A study of detonation timing and fragmentation using 3-D finite element techniques and a damage constitutive model Page: 3 of 11
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umes of different cracks. To this end they define F
by
F = 1-exp(-aCd) 2.1
where a = 16/9.
In order to relate stress to strain we will generate a
system of equations which can be solved for the
effective elastic moduli of the cracked medium. It is
convenient to introduce a damage parameter, D,
defined byD = f(ve) F
2.2
where ve is the effective Poisson's ratio. Ke is the
effective bulk modulus of a cracked medium and is
given in terms of the undamaged bulk modulus K byKe = (1 -D)K
2.3
The crack density, Cd, can be related to an average
flaw size, a, byCd = TNa3
2.4
where N is the number of active cracks and Y is a
proportionality ratio.
At this point, it should be noted that there is variety
of assumptions which can be made as to the form of
N, and there is almost no agreement as to the proper
form for a. Kipp and Grady, 1980, and Kuszmaul,
1987a, assume that the number of cracks activated at
a volumetric strain s is described by a Weibull dis-
tribution of the form.
N = km 2.5
where k and m are material dependent constants and
the volumetric strain, a, is one third of the time inte-
gral of the trace of the deformation tensor, d with s
being positive in tension. Equations 2.5 and 2.4
implyCd = ksma3
2.6
Based on energy considerations at high strain rates,
Grady, 1983 derives the following expression for the
nominal fragment radius, r, for dynamic fragmenta-
tion of a brittle materialr / K 3
r =2 pcR _2.7
Here KIC, p and c are the fracture toughness, den-
sity and sound speed of the undamaged material and
R is the strain rate, which is assumed in the deriva-
tion to be both constant and large. This is an average
fragment radius for the global response of a uni-
formly expanding sphere. We will assume that the
local average flaw size, a, is proportional to the
value of r appropriate to the local strain rate.
In order to apply equation 2.7 to the case where the
strain rate is not constant, Taylor, Chen and Kusz-
maul, 1986, replace the constant strain rate, R, in
equation 2.7 with the maximum strain rate, Rmax,
which the material has experienced. Making some
assumptions about the maximum strain rate and
combining equations 2.6 and 2.7 yields an expres-
sion for Cd based on measurable material parame-
ters.C _ 5kem ~ KIC 2
Cd 2 _ p cRmax2.8
In the constitutive model implementation, equations
2.1 through 2.8 form the basis for derivation of a
coupled system of ordinary differential equations
which can be integrated to define the response of the
damaged material.
The material parameters for granite were measured
by Olsson, 1989, and Chong et al, 1988, are listed in
Table 2.1.
Table 2.1: Granite Material Properties
Density p = 2680 kg/m3
Youngs' Modulus E = 62.8 GPa
Poissons' Ratio v = 0.29
k k = 5.3 x 1026/m3
m 6.0
Fracture Toughness KIc = 1.68 MPa F
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Preece, D.S. & Thorne, B.J. A study of detonation timing and fragmentation using 3-D finite element techniques and a damage constitutive model, article, March 1, 1996; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc665332/m1/3/: accessed April 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.