Three-Dimensional Eplicit Parallel Finite Element Analsis of Functionall Graded Solids under Impact Loading Ganesh Anandaumar and Jeong-Ho Kim Department of Civil and Environmental Engineering, Universit of Connecticut, 6 Glenbroo Rd, U-7, Storrs, CT 669, U.S.A. Januar, 9 Abstract This paper presents two numerical eamples to investigate the behavior of three-dimensional (D) functionall graded (FG) solids under dnamic loading using eplicit parallel finite element method. In the first eample, wave propagation in a D FG bar under transient sinusoidal loading is investigated. Material gradation and thicness (D) effects are seen in the dnamic stress behavior of the FG bar. In the second eample, a three-point bending beam made of epo and glass phases under velocit impact is studied. Bending stress histor for beam with higher values of material properties at the loading edge is consistentl higher than that of the homogeneous beam and the beam with lower values of material properties at the loading edge. Larger bending stresses for the foremost beam ma indicate earlier crac initiation time than the other two beams which was proven b eperiments performed b other researchers. Kewords: functionall graded material (FGM), dnamic analsis, three-dimensional wave propagation, finite element methods (FEM), parallel computing. Introduction Functionall graded material (FGM) is a material solution & concept used for a new advanced class of composites. is characteried b a gradual variation in composition, microstructure and material properties. eperiences severe dnamic thermal and mechanical loadings. tpicall fails through cracing and spallation. therefore requires a detailed dnamic analsis to determine the effect of material gradation and obtain locations of pea values of stresses. Several numerical methods have been used to investigate FGMs, including integral equations (Otur and Erdogan, 997), boundar element methods (Sutradhar et al., ), finite element methods (Santare and Lambros, ), etc. In this stud, we used the displacement-based FEM and graded finite elements (Kim and Paulino, ) to model FGMs using the direct Gaussian integration formulation. Parallel eplicit FEM is used for obtaining the dnamic response of D FG solids using Message Passing Interface (MPI) standard (MPI, 997). The following are the novelties of this stud: Wave propagation analsis of a FG D bar using FEM Dnamic analsis of a FG D beam under velocit impact load using FEM Parallel Eplicit Dnamic FEA using Newmar-β method The steps involved in the eplicit Newmar-β (γ=.5 and β=) method (Newmar, 959) are given below. Corresponding Author, e-mail: anandg@engr.uconn.edu
calculate velocities at time t n : u n = (ü un n +.5 t n + ü n ) calculate displacements at time t n+ : u n+ = u n + t n u n +.5 t nü n compute effective force vector (EFV) at time t n+ : solve for accelerations at time t n+ : ü n+ = M fn+ n+ f = fet n+ fint n+ where u, u, and ü represent displacement, velocit, and acceleration vectors, respectivel, M is the lumped mass matri, fet is the eternal force vector, n is the time step number, fint(=ku) is the internal force vector, and denotes degree of freedom (DOF). Figure shows a flowchart of the steps involved in the parallel eecution of the dnamic FE code using a master-slave approach. Tpicall, the echange of EFV needs to be done onl at processors that share the node through which partition is done. But this leads to man small messages being sent from one processor to another and ma lead to increase in communication time which is undesirable. To overcome this, the EFV at the slaves are sent to the master for assembl and the master returns the assembled EFV bac to the slaves for calculating acceleration vector. Initialie number of processors using MPI_INIT and MPI_COMM_RANK Slaves send EFV to master for assembl using MPI_SEND and MPI_RECV Partition FE mesh in to sub meshes manuall Calculate K & M matrices for local elements Master returns assembled force vector to slaves. Calculate acceleration vector. Loop over number of timesteps for time integration using Newmar beta method Repeat time integration for the rest of the simulation b this echange algorithm Calculate displacement, velocit, and effective force vector (EFV) at slaves & master Eit parallel code using MPI_FINALIZE Numerical Eamples Figure : Flow chart of the parallel eecution of the eplicit FE code. Eample : Wave Propagation in a D FG Bar: Wave propagation in a fied-free bar with graded materials in the direction is simulated to obtain the effect of material gradation. Consider a fied-free square bar (Figure (a)) of length L =m and height H =.5 m under a transient sinusoidal load (Figure (b)) applied at the free end of the bar. W=.5 m H=.5 m L =. m (a) P f(t) f(t).5 5 5 75 t(µs) (b) Figure : Eample : Wave propagation in FG D bar (a) schematic; (b) sinusoidal load. The material properties var linearl from alumina in the bottom to steel at the top. The material properties of steel and alumina are shown in Table. Due to fine mesh criterion, the D FE mesh is discretied into 5 5 quads, each quad divided into four 5-node wedge elements, totaling 7, elements and 75,86 nodes.
Table : Eample : Material properties of steel and alumina considered Material E(GP a) ν ρ(g/m ) C d (m/s). 78 69 9. 95 67 = =.5L =L =.5L =L Midplane σ /σ L σ /σ L.5.5.5.5 Midplane H = L =.5L = L 5 6 t*(c d ) /L 5 6 t*(c d ) /L (a) (b) Figure : Eample : Normalied longitudinal stress histor of 9 points (see the insert) on a graded bar subjected to transient sinusoidal loading at (a) =.5 m and (b) =m. Solid, dashed, and dash-dot lines indicate points at =, =.5L, and = L, respectivel. Thic, intermediate-thic and thin lines indicate -rich side, mid-plane and -rich side points, respectivel. Figure shows the stress histor at 9 locations for the FG bar at (a) =.5 m and (b) =m obtained using the parallel FE code. Despite the gradation, the stress wave remains the same for the three locations at = L, as it is the prescribed initial condition. At other locations ( =.5L and = ), we see that the stress wave gets distorted in time and the magnitude differs considerabl along the direction due to differences in material properties and wave speeds. The alumina side at different locations undergoes higher stresses when compared with the steel side, more so at the fied end than at other locations. It is interesting to see that the stresses at the fied end = is considerabl higher at =m than at =.5m. This is because of the D discretiation and the fied boundar condition at that location of the bar. This outcome is a novelt of this stud. Eample : An FG D Beam Subjected to Velocit Impact: A three point bending beam (TPBB) under velocit impact is studied to understand the influence of material gradation on the beam bending behavior. The beam, which is a real FGM sstem, is made of glass/epo phases. The dnamic fracture eperiments of the linearl graded specimens has been conducted b Rousseau and Tippur (). This stud offers nowledge of the dnamic behavior of this material sstem and understanding of the stress field which helps to predict fracture initiation times in various graded specimens. Consider a TPBB under velocit impact load of m/s applied at the top as shown in Figure (a). Due to the smmetr of the geometr and the loading conditions, onl one-fourth of the beam is modeled for numerical analsis (Figure (b)). The D FE mesh (Figure (c)) has an uniform element sie of 9.5 µm along the loaded edge. Point P (,.W ) is of significance because it corresponds to the location of the crac tip in the dnamic fracture analsis of the beam (Rousseau and Tippur, ) and is also the location at which stress results are obtained in this stud. Three material gradation cases are considered for the dnamic analsis of the TPBB:
V = m/s V = m/s V = m/s E E E W = 7 mm W = 7 mm W = 7 mm E L = 5 mm B = 6 mm P.W L = 76 mm E B = mm P.W L = 76 mm (a) (b) (c) (d) E B = mm Figure : Eample : Epo/glass beam subjected to velocit impact; (a) geometr and boundar conditions; (b) line load case; (b) point load case; (d) D FE mesh of one-quarter model with 85 5- node wedge elements and 87 nodes. Homogeneous beam (Homog, E = E ) Beam stiffer at the impacted surface (StiffT op, E >E ) Beam softer at the impacted surface (StiffBot, E <E ) where subscripts and denote bottom and top surfaces of the beam, respectivel. The material properties for the beam are obtained from Zhang and Paulino (7). Two tpes of smmetric loadings are considered in this stud. The first (Figure (b)) being the load being applied throughout the thicness of the beam (line load) and the second (point load) where the load is applied onl at a central node (Figure (c)). The point load case can onl be analed using D finite elements and is a novelt of this stud. 5 point load σ (MPa) 5 σ (MPa) line load 5 line load point load 5 5 5 6 t*c /W d(avg) 6 5 6 t*c /W d(avg) Figure 5: Eample : Stress histories σ and σ at location P(,.W ) for homogeneous and graded beams subjected to impact velocit of m/s as a line load (thic lines) and point load (thin lines). Solid, dashed, and dash-dot lines correspond to Stif f T op, Homog, and Stif f Bot beams, respectivel. Figure 5 shows the comparison of σ and σ at point P for the three beams under impact velocit of V = m/s applied as line load and as point load. We see that the stresses for the latter case is much lower compared to the former. Maimum tensile stress (σ ) is eperienced b the StiffT op beam followed b the Homog beam and the StiffBot beam, for both loading scenarios. This ma indicate earlier crac initiation time for StiffT op beam when compared to other two beams which was proven b eperiments performed b Rousseau and Tippur (). Figure 6 shows the stress σ contour at t = tc d(avg) /W 6 for the three graded beams and two load cases. Since the stress magnitudes for the two load cases are different, we use different ranges for the contour plot. There is quite a different behavior of stress waves in the top quarter region when comparing the si cases.
(a) (b) (c) (d) (e) (f) Figure 6: Eample : Stress contour (σ, MPa) along the direction at t 6. StiffBot: (a) line load; (d) point load; Homog: (b) line load; (e) point load; StiffT op: (c) line load; (f) point load. Concluding Remars Dnamic behavior of D FG solids is investigated using eplicit parallel FE formulation. Thicness (D) and gradation effects are seen in the stress behavior of the FG bar. Material gradation considerabl affects the dnamic stress behavior of the beam. Tensile stress is maimum for StiffT op beam at the imaginar crac-tip location indicating that crac initiation will occur earlier for this beam which was verified b Rousseau and Tippur (). References [] J.-H. Kim and G. H. Paulino. Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials. Journal of Applied Mechanics, Transactions ASME, 69():5 5,. [] MPI-:. Etension to the Message Passing Interface. Universit of Tennessee, Knoville, Tennessee, 997. [] N. M. Newmar. A method of computation for structural dnamics. Journal of Engineering Mechanics (ASCE), 85:67 9, 959. [] M. Otur and F. Erdogan. Mode I crac problem in an inhomogeneous orthotropic medium. International Journal of Engineering Science, 5(9):869 88, 997. [5] C.-E. Rousseau and H. V. Tippur. Dnamic fracture of compositionall graded materials with cracs along the elastic gradient: eperiments and analsis. Mechanics of Materials, :, a. [6] M. H. Santare and J. Lambros. Use of graded finite elements to model the behavior of nonhomogeneous materials. Journal of Applied Mechanics, Transactions ASME, 67():89 8,. [7] S. Sutradhar, G. H. Paulino, and L. J. Gra. Transient heat conduction in homogeneous and nonhomogeneous materials b the Laplace transform galerin boundar element method. Engineering Analsis with Boundar Elements, 6():9,. [8] Z. Zhang and G. H. Paulino. Wave propagation and dnamic analsis of smoothl graded heterogeneous continua using graded finite elements. International Journal of Solids and Structures, :6 66, 7. 5