Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

242

Given the range of physical characteristics of copper and its alloys ( E 0 = 120 − 150 [GPa] and σ m = 250 − 1000 [MPa]) it is obvious that, among the three options offered in Table 6.3, the most favorable is ( p , q ) = ( 7 , 14 ) . It is important to emphasize that this selection is not unique: the similar values of E 0 and σ m could be obtained with slightly different combinations of parameters ( p , q ) from which, on the other hand, different values of fracture toughness of materials on the macro-scale can be derived. Accordingly, the analyst has an additional "degree of freedom" to adjust the physical and mechanical properties of the target material to a certain extent. Another potential that has found wide application in PD [62], for simulations of the behavior of brittle materials with random mesostructure, is a combination of Born-Mayer (6.26) 1 and Hookean (6.26) 2 potentials.

k i j r 2 0 i j ( B − 2 ) k i j r 0 i j ( B − 2 ) 1 2 k i j r

i j , ¯ r i j < 1;

1 B

φ r ( ¯ r

e B ( 1 − ¯ r i j ) − ¯ r − 1

i j ) =

φ a ( ¯ r

2 0 i j ( ¯ r i j − 1 )

2 , ¯ r

i j ≥ 1;

i j ) =

(6.26)

i j , ¯ r i j < 1;

f r ( ¯ r

e B ( 1 − ¯ r i j ) − ¯ r − 2

i j ) =

f a ( ¯ r

i j ) = k i j r 0 i j ( ¯ r i j − 1 ) , ¯ r i j ≥ 1 .

In expression for this hybrid potential (6.26), superscripts r and a designate, respec tively, a repulsive and attractive branch of interaction, k i j bond stiffness (related to modulus of elasticity by (6.27), while the adjusting parameter B defines the slope (steepness) of the repulsive wall (Figure 6.13, on page 243). The parameter B is, in principle, identifiable from shock experiments (e.g., it is conceptually related to the particle-velocity multiplier in the linear form of the ballistic equation of state) [62].

Table 6.3: Parameters P and Q of potential (6.24) 1 , modulus of elasticity and tensile strength determined for different pairs of ( p , q ) parameters shown in Figure 6.12. The reference [61] offers a more precise estimate of these quantities as well as a change in the modulus of elasticity with increasing initial interparticle distance in a given range which is conveyed here in the last line.

( p , q )

(3,5)

(5,10)

(7,14)

2 . 5 · 10 7 1 . 8 · 10 6 1 . 1 · 10 5

P

Q 9 . 9 · 10 5 5 . 7 · 10 2

1 . 4

E 0 [ GPa ] σ m [ MPa ]

15. 90.

70 .

150 . 440 .

270 .

∆ E 0 [ % ] r 0 : ( 0 . 1 → 0 . 5 ) cm

-15.

-16.

-16.