Functional Grading of Rubber-Elastic Materials: From Chemistry to Mechanics

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1 Functional Grading of Rubber-Elastic Materials: From Chemistry to Mechanics Barry Bernstein Hamid Arastoopour Ecevit Bilgili Department of Chemical and Environmental Engineering llinois nstitute of Technology Chicago USA

2 Outline of the Presentation Overview of rubbers and basic concepts Experimental results: potential of grading A theoretical study on the effects of material non-homogeneity temperature gradients functional grading How to manufacture FGREMs? modeling the structure development Conclusions

3 Overview of Rubber-like Materials Rubbers widely used in industry and daily life Applications: tires bearings bushings sealants etc. Relatively soft light-weight and flexible. Undergo large strains energy dissipation sub-ambient T g. -D network structure of long flexible polymeric chains: RUBBER MOLECULES SULFUR VULCANZATON SOL FRACTON OF RUBBER CROSSLNK

localizations and temperature gradients controlling the

4 Functional Grading Concept Tailoring the material non-homogeneity P(X) for reducing stress-strain localizations and temperature gradients controlling the dynamic properties surface properties and the permeability to fluids

5 Graded Rubbers: Layering Rubber Slabs SBR-B: 4 phr Sulfur Laminate Molding 5 o C 4 min Graded Rubber FGR-AB.9 mm SBR-A: phr Sulfur SBR-A SBR-B Laminate Molding 5 o C 4 min Graded Rubber FGR-ABA.9 mm SBR-A keda Y.. J. Appl. Polym. Sci. 87: 6-67.

6 Milestones in Rubber Thermoelasticity Entropic origin for the stress (Anthony et al. 94; Meyer and van der yk 946). Large multi-axial isothermal deformations (Treloar 944; Rivlin and Saunders95) Universal controllable deformations of isotropic elastic materials (Ericksen954955). Universal deformations within the context of finite thermoelasticity (Petroski and Carlson 96897). Generalized thermoelastic models: Chadwick (974) Chadwick and Creasy (984) Ogden (99). Many BVPs solved in the last decade (e.g. Horgan Rajagopal Saccomandi ineman). Non-homogeneity???

7 Consequences of Material Non-Homogeneity Material homogeneity is simply assumed in the rheological characterization of rubbers. hat do standard tests really measure? Three fundamental issues to be addressed: isothermal/non-isothermal behavior of non-homogeneous rubbers (forward problem) tailoring the non-homogeneity characterization of non-homogeneous rubbers (inverse problem)

8 Kinematics with volume dv with volume dv da V p* da x n dr p X N P* r x R dr P i i i ds ds dr dr dr dr dx dx k K dx k dx K X x x Field Variables x ( X t ) and ( xt) X Reference Configuration Deformed Configuration

9 The deformation gradient F Deformation Measures F x x X The left (right) Cauchy-Green deformation tensor B (C) is obtained via polar decomposition of F: T F RU VR B V FF and C U F T F Material isotropy requires that the response functions depend on the principal invariants of B i.e. and : [( ) ] tr B tr B tr B ( tr B) + 6 [ ] tr B tr B B tr

10 Finite Thermoelasticity Thermo-mechanical behavior of non-homogeneous isotropic non-linearly elastic materials can be described by The Cauchy stress T and the heat flux vector q are given by ψ η and ε : specific Helmholtz free energy entropy and internal energy Γ i : thermo-mechanical response functions ρ and : a reference density and temperature ( ) η ψ ε ψ η ψ ψ + X ( ) ( ) ( ) Γ Γ Γ Γ Γ ψ ψ ψ ψ ρ or where B B X B B K q B B X T i i i

11 Entropic Finite Thermoelasticity A constrained thermoelastic model: The Cauchy stress T and the heat flux vector q are given by Clausius-Duhem nequalities: : isothermal strain energy function; Constraint: ( ) ( ) ( ) ( ) ( ) ( ) φ ρ π ρ ψ ψ d c X X X X X ( ) ( ) ( ) [ ] B q B B T K K K K p φ φ Γ + Γ Γ T q ( ) φ ρ ρ

12 Local Balance Equations Balance of Mass: D ρ + ρ v or ρ t Dt ( x ) ( X) ρ ( X) ρo det F o Balance of Linear Momentum: D v ρ Dt T ρb Balance of Thermal Energy: Dε ρ T : d q ρs Dt

13 sothermal Deformation of Homogeneous Rubbers Neo-Hookean model: Mooney model: Yeoh model: Gent model: Shear Stress T xy (MPa) 5 4 ( ) A( ) ( ) A( ) + A( ) ( ) A( ) + A( ) + A( ) ( ) µ J ln[ ( ) ]/ Simple shear data (Seki et al. 987) Yeoh model: A 48 kpa A.484 kpa A. kpa Mooney model: A kpa A 5 kpa Gent model: G 455 kpa ( G) m Shear Strain J m Strain-nduced Stiffening

14 Rectilinear Shearing of Rubber Slabs () H Y Reference Configuration Undeformed Slab Postulated Deformation: x X + f (Y) y Y z Z Postulated Temperature: (y) (Y) X y Current (Deformed) Configuration Displaced Boundary: f (H) U (H) H Simple Shear H nhomogeneous Shear Stationary Boundary: f () () C x

15 Rectilinear Shear (): Deformation Measures The deformation measures are implicitly given by Due to det F incompressibility constraint is satisfied. The strain inhomogeneity is determined from: ( ) ( ) [ ] ( ) det tr tr tr + + B B B B f f ( ) + T f f f f FF B X x F Z Y X M J z y x i f X F X F M ij M ij and : F f and f ' : shearing displacement and shear strain xf ''x: absolute value of the shear strain gradient

16 Rectilinear Shear (): Field Equations The Cauchy s equations of motion can be recorded as The local energy balance equation with Fourier s law: ( ) Y H Y C H C + Z Y X M z y x j i x X X T x T j M M ij j ij and T ( ) ( ) ( ) S H T U H f f f dy d dy dt xy xy + or and o ( ) ( ) H C and H k dy d k dy d q

17 Strain Gradient f '' (cm ) Shear Strain f ' C 98 K 8 6 Rectilinear Shear (V): Neo-Hookean Model K 4 K 5 K 77 K K 4 K 5 K 77 K Position Y (cm) Non-isothermal Deformation of a Homogeneous Slab: ( ) A ( ) An increase in temperature gradient enhances the strain inhomogeneity. The strain inhomogeneity is more pronounced near the colder boundary. H cm U 4 cm 96 K C 98 K A 5 kpa H varied

18 Shear Strain f ' C 98 K Rectilinear Shear (V): Yeoh Model K 4 K 5 K 77 K Non-isothermal Deformation of a Homogeneous Slab: ( ) A( ) + A( ) + A ( ) Strain Gradient f '' (cm ) K 4 K 5 K 77 K Position Y (cm) The strain-induced stiffening homogenizes the strain field. H cm U 4 cm 96 K C 98 K A 5 kpa A kpa A. kpa H varied

19 Shear Stress T xy (MPa) Normal Stress T xx (MPa) Rectilinear Shear (V): Neo-Hookean Model C 98 K K 4 K 5 K 77 K K 4 K 5 K 77 K Position Y (cm) Non-isothermal Deformation of a Homogeneous Slab: ( ) A ( ) Stress increase due to temperature-induced stiffening Colder boundary subjected to a higher normal stress! H cm U 4 cm 96 K C 98 K A 5 kpa H varied

20 Normal Stress T xx (MPa) Shear Stress T xy (MPa) 7 C 98 K Rectilinear Shear (V): Yeoh Model K 4 K 5 K 77 K K 4 K 5 K 77 K Position Y (cm) Non-isothermal Deformation of a Homogeneous Slab: ( ) A( ) + A( ) + A ( ) The stresses are higher and less inhomogeneous in the Yeoh slab than in the Neo- Hookean slab. H cm U 4 cm 96 K C 98 K A 5 kpa A kpa A. kpa H varied

21 Shear Modulus µ (MPa) Norm of F f '' (mm ) Rectilinear Shear (V): AB Type Graded Rubbers Position Y (mm) µ max µ min (MPa) Position Y (mm) Shear Strain f' Normal Stress T xx (MPa) sothermal Deformation of a Non-Homogeneous Slab: µ FGR-AB max min ( Y ) µ max ( Y Y )/ H H mm S. MPa 96 K φ 5 Y inf mm Λ.9 MPa µ + µ φ ( Y ) ln µ inf [ Λµ ( Y )( ) ] Λ

22 Shear Modulus µ (MPa) Norm of F f'' (mm ) Rectilinear Shear (X): AB Type Graded Rubbers Position Y (mm) Λ (MPa ) Self- Homogenization Position Y (mm) Shear Strain f' Normal Stress T xx (MPa) sothermal Deformation of a Non-Homogeneous Slab: µ FGR-AB µ + max min ( Y ) µ max ( Y Y )/ H µ φ ( Y ) ln H mm S. MPa µ 96 K φ 5 Y inf mm µ max.8 MPa µ min.8 MPa inf [ Λµ ( Y )( ) ] Λ

23 Shear Strain f ' Strain Gradient f '' (cm ) Rectilinear Shear (X): Functional Grading?. Colder Boundary at C 96 K.8 % -% -5% -% -5% Position Y (cm) Hotter Boundary at 444 K Non-isothermal Deformation of a Graded Slab: µ ( Y ) ( Y ) µ B + ln An optimum grading exists! H cm U 4 cm C 96 K H 444 K 5. µ B 455 kpa.64 MPa λ ( + κ ) Y ( ) κh + Y [ Λµ ( Y )( ) ] Λ

24 Normal Stress T xx * Rectilinear Shear (X): Functional Grading? T xy * Colder Boundary at C 96 K Position Y (cm) % -% -5% -% -5% Hotter Boundary at 444 K Non-isothermal Deformation of a Graded Slab: µ ( Y ) µ B + ( Y ) ln Grading leads to a reduction in stresses. H cm U 4 cm C 96 K H 444 K 5. µ B 455 kpa.64 MPa λ ( + κ ) Y ( ) κh + Y [ Λµ ( Y )( ) ] Λ

25 A Perspective in Materials Engineering PARADGM Processing Properties (Structure) Performance How to generate the spatial variation of the crosslink density V(X)? RUBBER MOLECULES SULFUR VULCANZATON SOL FRACTON OF RUBBER CROSSLNK

26 How to Produce FGREMs? Construction-based methods: layers with different contents of curing agents (keda ) secondary phase (fillers recycled rubber powder) rubber blends Transport-based methods: Thermally-induced structure (crosslink density) Performance & Grading Criteria Desired Material Structure Fabrication Processes

27 Molding of a Rubber Tube () Geometry r r r rt 4 oc Rubber: black Gray: Metal mold Rubber with curing agents is heated by the steam-jacketed metal mold. The spatio-temporal development of temperature and crosslink density was simulated. Long tube insulated ends constant physical properties except λ λ λ T r ( 7.5) r (E. Bilgili AChE Annu. Meet. )

28 Temperature T r (K) Molding Simulation (): Rubber Temperature Molding Time t (min) Radial Position R (cm) The rubber temperature rises fast near the outer mold (about 4 K) which is heated by steam. t monotonically increases and evolves to a steady state.

29 Molding Simulation (): Crosslink Density Crosslink Density V (mol/m ) Colder Region 9 Molding Time t (min) Radial Position R (cm) The evolution of crosslink density V is non-monotonic due to reversion. Transport-based grading of rubbers is possible via control of the process.

30 Conclusions Rubbers can be graded in a variety of ways. Functional grading can reduce stress-strain localizations. Sophisticated process models enable us to generate a desired structure. An attempt to connect processing-properties-performance aspects of rubbers has been made.

31 Publications E. Bilgili "Functional Grading of Rubber Tubes within the Context of a Molecularly nspired Finite Thermoelastic Model" Acta Mech (4). E. Bilgili "A Note on the Mechanical Characterization of Non-homogeneous and Graded Vulcanized Rubbers" Polym. Polym. Compos. -4 (4). E. Bilgili "A Parametric Study of the Circumferential Shearing of Rubber Tubes: Beyond sothermality and Material Homogeneity" Kaut. Gummi Kunstst (). E. Bilgili "Controlling the Stress Strain nhomogeneities in Axially Sheared and Radially Heated Hollow Rubber Tubes via Functional Grading" Mech. Res. Commun (). E. Bilgili B. Bernstein H. Arastoopour "Effect of Material Non-homogeneity on the nhomogeneous Shearing Deformation of a Gent Slab Subjected to a Temperature Gradient" nt. J. Non-Linear Mech (). E. Bilgili "Computer Simulation as a Tool to nvestigate the Shearing Deformation of Non- Homogeneous Elastomers" J. Elastom. Plast (). E. Bilgili B. Bernstein H. Arastoopour "nfluence of Material Non-Homogeneity on the Shearing Response of a Neo-Hookean Slab" Rubber Chem. Technol (). E. Bilgili B. Bernstein H. Arastoopour "nhomogeneous Shearing Deformation of a Rubber-like Slab within the Context of Finite Thermoelasticity with Entropic Origin for the Stress" nt. J. Non- Linear Mech ().

32 Acknowledgments Support from the National Science Foundation EB s collaboration with Dr. Yuko keda Kyoto nstitute of Technology

Stress Analysis of Thick Spherical Pressure Vessel Composed of Transversely Isotropic Functionally Graded Incompressible Hyperelastic Materials

407 Stress Analysis of Thick Spherical Pressure Vessel Composed of Transversely Isotropic Functionally Graded Incompressible Hyperelastic Materials Abstract In this paper closed form analytical solution