Page 288 - Handbook of Materials Failure Analysis

284 CHAPTER 12 A nonlocal damage-mechanics-based approach

where D and σ k are the parameters of the Rousselier’s model and are constants for a
material, p is hydrostatic pressure, q is von-Mises equivalent stress, R(ε eq ) is material
resistance (which is a function of von-Mises equivalent strain ε eq ). With loading, the
void volume fraction evolves from the initial void volume fraction f 0 (volume frac-
tion of eligible second-phase particles responsible for nucleation of voids upon plas-
tic deformation) in the material. At a critical void volume fraction f c , the voids
coalesce with each other and at the final void volume fraction f f , the material points
loses its stress carrying capability. Hence, the above three void parameters (i.e., f 0 , f c ,
and f f ) are also the material properties of the damage model. For solving the boundary
value problem of the nonlocal damage continuum, one needs to solve the partial dif-
ferential Equation 12.3 along with the mechanical equilibrium equation
r σ + f b ¼ 0 (12.5)
and the associated boundary conditions, where σ is the Cauchy stress tensor and f b is
the body force per unit volume. For the nonlocal damage degree of freedom, the addi-
tional (i.e., Neumann) boundary condition is used and is expressed as
_
rd n ¼ 0 (12.6)
Γ d
where nj is normal to the boundary Γ f . By discretizing the weak forms of the gov-
Γ f
erning differential Equations 12.3 and 12.5, we obtain the FE equations in matrix
form as [17]
ext int
K uu + K NL K ud Δ^ u ¼ f m f m (12.7)
K du K dd Δd ^ f int
d
It may be noted that the stiffness terms K ud , K du ,and K dd in the element stiffness matrix
are contributions of the nonlocal formulation. The matrix K uu represents the conven-
tional stiffness of the FE and it represents the mechanical stiffness, which corresponds
to the relationship of the nodal mechanical forces and nodal displacement vectors.
Similarly, K ud represents the matrix, which produces nodal damage vector cor-
responding to applied mechanical force vector. The matrix K du produces nodal dis-
placement vector corresponding to an applied damage force vector and K dd produces
int
nodal damage vector corresponding to applied damage force vector, respectively. f m
ext
and f m are the internal and external mechanical force vectors, respectively, whereas
int
f d is the internal damage force vector. The above-described nonlocal formulation of
the Rousselier’s damage model was implemented in an in-house FE-based code and
used for numerical simulation of fracture behavior of various types of specimens as
discussed in the later sections. For simulation of probability of cleavage fracture in
the DBT regime, the nonlocal damage model was combined with Beremin’s model
for cleavage fracture which is described briefly in the following section.

3 BEREMIN’S MODEL FOR CLEAVAGE FRACTURE

Beremin’s model [20] for cleavage fracture is based on the weakest link concept
where the probability of fracture can be represented in general as follows.