Mechanical and Thermal Buckling of Functionally Graded Plates Roman Arciniega and J. N. Reddy Department of Mechanical Engineering Texas A&M University College Station, TX 77843-3123, USA US-South American Workshop: Mechanics and Advanced Materials − Research and Education August 2-6, 2004
CONTENTS OF THE LECTURE • Background • Theoretical Formulation • FE Model • Numerical Results • Closing comments
BACKGROUND BACKGROUND � Functionally graded materials are inhomogeneous materials in which the material properties are varied continuously from point to point to eliminate interface problems and thus the stress distributions are smooth and uniform. � For example, a plate structure used as a thermal barrier may be graded through the plate thickness from ceramic on the face of the plate that is exposed to high temperature to metal on the other face.
FGM MATERIALS FGM MATERIALS P ( z ) = ( P c − P m ) f ( z ) + P m P c (ceramic) f ( z ) = [(2 z + h )/2 h )] n z ceramic ceramic x metal metal metal P c (ceramic) P m (metal) z y h x
Through- -thickness thickness- -Graded Graded Through Material Plates Material Plates � This is achieved by varying the volume fraction of the constituents i.e., ceramic and metal in a predetermined manner. The ceramic constituent of the material provides the high temperature resistance due to its low thermal conductivity. The ductile metal constituent, on the other hand, prevents fracture caused by stresses due to high temperature gradient in a very short period of time.
− Ceramic FGM A Typical Metal − Ceramic FGM A Typical Metal High temperature Ceramic Heat resistant; side good anti-oxidant property; low thermal conductivity Low temperature Metal Mechanical strength; side high thermal conductivity; high fracture toughness In between Ceramic Effective thermal stress & metal relaxation throughout z P c (ceramic) x P m (metal)
Aluminum and Zirconia Zirconia Aluminum and Material Properties Material Properties Aluminum E 1 = 70 GPa, ν = 0.3, ρ = 2707 kg/m 3 , k = 204 W/(m.K), α = 23 × 10 -6 / o C Zirconia E 1 = 151 GPa, ν = 0.3, ρ = 3000 kg/m 3 , k = 2.09 W/(m.K), α = 23 × 10 -5 / o C
The temperature distribution over the thickness is determined by solving the energy equation ⎛ ⎞ d dT ⎜ ⎟ − = k ( z ) 0 ⎝ ⎠ dz dz Non-dimensional parameters w Center deflection: w = h 4 q a Load parameter: = o P 4 E h m
Temperature variation through the Temperature variation through the aluminum- -zirconia zirconia FGM plate thickness FGM plate thickness aluminum
SUMMARY � The mechanical and thermal buckling of functionally graded ceramic-metal plates is investigated. � The third-order shear deformation theory for plates is used. � A displacement finite element model of the third-order theory is developed using c 0 -continuity with a family of high-order Lagrange interpolation functions to avoid shear locking. � The stability equations are derived using the Trefftz criterion. � Numerical results are compared with those of the FSDT.
RELATED RESEARCH � Most of research studies in FGMs had more focused on thermal stress analysis and fracture mechanics (Paulino and his colleagues). � Limited work has been done to study the buckling and vibration response of FGM structures. � Javaheri and Eslami derived the stability equations of FGM plates under thermal loads, based on the CLPT. � Shen carried out a postbuckling analysis for FGM panels and plates subjected to axial compression in thermal environments. � Na and Kim presented a 3D finite element solution for thermal buckling of functionally graded plates.
THEORETICAL FORMULATION KINEMATICS z,u 3 y,u 2 z,w Ceramic h/ 2 b x,u 1 x,u h/ 2 h Metal a ϕ ϕ ( , , ) ( , ) u x y z u z k w = + + + Displacement 1 1 x 1 Field of TSDT ϕ ϕ u ( x , y , z ) v z k ( w , ) = + + + 2 2 2 y u ( x , y , z ) w = 3
KINEMATICS (continued) To relax the continuity in the finite element formulation, we introduce the following auxiliary variables ϕ ψ u ( x , y , z ) u z k = + + ψ = + ϕ 1 1 1 w , 1 1 x ϕ ψ u ( x , y , z ) v z k = + + 2 2 2 ψ = + ϕ w , 2 y 2 u ( x , y , z ) w = 3 The strains are o 1 3 3 ε ε k z k z = ( 1 , 2 , 6 ) i = + + i i i i = 2 2 o ( m 4 , 5 ) ε ε k z = + m m m
KINEMATICS (continued) where 1 1 o 2 o 2 o ε = + ε = u w , v + w , , ε = + + u v w w 1 , x , x 2 , y , y 6 , y , x , x , y 2 2 o o ε = + ϕ ε = + ϕ w , w 4 , y 2 5 , x 1 and 1 1 1 = ϕ = ϕ = ϕ + ϕ k , k , k , 1 1 , 2 2 , 6 1 , 2 , x y y x 3 3 3 = ψ = ψ = ψ + ψ k k , k k , k k ( ), 1 1 , x 2 2 , y 6 1 , y 2 , x 2 2 = ψ = ψ k 3 k , k 3 k 4 2 , y 5 1 , x
EQUILIBRIUM EQUATIONS ∂ ∂ M M 1 6 δϕ : - 0 Q + = 1 1 ∂ x ∂ y ∂ N ∂ N 1 6 δ u : 0 + = ∂ M ∂ M ∂ ∂ x y 6 2 δϕ : - Q 0 + = 2 2 ∂ x ∂ y ∂ N ∂ N 6 2 δ v : 0 + = ∂ x ∂ y ∂ P ∂ P 1 6 δψ : - 3 R 0 + = 1 1 ∂ x ∂ y ∂ Q ∂ Q 1 2 δ w : q N 0 + + + = ∂ x ∂ y ∂ P ∂ P 6 2 δψ : - 3 R 0 + = 2 2 ∂ x ∂ y h 2 3 = σ = ( N , M , P ) ( 1 , z , z ) dz ( i 1 , 2 , 6 ), ∫ i i i i − h 2 h 2 h 2 2 2 = σ = σ ( Q , R ) ( 1 , z ) dz , ( Q , R ) ( 1 , z ) dz ∫ ∫ 1 1 5 2 2 4 − − h 2 h 2
MECHANICAL CHARACTERIZATION OF FUNCTIONALLY GRADED PLATES α α α E ( z ) E f E f ( z ) f f = + = + c c m m c c m m α α f E f E = + = + cm c m cm c m Volume Fractions: n + 1 z = = − f f 1 f c m c h 2 ν ν ( z ) Constant Poisson’s coefficient =
CONSTITUTIVE EQUATIONS: ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ σ ε Q ( z ) Q ( z ) 0 1 11 12 1 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎨ σ ⎬ = ⎨ ε ⎬ Q ( z ) Q ( z ) 0 ⎢ ⎥ 2 12 22 2 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ σ ε 0 0 Q ( z ) ⎩ ⎭ ⎣ ⎦ ⎩ ⎭ 6 66 6 ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ σ ε Q ( z ) 0 4 44 4 = ⎨ ⎬ ⎨ ⎬ ⎢ ⎥ σ ε ⎩ ⎭ ⎣ 0 Q 55 ( z ) ⎦ ⎩ ⎭ 5 5 WHERE: ν E ( z ) E ( z ) = = = Q ( z ) Q ( z ) , Q ( z ) , 11 22 12 2 2 − ν − ν 1 1 E ( z ) = = = Q ( z ) Q ( z ) Q ( z ) 66 44 55 + ν 2 ( 1 )
STRESS RESULTANTS: 0 1 3 = ε + + N A B k E k i ij j ij j ij j = ( , 1 , 2 , 6 ) i j 0 1 3 = ε + + M B D k F k i ij j ij j ij j 0 1 3 = ε + + P E F k H k i ij j ij j ij j 0 2 0 2 = ε + = ε + Q A D k , Q A D k 1 5 j j 5 j j 2 4 j j 4 j j = ( i , j 4 , 5 ) 0 2 0 2 = ε + = ε + R D F k , R D F k 1 5 j j 5 j j 2 4 j j 4 j j THE MATERIAL STIFFNESS COEFFICIENTS FOR FGMs: h 2 cm m 2 3 4 6 = + ( A , B , D , E , F , H ) ( Q f Q )( 1 , z , z , z , z , z ) dz ∫ ij ij ij ij ij ij ij c ij − h 2 h 2 cm m 2 4 = + ( A , D , F , ) ( Q f Q )( 1 , z , z ) dz ∫ ij ij ij ij c ij − h 2
cm c m = − Q Q Q RECALL: ij ij ij THE THERMAL STRESSES ARE: T ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ σ α Q ( z ) Q ( z ) 0 ( z ) 1 11 12 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎨ σ ⎬ = − ⎨ α ⎬ Q ( z ) Q ( z ) 0 ( z ) T ( x , y , z ) ⎢ ⎥ 2 12 22 ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ σ ⎩ ⎭ ⎣ 0 0 Q ( z ) ⎦ ⎩ 0 ⎭ 6 66 THERMAL STRESS RESULTANTS: h 2 1 T T T 2 ) = − + α + α ( N , M , P ) ( E f E )( f ) T ( x , y , z ) ( 1 , z , z dz ∫ − cm c m cm c m − ν ( 1 ) h 2
STABILITY ANALYSIS The potential energy increment may be written in the form 4 1 F F I n + − Π δ Π = Π ( U ) = n δ Π ( U U ) ∑ ! = n 1 F U : Fundamental prebuckling solution I : Incremental solution U δ Π 0 Necessary condition =
STABILITY ANALYSIS (continued) The critical load is defined as the smallest load for which the second variation is no longer positive definite. The limit of positive-definiteness for a continuous system can be expressed as 2 δ [ δ Π ] = 0 (THE TREFFTZ CRITERION) The fundamental solution (prebuckling state) is considered as a pure membrane state where bending and rotations are neglected. Then ⎧ 0 1 0 1 1 = + = + = u u u , v v v , w w , ⎪ ⎪ 1 1 → ⎨ ϕ = ϕ ϕ = ϕ U , , 1 1 2 2 ⎪ 1 1 ψ = ψ ψ = ψ ⎪ , ⎩ 1 1 2 2
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