# A note on the potentials of probabilistic and fuzzy logic

Anahita Jamshidnejad A. Jamshidnejad is with the Department of Control and
Operations, Faculty of Aerospace Engineering, Delft University of Technology,
* Corresponding author<EMAIL_ADDRESS>
###### Abstract

This paper mainly focuses on (1) a generalized treatment of fuzzy sets of type
$n$, where $n$ is an integer larger than or equal to $1$, with an example,
mathematical discussions, and real-life interpretation of the given
mathematical concepts; (2) the potentials and links between fuzzy logic and
probability logic that have not been discussed in one document in literature;
(3) representation of real-life random and fuzzy uncertainties and ambiguities
that arise in data-driven real-life problems, due to uncertain mathematical
and vague verbal terms in datasets.

###### Index Terms:

Type-$n$ fuzzy sets; fuzzy logic; probability theory.

##  I Introduction

In this paper an extensive treatment of type-2 fuzzy sets and membership
functions is presented that is more general than the ones that can be found in
literature. Moreover, the potentials of fuzzy logic and probability theory are
discussed, as well as how real-life random and fuzzy uncertainties may be
represented mathematically. The paper is organized as follows: Section II
provides an extensive discussion about fuzzy logic and the potential of this
logic in representing linguistic uncertainties. Section III discusses the
probability theory. Section IV represents various formulations of random and
fuzzy uncertainties.

##  II Fuzzy logic

Two main concepts that mathematical logic deals with are _sets_ and
_propositions_ [1]. Fuzzy logic was introduced by Zadeh in the 1960s [2, 3] to
extend the concepts of set theory and propositional calculus, which by then
were analyzed only through classical logic. Fuzzy logic is a continuous multi-
valued logic system. fuzzy logic may be considered as a generalization to the
classical logic. Linguistic variable, a concept unique to fuzzy logic, allows
this logic to serve as a basis for computations based on verbal information
[4, 5, 6]. Moreover, inspired by the procedure of reasoning of humans upon
imprecise information, FL maps an imprecise concept into one with a higher
precision [7].

A set in classical logic is a crisp concept: a mathematical object (e.g., a
number, partition, matrix, variable, …) either “belongs to” the set, i.e., the
degree of membership of the mathematical object to the set is $1$, or “does
not belong to” the set, i.e., the degree of membership of the mathematical
object to the set is $0$. Therefore, a crisp set $\pazocal{C}$ may be
expressed as a collection of mathematical objects, e.g.,

$\displaystyle\pazocal{C}=\left\\{x_{1},x_{2},\ldots,x_{n}\right\\}.$ (1)

Similarly, a proposition in classical logic is either “true” (may be
quantified by a crisp value $1$) or “false” (may be quantified by a crisp
value $0$).

In fuzzy logic, sets are fuzzy concepts [8]. Our main focus is on the general
case of type-$n$ fuzzy sets, with $n=1,2,3,\ldots$. To motivate this, we start
with an example.

### Opening example

Figure 1: Using crisp sets for quantifying _young_.

Figure 2: Using type-$1$ fuzzy sets for quantifying _young_.

Figure 3: Using type-2 fuzzy sets for quantifying _young_.

Figure 4: 3D representation of the type-2 membership function for quantifying
_young_ , represented for the specific age of 27.

$\rhd$ Suppose that based on the information “Felix is 27”, we would like to
answer to the question: _(To what extent) is Felix young?_ Since _young_ is a
qualitative concept, while $27$ is quantitative, these concepts should be
bridged by quantifying _young_ and corresponding the quantitative information
about Felix’s age with the quantified definition of _young_.

One may define _young_ with the graph represented in Figure 4, which
corresponds to a definition that categorizes people in two crisp sets, _young_
and _not young_ , i.e., people at an age below $40$ belong to the set _young_
and otherwise they are _not young_. Then, in quantitative terms, “Felix
belongs to the set _young_ with a membership degree of $1$”, and in
qualitative terms, “Felix is certainly young”.

Alternatively, _young_ may be quantified via the graph shown in Figure 4,
where instead of _certainly young_ and _certainly old_ , ages vary in a
spectrum, i.e., the degree of membership to the set _young_ varies in $[0,1]$
instead of $\\{0,1\\}$. Then, quantitatively, “Felix belongs to the set
_young_ with a membership degree of $0.9$”. Qualitatively, “Felix is to a high
extent young”. _Young_ is then quantified using a type-$1$ fuzzy set.

Next, suppose that the border of the curve that defines _young_ is not
strictly known. Figure 4 is an example where the intensity of the color black
indicates our certainty about where any point at the border of the separating
curve may lie in the given 2-dimensional plane. By just looking at the
vertical axis (which is called the _primary_ membership degree), one may reply
“Felix belongs to the set _young_ with a _primary_ membership degree in the
interval $[0.57,0.98]$”, or “Felix may to a high extent be young”. Now
consider a third dimension that quantifies the intensity of the color black
with real values between $0$ and $1$, i.e., black corresponds to $1$ and white
corresponds to $0$ (see figure 4). This dimension represents the _secondary_
membership degree. Then one can respond “Felix is young with a _primary_
membership degree varying in the interval $[0.57,0.98]$, and a _secondary_
membership degree varying in the interval $[0,1]$. For instance, “Felix’s age
corresponds to a secondary membership degree of $0.83$ for the primary
membership degree of $0.88$”. A type-$2$ fuzzy set (see [9, 10, 11] for more
details on this type of fuzzy sets) has been used to quantify _young_ for this
case.

More generally, if one interprets the word _young_ with a curve with blurry
borders that extend in $3$ (instead of $2$) dimensions, i.e., the graph of
young can be re-illustrated in $4$ dimensions considering _primary_ ,
_secondary_ , and _tertiary_ membership degrees, the qualitative word young
has actually been quantified using a type-$3$ fuzzy set. This concept can
further be extended to type-$n$ fuzzy sets for which the corresponding
membership functions can be illustrated in $n+1$ dimensions, and there are $n$
membership degrees involved. $\lhd$

### Mathematical Discussion

Next, the concept of type-$n$ fuzzy sets will be detailed mathematically. In
contrast to a crisp set, a mathematical object belongs to any type-$n$ fuzzy
set, with $n=1,\,2,\,\ldots$, with a primary, secondary, …membership degree
varying in $[0,1]$. Similarly, in fuzzy logic a proposition may “to a certain
extent” be true or false. To link the two theories of fuzzy propositions and
fuzzy sets, one may consider two fuzzy sets, the “set of true propositions”
and the “set of false propositions”. Any new proposition, which corresponds to
one or several mathematical variables called membership degrees, can belong to
both these sets at the same time with certain degrees of membership.

Generally, a membership function
$f^{[n]}:\textrm{dom}\left(f^{[n]}\right)\rightarrow[0,1]$ together with its
domain, $\textrm{dom}\left(f^{[n]}\right)$, which itself can be a fuzzy set,
characterizes a fuzzy set. For a type-1 fuzzy set,

$\displaystyle\pazocal{F}^{[1]}=\bigg{\\{}\Big{(}x_{1},\mu^{[1]}_{1}\Big{)},\Big{(}x_{2},\mu^{[1]}_{2}\Big{)},\ldots,\Big{(}x_{n},\mu^{[1]}_{n}\Big{)}\bigg{\\}},$
(2)

the corresponding type-1 membership function111A membership function is named
after the corresponding fuzzy set, i.e., they both are of the same type.
$f^{[1]}$ generates the degree of membership of mathematical objects $x_{i}$
within $\pazocal{C}$ to the fuzzy set $\pazocal{F}^{[1]}$, i.e.,

$\displaystyle
f^{[1]}:\Big{\\{}x_{1},x_{2},\ldots,x_{n}\Big{\\}}\rightarrow[0,1]:\
x_{i}\mapsto\mu^{[1]}_{i}$ (3)

or equivalently,

$\displaystyle\mu^{[1]}_{i}=f^{[1]}(x_{i}),\quad i=1,2,\ldots,n.$ (4)

In comparison with a crisp set, a type-1 fuzzy set $\pazocal{F}^{[1]}$ is
composed of pairs of mathematical objects and their degrees of membership to
$\pazocal{F}^{[1]}$ (see (2)).

###### Remark 1

One may generalize the above discussion by re-defining the crisp set
$\pazocal{C}$ in (1) as a special case of a type-1 fuzzy set, where
$\mu^{[1]}_{i}=1$, $i=1,2,\ldots,n$. This means that the corresponding type-1
membership function of $\pazocal{C}$ is the unit function. This definition may
even further be generalized (see the following discussions for more details),
i.e., a crisp set may be re-defined as a type-$n$ fuzzy set, with
$n=1,2,\ldots$, with all the membership functions of order $1,\ldots,n$ the
unit function.

For data-driven approaches in engineering problems, large amount of
information that are expressed in the human language, may be used to generate
fuzzy sets. The degree to which a fuzzy set can handle the uncertainties and
vagueness that exist in this information, depends on the type of the fuzzy
set. The main difference between type-1 fuzzy sets and fuzzy sets of type 2
and larger is in the domain of their corresponding membership functions, i.e.,
the domain of a type-1 membership function is a crisp set $\pazocal{C}$ (see
(2)), or equivalently, a type-1 fuzzy set with the unit function as its
membership function, while the domain of a type-$n$ fuzzy set with
$n=2,3,\ldots$ is a union of various fuzzy sets of type $n-1$.

More specifically, a type-2 fuzzy set $\pazocal{F}^{[2]}$,

$\displaystyle\pazocal{F}^{[2]}=\Bigg{\\{}\bigg{(}\Big{(}x_{1},\mu^{[1]}_{1,1}\Big{)},\mu^{[2]}_{1,1}\bigg{)},$
$\displaystyle\bigg{(}\Big{(}x_{1},\mu^{[1]}_{2,1}\Big{)},\mu^{[2]}_{2,1}\bigg{)},$
$\displaystyle\ldots,$
$\displaystyle\bigg{(}\Big{(}x_{1},\mu^{[1]}_{m_{1},1}\Big{)},\mu^{[2]}_{m_{1},1}\bigg{)},$
$\displaystyle\vdots$
$\displaystyle\bigg{(}\Big{(}x_{n},\mu^{[1]}_{1,n}\Big{)},\mu^{[2]}_{1,n}\bigg{)},$
$\displaystyle\bigg{(}\Big{(}x_{n},\mu^{[1]}_{2,n}\Big{)},\mu^{[2]}_{2,n}\bigg{)},$
$\displaystyle\ldots,$
$\displaystyle\bigg{(}\Big{(}x_{n},\mu^{[1]}_{m_{n},n}\Big{)},\mu^{[2]}_{m_{n},n}\bigg{)}\Bigg{\\}},$
(5)

corresponds to a type-2 membership function $f^{[2]}$, which generates the
secondary degree of membership of any mathematical object
$\Big{(}x_{i},\mu^{[1]}_{j,i}\Big{)}$ within the type-1 fuzzy set
$\pazocal{C}\times\pazocal{R}$ to the type-2 fuzzy set $\pazocal{F}^{[2]}$,
where $\pazocal{R}\subseteq[0,1]$, and $i=1,\ldots,n$, $j=1,\ldots,m_{i}$.
Therefore,

$\displaystyle f^{[2]}:\bigg{\\{}$
$\displaystyle\Big{(}x_{1},\mu^{[1]}_{1,1}\Big{)},\Big{(}x_{1},\mu^{[1]}_{2,1}\Big{)},\ldots,\Big{(}x_{1},\mu^{[1]}_{m_{1},1}\Big{)},$
$\displaystyle\ldots$
$\displaystyle\Big{(}x_{n},\mu^{[1]}_{1,n}\Big{)},\Big{(}x_{n},\mu^{[1]}_{2,n}\Big{)},\ldots,\Big{(}x_{n},\mu^{[1]}_{m_{n},n}\Big{)}\bigg{\\}}\rightarrow[0,1]:$
$\displaystyle\left(x_{i},\mu^{[1]}_{j,i}\right)\mapsto\mu^{[2]}_{j,i},$ (6)

or equivalently,

$\displaystyle\mu^{[2]}_{j,i}=f^{[2]}\Big{(}\left(x_{i},\mu^{[1]}_{j,i}\right)\Big{)},\qquad
i=1,\ldots,n,\quad j=1,\ldots,m_{i}.$ (7)

We can reformulate the domain of $f^{[2]}$ as:

$\displaystyle\textrm{dom}\left(f^{[2]}\right):=\bigg{\\{}$
$\displaystyle\Big{(}x_{1},\mu^{[1]}_{1,1}\Big{)},\
\ldots,\Big{(}x_{n},\mu^{[1]}_{1,n}\Big{)}\Big{\\}}\cup$ (8)
$\displaystyle\bigg{\\{}$ $\displaystyle\Big{(}x_{1},\mu^{[1]}_{2,1}\Big{)},\
\ldots,\Big{(}x_{n},\mu^{[1]}_{2,n}\Big{)}\Big{\\}}\cup$ $\displaystyle\vdots$
$\displaystyle\bigg{\\{}$
$\displaystyle\Big{(}x_{1},\mu^{[1]}_{m_{1},1}\Big{)},\
\ldots,\Big{(}x_{n},\mu^{[1]}_{m_{1},n}\Big{)}\Big{\\}}\cup$
$\displaystyle\vdots$ $\displaystyle\bigg{\\{}$
$\displaystyle\Big{(}x_{k},\mu^{[1]}_{m_{k},1}\Big{)}\Big{\\}}=\bigcup_{i=1}^{n}\bigcup_{j=1}^{m_{i}}\left\\{\left(x_{i},\mu^{[1]}_{j,i}\right)\right\\},$

assuming that

$\displaystyle\operatorname*{arg\,max}_{i}\ \\{m_{i}\\}=k.$

This shows the domain of the type-$2$ membership function $f^{[2]}$ is the
union of $m_{k}$ type-$1$ fuzzy sets. Therefore, a type-$2$ fuzzy set also
corresponds to $m_{k}$ type-$1$ membership functions $f^{[1]}_{j}$ that
generate the primary membership degrees $\mu^{[1]}_{j,i}$, i.e.,

$\displaystyle\mu^{[1]}_{j,i}=f^{[1]}_{j}(x_{i}),\qquad i=1,\ldots,n,\quad
j=1,\ldots,m_{i}.$ (9)

The domains of these type-$1$ membership functions are $\pazocal{C}$ or a
subset of $\pazocal{C}$.

In summary, to identify a type-$2$ fuzzy set $\pazocal{F}^{[2]}$ on a domain
$\pazocal{C}$ of the independent variables $x_{i}$, one needs to know the
corresponding type-$2$ membership function $f^{[2]}$ and the domain of this
function. Equivalently, one should know $f^{[2]}$ and the $m_{k}$ type-$1$
membership functions $f^{[1]}_{j}$. From (II), a type-2 fuzzy set
$\pazocal{F}^{[2]}$ is composed of pairs, where each pair itself includes a
pair consisting of the independent variable $x_{i}$ and a primary membership
degree $\mu^{[1]}_{j,i}$ of $x_{i}$, and a secondary membership degree.

Generally speaking, a type-$n$ fuzzy set $\pazocal{F}^{[n]}$ includes
mathematical objects of the form:

$\Bigg{(}\ldots\bigg{(}\left(x_{i},\mu^{[1]}_{j_{n-1},\ldots,j_{1},i}\right),\mu^{[2]}_{j_{n-1},\ldots,j_{1},i}\bigg{)}\ldots\bigg{)}\mu^{[n]}_{j_{n-1},\ldots,j_{1},i}\Bigg{)}$

with $j_{1}\in\left\\{1,\ldots,m_{i,1}\right\\}$, …,
$j_{n-1}\in\left\\{1,\ldots,m_{i,n-1}\right\\}$, and corresponds to a type-$n$
membership function $f^{[n]}$, which generates the $n^{\textrm{th}}$
membership degrees $\mu^{[n]}_{j_{n-1},\ldots,j_{1},i}$ of any mathematical
object that belongs to the union of $m_{k_{n}}$ fuzzy sets of type $n-1$, with

$k_{n}=\displaystyle\operatorname*{arg\,max}_{i}\\{m_{i,n-1}\\}.$

A type-$n$ fuzzy set corresponds to
$\operatorname*{arg\,max}_{i}\\{m_{i,n-1}\\}$ membership functions of type
$n-1$, to $\operatorname*{arg\,max}_{i}\\{m_{i,n-2}\\}$ membership functions
of type $n-2$, …, and to $\operatorname*{arg\,max}_{i}\\{m_{i,1}\\}$ type-$1$
membership functions. One should know all these membership functions, as well
as the unique type-$n$ membership function to identify $\pazocal{F}^{[n]}$.

### Interpretation for real-life problems

Based on the discussions given above, the following conclusions can be made.
For an independent variable $x_{i}\in\pazocal{C}$, the _extent_ that it
belongs to a type-$1$ fuzzy set $\pazocal{F}^{[1]}$ is characterized by one
uncertainty, which is specified by the degree of membership of $x_{i}$ to
$\pazocal{F}^{[1]}$.

Any type-$2$ fuzzy set $\pazocal{F}^{[2]}$ corresponds to $m_{k}$ type-$1$
membership functions and hence, the $m_{k}$ corresponding type-$1$ fuzzy sets.
Therefore, $\forall x_{i}\in\pazocal{C}$, the _extent_ that $x_{i}$ belongs to
$\pazocal{F}^{[2]}$ is characterized by two uncertainties:

  * •

Uncertainty about the extent to which $x_{i}$ belongs to the union or either
of the $m_{k}$ corresponding type-$1$ fuzzy sets. This is quantified by the
primary membership degrees.

  * •

Uncertainty about the extent to which $x_{i}$, which belongs with specific
primary membership degrees to the type-$1$ fuzzy sets, belongs to
$\pazocal{F}^{[2]}$. This is quantified by the secondary membership degree.

Generally speaking, the _extent_ that $x_{i}\in\pazocal{C}$ belongs to a
type-$n$ fuzzy set $\pazocal{F}^{[n]}$ is characterized by $n$ uncertainties:

  * •

Uncertainty about the extent to which $x_{i}$ belongs to the union or either
of the corresponding $\operatorname*{arg\,max}_{i}\\{m_{i,1}\\}$ type-$1$
fuzzy sets. This is quantified by the primary membership degrees.

  * ⋮
  * •

Uncertainty about the extent to which $x_{i}$ belongs to the union or either
of the corresponding $\operatorname*{arg\,max}_{i}\\{m_{i,n-1}\\}$ fuzzy sets
of type $n-1$. This is quantified by the $(n-1)^{\textrm{th}}$ membership
degrees.

  * •

Uncertainty about the extent to which $x_{i}$ belongs to $\pazocal{F}^{[n]}$.
This is quantified by the $n^{\textrm{th}}$ membership degree.

##  III Probabilistic logic

Figure 5: The sets of true $\pazocal{T}$ and false $\pazocal{F}$ propositions
in fuzzy logic may have an overlap, i.e.,
$\pazocal{T}\cap\pazocal{F}\neq\emptyset$. Moreover,some mathematical objects
that belong to $\pazocal{F}\backslash\pazocal{T}$ or
$\pazocal{T}\backslash\pazocal{F}$ may not necessarily have a membership
degree of $1$.

In this section, probabilistic logic [12] versus fuzzy logic is discussed.
With the analogy explained before, a proposition in mathematical logic may
correspond to a mathematical object that can belong to either (crisp or fuzzy)
sets of true or false propositions. For the sake of brevity, the discussions
of this section are represented using the concept of sets only. To notify the
correspondence of sets and propositions, the notations $\pazocal{T}$
(referring to _true_) and $\pazocal{F}$ (referring to _false_) are used for
the sets that build up the possible world222Possible world of the event $E$ is
a world set that embraces all possible realizations of $E$., $\pazocal{W}$, of
an event $E$.

Figure 6: The sets of true $\pazocal{T}$ and false $\pazocal{F}$ propositions
in probabilistic logic do not have any overlap, and any mathematical object
that does not belong to either $\pazocal{F}$ or $\pazocal{T}$, belongs to the
other set with a probability of $1$.

A main difference between fuzzy logic and probabilistic logic is that for the
former, $\pazocal{F}$ and $\pazocal{T}$ are fuzzy sets, while for the latter,
they are crisp sets. Therefore, the borders of $\pazocal{F}$ and $\pazocal{T}$
in fuzzy logic are exposed to uncertainties. Hence, there may be an overlap
between these two sets, i.e., we may have
$\pazocal{T}\cap\pazocal{F}\neq\emptyset$ (see Figure 5), while in
probabilistic logic we necessarily have $\pazocal{T}\cap\pazocal{F}=\emptyset$
(see figure 6).

Any realization of the repeatable event $E$ can correspond to a mathematical
object $x$. In probabilistic logic, in case in $100$ experiments, where $E$ is
repeated, for $\Pi_{1}$ experiments $x$ goes to $\pazocal{F}$ and for
$\Pi_{2}$ experiments $x$ goes to $\pazocal{T}$, then $\Pi_{1}+\Pi_{2}=100$.
The normalized values $\pi_{1}$ and $\pi_{2}$ of the natural values $\Pi_{1}$
and $\Pi_{2}$ are called the probability of $x$ belonging to $\pazocal{F}$ and
$\pazocal{T}$, respectively. From $100$ experiments in fuzzy logic, if in
$M_{1}$ experiments $x$ goes to $\pazocal{F}$ and in $M_{2}$ experiments $x$
goes to $\pazocal{T}$, then $M_{1}+M_{2}$ is not necessarily $100$ (may be
larger than $100$). The reason is that for some experiments $x$ may go to the
overlap $\pazocal{T}\cap\pazocal{F}$, i.e., the experiment is doubly counted
in both $M_{1}$ and $M_{2}$. The normalized values $\mu_{1}$ and $\mu_{2}$ of
the real values $M_{1}$ and $M_{2}$ are called the degrees of membership of
$x$ to the fuzzy sets $\pazocal{F}$ and $\pazocal{T}$, respectively.

To generalize, the summation $\sum_{i=1}^{s_{\textrm{c}}}\pi_{i}$ of the
probabilities $\pi_{i}$ of $x$ belonging to $s_{\textrm{c}}$ crisp sets that
build up the possible world $\pazocal{W}$ of the event $E$ is necessarily $1$,
while the sum $\sum_{i=1}^{s_{\textrm{f}}}\mu_{i}$ of the degrees of
membership of $x$ to $s_{\textrm{f}}$ fuzzy sets that build up $\pazocal{W}$
may be smaller than, equal to, or larger than $1$.

##  IV  Mathematical formulation of uncertainties in real-life events

In real life, an event $E$ may be prone to various types of imprecision and
hence, uncertainties.

### Uncertainties before realization of an event

Suppose that the world set $\pazocal{W}$ of possible realizations of an event
$E$ is known, i.e., we know and can measure (or estimate the value of) all the
characteristics that can define and distinguish any specific position within
$\pazocal{W}$ (this is called a _state_ in real-life engineering problems).

Before $E$ is realized, there is always uncertainty about which possible
realization is going to occur. If the realization of $E$ is measurable (e.g.,
a temperature, which can be measured directly using a thermostat), the
realization is _precise in value_ , is determined by the same measurable
characteristics as those that define the world set, and hence its position in
the world set can strictly be distinguished (see the left-hand side plot in
Figure 7). Therefore, the uncertainty is one-fold, i.e., uncertainty about the
possible measurements from the world set that will be realized in occurrence
of $E$. In this case, if one knows and can measure (or estimate the value of)
all the effective factors (these are called the controllable and
uncontrollable inputs in real-life engineering problems) that may play a role
in any realization of $E$ (or equivalently in any realization of the
characteristics that define the world set $\pazocal{W}$), then probabilistic
logic can determine the probability of occurrence of any possible realization.
In short, when all the concepts and characteristics involved in the procedure
of realization of an event are measurable or _precise in value_ , then
probabilistic logic can handle the uncertainties.

In case the realization of $E$ is non-measurable (e.g., the comfort, which
cannot be measured directly using a measurement device), this realization is
_imprecise in value_. Then the uncertainty is two-fold, i.e., uncertainty
about which possible non-measurable realization in the world set is going to
occur, and uncertainty about the exact position of the realization in the
world set. Systems in real life work with measurable concepts. For instance,
the non-measurable instruction “when the room’s comfort is low, decrease the
temperature”, should be transformed into a measurable instruction for an air
conditioning system, e.g., “when the room’s temperature is between $23$ and
$25$ degrees of Celsius, decrease the temperature for $3$ degrees of Celsius”.
This transforms a concept that is _imprecise in value_ into one that is
_precise in meaning_ , and reduces the original uncertainty about the exact
position of the concept in the world set $\pazocal{W}$ to an uncertainty about
the exact position of the concept in a known subset of $\pazocal{W}$. Fuzzy
logic can transform a realization that is imprecise in value into one that is
precise in meaning by assigning a membership function (of, generally, type
$n$) to the characteristics that are imprecise in value.

Figure 7: When the realization of an event is measurable, it will take an
exact position (illustrated by the black dot in the left-hand side plot) in
the world set. When the realization of an event is non-measurable, it will
take position in a subset (colored in grey in the right-hand side plot) of the
world set, while its exact position is uncertain. The corresponding subsets
are defined to make the non-measurable realization, precise in meaning.
Although the subsets illustrated in this plot are crisp, the may in general be
fuzzy.

It is important to note that although the transformation from a realization
that is imprecise in value into one that is precise in meaning is with the aim
of reducing the uncertainty from the entire world set to a subset of it,
further uncertainties may or may not exist in the exact position of the
borders of these subsets. For instance, when a specific type of search-and-
rescue robot is deployed to a burning building, the domains of temperature for
which the robot is functional and dysfunctional should be determined. Knowing
the materials, sensors, … used in the construction of the robot, one can
determine the temperature at which this robot or any other robot of this type
will become dysfunctional. In this case, the subsets functional and
dysfunctional are crisp. In this case the transformation to a realization that
is precise in meaning using membership functions (fuzzy logic) is identical to
the transformation using probability functions (probabilistic logic).
Distinguishing the exact borders of these two subsets for a human search-and-
rescuer is more challenging, since humans are not as _homogeneous_ as
identical robots produced in a factory. Therefore, these borders may vary from
person to person and the resulting subsets of functional and dysfunctional can
become fuzzy. Then the only right tool for transforming the non-measurable
realization into one that is precise in meaning is fuzzy logic.

### Uncertainties after realization of an event

After $E$ is realized, in case the realization of $E$ is measurable, e.g., a
temperature, which can be measured directly using a thermostat, the
realization is _precise in value_ and its position in the world set can
strictly be distinguished. When the realization of $E$ is non-measurable,
e.g., the comfort, which cannot be measured directly using a measurement
device, this realization is imprecise in value. Therefore, there is
uncertainty about its position in the world set. The realization should first
be quantified to become precise in meaning, and only then, a subset of the
world set that embeds all the possible positions of the realization in the
world set, together with the degree to which the realization is positioned at
any of these possible positions can be determined.

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# High Pressure Suppression of Plasticity due to Over-Nucleation of Shear
Strain

Brenden W. Hamilton  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA<EMAIL_ADDRESS>Timothy C. Germann

(October 2022)

###### Abstract

High pressure shear band formation is a critical phenomenon in energetic
materials due to its influence on both mechanical strength and mechanochemical
activation. While shear banding is know to occur in a variety of these
materials, the governing dynamics of the mechanisms is not well defined for
molecular crystals. We conduct molecular dynamics simulations of shock wave
induced shear band formation in the energetic material
1,3,5-trinitroperhydro-1,3,5-triazine (RDX) to assess shear band nucleation
processes. We find, that at high pressures, the initial formation sites for
shear bands ’over-nucleate’ and rapidly lower deviatoric stresses prior to
shear band formation and growth. This results in the suppression of plastic
deformation. A local cluster analysis is used to quantify and contrast this
mechanism with a more typical shear banding seen at lower pressures. These
results demonstrate a mechanism that is reversible in nature and that
supersedes shear band formation at increased pressures. We anticipate that
these results will have a broad impact on the modeling and development of high
strain rate application materials such as those for high explosives and
hypersonic systems.

##  1 Introduction

Shear bands, regions of localized plasticity induced from high shear stresses,
form to accommodate further deformation in materials and can lead to the
buildup of material damage, as well as a potential decrease in flow stress1.
Shear banding has been known to occur in a variety of materials classes,
including crystalline metals2, 3, 4, ceramics5, 6, 7, metallic glasses8, 9,
10, 11, 12, 13, 14, and molecular materials15, 16, 17.

Overall, shear banding generally occurs when there is a local shear induced
softening of the material, creating regions that are much more susceptible to
a rapid material flow9. This leads to a localized, plastic instability that
lowers the deviatoric stress components of the material and allows for
continued deformation. Shear banding has been of more recent interest due to
its relevance in metallic glasses, as well as the observance of shear band
formation without prior defect accumulation18. In the latter case, these shear
bands were found to allow for continued deformation in the absence of
dislocations without inducing cavitation or material failure.

A recent study has shown that shear bands nucleate to accommodate deformation
(without defect accumulation) when the material has a low difference in energy
between the crystal and amorphous phases, such that forming amorphous bands is
energetically favorable to other mechanisms2. Shear banding sans crystalline
defects is also well studied in polymers, in which glassy and entangled
molecules can flow, even below the glass transition state17, 19. Molecular
solids, which are often utilized at ultra-high strain rates, pose a grand
challenge in studying shear band formation due to their complex crystal
packing and limited crystalline defect formation mechanisms.

In general, for energetic materials, shear bands can induce a mechanochemical
acceleration of kinetics20, 21, 22, making shear banding a highly relevant and
complex chemical initiation mechanism. Mechanochemistry in energetic molecular
solids typically occurs via a shock wave creating large intra-molecular
strains via local material flow23, 24, 25, 26, 27. These strains then lead to
a lowered activation barrier and accelerated kinetics in areas such as pore
collapses and shear bands28, 29, 30, 31, and they can occur in other materials
such as graphite32, 33 and mechanophores34, 35.

Here, we study shear banding in a molecular crystal, specifically the
energetic material 1,3,5-trinitroperhydro-1,3,5-triazine (RDX). Shear banding
is known to occur in RDX15, and occurs due to local thermal softening, not
defect buildup36. These shear bands form out of regions that initially show a
rise in molecular conformation changes that leads to local softening37, and
the associated free energy changes are indicative of a local first-order
structural phase transition37. Here, we show an additional mechanism in which
higher pressure shear bands are suppressed due to an over-formation of shear
band nucleation sites that significantly lowers the deviatoric stress state.
This rapidly reduces the driving force for shear band growth and no
significant plasticity occurs at higher pressures, in contrast to the
significant plasticity seen at lower shock pressures.

##  2 Results and Discussion

We conduct shock compression simulations using all-atom molecular dynamics,
compressing RDX along the [100] crystal direction for particle velocities
($U_{p}$) ranging from 0.7 to 1.2 km/s. Figure 1 shows molecular (center of
mass) renderings of these simulations, colored by the molecular shear strain
(von Mises invariant of the Green-Lagrange strain tensor), at both the time of
maximum compression when the shockwave reaches the far surface, and 50ps
later, with the system held at maximum compression using shock absorbing
boundary conditions (SABCs)15, 38, 25.

These localizations of shear strain, in all cases, form a band-like structure.
For the weaker shock strengths, this manifests in a few thin bands of high
shear strain, illustrated by the orange and red regions in Figure 1.
Conversely, in the stronger shock cases, where resolved shear stresses are
heightened, the shear strain localizations strikingly lack intensity, but are
also considerably more pervasive. Supplemental Materials section SM-3 presents
a 1D binning (along the shock direction) of the shear strain, displaying that
the stronger shock cases result in a greater average shear strain per molecule
(or volume), despite a lack of appreciable shear banding. In the stronger
shock cases, running the simulation for significantly longer times does not
result in more localized shear or any shear band formation (see Supplemental
Materials section SM-2), such that, by 50ps after maximum compression, the
system strain is predominately static, regarding any continual structural
evolution.

Figure 1: Molecular renderings of the X-Z plane for the time of maximum
compression (left) and 50ps later, held at that compression (right), molecules
are colored by the shear strain (von Mises invariant of the Green-Lagrange
strain tensor). A figure of the same frames with a narrower color bar range is
available in Supplemental Materials section SM-2.

Hence, here we observe conditions in which the lower pressure systems exhibit
significant, localized plasticity, however, increases in the uniaxial
compression rate result in little to no plasticity, just a nominal shear
deformation to a significant number of molecules, suppressing plasticity with
increased pressure. Additionally, for the weaker shocks, the shear banding
occurs significantly behind the shock front, whereas the more homogeneous
deformation of the stronger shock cases occurs nearly adjacent to the rise in
pressure from the shock. Furthermore, these stronger shock cases result in the
retainment of the crystal structure, not a bulk plasticity event.

Figure 2 shows a time history of the von Mises (VM) stress for a planar slice
of material 2.5nm wide, beginning at 10.0nm from the piston. For 0.7 km/s,
which does not shear band in the time of the shock passage of the entire
sample, which is 24 ps for this case, the VM stress does not significantly
change. Shifting to the prompt shear banding cases of 0.8 and 0.9 km/s, the VM
stress sits at its peak value for 5-10ps, before slowly decaying as plasticity
begins to materialize, resulting in a much more hydrostatic state during the
maximum compression hold.

For the strongest shock cases, a rapid decrease in VM stress is observed,
occurring on the timescale of the shock rise, decreasing so promptly that the
peak value shown lowers with rising shock strength. However this is an
artificial smoothing due to the bin size and temporal spacing of the points,
where material in the upstream side of the bin commences decaying its VM
stress prior to total compression of the bin. This rapid decrease occurs, sans
plasticity of any meaningful quantity, and the driving force for shear band
formation, large deviatoric stresses, fails to be sustained in a requisite
intensity to induce shear band formation. Hence, this deformation mechanism
that does occur usurps the shear banding mechanism and nullifies the driving
force before significant plasticity can occur.

Rapid VM stress decays like that show here for strong shocks may also indicate
a phase transformation. Based on experimental phase diagrams39 for RDX, all
the states studied here should exist in the $\gamma$ phase (the highest shock
temperature reached here is 420 K). Supplemental Materials section SM-4 shows
radial distribution function g(r) curves for the initial state and all shocked
states, further showing that no phase transition occurs at the stronger
shocks, relative to the weaker shocks. Supplemental Materials section SM-4
also presents molecular maps of molecular rotations, showing that no other
explicit phase change mechanism is occurring.

Figure 2: Time history of the von Mises stress of a 1D slice of material at
10.0 to 12.5 nm from the piston for all particle velocities.

Figure 3 shows molecular renderings of a release of the SABCs, permitting the
material to expand back towards ambient density, for a shear band and high
pressure no shear band case, 0.8 and 1.1 km/s, respectively. The SABC on the
downstream face of the material is lifted after 50ps of continuous
compression, and the material begins to uniaxially expand in a rarefaction
process. In the 0.8 km/s case, the shear bands neither recover to a
crystalline phase nor initiate failure via void formation/cavitation. The
shear bands persist throughout the total release of the material (bottom row,
+80ps) and continue to exhibit some strength and binding of the material.

Conversely, with the 1.1 km/s case, the regions of deformation, which can be
discerned as the regions of increased color/darkness due to obfuscation of the
white background, present an entirely alternative release response. As the
release wave alleviates the latent stress of the compressed system, a
substantial majority of these local deformations undergo an annealing-like
processes and reverts to the original crystal structure. Supplemental
Materials section SM-5 shows the decrease rate in sheared molecules for both
cases in Figure 3. These results further propound the idea of plasticity
suppression at higher shock strengths, such that the release wave rapidly
eliminates the presence of sheared molecules in the 1.1 km/s case.

Overall, this reveals the high pressure mechanism to be reversible in nature,
and, therefore, by definition, non-plastic. This non-plastic shear deformation
occurs at high enough density and has broad enough spatial tendencies to drive
down the local deviatoric stresses enough to suppress plasticity, preventing
the nucleation site from overcoming the high barrier for shear band formation
and growth. We spend the rest of this work attempting to characterize this
high pressure mechanism.

Figure 3: Molecular center of mass renderings for the release of the
compressed systems, starting from 50ps after maximum compression, until full
release, for the 0.8 and 1.1 km/s cases.

To arbitrarily classify and study only the highly sheared regions of the
material, we remove all molecules that have a shear strain of less than 0.15.
Figure 4 displays molecular renderings of this reduced system at 50ps after
maximum compression. The weaker shocks (top row) develop clear bands of
amorphous material that span the width of the sample and consist of both
’upwards’ and ’downwards’ propagating shear bands. These bands interweave with
one another to form a shear band network. By 1.0 km/s, these bands begin to
increase in density sufficiently such that they begin to coalesce, resulting
in a less defined network structure.

In the 1.1 and 1.2 km/s cases, no well-defined structure of the sheared
material exists, and the magnitude of the shear strain is considerably lower
on average. For these cases, sheared material appears rapidly at the shock
front, en masse, but never forms the defined bands of the weaker shock cases.
This is due to an ’over-nucleation’ of shear band sites that never grow into
plastic deformation. Small amounts of molecules in various locations begin to
shear, just as occurs for the shear bands that nucleate in the weaker shocks.
However, the number of these ’nucleation sites’ is far greater and they occur
much closer to the shock front. This rapidly lowers the VM stress (as shown in
Figure 2) and reduces the driving force for the shear band formation,
preventing the nucleation sites from growing into shear bands, suppressing the
formation of plasticity.

Supplemental Materials section SM-6 has heat map plots of the XZ Component of
the Deformation Gradient Tensor (Green-Lagrange) vs the Shear Strain, in which
the shear band cases show significant structure, but the suppressed plasticity
cases show no differentiation between the sheared molecules and the rest of
the system. While understanding the states in which plasticity occurs is
paramount for understanding continued deformation and failure of materials in
general, for energetic materials such as RDX, there is the further pertinence
in that shear bands tend to generate a mechanochemical effect that can greatly
alter the material’s overall reactivity20, 25. The high pressure lack of
plastic deformation has significant ramifications on the material strength and
reactivity at its intended usage conditions.

Figure 4: Molecular center of mass renderings at 50ps after maximum
compression for all particle velocities. Only molecules with a shear strain
above 0.15 are rendered. Molecules are colored by shear strain.

To numerically assess the structure of the molecules with a shear strain
greater than 0.15, as shown in Figure 4, we perform a cluster analysis with a
distance based cutoff of 6 A. Table 1 shows the results for each of the
particle velocities with the total number of molecules considered, the number
of clusters with at least 10, 25, and 100 molecules, the size of the largest
cluster, and what percentage of the considered molecules are in the largest
cluster. While we conduct this analysis here on the 50ps after maximum
compression frame, we find this to also be an extremely elucidating form of
analysis for early time nucleation sites of shear bands in which we find
similar results.

Starting with the maximum cluster size and percent, for the cases that shear
band, the largest cluster is over 90% of all sheared atoms. This results from
the various shear bands intersecting into a network to form one large cluster.
In the 0.9 km/s case, a number of small ($N>10$ molecules) clusters begin to
form and do not become part of the shear banding network. By 1.1 km/s there
are several hundred small clusters that fail to form a network of sheared
material or result in plasticity, with the largest cluster in the 1.2 km/s
case only accounting for 26.2% of all sheared material. Supplemental Materials
section SM-7 shows time history plots of the max cluster size and number of
clusters larger than N=25 for the initial shock. The latter shows the much
larger increases for the non-shear banding cases, as well as a significant
reduction in clusters after the material begins to release, showcasing the
reversible (non-plastic) nature of the mechanism.

Table 1: Clustering results for a cluster analysis of the high shear strain molecules shown in Figure 4. Pop. is the total molecules used per frame. $N_{X}$ is number of clusters with at least X molecules. $C_{Max}$ is maximum cluster size, and $C_{Max\%}$ is the percentage of molecules in the max cluster. $U_{p}$ | Pop. | $N_{10}$ | $N_{25}$ | $N_{100}$ | $C_{Max}$ | $C_{Max\%}$
---|---|---|---|---|---|---
0.7 | 16145 | 3 | 1 | 1 | 15401 | 95.3
0.8 | 92627 | 7 | 1 | 1 | 88787 | 95.8
0.9 | 109401 | 40 | 2 | 1 | 103225 | 94.3
1.0 | 107424 | 85 | 8 | 4 | 98328 | 91.5
1.1 | 97802 | 193 | 37 | 18 | 80091 | 81.9
1.2 | 52902 | 372 | 102 | 65 | 13854 | 26.2

To better assess when a region has been shocked, we move to a 1$nm^{3}$ 3D
Lagrangian binning, in which each bin only contains 3-4 molecules, resulting
in little smoothing of the shear strain states. Figure 5 shows the average
shear strain of each bin at 2.5 ps after the bin molecules experience the
shockwave in the x-axis, with the shear strain after 50ps held at maximum
compression in the y-axis. For the weak shocks that shear band, little
shearing occurs at early times and results in the data having a highly
vertical trend. This lack of correlation is due to a majority of the shear
bands growing from smaller nucleation sites. In the homogeneous shearing cases
of 1.1 and 1.2 km/s, the data follows a trend of the slope just greater than
1, showing almost no increase in shearing over time or any rearrangement in
what areas are sheared more. The result data trends have Pearson correlation
coefficients of 0.083, 0.443, 0.674, 0.803, and 0.804 for 0.8, 0.9, 1.0, 1.1,
and 1.2 km/s, respectively. Other than the initial nucleation sites, there is
little to no correlation between the initial and final states of the shear
banding case. However, the shear states of the higher pressure cases shows
little temporal evolution, as the non-plastic mechanism lowers the deviatoric
stress components more than the shear banding, allowing the system to remain
static in this state, resulting in a high pressure suppression of plasticity.

Figure 5: Correlations between the shear strain at 2.5ps after a material
section is shocked vs the shear strain of that material at 50ps after maximum
compression. Each point represents a 1$nm^{3}$ Lagrangian bin.

##  3 Conclusions

Through molecular dynamics simulations of the shock compression of the
energetic material RDX, we find a high pressure mechanism for the suppression
of plasticity via the over formation of shear band nucleation sites. Upon
release of the compression state, these nucleation sites heal, resulting in a
reversible processes. This highly differs from lower pressure shocks in which
plastic deformation results in significant shear band growth that remains upon
release. In both cases, the deviatoric stress components are lowered, however,
the non-plastic, high pressure mechanism results in a more rapid decay of the
von Mises stress. Hence, the deformation mechanism that occur at higher
pressures supplants the shear banding mechanism and curtails its driving force
prior to any plasticity.

A cluster analysis of sheared molecules shows the shear bands to be one large,
interconnected network, whereas the high pressure system yields hundreds of
small, independent clusters that do not significantly grow in size or shear
magnitude over time, yet nevertheless lower the shear stress drivers for
plasticity. Overall, the lack of high pressure shear banding is critical to
RDX for both mechanical strength and mechanochemical effects on kinetics.
Hence, future work will assess the broad nature of this mechanism in RDX for a
variety of thermal, microstructure, and mechanical states, as well as non-
shock high-strain rate conditions and its prevalence in a variety of materials
classes where high strain rate plasticity is of great importance, such as
polymer binders and materials for hypersonic applications.

##  4 Methods

We use all-atom molecular dynamics using the LAMMPS package40, 41. Atomic
forces and energies for RDX are calculated with the non-reactive potential
from Smith and Bharadwaj 42. A 1.0 fs timestep was used in all simulations.
Cells were constructed by replicating the alpha-RDX crystal structure along
the [100], [010] and [001] directions, aligned with the x, y, and z Cartesian
axis respectively, to lengths of 118.2nm, 34.3nm, and 31.8nm (30x10x10 unit
cells), respectively.

5.0nm of material was removed, by whole molecules, at either end of the
x-direction to break periodicity and form a free surface. The system was then
equilibrated for 500ps at 300 K using a Nosé-Hoover thermostat43, to allow
breathing modes from the free surface creation to attenuate. Shock simulations
were conducted using the reverse ballistic approach along the [100]
direction44. The bottom 2.5nm, by whole molecules, was held rigid to form an
infinitely massive piston. When each shock reaches maximum compression, shock
absorbing boundary conditions are applied to extend the simulation
indefinitely15, 38, 25. All simulation analysis was conducted on a molecular
center of mass framework. Additional methods and all specific analysis details
are available in Supplemental Materials section SM-1.

##  5 Acknowledgments

The authors thank Marc Cawkwell for useful discussions regarding RDX shear
banding and phase transformations. Funding for this project was provided by
the Director’s Postdoctoral Fellowship program at Los Alamos National
Laboratory, project LDRD 20220705PRD1. Partial funding was provided by the
Advanced Simulation and Computing Physics and Engineering Models project (ASC-
PEM). This research used resources provided by the Los Alamos National
Laboratory (LANL) Institutional Computing Program. This work was supported by
the U.S. Department of Energy (DOE) through LANL, which is operated by Triad
National Security, LLC, for the National Nuclear Security Administration of
the U.S. Department of Energy (Contract No. 89233218CNA000001). Approved for
Unlimited Release LA-UR-21177.

##  6 Author Contributions

BWH and TCG conceived the study. BWH conducted all simulations and wrote the
manuscript. TCG helped with devising analysis techniques. BWH and TCG
contributed to discussions and revisions of the results and manuscript.

##  7 Supplemental Material

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