东南大学：《固体力学基础》课程教学课件（英文讲稿）06 Hyper-elastic Materials

Hyper-elastic Materialsmi@se.ed.cn

Hyper-elastic Materials

Outline·Introduction（引言）·Mechanical behavior of rubbers（橡胶性能）·Mechanical behaviorofpolymericfoams（泡沫性能）·Strainmeasure（应变度量）·Stress measure（应力度量）·Generalized constitutive law（一般本构关系）·Incompressibility（不可压缩性）·Polynomialmodels for rubbers（橡胶多项式本构）·More sophisticated rubber models（复杂本构）·Foam constitutive models（泡沫本构）·Calibrating nonlinearelastic models（模型校准）2

Outline • Introduction（引言） • Mechanical behavior of rubbers（橡胶性能） • Mechanical behavior of polymeric foams（泡沫性能） • Strain measure（应变度量） • Stress measure（应力度量） • Generalized constitutive law（一般本构关系） • Incompressibility（不可压缩性） • Polynomial models for rubbers（橡胶多项式本构） • More sophisticated rubber models（复杂本构） • Foam constitutive models（泡沫本构） • Calibrating nonlinear elastic models（模型校准） 2

Introduction: Main applications of the theory are (1) to model therubbery behavior of a polymeric material and (2) to modelpolymeric foams that can be subjected to large reversibleshape changes (e.g., a sponge). In general, the response of a typical polymer is stronglydependent on temperature, strain history, and loading rateShear modulus (N/m?2)ViscoelasticGlassy109RubberyMelt105Glass transitiontemperatureTTemperature3

Introduction 3 Shear modulus (N/m2 ) • Main applications of the theory are (1) to model the rubbery behavior of a polymeric material and (2) to model polymeric foams that can be subjected to large reversible shape changes (e.g., a sponge). • In general, the response of a typical polymer is strongly dependent on temperature, strain history, and loading rate

Introduction. Rubbery behavior: the response is elastic, the stress does not dependstrongly on strain rate or strain history, and the modulus increaseswithtemperature. Heavily cross-linked polymers (elastomers) are the most likely toshow ideal rubbery behavior.Hyperelastic constitutive laws are intended to approximate thisrubbery behavior.Shearmodulus (N/m?)ViscoelasticGlassy109RubberyMelt105Glass transitiontemperatureTTemperature4

Introduction 4 Shear modulus (N/m2 ) • Rubbery behavior: the response is elastic, the stress does not depend strongly on strain rate or strain history, and the modulus increases with temperature. • Heavily cross-linked polymers (elastomers) are the most likely to show ideal rubbery behavior. • Hyperelastic constitutive laws are intended to approximate this rubbery behavior

Mechanical Behavior of Rubbers. Features of the behavior of a solid rubber:> The material is close to ideally elastic> The material strongly resists volume changes. The bulk modulus iscomparable with that of metals> The material is very compliant in shear: shear modulus is of theorder of 10-5 times that of most metals> The material is isotropic.> The shear modulus is temperature dependent: the material becomesstiffer as it is heated, in sharp contrast to metals5

• Features of the behavior of a solid rubber: Mechanical Behavior of Rubbers 5  The material is close to ideally elastic.  The material strongly resists volume changes. The bulk modulus is comparable with that of metals.  The material is very compliant in shear: shear modulus is of the order of 10−5 times that of most metals.  The material is isotropic.  The shear modulus is temperature dependent: the material becomes stiffer as it is heated, in sharp contrast to metals

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 stress￾true 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

Stress Measure and General Constitutive Law. Stress measure: dp(l) = dAn,j? Strain energy density:W(F)=U(B)=U(I1,I2,I)=U(I,I2,I)=U(, 2,) Cauchy stress in terms of deformation gradient F=y+y=aw111F.SFikSHK=a0:JaFJjk9

• Stress measure: Stress Measure and General Constitutive Law 9   j i ij dP dAn   n • Strain energy density: W U U I I I U I I I U F B         1 2 3 1 2 3 1 2 3 , , , , , ,        • Cauchy stress in terms of deformation gradient F 0 0 0 0 0 1 1 1 1 2 n i i i i i kj jk j ji i i S V V k V ij ik i d W T v dS Fv dV S F d V v F F J v d W S F V dt J J                  σ F S

General Constitutive Law: Cauchy stress in terms of invariants of Bau alawaual2aual11F.HikikaFTal, aFVIal,aFal,aFikjk3ik. Derivatives of B w.r.t. F:aBaFkkkmB=F.FTB,=FF.=FF= Bu2F2Fkkkmpkkmpm/kmkm1aFaF.ijiaFaBaFpkpmkmF+FSSF.+F.S..=8F.+FSkmkpmmjkmpmkimipiKppaFaFaF1i公aB,ikS, F, +F,Oh =2FhaFij=110

• Cauchy stress in terms of invariants of B General Constitutive Law 10   1 , 2 2 2 T kk km pk pm km kk km km km ij ij ij pk pm km km pm pi mj km pm ki mj pi kj pj ki ij ij ij ik kj ij ki kj ij ii B F B F F B F F F F F F B F F F F F F F F F F F B p i F F F F                                          B F F 1 2 3 3 1 2 3 1 1 1 ij kj jk ik ik ik jk jk jk UUU I I I F F F W S J J I F I F I F F I                           • Derivatives of B w.r.t. F: