For the modelling of bulk material with very randomly varying internal
structure, reliance on the RVE approach can be misleading. By defining a
particular RVE, then the internal heterogeneous material structure of
that RVE is defined, and then, as a result of the boundary condition
symmetry assumptions, it is replicated across the infinite material
domain. Thus the model is one of a patterned structure and not a random
one. The same is also true for the case of a mixed dimensional problem,
one concerned with bulk properties and also surface features or lead
crack propagation, the assumptions made for boundary conditions in RVE
modelling no longer apply. The reason for this becomes obvious if one
again considers the two neighbouring notional RVEs: one is comprised
only of bulk material, but the other includes a free surface. It is now
clear that the assumption of similar levels of loading and deformation
in these two RVEs is inappropriate, and so to make any assumption about
stress-strain transfer at the boundary would be erroneous. In such a
case, it is necessary to make a much larger domain model, and pad the
boundaries with excess material, so that the areas of interest within
the model are a long way away from the influence of any inappropriate
boundary condition [56-58].
Surface roughness and surface effect
representation
It is well-known that in fatigue coupon testing, the quality of the
surface finish has a significant effect on the fatigue life achieved
[33-34]. In classical stress analysis, it is well-known that surface
notches create stress concentrations or raisers, and solutions for many
particular geometrical shapes have been tabulated [59]. More
recently, it has been suggested that surface roughness can be assessed
using the same approach used for short cracks [60].
In order to replicate these observations in a model, it is necessary to
have a means to characterise the surface texture of a typical
engineering component [61]. Other authors have measured surface
texture directly using a variety of methods, the highest precision
method currently being Atomic Force Microscopy [62]. A recent review
of surface texture of additively manufactured materials has been
reported by Townsend et al. [35], providing many images of
many surface textures pre- and post-finishing processing. Another recent
report, by Triantaphyllou et al. , [36] focusses on metrology
methods and provides some contrasting information. It is believed that
surface roughness can be considered to be fractal, with similar
geometric features appearing at different length-scales, and this is
certainly a useful starting point for generating models of surface
roughness [63-64]. Thus, although it is recognised that surface
roughness is particularly significant, it is not entirely clear how
fractal, or sub-surface phenomenology, such as surface braking cracks or
porosity, should be included.
Material grain structure and methods of
manufacture
It is reasonable to consider that the final surface texture of an
engineering component might be significantly influenced by the
particular material in question, the materials processing, the method of
manufacture, any finishing techniques applied, and also any
environmental effects to which it might have been subjected. This very
rapidly leads an unwieldly set of parameters, each of which might have a
relatively greater or lesser influence on the actual surface texture.
In conventional metallic component manufacturing, the materials
processing leads to the development of a particular crystal grain
structure. It is well-known that particular materials with particular
grain structures have better fatigue performance than others that are
chemically similar but structurally different [34]. There is also a
consequence to the surface texture: the nature and typical size of the
grain structure will have an effect on the particular surface finish
that is obtained following subtractive processes such as machining,
grinding or polishing. Thus, it is easy to see how the connection
between fatigue performance and surface finish can become conflated with
fatigue performance and grain structure. As a result, there are a number
of authors developing detailed FEA models of crystal grain structure,
but without including surface texture modelling [53-55]. The results
of these analyses are qualitatively interesting and suggest
phenomenological processes in the development of failure, but perhaps
show only a part of the overall picture.
In Additive Manufacturing (AM), the process is quite different to the
conventional processes, and it is recognised [65] that AM processes
produce microstructures that are different from those of conventionally
manufactured materials. There are many different forms of AM, so it
might be expected that the material produced would be quite different;
however, it does seem from the consistency in the test evidence [5-12,
15-22, 66] that there is an implicit connection between microstructure
and surface texture. Therefore, differences might be accounted for by
parameter choice, rather than being due to significant differences in
physics.
Surface geometry representation in fatigue
modelling
Conventional fatigue theory is often based on the assumption of an
initial crack, like EIDS. Gorelik suggests that the surface geometry can
stand in place of the initial crack [60], but, as discussed above,
it is still not clear quite what surface measurement would provide the
associated EIDS. Describing a surface or a crack surface as“fractal” implies a non-integral dimension: something between a
surface and a volume. The immediate sub-surface of a piece of material
may contain flaws, porosity, and perhaps immediate sub-surface cracks.
Considering the latter, before the surface is broken, these would be
sub-surface phenomena, but as soon as the crack breaks the surface, the
entire crack becomes part of the surface. Physically, these two
situations are similar, so it seems that our understanding and
representation of “surface geometry” might need to include the
phenomenology of the immediate sub-surface porosity.
It has also been suggested [67] that, for the damage tolerant design
of an AM part, the use of an EIDS of 1.27 mm is sufficiently large that
the effect of surface roughness and near surface porosity can be
ignored.
In the description here, we follow the ASTM E647-13a [68]
definitions of “small” and “long” cracks. For small
naturally occurring cracks, the influence of the microstructural size on
crack growth has been found to be minimal [15]. For long cracks, the
grain size can influence the crack growth rate significantly; however,
since the focus in the present paper is on the surface geometry
representation, and since, for small cracks, the effect of grain size is
generally small [15], this suggests that modelling individual grains
within the sub-surface would be unnecessary, at least in the first
instance.