# Sensitivity of void mediated failure to geometric design features of porous
metals

G.H. Teichert  University of Michigan, Ann Arbor, MI 48109 M. Khalil  Sandia
National Laboratories, Livermore, CA 94551 C. Alleman  Sandia National
Laboratories, Livermore, CA 94551 K. Garikipati  University of Michigan, Ann
Arbor, MI 48109 R.E. Jones  Sandia National Laboratories, Livermore, CA 94551

###### Abstract

Material produced by current metal additive manufacturing processes is
susceptible to variable performance due to imprecise control of internal
porosity, surface roughness, and conformity to designed geometry. Using a
double U-notched specimen, we investigate the interplay of nominal geometry
and porosity in determining ductile failure characteristics during monotonic
tensile loading. We simulate the effects of distributed porosity on plasticity
and damage using a statistical model based on populations of pores visible in
computed tomography scans and additional sub-threshold voids required to match
experimental observations of deformation and failure. We interpret the
simulation results from a physical viewpoint and provide statistical models of
the probability of failure near stress concentrations. We provide guidance for
designs where material defects could cause unexpected failures depending on
the relative importance of these defects with respect to features of the
nominal geometry.

###### keywords:

plasticity , failure , damage , porosity , additive manufacturing ,
microstructural variability.

††journal: IJSS

##  1 Introduction

Metal additive manufacturing (AM) is a relatively recent and now widespread
manufacturing process that enables new levels of geometric complexity in
designs. In its current state of development, issues with uncontrolled
porosity and surface roughness [1] prevent AM from being adopted for certain
high strength and high consequence applications. Furthermore, depending on the
scale of a part relative to the particles comprising the granular feedstock,
adherence to the designed geometry can be poor [2, 3]. For example, surface
roughness can be comparable to small, high curvature geometric features that
result from sophisticated topology optimization and more mundane corners and
fillets [4, 5, 6, 7, 5, 8], preventing the precise manufacture of some
designs.

In our previous work [9] we developed a model of porous metal deformation and
failure tuned to experimental data from AM 17-4H steel [10]. That work
employed Bayesian techniques to represent the observed variable mechanical
response due to porosity and other microstructural effects. In this effort we
utilize the previously calibrated model together with the process we developed
to generate porosity realizations to examine how stress concentrations due to
geometric design features interact with material porosity. To establish
quantitative relationships between geometric and material effects, we focus on
a double U-notched specimen, a test geometry commonly used to explore the
influence of designed geometric features and inherent defects on ductile
failure. With a porous material there is a competition among the stress
concentrations induced by notches and other geometric features and those due
to the distributed porosity, which adds complexity to the design process. In
this work we quantify the probability of unexpected failure using statistical
models based on a large sample of the porosity realizations. We quantify the
probability of failure as a function of the notch geometry, associating
certain features of the failure distribution with the stress-inducing notches
and others with the background density of pores.

Although we do not apply reliability methods to the data, the work falls in
the general context of design under uncertainty. There is a large body of
literature on the probabilistic failure of brittle materials [11, 12, 13, 14,
15], in large part focused on the applicability of the Weibull and related
distributions. This is a mature field of study that has recently been applied
to advanced materials, such as silicon based micro-electro-mechanical systems
(MEMS) [16, 17, 18, 8]. There has also been focus on the uncertainty due to
the ductility and porosity of metals, AM or otherwise. In fact, the
uncertainty of the mechanical response of porous metals such as cast iron has
long been been a topic of interest [19, 15, 20]. Notably, Yin et al. [21]
presented an integrated framework to perform design under material
uncertainty. They propagated the mean and standard deviation of damage based
on a random field of initial damage and demonstrated a reliability-based
method with the design of a cast control arm. Gurson-type constitutive models
that they, and we, have employed have shown success in representing the
complex failure process [15, 22, 23]. For instance, Becker et al. [15]
compared predictions of a micro-mechanically based model to tensile data for
steel U-notched cylinders. The model predictions up to and including failure
compared favorably to the experimental data. Allison et al. [22] collected
comprehensive experimental data on the effects of porosity on the plasticity
and damage of FC-0205 steel alloy made with powder metallurgy. The associated
modelling effort focused on the complex behavior of nearby void coalescence
leading to failure. Shakoor et al. [23] studied the influence of particle
debonding and fragmentation on void coalescence. Some other studies have
focused on the statistical aspects of failure of porous metals. Boyce et al.
[24] were able to perform tension tests of 1000 nominally identical specimens
of AM material. With this large statistical sample they were able to capture
rare failures associated with initial pore forming networks making the sample
more susceptible to failure. Solberg et al. [7] investigated the fatigue
behavior of AM 317 stainless steel with tension experiments. They discovered a
transition from surface initiated fatigue to internally initiated fatigue.
They were able to relate the trends in failure locations and behavior to the
geometry of the specimen notch and the resulting stress concentration factors.
In a related effort, we use thousands of realizations of porous AM 17-4 PH
specimens to explore the interplay of geometric design features and explicitly
modeled pores in this work. With this extensive exploration, which is
necessary to sample rare failure events, we also find a transition in failure
behavior and location with geometry and stress concentration. We utilize these
observations in a statistical model that characterizes the contributions of
the stress concentration versus the background of randomly located pores. This
model enables estimation of the likelihood of failure, either as expected near
a stress concentration versus randomly due to a critical pore not near the
stress concentration. This information has significant value in design. We
find that a significant likelihood of unexpected failures only occurs in a
limited and bounded regime of mild stress concentrations.

In Sec. 2 we review the computational model we developed and calibrated in
Ref. [9]. In Sec. 3 we describe the results of parametric study of notches of
various widths and depths and examine failure mechanisms. Then in Sec. 4 we
develop a statistical model of the failures observed across a range of notch
geometries. Finally, in Sec. 5, we summarize the findings and design
guidelines.

##  2 Simulation Method

As discussed in the Introduction, the performance of AM materials can be
significantly degraded by uncontrolled porosity. This porosity arises from a
variety of physical phenomena active in the AM process, giving rise to
distributions of defects of varying size and character. Thus it is difficult
to represent the effects of this porosity with a bulk quantity such as overall
material density. As in our previous work [9], we employ a model of
distributed porosity which accurately reproduces the sizes and spatial
correlations of defects observed in the material. This two-level model
represents pores above a given size threshold explicitly in the finite element
meshes, and sub-threshold pores are modeled with an initial damage field. The
threshold is determined by the size of pores visible with current computed
tomography (CT) resolution. In this model sub-threshold porosity evolves as
damage; and, highly damaged elements act like voids and therefore add to the
influence of the explicit porosity. In the following we will refer to
explicitly represented pores as pores and the porosity implicit in the
constitutive model as voids.

###  2.1 Constitutive Model

In the previous study [9] we calibrated an isothermal variant of the Bammann-
Chiesa-Johnson (BCJ) viscoplastic damage model [25, 26, 27, 28] to
experimental data on AM 17-4 PH stainless steel tension specimens loaded to
failure. We employ that calibrated model to represent the deformation and
failure behavior of the specimens in this study. In the BCJ constitutive
model, stress $\bm{\sigma}$ follows a linear elastic rule:

$\bm{\sigma}=(1-\phi)\mathbb{C}(\bm{\epsilon}-\bm{\epsilon}_{p})\ ,$ (1)

where $\mathbb{C}$ is the isotropic elastic modulus tensor with components

$[\mathbb{C}]_{ijkl}=\frac{E}{(1+v)}\left(\frac{v}{(1-2v)}\delta_{ij}\delta_{kl}+\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\right)$
(2)

which depend on Young’s modulus $E$ and Poisson’s ratio $v$. Here
$\bm{\epsilon}$ is the total strain, $\bm{\epsilon}_{p}$ is the plastic
strain, and $\bm{\epsilon}-\bm{\epsilon}_{p}$ is defined as the elastic
strain. The void fraction is denoted by $\phi$, and its evolution effects a
reduction of stiffness of the material via Eq. (1).

The plastic strain ${\bm{\epsilon}}_{p}$ evolves according to the viscoplastic
flow rule:

$\dot{\bm{\epsilon}}_{p}=\sqrt{\frac{3}{2}}f\sinh^{n}\left(\frac{\sigma/(1-\phi)-\kappa}{Y}-1\right)\frac{\mathbf{s}}{\|\mathbf{s}\|}\
.$ (3)

The evolution of the isotropic hardening variable $\kappa$ is governed by:

$\dot{\kappa}=(H-R\kappa)\,\dot{\epsilon}_{p}$ (4)

with $\kappa(t=0)=\kappa_{0}$. Here,
$\mathbf{s}=\operatorname{dev}\bm{\sigma}$ is the deviatoric stress,
$\sigma_{\text{vm}}\equiv\sqrt{\frac{3}{2}\mathbf{s}\cdot\mathbf{s}}$ is the
von Mises stress, and
$\dot{\epsilon}_{p}=\sqrt{\frac{2}{3}\dot{\bm{\epsilon}}_{p}\cdot\dot{\bm{\epsilon}}_{p}}$
is the equivalent plastic strain rate. The model parameters are the yield
stress $Y$, the flow coefficient $f$, the flow exponent $n$, the hardening
modulus $H$, and the recovery coefficient $R$.

The damage due to sub-threshold porosity $\phi$ also evolves. In the
formulation, the implicit void volume fraction $\phi$ is related to the void
concentration $\eta$ and the average void size $\nu$ by
$\phi\equiv\eta\nu/(1+\eta\nu)$. To model nucleation, the void concentration
$\eta$, a number density, evolves according to:

$\dot{\eta}=\left(N_{1}\left(\frac{2^{2}}{3^{3}}-\frac{J_{3}^{2}}{J_{2}^{3}}\right)+N_{3}\frac{p}{\sigma_{\text{vm}}}\right)\eta\,\dot{\epsilon}_{p}$
(5)

where $p=1/3\,\bm{\sigma}\cdot\mathbf{I}$ is the pressure,
$J_{2}=1/2\operatorname{tr}\mathbf{s}^{2}=1/3\,\sigma_{\text{vm}}^{2}$ and
$J_{3}=1/3\operatorname{tr}\mathbf{s}^{3}$. The $N_{1}$ component responds to
shear, and the $N_{3}$ component corresponds to nucleation due to triaxiality
$p/\sigma_{\text{vm}}$. The evolution relation for $\phi$,

$\dot{\phi}=\sqrt{\frac{2}{3}}\dot{\epsilon}_{\text{p}}\frac{1-(1-\phi)^{m+1}}{(1-\phi)^{m}}\sinh\left(\frac{2(2m-1)}{2m+1}\frac{p}{\sigma_{\text{vm}}}\right)+(1-\phi)^{2}\dot{\eta}\nu_{0}\
,$ (6)

includes nucleation and growth of voids, where the damage exponent $m$, and
the volume of newly nucleated voids $\nu_{0}$, are additional model
parameters. Lastly, once the void fraction $\phi$ exceeds a threshold
$\phi_{\text{max}}$ the material, as discretized by finite elements, is
considered completely failed and acts as a pore.

The calibrated parameters are summarized in Table 1 and further details can be
found in Ref. [9].

Parameter |  | Value
---|---|---
Young’s modulus (GPa) | $E$ | 240
Poisson’s ratio | $v$ | 0.27
Yield strength (MPa) | $Y$ | 600
Initial void size ($\mu$m3) | $\nu_{0}$ | 0.1
Initial void density ($\mu$m-3) | $\eta_{0}$ | 0.001
Flow exponent | $n$ | 10
Damage exponent | $m$ | 2
Flow coefficient | $f$ | 10
Isotropic dynamic recovery | $R$ | 4
Isotropic hardening (GPa) | $H$ | 5
Initial hardening (MPa) | $\kappa_{0}$ | 460
Shear nucleation | $N_{1}$ | 10
Triaxiality nucleation | $N_{3}$ | 13
Maximum damage | $\phi_{\text{max}}$ | 0.5
Table 1: Constitutive model parameters.

###  2.2 Model geometry

As discussed in the Introduction, we selected the double U-notched tension
specimen shown in Fig. 1 since this a common geometry for mechanical tests
[29, 30, 18] and it allows us to investigate the effects of stress
concentration with a few geometric parameters, namely the notch depth $D$ and
notch radius $R$. The overall dimensions of all the specimens were fixed at
1$\times$1$\times$4 mm3, as was element size ($\approx$ 0.05 mm). As discussed
in Ref. [9], fixing the element size across all the mesh realizations
ameliorates the impact of mesh dependence due to softening on the quantities
of interest. We varied the notch radius $R$ in the range 0.05–0.45 mm and
depth $D$ over the range 0.15–0.45 mm to create a set of nominal geometries.
As will be seen in the results in Sec. 3, notch radius largely determines the
height of the potential localization zone and the acuteness of the stress
concentration, whereas notch depth influences the width of the potential
localization zone.

Explicit pores were added to each of the nominal geometries. We generated
realizations of the explicit porosity using a Karhunen–Loève process that was
tuned to reproduce the experimentally observed average porosity $\bar{\phi}$ =
0.008 and spatial correlation length $\approx$ 0.05 mm as described in detail
in Ref. [9]. The largest pore in the ensemble was 81 elements (mean radius of
0.13 mm), and the average pore size was 4.1 elements (mean radius of 0.05 mm)
across 10,000 realizations. Since this process was applied to the mesh, the
smallest explicit pores were one element. Pores were allowed reside at the
surface of the nominal geometry and acted as surface imperfections. Fig. 1
shows the nominal geometry and Fig. 2 shows sample realizations of the
explicit porosity on variations in nominal notch geometry.

Tensile loading was effected with minimal displacement boundary conditions on
the top and bottom surfaces (refer to Fig. 1) such that these boundary
surfaces remain planar but can latterly expand or contract freely. The
displacement loading rate was held fixed at 8 $\mu$m/s (0.002 1/s strain
rate).

A number of dimensionless scales are apparent in this model. The analytic
expressions for stress concentration provide a few based on the nominal
geometry, shown in Fig. 1, such as the gauge ratio $D/W$ and notch aspect
ratio $R/D$. The size of the explicit pores relative to the geometry of the
notch suggest others such as the ratio of the notch radius $R$ to that of the
largest pore.

The stress concentration $K\equiv{\sigma_{\text{max}}}/{\bar{\sigma}}$ induced
by notches of this type have been estimated analytically [31, Chart 2.4] and
takes the form:

$K_{\text{notch}}=K_{\text{notch}}(R/W,D/R)$ (7)

where $\sigma_{\text{max}}$ is the maximum stress, $\bar{\sigma}$ is the
nominal stress (without the notch), and $W$ is the width of the specimen. Our
voids are coarsely discretized and three dimensional, nevertheless the stress
concentration due to cylindrical holes [31, Chart 4.3] is relevant as an
indication of scaling of the stress concentration. These stress concentrations
are of the form

$K_{\text{void}}=K_{\text{void}}(r/W,d/r))$ (8)

where $r$ is radius of the hole, $d$ is the distance from the hole to the
surface, and $W$ is the width of the rectangular specimen. Following Fig. 3.10
in Ref. [32] for semicircular double notches, Fig. 3 shows the competing
effects of U-notches and sub-surface pores. Taken qualitatively, this analysis
shows that sharp, deep notches $R/D\ll 1$ will be the dominant effect in a
porous specimen with small pores; however, there is a cross-over point where a
large pore far from the surface can dominate. Conversely, pores play a
dominant role as stress concentrations in the regime where the notch is wide
and shallow $R/D\gg 1$. Although not quantitatively predictive for the
simulations where strong pore-pore effects are at play, these observations do
provide expectations that bear out in our simulations discussed in the next
section. While not present in the analytical solutions, the length of the
specimen $L$ relative to the dimensions of the notch also plays a role, albeit
a less significant one, as the simulations discussed in Sec. 2 will
illustrate.

Figure 1: Schematic of notch geometry.

(a) $R=$ 0.15 mm, $D=$ 0.05 mm

(b) $R=$ 0.45 mm, $D=$ 0.05 mm

(c) $R=$ 5 mm, $D=$ 0.05 mm

Figure 2: Representative realizations of notched geometries with explicit
pores  Figure 3: Elastic stress concentrations due to holes (at various
locations $d/W$) and notches (of various aspect ratios $R/D$) in a thin plate.
Horizontal axis is radius of the U-shaped notch $R$ or cylindrical pore $r$
relative to the width of the specimen $W$.

##  3 Plasticity-Damage-Porosity Simulations

With the computational model described in Sec. 2, we ran 1000–10,000
realizations of nominal geometries with different notch depth $D$ and notch
radius $R$ to explore the effects of specific porosity configurations on
failure. The simulations display a complex interplay between explicit pores
and stress concentrations caused by the geometry. We isolate physical effects
by: changing notch dimensions to alter the stress concentrations and necking
zone; and including or omitting the explicit porosity which engenders random
small scale stress concentrations. The behavior of the ensemble of
realizations is described in this section and a descriptive statistical model
of the behavior in the subsequent section.

###  3.1 Rectangular specimen

To determine the baseline effect of pores without stress concentrations we ran
10,000 explicit porosity realizations of the nominal geometry without the
double U-notch. Fig. 4 illustrates the variation of specimen stress-strain
response due to variations in 50 of the explicit porosity realizations. Given
that the average porosity is fixed, the elastic response shows little
variability across the realizations. Nevertheless there are outliers. In fact,
one rectangular specimen with several large surface voids failed at a
noticeably lower force and displacement than the others shown. There is some
variation in yield and hardening among the other specimens, but the dominant
variability is in failure strain. In our previous work [9] we explored yield
strength, ultimate strength and several quantities related to failure as
variable features of the stress-stain response. Here we focus on the failure
location as the main quantity of interest in order to understand the
phenomenology driving the variability in failure strain. The failure location
can be identified by a localized region of highly damaged material in the
model. This localization is precipitated by a rapid, approximately exponential
rise in the damage field culminating in a cascade of complete element failures
as element damage reaches the specified threshold of $\phi_{\mathrm{max}}$
nearly simultaneously across the region. At the final loading step, we
identify this spike in damage by locating all elements with damage values
within 10% of the specimen’s maximum damage. We then define the failure
location as the average of the $z$-coordinate values of these highly damaged
elements. This threshold based metric provided a more complete and sharper
picture of the failure surface than considering only elements that met the
maximum damage $\phi_{\text{max}}$ criterion.

Fig. 5 indicates that without a notch there is a nearly uniform distribution
in failure location across the specimens with slight but apparent boundary
effects. These are effects are evidently due to a reduction of spatial-
correlation effects for pores near the boundaries, which are subject to
kinematic constraints that enforce that the specimen ends remain planar and
parallel. We found that the failure locations were not highly correlated with
any obvious aspect of the porosity field, e.g. the location of the largest
void or the void closest to the surface. In fact, the correlation coefficient
of the location of the largest void and failure was only 0.26. In Fig. 6, the
failure locations have been superimposed on the explicit porosity as seen
through the thickness of the specimen, clearly illustrating the difficulty
with simple correlative analyses. While individual specimens are seen to fail
near large pores, in regions with multiple pores, or where pores are adjacent
to the specimen surface, detailed statistical analysis reveals that none of
these observations have significant explanatory power in a statistical sense.
Furthermore, analysis of the local fields in the specimens does not offer a
clear picture of any qualitative differences among the failure phenomenology
in different specimens. For example, Fig. 7 shows the failure modes for a
realization that fails at the largest concentrations of pores and one that
does not. It is evident that the interactions among the pores are strong and
multiplex, and that the stress fields are qualitatively similar from one
realization to the next. However, there is little to suggest a simple
mechanistic explanation for why one realization fails where pore concentration
is high, and the other does not. The evident difficulty in devising a simple
mechanistic predictor of failure location motivates the statistical treatment
developed in Sec. 4.

Figure 4: Force-displacement curves for 50 realizations of porosity in un-
notched, rectangular specimens. The inset shows a zoomed-in portion of the
force-displacement plot to better indicate where failure occurs.  Figure 5:
Histogram of failure locations with no notch for approximately 10,000
realizations of explicit porosity.  Figure 6: Failure surface and explicit
pores for six different realizations of explicit pores with no notch. The
failure surface is represented by the red line and is constructed from a
rolling average along the $x$-axis of the location of high-damage elements.
The high-damage elements are correlated with the failure location. The
grayscale heatmap represents the 2D distribution of explicit pores, with the
number of void voxels summed along the $y$-axis. The blue histograms shows the
distribution of voids along the $z$-axis.

(a) Realization 2

(b) Realization 6

Figure 7: Evolution of the von Mises stress field for two realizations where
(a) failure occurs at the largest void concentration, and (b) failure does not
occur at the largest void concentration (compare with Fig. 6). The total
displacement of the top surface at each step is shown in parentheses.

###  3.2 Notched specimens

In this section, we survey the influence of varying the nominal geometric
features of the U-notch. In contrast to the stochastic nature of the failure
processes observed in the nominally uniform specimens of the previous section,
we expect to observe effectively deterministic effects of stress concentration
due to the double U notch on the failure location. We explored a wide range of
the geometric notch parameters $R$ and $D$, from a true notch $R/D\ll 1$ with
a significant stress concentration factor to a taper $R/D\gg 1$ that only
slightly focuses the stress field. As mentioned in Sec. 2.2, the radius of
the smallest notches we examine is comparable in size to the largest explicit
voids.

Fig. 8 shows the force-displacement response of the ensemble of specimens. The
response of shallowest notches resembles those of the specimens without
notches (Fig. 4) in that there is minimal variation except in the failure
strain. A trend to fail earlier with decreasing notch radius (increasing
stress concentration and triaxiality) is apparent, and this trend is evident
for all notch depths. With increasing notch depth, in addition to experiencing
more rapid damage evolution, specimens with sharper notches also experience an
apparent delay in the onset of yield due to the higher triaxiality of the
stress field and some apparent hardening due to a transition toward lower
triaxiality during concentrated plastic deformation. For deeper notches, yield
and hardening respond more strongly to the particular arrangement of pores in
the realization as the cross-sectional area of the effective gauge section
decreases and approaches the average pore size. In general, it is apparent
that the notch depth sets the minimum cross-section and hence the mean maximum
stress and width of the plastic zone. The notch radius controls the intensity
and spatial extent of the stress concentration and hence the height and
concentration of the plastic zone and the number of voids experiencing these
conditions.

To understand the competition of the nominal geometry and material porosity in
driving failure, we use distributions of failure location as surrogates.
Intuitively, high variance in the failure location indicates a dominance of
the random porosity, while low variance indicates a dominance of the nominal
geometry. Fig. 9 shows the distribution of the failure locations across
realizations for a sampling of the nominal geometries. For most geometries the
distribution of failure is centered at the mid-plane with a roughly lenticular
locus. The failures that lie outside this region tend to be horizontally
oriented and not spatially clustered. Consistent with the interpretation of
the analytic stress concentrations illustrated in Fig. 3, we found that only
the shallowest notches had a significant likelihood of failure outside the
vicinity of the notch. In fact, Fig. 9 demonstrates that deep and/or narrow
notches with high stress concentration have little uncertainty in failure
location. This analysis suggests that the critical parameter for determining
the influence of the material porosity on failure strain is the ratio of the
notch depth to the average pore size, which is an indicator of the relative
stress concentrations as well as the extent of the region of increased stress.
If this parameter is much less than unity, the material porosity is
significant influence on failure location and should be considered wherever
failure analysis is required. For larger values of the ratio of the notch
depth to the average pore size, we will show analysis based solely on the
stress concentration of the notches is sufficient.

Figure 8: Force-displacement curves for range of $R$ and $D$. Note the range
of vertical and horizontal axis changes across the panels.  Figure 9: Plot
presenting the range of failure surfaces for 50 explicit void realizations for
each notch geometry. Each colored line represents a different realization and
is constructed from a rolling average along the x-axis of the location of
high-damage elements. The high-damage elements are correlated with the failure
location. The histogram in each plot shows the distribution of high-damage
elements across all realizations for each geometry.  Figure 10: Ensemble
averages of the final damage field for notches with various radii ($D=$ 0.05
mm), averaged over approximately 1,000 realizations of explicit porosity. The
damage field was mirror-symmetrized about the horizontal midline of the
specimen to mitigate slight biases in the results due to asymmetries in the
top and bottom halves of the meshes.

We now shift focus to specimens with shallow notches/tapers where the failure
location is more unpredictable. For the collection with $D=$ 0.05 mm, Fig. 10
shows the ensemble averaged damage fields after failure has occurred for
various geometries and how the localized zone responds to the width of the
notch. For sharp (small $R$) notches the damage zone is localized. As the
notches widen so does the damage zone but only up to a point, at which the
zone remains effectively constant width. This can also be observed in the
progression of histograms shown in Fig. 11. The failures are distributed over
the specimen for the notches with smaller radii, but are generally contained
within the damage zone as the radii get larger, $R\geq 7.5$ mm.

Fig. 11 shows the reduction of failure locations to a one-dimensional
distribution along the long/$z$ coordinate of the specimens. Generally
speaking the distribution is narrow and peaked for the narrow notches and
correspondingly broad for the wide notches. However, for the narrowest notches
there is a significant background distribution of failures outside the notch.
As with the damage field for the broadest notches the distribution width does
not directly scale with the width of the notch and eventually remains
essentially constant for the widest notches/tapers. In fact the transformation
of the distributions displays a complex dependence on the geometric factor
$R$. For intermediate notch widths the central peak superimposed on a fairly
uniform background transitions to a tri-modal distribution for intermediate
notch radii. For the widest notches the distribution is bell-shaped with no
failures outside the notch.

We can interpret this behavior in the context of energy dissipation. If the
notch creates only a narrow zone of damage, the power due to external loading
needs to dissipate outside this zone, and does so by creating localized
failure outside the notch. However, a wider notch creates a larger damage zone
that is sufficient to dissipate the external power supplied by the boundary
loading. Consequently, on average there is no excess of power to be dissipated
in failures outside this zone, and therefore they are unlikely to arise. In
this case, the localization stabilizes in a constant width zone sufficient to
dissipate the prescribed external power.

Clearly, by adding the notch we impose a background stress field that makes
failure more likely at certain pores. Given the non-linearity of the model the
effect is not strictly a superposition of the fields due to the notch and to
the porosity, as assumed in classical analytical models. Nevertheless it is
illuminating to try to decompose the effects by simulating the same nominal
geometry without pores. Fig. 12 illustrates how the pressure and von Mises
stress change with geometry (the fields are shown at nominal strain of 4.7%).
It is apparent that the damage distribution at this stage of the deformation
resembles the von Mises stress field for narrow notches but transitions to
resemble the diffuse, fairly uniform distribution of pressure for wider
notches. In particular the diamond pattern of the von Mises stress field for
the narrow notches appears to be correlated with the damage shown in Fig. 10
and the failure pattern in Fig. 9. This superficial resemblance, while hinting
at the complex interactions of plasticity and void evolution represented in
models such as Eq. (6), is insufficient to explain the complex, history-
dependent behavior of ductile materials under low-triaxiality loading. In
fact, the well-established concept of stress concentration, which reduces
spatial distributions of stress to a single scalar for failure analysis, tells
a conflicting story. Fig. 13a shows the range of magnitudes of the pressure
and von Mises stress over the collection of shallow notch geometries. As
expected, the extremes of the stress fields become more pronounced as $R$
decreases. We can also obtain empirical estimates of stress concentration,
plotted in Fig. 13b, where it is apparent that even the shallowest notches
exhibit noticeable concentrations. (Note that these empirical stress
concentrations are based on the elastic response, post yield these measures
can change significantly.) From this figure, it is clear that the triaxiality
(the ratio $p/\sigma_{\text{vm}}$) increases for smaller $R$, consistent with
the reduced ductilities shown in Fig. 8.

Thus, on one hand it is clear that the pressure is important in determining
the global performance of the specimen, while local behavior can be more
closely correlated with the von Mises stress. To reconcile these observations,
we develop a statistical model for the interaction of the structural geometry-
induced stress fields and the material porosity. We relate the statistical
model to design parameters for a particular geometry, demonstrating a
practical application of the model; and, we use empirical stress
concentrations, such as those plotted in Fig. 13b, to link the statistical
model back to classical concepts in damage mechanics to establish the
generality of the model.

Figure 11: Histograms of failure locations with a notch depth of 0.05 mm and
various notch radii. Each geometry has approximately 1,000 realizations of
explicit porosity. The dashed lines mark the edge of the notch.

(a) pressure

(b) von Mises stress

(c) damage

Figure 12: Fields without explicit pores for $R$ = 0.15, 0.45, 0.75, 5.0, 10.0
mm at 4.7% strain. For pressure: 1.1 MPa (blue), 4.9 MPa (red); for von Mises
stress: 6.6 MPa (blue), 11.5 MPa (red); and for damage: 0.08 (blue), 0.10
(red).

(a) mean, minimum and maximum stress (b) stress concentration

Figure 13: Stress variance and concentration for the pressure $p$ and von
Mises stress $\sigma_{\text{vm}}$. The stress concentration for the von Mises
stress is approximately $\log K=0.0069\log^{2}R-0.0870\log R+0.2602$ and for
the pressure $\log K=0.0187\log^{2}R-0.1530\log R+0.3572$.

##  4 Statistical models of failure locations

Since the explicit pore locations are unlikely to be available for design, we
develop statistical models for a probabilistic characterization of failure
location. We reduce the three-dimensional failure location to a one-
dimensional random variable representing the axial location of the centroid of
the failure region to enable robust statistical analysis with the amount of
data collected. Gaussian mixture models (GMM) are widely-used finite mixture
models used in capturing the underlying distribution of observed data in
various fields of applications [33, 34, 35, 36, 37]. However, Gaussian
distributions are symmetric and have infinite support. Here, the failure
location, normalized by the specimen length, falls within the unit interval,
and its distribution is not generally symmetric, as observed from analysis
based on kernel density estimation. Thus, we employ a mixture of beta
distributions, also known as a beta mixture model (BMM) [38, 39]. The BMM,
once calibrated with the data provided by the set realizations described in
Sec. 2, is subsequently used to distinguish between the different sources of
failure. As conveyed in Sec. 4.2, we are able to provide an estimate for the
likelihood of the failure having been caused or initiated by the presence of a
notch, as opposed to being associated with the explicit porosity.

###  4.1 Methodology

Finite mixture models [36] utilize a mixture of parametric distributions to
capture the scatter in the observed data. Although we are dealing with uni-
variate data, namely the failure location along the length of the specimen for
this application, this analysis can be directly extended to model multivariate
data, such as the full three dimensional failure location.

Denoting by $z$ the failure location normalized by $L=4$ mm to the unit
interval, we define the $K$-component uni-variate BMM as

$f(z\ ;{\bf W},{\bf A},{\bf B})=\sum_{k=1}^{K}w_{k}\ {\rm Beta}(z\
;a_{k},b_{k})\ ,$ (9)

where

${\bf W}=\\{w_{1},\ldots,w_{K}\\},\ {\bf A}=\\{{\bf a}_{1},\ldots,{\bf
a}_{K}\\},\ {\bf B}=\\{{\bf b}_{1},\ldots,{\bf b}_{K}\\}$ (10)

denote the collection of component weights (or component coefficients),
$w_{k}$, and beta PDF hyperparameters (or component parameters) $a_{k}$ and
$b_{k}$. A beta PDF is given by

${\rm Beta}(z;a,b)=\frac{1}{{\rm beta}(a,b)}z^{a-1}(1-z)^{b-1},\ a,b>0\ ,$
(11)

where ${\rm beta}(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$ is the beta function,
and $\Gamma(\cdot)$ is the gamma function. The shape of the beta distribution
depends on two hyperparameters, $a$ and $b$, which determine whether the
distribution is symmetric or highly skewed. See Fig. 14 for an example.

The calibration of the BMM of the $z$ failure locations reduces to estimating
W, A and B, subject to the normalization constraint $\sum_{i=1}^{K}w_{k}=1$.
We handle this constraint by explicitly expressing the one coefficient
$w_{K}=1-\sum_{i=1}^{K-1}w_{k}$ in terms of the others for $K>1$ and $w_{1}=1$
for $K=1$, therefore reducing the number of unknowns by 1 coefficient. We will
estimate the unknown parameters from the available $n\approx 10^{5}$
independent samples of failure location, $z_{j}$, $j=1,\ldots,n$, using the
maximum likelihood estimation (MLE) technique [40, 41], given by

$\displaystyle\left\\{{\bf W,A,B}\right\\}_{\rm MLE}$
$\displaystyle=\operatorname*{arg\,max}_{{\bf W,A,B}}\mathcal{L}\left({\bf
W},{\bf A},{\bf B}\ ;z_{1}\ldots,z_{n}\right)$
$\displaystyle=\operatorname*{arg\,max}_{{\bf
W,A,B}}\prod_{j=1}^{n}\mathcal{L}\left({\bf W},{\bf A},{\bf B}\ ;z_{j}\right)$
$\displaystyle=\operatorname*{arg\,max}_{{\bf W,A,B}}\prod_{j=1}^{n}f(z_{j}\
;{\bf W},{\bf A},{\bf B})\ ,$ (12)

or, equivalently, through maximizing the log-likelihood given by

$\displaystyle\left\\{{\bf W,A,B}\right\\}_{\rm MLE}$
$\displaystyle=\operatorname*{arg\,max}_{{\bf W,A,B}}\sum_{j=1}^{n}{\rm log}\
f(z_{j}\ ;{\bf W},{\bf A},{\bf B})\ ,$ (13)

where $f$ is given by Eq. (9).

In this context of calibrating a beta mixture model, MLE generally results in
a non-concave optimization problem with several local maxima [36]. We tackle
this optimization issue with a multi-start approach [42] whereby the Nelder-
Mead simplex method [43] is run several times (100 in this investigation) with
random starting values for the parameters. Another challenge with the BMM
arises when dealing with the $z$ samples that are 0 or 1 in value with
corresponding ill-defined likelihood function for certain parameter
realizations. This is not relevant to this investigation since the $z$ failure
locations are strictly greater than zero and less than one with all
probability due to the boundary conditions on the tension specimens.

Another practical issue with the use of finite mixture models relates to the
choice of number of components, $K$. Akaike information criterion (AIC) [44]
and Bayesian information criterion (BIC) [45] have been used to decide the
number of components for finite mixture models [46]. More generally, Bayesian
model selection may also be used, albeit with greater computational demands
[47]. In this investigation, the initial choice of $K$ was informed by AIC
and/or BIC for the model selection problem relating to the optimal number of
beta components. These criteria are readily available since they utilize the
likelihood value corresponding to the MLE estimates. The number of optimal
components varied from 1 to 5 across the various geometries ($R$ in 0.15–15.0
mm) investigated. Ultimately, we deviated slightly from the AIC/BIC results in
order to arrive at an interpretable representation of the data [48] at the
expense of a relatively small decrease in data-fit. Specifically, we decided
to apply AIC to the set of mixture models that are constructed with two
symmetric/centered (i.e. $a=b$) components and two asymmetric/off-center (i.e.
$a\neq b$) components. There is an additional constraint on the two
asymmetric/off-center components in that they are mirrored versions of each
other, with the weights being equal and the hyper-parameters $a$ and $b$ of
the first asymmetric component are equal to $b$ and $a$ of the second
component, respectively. The symmetry of the combined off-center modes is
motivated by the data and the boundary value problem. Therefore, the maximum
number of unknowns to be determined through optimization in Eq. (4.1) is 6 (4
component parameters and 2 component coefficients/weights), low enough to
justify the use of a multi-start globalization scheme of the Nelder-Mead
simplex method to solve the associated non-concave problem.

Fig. 15 shows the fitted mixture model superimposed over the histogram of
failure locations for various notch radii. The four component BMM represents
the transition from the predominance of failure for sharply notched specimens
to the nearly uniform expectation of failure for the no notch case.
Furthermore, both the broad, background propensity for failure due to the
nearly-uniform distribution of pores/voids and secondary side modes that
appear for intermediate notch radii are well captured. For illustration, the
components of the calibrated BMM, scaled by the component
coefficients/weights, are shown in Fig. 14 for the notch of radius $R=$ 0.75
mm. In what follows, we use the the term “broad” to refer to the first
symmetric component; this component is sufficient to model the no notch case,
and is thus associated with the background density. We use the term “narrow”
to refer to the second symmetric component, with parameters $a=b$ of greater
magnitude than the broad component; this component is associated with the
high-triaxiality stress concentration aligned with the center/deepest part of
the notch, as seen in Fig. 12a. We use the term “asymmetric” to refer to the
mirrored off-center components associated with the shear-dominant/inclined
stress concentration bands seen in Fig. 12b emanating from the notches. The
competitive action of these three drivers determines the evolution of the
damage fields shown in Fig. 12c. We quantify this competition in terms of the
components of the BMM in the following section.

Figure 14: Calibrated Beta mixture model components for the notch of radius
$R=0.75$ mm, refer to Fig. 15d. We associate the broad first component with
the background density of pores, the narrow second component with stress
concentration at the deepest part of the notches, and the combined
asymmetric/off-center components with the inclined, shear-dominant
concentrations emanating from the notches.

(a) No notch (b) Notch with 0.15 mm radius

(c) Notch with 0.45 mm radius (d) Notch with 0.75 mm radius

(e) Notch with 1.25 mm radius (f) Notch with 2.5 mm radius

(g) Notch with 3.75 mm radius (h) Notch with 5 mm radius

(i) Notch with 7.5 mm radius (j) Notch with 10 mm radius

(k) Notch with 12.5 mm radius (l) Notch with 15 mm radius

Figure 15: Fitted BMMs for various geometries with varying notch radius.

###  4.2 Physical interpretation

With the mixture model we can answer the engineering design question, “What is
the likelihood of failure outside the notch region as a function of the stress
intensities induced by the geometric features?” The four-component mixture
model was selected with the features of the empirical distributions shown in
Fig. 11 in mind, namely: (a) a broad, nearly uniform distribution, (b) a
narrow, peaked distribution centered at the notch, and (c) two minor, mirrored
off-center peaks near the notch edges. These component distributions can be
easily interpreted as representative of three competing physical mechanisms:
(a) statistically homogeneous stress concentrations arising from the material
porosity, (b) a high-triaxiality stress concentration at the center of the
notch driving a triaxial-nucleation$\to$growth$\to$coalescence failure
process, and (c) shear-dominant stress concentrations emanating from the notch
edges driving a shear-nucleation$\to$coalescence failure process.

Fig. 16 shows the weights associated with the three mixture components,
representing the relative importance of the three mechanisms of stress
concentration in the failure processes for the different specimen geometries.
The weights are plotted as functions of both notch radius and stress
concentration, which is inversely proportional to the notch radius per the
empirical relation shown in Fig. 13. A number of trends are evident in the
plots in this figure; the conclusions are consistent whether the results are
cast in terms of specimen geometry or in terms of stress concentration.

First, as the notch broadens the narrow component dominates while the broad
and asymmetric/off-center components become negligible as the stress
concentrations at the notch edges fade. At the same time, the volume of the
notch region grows which involves sufficient material to ensure that the
failure triggering pore/s will be in this higher stress region. In effect,
there is increasing certainty that the specimen will fail within the notch as
it becomes a taper; although some uncertainty in the precise location persists
due to the stochastic nature of the explicitly represented porosity.

Second, for increasingly sharp notches there is a trend for the narrow
component to dominate, but with significant contributions from the asymmetric
component. Although not as well resolved due to limitations imposed by the
mesh size we selected at the outset, this trend is clear for the sharper
notches. The stress concentrations at the notch edges are significant drivers
for all but the sharpest notch, where the stress concentrations at the notch
center becomes entirely dominant.

Third, there appears to be a transition regime where the notch is small enough
relative to the specimen that failures occur with significant frequency
outside the notch. This regime is where $R/L$ is on the order of $1/10$–$1$
for the notch depth of $D=0.05$ mm analyzed in this section, which translates
to stress intensities $\approx$ 1.2–1.5. In much of this regime the off-center
component is also significant, indicating that the effects of the
secondary/shear stress concentrations at the notch edges are of comparable
influence to the effects the material porosity. From a design perspective Fig.
16d indicates the likelihood of failure outside the expected region as a
function of stress concentration. Clearly this can be as high as 40%, but the
effect is confined to a finite part of the design space as parameterized by
the notch depth and radius.

These conclusions are based on a first order, additively decomposed BMM model
where failures are categorized as either due to the notches’ stress
concentrations or the background of explicit pores. A more sophisticated model
could account for correlated effects; this development is left for future
work. The calibrated PDFs themselves have great utility, for instance they can
be be integrated over any region of interest to estimate a probability of
failure in that region. Our technique could be used to analyze more general
geometries, especially those where the dominant stress concentrating features
are not known a priori . The continuity of the trends observed in the
components of the BMM and how their features track the geometric features
supports our claim that the association of the components of the BMM with
physical characteristics of the specimens is valid. In particular the standard
deviation of the component representing the primary peak becomes essentially
constant for $R/L>1.5$ in concert with our observations and energy dissipation
argument of Sec. 3.2. Since the main reason we did not apply this technique
to the full three dimensional failure location was insufficient data to
resolve a three dimensional PDF, we believe the application of the technique
in a more general setting will be straightforward.

(a) Weight of the broad, background component vs. notch radius (b) Weight of
the broad, background component vs. stress concentration

(c) Weight of the narrow, centered component vs. notch radius (d) Weight of
the narrow, centered component vs. stress concentration

(e) Weight of asymmetric, off-center component vs. notch radius (f) Weight of
the asymmetric, off-center component vs. stress concentration

Figure 16: Weights of BMM components: (a-b) broad component, being symmetric,
is intended to capture background scatter in failure location due to explicit
porosity, (c-d) peaked component, also symmetric, captures failure due to
notch and (e-f) asymmetric/off-center components capturing local cluster of
failures around notch edge. The weight reported for (e-f) is the combined
weight of the two mirrored asymmetric components. Note the notch radius and
stress concentration are plotted on a log scale.

##  5 Conclusion

We performed an extensive statistical study to investigate how a background,
uniform porosity interacts with stress concentrations due to as-designed,
nominal geometry in a ductile AM metal. Although we focused on our study on a
single specimen geometry, the U-notched tension specimen is a commonly
employed geometry and we generalized the results using the classical stress
concentration concept. The trends in the spatial distribution of failure
locations are complex, and exhibit two asymptotic regimes: (a) where the
stress concentration is high but the high stress region is small relative to
the sample, and (b) where the higher stress region is large but the stress
differences are relatively small. In the first case, a significant fraction of
the failures are outside the stress concentration region due to the intrinsic
stress concentrations of the pores themselves. In the second case, effectively
all failures occur in the notch region and the distribution of failure
locations appears insensitive to geometry for specimens with well-separated
geometric and material length scales. Between these two regimes is an
interesting regime where multi-modal distributions arise associated with the
primary stress concentration at the notch root as well as secondary stress
concentrations at the notch edges. An interpretable statistical model fitted
to the data allowed us to make these observations quantitative. We found that
the distribution of failures can be accurately described with a mixture of
four beta distributions: (a) a broad one associated with the uniform density
of pores, (b) a narrow one due to the strongest stress concentration at the
notch root, and (c) two off-center, less dominant distributions associated
with shear stress concentration regions near edges of the notch. This work
clearly demonstrates, for a relevant specimen geometry, the existence and
extent of regimes where the likelihood of unexpected failure due to latent
material defects is significant, and suggests paths to generalize these
results in ways that could be used to establish quantitative design
guidelines.

## Acknowledgments

The authors would like to acknowledge insightful discussions with Jay Foulk
(Sandia) and Brad Boyce (Sandia). Sandia National Laboratories is a
multimission laboratory managed and operated by National Technology and
Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell
International, Inc., for the U.S. Department of Energy’s National Nuclear
Security Administration under contract DE-NA-0003525. The views expressed in
the article do not necessarily represent the views of the U.S. Department of
Energy or the United States Government.

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