Application of Neural Network and Finite Element Method for Multiscale Prediction of Bone Fatigue Crack Growth in Cancellous Bone - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 16

used in a compressive region, it is necessary to model the self-contact in the gap
created the element removal. The alternative is to keep the elements, but to reduce
their stiffness to a low, but not null value. In compression, the critical damage
value at fracture was set to D c ¼ 0 : 5[ 19 , 20 ]. The kill element technique was
implemented into Abaqus/Standard via the user subroutine UMAT.
With the concept of CDM, there is no difference between crack initiation and
propagation. Both result from the failure of an element with a characteristic
dimension (typical size of a crack). Thus, crack initiation and propagation are
studied in a unified approach [ 6 , 34 ]. Moreover, the CDM has no intrinsic material
characteristic length in the constitutive law which leads to mesh size dependence
results i.e. the crack growth depends on the FE mesh size. The average crack
lengths found in bones are typically 100 lm[ 4 , 46 ]. In the present work, this size
corresponds to the mesh characteristic length at trabeculae level of about five finite
elements (20 lm/element). Hence, to prevent mesh dependence that generally
affects the damage propagation rate, numerical fatigue fracture occurs when the
damage value reaches a critical value at a set of five serial broken elements
(a crack length of about 100 lm). Concerning the whole specimen in the case of
compressive cyclic tests, the definition of the apparent critical damage value at
failure (D c ) is rather arbitrary, varying between 0.1 and 0.5 [ 2 , 45 ]. In this study,
the apparent failure criterion was set to D c ¼ 0 : 4[ 45 ].
A major drawback of cumulative damage models is the computational cost
associated with modeling every loading cycle. In order to reduce the computation
time, the integration of the damage growth rate was based on the cycle blocks
approach. In this case, the real cycle number is reduced (divided) into equivalent
cycle blocks. Damage accumulation is computed over the cycle blocks and
extrapolated over the real corresponding cycles.
Within the framework of the cycle blocks approach, fatigue damage evolution
can be obtained by:

D n þ p ¼ D n þ DD ð 16 Þ

where D n þ p is the damage at iteration n ? p where p denotes the number of
cycles in one block set, D n is the damage at iteration n and DD is the damage
increment computed for one jump of p cycles. Further, to compute the damage at
every cycle, it is possible to extrapolate the damage state using:

D n þ 1 ¼ D n þ DD
p

ð 17 Þ

To ensure convergence of the Newton-Raphson iterations of the numerical
calculation and to avoid discontinuities in the responses (NN computation), the
compressive stress applied was decomposed into several increments. In this way,
for all these increments an equilibrium solution of the system was found.