Mechanical Behavior of Polymeric Foams? Polymeric foams:> Polymeric foams are close to reversible and show little rate orhistory dependence.> In contrast to rubbers, most foams are highly compressible: bulkand shear moduli are comparableStress-strainresponseoffoam>Foams have a complicated true stress-true strain response. The finite strainSresponse of the foam in compressionis quite different from that in tensionbecause of buckling in the cell walls> Foams can be anisotropic depending on their cell structure. Foamswith a random cell structure are isotropic6
• Polymeric foams: Mechanical Behavior of Polymeric Foams 6 Polymeric foams are close to reversible and show little rate or history dependence. In contrast to rubbers, most foams are highly compressible; bulk and shear moduli are comparable. Foams have a complicated true stresstrue strain response. The finite strain response of the foam in compression is quite different from that in tension because of buckling in the cell walls. Foams can be anisotropic depending on their cell structure. Foams with a random cell structure are isotropic
Strain Measure? Define the stress-strain relation for the solid by specifyingits strain energy density as a function of deformationgradient tensor: W = W(F). The general form of the strainenergy density is guided by experiment.. If W is a function of the left Cauchy-Green deformationtensor B = F-FT, the constitutive equation is automaticallyisotropic.dp(n).Invariants of Bdp(n)= BkdAu(x)(B,Bux - BikRDeformedOriginaI; = det[ B, ]= J2configurationconfiguration7
• Define the stress-strain relation for the solid by specifying its strain energy density as a function of deformation gradient tensor: W = W(F). The general form of the strain energy density is guided by experiment. • If W is a function of the left Cauchy-Green deformation tensor B = F∙FT , the constitutive equation is automatically isotropic. • Invariants of B: Strain Measure 7 1 2 2 1 2 3 1 1 2 2 det kk ii kk ik ki ik ki ij I B I B B B B I B B I B J
Strain Measure1,Bk An alternative set ofI7J2/3J213invariants of B more11(T-BT2-B.12. J4/3J4/3convenient for models ofI, = det[ B, ]= J2nearly incompressiblematerials[元?00[B,]=2200= I,=Bu=3元2,. Note that the first two0012invariants remain constantI,=(P-BBu)=324, I,=det[B,]==26under a pure volume change.→=3,7-0-3? Principal stretches and principal directions[2222B=2b, @b,+b2 ?b2 +b, @b3, B, =228
• An alternative set of invariants of B more convenient for models of nearly incompressible materials • Note that the first two invariants remain constant under a pure volume change. Strain Measure 8 1 1 2 3 2 3 2 2 2 2 1 1 4 3 4 3 4 3 2 3 1 1 2 2 det kk ik ki ik ki ij I B I J J I B B I I B B I J J J I B J 2 2 2 1 2 2 4 2 6 2 1 3 1 2 1 2 2 3 4 3 0 0 0 0 3 , 0 0 1 3 , det 2 3, 3. ij kk ik ki ij B I B I I B B I B J I I I I J J • Principal stretches and principal directions 2 1 2 2 2 2 1 1 1 2 2 2 3 3 3 2 2 3 , Bij B b b b b b b