Issue 50

Ch. F. Markides, Frattura ed Integrità Strutturale, 50 (2019) 451-470; DOI: 10.3221/IGF-ESIS.50.38

 1 ( ) ρ R

( , C E Z

)

f

, o f

J

 , ) 2



frame F E P R Z Z , o f ( , , , f J

1

i

frame λ Z Z K E R P R E C E Z ρ R λ Z Z K E R P R E C E Z  ρ R K E R P R E frame  1 ( )  , ) ( ) (  , ) ( ) ( , m f , o f , o f , m f , o f , o t , m t λ Z Z , o t ( , , ) ( , ) ( , , ) ( , ) ( , ) i J r J i J t i frame  1 ( ) J  ( ) ( , , ) J

(11)

 , ) 2



frame F E P R Z Z , o f ( , , , r J

1

i

 , ) 2

frame F E P R Z Z , o t ( , , , t J

1

i

where

  C E Z Z t c E C E Z ( , ) ( ), ( ,

 

  Z t c E C E Z Z t c E η , o f , o t , o t ( ), ( , ) ( ) r t t

)

2

f

, o f

, o f

f

r

, o f

ν

o



  c E kc E c E c E c E   ( ) ( ) ( ), ( ) ( )

c E

( )

,

f

r

f

t

r

f

E

2

 Z Z , o f

 Z Z , o t

(12)

i

i

 λ Z Z λ Z Z ( , ) ( , )





, m t λ Z Z , o t (

,

, )

, m f

, o f

i

, m r

, o f

i

i

Z

Z

i

i

κ

+1

 ( +1)(1 ) κ ν

K E RP

2 ( )

, R ρ R K E R ( )

J

frame







frame P R E , , ) J

+ ,

(

( )



J

 π [1 ( )] t ρ R

E

μ

μ

4

2

J

J

J

Under the above assumptions, it is seen that six independent parameters, namely, E , P frame , R J , Z o,f , Z o,t and Z i , entering Eqs. (11), influence the development of a double initial curve. The degree of their influence is shown in Fig.5, where the quantities F f , F r , and F t are plotted (in juxtaposition) against each one of these six parameters. To draw this figure, a reference set-up has been considered as a basis, namely, that concerning a disc of R =5 cm, t =1cm, made of Poly-Methyl Meth-Acrylate (PMMA) ( E =3.20 GPa, ν =0.38) with k =3.12 [48], squeezed between the ISRM’s curved steel jaws ( R J =1.5 R , E J =210 GPa, ν J =0.30) for the implementation of the Brazilian-disc test [47], by an overall load P frame =15 kN. The disc was under plane stress- and the jaw under plane strain-conditions. In addition, it was assumed that Z o,f =1.5 m, Z o,t =0.7 m and Z i =1.0 m, with the second lens’ focus point located before the disc, i.e., a divergent incident light beam was considered ( E , P frame , R J , Z o,f , Z o,t and Z i were put in bold for clarity). Then, in each of the plots from Fig.5(a-f), keeping four out of the five different independent parameters appearing each time in F f , F r and F t , fixed and letting the fifth one vary within a reasonable interval, the variations of F f (red color), F r (blue color) and F t (green color) are plotted. In this context, keeping P frame , R J , Z o,f , and Z i constant, equal to the respective values of the reference set-up, Fig.5a shows the variations of F f , F r and F t against the modulus of elasticity E ; the F f,r,t =1 line is also shown as the critical value below which a double initial curve appears. Thus, for an E less than 1.25 GPa, then F f is less than 1 and a double front initial curve occurs. For an E less than 0.4 GPa, then F t is less than 1 and a double rear transmission initial curve is generated. Regarding the rear reflective initial curve, it is seen that only in the case of a material with very high compliance, i.e., with an E less than 0.02 GPa, a double initial curve appears. In Fig.5b, fixing E , R J , Z o,f , and Z i , the variation of F f , F r and F t versus P frame is shown together with the limiting line F f,r,t =1. Starting from the reflective and the transmitting initial curves on the disc’s rear face, it is seen that only an extraordinary P frame could make them split into two parts while a P frame of a value 38 kN upwards would result in a double front initial curve. In Fig.5c, fixing E , P frame , Z o,f , and Z i , the variation of F f , F r and F t is plotted versus R J . It is definitely seen, for example, that for the chosen reference set-up values, i.e., for the ISRM’s jaw with R J =1.5 R =0.075 m, a double initial curve cannot happen, as in that case all three F f , F r and F t values are bigger that 1; but as R J decreases a double front initial curve appears for R J =0.067 m, a double transmitting initial curve occurs for R J =0.06 m while a double rear reflective curve is generated for R J =0.054 m. In Fig.5d, fixing E , P frame , R J , Z o,f and Z o,t , the variation of F f , F r and F t is plotted versus Z i . It is seen that for a value Z i =0.07 m downwards a double rear reflective initial curve appears, for a value Z i =0.19 m downwards a double transmission initial curve appears while for a value Z i =0.52 m downwards a double front initial curve occurs. In Fig.5e, fixing E , P frame , R J and Z i , the variation of F f and F r is plotted against Z o,f . As it is seen, placing the front reference screen at a distance Z o,f less than 0.6 m from the disc’s middle section, results always in a double front initial curve whereas a rather unmaterialized distance Z o,f , less than 0.07 m, is required to pump out a rear double reflective initial curve. Finally, in Fig.5f, fixing E , P frame , R J and Z i , the variation of F t is plotted against Z o,t . It is seen that placing the rear reference screen at a distance Z o,t less than 0.18 m from the disc’s middle section, results always in a double rear initial

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