Calibration of an Elastic-Plastic Material Model for Tire Shreds

1. Calibration of an Elastic-Plastic Material Model for Tire Shreds Kallol Sett

2. Tire Shreds Material • Automobile tires shredded into pieces of sizes varying from long thin pieces (~ 30 cm X 5 cm X 0.8 cm) to massive rubber (~ 10 cm X 5 cm X 5 cm) to tire chips (~3 cm X 2 cm X 0.8 cm). • Civil engineering applications is a growing market for scrap tires because of low unit weight, high permeability and good insulating properties. • Scrap tires has been successfully used as lightweight fill, insulation beneath road, lightweight backfill for retaining walls and overlay of underground ramps or storage space.

3. 3 1 2 (1-2 is the plane of isotropy) Elastic Constitutive Law • The placement and compaction of tire shreds leads to a layered material and hence its elastic behavior can be best described by cross-anisotropic constitutive law. • Cross anisotropic material can be characterized by 5 independent material constants, E11, E33, G12, G13 and G31 (Drescher 1999)

4. Wave Propagation and Governing Moduli 3 Wave propagation in the 1/2-direction with vertical polarization (Governing modulus G13) Wave propagation in the 3-direction with horizontal polarization (Governing modulus G31) Wave propagation in the 1/2-direction with horizontal polarization (Governing modulus G12) 1 2 Wave propagation in the 3-direction with polarization in the 3-direction (Governing modulus E33) Wave propagation in the 1/2-direction with polarization in the 1/2-direction (Governing modulus E11)

5. 3 1 2 Elastic Constants • E11 = rVci2 (i = 1,2) • E33 = rVc32 • G12 = r (VsiH)2 (i = 1,2) • G31 = r (Vs3)2 • G13 = r (VsiV)2 (i = 1,2) • Redundant Constants: • n12 = (E11/2G12) -1 • n31 = (E33/2G31) -1 • n13 = (E11/2G13) -1

6. Elastic-Plastic Constitutive Law Using Hooke’s law and additive decomposition of strain tensors in elastic and plastic parts (Jeremic and Sture 1997): Where, K = Bulk modulus G = Shear modulus f = Yield function

7. Yield Function Salient features: • Based on Drucker-Prager yield surface, having fixed friction angle. • Yield surface has a shape of distorted (to model the cross anisotropy) and rotated (to model the Bauschinger effect) cone. • The plastic flow is assumed to be perpendicular to the yield surface.

8. Principal Stress Space depij q p P- Plane a where, J2 = Second invariant of the deviatoric stress tensor I1 = First invariant of the hydrostatic stress tensor a and k = Drucker-Prager material constant • a can be related to the friction angle (f) as (Chen 1988),

9. Elastic-Plastic Material Model The general form of Drucker-Prager elastic-perfectly plastic model is, which can be simplified to any direction of interest e.g. in 3-1 direction it reduces to: where

10. Elastic-Plastic Material Model • Knowing E11, E33, n12, n31 (From wave propagation tests) and f = 20o (Tweedie et al. 1998) the elastic-plastic model was solved incrementally to get the stress-strain relationship in 3-1 direction. • Another important benefit of this model is that it represents the variation of G/Gmax which is usually found in traditional analysis of dynamics of soils.

11. 1 t31 g31 3 Elastic Elastic-Plastic Predicted Elastic and Elastic-Plastic Stress-Strain Relationship

12. 1 t31 g31 3 Elastic Elastic-Plastic Predicted Variation of Shear Modulus with strain

13. Conclusions • 3-D P- and Shear- wave propagation tests can be used to determine 5 elastic constants necessary for modeling the behavior of cross-anisotropic tire shreds. 2. The data obtained from 3-D P- and Shear- wave propagation tests can be used to calibrate an elastic-plastic material model for the tire shreds. This model also represented the variation of G/Gmax, that is usually found in traditional analysis of dynamics of frictional media.