PSI - Issue 42

Monika Středulová et al. / Procedia Structural Integrity 42 (2022) 1537– 1544 M. Strˇedulova´ et al. / Structural Integrity Procedia 00 (2019) 000–000

1540

4

The model has 4 parameters. The elastic parameters are normal elastic modulus E 0 and the ratio between tangential and normal elastic moduli α . The inelastic behavior depends on two parameters only: the tensile strength f t and fracture energy in tension G t . The solver is implicit quasi-static featuring Newton–Raphson iteration scheme.

2.1. Friction boundary conditions

In order to simulate and observe the e ff ect of friction during compressive testing, the in-house software has been extended by a friction boundary condition. The implementation follows regularized Coulomb law as defined for ex ample in Ref. (Wriggers and Moftah, 2006). The movement of a particle in tangential direction is described by vector u T , the tangential force is f T and the normal force magnitude is f N . As the simulation moves to the next time step, the displacements changes by ∆ u T . This invokes change in the tangential force

∆ f T = − K T ∆ u

(1)

where K T is a user defined tangential sti ff ness. The shear force magnitude is however limited by a friction coe ffi cient µ . Whenever | f T | > µ f N , then

µ f N | f T |

f T , updt = f T

(2)

Naturally the tangential force is applied on the particles adjacent to the bases of the cylinder, where the interface be tween specimen and loading platens exists (Fig. 3). The initial elastic phase is a regularized ”stick” phase of the move ment, which physically represents deformation on a micro level and precedes the development of a friction slip, de scribed by the standard Coulombs law (Wriggers and Moftah, 2006). Parameter K T was chosen as high as possible (too high values might corrupt solution of the system of linear equations), value 10 13 N / m is used in all the presented simulations. Friction parameter µ was left for identification from the experimental data. Besides the case where friction model from Eq. (2) is employed, we studied also two extreme cases. The full friction case completely restricts any tangential movement at the top and bottom loading platens, while the friction free case assumes unrestricted free slips at these planes. Necessary part of any model preparation is the identification of material parameters and calibration. Separate set of compressive experiments was done to be able to perform such identification. Three cylinders of height 125 mm and diameter 50 mm were subjected to the uniaxial compressive test. Slenderness ratio of 2.5 has been chosen on purpose to eliminate the e ff ect of friction. However, to ensure that the premise of a friction elimination is fulfilled, both models with no friction present as well as models with full friction (no lateral movement of bases particles allowed) were used for the calibration. During the experiments, load was recorded by the loading machine, while strains were obtained from two strain gauges placed opposite each other on a cylinder. In addition, displacement of the upper loading platen was recorded using two LVDT transducers as a controlling measure. Based on load–displacement (LD) curves obtained from these experiments, material characteristics were derived. The macroscopic elastic modulus of the material has been directly extracted from the slope of the averaged experimen tal LD curve. Poisson’s ratio was considered to be 0.18. Based on these numbers, mesoscale elastic parameters were determined from equations derived in Refs. (Cusatis et al., 2011a; Elia´sˇ, 2020) to be E 0 = 35 . 5 GPa and α = 0 . 24. The mesoscale tensile strength f t = 2 . 1MPa and fracture energy G t = 50 N / m were obtained via a trial and error procedure by fitting the experimental curves. The distance l min , which brings the aggregate fraction characteristic dimension into the model, has been chosen to equal 10 mm. Load–displacement curves from the experiments and averaged curve for both friction free and full friction case as obtained after the calibration of the model are shown in Fig. 2. 2.2. Identification of material parameters