Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 1, pp. 207-217, Warsaw 2010 NON-LINEAR ANALYSIS OF DYNAMIC STABILITY OF METAL FOAM CIRCULAR PLATE Ewa Magnucka-Blandzi Poznan University of Technology, Institute of Mathematics, Poznań, Poland e-mail: ewa.magnucka-blandzi@put.poznan.pl The subject of the paper is a circular plate under radial compression. The plate is made of metal foam. Properties of the plate vary across its thickness. Themiddle plane of the plate is its symmetry plane. The field of displacement of any cross section of the plate, nonlinear components of the strainfield and the stress field aredefined.Basing on theHamilton principle, a system of differential equations of dynamic stability of the plate is formulated. This basic system of equations is approximately solved. The results of the studies are compared to the homogeneous circular plate and shown in figures. Key words: metal foam plate, critical load, dynamic 1. Introduction Contemporary structures are manufactured, among other things, of metal fo- am. Banhart (2001) presented themanufacture, characterisation and applica- tion of cellular metals and metal foams for structures. Carrera (2000, 2001, 2003) reviewed the broken-line hypothesis and generalized it to multi-layered constructions. Carrera et al. (2008) described the static analysis of functional- ly graded material plates subjected to transverse mechanical loadings. Wang et al. (2000) discussed in details the shear deformable effect of beams and plates and described some special theories of them.Magnucka-Blandzi (2006, 2008, 2009) carried out analytical investigations of strength, stability and free vibrations of porous-cellular plates with consideration of a non-linear hypo- thesis of deformation of flat cross section of the structures. Volmir (1972) collected and presented problems of non-linear vibrations and dynamic sta- bility of thin plates and shells. Moreover, the dynamic criterion of stability was formulated. Kowal-Michalska (2006) described dynamic stability of plates 208 E. Magnucka-Blandzi including criterions of: Volmir, Budiansky-Hutchinson, Ari-Gur and Weller, Petry and Fahlbush. InMagnucka-Blandzi (2009), linear analysis of theplatewas studied. Inste- ad, thispaper is concernedwithnon-linear analysis of theporouscircular plate. The paper is an improvement and continuation of the papers by Magnucka- Blandzi (2006, 2008, 2009). The plate with radius R and thickness h carries a radial compressive force N(t). 2. Physical model of the plate An isotropicporouscircularplatewith the clampededgeunder radial compres- sion is studied. The plate with a simply supported edge can also be analysed in a similar way. The plate is made of the metal foam. The plate is porous inside and the material is of continuous mechanical properties varying in the normal direction (Fig.1).Adegree of porosity andYoung’smodulusvary thro- Fig. 1. Schemes of the plate and deformation of its plane cross section ugh the thickness of the plate. The minimal values are in the middle surface of the plate. The maximal values occur at its top and bottom surfaces. For such a case, the Kirchhoff and Mindlin plate theories do not correctly deter- mine displacements of the plate cross-section. Wang et al. (2000) discussed in details the effect of non-dilatational strain of middle layers on bending of plates subject to various load cases. Magnucka-Blandzi andMagnucki (2007), Non-linear analysis of dynamic stability... 209 Magnucka-Blandzi (2006, 2008, 2009) thoroughly described the non-linear hy- pothesis of deformation of the plate cross section. Themoduli of elasticity and mass density are defined as follows E(z)= E1[1−e0cos(πζ)] G(z) =G1[1−e0cos(πζ)] (2.1) ̺(z)= ̺1[1−emcos(πζ)] where e0 – porosity coefficient of elasticity moduli, e0 =1−E0/E1 em – dimensionless parameter of mass density, em =1−̺0/̺1 E0,E1 – Young’s modulus at z =0 and z =±h/2, respectively G0,G1 – shear modulus for z =0 and z =±h/2, respectively Gj – relationship between moduli of elasticity for j = 0,1,, Gj = Ej/[2(1+ν)] ν – Poisson’s ratio (constant for the entire plate) ̺0,̺1 – mass densities for z =0 and z =±h/2, respectively ζ – dimensionless coordinate, ζ = z/h h – thickness of the plate. Choi and Lakes (1995) presented mechanical properties for porous materials. Basing on their results, the following relationship is defined: em = 1− √ 1−e0. Magnucka-Blandzi and Magnucki (2007), Magnucki et al. (2006), Magnucki and Stasiewicz (2004a,b) proposed a non-linear hypothesis of the cross-section deformation of the structurewall. Deformation of any pla- ne cross section is shown inFig.1.Applying this hypothesis, the displacements are assumed in the same form as inMagnucka-Blandzi (2009) u(r,z,t) = (2.2) +−h { ζ ∂w ∂r − 1 π [ψ1(r,t)sin(πζ)+ψ2(r,t)sin(2πζ)cos 2(πζ)] } where ψ1(r,t), ψ2(r,t) are dimensionless functions of displacements. If ψ1(r,t) = ψ2(r,t) = 0, the field of displacement u is the linear Kirchhoff- Love hypothesis. 210 E. Magnucka-Blandzi The nonlinear geometric relationships, i.e. components of the strain are εr = ∂u ∂r + 1 2 (∂w ∂r )2 = =−h { ζ ∂2w ∂r2 − 1 π [∂ψ1 ∂r sin(πζ)+ ∂ψ2 ∂r sin(2πζ)cos2(πζ) ]} + 1 2 (∂w ∂r )2 εϕ = u r = (2.3) =−h {1 r ζ ∂w ∂r − 1 π [1 r ψ1(r,t)sin(πζ)+ 1 r ψ2(r,t)sin(2πζ)cos 2(πζ) ]} γrz = ∂u ∂z + ∂w ∂r = ψ1(r,t)cos(πζ)+ψ2(r,t)[cos(2πζ)+cos(4πζ)] where εr is the normal strain along the r-axis, εϕ is the circular strain, and γrz – the shear strain. Basing on Hooke’s law, the stresses were defined. 3. Mathematical model of the plate 3.1. Potential energy and work of the load Equations of dynamic stability are formulated basing on Hamilton’s prin- ciple δ t2∫ t1 (T −Uε+W) dt =0 (3.1) where T denotes kinetic energy, which is approximately formulated. It only includes deflections w(r,t) without tangent displacements u(r,t). T = πh R∫ 0 1 2∫ − 1 2 r̺(ζ) (∂w ∂t )2 dζdr Uε is the energy of elastic strain Uε = πh R∫ 0 1/2∫ −1/2 r(σrεr+σϕεϕ+ τrzγrz) dζdr W is the work which follows from the compressive force W = πN(t) R∫ 0 r (∂w ∂r )2 dr Non-linear analysis of dynamic stability... 211 R is the radius of the plate, ̺ – mass density of the plate, t1, t2 – initial and final times, N(t) – intensity of the compressive force. In the numerical calculations, the intensity of compressive force is assumed as follows N(t)= N0 sin 2 (1 2 θt ) or N(t)= N0 t t0 where θ = π/t0, t0 – the initial time. These forces have unchanging direction, whereas the first one is an impulsive compressive force and the second one steadly increases. 3.2. Equations of stability Taking into account principle (3.1), the system of three stability equations ofmotion for theporousplate under compression is formulated in the following form (δw) ∂ ∂r { r ∂ ∂r {1 r ∂ ∂r [ r ( c0 ∂w ∂r − c1ψ1− c2ψ2 )]}} − 1 h2 c9 ∂ ∂r [ r (∂w ∂r )3] + +4 1−ν2 E1h3 [ πN(t) ∂ ∂r ( r ∂w ∂r ) + c10̺1rh ∂2w ∂t2 ] =0 (δψ1) ∂ ∂r {1 r ∂ ∂r [ r ( c1 ∂w ∂r − c3ψ1− c4ψ2 )]} + 1−ν h2 (c5ψ1+ c6ψ2)= 0 (3.2) (δψ2) ∂ ∂r {1 r ∂ ∂r [ r ( c2 ∂w ∂r − c4ψ1− c7ψ2 )]} + 1−ν h2 (c6ψ1+ c8ψ2)= 0 where c0 = π3−6e0(π2−8) 3π2 c1 = 8−πe0 π2 c2 = 225π −512e0 300π2 c3 =2 3π−4e0 3π2 c4 = 64−15πe0 30π2 c5 = 3π−8e0 3 c6 = 32−15πe0 30 c7 = 1575π −4096e0 2520π2 c8 =2 315π −832e0 315 c9 =2(π−2e0) c10 = π−2em 3 212 E. Magnucka-Blandzi The boundary conditions for the plate with the clamped edge are w(R,t)= 0 ψ1(0, t)= ψ2(0, t) = 0 ∂w ∂r ∣∣∣ r=R =0 ∂w ∂r ∣∣∣ r=0 =0 (3.3) where Mr = ∫h/2 −h/2 zσr dz is the radial bendingmoment. The system of diffe- rential equations (3.2) includes three unknown functions, which are assumed in forms ψ1(r,t) =−6ψa1 [( r R ) − ( r R )2] ψ2(r,t) =−6ψa2 [( r R ) − ( r R )2] (3.4) w(r,t) = wa(t) [ 1−3 ( r R )2 +2 ( r R )3] These functions satisfy boundary conditions (3.3). Substituting them into sys- tem(3.2) andusingGalerkin’smethod, oneobtainsa systemof three equations in the form [ c0− 4π(1−ν2)R2 15E1h3 N(t) ] wa(t)− c1Rψa1 − c2Rψa2 + 4 35 c9 1 h2 w3a(t)+ +c10̺1 12(1−ν2)R4 105E1h2 d2wa dt2 =0 c1wa− c13Rψa1 − c12Rψa2 =0 (3.5) c2wa− c12Rψa1 − c11Rψa2 =0 where c11 = c7+ c8c14 c12 = c4+ c6c14 c13 = c3+ c5c14 c14 = (1−ν)R2 15h2 From the second and third equations of system (3.5), ψa1, ψa2 functionsmay be calculated, namely ψa1 = ψ̃a1 wa R ψa2 = ψ̃a2 wa R (3.6) where ψ̃a1 = c1c11− c2c12 c13c11− c212 ψ̃a2 = c2c13− c1c12 c13c11− c212 Non-linear analysis of dynamic stability... 213 Substitution of functions (3.6) into the first equation of system (3.5) yields the second order nonlinear differential equation of motion in the following form d2wa dt2 + c9 c10(1−ν2) E1 ̺1R4 w3a(t)+ 7πNcr 3c10̺1hR2 ( 1− Nt Ncr ) wa(t)= 0 (3.7) where Ncr = 15E1h 3 4π(1−ν2)R2 (c0− c1ψ̃a1− c2ψ̃a2) is the intensity of the critical force (Ncr [N/mm]). Galerkin’s method allo- wed one to reduce this problem of a continuous structure, circular plate, to a discrete problemwith a single degree of freedom. In a particular case, the static equilibriumpath follows fromequation (3.7) in the form N(t)= 15 4π(1−ν2) [ c0−c1ψ̃a1−c2ψ̃a2+ 4 35 c9 (wa h )2]E1h3 R2 4. Numerical calculations Some examples will be given below for a family of plates with height h = 10mm, radius R = 1500mm, Young’s modulus E1 = 7100MPa and mass density ̺1 =2.7 ·10−7kg/mm3. The influence of porosity coefficient of elasticity moduli and the influence of compressive force on the amplitude of displacement is studied. Two kinds of radial compressive forces are assumed. Their plots are shown in Fig.2. These two loads are in forms N(1)(t)= N0 t t0 N(2)(t)= N0 sin 2 (1 2 θt ) where θ = π/t0. In thefirstexample, the intensity of compressive force is linear andassumed in form N(t)= N0t/t0, where N0 = Ncr, t0 =3 (the initial time). Static and dynamic equilibrium paths are presented in Fig.3 for the homogeneous plate (e0 =0) and for the non-homogeneous plate (e0 =0.8). In the second example, only the homogeneous plate is considered. The intensity of compressive force is assumed as a pulsating compressive force in form N(t) = N0 sin 2(θt/2), where θ = π/t0, t0 = 3 and N0 = kNcr (k = 1.2,1.5,1.8). Static and dynamic equilibrium paths are presented in 214 E. Magnucka-Blandzi Fig. 2. Radial intensity of compressive forces Fig. 3. Amplitudes of deflections for homogeneous and non-homogeneous plates Fig. 4. Amplitudes of deflections for the homogeneous plate Fig.4. The influence of the pulsating compressive force on the amplitude of displacement is shown. In the last example, homogeneous and non-homogeneous plates are com- pared. In Fig.5, the plots of equilibrium paths are shown. The pulsating com- pressive force is the same as previously, but N0 =1.8Ncr. It could be noticed that in the post-buckling state vibrations of the plates around the static equilibrium paths for homogeneous and non-homogeneous Non-linear analysis of dynamic stability... 215 Fig. 5. Amplitudes of deflections for homogeneous and non-homogeneous plates plates appear as well. The above results of numerical analysis issue from the simplified circular platemodel. Despite of this simplification, the behaviour of the plate under dynamic loads could be useful in practice. 5. Conclusions • The metal foam circular plate is a generalization of sandwich or multi- layer plates. • Correct hypotheses of plane cross sections for homogeneous plates are useless in the case of a porous-cellular plate as elastic constants vary considerably along its depth. • The non-linear hypothesis of deformation of the flat cross section of the plate is optional tohypothesispresentedbyWang et al. (2000) orCarrera (2001, 2003) and Carrera et al. (2008) and it includes: – linear hypothesis for homogeneous plates – shear deformable effect. • Themathematical model of dynamic stability of themetal foam circular plate could be reduced to a single differential equation of motion. • The dynamic equilibrium path is the solution to differential equation of motion (3.7). • In a particular case, the static equilibriumpath follows from equation of motion (3.7). 216 E. Magnucka-Blandzi References 1. Banhart J., 2001, Manufacture, characterisation and application of cellular metals andmetal foams,Progress in Materials Science, 46, 559-632 2. Carrera E., 2000, An assessment of mixed and classical theories on global and local response ofmultilayeredorthotropicplates,Composite Structures,50, 183-198 3. Carrera E., 2001, Developments, ideas, and evaluations based upon Reis- sner’s mixed variational theorem in the modelling of multilayered plates and shells,Applied Mechanics Reviews, 54, 301-329 4. 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WangC.M., Reddy J.N., LeeK.H., 2000,Shear deformable beams and pla- tes, Elsevier, Amsterdam, Lousanne, New York, Oxford, Shannon, Singapore, Tokyo Analiza nieliniowa stateczności dynamicznej płyty kołowej wykonanej z piany metalowej Streszczenie Przedmiotem pracy jest płyta kołowa obciążona promieniowo. Płyta wykonana jest z piany metalowej. Właściwości mechaniczne płyty są zmienne na jej grubości. Płaszczyzna środkowa płyty jest jej płaszczyzną symetrii. Zdefiniowano pole prze- mieszczeń dla dowolnego przekroju poprzecznego płyty, nieliniowe odkształcenia oraz naprężenia. Układ równań różniczkowych stateczności dynamicznej płyty zdefinio- wano na podstawie zasady Hamiltona. Układ ten rozwiązano w sposób przybliżony. Wyniki badań numerycznych tej płyty porównano z odpowiednimi wielkościami dla płyty jednorodnej i przedstawiono na rysunkach. Manuscript received March 13, 2009; accepted for print August 12, 2009