multiply connected) is reduced to the analysis of the deformation of one
point. Speed up of online nonlinear stage for this approach, and their
advanced modifications (see, e.g., [184]) over FEA can reach $10^{4}$.

The expressions of the effective properties and their consistent conditions
found in this Subsection are general, exact, and obtained for any number and
any kind (i.e. the domains $\Omega^{J}$ need not be spatially connected) and
shape of domains $\Omega^{J}$ ($J=1,\ldots,K$) with no restriction on the
strain-stress field inhomogeneity in $\Omega^{J}$ (e.g. even the field
singularity is also possible in $\Omega^{J}$). A single restriction on
mechanical properties is a homogeneity of
$\mbox{\boldmath$\beta$}(\mbox{\boldmath$\bf x$})$ inside each domain
$\mbox{\boldmath$\bf x$}\in\Omega^{J}$ ($J=1,\ldots,K$).

###  7.2 Clustering discretization method in peridynamic micromechanics

We perform step-by-step LM representations (4.4)-(4.17) recast to the
corresponding counterparts of PM. So, we present the total relative
displacement and stress in the form of additive decomposition
($\mbox{\boldmath$\bf x$}\in\Omega^{J}$)

$\displaystyle\mbox{\boldmath$\eta$}_{J}(\mbox{\boldmath$\xi$})=\mbox{\boldmath$\eta$}_{J}^{\rm
lin}(\mbox{\boldmath$\xi$})+\mbox{\boldmath$\eta$}_{J}^{\rm
non}(\mbox{\boldmath$\xi$}),\ \ \
\mbox{\boldmath$\sigma$}_{J}(\mbox{\boldmath$\bf x$})$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}^{\rm lin}_{J}(\mbox{\boldmath$\bf
x$})+\mbox{\boldmath$\sigma$}^{\rm non}_{J}(\mbox{\boldmath$\bf x$}).$ (7.8)

Hereafter, $\mbox{\boldmath$\eta$}_{J}^{\rm lin}(\mbox{\boldmath$\bf x$})$ and
$\mbox{\boldmath$\sigma$}^{\rm lin}_{J}(\mbox{\boldmath$\bf x$})$ are the
linear elastic fields related by the constitutive equations (2.21) and (2.26)
whereas the inelastic parts $\mbox{\boldmath$\eta$}_{J}^{\rm
non}(\mbox{\boldmath$\bf x$})$ and $\mbox{\boldmath$\sigma$}^{\rm
non}_{J}(\mbox{\boldmath$\bf x$})$ are produced by applied phase stress and
strain, respectively (analogous to Eqs. (7.5)). It is assumed homogeneity of
the stress $\mbox{\boldmath$\sigma$}_{J}(\mbox{\boldmath$\bf x$})$ as well as
the linearity of the relative displacements

$\displaystyle\mbox{\boldmath$\eta$}_{J}(\mbox{\boldmath$\xi$})=\mbox{\boldmath$\varepsilon$}_{\eta}\cdot\mbox{\boldmath$\xi$},\
\ \ \mbox{\boldmath$\eta$}_{J}^{\rm
lin}(\mbox{\boldmath$\xi$})=\mbox{\boldmath$\varepsilon$}_{\eta}^{\rm
lin}\cdot\mbox{\boldmath$\xi$},\ \ \ \mbox{\boldmath$\eta$}_{J}^{\rm
non}(\mbox{\boldmath$\xi$})=\mbox{\boldmath$\varepsilon$}_{\eta}^{\rm
non}\cdot\mbox{\boldmath$\xi$}$ (7.9)

inside the cluster ${\cal H}_{\bf x}\subset\Omega^{J}$. For convenience, we
introduce the linear operator $\mathbb{L}$ returning the coefficients of the
linear approximations (7.9)

$\displaystyle\mbox{$\mathbb{L}$}(\mbox{\boldmath$\eta$}_{J}(\mbox{\boldmath$\xi$}))=\mbox{\boldmath$\varepsilon$}_{J},\
\ \ \mbox{$\mathbb{L}$}(\mbox{\boldmath$\eta$}_{J}^{\rm
lin}(\mbox{\boldmath$\xi$}))=\mbox{\boldmath$\varepsilon$}_{J}^{\rm lin},\ \ \
\mbox{$\mathbb{L}$}(\mbox{\boldmath$\eta$}_{J}^{\rm
non}(\mbox{\boldmath$\xi$}))=\mbox{\boldmath$\varepsilon$}_{J}^{\rm non}.$
(7.10)

Equations (2.1) at $\mbox{\boldmath$\beta$}^{J}\equiv{\bf 0}$ ($J=1,\ldots,K$)
is used for estimation of the piecewise linear functions
$\mbox{\boldmath$\eta$}_{J}^{\rm lin}(\mbox{\boldmath$\bf x$})$ of
$\mbox{\boldmath$\bf x$}\in\Omega^{J}$, which is exploited for the evaluation
of the piecewise constant functions $\mbox{\boldmath$\sigma$}^{\rm
lin}_{J}(\mbox{\boldmath$\bf x$})\equiv$const. ($\mbox{\boldmath$\bf
x$}\in\Omega^{J},\ J=1,\ldots,L$) by Eqs. (2.26). The linear parts
$\mbox{\boldmath$\eta$}_{J}^{\rm lin}(\mbox{\boldmath$\bf x$})$,
$\mbox{\boldmath$\sigma$}_{J}^{\rm lin}(\mbox{\boldmath$\bf x$})$ of the
fields (7.8) and nonlinear ones $\mbox{\boldmath$\eta$}_{J}^{\rm
non}(\mbox{\boldmath$\bf x$})$, $\mbox{\boldmath$\sigma$}_{J}^{\rm
non}(\mbox{\boldmath$\bf x$})$ are related by the linear constitutive
relations

$\displaystyle\mbox{\boldmath$\sigma$}^{\rm lin}_{J}(\mbox{\boldmath$\bf x$})$
$\displaystyle=$ $\displaystyle\mbox{\boldmath$\cal L$}^{\sigma(J)}_{\rm
lin}(\mbox{\boldmath$\eta$}_{J}^{\rm lin}),\ \ \
\mbox{\boldmath$\eta$}_{J}^{\rm
lin}(\mbox{\boldmath$\xi$})=\mbox{\boldmath$\cal M$}^{\eta(J)}_{\rm
lin}(\mbox{\boldmath$\sigma$}_{J}^{\rm lin})(\mbox{\boldmath$\xi$}),$ (7.11)
$\displaystyle\mbox{\boldmath$\sigma$}^{\rm non}_{J}(\mbox{\boldmath$\bf x$})$
$\displaystyle=$ $\displaystyle-\mbox{\boldmath$\cal L$}^{\sigma(J)}_{\rm
lin}(\mbox{\boldmath$\eta$}_{J}^{\rm non}),\ \ \ \mbox{\boldmath$\eta$}^{\rm
non}_{J}(\mbox{\boldmath$\bf x$})=-\mbox{\boldmath$\cal M$}^{\eta(J)}_{\rm
lin}(\mbox{\boldmath$\sigma$}_{J}^{\rm non})(\mbox{\boldmath$\bf x$}),$ (7.12)

where the linear operator $\mbox{\boldmath$\cal M$}^{\eta(J)}_{\rm
lin}:=[\mbox{\boldmath$\cal L$}^{\sigma(J)}_{\rm lin}]^{-1}$ (defining the
linear functions of $\bf x$ in (7.111)) can be found as a solution of Eq.
(7.111). The operators $\mbox{\boldmath$\cal L$}^{\sigma(J)}_{\rm lin}$ and
$\mbox{\boldmath$\cal M$}^{\eta(J)}_{\rm lin}$ are defined on the extended
cluster $\Omega^{Jl}:=\Omega^{J}\oplus{\cal H}_{\bf 0}$ whereas
$\mbox{\boldmath$\eta$}_{J}(\mbox{\boldmath$\xi$})$ and
$\mbox{\boldmath$\sigma$}_{J}(\mbox{\boldmath$\bf x$})$ ($\mbox{\boldmath$\bf
x$}\in\Omega^{Jl}\setminus\Omega^{J}$) is prolongated as a linear function and
a constant of $\mbox{\boldmath$\eta$}_{J}(\mbox{\boldmath$\xi$})$ and
$\mbox{\boldmath$\sigma$}_{J}(\mbox{\boldmath$\bf y$})$, respectively, defined
on $\mbox{\boldmath$\bf y$}\in\Omega^{J}$. For the prescribed structures of
the linear $\mbox{\boldmath$\eta$}_{J}(\mbox{\boldmath$\xi$})$ and constant
$\mbox{\boldmath$\sigma$}_{J}(\mbox{\boldmath$\bf y$})$ fields, the action of
the operators $\mbox{\boldmath$\cal L$}^{\sigma(J)}_{\rm lin}$ and
$\mbox{\boldmath$\cal M$}^{\eta(J)}_{\rm lin}$, respectively, are reduced to
the tensor multiplication (analogous to the tensors
${}^{L}\\!\mbox{\boldmath$\bf L$}^{I}$ and ${}^{L}\\!\mbox{\boldmath$\bf
M$}^{I}$ (2.2)), which can be found only one time at the offline stage.

Nonlinear parts of phase relative displacement $\mbox{\boldmath$\eta$}^{\rm
non}_{J}$ can be expressed through the inverse stress operator
$\mbox{\boldmath$\cal M$}^{\eta(J)}:=(\mbox{\boldmath$\cal
L$}^{\sigma(J)})^{-1}$ (2.18), (2.17) and the linearized inverse stress
operator $\mbox{\boldmath$\cal M$}^{\eta(J)}_{\rm lin}$ (2.18), (2.21)

$\displaystyle\mbox{\boldmath$\eta$}^{\rm non}_{J}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\cal
M$}^{\eta(J)}(\mbox{\boldmath$\sigma$}_{J})(\mbox{\boldmath$\xi$})-\mbox{\boldmath$\cal
M$}^{(J)}_{\rm lin}(\mbox{\boldmath$\sigma$}_{J})(\mbox{\boldmath$\xi$}).$
(7.13)

presenting peridynamic counterpart of $\mbox{\boldmath$\varepsilon$}_{J}^{\rm
in}={\mbox{\boldmath$\mathfrak{f}$}}(\mbox{\boldmath$\sigma$}_{J})$ (7.7). The
action of the operator $\mathbb{L}$ on both sides of Eq. (7.13) leads to the
peridynamic analog of Eq. (7.7)

$\displaystyle\mbox{\boldmath$\varepsilon$}^{\rm non}_{J}$ $\displaystyle=$
$\displaystyle\mbox{$\mathbb{L}$}[\mbox{\boldmath$\cal
M$}^{\eta(J)}(\mbox{\boldmath$\sigma$}_{J})(\mbox{\boldmath$\xi$})]-\mbox{$\mathbb{L}$}[\mbox{\boldmath$\cal
M$}^{(J)}_{\rm lin}(\mbox{\boldmath$\sigma$}_{J})(\mbox{\boldmath$\xi$})].$
(7.14)

Substitution of Eq. (7.14) into Eq. (7.4) provides the following ROM system of
governing equations for evaluation of the local stresses ($I=1,\ldots,K)$

$\displaystyle\mbox{\boldmath$\sigma$}_{I}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\bf
A$}_{\sigma\varepsilon}^{I}\mbox{\boldmath$\varepsilon$}^{w\Gamma}+\sum_{J=1}^{K}\mbox{\boldmath$\bf
D$}^{IJ}_{\sigma}(\mbox{\boldmath$\bf x$})\mbox{\boldmath$\eta$}_{J}^{\rm
non},$ (7.15)

which is similar to Eq. (7.72) for the local micromechanics. For the PM, the
estimations of $\mbox{\boldmath$\bf A$}_{\sigma\varepsilon}^{I}$ and
$\mbox{\boldmath$\bf D$}^{IJ}_{\sigma}(\mbox{\boldmath$\bf x$})$ are presented
in [66]. The nonlinear equations (7.15) are recast in residual form in each
cluster $\Omega^{I}$ ($I=1,\ldots,K)$, which can be solved by the Newton-
Raphson method with a return-mapping technique [265] (see for details [66]).
Averaging of the found $\mbox{\boldmath$\sigma$}_{I}$ ($I=1,\ldots,K$) as the
functions of $\mbox{\boldmath$\varepsilon$}^{w\Gamma}$ leads to the definition
of the overall stiffness (locally elastic) $\mbox{\boldmath$\bf L$}^{*}$:
$\mbox{$\langle$}\mbox{\boldmath$\sigma$}\mbox{$\rangle$}=\mbox{\boldmath$\bf
L$}^{*}\mbox{\boldmath$\varepsilon$}^{w\Gamma}$, which differs from the linear
approximation $\mbox{\boldmath$\bf
L$}^{*}=\sum_{r=0}^{N}c^{(r)}\mbox{\boldmath$\bf A$}^{r}_{\sigma\varepsilon}$.

The nonlinear deformation problem for a multiphase CM is reduced to the
evaluation of the mechanical and transformation factor tensors (7.4) and
(7.15) only one time for the linear offline stage problem (see for details
[60]), and integration of the ROM system (7.14) for any nonlinear loading in
an online stage. The nonlinear problem at each $i$-the iteration is reduced to
the estimation of actions of the operators $\mbox{\boldmath$\cal
M$}^{\eta(J)}$, $\mbox{\boldmath$\cal M$}^{\eta(J)}_{\rm lin}$ and
$\mbox{\boldmath$\cal L$}^{\sigma(J)}_{\rm lin}$ on the constant and linear
functions, respectively, in each cluster $\Omega^{(J)}$. In so doing, the
actions of the linear operators $\mbox{\boldmath$\cal L$}^{\sigma(J)}_{\rm
lin}$ and $\mbox{\boldmath$\cal M$}^{\eta(J)}_{\rm lin}$ can be estimated only
one time at the offline stage. Only one micro nonlinear problem for the
operator $\mbox{\boldmath$\cal M$}^{\eta(I)}$ (for each $I=1,\ldots,K$) is
solved in the point $\mbox{\boldmath$\bf x$}\in\Omega^{I}$ (2.6) which is a
center of the horizon’s ${\cal H}_{\bf x}$. I.e. the cluster $\Omega^{I}$ is
reduced to one family instead of one point in the LM. The cluster
$\mbox{\boldmath$\bf x$}\in\Omega^{I}\subset\mathbb{R}^{d}$ is replaced by a
full space $\mathbb{R}^{d}$ with prescribed linear function
$\mbox{\boldmath$\eta$}(\mbox{\boldmath$\bf x$})$, and its response is
assigned to all points of the cluster $\Omega^{J}$. Since the number of
clusters $K$ is much smaller than the number of elements $n$ ($K\ll n)$ of
fine-scale, computational efficiency is greatly improved.

###  7.3 Data-driven modeling and machine learning based on CDM

The presented ROM contains a few stages. The offline stage of material
modeling consists of three steps: a) data collection, b) unsupervised learning
(i.e., clustering), and c) preprocessing (e.g., estimation of cluster
interactions). Since the numbers of clusters are much smaller than the number
of fine data points, computational efficiency is greatly increased, and, in
such a way that CDM fulfills the role of data compression. The central
ingredient of the nonlinear online stage, a material subroutine (7.13) is
reduced to only $K$ uncoupled integrals (inversed to (2.17) and (2.18)) on
constant functions $\mbox{\boldmath$\sigma$}(\mbox{\boldmath$\bf x$})$ over
${\cal H}_{\bf x}$. The computational complexity of estimations of these
integrals is comparable with the computational complexity of evaluations of
$K$ nonlinear equations of locally elastic counterparts (7.7).

The next step of multi-stage design is usually addressed to the methods of
machine learning (ML) and neural networks (see for reference and details
[133],[176], [246]) for providing real-time prediction of materials’
properties and multi-stage design de novo materials. Neural network techniques
are based, loosely speaking, on a generalization of linear regression
analysis. So, in the simplest feedforward neural networks (FFNNs, see, e.g.,
[41]) each layer consisting of a few neurons computes the output to the next
layer $l$ based on the input from the previous $(l-1)$th layer. The neurons in
the same layer are not connected whereas every pair of neurons in neighboring
layers are assigned by network parameters such as a weight ${\bf w}^{l}$ and a
bias $\mbox{\boldmath$\bf b$}^{l}$ so that each layer creates data for the
next layer through the following nested transformations:

$\mbox{\boldmath$\bf z$}^{l}={\cal A}^{l}({\bf w}^{l}\cdot\mbox{\boldmath$\bf
z$}^{l-1}+\mbox{\boldmath$\bf b$}^{l}),\ \ \ l=1,\ldots,L,$ (7.16)

where $\mbox{\boldmath$\bf z$}^{0}$ and $\mbox{\boldmath$\bf z$}^{L}$ are the
input and output of the model. The functions ${\cal A}^{l}$ are called
activation functions and make the network nonlinear to the inputs. The weights
of parameters used for calculation for each neuron are optimized to minimize
the loss function. Linear regression supports also other ML approaches such as
convolutional neural networks (CNNs) [132] and recurrent neural networks
(RNNs) which are generally used for supervised learning.

Running ROMs (such as, e.g., CDM) for RVEs can still be time-consuming in
design optimization To avoid this difficulty, one usually replaces the ROMs
with neural networks (such as, e.g., FFNNs, CNNs, etc.) trained on the RVE
responses computed with SCA for micro and macro fields. The training procedure
is formulated as an optimization problem with the definition of the loss
function as Mean Square Error (MSE) among the estimated fields in both the ML
approach (7.16) and CDM. These schemes were realized by WK Liu and others for
a wide class of nonlinear problems (nonlinear elasticity, [176], [298];
elasto-plastic [146], elasto-viscoplasticity, [306]; damage analysis, [137])
of local micromechanics.

Comment 7.1. We describe the differences and similarities between using CDM in
the LM and PM by the example of Fig. 8. So, the goal of the linear Offline
stage (Block 1) is estimations of the concentration factors (4.13) for the LM
and PM (although the clusters and cluster interactions in the LM and PM are
different simply because DNS in the LM and PM are different). The linear
solution of the Offline stage (Block 1) is used for the subsequent solution of
a nonlinear system of algebraic equations (nonlinear Online stage, Block 2)
corresponding to $K$ pairs of points
$(\mbox{\boldmath$\varepsilon$}_{I},\mbox{\boldmath$\sigma$}_{I})$
($I=1,\ldots,K$) in the LM, whereas a nonlinear PM counterparts
$(\mbox{\boldmath$\cal L$}^{\sigma(J)},\mbox{\boldmath$\cal M$}^{\eta(J)})$
are defined at $K$ families. Block 3 (Machine learning, see Eq. (7.16)) is
identical for both the LM and PM (this is a reason why we did not discuss PM’s
application in Block 3 in more detail).

Comment 7.2. We considered in Section 8 only one type of nonlinear
peridynamics, precisely nonlinear elasticity. However, the discovered formal
similarity of DSA and CDM opens wide opportunities for a straightforward
generalization of well-developed LM theory of data-driven ML approaches [60],
[136], [137], [158], [176], [186], [298], [306] (interested readers are
referred to these references to achieve a deeper understanding of these
topics) to their peridynamic counterparts including different physical
phenomena (e.g., state-based models, diffusion, viscosity,
thermoelastoplasticity, damage accumulation, debonding, plastic localization,
wave propagation) for CMs with different periodic structures (e.g.,
polycrystals, fiber networks, hybrid structures, foam materials).
Consideration of these problems in more detail is beyond the scope of the
current presentation.

Fig. 8: The scheme of application of CDM in PM

###  7.4 Proper Orthogonal Decomposition in PM

Apparently most suitable POD application in PD for reformulation in the
framework of the current review is presented in [101] (see also [23]) which is
shortly reproduced below. So, to solve the difficulty caused by the nonlocal
boundary conditions in the process of ROM construction, a boundary layer
separation approach is presented, and the global equilibrium equations are
symbolically reorganized in a block form (compare with Eq. (3.15))

$\displaystyle\\!\begin{bmatrix}&\\!\\!\mathbb{K}_{bb}&\mathbb{K}_{bi}\\\
&\\!\\!\mathbb{K}_{ib}&\mathbb{K}_{ii}\end{bmatrix}\begin{bmatrix}&\\!\\!\mathbb{U}_{b}\\\
&\\!\\!\mathbb{U}_{i}\end{bmatrix}=\begin{bmatrix}&\\!\\!\mathbb{B}_{b}\\\
&\\!\\!\mathbb{B}_{i}\end{bmatrix},$ (7.17)

where $\mathbb{U}_{b}$ and $\mathbb{U}_{i}$ are the displacement vectors of
the boundary layer points $w_{\Gamma}$ and internal points $w$, respectively.
Similarly the PD stiffness matrix $\mathbb{K}_{\alpha\beta}$
($\alpha\beta=b,i$) and the loading vectors $\mathbb{B}_{b}$, $\mathbb{B}_{i}$
are defined. DNS displacement solutions under various loading conditions
$\mathbb{U}_{ij}$ ($j=1,\ldots,m$) are collected to a training database called
snapshot $\mbox{\boldmath$\bf S$}=\\{\mathbb{U}_{ij}\\}^{m}_{j=1},\
\mbox{\boldmath$\bf S$}\in\mathbb{R}^{N_{i}\times m},\
\mathbb{U}_{ij}\in\mathbb{R}^{N_{i}\times 1},$ where $N_{i}$ is the total
number of DoFs of internal points. The covariance matrix $\bf P$ is defined as
$\mbox{\boldmath$\bf P$}=\mbox{\boldmath$\bf S$}^{\top}\mbox{\boldmath$\bf
S$}$ ($\mbox{\boldmath$\bf P$}\in\mathbb{R}^{m\times m}$). A set of basis
vectors $\mbox{\boldmath$\phi$}_{j}$ is generated by solving an eigenvalue
problem $\mbox{\boldmath$\bf
P$}\mbox{\boldmath$\phi$}_{j}=\lambda_{j}\mbox{\boldmath$\phi$}_{j}$
($\mbox{\boldmath$\phi$}_{j}\in\mathbb{R}^{m\times 1}$, $j=1,\ldots,k,$
$\lambda_{1}\geq\ldots\geq\lambda_{k}>0$) where $\mbox{\boldmath$\phi$}_{j}$
is the eigenvector of $\bf P$, $\lambda_{j}$ is the eigenvalue of $\bf P$, $k$
is the number of nonzero eigenvalues. The POD orthogonal basis functions are
expressed as
$\mbox{\boldmath$\Phi$}_{j}=\sqrt{\lambda}_{j}^{-1}\mbox{\boldmath$\bf
S$}\mbox{\boldmath$\phi$}_{j}$, where
$\mbox{\boldmath$\Phi$}_{j}\mbox{\boldmath$\Phi$}_{l}=\delta_{jl}$. To
simplify the system, only $n_{i}\gg N_{i}$ basis vectors
$\mbox{\boldmath$\Phi$}_{j}$ ($j=1,\ldots,n_{i}$) containing the majority of
the information of the snapshot are selected. Finally, the displacement field
can be reconstructed by using the Galerkin projection method:

$\displaystyle\mathbb{U}_{i}=\overline{\mbox{\boldmath$\Phi$}}\mbox{\boldmath$\bf
a$}_{i},$ (7.18)

where
$\overline{\mbox{\boldmath$\Phi$}}=[\mbox{\boldmath$\Phi$}_{1},\ldots,\mbox{\boldmath$\Phi$}_{n_{i}}]\in\mathbb{R}^{N_{i}\times
n_{i}}$ and $\mbox{\boldmath$\bf a$}_{i}=[\mbox{\boldmath$\bf
a$}_{ij},\ldots,\mbox{\boldmath$\bf a$}_{ij}]^{\top}\in\mathbb{R}^{n_{i}\times
1}$ are the POD basis and the corresponding coefficient vector, respectively,
of the reduced DoFs of internal points, $n_{i}$. Substitution of Eq. (7.18)
into Eq. (7.17) leads to the reduced PD equations

$\displaystyle\\!\begin{bmatrix}&\\!\\!\mathbb{K}_{bb}&\,\,\,\,\mathbb{K}_{bi}\overline{\mbox{\boldmath$\Phi$}}\\\
&\\!\\!\overline{\mbox{\boldmath$\Phi$}}^{\top}\mathbb{K}_{ib}&\,\,\,\,\overline{\mbox{\boldmath$\Phi$}}^{\top}\mathbb{K}_{ii}\overline{\mbox{\boldmath$\Phi$}}\end{bmatrix}\begin{bmatrix}&\\!\\!\mathbb{U}_{b}\\\
&\\!\\!\mbox{\boldmath$\bf
a$}_{i}\end{bmatrix}=\begin{bmatrix}&\\!\\!\mathbb{B}_{b}\\\
&\\!\\!\overline{\mbox{\boldmath$\Phi$}}^{\top}\mathbb{B}_{i}\end{bmatrix}.$
(7.19)

$\mathbb{U}_{b}$ and $\mbox{\boldmath$\bf a$}_{i}$ can be found from Eq.
(7.19), and the displacement field of the internal computational domain
$\mathbb{U}_{i}$ can be reproduced by using Eq. (7.18). In this way, the
boundary conditions are accurately reproduced, and the number of DoFs of the
PD model dramatically decreases from $N=N_{i}+N_{b}$ to $N_{b}+n_{i}$.

We now turn our attention to the schematic representation of possible
reformulation of the known POD method (7.17)-(7.19). So, the VBC (2.3) are
intristically incorporated into the first line of Eq. (7.17). For the periodic
structure CM, we need to replace the VBC VBC (2.3) by the VPBC VBC (2.29) and
VBC (2.30). For the periodic counterpart of Eq. (7.17), it leads to both
elimination of the first line in Eq. (7.17) (i.e. elimination of
$\mathbb{K}_{bb}$, $\mathbb{K}_{bi}$, and $\mathbb{K}_{ib}$) and correction of
the rigidity matrix $\mathbb{K}_{ii}$ (see for details Chapter 19 in [60])
which also depends on the the properties of both the matrix and inclusions in
the unit cell $w$ with the numbers $N=N_{i}$ of DoFs; corrections of Eqs.
(7.18) and (7.19) are straightforward.

For statistically homogeneous random structure CMs, estimation of the
effective moduli $\mbox{\boldmath$\bf L$}^{*}$ (4.29) is reduced to a solution
of basic problem (3.29) for one inclusion $v_{i}$ inside the infinite
homogeneous matrix subjected to some homogeneous effective field. The
truncation method is used for the modeling of the infinite medium by an
increased-size sample $\overline{\overline{w}}=w\cup w_{\Gamma}$. The scheme
(7.17)-(7.19) is not altered whereas Eq. (7.18) is used for estimation of the
perturbator $\mbox{\boldmath$\bf L$}_{j}^{u\epsilon}(\mbox{\boldmath$\bf
s$}-\mbox{\boldmath$\bf x$}_{i})$ $(\mbox{\boldmath$\bf s$}\in v_{i}^{l}$) and
the concentration factors $\mbox{\boldmath$\bf R$}_{i}^{\l\omega}$ and
$\mbox{\boldmath$\bf A$}_{i}^{\l\omega}$ which, in turn, are exploited for
estimation of effective moduli $\mbox{\boldmath$\bf L$}^{*}$ by either the EFM
(4.37), (4.38) or the MTA (4.40), (4.41).

Comment 7.3. It is interesting that for random structure CMs subjected to body
force with compact support (see Subsection 4.2), Eqs. (7.17)-(7.19) of POD
method are simplified. So, we need to eliminate the first line in Eq. (7.17)
(i.e. to eliminat $\mathbb{K}_{bb}$, $\mathbb{K}_{bi}$, and $\mathbb{K}_{ib}$)
wherias the rigidity matrix $\mathbb{K}_{ii}$ is not altered and the body
force $\mathbb{U}_{i}$ has a compact support.In so doing, Block 3 (Machine
learning, see Eq. (7.16) and Fig. 8) is identical for both the LM and PM of
both the random and periodic structure peridynamic CM analyzed by POD methods.

##  8 Peridynamic micromechanics of laminated structure composites

In the subsequent presentation, the inclusions are considered as identical
aligned layers $v_{i}=\\{\mbox{\boldmath$\bf x$}=(x^{1},x^{2},x^{3})^{\top}|\
|x^{1}|\leq h\\}$ with the sickness $2h$ and a statistically homogeneous
distribution in the space. For applying the Mori–Tanaka [216] approach, the
presence of reference material is required; the layer $v^{(0)}$ is chosen to
represent the “matrix” phase. For linear state-based model, Eq. (3.22) is
presented in the form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{\cal H_{\bf
x}}\\!\\!\\!\mbox{\boldmath$\bf C$}(\mbox{\boldmath$\bf
x$},\mbox{\boldmath$\bf q$})[\mbox{\boldmath$\bf u$}(\mbox{\boldmath$\bf
q$})-\mbox{\boldmath$\bf u$}(\mbox{\boldmath$\bf
x$})-\overline{}\mbox{\boldmath$\varepsilon$}(\mbox{\boldmath$\bf
q$}-\mbox{\boldmath$\bf x$})]~{}d\mbox{\boldmath$\bf q$}$ $\displaystyle=$
$\displaystyle-\\!\\!\\!\int_{\cal H_{\bf x}}\\!\\!\\!\mbox{\boldmath$\bf
C$}_{1}(\mbox{\boldmath$\bf x$},\mbox{\boldmath$\bf
q$})\overline{}\mbox{\boldmath$\varepsilon$}(\mbox{\boldmath$\bf
q$}-\mbox{\boldmath$\bf x$})]~{}d\mbox{\boldmath$\bf q$}.$ (8.1)

Owing to the form of structure, the displacement field can be obtained one
from another by parallel transformation

$\displaystyle\mbox{\boldmath$\bf u$}(\mbox{\boldmath$\bf
x$})=\mbox{\boldmath$\bf u$}({\mbox{\boldmath$\bf
x$}_{0}})+\overline{}\mbox{\boldmath$\varepsilon$}^{t}(\mbox{\boldmath$\bf
x$}-\mbox{\boldmath$\bf x$}_{0}),\ \ \
\overline{}\mbox{\boldmath$\varepsilon$}=\overline{}\mbox{\boldmath$\varepsilon$}^{n}+\overline{}\mbox{\boldmath$\varepsilon$}^{t},$
(8.2)

where $\mbox{\boldmath$\bf x$}=(x^{1},x^{2},x^{3})^{\top}$,
$\mbox{\boldmath$\bf x$}_{0}=(x^{1}_{0},0,0)^{\top}$, and
$\overline{\varepsilon}_{ij}$ is an arbitrary symmetric tensor decomposed on
the normal
$\overline{}\mbox{\boldmath$\varepsilon$}^{n}=\overline{\varepsilon}^{n}_{ij}=\overline{\varepsilon}_{ij}(\delta_{i1}+\delta_{j1}-\delta_{1i}\delta_{1j})$
and transversal
$\overline{}\mbox{\boldmath$\varepsilon$}^{t}=\overline{\varepsilon}^{t}_{ij}=\overline{\varepsilon}_{ij}(1-\delta_{i1})(1-\delta_{j1})$
constituents. We introduce a cross-section of the horizon region ${\cal
H}_{\bf x}$ ($\mbox{\boldmath$\bf x$}=(x^{1},0,0)^{\top})$ by the plane
$q^{1}=$const. ($x^{1}-l_{\delta}\leq q^{1}\leq x^{1}+l_{\delta})$ ${\cal
S}^{q^{1}}_{x^{1}}={\cal H}_{\bf x}\cap\\{\mbox{\boldmath$\bf
q$}|\mbox{\boldmath$\bf q$}=(q^{1},0,0)^{\top},\ x^{1}-l_{\delta}\leq
q^{1}\leq x^{1}+l_{\delta}\\}.$ Then Eq. (8.1) is reduced to the following
equation

$\displaystyle\int_{{\cal H}_{x^{1}}}\mbox{\boldmath$\bf
C$}^{1}(x^{1},q^{1})\big{[}\mbox{\boldmath$\bf u$}(q^{1})-\mbox{\boldmath$\bf
u$}(x^{1})-\overline{}\mbox{\boldmath$\varepsilon$}^{n}(q^{1}-x^{1})\big{]}~{}dq^{1}=\hat{}\mbox{\boldmath$\bf
b$}(x^{1})\overline{}\mbox{\boldmath$\varepsilon$},$ (8.3)

where

$\displaystyle\\!\\!\\!\\!\\!\\!\mbox{\boldmath$\bf
C$}^{1}(x^{1},q^{1})=\\!\int_{{\cal
S}^{q^{1}}_{x^{1}}}\\!\\!\mbox{\boldmath$\bf C$}(\mbox{\boldmath$\bf
x$},\mbox{\boldmath$\bf q$})~{}dq^{2}dq^{3},\ \ \widehat{\mbox{\boldmath$\bf
b$}}(x^{1})=-\\!\int_{\cal H_{\bf x}}\\!\\!\mbox{\boldmath$\bf
C$}_{1}(\mbox{\boldmath$\bf x$},\mbox{\boldmath$\bf
q$})\otimes(\mbox{\boldmath$\bf q$}-\mbox{\boldmath$\bf
x$})d\mbox{\boldmath$\bf q$}$ (8.4)

and the right-hand side of Eq. (8.3) has a compact support $|x^{1}|\leq
h+l_{\delta}$. In Fig. 9, we present a scheme of obtaining the transformed
micromodulus $\mbox{\boldmath$\bf C$}^{1}(x^{1},q^{1})$ (8.41) by the use of
the cross-section ${\cal S}^{q^{1}}_{x^{1}}$ of the $3D$ horizon region ${\cal
H}_{\bf x}$ by the plane $\\{\mbox{\boldmath$\bf q$}|\mbox{\boldmath$\bf
q$}=(q^{1},0,0)^{\top},\ x^{1}-l_{\delta}\leq q^{1}\leq x^{1}+l_{\delta}\\}$.
It is interesting that for the constant micromodulus $\mbox{\boldmath$\bf
C$}(\mbox{\boldmath$\bf x$},\mbox{\boldmath$\bf q$})\equiv$const.
($\mbox{\boldmath$\bf q$}\in{\cal H}_{\bf x})$ (horizon region ${\cal H}_{\bf
x}\subset R^{3}$ is colored by one color), the transformed micromodulus
$\mbox{\boldmath$\bf C$}^{1}(x^{1},q^{1})$ (8.41) is an inhomogeneous function
of $q^{1}\in{\cal H}_{x^{1}}$ (because of this, the $1D$ horizon region ${\cal
H}_{x}\subset R^{1}$ in Fig. 9 is colored by the same color of varying
intensity).

Fig. 9: Schematic transformation of $3D$ horizon region ${\cal H}_{\bf x}$
into $1D$ horizon region ${\cal H}_{x^{1}}$

It should be also mentioned that Eq. (8.3) coincides (with accuracy to
notations) with the corresponding $1D$ equation for one inclusion in the
infinite homogeneous bar (we need only to replace the matrix
$\mbox{\boldmath$\bf C$}^{1}$ and vectors $\bf u$,
$\overline{}\mbox{\boldmath$\varepsilon$}$,
$\overline{}\mbox{\boldmath$\varepsilon$}^{n}$ by the corresponding scalar
variables):

$\displaystyle\int_{{\cal
H}_{x^{1}}}C(x^{1},q^{1})\big{[}u(q^{1})-u(x^{1})-\overline{\varepsilon}(q^{1}-x^{1})\big{]}~{}dq^{1}=\hat{b}(x^{1})\overline{\varepsilon}.$
(8.5)

It means that for Eq. (8.3), we can repeat the development of generalized
methods of the EFM and MTM (and computational homogenization for periodic
structures) obtained for 1D case (8.3). In particular, the solution of Eq.
(8.3) can be presented in the form ($\mbox{\boldmath$\bf
x$}=(x^{1},0,0)^{\top}\in v_{i}^{l}$) $\mbox{\boldmath$\bf
u$}(\mbox{\boldmath$\bf x$})=\widetilde{}\mbox{\boldmath$\bf
B$}(\mbox{\boldmath$\bf
x$})\widetilde{\overline{}\mbox{\boldmath$\varepsilon$}}$, where
$\widetilde{}\mbox{\boldmath$\bf B$}(\mbox{\boldmath$\bf
x$})=\big{(}\mbox{\boldmath$\bf B$}^{n}(\mbox{\boldmath$\bf
x$}),\mbox{\boldmath$\bf B$}^{t}(\mbox{\boldmath$\bf x$})\big{)}$ and
$\widetilde{\overline{}\mbox{\boldmath$\varepsilon$}}=(\overline{}\mbox{\boldmath$\varepsilon$}^{n},\overline{}\mbox{\boldmath$\varepsilon$}^{t})^{\top}$
are the decompositions of the corresponding value on the normal and tangential
parts. After that, we can estimate the stresses
$\mbox{\boldmath$\sigma$}(\mbox{\boldmath$\bf x$})$ (2.26) at the interval
$l^{\delta}_{0}=\\{\mbox{\boldmath$\bf x$}|\mbox{\boldmath$\bf
x$}=(x^{1},0,0)^{\top},\ -h-l_{\delta}\leq x^{1}\leq h+l_{\delta}\\}$ and the
effective moduli $\mbox{\boldmath$\bf L$}^{*}$ (3.47) (see for details [60],
[68]).

Comment 8.1. The generalization of the present approach (obtained in [68] for
the linear bond-based model) to the generalized MTM and EFM for multilayered
CM with linearized state-based peridynamic (as in [64]) properties is
straightforward. It is expected that the potential exploitation of these
methods (MTM, EFM, and periodic homogenization method) for the estimation of
effective moduli of peridynamic multilayered CM will lead to different
results. It can be considered the prescribed random orientation of the
anisotropic layers (with the same normals $\mbox{\boldmath$\bf n$}=(\pm
1,0,0)^{\top}$). The analysis of clustered CMs, where the aligned hierarchical
clusters contain some anisotropic layers, is straightforward.

Comment 8.2. It should be mentioned that the fields $\mbox{\boldmath$\bf
u$}(\mbox{\boldmath$\bf x$})$ and
$\mbox{\boldmath$\sigma$}(\mbox{\boldmath$\bf x$})$, see Eq. (4.11) are
assumed to be invariant to the parallel transformations along the layers. It
means, that a number of popular loading (e.g. impact, torsion, deflections,
and buckling from both a static and dynamic point of view) can not be reduced
from 3D to 1D problems by the method proposed (for both local micromechanics
and PD micromechanics). On another side, we did not exploit any additional
assumption, such as, e.g. the scale separation hypothesis $l_{\delta}/2h\to 0$
or other hypotheses used in [10], [12], [17], [97], [143], [145], [175],
[197], [199], [200], [240], [295] which are based on the consideration of PD
bounds as the structural elements. Thus, the Mori-Tanaka method (4.39) is
reduced to Eq. (8.3) with 1D argument for one (segment) inclusion in the
infinite matrix (see Fig. 10a). The method proposed can be slightly modified

Fig. 10: Schematic representation of the field of layers $v_{i}$ to the
periodic infinite field of layers (which, perhaps, are not identical). In such
a case the problem for 1D inclusion inside an infinite matrix (see Fig. 10a)
should be replaced by the 1D problem at the unit cell
$[-l^{\Omega},l^{\Omega}]$ (see Fig. 10b) with the VPBC [52]. Furthermore, in
the framework of the hypothesis (8.2), a finite number of layers of
deterministic structures $(L^{\Omega}\leq x^{1}\leq L^{\Omega}$) of the 3D
problem is reduced to the 1D problem with the voluetric boundary conditions
(see Fig. 10c). In such a case, free edge effect can be easily considered and
application of the higher-order homogenization theories, or plate-type
homogenization strategies [159] and [293] are not required.

Comment 8.3. The problems for nonlinear peridynamic laminated structure (with
the possible imperfect interface model, as in, e.g., Kapitza thermal
resistance model [283], [316]) are also affortable. So, a problem linearity
was only exploited in the decomposition of the integrand in Eqs. (8.3) for
separate integration of the micromodulus in Eq. (8.41) over the cross-sections
${\cal S}^{q^{1}}_{x^{1}}$. For nonlinear model, a bond force density
$\mbox{\boldmath$\bf f$}^{\rm
bond}(\mbox{\boldmath$\eta$},\mbox{\boldmath$\xi$},\mbox{\boldmath$\bf x$})$
(2.18) is not decomposed as in Eq. (2.21). But we can estimate the averages of
$\mbox{\boldmath$\bf f$}^{\rm
bond}(\mbox{\boldmath$\eta$},\mbox{\boldmath$\xi$},\mbox{\boldmath$\bf x$})$
over the cross-sections ${\cal S}^{q^{1}}_{x^{1}}$. Owing to Eq. (8.3), we
obtain a nonlinear version of Eq. (8.3) with $1D$ argument $x^{1}$ which can
be solved, e.g., by the fixed-point iterative method (as in Eq. (3.20)).

##  9 Conclusion

Basic concepts, methods, their interdependence, and potential generalizations
in the LM are straightforwardly generalized to the PM. In the sense of
generality of the hypotheses exploited, 70 years of the progress of the EFM in
the LM [from Foldy(1945) [122] to Buryachenko (2015) [47], [49]) is compressed
to a few years of their development in the PM. The proposed universal tool
(called CAM) is sufficiently flexible and based on physically clear hypotheses
that can be modified and improved if necessary (up to abandonment of these
hypotheses that are forced by challenging achievement rather than only by
pursuing an abstruse theoretic exercise) in the framework of a unique scheme
for analyses of a wide class of micromechanical problems, e.g., statistically
homogeneous and inhomogeneous media, inhomogeneous loading (inhomogeneous body
force is included), nonlinear and nonlocal constitutive laws of phases, and
coupled physical phenomena. Although PM of both the random and periodic
structure CMs was historically developed as a generalization of the
corresponding methods of the LM, we also demonstrated that some methods
developed in PM initiate the new methods of LM unknown before (i.e. the
methods of both the LM and PM mutually enrich one another). On the other side,
for the presentation of PM as a unified theory, we describe PM as the
formalized schemes of blocked (or modular) structures so that the experts
developing one block need not be experts in the underlying another block (this
is a good background for effective collaborations of different teams in so
many multidisciplinary areas as PM). The opportunity for the creation of this
blocked structure of the PM is supported by a critical generalization of CAM
which is extremely flexible, robust, and general. Lastly, this review has a
dual goal and is intended for the target auditorium, which can be subdivided
into two groups. It can be used as guidance for practitioners in the selection
of appropriate PM methods based on the characteristics and needs of the
problem being considered. On the other side, it can be also exploited by
developers in the choice of attack directions in broad prospective problems of
the new field of peridynamic micromechanics.

Acknowledgements:

The authors acknowledge Dr. Stewart A. Silling for the fruitful personal
discussions, encouragements, helpful comments, and suggestions.

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