency of this approach. Finally, conclusions and further research are treated in Section 8.
II. MESHLESS METHOD (EFG)
A. Oveview of the MLS Approximation
In the MLS approximation, the local approximation at a point x is defined as
𝑢ℎ(𝑥) = ∑ 𝑝𝑖(𝑥)𝑎𝑖(𝑥) = 𝑝𝑇(𝑥)𝑎(𝑥), 𝑚
𝑖=1
(1)
where 𝑚 is the number of terms in the basis, 𝑎𝑖(𝑥) are the coefficients of the base functions, and 𝑝𝑖(𝑥) are monomial functions.
We consider 𝑛 nodes situated in the positions
𝑥1 , 𝑥2 , … . , 𝑥𝑛 where the values of
displacement 𝑢1 , 𝑢2, … . 𝑢𝑛 are known. The approximation in the point 𝑥𝑖 can be built by
𝑢ℎ(𝑥, 𝑥
𝑖) = 𝑝𝑇(𝑥𝑖)𝑎(𝑥), 𝑖 = 1,2, … . , 𝑛. (2)
Weight discrete square norm is defined as
𝐽(𝑥) = ∑ 𝜔(𝑥 − 𝑥𝑖) 𝑛
𝑖=1
[𝑢ℎ(𝑥, 𝑥
𝑖) − 𝑢(𝑥𝑖)]2
= ∑ 𝜔𝑖 𝑛
𝑖=1
[𝑝𝑇(𝑥
𝑖)𝑎(𝑥) − 𝑢(𝑥𝑖)]2 }
(3)
where 𝜔𝑖 is a weight function associated with node i. The MLS approximation of displacement will be
𝑢ℎ(𝑥) = ∑ ∑ 𝑝
𝑗(𝑥) 𝑚
𝑗
(𝐴−1 (𝑥)𝐵(𝑥)) 𝑗𝑖𝑢𝑖 𝑛
𝑖
, (4)
where
𝐵(𝑥) = [𝐵1 , 𝐵2 , … . . , 𝐵𝑛], (𝐵𝑖= 𝜔𝑖(𝑥)𝑝(𝑥𝑖))
or
𝑢ℎ(𝑥) = ∑ ∅
𝑖(𝑥)𝑢𝑖, (5) 𝑛
𝑖
where
∅𝑖(𝑥) = ∑ 𝑝𝑗(𝑥) 𝑚
𝑗
∅𝑖(𝑥) are the shape functions of the MLS approximation corresponding to nodal point xi .The matrix 𝐴(𝑥) is a square
matrix and its size is equal to the size of the vector 𝑝.
B. Imposition of the Boundary Conditions
Since the shape functions constructed according to MLS do not possess Kronecker delta property, the coefficients of the interpolants are not equal to the nodal values. Therefore, the essential boundary conditions cannot be imposed directly. Several techniques, including Lagrange multipliers method, penalty method [19], collocation method [20], and coupled meshless-finite element method [21] have been developed to enforce the boundary conditions in meshless methods. In this paper, the essential boundary condition is imposed by using the Lagrange multipliers method.
The Galerkin weak form will be
∫ 𝛿(𝜀)𝑇(𝜎)𝑑𝛺 − ∫ 𝛿u𝑇𝑏𝑑𝛺 − ∫ 𝛿uT𝑡̅𝑑Г
Г𝑡 Ω
Ω
− ∫ 𝛿𝜆𝑇 (u − u̅)𝑑Г Г𝑢
− ∫ 𝛿u𝑇 𝜆𝑑Г Г𝑢
= 0 (7)
where u, 𝜎, b, 𝜆, are the displacement field, the stress tensor, the body forces and the vector of Lagrange multipliers, respectively. 𝑢̅ and 𝑡̅ represent the given displacements and tractions, respectively, on the displacement boundary Гu and
the traction boundary Гt .
The unknown Lagrange multipliers are approximated through the outline
𝜆(𝑥) = ∑ 𝑁𝑖(𝑠)𝜆𝑖 𝑛𝜆
𝑖
𝑥 ∈ Г𝑢 (8)
where 𝜆𝑖 is Lagrange multiplier in the node 𝑖. The system of equations of problem is given by
[ 𝐾 𝐺
𝐺𝑇 0] {𝑈𝜆} = {𝐹𝑞}, (9)
where
𝐾𝑖𝑗= ∫ B𝑖T
𝛺 𝐶 B𝑗𝑑𝛺 , (10)
𝐺𝑖𝑗 = − ∫ ∅𝑖𝑇N𝑗𝑑Г Г𝑢
, (11)
𝐹𝑖= ∫ ∅𝑖𝑇𝑏𝑑𝛺 + ∫ ∅𝑖𝑇𝑡̅𝑑Г Г𝑡
, 𝛺
(12)
𝑞𝑖= − ∫ N𝑖 𝑇 u̅𝑑Г
Г𝑢 . (13)
C. Numerical Integration
In the MLS method, the concept of element does not exist, and the shape functions are not polynomial, so we cannot evaluate the integrals as for the finite element method. However, we can use a direct integration nodes or an underlying grid that serves only in the numerical integration, and does not interfere in the approximation scheme. In this work we used the second method [3].
D. Resolution of the System
The system of equations of problem (9) is solved by mathematical programming [18].
A quadratic program (QP) is a special type of optimization problem in which a quadratic objective function is minimized or maximized subject to linear inequality constraints. QPs may be stated in many equivalent forms and we define a QP in the most general form to be
P𝑟𝑜𝑏𝑙𝑒𝑚:
{
𝑀𝑖𝑛
𝑥 (
1
2𝑥𝑇𝐻𝑥 + 𝑥𝑇𝑓)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑖𝑛: {
𝐴. 𝑥 ≤ 𝑏 𝐴𝑒𝑞. 𝑥 = 𝑏𝑒𝑞
𝐿𝐵≤ 𝑥 ≤ 𝐿𝑈
. (14)
For the solution, we call on to the function Quadprog
(quadratic programming) [22] available in
Toolbox/Optimisation/ Matlab by using the command
𝑥 = 𝑞𝑢𝑎𝑑𝑝𝑟𝑜𝑔 (𝐻, 𝑓, 𝐴, 𝑏, 𝐴𝑒𝑞, 𝑏𝑒𝑞, 𝐿𝐵, 𝐿𝑈) (15)
Exploiting this command in our problem (33), (35) we have
𝑋 = 𝑞𝑢𝑎𝑑𝑝𝑟𝑜𝑔 (𝐾, 𝐹, [ ], [ ], 𝐺𝑇, 𝑞, [ ], [ ]). (16)
We have no inequalities and no bounds, then we use empty matrices: A = [ ], b = [ ], LB = [ ], and LU = [ ],
The equality: GTU=q, gives Aeq=GT, beq=q.
III. ELASTIC PERFECTLY PLASTIC BEHAVIOR The shape of the curve stress-strain corresponding to materials with elastic perfectly plastic behavior is represented on Fig.1 in the case of a uniaxial load.
Fig.1. Elastic perfectly plastic behavior in uniaxial load.
The plastic flow law can be written as follow
𝑖𝑓 𝑖𝑓 𝑖𝑓
|𝜎| < 𝜎𝑦 𝑡ℎ𝑒𝑛 𝜀̇𝑝= 0 𝜎 = 𝜎𝑦 𝑡ℎ𝑒𝑛 𝜀̇𝑝> 0 𝜎 = −𝜎𝑦 𝑡ℎ𝑒𝑛 𝜀̇𝑝< 0
(17)
𝜎𝑦 is the yield stress and 𝜀̇𝑝 is the plastic strain.
The inverse law can be described as
𝑖𝑓 𝑖𝑓 𝑖𝑓
𝜀̇𝑝= 0 𝑡ℎ𝑒𝑛 |𝜎| < 𝜎 𝑦 𝜀̇𝑝> 0 𝑡ℎ𝑒𝑛 𝜎 = 𝜎𝑦 𝜀̇𝑝< 0 𝑡ℎ𝑒𝑛 𝜎 = −𝜎 𝑦
(18) 𝝈𝒚
𝝈
IV. EVOLUTION PROBLEMS
A. Elasto-plastic Analysis
The classical hypothesis of elastoplastic decomposition of the total strain rate in two parts is considered:
𝜀̇ = 𝜀̇𝑒+ 𝜀̇𝑝 (19)
where 𝜺̇𝒆 is the elastic part verifying the generalized Hooke’s law and 𝜺̇𝒑 the plastic part given by the classical normality law. Using the last decomposition the couple (𝝈,𝜀̇𝑝) associated to the history of strain 𝜺̇(𝝉)may be considered as the solution of the following system of differential equations of the first order:
𝑺. 𝝈̇ + 𝜺̇𝒑= 𝜺̇(𝝉) (20)
where 𝝉 is the variable of time evolution and S the matrix of elastic stiffness.
The plastic flaw is a quasistatic problem. Consequently, the time plays the role of a simple parameter of evolution. The stress state does not depend on the intensity of the velocity. By applying the implicit integration scheme introduced first by Moreau [15], the time will be eliminated. For the incremental formulation, the incremental energy of deformation will be indicated by ∆𝑉, and the following notations will be used:
∆𝜎 = 𝜎1− 𝜎0 , ∆𝜀 = 𝜀1− 𝜀0 (21) the same for ∆εe and ∆εp
.
where the index 0 (resp.1) is relative to the beginning (resp. to the end) of the step.
According to the implicit integration, the plastic strain increment is given by
∆𝜀𝑝= ∆𝜏𝜀
1𝑝 (22)
In the frame of convex analysis, the incremental law and its inverse will be expressed by the following relations
∆𝜀𝑝∈ 𝜕∆𝜎∆𝑊
𝑝(∆𝜎)
∆𝜎 ∈ 𝜕∆𝜀𝑝∆𝑉𝑝(∆𝜀𝑝) (23)
where
∆𝑉𝑝 : the incremental energy of plastic deformation. ∆𝑊𝑝 : the incremental complementary energy.
B. Uniaxial Load
We suppose that the incremental superpotential is divided to elastic ∆𝑉𝑒 and plastic ∆𝑉𝑝 :
∆𝑉𝑒 is the elastic energy of deformation defined by
∆𝑉𝑒(∆𝜀𝑒) = 1
2𝐸(∆𝜀𝑒)² = 1
2𝐸(∆𝜀 − ∆𝜀𝑝)2 (24) ∆𝑉𝑝 can be calculated from the so-called plastic dissipation
which is given by the following expression
∆𝑉𝑝(∆𝜀𝑝) = 𝜎𝑦|∆𝜀𝑝| − 𝜎0∆𝜀𝑝 (25)
Finally, to determine the incremental elasto-plastic superpotential ∆𝑉(∆𝜀) that will be used in the variational formulation, we use the inf-convolution concept defined by
∆𝑉(∆𝜀) = 𝑖𝑛𝑓 ∆𝜀𝑝{
1
2𝐸(∆𝜀 − ∆𝜀𝑝)2+ 𝜎𝑦|∆𝜀𝑝| − 𝜎0∆𝜀𝑝} (26)
Using the stationary condition in the minimization problem (26), the following algorithm holds, for metals in uniaxial load:
If |∆𝜀| ≥ 𝜎𝑦/𝐸 then the plastic flow will be
∆𝑉(∆𝜀) =𝐸
2((∆𝜀)2− |∆𝜀𝑝|2) (27)
Else the elastic domain is
∆𝑉(∆𝜀) =𝐸
2(∆𝜀)2 (28)
V. VARIATIONAL PRINCIPLES
LetΩ be the domain that a solid occupying and ∂Г be its boundaries. During the time increment, Ω is subjected to imposed surface traction increments ∆𝑡̅ on the part 𝜕Гt of 𝜕Г and to imposed displacement increment ∆𝑢̅ on the part 𝜕Гu of
𝜕Г. ∆𝑏̅ is the body force.
A displacement increment field is said to be kinematically admissible (KA) if the following compatibility conditions are fulfilled:
∆𝜀 = 𝑔𝑟𝑎𝑑𝑠∆𝑢 in Ω
∆𝑢 = ∆𝑢̅ on 𝜕Гu (29)
where 𝑔𝑟𝑎𝑑𝑠 is the symmetric gradient.
The use of the incremental formulation leads to the following expression:
∆𝛽(∆𝑢, ∆𝜎) = ∫ ∆𝑉(∆𝜀(𝑢))𝑑𝛺 Ω
− ∫ ∆𝑏̅ Ω
. ∆𝑢𝑑𝛺 −
∫ ∆𝑡̅ 𝜕Гt
. ∆𝑢𝑑Г + ∫ ∆𝑊(∆𝜎)𝑑𝛺 Ω
−
∫ ∆𝑡
𝜕Гu . ∆𝑢̅𝑑Г (30)
The exact solution of the boundary value problem, defined by (29) and the constitutive laws (23) is the solution of the following variational principles:
inf ∆𝛽(∆𝑢, ∆𝜎) ,
∆𝑢 KA (31)
VI. EFGDISCRETIZATION
The KA formulation in terms of displacement being chosen to perform the numerical treatment of (30), therefore the stress field will not be considered as variable but only as parameter.
Then, separate terms containing the stress field will be eliminated from (30):
∆𝛽(∆𝑢) = ∫ ∆𝑉(∆𝜀(𝑢))𝑑𝛺 Ω
− ∫ ∆𝑏̅ Ω
. ∆𝑢𝑑𝛺
− ∫ ∆𝑡̅
𝜕Гt . ∆𝑢𝑑Г (32)
we have
∆𝑢(𝑥) = ∅(𝑥)∆𝑈 , ∆𝜀 = 𝐵(𝑥)∆𝑈 (33)
By replacing the terms in (32) we find
∆𝛽(∆𝑈) = ∫ ∆𝜎𝐵∆𝑈𝑑𝛺 Ω
− ∫ ∆𝑏̅ Ω
∅∆𝑈𝑑𝛺
− ∫ ∆𝑡̅
𝜕Гt ∅∆𝑈𝑑Г (34)
we have to solve the following system:
∫ 𝐵𝑇∆𝜎𝑑𝛺
Ω
− ∫ ∅𝑇∆𝑏̅ Ω
𝑑𝛺 − ∫ ∅𝑇∆𝑡̅ 𝜕Гt
𝑑Г = 0 (35)
with the condition :
∆𝜎 =𝜕𝑉(𝐵∆𝑈)
𝜕∆𝜀 (36)
(36) must be satisfied everywhere in the domainΩ . VII. NUMERICAL EXAMPLE:ACANTILEVER BEAM
SUBJECTED TO UNIAXIAL LOAD
In the numerical example presented in this section, the linear basis function and the cubic spline weight function are used in the MLS approximation. In each integration cell, 4×4 Gauss points are used for Gaussian quadrature. The problem is considered to be in plane strain state.
A cantilever beam subject to a uniaxial load at the free end is considered (Fig.2). The geometry of the problem, with the material properties are
- 𝐿 = 100 , 𝐷 = 10, - 𝐸 = 2,1. 105𝑀𝑃𝑎,
- 𝜐 = 0,3,
- 𝜎𝑦= 900 𝑀𝑃𝑎.
The nodal discretization is shown in Fig.3, the nodes are concentrated at the longitudinal section of the beam.
Fig.2. Cantilever beam subject to uniaxial load.
Fig.3. Nodal discretization.
A. Elastic Analysis
A displacement of 2 mm is applied at the free end of the beam.
The displacements are computed for the nodes following x (Fig.4).
Fig.4. Displacements following x for the nodes at the longitudinal beam
B. Elasto-plastic Analysis
a. Material Behavior Law
In this case 0,4 mm of displacement u is applied to the beam in 80 steps.
The first deformed configuration (u = 0,005) and the final deformed configuration (at the final displacement of 0,4 mm) are shown, in Fig.5.
a) Deformation at the beginning of the step.
0 0,5 1 1,5 2 2,5
0 5 10 15 20 25
d
isp
la
ce
m
e
n
t
fo
ll
o
w
in
g
x
(m
m
)
node i
𝑢 𝐷
b) Deformation at the end of the step.
Fig.5. Visualization of deformation.
Stresses / strains plot is given in Fig.6, it can be seen a perfectly plastic form from the curve and it is clear that the stresses do not exceed the yield stress 𝜎𝑦
Fig.6. Stresses (σ) vs. strains (ε).
Plastic strain over the final deformed configuration is shown in Fig.7, which clearly shows the most and the least
constrained areas in the beam.
Fig.7. Plastic strain (𝜀𝑝) over the final deformed configuration.
b. Influence of Mechanical Properties on the
Material’s Behavior
The purpose of this study is to show the influence of mechanical properties on the material’s behavior.
The problem above is used with the same geometry and nodal discretization.
Materials concerned by the study are:
E (MPa) 𝝊 𝝈𝒚 (MPa)
Steel 2,1. 105 0,3 300 Aluminum 1,1. 105 0,3 200 Titanium 10,5. 104 0,3 870
0 100 200 300 400 500 600 700 800 900 1000
0 0,002 0,004 0,006 0,008 0,01
sig
m
a
(
σ)
A total displacement of 1mm is applied over 100 equal steps.
The plots of the constitutive law for the different materials are shown in Fig.8. It’s clearly observed that when the yield
stress 𝜎𝑦 of the material is higher, the margin of elastic domain of the latter is important.
Fig.8. Constitutive law for different materials.
VIII. CONCLUSION AND FUTURE WORK
In this paper, the elastic perfectly plastic behavior has been treated. An algorithm for solving the elasto-plastic evolution problem using a meshless method based on the Moving Least Square approximation has been presented. A numerical example is presented to prove the efficiency of the method.
We have treated only a case of uniaxial load. This work can be extended to a multiaxial load and by taking into account others parameters, different behaviors can be treated as well.
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0 100 200 300 400 500 600 700 800 900 1000
0 0,002 0,004 0,006 0,008 0,01 0,012
Sig
m
a
(
σ)
Epsilon (ε)
Steel
Aluminium
