# A Unified Empirical Equation for Determining the Mechanical Properties of
Porous NiTi Alloy: From Nanoporosity to Microporosity

B.N. Galimzyanov  G.A. Nikiforov  S.G. Anikeev  N.V. Artyukhova  A.V. Mokshin
<EMAIL_ADDRESS>Kazan Federal University, 420008 Kazan, Russia  Tomsk
State University, 634050 Tomsk, Russia

###### Abstract

The mechanical characteristics of a monolithic (non-porous) crystalline or
amorphous material are described by a well-defined set of quantities. It is
possible to change the mechanical properties by introducing porosity into this
material: as a rule, the strength values decrease with the introduction of
porosity. Thus, porosity can be considered as an additional degree of freedom
that can be used to influence the hardness, strength and plasticity of the
material. In the present work, using porous crystalline NiTi as an example, it
is shown that the mechanical characteristics such as the Young’s modulus, the
yield strength, the ultimate tensile strength, etc. demonstrate a pronounced
dependence on the average linear size $\bar{l}$ of the pores. For the first
time, an empirical equation is proposed that correctly reproduces the
dependence of the mechanical characteristics on the porosity $\phi$ and on the
average linear size $\bar{l}$ of the pores in a wide range of sizes: from
nano-sized pores to pores of a few hundred microns in size. This equation
correctly takes into account the limit case corresponding to the monolithic
material. The obtained results can be used directly to solve applied problems
associated with the design of materials with the necessary combination of
physical and mechanical characteristics, in particular, porous metallic
biomaterials.

###### keywords:

NiTi; porous NiTi; mechanical properties; porous structure

##  1 Introduction

Porous materials have a unique combination of physical and mechanical
properties that is not found in their monolithic analogues [1, 2, 3, 4, 5].
For example, the presence of percolating voids and a significant specific
surface area typical of porous materials provide their lower density, lower
thermal conductivity and high catalytic activity compared to non-porous
analogues [6]. The potential functional applicability of porous materials is
largely determined by their mechanical properties $\mathcal{M}$ [7, 8, 9, 10,
11]. Here the quantity $\mathcal{M}$ denotes a set of properties such as the
Young’s modulus $E$, the yield stress $\sigma_{y}$, the shear modulus $K_{s}$,
the bulk modulus $G_{b}$, etc., i.e.

$\mathcal{M}=\\{E,\,\sigma_{y},\,K_{s},\,G_{b},\,\dots\\}.$ (1)

It is well known that the mechanical properties $\mathcal{M}$ of the
monolithic crystalline or amorphous material are mainly determined by
thermodynamic conditions: the pressure $p$ and the temperature $T$ [12, 13,
14, 15].

Let us assume that there is a bulk porous material whose characteristic
dimensions – length, width and height – are comparable. In this case, the
linear size of the material can be estimated as $\sqrt[3]{V}$, where $V$ is
the volume of the material. Obviously, such specific porous materials as
porous rods, membranes, porous thin films are not included in this category.
The porosity of the bulk material can be quite unambiguously characterized by
two quantities: the pore size distribution function $P(l)$ and the interpore
partition size distribution function $P(g)$. Other characteristics of the
porous structure – such as the most probably pore shape, the type of pores,
the average pore surface area, etc. – should be directly or implicitly related
to $P(l)$ and $P(g)$. On the other hand, it is convenient to characterize the
porous structure in terms of scalar quantities. Therefore, the functions
$P(l)$ and $P(g)$ can be replaced by the pair of quantities with some loss of
uniqueness characterization:

$\bar{l}=\frac{\int l\,P(l)\,dl}{\int P(l)\,dl}\hskip
28.45274pt\text{and}\hskip 28.45274pt\bar{g}=\frac{\int g\,P(g)\,dg}{\int
P(g)\,dg},$ (2)

where $\bar{l}$ is the average linear size of the pores, $\bar{g}$ is the
average linear size of the interpore partitions. In practice, both $\bar{l}$
and $\bar{g}$ can be estimated by statistical analysis of “patterns” (pore
contours) on thin sections of the material. In addition, these two quantities
correlate with a well-known quantity such as the total porosity of the
material $\phi=V_{0}/V$, where $V_{0}$ is the volume of empty space in the
material, the total volume of which is $V$. The porosity $\phi$ is
dimensionless and $\phi\in(0;\,1)$; for $\phi=0$ we have the case of the
monolithic material. It is therefore clear that it is convenient to use the
porosity $\phi$ instead of one of the two quantities $\bar{l}$ and $\bar{g}$
to characterize a porous material [16, 17, 18, 19].

In fact, one has two physical quantities: (i) the average linear size of a
pore and (ii) the average linear pore size for a whole system. First, the
average linear size of a pore is defined as the average of all possible
distances between opposite walls of a considered single pore. The distance
between opposite walls means the length of the segment that is common to the
perpendiculars drawn from the opposite walls. Second, the average linear pore
size $\bar{l}$ for a whole system is just an average characteristic for all
the pores of the material. It can be simply defined as the normalized first
moment of the pore size distribution function $P(\bar{l})$ [see Eq. (2)]. In
the general case, we have

$\bar{l}\in(0;\,\sqrt[3]{V})\hskip 8.5359pt\text{or}\hskip
8.5359pt\bar{l}\in[\bar{l}_{0};\,\sqrt[3]{V}).$

Here, $\bar{l}_{0}$ is the quantity characterising the average interatomic
distance in the monolithic material. The value of this quantity can
approximate the size of point defects. If we define the average size of the
atoms forming the alloy as $d$, then the value of $\bar{l}_{0}$ can be
estimated as $\bar{l}_{0}\simeq(1/3)d$. For $\bar{l}=\bar{l}_{0}$ we have the
monolithic material; and for $\bar{l}\to\sqrt[3]{V}$ there is no material as
such. The range of possible values of $\bar{l}$ is limited by the value of the
porosity $\phi$ according to the following inequality:

$\bar{l}-\bar{l}_{0}\leq\sqrt[3]{\phi V}.$ (3)

On the ($\bar{l}$, $\phi$)-diagram, a porous material can only be realized
within a strictly defined range of values of these quantities, as shown in
Figure 1.

Figure 1: Diagram demonstrating the possible range of $\bar{l}$ and $\phi$ for
which a porous material is realized. Here, the average linear size $\bar{l}$
of the pores is measured in the reduced unit $\sqrt[3]{V}$ and takes values in
the range $(\bar{l}-\bar{l}_{0})/\sqrt[3]{V}\in(0;\,1)$; $V$ is the volume of
the material. The thick red line indicates the theoretical limit, which
results from equation $\bar{l}-\bar{l}_{0}=\sqrt[3]{\phi V}$. A porous
material can only be realized in the coloured region, where the condition
$\bar{l}-\bar{l}_{0}\leq\sqrt[3]{\phi V}$ is satisfied. Limit values $\phi=0$
and $(\bar{l}-\bar{l}_{0})/\sqrt[3]{V}=0$ correspond to the monolithic
material, while $\phi=1$ and $(\bar{l}-\bar{l}_{0})/\sqrt[3]{V}=1$ material is
not realized.

In the case of a porous material with non-zero values of $\bar{l}$ and $\phi$,
an arbitrary mechanical characteristic from the set $\mathcal{M}$ [see
relation (1)] will be a function of $\bar{l}$ and $\phi$, i.e.

$\mathcal{M}=\mathcal{M}(\bar{l},\,\phi).$ (4)

Almost all strength properties of the material decrease with increasing
$\bar{l}$ and $\phi$ [20, 21, 22, 23]. In general, the following conditions
can be formulated

$\frac{\partial\mathcal{M}(\bar{l},\,\phi)}{\partial\phi}\leq 0,$ (5a)
$\frac{\partial\mathcal{M}(\bar{l},\,\phi)}{\partial\bar{l}}\leq 0,$ (5b)

which characterize the decreasing strength properties with increasing porosity
and/or pore size.

The condition (5a) is taken into account by almost all known models. So, the
example of such a model is the Bert’s power law equation for porous materials
with elliptically shaped pores [24, 25, 26]:

$\mathcal{M}(\phi)=\mathcal{M}_{0}\left(1-\frac{\phi}{\phi_{m}}\right)^{K\phi_{m}},$
(6)

where $K=0.75+1.25(b/c)$. Here, the parameter $c$ is the average linear size
of the pores along the applied stress; $b$ is the average linear size of the
pores perpendicular to the direction of the applied stress; $\mathcal{M}_{0}$
is the mechanical properties of the monolithic material; $\phi_{m}$ is the
maximum possible porosity for the material. Equation (6) is a modification of
the linear Rossi expression [27], and this expression is mainly used to
describe the dependence of elastic properties on porosity. In Eq. (6), the
degree function is required for a more accurate description of the mechanical
properties at porosity $\phi>0.2$. In addition, in the case of an isotropic
porous system with isolated pores whose sizes have a random distribution, the
dependence of mechanical properties on porosity can be described by
Mackenzie’s semi-empirical equation [28]:

$\mathcal{M}(\phi)=\mathcal{M}_{0}\left(1-d\phi+g\phi^{2}\right).$ (7)

Here, the parameters $d$ and $g$ characterize the shape of the closed pores.
At the same time, Eq. (7) incorrectly describes the quantity
$\mathcal{M}(\phi)$ in the case of a porous system with open pores, due to the
difficulties associated with the determination of the linear dimensions of the
pores. There are empirical model that do not explicitly take into account the
morphology of a porous system. The parameters in such models are fitted
through approximation of empirical data for a sample with a specific synthesis
protocol, which makes the application of such models more universal. An
example of such a model is the Ryshkewitch-Duckwoth exponential equation [29,
30]:

$\mathcal{M}(\phi)=\mathcal{M}_{0}\exp(-B\phi),$ (8)

where the exponent $B$ is the empirical constant. It is important to note that
this equation correctly describes the behaviour of mechanical properties at
porosity values up to $0.6$. Bal’shin power law equation is suitable for
reproducing the mechanical properties of porous materials over a wide range of
porosities [31]:

$\mathcal{M}(\phi)=\mathcal{M}_{0}(1-\phi)^{n},$ (9)

where the exponent $n$ is the parameter that depends on the type of material
and on the deformation protocol of the material. Thus, all the above equations
take into account the fact that the introduction of porosity into the material
leads to a significant decrease in the elastic, plastic and strength
properties according to the exponential or power law.

Let the porosity $\phi$ be fixed. The average linear size $\bar{l}$ of the
pores can vary over a fairly wide range, and the maximum possible pore size
$\bar{l}_{\mathrm{max}}$ cannot exceed the size of the material. For such a
fixed porosity $\phi$, the maximum possible pore size $\bar{l}_{\mathrm{max}}$
can be realized in a material with a single large pore; and a porous material
with the minimum possible pore size $\bar{l}_{\mathrm{min}}$ is manifested by
the presence of a certain number of pores with an extremely small non-zero
size, i.e. $\bar{l}_{\mathrm{min}}\neq 0$. The aim of the present work is to
find out how the mechanical strength properties of the porous material depend
on the pore size. It is assumed that a fixed porosity ($\phi=$const) can be
provided by both millimeter-sized pores and extremely small pores of the order
of tens of nanometres. This problem is solved using the example of such a
well-known metal alloy as NiTi under normal conditions (temperature $T=300$ K
and pressure $p=1$ atm).

The NiTi alloy is considered as a promising material for use in engineering
and medicine due to its unique combination of physical properties, including
shape memory effect, superelasticity, high corrosion resistance and
biocompatibility [18, 32, 33]. Materials based on this alloy have found wide
application as actuators in microelectromechanical devices [34], in the
aerospace industry [35, 36], in the manufacture of medical implants [37, 38].
Porous NiTi has a large specific surface area, which allows it to be used to
treat or replace damaged human organs due to its ability to penetrate and
implant biological tissue into the pore space, where this tissue develops
naturally [39]. At the same time, porous NiTi, like most other porous
materials, has a lower resistance to deformation than its monolithic analogue
[16]. It should be noted that the introduction of porosity into NiTi alloy can
change its microstructure limiting phase transitions associated with
austenitic-martensitic transitions [33]. The presence of defects and free
surface due to porosity as well as the formation of a percolating branched
structure by the crystalline matrix can limit the superelasticity and shape
memory effect [40, 41]. In the present work, we do not study the effect of
porosity on the austenitic-martensitic phase transitions because the porosity
of the considered porous NiTi samples is fixed.

The NiTi alloy is an intermetallic compound whose crystal structure has been
identified as the high-temperature B2 cubic phase (austenite). The NiTi alloy
can undergo a martensitic transition to the low-temperature orthorhombic B19’
structure (martensite) with monoclinic distortion through an intermediate
rhombohedral R-phase. This transition can be induced by heat treatment and
forces such as tension, compression and bending. The structural transition
occurs on cooling through the scheme B2$\rightarrow$R$\rightarrow$B19’, while
the reverse martensitic transition B19’$\rightarrow$R$\rightarrow$B2 is
realized on heating [39]. Due to the austenitic-martensitic phase transitions,
NiTi alloy has a shape memory effect and superelasticity - the ability to
fully recover its original shape after removal of an external load, which can
significantly exceed the yield strength of the material [42, 43]. At normal
atmospheric pressure, NiTi has the relatively high liquidus temperature
$T_{l}\simeq 1580$ K, which allows this alloy to be classified as heat
resistant [44]. Therefore, NiTi is widely used in the aerospace, automotive,
electronics and biomedical industries, for example, in the manufacture of
self-sealing thermomechanical joints, thermally sensitive mechanical
actuators, robotic elements, electronic device actuators, instruments and
implants for cranio-cerebral and maxillofacial surgery, traumatology,
orthopaedics, etc. [4, 45].

##  2 Materials and Methods

To solve the problem formulated above, we propose to consider monolithic NiTi
alloy without pores as well as porous NiTi with micron-sized pores prepared by
powder sintering method [Figure 2(a)] and nanoporous NiTi obtained through
simulation [Figure 2(b)]. The following conditions are considered here:

Figure 2: (a) Thin section of porous NiTi with micron-sized pores obtained by
powder sintering. The pores and the solid matrix are identified by scanning
electron microscopy through analysis of the thin sections of porous samples,
where the pores are black because they are filled with impregnating resin and
the solid frame is grey in color. (b) Snapshot of nanoporous NiTi obtained by
simulation using OVITO software [46]. The solid matrix and the pores are
indicated by arrows.

(I) Monolithic material. This case can be considered as the limit case of a
porous material, where the average linear pore size $\bar{l}$ approximates the
linear sizes of the point defects, i.e. $\bar{l}\simeq\bar{l}_{0}\simeq 0.09$
nm. The mechanical properties of monolithic NiTi depend on the type of crystal
lattice. For example, for B2 phase monolithic NiTi the Young’s modulus $E$,
which characterizes the ability of the material to resist tensile stress under
elastic deformation, is $E\simeq 75\pm 10$ GPa, whereas $E$ for the B19’ phase
is $E\simeq 32\pm 9$ GPa [see Table 1]. The yield strength, which determines
the limit of elastic behavior and the onset of plastic deformation on the
stress-strain curve, is $\sigma_{y}\simeq 0.8\pm 0.1$ GPa for austenite and
$\sigma_{y}\simeq 0.1\pm 0.03$ GPa for martensite. The ultimate strength,
which is the stress at which the material begins to fracture, is
$\sigma_{f}\simeq 1.9\pm 0.3$ GPa [47, 48, 49]. The ultimate strain
$\epsilon_{f}$, which corresponds to ultimate strength, is $\epsilon_{f}>15$ %
and depends on the quality of the raw material as well as on the production
protocol of the crystalline alloy (i.e. on the degree of mixing of the alloy
components, on the heating temperature, on the cooling rate of the melt).

Figure 3: Pore distributions $P(l)$ over linear sizes obtained for nanoporous
NiTi (through simulation) and for porous NiTi with micron-sized pores (by
experiment) with the porosity $\phi=0.55$ at the temperature $T=300\,$K.
Inset: image of the thin section obtained by powder sintering, where interpore
partitions (light areas) and voids (dark areas) are shown. The linear pore
size $l$ is represented as the reduced quantity $l/\bar{l}$, where $\bar{l}$
is the average linear size of the pores.

(II) Nanoporous alloy ($\phi=0.55$, $\bar{l}\simeq 4.4$ nm). According to the
classical definition [50], the nanoporous material is a porous material with
characteristic pore sizes of less than $100$ nm. In this case, we propose to
consider NiTi with the porosity $\phi=0.55$, where the average linear pore
size is $\bar{l}\simeq 4.4$ nm. This value of the quantity $\bar{l}$ follows
from the pore distribution $P(l)$ over linear sizes obtained by molecular
dynamics simulations of the nanoporous alloy and shown in Figure 3. The
obtained distribution $P(l)$ is the result of averaging over five independent
molecular dynamics simulations performed from different initial configurations
under identical thermodynamic conditions. Non-equilibrium molecular dynamics
simulations of this nanoporous alloy have been used to determine stress-strain
curves and to identify key mechanical properties. Figure 4(a) shows the
stress-strain curves for uniaxial tensile strain obtained for different porous
systems from independent molecular dynamics calculations. These stress-strain
curves vary widely due to the inhomogeneous structure of the porous systems.
Therefore, these curves are averaged and the most probable stress-strain curve
is found [in Figure 4(a) this curve is shown in red]. The Young’s modulus $E$
is determined as the slope of the linear part of this curve in the low strain
region corresponding to the elastic regime. The yield strength (or
conventional yield strength) $\sigma_{y}$ is defined as the stress
corresponding to $0.2$% plastic deformation. The ultimate strength
$\sigma_{f}$ is the maximum stress on the averaged stress-strain curve. Thus,
for uniaxial tensile strain, we find that nanoporous NiTi is characterized by
the Young’s modulus $E\simeq 17\pm 3$ GPa, the yield strength
$\sigma_{y}\simeq 0.38\pm 0.15$ GPa and the ultimate strength
$\sigma_{f}\simeq 1.1\pm 0.3$ GPa. The limit value of the deformation
$\epsilon_{f}$, at which the destruction of the nanoporous alloy is initiated,
is $\epsilon_{f}\sim 13.4$ %, which is comparable to the tensile limit of
monolithic NiTi. Details of the molecular dynamics calculations associated
with the preparation of nanoporous NiTi and the determination of mechanical
properties are given in “Appendix: Experimental and simulation details”.

Table 1: Mechanical properties and parameters of monolithic NiTi and porous NiTi: porosity $\phi$, average linear size $\bar{l}$, Young’s modulus $E$, yield strength $\sigma_{y}$, ultimate strength $\sigma_{f}$, strain at the yield strength $\epsilon_{y}$ and strain at the ultimate strength $\sigma_{f}$. The experimental values of the mechanical properties for monolithic NiTi are taken from Refs. [47, 48, 49], whereas the mechanical properties of porous NiTi with micron-sized pores and nano-sized pores are defined in the present study. Properties | Exp (monolithic) | MD (nano-sized pores) | Exp (micron-sized pores)
---|---|---|---
$\phi$ | $0$ | $0.55$ | $0.55$
$\bar{l}$, nm | $\bar{l}_{0}\simeq 0.09$ | $4.4\pm 0.6$ | $(78\pm 7)\times 10^{3}$
$E$, GPa | $75\pm 10$ | $17\pm 3$ | $5.85\pm 0.8$
$\sigma_{y}$, GPa | $0.8\pm 0.11$ | $0.38\pm 0.15$ | $(4.2\pm 3)\times 10^{-3}$
$\sigma_{f}$, GPa | $1.9\pm 0.3$ | $1.1\pm 0.3$ | $(17.8\pm 8)\times 10^{-3}$
$\epsilon_{y},\leavevmode\nobreak\ \%$ | – | $4.7$ | $0.56$
$\epsilon_{f},\leavevmode\nobreak\ \%$ | $>15$ | $13.4$ | $1.32$

(III) Porous alloy with micron-sized pores ($\phi=0.55$, $\bar{l}\simeq 78$
$\mu$m). In this case, we propose to consider NiTi with the porosity
$\phi=0.55$ and whose characteristic pore size is $\bar{l}\simeq 78$ $\mu$m.
This alloy was prepared by powder sintering (experimental details are given in
the section “Appendix: Experimental and simulation details”). The value
$\bar{l}\simeq 78$ $\mu$m was obtained from the pore size distribution $P(l)$,
which, in turn, was determined using the stereometric method of random section
planes. The pore size distribution shown in Figure 3 is the result of
averaging over different thin sections. The obtained samples show the
percolating porous structure, where the voids are in the form of channels of
approximately the same width, mainly due to the peculiarities of sintering the
mixture of powdered raw material. The mechanical properties of porous NiTi
with micron-sized pores are determined from the stress-strain curve obtained
by averaging the experimental results for different samples under identical
thermodynamic and deformation conditions. Figure 4(b) shows that the
transition region from elastic to plastic deformation is blurred at strains
$<0.5$%. This is mainly due to the fact that structural transitions between
the austenitic B2-phase and the martensitic B19’-phase occur at the tensile
strain, which causes the pseudoplastic behavior of the porous material at
strains from $0.1$ to $0.3$%. As a result, it was found that for porous NiTi
alloys the Young’s modulus is $E\simeq 5.85\pm 0.8$ GPa, the yield strength is
$\sigma_{y}\simeq(4.2\pm 3)\times 10^{-3}$ GPa and the ultimate strength is
$\sigma_{f}\simeq(17.8\pm 8)\times 10^{-3}$ GPa [see Table 1]. Such relatively
low strength characteristics in the case of samples with micron-sized pores
can be related to the quality of the initial powdered raw material and the
powder sintering conditions [51]. Namely, in the case of porous NiTi, the use
of powder metallurgy methods can lead to the formation of defects inherent in
the crystal structure, as well as the formation of nano-sized cracks at the
places of powder sintering, which significantly reduce the strength
characteristics of the porous alloy [52, 53].

Figure 4: Stress-strain curves of porous NiTi under uniaxial tension obtained
for samples with (a) nano-sized pores and (b) micron-sized pores by means of
non-equilibrium molecular dynamics (MD) simulations and experimental studies
(Exp.). Insets: thin sections of nanoporous NiTi and porous NiTi with the
characteristic pore size $\bar{l}\simeq 78$ $\mu$m, obtained from MD
simulation and from experiment, respectively. The red solid curves represent
the averaged results.

##  3 Results and discussion

Let us define the general conditions for an arbitrary mechanical property from
the set $\mathcal{M}$ as a quantity depending on the average linear size
$\bar{l}$ and on the porosity $\phi$. First, it must satisfy the conditions
(5) characterising that the value of $\mathcal{M}$ decreases with increasing
$\bar{l}$ and $\phi$. Second, in the case of the monolithic material, we have

$\bar{l}=\bar{l}_{0},\hskip 14.22636pt\phi=0\hskip 14.22636pt\text{and}\hskip
14.22636pt\mathcal{M}(\bar{l},\,\phi)=\mathcal{M}_{0}.$ (10a) In addition, if
there is only one void or an extremely small number of voids with the linear
size $\bar{l}$, which is much smaller than the size of the material
$\sqrt[3]{V}$, then the values of the mechanical properties will also approach
the values of the monolithic material:
$\frac{\bar{l}-\bar{l}_{0}}{\sqrt[3]{V}}\to 0,\hskip 14.22636pt\phi\to 0\hskip
14.22636pt\text{and}\hskip
14.22636pt\mathcal{M}(\bar{l},\,\phi)\to\mathcal{M}_{0}.$ (10b) Furthermore,
we can formulate the following boundary condition:
$\bar{l}\to\bar{l}_{0},\hskip 14.22636pt\phi\in(0;\,1)\hskip
14.22636pt\text{and}\hskip
14.22636pt\mathcal{M}(\bar{l},\,\phi)\to\mathcal{M}_{0},$ (10c) which
corresponds to the monolithic material with a set of extremely small voids.
And, finally, for the limit situation, where the material is completely
absent, we have $\bar{l}=\sqrt[3]{V},\hskip 14.22636pt\phi=1\hskip
14.22636pt\text{and}\hskip 14.22636pt\mathcal{M}(\bar{l},\,\phi)=0.$ (10d)

For the $\phi$-dependence of an arbitrary mechanical characteristic from the
set $\mathcal{M}$, we propose to use the empirical Bal’shin equation (9),
which ensures the fulfillment of the conditions (10a) and (10d). On the other
hand, we obtain that the $\bar{l}$-dependence of an arbitrary mechanical
characteristic is reproduced by the power law of the form

$\mathcal{M}(\bar{l})\propto\left(\frac{\bar{l}_{0}}{\bar{l}}\right)^{\alpha},$
(11)

where the exponent $\alpha$ depends on the corresponding quantity from the set
$\mathcal{M}$. The expression (11) correctly reproduces the values of the
mechanical properties for the cases considered above – the monolithic alloy,
the nanoporous alloy and the alloy with micron-sized pores, as shown in Figure
5. Then, taking into account the conditions (5) and (10) as well as Eqs. (9)
and (11), we find the following general equation

$\mathcal{M}(\bar{l},\,\phi)=\mathcal{M}_{0}\left(\frac{\bar{l}_{0}}{\bar{l}}\right)^{\gamma\phi}(1-\phi)^{n\left[1-\left(\bar{l}_{0}/\bar{l}\right)^{\lambda}\right]}$
(12a) on the complete fulfilment of the condition
$\bar{l}-\bar{l}_{0}\leq\sqrt[3]{\phi V},$ (12b)

which is performed for the full set of mechanical properties
$\mathcal{M}=\\{E,\,\sigma_{y},\,\sigma_{f}\\}$. In Eq. (12), the quantity $V$
is the volume of the system.

Figure 5: Dependence of the mechanical properties of NiTi on the reduced
average linear pore size $\log(\bar{l}/\bar{l}_{0})$. The solid curve is the
result of equation
$\mathcal{M}=\mathcal{M}_{0}(\bar{l}_{0}/\bar{l})^{\alpha}$, where
$\alpha\simeq 0.43$ for the Young’s modulus $E$ and $\alpha\simeq 0.22$ for
the yield strength $\sigma_{y}$ and the ultimate strength $\sigma_{f}$.

We are not aware of any models that produce an equation for the mechanical
characteristic as a function of two parameters: porosity $\phi$ and average
linear pore size $\bar{l}$, as this is accounted for in the proposed equation
(12). The closest in its physical meaning to our model (12) is the Knudsen
model with equation

$\mathcal{M}(G,\,\phi)=\frac{K}{G^{a}}\exp\left(-b\phi\right),$ (13)

which is applied to porous ceramics with polycrystalline matrix [54]. Equation
(13) allows us to approximate the dependence of the strength properties taking
into account the porosity $\phi$ and the parameter $G$ characterising the size
of crystalline grains in the material. Here, $K$, $a$ and $b$ are empirical
parameters. Equation (13) is given as the product of two contributions, where
the first contribution is defined as a power-law dependence on the grain size
$G$, and the second contribution is defined as an exponentially decaying
dependence on the porosity $\phi$. However, it should be noted that the
Knudsen model does not apply to porous alloys, but to porous ceramics [55,
56]. The Bal’shin model with equation (9) does not take into account the
dependence of $\mathcal{M}$ on the average linear pore size $\bar{l}$.
Nevertheless, it can be generalised to the following form:

$\mathcal{M}(\phi)=\mathcal{M}_{0}(\bar{l})\left(1-\phi\right)^{n(\bar{l})}.$
(14)

Comparing (12) and (14), we can see that the generalization of the Bal’shin
model represents only a special case accounted by equation (12), where
$\mathcal{M}_{0}({\bar{l}_{0}}/{\bar{l}})^{\gamma\phi}$ has a weak dependence
on $\phi$.

Equation (12) has the following features. First of all, this equation
reproduces the nature of the change in mechanical properties with the change
in the porosity $\phi$ and in the average linear pore size $\bar{l}$. In this
case, the equation depends on the range of values of $\bar{l}$ and $\phi$
defined by relation (12b). Here, the quantity $\mathcal{M}_{0}$ is the value
of the corresponding mechanical property for the case of the monolithic alloy,
and the quantity $\bar{l}_{0}$ is the average interatomic distance, as
mentioned above. In the case of monolithic NiTi alloy, we have
$\bar{l}_{0}\simeq 0.09$ nm. Equation (12) contains three dimensionless
parameters $n$, $\gamma$ and $\lambda$ which take positive values. The
parameter $n$ is identical to the parameter in the Bal’shin equation, and in
the case of NiTi it takes the value $n=3$ for all considered mechanical
properties. Furthermore, the parameters $\gamma$ and $\lambda$ characterize
how the mechanical property of the material decreases with increasing void
size: the larger the value of the parameters $\gamma$ and $\lambda$, the
faster the value of the mechanical property decreases. In this case, the
parameter $\gamma$ corrects the value of the mechanical characteristic mainly
in the limit of large values of the average linear pore size $\bar{l}$. The
parameter $\lambda$ is responsible for changing the mechanical characteristic
in the limit of small values of $\bar{l}$. It is noteworthy that in the limit
of large values of $\bar{l}$, and also when $\gamma\to 0$ and $\lambda>>0$,
Eq. (12) tends to the Bal’shin equation (9). In the case of NiTi, we find that
$\gamma=0.028\pm 0.002$ and $\lambda=0.25\pm 0.03$ for the Young’s modulus,
while $\gamma=0.65\pm 0.04$ and $\lambda=0$ for the yield strength
$\sigma_{y}$ and the ultimate strength $\sigma_{f}$.

Figure 6: Dependence of the mechanical properties of porous NiTi on the reduced average linear pore size $\log(\bar{l}/\bar{l}_{0})$ and on the porosity $\phi$: (a) Young’s modulus $E$, (b) yield strength $\sigma_{y}$ and (c) ultimate strength $\sigma_{f}$. In all these figures, the curved surface is the result of Eq. (12) obtained with different values of $\phi$ and $\bar{l}$. The red curve is the slice of the $\mathcal{M}(\bar{l},\,\phi)$ surface at the porosity $\phi=0.55$. The round markers indicate the mechanical properties found for porous NiTi with nano-sized pores and for NiTi with micron-sized pores. Table 2: Results of Eq. (12) for the mechanical properties of porous NiTi with nano- and micron-sized pores as well as the mean relative error (MRE) $(|\mathcal{M}_{pred}-\mathcal{M}_{calc}|/\mathcal{M}_{calc})\times 100\%$, where $\mathcal{M}_{calc}$ are the values of the calculated mechanical properties from Table 1, $\mathcal{M}_{pred}$ is the result of Eq. (12). Sample |  $E$, GPa |  $\sigma_{y}$, GPa |  $\sigma_{f}$, GPa
---|---|---|---
Nano-sized pores | $15.9$ | $0.183$ | $0.504$
Micron-sized pores | $5.989$ | $4.5\times 10^{-3}$ | $17.9\times 10^{-3}$
MRE, % | $\approx 4.3$ | $\approx 28$ | $\approx 28$

Figure 6 shows the Young’s modulus $E$, the yield stress $\sigma_{y}$ and the
ultimate strength $\sigma_{f}$ as functions of the porosity $\phi$ and the
average linear pore size $\bar{l}$ calculated for NiTi alloy using Eq. (12).
It can be seen from this figure that the values of the mechanical properties
approach the strength parameters of monolithic NiTi in the limit of small
$\phi$ and $\bar{l}$. On the other hand, when the pore sizes become comparable
to the size of the system, i.e. $\bar{l}-\bar{l}_{0}=\sqrt[3]{\phi V}$, the
theoretical limit is reached, which satisfies the condition (3), under which
the material does not physically exist. At the same time, the change in the
mechanical properties is more pronounced at the porosity $\phi\in(0,\,0.6]$
and/or at the average linear pore sizes $\bar{l}<10$ nm. The same result is
obtained from the analysis of Eq. (12) with the parameters $n$, $\gamma$ and
$\lambda$ defined for NiTi. Thus, the introduction of insignificant porosity
into an almost homogeneous monolithic material leads to a significant
reduction in its mechanical properties. In the limit of high porosity (i.e. at
$\phi>0.6$) and the average linear pore size corresponding to a highly porous
material with micron or millimeter sized pores, the mechanical properties are
weakly dependent on $\phi$ and $\bar{l}$. Such porosity conditions correspond
to a material in which the porous structure is percolating, and the increase
in porosity is mainly due to the thinning of the interpore partitions.

The mean relative error (MRE) between the calculated values of the mechanical
properties and the result of Eq. (12) is $\approx 4.3$ % in the case of the
Young’s modulus $E$ [see Table 2]. For the yield strength $\sigma_{y}$ and the
ultimate strength $\sigma_{f}$, the MRE takes the value $\sim 28$ %, which is
to be expected since the values of these quantities can vary over a wide range
[48, 49, 57]. In this case, the relative error can be reduced by analyzing a
large set of empirical and simulation data for NiTi samples with different
porosity and with different average linear pore size. In addition, given the
recent advances in the application of the artificial intelligence methodology
to solving physical problems, it is expected that the problem of increasing
the amount of data for the mechanical properties can be solved using machine
learning methods [58, 59].

##  4 Conclusions

Thus, in the present work, a comprehensive experimental and simulation study
of the fracture process of porous crystalline NiTi under uniaxial stress has
been investigated. In the experimental part of the work, porous NiTi with
porosity $\phi=0.55$ and micron-sized pores were synthesized. By structural
analysis of the porous samples, the distribution of pores by linear size was
obtained and the average linear pore size $\bar{l}$ was estimated. In
addition, the mechanical properties of the porous NiTi samples were calculated
in tensile tests. The statistical treatment of the obtained empirical data
allowed to determine the most probable stress-strain curve and to correctly
estimate the Young’s modulus, the yield strength and the ultimate tensile
strength. In the simulation part, the fracture process of porous crystalline
NiTi under tensile stress was considered by the method of non-equilibrium
molecular dynamics simulation. For the microscopic scales, which are related
to the scales on which the interparticle interaction potential acts, the
information about the mechanical properties has been obtained. The porous
system has the porosity $\phi=0.55$, and this system consists of nano-sized
pores. For this system, the most probable stress-strain diagram is determined
by averaging the results obtained from different independent molecular dynamic
simulations. In addition, the general methodology for determining key
characteristics of porous materials such as pore size distribution, interpore
partition size distribution, average linear pore size, average partition
thickness and porosity is discussed. It is shown that these characteristics
are related to the mechanical properties within the framework of well-known
models such as the Bal’shin and Ryshkewitch-Duckworth equations.

The analysis of the obtained results showed that the average linear pore size
$\bar{l}$ is an important factor in addition to the porosity $\phi$ in the
estimation of the mechanical properties of porous materials. A condition is
formulated which shows that a porous material can be realized only in a
strictly defined range of values of $\phi$ and $\bar{l}$. The results of the
present work have been supplemented with data for the case of a monolithic
NiTi alloy, which allowed us to obtain a completely new general equation (12),
which defines the dependence of an arbitrary mechanical characteristic on the
average linear pore size and on the porosity. Equation (12) correctly
considers the limiting case corresponding to a monolithic material (i.e., at
the porosity $\phi\rightarrow 0$) as well as the case of no material (i.e., at
the porosity $\phi\rightarrow 1$). Using porous NiTi as an example, it was
shown that Eq. (12) correctly reproduces the dependence of the Young’s
modulus, the yield strength and the ultimate tensile strength of porous NiTi
on the average linear pore size. This equation can be adapted to predict the
strength properties of porous alloys with different compositions and porosity
parameters if sufficient empirical and simulation data are available.

The results obtained in the present study allow us to approach the solution of
actual problems of fundamental character, which may include: (i) understanding
the mechanisms by which crack formation and growth occur in porous metal
alloys; (ii) understanding how porous structure parameters, such as pore size
distribution, interpore partition size distribution, pore shape, pore wall
surface area, porosity, etc., affect the elastic and plastic properties and
strength of porous metal alloys; (iii) explanation of the influence of
structural heterogeneity arising due to the non-sphericity of pores, their
orientation and distribution in the system on the mechanical properties of the
porous alloy. The solution of these problems is only possible by combining
experimental results for macroscopic mechanical properties with the results of
molecular dynamics simulations of non-equilibrium processes on microscopic
scales.

## Acknowledgment

The work was supported by the Russian Science Foundation (project No.
19-12-00022,
https://rscf.ru/project/19-12-00022/, accessed on 06 December 2023). S.G.A.
and N.V.A. are grateful to the Kazan Federal University Strategic Academic
Leadership Program (PRIORITY-2030), which supported the experimental part of
the work.

## Appendix: Experimental and simulation details

### Experimental synthesis of porous NiTi samples

Porous NiTi samples were obtained by single sintering using NiTi powders of
grade PV-N55T45 (manufacturer: Polema, Russia) with the fraction $100$-$200$
$\mu$m, which have a two-phase state: B2-austenite ($30$ wt. %) and
B19’-martensite ($34$ wt. %) [38, 33]. The powders contain phases enriched in
titanium Ti2Ni ($26$ wt. %) and nickel – TiNi3 ($3$ wt. %), as well as
metastable phases Ti3Ni4 ($7$ wt. %). Single sintering was carried out in the
electric vacuum furnace SNVE-1.31/16-I4 (manufacturer: VARP Vacuum, Russia)
for $15$ min at the temperature $1528\pm 5$ K. The temperature was controlled
by means of a tungsten-rhenium alloy thermocouple. The specified temperature-
time regime allows one to obtain experimental samples with an optimal degree
of sintering with a high quality of interparticle contacts, a regular porous
structure and the lowest possible degree of shrinkage. Sintering was carried
out at the pressure $6.65\cdot 10^{-4}$ Pa with the average heating rate $10$
K/min. The porosity of the samples was determined by the weighing method and
is equal to $\phi=0.55$. The porous samples have a cylindrical shape with the
diameter $11$–$13$ mm and the length $65$–$80$ mm. The sizes of the pores and
the interpore partitions were determined by combination of the cutting plane
method. The study of physical and mechanical properties was carried out by the
tensile method on the universal electromechanical testing machine Instron 3369
(manufacturer: Instron, USA) at a strain rate not higher than $7.5\times
10^{-6}$ m/sec. The surface of the pore walls, the microstructure and the
fracture fractograms of the samples were studied by using Quanta 3D scanning
electron microscopes (manufacturer: FEI Company, USA) in the secondary
electron mode in high vacuum and at the accelerating voltages from $20$ to
$30$ kV and the beam sizes $5$–$20$ nm.

### Computational details and methods

Molecular dynamics simulation of crystalline NiTi has been carried out for the
sample with the B2 crystal lattice consisting $12\,577$ atoms of Ni and
$12\,577$ atoms of Ti. The atoms have been placed in a cubic system with the
linear size $L\simeq 9$ nm. The interaction between Ni and Ti atoms is
determined using the second nearest neighbour modified embedded atom method
(2NN MEAM) [60, 61, 62]:

$E=\sum_{i=1}^{N}\left[F_{i}(\rho_{i})+\frac{1}{2}\sum_{j\neq
i}^{N}S_{ij}\phi_{ij}(r_{ij})\right].$ (15)

This potential reproduces the physical properties of NiTi for a wide
thermodynamic region, as has been shown previously in the works [63, 64, 65].
In Eq. (15), $E$ is the total energy of the system; $F_{i}(\rho_{i})$ is the
embedding function for the atom $i$ within the background electron density
$\rho_{i}$; the pair potential $\phi_{ij}(r_{ij})$ and the screening function
$S_{ij}$ are evaluated at the distance $r_{ij}$ between atoms $i$ and $j$; the
cutoff radius of this potential is $5.0$ Å for the considered binary system.

Porous NiTi samples were formed by cutting out the voids through by removing
some of the atoms from the monolithic system. Only those atoms that were
located inside imaginary ellipsoids were removed from the system. The
positions and spatial orientations of these ellipsoids were determined
randomly. At the same time, the ellipsoids can intersect with each other to
form open voids. The values of the semi-axes of the ellipsoids were set
randomly and were chosen from the range $[2.0;\,6.0]\,$nm. Changing the semi-
axes of the ellipsoids in the considered range makes it possible to obtain an
open-type porous structure with the porosity $\phi=0.55$, which coincides with
the porosity of the experimental samples. After cutting out the voids, the
sample was brought to a state of thermodynamic equilibrium at the temperature
$T=300\,$K and the pressure $1$ atm for the time $60$ ps. This time is
sufficient to minimise the free energy of the void surfaces and to obtain a
stable porous structure. The molecular dynamics simulation was performed using
the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)
software (Sandia National Laboratories, USA) [66]. The 3D visualisation of the
simulation results and the analysis of the porous structure were performed
using the OVITO software (OVITO GmbH, Germany) [46].

The resulting porous samples were subjected to uniaxial tensile stress along
the OX axis at the constant rate $1.0\cdot 10^{10}$ s-1. It is important to
note that such ultra-high strain rates are typical for the molecular dynamics
simulation method used to study the process of material destruction under high
dynamic loading [67, 68]. This rate differs from the experimental strain rate,
which can lead to overestimation of mechanical properties in the case of
molecular dynamics simulation, for example, as shown in Refs. [69, 70]. The
value of the stress $\sigma_{xx}$ was determined using the Irwin-Kirkwood
equation [71]:

$\sigma_{xx}=-\frac{1}{V}\left(\sum_{i=1}^{N}m_{i}\vartheta_{xi}^{2}+\sum_{i=1}^{N}\sum_{j>i}^{N}r_{ijx}F_{ijx}\right).$
(16)

Here, $V$ is the volume of the system; $m_{i}$ is the mass of the $i$th atom;
$\vartheta_{xi}$ is the $x$ velocity component of the $i$th atom; $F_{ijx}$ is
the $x$ component of the force between the particles with the labels $i$ and
$j$; $r_{ijx}$ is the $x$ component of the distance between the particles $i$
and $j$. The tensile strain $\epsilon$ was determined by the relation
$\epsilon(t)=(L_{x}(t)-L_{x}^{(init)})/L_{x}^{(init)}$, where,
$L_{x}^{(init)}$ is the length of the simulation box before deformation, and
$L_{x}(t)$ is this length at time $t$ after the start of deformation.

## References

  * [1] Thomas, A. Much ado about nothing – a decade of porous materials research. Nature Communications 2020, 11, 4985.
  * [2] Yan, M.; Xiao, Z.; Ye, J.; Yuan, X.; Li, Z.; Bowen, C.; Zhang, Y.; Zhang, D.; Porous ferroelectric materials for energy technologies: current status and future perspectives. Energy & Environmental Science 2021, 14, 6158–6190.
  * [3] Orellano, M.S.; Sanz, O.; Camarero-Espinosa, S.; Beloqui, A.; Calderón, M.; Recent advances and future perspectives of porous materials for biomedical applications. Nanomedicine 2022, 17, 197–200.
  * [4] Zadpoor, A.A. Additively manufactured porous metallic biomaterials. Journal of Materials Chemistry B 2019, 7, 4088–4117.
  * [5] Anikeev, S.G.; Kaftaranova, M.I.; Hodorenko, V.N.; Ivanov, S.D.; Artyukhova, N.V.; Shabalinam A.V.; Kulinich, S.A.; Slizovsky, G.V.; Mokshin, A.V.; Gunther, V.E.; TiNi-Based Material with Shape-Memory Effect for Surgical Treatment of Diseases of Small Intestine in Newborn and Young Children. Journal of Functional Biomaterials 2023, 14, 155.
  * [6] P.S. Liu, G.F. Chen, Porous Materials. Processing and Applications, Butterworth-Heinemann, Elsevier, 2014; pp 113–188.
  * [7] Kruzic, J.J. Bulk Metallic Glasses as Structural Materials: A Review. Advanced Engineering Materials 2016, 18, 1308–1331.
  * [8] Monogenov, A.N.; Marchenko, E.S.; Baigonakova, G.A.; Yasenchuk, Y.F.; Garin, A.S.; Volinsky, A.A. Improved mechanical properties of porous nitinol by aluminum alloying. Journal of Alloys and Compounds 2022, 918, 165617.
  * [9] Brothers, A.H.; Dunand, D.C. Amorphous metal foams. Scripta Materialia 2006, 54, 513–520.
  * [10] Galimzyanov, B.N.; Nikiforov, G.A.; Mokshin, A.V. Effect of ultrafast cooling on pore formation in amorphous titanium nickelide. Acta Physica Polonica A, 2020, 137, 1149–1152.
  * [11] Filipe, H.A.L.; Loura, L.M.S. Molecular Dynamics Simulations: Advances and Applications. Molecules 2022, 27, 2105.
  * [12] Wu, Z.; Bei, H.; Pharr, G.M.; George, E.P. Temperature dependence of the mechanical properties of equiatomic solid solution alloys with face-centered cubic crystal structures. Acta Materialia 2014, 81, 428–441.
  * [13] Elkenany, E.B.; Degheidy, A.R.; Alfrnwani, O.A. Pressure Dependence of Mechanical Properties in AlP and AlSb Semiconductors. Silicon 2019, 11, 919–924.
  * [14] Galimzyanov, B.N.; Doronina, M.A.; Mokshin, A.V. Excellent glass former Ni62Nb38 crystallizing under combined shear and ultra-high pressure. Journal of Non-Crystalline Solids 2021, 572, 121102.
  * [15] Galimzyanov, B.N.; Mokshin, A.V. A novel view on classification of glass-forming liquids and empirical viscosity model. Journal of Non-Crystalline Solids 2021, 570, 121009.
  * [16] Ji, S.; Gu, Q.; Xia, B. Porosity dependence of mechanical properties of solid materials. Journal of Materials Science, 2006, 41, 1757–1768.
  * [17] Hönicke, I.M.; Senkovska, I.; Bon, V.; Baburin, I.A.; Bönisch, N.; Raschke, S.; Evans, J.D.; Kaskel, S. Balancing mechanical stability and ultrahigh porosity in crystalline framework materials. Angewandte Chemie International Edition 2018, 57, 13780–13783.
  * [18] Galimzyanov, B.N.; Mokshin, A.V. Mechanical response of mesoporous amorphous NiTi alloy to external deformations, International Journal of Solids and Structures 2021, 224, 111047.
  * [19] Liu, B.; Pan, Y. Effect of Pore Shape on Mechanical Properties of Porous Shape Memory Alloy. Micromachines 2022, 13, 566.
  * [20] Kolesnikova, A.S. The dependence of the mechanical properties on sizes of micropore of sorbents. Reporters, Markers, Dyes, Nanoparticles, and Molecular Probes for Biomedical Applications IX 2017, 10079, 100790Q.
  * [21] Torres-Sanchez, C.; Al Mushref, F.R.A.; Norrito, M.; Yendall, K.; Liu, Y.; Conway, P. P. The effect of pore size and porosity on mechanical properties and biological response of porous titanium scaffolds. Materials Science and Engineering: C 2017, 77, 219–228.
  * [22] Su, S.-l.; Rao, Q.; He, Y.; Xie, W. Effects of porosity on tensile mechanical properties of porous FeAl intermetallics. Transactions of Nonferrous Metals Society of China 2020, 30, 2757–2763.
  * [23] Galimzyanov, B.N.; Mokshin, A.V. Cavity nucleation in single-component homogeneous amorphous solids under negative pressure. Journal of Physics: Condensed Matter 2022, 34, 414001.
  * [24] Bert, C.W. Prediction of elastic moduli of solids with oriented porosity. Journal of Materials Science 1985, 20, 2220–2224.
  * [25] Choren, J.A.; Heinrich, S.M.; Silver-Thorn, M.B. Young’s modulus and volume porosity relationships for additive manufacturing applications. Journal of Materials Science 2013, 48, 5103-5112.
  * [26] Ji, S.; Gu, Q.; Xia, B. Porosity dependence of mechanical properties of solid materials. Journal of Materials Science 2006, 41, 1757–1768.
  * [27] Rossi, R.C. Prediction of the Elastic Moduli of Composites. Journal of the American Ceramic Society 1968, 51, 433.
  * [28] Mackenzie, J.K. The Elastic Constants of a Solid containing Spherical Holes. Proceedings of the Physical Society B 1950, 63, 2.
  * [29] Ryshkewitch, E. Compression strength of porous sintered alumina and zirconia. Journal of the American Ceramic Society 1953, 36, 65–68.
  * [30] Duckworth, W. Discussion of Ryshkewitch Paper, Journal of the American Ceramic Society 1953, 36, 68.
  * [31] Bal’shin, M.Y. Relation of mechanical properties of powder metals and their porosity and the ultimate properties of porous metal-ceramic materials. Doklady Akademii Nauk SSSR, 1949, 67, 831–834.
  * [32] Greiner, C.; Oppenheimer, S.M.; Dunand, D.C. High strength, low stiffness, porous NiTi with superelastic properties. Acta Biomaterialia 2005, 1, 705–716.
  * [33] Anikeev, S.G.; Artyukhova, N.V.; Shabalina, A.V.; Kulinich, S.A.; Hodorenko, V.N.; Kaftaranova, M.I.; Promakhov, V.V.; Gunter, V.E. Preparation of porous TiNi-Ti alloy by diffusion sintering method and study of its composition, structure and martensitic transformations. Journal of Alloys and Compounds 2022, 900, 163559.
  * [34] Stachiv, I.; Alarcon, E.; Lamac, M. Shape Memory Alloys and Polymers for MEMS/NEMS Applications: Review on Recent Findings and Challenges in Design, Preparation, and Characterization Metals 2021, 11, 1–3.
  * [35] Hartl, D.J; Lagoudas, D.C. Aerospace applications of shape memory alloys. Proceedings of the Institution of Mechanical Engineers. Part G 2007, 221, 540–548.
  * [36] Costanza, G.; Tata, M.E. Shape Memory Alloys for Aerospace, Recent Developments, and New Applications: A Short Review. Materials 2020, 13, 1–16.
  * [37] Kapoor, D. Nitinol for Medical Applications: A Brief Introduction to the Properties and Processing of Nickel Titanium Shape Memory Alloys and their Use in Stents. Johnson Matthey Technology Review 2017, 61, 66–76.
  * [38] Anikeev, S.G.; Garin, A.S.; Artyukhova, N.V.; Khodorenko, V.N.; Gunther, V.E. Structural and Morphological Features of TiNi-Based Powder Manufactured by the Method of Hybrid-Calcium Reduction. Russian Physics Journal 2018, 61, 749–756.
  * [39] Aihara, H.; Zider, J.; Fanton, G.; Duerig, T. Combustion Synthesis Porous Nitinol for Biomedical Applications. International Journal of Biomaterials 2019, 2019, 4307461.
  * [40] Liu, B.; Li, Z.; Li, W.; Pan, Y.; Wu, W. Molecular dynamics simulation of the porosity effect on transformation mechanism of nanocrystalline porous NiTi shape memory alloy. Materials Today Communications 2023, 34, 105320.
  * [41] Hosseini, S.A.; Sadrnezhaad, S.K.; Ekrami, A. Phase transformation behavior of porous NiTi alloy fabricated by powder metallurgical method. Materials Science and Engineering C 2009, 29, 2203–2207.
  * [42] Sadiq, H.; Wong, M. B.; Al-Mahaidi, R.; Zhao, X. L. The effects of heat treatment on the recovery stresses of shape memory alloys. Smart Materials and Structures 2010, 19, 035021.
  * [43] Casati, R.; Saghafi, F.; Biffi, C.A.; Vedani, M.; Tuissi, A. Improved Functional Properties and Efficiencies of Nitinol Wires Under High-Performance Shape Memory Effect (HP-SME). Journal of Materials Engineering and Performance 2017, 26, 4964–4969.
  * [44] Gao, S.; Bodunde, O.P.; Qin, M.; Liao, W.-H.; Guo, P. Microstructure and phase transformation of nickel-titanium shape memory alloy fabricated by directed energy deposition with in-situ heat treatment. Journal of Alloys and Compounds 2022, 898, 162896.
  * [45] Huang, X.; Ford, M.; Patterson, Z.J.; Zarepoor, M.; Pana,C.; Majidi, C. Shape memory materials for electrically-powered soft machines. Journal of Materials Chemistry B 2020, 8, 4539–4551.
  * [46] Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO – the Open Visualization Tool. Modelling and Simulation in Materials Science and Engineering 2010, 18, 015012.
  * [47] AL-Mangour, B.; Mongrain, R.; Yue, S. Coronary Stents Fracture: An Engineering Approach (Review). Materials Sciences and Applications 2013, 4, 606–621.
  * [48] Hellberg, S.; Humme, J.; Krooß, P.; Niendorf, T.; Böhm, S. Microstructural and mechanical properties of dissimilar nitinol and stainless steel wire joints produced by micro electron beam welding without filler material. Welding in the World 2020, 64, 2159–2168.
  * [49] Jalaeefar, A.; Asgarian, B. Experimental Investigation of Mechanical Properties of Nitinol, Structural Steel, and Their Hybrid Component. Journal of Materials in Civil Engineering 2013, 25, 1498–1505.
  * [50] Max Lu, G.Q.; Zhao, X.S. Nanoporous Materials: Science And Engineering, Imperial College Press, UK, London, 2004.
  * [51] Dharmalingam, G.; Mariappan, R.; Arun Prasad Murali Effect of Sintering Parameters on Mechanical Properties of 17-Cr Ferritic ODS Steel Through Taguchi Approach. Journal of The Institution of Engineers: Series D 2019, 100, 163–176.
  * [52] Taheri, A.M.; Saedi, S.; Turabi, A.S.; Karamooz, M.R.; Haberland, C.; Karaca, H.E.; Elahinia, M. Mechanical and shape memory properties of porous Ni50.1Ti49.9 alloys manufactured by selective laser melting. Journal of the Mechanical Behavior of Biomedical Materials 2017, 68, 224–231.
  * [53] Jian, Y.-T.; Yang, Y.; Tian, T.; Stanford, C.; Zhang, X.-P.; Zhao, K. Effect of Pore Size and Porosity on the Biomechanical Properties and Cytocompatibility of Porous NiTi Alloys. PLOS ONE 2016, 10, e0128138.
  * [54] Knudsen, F.P. Dependence of Mechanical Strength of Brittle Polycrystalline Specimens on Porosity and Grain Size. Journal of the American Ceramic Society 1959, 42, 376–387.
  * [55] Gu, S.; Qi, Q.; Zhang, Y.; Ji, B.; Zhou, H.; Luo, G.-N. Mechanical strength stability of tritium breeding materials under high temperature deuterium conditions. Ceramics International 2020, 46, 1195–1202.
  * [56] Guével, A.; Rattez, H.; Veveakis, E. Morphometric description of strength and degradation in porous media. International Journal of Solids and Structures 2022, 241, 111454.
  * [57] Galimzyanov, B.N.; Doronina, M.A.; Mokshin, A.V. Neural Network as a Tool for Design of Amorphous Metal Alloys with Desired Elastoplastic Properties. Metals 2023, 13, 812.
  * [58] Galimzyanov, B.N.; Doronina, M.A.; Mokshin, A.V. Arrhenius Crossover Temperature of Glass-Forming Liquids Predicted by an Artificial Neural Network. Materials 2023, 16, 1127.
  * [59] Galimzyanov, B.N.; Doronina, M.A.; Mokshin, A.V. Machine learning-based prediction of elastic properties of amorphous metal alloys. Physica A: Statistical Mechanics and its Applications 2023, 617, 128678.
  * [60] Tsygankov, A.A., Galimzyanov, B.N.; Mokshin, A.V. Porous amorphous nitinol synthesized by argon injection: a molecular dynamics study. Journal of Physics Condensed Matter 2022, 34, 414003.
  * [61] Lee, B.-J.; Baskes, M.I. Second nearest-neighbor modified embedded-atom-method potential. Physical Review B 2001, 62, 8564.
  * [62] Ko, W.-S.; Grabowski, B.; Neugebauer, J. Development and application of a Ni-Ti interatomic potential with high predictive accuracy of the martensitic phase transition. Physical Review B 2015, 92, 134107.
  * [63] Chen, J.; Huo, D.; Yeddu, H.K. Molecular dynamics study of phase transformations in NiTi shape memory alloy embedded with precipitates. Materials Research Express 2021, 8, 106508.
  * [64] Lee, J.; Shin, Y.C. Molecular Dynamics Study of Bulk Properties of Polycrystalline NiTi. Metals 2021, 11, 1237.
  * [65] Guo, Y.; Zeng, X.; Chen, H.; Han, T.; Tian, H.; Wang, F. Molecular Dynamics Modeling of the Effect of Nanotwins on the Superelasticity of Single-Crystalline NiTi Alloys. Advances in Materials Science and Engineering 2017, 2017, 7427039.
  * [66] Thompson, A.P.; Aktulga, H.M.; Berger, R.; Bolintineanu, D.S.; Brown, W.M.; Crozier, P.S.; in ’t Veld, P.J.; Kohlmeyer, A.; Moore, S.G.; Nguyen, T.D.; Shan, R.; Stevens, M.J.; Tranchida, J.; Trott, C.; Plimpton, S.J. LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Computer Physics Communications 2022, 271, 10817.
  * [67] Li, J.; Lu, B.; Zhou, H.; Tian, C.; Xian, Y.; Hu, G.; Xia, R. Molecular dynamics simulation of mechanical properties of nanocrystalline platinum: Grain-size and temperature effects. Physics Letters A 2019, 383, 1922–1928.
  * [68] Zeng, Q.; Wang, L.; Jiang, W. Molecular Dynamics Simulations of the Tensile Mechanical Responses of Selective Laser-Melted Aluminum with Different Crystalline Forms. Crystals 2021, 11, 1388.
  * [69] Herman, J.; Govednik, M.; Patil, S.P.; Markert, B. Molecular Dynamics Simulation Study of the Mechanical Properties of Nanocrystalline Body-Centered Cubic Iron. Surfaces 2020, 3, 381–391.
  * [70] Piao, M.; Huh, H.; Lee, I.: Ahn, K.; Kim, H.; Park, L. Characterization of flow stress at ultra-high strain rates by proper extrapolation with Taylor impact tests. International Journal of Impact Engineering 2016, 91, 142–157.
  * [71] Tsai, D.H. The virial theorem and stress calculation in molecular dynamics. The Journal of Chemical Physics 1979, 70, 1375–1382.