Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 2, pp. 439-455, Warsaw 2011 MATHEMATICAL MODELLING OF A RECTANGULAR SANDWICH PLATE WITH A METAL FOAM CORE Ewa Magnucka-Blandzi Poznań University of Technology, Institute of Mathematics, Poznań, Poland e-mail: ewa.magnucka-blandzi@put.poznan.pl The subject of the paper is a simply supported rectangular sandwich plate. The plate is compressed in plane. It is assumed that the plate under consi- deration is symmetrical in build and consists of two isotropic facings and a core. Themiddle plane of the plate is its symmetry plane. The core is made of a metal foam with properties varying across its thickness. The porous- cellular metal as a core of the three layered plate is of continuous structure, while its mechanical properties are isotropic. Dimensionless coefficients are introduced to compensate for this. The field of displacements and geometric relationships are assumed. This non-linear hypothesis is generalization of the classical hypotheses, in parti- cular, the broken-line hypothesis. The principle of stationarity of the total potential energy of the compressed sandwich plate is used and a system of differential equations is formulated. This system is approximately solved. The forms of unknown functions are assumed,which satisfy boundary condi- tions for supports of the plate. Critical loads for a family of sandwich plates are numerically determined. Results of the calculation are shown in figures. Key words: sandwich plate, critical load, metal foam core 1. Introduction In the last years, composite beams, plates and shells are applied in mecha- nical engineering, particularly in vehicles and building engineering. Strength and buckling problems of sandwich structures are studied in practice since the mid of the 20th century. There are monograph works devoted to this topic, e.g. Plantema (1966), Volmir (1967), Grigolyuk and Chulkov (1973), Noor et al. (1996), Wang et al. (2000), Magnucki and Ostwald (2001). These monograph papers demonstrate the development of research of strength and buckling of classical sandwich beams, plates, and shells with homogeneous 440 E. Magnucka-Blandzi cores. Contemporary studies of the strength and stability problems of classi- cal sandwich structures are presented by Kotełko and Mania (2005) or Ohga et al. (2005). The sandwich structures with metal foam cores are only rare- ly dealt within such a wide field of investigation. Magnucki and Stasiewicz (2004a,b), Malinowski and Magnucki (2005), Magnucki et al. (2006), carried out analytical investigations of strength and stability of porous-cellular be- ams, plates and cylindrical shellswith consideration of a non-linear hypothesis of the deformation of flat cross section of the structures. The first hypothe- sis of displacements and equilibrium equations of three-layered constructions were formulated in the middle of 20th century and it was presented by Gri- golyuk and Chulkov (1973). Wang et al. (2000) discussed the higher order hypotheses including shear deformation of beams and plates. Carrera (2000, 2001, 2003) formulated the zig-zag hypotheses for multilayered plates. Carre- ra et al. (2008) presented the static analysis of functionally graded material plates subjected to transverse mechanical loadings. Debowski and Magnucki (2006) formulated a nonlinear hypothesis of deformation for porous rectangu- lar plates with using trygonometric functions. Kasprzak and Ostwald (2006) presented a generalization of the hypotheses of deformations. Banhart (2001), Bart-Smith et al. (2001), andHohe andBecker (2002) presented themanufac- ture, characterization and application of cellular metals and metal foams for sandwich structures. Magnucka-Blandzi and Magnucki (2007) and Magnucki and Magnucka-Blandzi (2006) described the strength and stability problems of a sandwich beamwith a porous-cellular core and its effective design. Pandit et al. (2008) presented an improved higher order zigzag theory and applied it to study the buckling of laminated sandwich plates. The variation of in-plane displacements through the thickness direction is assumed to be cubic for both the face sheets and the core, while transverse displacement is assumed to va- ry quadratically within the core but it remains constant over the face sheets. Apetre et al. (2008) investigated several available sandwich beam theories for their suitability of application to one-dimensional sandwich plates with func- tionally graded cores. Two equivalent single-layer theories based on assumed displacements, a higher-order theory, and the Fourier-Galerkin method were compared. The variation of coreYoung’smoduluswas presented by a differen- tiable function in the thickness coordinate, but the Poisson’s ratio was kept constant. The subject of the paper is a simply-supported rectangular sandwich plate with ametal foam core. The paper is an improvement and continuation of the papers byMagnucka-Blandzi andMagnucki (2007),Magnucki andMagnucka- -Blandzi (2006), Magnucka-Blandzi (2008, 2009), Magnucka-Blandzi andWa- silewicz (2009) andMagnucka-Blandzi (2010). Mathematical modelling of a rectangular sandwich plate... 441 The plate with sizes a, b and the thickness 2tf + tc carries a uniform compressive forces N0x, N 0 y (Fig.1). Fig. 1. Scheme of the sandwich plate under compression 2. Physical model of the sandwich plate The sandwich plate with a metal foam core is studied. Metal faces of thick- ness tf are isotropic of Young’s modulus Ef and Poisson’s ratio νf . The metal foam core of thickness tc is assumed as isotropic with varyingmechani- cal properties (Fig.2), but Poisson’s ratio νc is kept constant. Fig. 2. Scheme of deformation of a plane cross section of the plate 442 E. Magnucka-Blandzi The minimal value of Young’s modulus occurs in the middle plane of the plate and themaximal value at its topandbottomsurfaces of the core.The co- re is porous insidewith the degree of porosity varying in the normal direction. Themoduli of elasticities are defined as follows Ec(ζ)= Ec1[1−e0cos(πζ)] Gc(ζ)= Gc1[1−e0cos(πζ)] (2.1) where e0 – coefficient of the core porosity, e0 =1−Ec0/Ec1 Ec0,Ec1 – Young’s moduli at z =0 and z =±tc/2, respectively Gc0,Gc1 – shear moduli for z =0 and z =±tc/2, respectively Gcj – relationship between the moduli of elasticity for j = 0,1, Gcj = Ecj/[2(1+ν)] νf,νc – Poisson’s ratios for faces and the core ζ – dimensionless coordinate, ζ = z/tc tf – thickness of each face tc – thickness of the core Displacements of points laying on the cross-section of the plate arise from the assumed hypothesis of deformation (Fig.2). The field of displacement is defined: — the upper face: −(0.5+x1)¬ ζ ¬−0.5 u(x,y,ζ) =−tc [ ζ ∂w ∂x +ψ0(x,y)− 1 π ψ1(x,y) ] (2.2) v(x,y,ζ) =−tc [ ζ ∂w ∂y +φ0(x,y)− 1 π φ1(x,y) ] where ψ1(x,t)= u1(x,t)/tc —the core: −0.5¬ ζ ¬ 0.5 u(x,y,ζ)=−tc { ζ [∂w ∂x −2ψ0(x,y) ] + 1 π ψ1(x,y)sin(πζ) } (2.3) v(x,y,ζ)=−tc { ζ [∂w ∂y −2φ0(x,y) ] + 1 π φ1(x,y)sin(πζ) } —the lower face: 0.5¬ ζ ¬ 0.5+x1 u(x,y,ζ) =−tc [ ζ ∂w ∂x −ψ0(x,y)+ 1 π ψ1(x,y) ] (2.4) v(x,y,ζ) =−tc [ ζ ∂w ∂y −φ0(x,y)+ 1 π φ1(x,y) ] where x1 = tf/tc. Mathematical modelling of a rectangular sandwich plate... 443 There are five unknown autonomous functions: w(x,y) – deflection, ψ0(x,y), ψ1(x,y), phi0(x,y), φ1(x,y) – dimensionless functions of displace- ments. In the particular case ψ0(x,y) = ψ1(x,y) = φ0(x,y) = φ1(x,y) = 0, the field of displacements u, v is linear the Kirchhoff-Love hypothesis. Func- tions ψ0(x,y),ψ1(x,y),φ0(x,y),φ1(x,y) extend the linear classical hypothesis. In the classical theory, thea shear force is equal to zero (it follows from this linear theory), but in the proposed non-linear hypothesis the shear force does not equal zero, which corresponds with the facts. The geometric relationships, i.e. components of the strain for each layer of the plate, are: — the upper face: −(0.5+x1)¬ ζ ¬−0.5 ε(f1)x = ∂u ∂x =−tc ( ζ ∂2w ∂x2 + ∂ψ0 ∂x − 1 π ∂ψ1 ∂x ) ε(f1)y = ∂v ∂y =−tc ( ζ ∂2w ∂y2 + ∂φ0 ∂y − 1 π ∂φ1 ∂y ) γ(f1)xz = 1 tc ∂u ∂ζ + ∂w ∂x =0 (2.5) γ(f1)yz = 1 tc ∂v ∂ζ + ∂w ∂y =0 γ(f1)xy = ∂u ∂y + ∂v ∂x = =−tc [ 2ζ ∂2w ∂x∂y + ∂ψ0 ∂y + ∂φ0 ∂x − 1 π (∂ψ1 ∂y + ∂φ1 ∂x )] —the core: −0.5¬ ζ ¬ 0.5 ε(c)x = ∂u ∂x =−tc [ ζ (∂2w ∂x2 −2 ∂ψ0 ∂x ) + 1 π ∂ψ1 ∂x sin(πζ) ] ε(c)y = ∂v ∂y =−tc [ ζ (∂2w ∂y2 −2 ∂φ0 ∂y ) + 1 π ∂φ1 ∂y sin(πζ) ] γ(c)xz = 1 tc ∂u ∂ζ + ∂w ∂x =2ψ0(x,y)−ψ1(x,y)cos(πζ) (2.6) γ(c)yz = 1 tc ∂v ∂ζ + ∂w ∂y =2φ0(x,y)−φ1(x,y)cos(πζ) γ(c)xy = ∂u ∂y + ∂v ∂x = =−tc [ 2ζ ( ∂2w ∂x∂y − ∂ψ0 ∂y − ∂φ0 ∂x ) + 1 π (∂ψ1 ∂y + ∂φ1 ∂x ) sin(πζ) ] 444 E. Magnucka-Blandzi —the lower face: 0.5¬ ζ ¬ 0.5+x1 ε(f2)x = ∂u ∂x =−tc ( ζ ∂2w ∂x2 − ∂ψ0 ∂x + 1 π ∂ψ1 ∂x ) ε(f2)y = ∂v ∂y =−tc ( ζ ∂2w ∂y2 − ∂φ0 ∂y + 1 π ∂φ1 ∂y ) γ(f2)xz = 1 tc ∂u ∂ζ + ∂w ∂x =0 (2.7) γ(f2)yz = 1 tc ∂v ∂ζ + ∂w ∂y =0 γ(f2)xy = ∂u ∂y + ∂v ∂x = =−tc [ 2ζ ∂2w ∂x∂y − ∂ψ0 ∂y − ∂φ0 ∂x + 1 π (∂ψ1 ∂y + ∂φ1 ∂x )] Stresses inall layers of theplate,with respect toHooke’s law, are as follows: — the upper or the lower face σ(fi)x = Ef 1−ν2 f ( ε(fi)x +νfε (fi) y ) σ(fi)y = Ef 1−ν2 f ( ε(fi)y +νfε (fi) x ) (2.8) τ(fi)xy = Gfγ (fi) xy —the core σ(c)x = Ec1 1−ν2c [1−e0cos(πζ)] ( ε(c)x +νcε (c) y ) σ(c)y = Ec1 1−ν2c [1−e0cos(πζ)] ( ε(c)y +νcε (c) x ) τ(c)xy = Gc1[1−e0cos(πζ)]γ (c) xy (2.9) τ(c)xz = Gc1[1−e0cos(πζ)]γ (c) xz τ(c)yz = Gc1[1−e0cos(πζ)]γ (c) yz The deflection for each layer of the plate is the same and does not depend on the z coordinate, which means w(x,y,z) = w(x,y) (2.10) Mathematical modelling of a rectangular sandwich plate... 445 3. Mathematical model of the sandwich plate Equations of stability are based on the principle of minimum of the total potential energy δ(Uε−W)= 0 (3.1) Uε is the energy of elastic strain, where Uε = U (f1) ε +U (c) ε +U (f2) ε U(f1)ε = tc 2 a ∫ 0 b ∫ 0 − 1 2 ∫ −(1 2 +x1) ( σ(f1)x ε (f1) x +σ (f1) y ε (f1) y + τ (f1) xy γ (f1) xy ) dζ dy dx U(c)ε = tc 2 a ∫ 0 b ∫ 0 1 2 ∫ − 1 2 ( σ(c)x ε (c) x +σ (c) y ε (c) y +τ (c) xy γ (c) xy + τ (c) xz γ (c) xz + τ (c) yz γ (c) yz ) dζ dy dx (3.2) U(f2)ε = tc 2 a ∫ 0 b ∫ 0 1 2 +x1 ∫ 1 2 ( σ(f2)x ε (f2) x +σ (f2) y ε (f2) y + τ (f2) xy γ (f2) xy ) dζ dy dx U (f1) ε – energy of the upper face, U (c) ε – energy of the core, U (f2) ε – energy of the lower face. W is the work of the compressive force W = 1 2 a ∫ 0 b ∫ 0 [ N0x (∂w ∂x )2 +N0y (∂w ∂y )2] dy dx (3.3) where N0x = kN0, N 0 y =(1−k)N0, (0¬ k ¬ 1). Basing on the principle ofminimumof the total potential energy, Eq. (3.1), a system of five differential stability equations is obtained (δw) Ec1t 3 c 1−ν2c [ (2α11c20+ c11) (∂4w ∂x4 + ∂4w ∂y4 ) + +(4α11νfc20+4α11c21+2c11) ∂4w ∂x2∂y2 − (α12c20+2c11) · · (∂3ψ0 ∂x3 + ∂3φ0 ∂y3 ) − (α12νfc20+α12c21+2c11)· (3.4) 446 E. Magnucka-Blandzi · ( ∂3ψ0 ∂x∂y2 + ∂3φ0 ∂x2∂y ) + (1 π α12c20+ c15 )(∂3ψ1 ∂x3 + ∂3φ1 ∂y3 ) + + (1 π α12νfc20+ 1 π α12c21+c15 )( ∂3ψ1 ∂x∂y2 + ∂3φ1 ∂x2∂y )] = =−N0x ∂2w ∂x2 −N0y ∂2w ∂y2 (δψ0) 2t2c 1−νc { (α12c20+2c11) ∂3w ∂x3 +(α12νfc20+α12c21+2c11) · · ∂3w ∂x∂y2 − (2x1c20+4c11) ∂2ψ0 ∂x2 − [x1c21+2c11(1−νc)] · · ∂2ψ0 ∂y2 + (2 π x1c20+2c15 )∂2ψ1 ∂x2 + [1 π x1c21+ c15(1−νc) ] · (3.5) · ∂2ψ1 ∂y2 − [2x1νfc20+x1x21+2c11(1+νc)] ∂2φ0 ∂x∂y + + [2 π x1νfc20+ 1 π x1c21+ c15(1+νc) ]∂2φ1 ∂x∂y } + +4c0ψ0− c16ψ1 =0 (δψ1) t2c 1−νc { − (2 π α12c20+2c15 )∂3w ∂x3 + − (2 π α12νfc20+ 2 π α12c21+2c15 ) ∂3w ∂x∂y2 + + (4 π x1c20+4c15 )∂2ψ0 ∂x2 + [2 π x1c21+2c15(1−νc) ]∂2ψ0 ∂y2 + − ( 4 π2 x1c20+2c18 )∂2ψ1 ∂x2 − [ 2 π2 x1c21+ c18(1−νc) ]∂2ψ1 ∂y2 + (3.6) + [4 π x1νfc20+ 2 π x1c21+2c15(1+νc) ]∂2φ0 ∂x∂y + − [ 4 π2 x1νfc20+ 2 π2 x1c21+ c18(1+νc) ]∂2φ1 ∂x∂y } + −c16ψ0+c19ψ1 =0 (δφ0) 2t2c 1−νc { (α12νfc20+α12c21+2c11) ∂3w ∂x2∂y +(α12c20+2c11) · · ∂3w ∂y3 − [2x1νfc20+x1c21+2c11(1+νc)] ∂2ψ0 ∂x∂y + Mathematical modelling of a rectangular sandwich plate... 447 + [2 π x1νfc20+ 1 π x1c21+ c15(1+νc) ]∂2ψ1 ∂x∂y + (3.7) −[x1c21+2c11(1−νc)] ∂2φ0 ∂x2 − (2x1c20+4c11) ∂2φ0 ∂y2 + + [1 π x1c21+ c15(1−νc) ]∂2φ1 ∂x2 + (2 π x1c20+2c15 )∂2φ1 ∂y2 } + +4c0φ0− c16φ1 =0 (δφ1) t2c 1−νc { − (2 π α12νfc20+ 2 π α12c21+2c15 ) ∂3w ∂x2∂y + − (2 π α12c20+2c15 )∂3w ∂y3 + + [4 π x1νfc20+ 2 π x1c21+2c15(1+νc) ]∂2ψ0 ∂x∂y + − [ 4 π2 x1νfc20+ 2 π2 x1c21+ c18(1+νc) ]∂2ψ1 ∂x∂y + (3.8) + [2 π x1c21+2c15(1−νc) ]∂2φ0 ∂x2 + (4 π x1c20+4c15 )∂2φ0 ∂y2 + − [ 2 π2 x1c21+ c18(1−νc) ]∂2φ1 ∂x2 − ( 4 π2 x1c20+2c18 )∂2φ1 ∂y2 } + −c16φ0+ c19φ1 =0 where α11 = x1(4x 2 1+6x1+3) 12 α12 = x1(x1+1) c0 =1− 2 π e0 c11 = 1 12 ( 1−6 π2−8 π3 e0 ) c12 = 1 π2 (1 2 − 8 9π e0 ) c13 = 1 π2 (1 8 − 4 15π e0 ) c14 = 1 2 − 14 15π e0 c15 = 1 4π3 ( 8−πe0 ) c16 = 1 π ( 4−πe0 ) c18 = 1 2π2 ( 1− 4 3π e0 ) c19 = 1 2 − 4 3π e0 c20 = e1 1−ν2c 1−ν2 f c21 = e1 1−ν2c 1+νf c22 =2α11c20+ c11 c23 =4α11c20+2c11 c24 = α12c20+2c11 448 E. Magnucka-Blandzi c25 = α12c20+2c11 c26 = 1 π α12c20+ c15 c27 = 1 π α12c20+ c15 c28 =2x1c20+4c11 c29 = x1c21+2c11(1−νc) c30 = 2 π x1c20+2c15 c31 = 1 π x1c21+ c15(1−νc) c32 = x1(2νfc20+ c21)+2c11(1+νc) c34 = 2 π2 x1c20+ c18 c33 = 1 π x1(2νfc20+ c21)+ c15(1+νc) c35 = 1 π2 x1c21+ 1 2 c18(1−νc) c36 = 1 π2 x1(2νfc20+c21)+ 1 2 c18(1+νc) c37 = 1 π x1c21+ c15(1−νc) e1 = Ef Ec1 The boundary conditions for the simply supported sandwich plate are w(0,y) = 0 w(a,y) = 0 w(x,0)= 0 w(x,b) = 0 Mg(0,y) = 0 Mg(a,y) = 0 Mg(x,0)= 0 Mg(x,b) = 0 (3.9) where Mg is the bendingmoment and w – deflection. 4. Analytical solution There are five unknown functions in the system of stability equations. Forms of them are assumed as follows w(x,y) = wa sin mπx a sin nπy b ψ0(x,y)= ψa0cos mπx a sin nπy b ψ1(x,y)= ψa1cos mπx a sin nπy b φ0(x,y) = φa0 sin mπx a cos nπy b φ1(x,y) = φa1 sin mπx a cos nπy b (4.1) where m,n ∈ N (N – the set of natural numbers), wa – the amplitude of deflection, ψa0, ψa1, φa0, φa1 – the amplitudes of dimensionless diceplacment functions. These functions,Eq. (4.1) satisfy boundary conditions, Eq. (3.9). Substitu- ting these above five functions, Eq. (4.1), into the systemof stability equations (3.4)-(3.8) a system of five algebraic homogeneous equations is obtained Mathematical modelling of a rectangular sandwich plate... 449        a11−Km a12 a13 a14 a15 a12 a22 a23 a24 a25 a13 a23 a33 a25 a35 a14 a24 a25 a44 a45 a15 a25 a35 a45 a55                wa tc ψa0 ψa1 φa0 φa1         =        0 0 0 0 0        (4.2) where a11 = t2c a2 (mπ)2[c22(1+β 4)+ c23β 2] a12 =− tc a mπ(c24+c25β 2) a13 = tc a mπ(c26+ c27β 2) a14 =− tc a mπ(c24β 3+ c25β) a15 = tc a mπ(c26β 3+c27β) a22 = c28+ c29β 2+2c0c38 a23 =− ( c30+ c31β 2+ 1 2 c16c38 ) a24 = c32β a25 =−c33β a33 = c34+ c35β 2+ 1 2 c19c38 a35 = c36β a44 = c29+ c28β 2+2c0c38 a45 =− ( c37+ c30β 2+ 1 2 c16c38 ) a55 = c34β 2+ c35+ 1 2 c19c38 c38 = a2 t2c 1−νc (mπ)2 β = a b n m Km = N0 Ec1tc [k+(1−k)β2](1−ν2c) Because of the homogeneous algebraic equations, themain determinant of the systemmust be equal to zero. So, the critical forces N0,cr =min m,n {N0(m,n)} (4.3) could be calculated from this equation. 5. Numerical calculations There are some examples considered below, where the influence of the core porosity is shown for a family of plateswith b =200mm, νf =0.34, νc =0.15, Ec1 = 7.1 · 10 3MPa. The dimensionless parameter k is connected with the compressive forces, which means N0x = kN0, N 0 y = (1−k)N0, (0 ¬ k ¬ 1). 450 E. Magnucka-Blandzi The thickness of the core is tc = b/20=10mm (Figs.3-5). The dimensionless parameter x1 = tf/tc =1/20 is in every example below. In Fig.3, the critical loads in the case k = 1, which means N0x = N0, N0y =0 for the plate with constantmechanical properties of the core (e0 =0) and for the plate with varying mechanical properties of the core (e0 6= 0, e0 = 0, 0.5, 0.8) and for different e1 = 10, 20, 30, where e1 = Ef/Ec1 are shown.The critical load increases when the dimensionless parameter e1 incre- ases or dimensionless coefficient of the core porosity e0 decreases. Fig. 3. Critical loads in the case N0 x = N0, N 0 y =0, k =1 In Fig.4, the critical loads are also shown, but for different compressive forces, which means for k = 1, then N0x = N0, N 0 y = 0, for k = 0.75, then N0x = 0.75N0, N 0 y = 0.25N0, for k = 0.5, then N 0 x = N 0 y = 0.5N0 and for the plate with constant mechanical properties of the core (e0 = 0). In this example, the influence of dimensionless parameter e1 is shown too. If the parameter k decreases then the critical load also decreases. In Fig.5, the critical loads are shown for different compressive forces (k = 1, 0.75, 0.5) as previously, but for the plate with varying mechanical properties of the core (e0 = 0.5 in Fig.5a and e0 = 0.8 in Fig.5b). Both of them are for the same value e1 =10. The last two examples are for different thickness of the plate core. The dimensionless parameter x1 = 1/20, so the thickness of each face also changes. Mathematical modelling of a rectangular sandwich plate... 451 Fig. 4. Critical loads for the plate with constant mechanical properties of the core under different compressive forces; e0 =0 Fig. 5. Critical loads of the plate under different compressive forces; (a) e0 =0.5 e1 =10, (b) e0 =0.8 e1 =10 In Fig.6, the influence of the thickness of the core on the critical load is shown while the parameter e1 changes. In this example the core of the plate has constant mechanical properties (e0 = 0) and the commpresive forces are N0x =N0, ,N 0 y =0 (k =1). Instead, in the last example, in Fig.7, the influence of the thickness of the plate core for the critical load is shown, but the parameter e1 is fixed and the coefficient of core porosity e0 is changing. 452 E. Magnucka-Blandzi Fig. 6. Critical loads of the plate with different thickness of the core; e0 =0, k =1 Fig. 7. Critical loads of the plate with different thickness of the core; e1 =10, k =1 6. Conclusions Thefield of displacements for the sandwichplate is a generalization of the clas- sical hypotheses. The non-linear hypothesis of deformation of the plane cross section for a sandwich plate includes the shear deformable effect. Themathe- matical model of the sandwich plate is without internal contradiction. The Mathematical modelling of a rectangular sandwich plate... 453 equations of equilibrium-stability are correct for thin or thick plates. The sys- tem of five differential stability equations can be reduced to a single equation. 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Pandit M.K., Singh B.N., Sheikh A.H., 2008, Buckling of laminated san- dwich plates with soft core based on an improved higher order zigzag theory, Thin-Walled Structures, 46, 1183-1191 27. Plantema F.J., 1966, Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates and Shells, New York: JohnWiley & Sons 28. VolmirA.S., 1967,Stability of Deformation Systems,Moscow:Nauka, Fizma- tlit [in Russian] 29. Wang C.M., Reddy J.N., Lee K.H., 2000, Shear Deformable Beams and Plates, Elsevier, Amsterdam, Lousanne,NewYork,Oxford, Shannon, Singapo- re, Tokyo Matematyczne modelowanie prostokątnej płyty trójwarstwowej z rdzeniem z pianki metalowej Streszczenie Przedmiotem pracy jest prostokątna płyta trójwarstwowa podparta przegubowo naczterechbrzegach i ściskanawpłaszczyźnie środkowej.Okładzinypłyty są izotropo- we i o takich samychwłaściwościachmechanicznych.Rdzeńwykonany z piankimeta- lowej jest również izotropowy, jego właściwości mechaniczne są zmienne na grubości. Płaszczyzna środkowa płyty jest jej płaszczyzną symetrii. Zdefiniowano pole prze- mieszczeń dla dowolnego punktu rdzenia oraz okładzin płyty. Sformułowano energię odkształcenia sprężystegopłyty i pracę obciążenia.Następnie z zasady stacjonarności całkowitej energii potencjalnej otrzymanoukład równań równowagi,który rozwiązano analitycznie w sposób przybliżony i wyznaczono obciążenie krytyczne płyty. Manuscript received July 5, 2010; accepted for print November 22, 2010