Predicting the constitutive behaviour of microsphere-filled elastomers: the influence of shell buckling and interaction

Transcription

1 Predicting the constitutive behaviour of microshere-filled elastomers: the influence of shell buckling and interaction William J. Parnell Professor of Alied Mathematics and EPSRC Research Fellow School of Mathematics, University of Manchester, UK Innovations in Rubber Design 7th December 2016 work with R. De Pascalis, M. Thore, G. Wyn Jones and I.D. Abrahams

2 Overview Background WICC - my research grou Problem of interest - elastomeric comosites

3 Overview Background WICC - my research grou Problem of interest - elastomeric comosites Constitutive modelling of microshere filled elastomers (syntactic foams) nonlinear load curve

4 Overview Background WICC - my research grou Problem of interest - elastomeric comosites Constitutive modelling of microshere filled elastomers (syntactic foams) nonlinear load curve microshere buckling models

5 Overview Background WICC - my research grou Problem of interest - elastomeric comosites Constitutive modelling of microshere filled elastomers (syntactic foams) nonlinear load curve microshere buckling models Wave roagation in re-stressed microshere comosites

6 Overview Background WICC - my research grou Problem of interest - elastomeric comosites Constitutive modelling of microshere filled elastomers (syntactic foams) nonlinear load curve microshere buckling models Wave roagation in re-stressed microshere comosites Persectives

7 Overview Background WICC - my research grou Problem of interest - elastomeric comosites Constitutive modelling of microshere filled elastomers (syntactic foams) nonlinear load curve microshere buckling models Wave roagation in re-stressed microshere comosites Persectives More general constitutive modelling

8 Overview Background WICC - my research grou Problem of interest - elastomeric comosites Constitutive modelling of microshere filled elastomers (syntactic foams) nonlinear load curve microshere buckling models Wave roagation in re-stressed microshere comosites Persectives More general constitutive modelling Metamaterials and tunable materials

9 Overview Background WICC - my research grou Problem of interest - elastomeric comosites Constitutive modelling of microshere filled elastomers (syntactic foams) nonlinear load curve microshere buckling models Wave roagation in re-stressed microshere comosites Persectives More general constitutive modelling Metamaterials and tunable materials Microshere comosites roject (oortunities)

10 Background A traditional modelling roblem (with Thales) Some wave concets Waves in re-stressed media Tunable band-gas

11 WICC - Waves in Comlex Continua Grou founded in Set 2010, led by Parnell and Abrahams (now Director of INI, Cambridge)

12 WICC - Waves in Comlex Continua Grou founded in Set 2010, led by Parnell and Abrahams (now Director of INI, Cambridge) General objective is to model, mathematically, comlex materials

13 WICC - Waves in Comlex Continua Grou founded in Set 2010, led by Parnell and Abrahams (now Director of INI, Cambridge) General objective is to model, mathematically, comlex materials Includes general constitutive modelling as well as wave roagation henomena

14 WICC - Waves in Comlex Continua Grou founded in Set 2010, led by Parnell and Abrahams (now Director of INI, Cambridge) General objective is to model, mathematically, comlex materials Includes general constitutive modelling as well as wave roagation henomena Large grou, around 25 members

15 WICC - Waves in Comlex Continua Grou founded in Set 2010, led by Parnell and Abrahams (now Director of INI, Cambridge) General objective is to model, mathematically, comlex materials Includes general constitutive modelling as well as wave roagation henomena Large grou, around 25 members Breadth of funding from EPSRC, Royal Society, Leverhulme Trust and Industry (Dyson, Thales, NNS)

16 Research interests, Inhomogeneous media models Industrial modelling zone.com WICC st.fr Phononics, metamaterials Biological tissues Coutinho et al (2007)

17 Background to Manchester and my research grou Constitutive modelling of microshere filled elastomers Wave roagation in re-stressed microshere comosites Persectives

18 Microshere comosites (syntactic foams) (a) SEM image (a) Hollow Glass Microshere, (b) surface/shell structure. Li et al. [2011]. (b)

19 Microshere comosites (syntactic foams) (a) SEM image (a) Hollow Glass Microshere, (b) surface/shell structure. Li et al. [2011]. (b) Reasons to use microsheres in matrix comosites: Add volume whilst reducing weight Imrove thermal and sound insulation Increase comressibility and reduce cost Isotroic, homogeneous rubber matrix filled with exancel (glassy) microsheres - distribution of shell thicknesses.

20 Industrial interest and relation to waves Need to understand how the material resonds acoustically when it is under significant re-stress. Shorter et al. (2008)

21 Industrial interest and relation to waves Need to understand how the material resonds acoustically when it is under significant re-stress. Shorter et al. (2008) To understand the acoustic resonse we need a model for the constitutive behaviour of the medium (strongly related).

22 Motivation Need to be able to redict the so-called ressure-volume curve associated with this material

23 Motivation Need to be able to redict the so-called ressure-volume curve associated with this material Exeriments very difficult and costly to erform

24 Motivation Need to be able to redict the so-called ressure-volume curve associated with this material Exeriments very difficult and costly to erform Reeatability is difficult

25 Motivation Need to be able to redict the so-called ressure-volume curve associated with this material Exeriments very difficult and costly to erform Reeatability is difficult Deendence on a number of arameters can be tested with the model

26 Motivation Need to be able to redict the so-called ressure-volume curve associated with this material Exeriments very difficult and costly to erform Reeatability is difficult Deendence on a number of arameters can be tested with the model Require careful models incororating all of the major effects

27 Geometry and assumtions In ractice the material resonse is strongly nonlinear - why?

28 Geometry and assumtions In ractice the material resonse is strongly nonlinear - why? We need a model for the nonlinear constitutive behaviour of the medium.

29 Geometry and assumtions In ractice the material resonse is strongly nonlinear - why? We need a model for the nonlinear constitutive behaviour of the medium. deformation

30 Single shell - model - dilute distribution S

31 Single shell - model - dilute distribution We need a relationshi S δv = δv() δv = relative volume change.

32 Single shell - model - dilute distribution We need a relationshi S δv = δv() δv = relative volume change. < c = c > c Pre-buckling Buckling Post-buckling Linear elasticity Fok-Allwright model Nonlinear elasticity δv() is then integrated resonse over all shell thicknesses.

33 Parameter studies and some lots Rubbery matrix. Glassy exancel microsheres. Initial volume fraction of microsheres = 5%, Gamma distribution of shell thicknesses. Neo-Hookean (dotted), Mooney-Rivlin (solid), nearly-comressible (dashed) and linear elasticity (dot-dash). De Pascalis, R., WJP and Abrahams, I.D., 2013, JMPS

34 Buckling - Thore PhD Investigated shell buckling inside elastomers. General method derived - energy minimization. Stiff, thin-shell limit derived: Hydrostatic buckling ressure of a thin, glassy shell relative to the Classical buckling ressure for an unembedded shell, 0 = 4µs(1+νs) (Zoelly-Van der Neut), for each buckling mode n. 3(1 νs)

35 Background to Manchester and my research grou Constitutive modelling of microshere filled elastomers Wave roagation in re-stressed microshere comosites Persectives

36 Other insiration Inhomogeneous soft materials occur frequently in natural and synthetic forms: Polymer based comosites Self-assembly based hydrogels Soft tissue

37 Other insiration Inhomogeneous soft materials occur frequently in natural and synthetic forms: Polymer based comosites Self-assembly based hydrogels Soft tissue Their dynamic resonse in re-stressed states is fundamental.

38 Other insiration Inhomogeneous soft materials occur frequently in natural and synthetic forms: Polymer based comosites Self-assembly based hydrogels Soft tissue Their dynamic resonse in re-stressed states is fundamental. Difficulties: Homogeneous deformations are not ossible

39 Other insiration Inhomogeneous soft materials occur frequently in natural and synthetic forms: Polymer based comosites Self-assembly based hydrogels Soft tissue Their dynamic resonse in re-stressed states is fundamental. Difficulties: Homogeneous deformations are not ossible Microstructure evolution

40 Other insiration Inhomogeneous soft materials occur frequently in natural and synthetic forms: Polymer based comosites Self-assembly based hydrogels Soft tissue Their dynamic resonse in re-stressed states is fundamental. Difficulties: Homogeneous deformations are not ossible Microstructure evolution Changes in material roerties

41 Other insiration Inhomogeneous soft materials occur frequently in natural and synthetic forms: Polymer based comosites Self-assembly based hydrogels Soft tissue Their dynamic resonse in re-stressed states is fundamental. Difficulties: Homogeneous deformations are not ossible Microstructure evolution Changes in material roerties Resonse is often strongly frequency deendent

42 Wave scattering in re-stressed microshere comosites Modify the classical Multile Scattering Theory of Mal and Bose (1974) as mechanism for evaluating effective wavenumber.

43 Wave scattering in re-stressed microshere comosites Modify the classical Multile Scattering Theory of Mal and Bose (1974) as mechanism for evaluating effective wavenumber. E.g. k s = F(φ,A 2,A 1 ) k s( ) = f(φ( ),a 2 ( ),a 1 ( ))

44 Wave scattering in re-stressed microshere comosites Modify the classical Multile Scattering Theory of Mal and Bose (1974) as mechanism for evaluating effective wavenumber. E.g. k s = F(φ,A 2,A 1 ) k s( ) = f(φ( ),a 2 ( ),a 1 ( )) which gives the effective (incremental) shear modulus at fixed re-stress µ = g(µ 1 /µ 0,φ, )

45 Background to Manchester and my research grou Constitutive modelling of microshere filled elastomers Wave roagation in re-stressed microshere comosites Persectives

46 General constitutive modelling Significant interest in nonlinear viscoelasticity,

47 General constitutive modelling Significant interest in nonlinear viscoelasticity, e.g. QLV Π(t) = t G(t s) dπe (C(s)) ds ds

48 General constitutive modelling Significant interest in nonlinear viscoelasticity, e.g. QLV Π(t) = t G(t s) dπe (C(s)) ds ds and extensions to accommodate strain deendent relaxation Π(t) = t G(t s,c) dπe (C(s)) ds ds

49 General constitutive modelling Significant interest in nonlinear viscoelasticity, e.g. QLV Π(t) = t G(t s) dπe (C(s)) ds ds and extensions to accommodate strain deendent relaxation Π(t) = t G(t s,c) dπe (C(s)) ds ds and secifically motivated by microstructural models, for e.g. tendon, filled olymers, etc. again informed by imaging. De Pascalis, Abrahams & WJP, 2014, Proc. Roy. Soc. A

50 Metamaterials & tuneable structures WJP, 2012, Proc. Roy. Soc. A Also interest in tuning band structure of eriodic elastomers via re-load, or erhas more interestingly retain them under re-load (invariance)?? See oster by Zhang and Parnell tomorrow

51 Microshere modelling roject Two year roject as art of my fellowshi 2 year PDRA - theoretical modelling (currently advertised) 2 year PDRA - XCT imaging (advertised shortly) Image-based modelling, articularly regarding behaviour under load [T. Lowe & S. Coban, Henry Moseley Centre, Manchester]

52 Acknowledgements EPSRC (Parnell NEMESIS Fellowshi EP/L018039/1) EPSRC (De Pascalis PDRA EP/H050779/1) Thales UK/EPSRC (Thore CASE PhD) Royal Society (Abrahams Wolfson)