东南大学：《固体力学基础》课程教学课件（英文讲稿）07 Viscoelastic Material Models

Viscoelastic Material Modelsmi@seu.edlu.cn

Viscoelastic Material Models

Outline·Introduction（引」言）·Creep compliance and relaxationmodulus（蠕变柔度与松弛刚度）·Retardedelasticity（滞／粘弹性）·Steady-state creep（稳态蠕变）·Elasticvs.viscoelasticmaterials（弹性与粘弹性材料）·Maxwellmaterials（Maxwell模型）·Kelvin-voigt materials（KV模型）·Pronyseriesrepresentation（Prony级数模型）·WLFtime/temperatureequivalence（WLF时间／温度相当）·Calibratingviscoelastic models（粘弹性模型校准）·More sophisticatedviscoelasticmodels（复杂粘弹性模型）·Generalizationto3D（向三维模型推广）·Responseto harmonic loading（周期载荷的响应）2

Outline • Introduction（引言） • Creep compliance and relaxation modulus（蠕变柔度与松弛刚 度） • Retarded elasticity（滞/粘弹性） • Steady-state creep（稳态蠕变） • Elastic vs. viscoelastic materials（弹性与粘弹性材料） • Maxwell materials（Maxwell模型） • Kelvin-voigt materials（KV模型） • Prony series representation（Prony级数模型） • WLF time/temperature equivalence（WLF时间/温度相当） • Calibrating viscoelastic models（粘弹性模型校准） • More sophisticated viscoelastic models（复杂粘弹性模型） • Generalization to 3D（向三维模型推广） • Response to harmonic loading（周期载荷的响应） 2

Introduction.Amorphouspolymersshowcomplextime-dependentbehaviorwhen subjectedtoahistoryofstressorstrain.Viscoelasticity theory was developed to approximate this behavior in polymersthat are subjected to modest strains (less than 0.5%): Polymers strongly resist volume changes at all temperatures. The bulk modulus iscomparabletothatofmetalsorcovalentlybondedsolidsThe shearresponse of a polymer is stronglytemperaturedependent.Shear modulus (N/m?)ViscoelasticGlassy109RubberyMelt105GlasstransitiontemperatureTTemperature3

• Amorphous polymers show complex time-dependent behavior when subjected to a history of stress or strain. • Viscoelasticity theory was developed to approximate this behavior in polymers that are subjected to modest strains (less than 0.5%). • Polymers strongly resist volume changes at all temperatures. The bulk modulus is comparable to that of metals or covalently bonded solids. • The shear response of a polymer is strongly temperature dependent. Introduction 3 Shear modulus (N/m2 )

Introduction· At temperatures near the glass transition, the shear modulus is strongly timedependent. The time dependent shear response can be measured by applying (1) astep load or (2) a harmonic (sinusoidal) load to the specimen.The time-dependent modulus of polymers is also temperature dependentReducing the temperature is qualitatively equivalent to increasing the strain rateMost amorphous polymers are isotropicShearmodulus (N/m?)ViscoelasticGlassyButyl109RubberyMelt105Glass transitiontemperatureTTemperature4

Introduction 4 Shear modulus (N/m2 ) • At temperatures near the glass transition, the shear modulus is strongly time dependent. The time dependent shear response can be measured by applying (1) a step load or (2) a harmonic (sinusoidal) load to the specimen. • The time-dependent modulus of polymers is also temperature dependent. Reducing the temperature is qualitatively equivalent to increasing the strain rate. • Most amorphous polymers are isotropic

Response to Step Loading8O? Creep compliance:the strain response toa unit constant stress30 Relaxation modulus:the stress response to aunit constant strain.. The results of such a test depend on the degree of cross-linking inthe polymer.. Heavily cross-linked materials show “retarded elastic" behavior.whereas un-cross-linked materials show steady-state creep5

• Creep compliance: the strain response to a unit constant stress • Relaxation modulus: the stress response to a unit constant strain. Response to Step Loading 5 • The results of such a test depend on the degree of cross-linking in the polymer. • Heavily cross-linked materials show “retarded elastic” behavior, whereas un-cross-linked materials show steady-state creep

Retarded Elasticity for Crosslinked Polymers. There is always an instantaneous strain in response to a step change in stress·At temperatures significantly below T., the solid is essentially elastic.Attemperatures significantlyaboveT.,the solid is very compliant: For a range of temperatures near T., the solid shows a slow transient response.The deformation is reversible, although this may take a very long timeComplianceJRubberyregimeT>>TgUnloadingTransitionregimeJ (T)GlassyregimeT<<TYTimet6

• There is always an instantaneous strain in response to a step change in stress. • At temperatures significantly below Tg , the solid is essentially elastic. • At temperatures significantly above Tg , the solid is very compliant. • For a range of temperatures near Tg , the solid shows a slow transient response. • The deformation is reversible, although this may take a very long time. Retarded Elasticity for Crosslinked Polymers 6

Steady-state Creep for uncrosslinked polymers: There is always an instantaneous strain in response to a step changein stress, exactly as in crosslinked polymers: At temperatures well below T., the solid is essentially elastic andhas a very low compliance, comparable with J.. At temperatures above T., the solid is very compliant. For mostpractical ranges of loading, the compliance will increase more orless linearly with time. The slope of compliance is stronglytemperature dependentComplianceJ个·Above the glass transitiontemperature T。, the deformationis irreversible.Unloading10Timet7

• There is always an instantaneous strain in response to a step change in stress, exactly as in crosslinked polymers. • At temperatures well below Tg , the solid is essentially elastic and has a very low compliance, comparable with Jg . • At temperatures above Tg , the solid is very compliant. For most practical ranges of loading, the compliance will increase more or less linearly with time. The slope of compliance is strongly temperature dependent. Steady-state Creep for uncrosslinked polymers 7 • Above the glass transition temperature Tg , the deformation is irreversible

Elastic ys. Viscoelastic Materials. For a constant loading actingfrom time O to Ta60: The strain responses of elasticmaterials:TEa=E二8=E: The strain responses of viscoelastic materials:n0= =n8

• For a constant loading acting from time 0 to T, • The strain responses of elastic materials: Elastic vs. Viscoelastic Materials 8 E E        • The strain responses of viscoelastic materials:        

Viscoelastic Models? Maxwell model (steady-state creep of uncrosslinkednpolymers)Ea0=6e+6,9En: Creep compliance due to a constant loading o acting0ot+00from time O to TOTn[t]11.一t-1TE6on: Relaxation modulus due to a constant strain co6EaEta+lnC =α[]=Ce-Eln= lng8 = = const二Enanna[1]Ee-Ei/n[0]= E =C =α[]= E6pe-Ei/n一69

• Maxwell model (steady-state creep of uncrosslinked polymers) Viscoelastic Models 9 E E                      0 0 0 0 const ln ln 0 Et Et Et E Et C t Ce E t E C t E e G t Ee                                             • Creep compliance due to a constant loading σ0 acting from time 0 to T • Relaxation modulus due to a constant strain ε0   0 0 0 0 0 , 0 0 , t t T E t T t T                                  0 t 1 1 J t t E       

Kelvin-Voigt Model (Retarded Elasticity of Crosslinked Polymers)E=,+=+Eg Creep compliance due to aFconstant loading o acting fromntime O to TE6%Et/n0T: +E=0n6-Et/neET/n[T] = De-ET/n = -oET/n-e-ET/n) = [t] = o-Et/n=J [4] 6EH

Kelvin-Voigt Model (Retarded Elasticity of Crosslinked Polymers)           E E • Creep compliance due to a constant loading σ0 acting from time 0 to T                       0 0 0 0 0 0 0 0 0 0 : 0 0 1 : 0 1 1 Et Et Et Et Et Et Et Et ET ET ET Et E t T E e e d e e t Ce dt E C C t e E E E E t T E t De e T De e t e e J t E E                                                                                                 1 ET e E   