Three-Dimensional Explicit Parallel Finite Element Analysis of Functionally

Graded Solids under Impact Loading

Ganesh Anandakumar! and Jeong-Ho KimDepartment of Civil and Environmental Engineering, University of Connecticut,

261 Glenbrook Rd, U-2037, Storrs, CT 06269, U.S.A.

January 30, 2009

Abstract

This paper presents two numerical examples to investigate the behavior of three-dimensional (3D)functionally graded (FG) solids under dynamic loading using explicit parallel finite element method.In the first example, wave propagation in a 3D FG bar under transient sinusoidal loading is inves-tigated. Material gradation and thickness (3D) e!ects are seen in the dynamic stress behavior ofthe FG bar. In the second example, a three-point bending beam made of epoxy and glass phasesunder velocity impact is studied. Bending stress history for beam with higher values of materialproperties at the loading edge is consistently higher than that of the homogeneous beam and thebeam with lower values of material properties at the loading edge. Larger bending stresses forthe foremost beam may indicate earlier crack initiation time than the other two beams which wasproven by experiments performed by other researchers.

Keywords: functionally graded material (FGM), dynamic analysis, three-dimensional wave prop-agation, finite element methods (FEM), parallel computing.

1 Introduction

Functionally graded material (FGM)• is a material solution & concept used for a new advanced class of composites.• is characterized by a gradual variation in composition, microstructure and material properties.• experiences severe dynamic thermal and mechanical loadings.• typically fails through cracking and spallation.• therefore requires a detailed dynamic analysis to determine the e!ect of material gradation and

obtain locations of peak values of stresses.

Several numerical methods have been used to investigate FGMs, including integral equations (Ozturkand Erdogan, 1997), boundary element methods (Sutradhar et al., 2002), finite element methods(Santare and Lambros, 2000), etc. In this study, we used the displacement-based FEM and gradedfinite elements (Kim and Paulino, 2002) to model FGMs using the direct Gaussian integration for-mulation. Parallel explicit FEM is used for obtaining the dynamic response of 3D FG solids usingMessage Passing Interface (MPI) standard (MPI2, 1997).

The following are the novelties of this study:• Wave propagation analysis of a FG 3D bar using FEM• Dynamic analysis of a FG 3D beam under velocity impact load using FEM

2 Parallel Explicit Dynamic FEA using Newmark-! method

The steps involved in the explicit Newmark-! ("=0.5 and !=0) method (Newmark, 1959) are givenbelow.

!Corresponding Author, e-mail: anandg@engr.uconn.edu

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• calculate velocities at time tn: !unk = !un"1

k + 0.5!tn"1!un"1k + un

k

"

• calculate displacements at time tn+1: un+1k = un

k +!tn !unk + 0.5!t2nu

nk

• compute e!ective force vector (EFV) at time tn+1: fn+1k = fextn+1

k " fintn+1k

• solve for accelerations at time tn+1: un+1k = M"1fn+1

k

where u, !u, and u represent displacement, velocity, and acceleration vectors, respectively, M is thelumped mass matrix, fext is the external force vector, n is the time step number, fint(=Ku) is theinternal force vector, and k denotes degree of freedom (DOF).

Figure 1 shows a flowchart of the steps involved in the parallel execution of the dynamic FE codeusing a master-slave approach. Typically, the exchange of EFV needs to be done only at processorsthat share the node through which partition is done. But this leads to many small messages being sentfrom one processor to another and may lead to increase in communication time which is undesirable.To overcome this, the EFV at the slaves are sent to the master for assembly and the master returnsthe assembled EFV back to the slaves for calculating acceleration vector.

MPI_INIT and MPI_COMM_RANKInitialize number of processors using

integration using Newmark!beta methodLoop over number of timesteps for time

Partition FE mesh in to sub!meshes manually

Calculate displacement, velocity, and effective

Calculate K & M matrices for local elements

Slaves send EFV to master for assemblyusing MPI_SEND and MPI_RECV

Master returns assembled force vector toslaves. Calculate acceleration vector.

Repeat time!integration for the rest of thesimulation by this exchange algorithm

Exit parallel code using MPI_FINALIZEforce vector (EFV) at slaves & master

Figure 1: Flow chart of the parallel execution of the explicit FE code.

3 Numerical Examples

Example 1: Wave Propagation in a 3D FG Bar: Wave propagation in a fixed-free bar withgraded materials in the y direction is simulated to obtain the e!ect of material gradation. Considera fixed-free square bar (Figure 2(a)) of length L = 1 m and height H = 0.05 m under a transientsinusoidal load (Figure 2(b)) applied at the free end of the bar.

yz x

H=0.05 m

W=0.05 m

P f(t)

L = 1.0 mAlumina

Steel

0 25 50 75 1000

0.5

1

t(µs)

f(t)

(a) (b)

Figure 2: Example 1: Wave propagation in FG 3D bar (a) schematic; (b) sinusoidal load.

The material properties vary linearly from alumina in the bottom to steel at the top. The materialproperties of steel and alumina are shown in Table 1. Due to fine mesh criterion, the 3D FE meshis discretized into 300#15#15 quads, each quad divided into four 15-node wedge elements, totaling270,000 elements and 725,836 nodes.

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Table 1: Example 1: Material properties of steel and alumina considered

Material E(GPa) ! "(kg/m3) Cd(m/s)Steel 210 0.31 7800 6109

Alumina 390 0.22 3950 10617

0 1 2 3 4 5 6!4

!3

!2

!1

0

1

2

3

4

t*(Cd)Alumina

/L

! x/!xL

x=0.5LAlumina

x=L

Steel Midplane

Steel

Alumina

x = 0

Hx

y

z

0.05

0 0

0.05

0.025

0.025L

x = Lx = 0.5L

Steel

Alumina

0 1 2 3 4 5 6!4

!3

!2

!1

0

1

2

3

4

t*(Cd)Alumina

/L

! x/!xL

x=L

x=0.5LAlumina

x=0 Alumina

Steel

Midplane

Steel

(a) (b)Figure 3: Example 1: Normalized longitudinal stress history of 9 points (see the insert) on a gradedbar subjected to transient sinusoidal loading at (a) z = 0.025 m and (b) z = 0 m. Solid, dashed, anddash-dot lines indicate points at x = 0, x = 0.5L, and x = L, respectively. Thick, intermediate-thickand thin lines indicate Alumina-rich side, mid-plane and Steel-rich side points, respectively.

Figure 3 shows the stress history at 9 locations for the FG bar at (a) z = 0.025 m and (b) z = 0 mobtained using the parallel FE code. Despite the gradation, the stress wave remains the same for thethree y locations at x = L, as it is the prescribed initial condition. At other locations (x = 0.5Land x = 0), we see that the stress wave gets distorted in time and the magnitude di!ers considerablyalong the y direction due to di!erences in material properties and wave speeds. The alumina sideat di!erent x locations undergoes higher stresses when compared with the steel side, more so at thefixed end than at other x locations. It is interesting to see that the stresses at the fixed end x = 0 isconsiderably higher at z = 0m than at z = 0.025m. This is because of the 3D discretization and thefixed boundary condition at that location of the bar. This outcome is a novelty of this study.

Example 2: An FG 3D Beam Subjected to Velocity Impact: A three point bending beam(TPBB) under velocity impact is studied to understand the influence of material gradation on thebeam bending behavior. The beam, which is a real FGM system, is made of glass/epoxy phases.The dynamic fracture experiments of the linearly graded specimens has been conducted by Rousseauand Tippur (2001). This study o!ers knowledge of the dynamic behavior of this material system andunderstanding of the stress field which helps to predict fracture initiation times in various gradedspecimens.

Consider a TPBB under velocity impact load of 10 m/s applied at the top as shown in Figure4(a). Due to the symmetry of the geometry and the loading conditions, only one-fourth of the beam ismodeled for numerical analysis (Figure 4(b)). The 3D FE mesh (Figure 4(c)) has an uniform elementsize of 92.5 µm along the loaded edge. Point P (0, 0.2W ) is of significance because it corresponds to thelocation of the crack tip in the dynamic fracture analysis of the beam (Rousseau and Tippur, 2001)and is also the location at which stress results are obtained in this study.

Three material gradation cases are considered for the dynamic analysis of the TPBB:

3

W = 37 mm

xz 0.2W

L = 76 mm

P

2L = 152 mm

W = 37 mm

x

yz

y

2

E1

V = 10 m/s

E

2B = 6 mm

V = 10 m/s

2E

1

W = 37 mm

0.2W

L = 76 mm

P

V = 10 m/s

E

E1

2E

B = 3 mm

B = 3 mm

(a) (b) (c) (d)

Figure 4: Example 2: Epoxy/glass beam subjected to velocity impact; (a) geometry and boundaryconditions; (b) line load case; (b) point load case; (d) 3D FE mesh of one-quarter model with 14085 15-node wedge elements and 44827 nodes.

• Homogeneous beam (Homog, E2 = E1)• Beam sti!er at the impacted surface (StiffTop, E2 > E1)• Beam softer at the impacted surface (StiffBot, E2 < E1)

where subscripts 1 and 2 denote bottom and top surfaces of the beam, respectively. The materialproperties for the beam are obtained from Zhang and Paulino (2007). Two types of symmetric loadingsare considered in this study. The first (Figure 4(b)) being the load being applied throughout thethickness of the beam (line load) and the second (point load) where the load is applied only at acentral node (Figure 4(c)). The point load case can only be analyzed using 3D finite elements and isa novelty of this study.

0 1 2 3 4 5 6!5

0

5

10

15

20

25

30

t*Cd(avg)

/W

! x (M

Pa)

line load

point load

0 1 2 3 4 5 6!6

!5

!4

!3

!2

!1

0

1

t*Cd(avg)

/W

! y (M

Pa)

point load

line load

Figure 5: Example 2: Stress histories #x and #y at location P(0, 0.2W ) for homogeneous and gradedbeams subjected to impact velocity of 10 m/s as a line load (thick lines) and point load (thin lines).Solid, dashed, and dash-dot lines correspond to StiffTop, Homog, and StiffBot beams, respectively.

Figure 5 shows the comparison of #x and #y at point P for the three beams under impact velocityof V = 10 m/s applied as line load and as point load. We see that the stresses for the latter caseis much lower compared to the former. Maximum tensile stress (#x) is experienced by the StiffTopbeam followed by the Homog beam and the StiffBot beam, for both loading scenarios. This mayindicate earlier crack initiation time for StiffTop beam when compared to other two beams whichwas proven by experiments performed by Rousseau and Tippur (2001). Figure 6 shows the stress#x contour at t# = tCd(avg)/W $ 6 for the three graded beams and two load cases. Since the stressmagnitudes for the two load cases are di!erent, we use di!erent ranges for the contour plot. There isquite a di!erent behavior of stress waves in the top quarter region when comparing the six cases.

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(a) (b) (c) (d) (e) (f)

Figure 6: Example 2: Stress contour (#x, MPa) along the z direction at t# $ 6. StiffBot: (a) line load;(d) point load; Homog: (b) line load; (e) point load; StiffTop: (c) line load; (f) point load.

4 Concluding Remarks

Dynamic behavior of 3D FG solids is investigated using explicit parallel FE formulation. Thickness(3D) and gradation e!ects are seen in the stress behavior of the FG bar. Material gradation consider-ably a!ects the dynamic stress behavior of the beam. Tensile stress is maximum for StiffTop beamat the imaginary crack-tip location indicating that crack initiation will occur earlier for this beamwhich was verified by Rousseau and Tippur (2001).

References

[1] J.-H. Kim and G. H. Paulino. Isoparametric graded finite elements for nonhomogeneous isotropicand orthotropic materials. Journal of Applied Mechanics, Transactions ASME, 69(4):502–514,2002.

[2] MPI-2:. Extension to the Message Passing Interface. University of Tennessee, Knoxville,Tennessee, 1997.

[3] N. M. Newmark. A method of computation for structural dynamics. Journal of EngineeringMechanics (ASCE), 85:67–94, 1959.

[4] M. Ozturk and F. Erdogan. Mode I crack problem in an inhomogeneous orthotropic medium.International Journal of Engineering Science, 35(9):869–883, 1997.

[5] C.-E. Rousseau and H. V. Tippur. Dynamic fracture of compositionally graded materials withcracks along the elastic gradient: experiments and analysis. Mechanics of Materials, 33:403–421,2001a.

[6] M. H. Santare and J. Lambros. Use of graded finite elements to model the behavior of non-homogeneous materials. Journal of Applied Mechanics, Transactions ASME, 67(4):819–822,2000.

[7] S. Sutradhar, G. H. Paulino, and L. J. Gray. Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform galerkin boundary element method. EngineeringAnalysis with Boundary Elements, 26(2):119–132, 2002.

[8] Z. Zhang and G. H. Paulino. Wave propagation and dynamic analysis of smoothly graded hetero-geneous continua using graded finite elements. International Journal of Solids and Structures,44:3601–3626, 2007.

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