Generalized Mori–Tanaka Approach in Micromechanics of Peristatic Random Structure Composites

Abstract

A statistically homogeneous random matrix medium with the bond-based peridynamic properties (see Silling, J Mech Phys Solids 48:175–209, 2000) of constituents is considered. For the media subjected to remote homogeneous volumetric boundary loading, one proved that the effective behavior of this media is described by conventional effective constitutive equation which is intrinsic to the local elasticity theory. It was made by the most exploitation of the popular tools and concepts used in conventional elasticity of CMs and adapted to peristatics. This is extraction from the material properties a constituent of the matrix properties. Effective properties moduli are expressed through the introduced local stress polarization tensor averaged over the extended inclusion phase rather than in an entire space. One proposes a generalization of a dilute approximation method to their peristatic counterpart in the sense that the volume fraction of the particles is small and their mutual interaction can be neglected. As in a classical approach, the essential assumption in the generalized Mori–Tanaka method (MTM) states that each extended inclusion behaves as an isolated one in the infinite matrix and subject to some effective strain field coinciding with the average strain in the truncated matrix. Comparative numerical analyses of both the dilute approximation and MTM method are performed for 1D case.

Introduction

Proposal of peridynamics by Silling [54] (see also [8, 38, 40, 57, 58]), a nonlocal theory of solid mechanics, initiated the explosive character of the progress in different physical phenomena based on the replacement of the classical partial differential equation of balance of linear momentum by the integral equation which is free of any spatial derivatives of displacement. In nonlocal peridynamic theory, the equilibrium of a material point is achieved by a summation of internal forces produced by surrounding points within a given distance called the horizon while in the classical theory such interactions are only exerted by adjacent points through the contact forces. Generally in peridynamics, the state-based approach permits the response of a material at a point to depend collectively on the deformation of all bonds connected to the point within its finite radius horizon [57, 58] via a response function that completely describes the interaction. It means that the forces between two peridynamic nodes depend also on deformations of other bonds surrounding these nodes within the horizon which is linked with a characteristic length scale of the material and of the considered phenomenon. The horizon can encompass discontinuities or different materials. The initial formulation of peridynamics (called bond-based peridynamics) is that in which interactions only occur between pairs of material points within a horizon. As is well known, a direct consequence of this assumption is that the Poisson’s ratio for isotropic linear materials is fixed at ν = 1/4 and ν = 1/3 for three-dimensional and two-dimensional models, respectively [54]. The major advantages of the state-based approach include a material response depending on collective quantities (like volume change or shear angle), which allows constitutive models from the conventional theory of solid mechanics to be incorporated directly within the peridynamic approach (see, for example, [1, 56]). However, this paper will use the bond-based approach as it is most suitable to the chosen implementation. The term “peristatics” is used analogously to Mikata [43] to differentiate the static problems considered in the current paper from the dynamic problems.

The mentioned achievements of peridynamics were mostly performed for either the initially homogeneous materials or the deterministic structures. However, estimation of macroscopic effective response of heterogeneous media with random structures in an averaged (or homogenized) meaning in terms of the mechanical and geometrical properties of constituents is a central focus of micromechanics denoted as micro-to-macro modeling (see for references, e.g., [11]). The general results establishing the links between the effective properties and the corresponding mechanical and transformation influence functions were inspired by Hill [26] for locally elastic composites. Some basic representations analogous to the mentioned above were generalized by Buryachenko [14, 17] to the thermoperistatics of CMs. The stress field estimations in the constituents, it its turn, are based on a substitution into the one or another micromechanical scheme of a solution (called basic problem, see for details [18]) for one inclusion inside the infinite matrix subjected to some effective field. So, for locally elastic random structure CMs, a number of micromechanical models inspired by Eshelby [24] (see for references [50, 65]) were proposed in the literature for describing the thermoelastic behavior of composites with ellipsoidal inclusions (see for references, e.g., [11, 61]). Buryachenko [13, 14, 16] established similarity of the general integral equations for both locally elastic CMs and peristatic ones that opens the opportunities for straightforward generalization of their solutions for locally elastic CMs (see [13, 14, 16]) to the peristatic ones (instead of the popular simplified methods such as, e.g., the mixture theory, see [4,5,6, 29]). Analogous similarity was also established in [13, 14] nonlocally elastic ones (in the sense of Eringen [23]).

At first, the scheme by the finite element analysis (FEA) for the locally elastic basic problem for one inclusion is generalized to the peristatic sample of increasing size subjected to the volumetric displacement loading. The infinite medium is modeled by an increased size sample with controlled stabilization of the solution for this sample. A considerable number of methods are known in the linear theory of composites that yield the effective elastic constants and stress field averages in the components. Appropriate, but by no means exhaustive, references are provided by the reviews [11, 36, 44, 47, 48, 61, 64]. It appears today that variants of the effective medium method [27, 32] and the Mori–Tanaka [45] method (MTM, [7]) are the most popular and widely used methods. The effective field hypothesis (EFH, also called the H1a hypothesis, p. 253 in [11]) is apparently the most fundamental and most exploited concept of micromechanics. The idea of this concept dates back to Mossotti [46], who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. Markov [41] presented comprehensive reviews of the 150-year history of this concept accompanied by some famous formulae with extensive references. When an infinite number of randomly distributed inhomogeneities are analyzed, the average field in the matrix can be considered as good approximations of the effective field that was pointed out by Benveniste [7] in a reformulation of the MTM for locally elastic CMs. This approach in accompany with the general results for the random structure peristatic CMs (see [14, 17]) is generalized to its peristatic counterpart in the current paper.

Namely, the paper is originated as follows. In Section 2, we give a short introduction into the peristatic theory of solids adapted for a subsequent presentation. For random structure peristatic CMs, some averages for the field parameters and general representations for the effective elastic moduli are presented in Section 3. In Section 4, one presents a scheme of the truncation method for one inclusion in the infinite matrix modeled by an increased size sample with controlled stabilization of the solution for this sample. Micromechanical models (generalized the dilute approximation and the Mori–Tanaka method, MTM) are obtained in Section 5. Numerical results for 1D example of the mentioned micromechanical models are presented in Section 6. A convergence of numerical results for the peristatic composite bar to the corresponding exact evaluation for the local elastic theory are shown.

Preliminaries: Basic Equations of Peristatics

Let a linear elastic body occupy an open simply connected bounded domain wRd with a smooth boundary Γ0 and with an indicator function W and space dimensionality d (d = 2 and d = 3 for 2-D and 3-D problems, respectively). The domain w contains a homogeneous matrix v(0) and a heterogeneities v with indicator functions V and bounded by the closed smooth surfaces Γ0 defined by the relations Γ(x) = 0 (xΓ0), Γ0(x) > 0 (xv), and Γ0(x) < 0 (xv). That is to say, we consider the methods for the matrix-inclusion type microtopologies, when the expression “composite” is often used in the present instead of the more general designation “heterogeneous material.” Initially no restrictions are imposed on the elastic symmetry of the phases or on the geometry of heterogeneities.

We first consider the basic equations of locally elastic composites

$$ \begin{array}{@{}rcl@{}} \nabla\boldsymbol{\sigma}(\mathbf{x})&=&-\mathbf{b}(\mathbf{x}), \end{array} $$
(1)
$$ \begin{array}{@{}rcl@{}} \boldsymbol{\sigma}(\mathbf{x})&=&\mathbf{L}(\mathbf{x})\boldsymbol{\varepsilon}(\mathbf{x}), \ \ \text{or}\ \ \ \boldsymbol{\varepsilon}(\mathbf{x})=\mathbf{M}(\mathbf{x})\boldsymbol{\sigma}(\mathbf{x}), \end{array} $$
(2)
$$ \begin{array}{@{}rcl@{}} \boldsymbol{\varepsilon}(\mathbf{x})&=&[\nabla {\otimes}\mathbf{u}+(\nabla{\otimes}\mathbf{u})^{\top}]/2, \ \ \nabla\times\boldsymbol{\varepsilon}(\mathbf{x})\times\nabla=\mathbf{0}, \end{array} $$
(3)

where ⊗ and × are the tensor and vector products, respectively, and (.) denotes matrix transposition. The body force tensor b can be generated, e.g., by either gravitational loads or a centrifugal load. L(x) and M(x)L(x)− 1 are the known phase stiffness and compliance fourth-order tensors, and the common notation for contracted products has been employed: [Lε]ij = Lijklεkl (i,j,k,l = 1,…,d). In particular, for isotropic constituents, the local stiffness tensor L(x) is given in terms of the local bulk modulus k(x) and the local shear modulus μ(x):

$$ \mathbf{L}(\mathbf{x})=(dk,2\mu)\equiv dk(\mathbf{x})\mathbf{N}_1+2\mu(\mathbf{x})\mathbf{N}_2, $$
(4)

N1 = δδ/d, N2 = IN1 (d = 2 or 3); δ and I are the unit second-order and fourth-order tensors. For all material tensors g (e.g., L,M), the notation \(\mathbf {g}_{1}(\mathbf {x})\equiv \mathbf {g}(\mathbf {x})-\mathbf {g}^{(0)}\) is used.

Substitution of Eqs. 2 and 31 into Eq. 1 yields a representation of the local equilibrium Eq. 1 in the form

$$ {}^L\hat{\mathcal{L}}({}^L\mathbf{u})(\mathbf{x})+\mathbf{b}(\mathbf{x})=\mathbf{0},\ \ \ {}^L\hat{\mathcal{L}}({}^L\mathbf{u})(\mathbf{x}):=\nabla[\mathbf{L}\nabla{}^L\mathbf{u}(\mathbf{x})], $$
(5)

where \(^{L}\hat {{\mathcal{L}}}(^{L}\mathbf {u})(\mathbf {x})\) is an elliptic differential operator of the second order. The superscripts L(⋅) will be corresponded hereafter to the local case.

In this section, we also summarize the linear peristatics model introduced by Silling [54] (see also [35, 55]). An equilibrium equation is free of any spatial derivatives of displacement (contrary to Eq. 5) and presented in the form of a Fredholm equation of second kind

$$ \hat{\mathcal{L}}(\mathbf{u})(\mathbf{x})+\mathbf{b}(\mathbf{x})=\mathbf{0}, \ \ \ \hat{\mathcal{L}}(\mathbf{u})(\mathbf{x}):={\int}_{\mathcal{H}_{\mathbf{x}}} \mathbf{f}(\mathbf{u}(\hat {\mathbf{x}})-\mathbf{u}(\mathbf{x}),\hat{\mathbf{x}}-\mathbf{x},\mathbf{x})d\hat {\mathbf{x}} , $$
(6)

where f is a pairwise force function (called also a bond force) whose value is the force vector that the point located at \(\hat {\mathbf {x}}\) (in the reference configuration) exerts on the point located at x (also in the reference configuration); the third argument x of f (6) can be dropped for the homogeneous media \( \mathbf {f}(\mathbf {u}(\hat {\mathbf {x}})-\mathbf {u}(\mathbf {x}),\hat {\mathbf {x}}-\mathbf {x},\mathbf {x})=\mathbf {f}(\mathbf {u}(\hat {\mathbf {x}})-\mathbf {u}(\mathbf {x}),\hat {\mathbf {x}}-\mathbf {x})\). The finite domain (termed horizon) of the integral \({\mathcal{H}}_{\mathbf {x}}:\{\hat {\mathbf {x}}\in w| |\mathbf {x}-\hat {\mathbf {x}}| \leq l_{\delta } \}\) means that the point x cannot “see” outside \({\mathcal{H}}_{\mathbf {x}}\) while a positive number ≤ lδ is called the horizon. The dimension of f is [f] = F/L2d where L and F denote the physical dimensions of length and force. Equations 51 and 61 have the same form for both local and non-local formulation with the different operators (52) and (62). The body force density function b(x) is assumed to be self-equilibrated

$$ \int \mathbf{b}(\mathbf{x})d\mathbf{x}=0 $$
(7)

and vanished outside some loading region: b(x) = 0 for |x| > lδ.

The notations \(\boldsymbol {\xi }=\hat {\mathbf {x}}-\mathbf {x}\) and \(\boldsymbol {\eta }=\mathbf {u}(\hat {\mathbf {x}})-\mathbf {u}(\mathbf {x})\) are introduced to represent the relative position and the relative displacement, respectively. The relative displacements are assumed to be small (see for details [58]). Only points \(\hat {\mathbf {x}}\) inside some neighborhood \({\mathcal{H}}_{\mathbf {x}}\) of x interact with x: \( \mathbf {f}(\boldsymbol {\eta },\boldsymbol {\xi },\mathbf {x})\equiv \mathbf {0}\ \ \ \forall \hat {\mathbf {x}}\not \in {\mathcal{H}}_{\mathbf {x}}\). The vector \(\boldsymbol {\xi }=\hat {\mathbf {x}}-\mathbf {x}\) (\(\hat {\mathbf {x}}\in {\mathcal{H}}_{\mathbf {x}}\)) is called a bond to x, and the collection of all bonds to x form the domain \({\mathcal{H}}_{\mathbf {x}}\). It is assumed that the neighborhood \({\mathcal{H}}_{\mathbf {x}}\) is spherical: \({\mathcal{H}}_{\mathbf {x}}=\{\hat {\mathbf {x}}:\ |\hat {\mathbf {x}}-\mathbf {x}| l_{\delta }\}\) and the horizon lδ does not depend on x (although, there is no restriction on the form of \({\mathcal{H}}_{\mathbf {x}}\) in Sections 35). The pairwise force function f is required to have the following properties (for ∀η,ξ and ∀xw):

$$ \mathbf{f}(-\boldsymbol{\eta},-\boldsymbol{\xi},\mathbf{x}+\boldsymbol{\xi})\equiv -\mathbf{f}(\boldsymbol{\eta},\boldsymbol{\xi},\mathbf{x}) \ \ \ \forall \boldsymbol{\eta},\boldsymbol{\xi} $$
(8)

meaning assurance of conservation of linear momentum, and

$$ (\boldsymbol{\eta}+\boldsymbol{\xi})\times\mathbf{f}(\boldsymbol{\eta},\boldsymbol{\xi},\mathbf{x})\equiv \mathbf{0} \ \ \ \forall \boldsymbol{\eta},\boldsymbol{\xi}, $$
(9)

which assures conservation of angular momentum and means that the force vector between these particles \(\hat {\mathbf {x}}\) and x is parallel to their current relative position. The function f contains all constitutive information about the materials.

A linearized version of the theory for a microelastic homogeneous material takes the form

$$ \mathbf{f}(\boldsymbol{\eta},\boldsymbol{\xi},\mathbf{x})=\mathbf{C}(\boldsymbol{\xi},\mathbf{x})\boldsymbol{\eta} \ \ \ \forall \boldsymbol{\eta},\boldsymbol{\xi}, $$
(10)

where C(ξ,x) ≡0 at |ξ| > lδ and the properties of the material’s micromodulus function C containing all constitutive information are discussed in detail in Silling [54]. For consistency with Newton’s third law (also following the requirement from Eq. 8), the micromodulus function C for the homogeneous materials must be symmetric with respect to its argument as well as with respect to its tensor structure C(−ξ) = C(ξ) = C(ξ) (∀ξ). Substitution of Eq. 10 into 8 leads to the following property of C(ξ,x): C(ξ,x) = C(−ξ,x + ξ).

For example, for the micromodulus functions with the step-function and triangular profiles, we have

$$ \mathbf{C}(\boldsymbol{\xi})= \mathbf{C} V(\mathcal{H}_{\mathbf{x}}), \ \ \mathbf{C}(\boldsymbol{\xi})=\mathbf{C}(1-|\boldsymbol{\xi}|/l_{\delta})V(\mathcal{H}_{\mathbf{x}}), $$
(11)

respectively, where \(V({\mathcal{H}}_{\mathbf {x}})\) is the indicator function of \({\mathcal{H}}_{\mathbf {x}}\). The peristatic solution of Eq. 6 with C described by Eq. 11 is in detail investigated by both numerical and analytical methods in 1D case (see [10, 43, 59, 62]) and 2D case (see [29, 30]). If we assume a linear microelastic material then, in general, the stiffness tensor for the linear microelastic materials can be shown to read as (see [20, 21, 54])

$$ \mathbf{C}(\boldsymbol{\xi})= \lambda (\boldsymbol{\xi})\boldsymbol{\xi}\otimes\boldsymbol{\xi}, $$
(12)

where a scalar function λ(ξ) depends on the specific material model and, for a linear isotropic material, λ depends on ξ only through ξ = |ξ|. For the special case of a linear microelastic materials

$$ \mathbf{C}(\boldsymbol{\xi})=c\frac{\boldsymbol{\xi}\otimes\boldsymbol{\xi}}{|\boldsymbol{\xi}|^3}V(\mathcal{H}_{\mathbf{x}}), \ \ \ \text{i.e} \ \ \ C_{ij}(\boldsymbol{\xi})=c\frac{\xi_i\xi_j}{ (\xi_k\xi_k)^{3/2}}V(\mathcal{H}_{\mathbf{x}}), $$
(13)

where a constant c depends again on the material but also on the dimension d of the problem. The proportionality factor c (13) (or C (11)) is to be determined in such a way that the deformation energy density (or the constitutive equation) of a homogeneous body under homogeneous loading arising from the peridynamic model coincides with the corresponding value from the classical linear elasticity theory (see, e.g., [20, 22, 58]). A direct consequence of the assumption (12) is that the Poisson’s ratio for isotropic linear microelastic materials is fixed at the value of ν = 1/4 in 3D or ν = 1/3 in 2D [54].

For two-phase composite medium containing any two points x and \(\hat {\mathbf {x}}\) in Rd, the micromodulus \(\mathbf {C}(\boldsymbol {\xi })=\mathbf {C}(\mathbf {x},\hat {\mathbf {x}})\) (\(\boldsymbol {\xi }=\hat {\mathbf {x}}-{\mathbf {x}}\)) is given by the formula

$$ \begin{array}{@{}rcl@{}} \mathbf{C}(\mathbf{x},\hat{\mathbf{x}})=\left\{\begin{array}{ll} \mathbf{C}^{(1)}(\mathbf{x},\hat{\mathbf{x}}), & \text{for} \ \mathbf{x},\hat{\mathbf{x}}\in v,\\ \mathbf{C}^{(0)}(\mathbf{x},\hat{\mathbf{x}}), & \text{for} \ \mathbf{x},\hat{\mathbf{x}}\in v^{(0)},\\ \mathbf{C}^{i}(\mathbf{x},\hat{\mathbf{x}}), & \text{for} \ \mathbf{x}\in v,\hat{\mathbf{x}}\in v^{(0)} \text{or}\ \ \mathbf{x}\in v^{(0)},\hat{\mathbf{x}}\in v, \end{array}\right. \end{array} $$
(14)

which can also be presented in the form

$$ \begin{array}{@{}rcl@{}} \mathbf{C}(\mathbf{x},\hat{\mathbf{x}}) &=& \mathbf{C}^{(1)}(\mathbf{x},\hat{\mathbf{x}})V^{(1)}(\mathbf{x})V^{(1)}(\hat {\mathbf{x}})+\mathbf{C}^{(0)}(\mathbf{x},\hat{\mathbf{x}})V^{(0)}(\mathbf{x})V^{(0)}(\hat {\mathbf{x}}) \\ && + \mathbf{C}^{i}(\mathbf{x},\hat{\mathbf{x}})[V^{(1)}(\mathbf{x})V^{(0)}(\hat {\mathbf{x}})+ V^{(0)}(\mathbf{x})V^{(1)}(\hat {\mathbf{x}})]. \end{array} $$
(15)

The material parameters C(1) and C(0) are intrinsic to each phase and can be determined through experiments. Bonds connecting points in the different materials are characterized by micromodulus Ci, which can be chosen such as, e.g.,

$$ \begin{array}{@{}rcl@{}} \mathbf{C}^i(\mathbf{x},\hat {\mathbf{x}})&=&(\mathbf{C}^{(0)}(\mathbf{x},\hat {\mathbf{x}})+\mathbf{C}^{(1)}(\mathbf{x},\hat {\mathbf{x}}))/2, \end{array} $$
(16)

(see [3, 55]) where \(\mathbf {x},\hat {\mathbf {x}}\in v_{\varGamma }\) is called the interaction interface interface (see for details [51]). Important to note that by this averaging, the interface is fuzzy, i.e., not sharp as it would be in a local formulation. Adaptive grid refinement technique was proposed in [9] for analysis of peristatic problems in the vicinity of interaction interface that involves a variable horizon size.

The peridynamic theory is traditionally based on the use of the displacement field u(x) rather than either the stress σ(x) or strain ε(x) fields which are not conceptually necessary. However, introduction of the notion of stress is helpful, as one can use it to formulate stress-strain relations for exploiting of well developed tool of classical elasticity theory in subsequent application of the present theory for heterogeneous materials. For subsequent convenience, one introduces a vector valued function \(\hat {\mathbf {f}}:\ R^{d}\times R^{d}\to R^{d}\) by

$$ \begin{array}{@{}rcl@{}} \hat{\mathbf{f}}(\mathbf{p},\mathbf{q})=\left\{\begin{array}{ll} \mathbf{f}(\mathbf{u}(\mathbf{p})-\mathbf{u}(\mathbf{q}),\mathbf{p}-\mathbf{q},\mathbf{q}), & \text{if} \ \mathbf{p},\mathbf{q}\in w,\\ \mathbf{0}, & \text{otherwise}. \end{array}\right. \end{array} $$
(17)

It is assumed that \(\hat {\mathbf {f}}(\hat {\mathbf {x}},\mathbf {x})\) is Riemann-integrable that does not imply the boundedness of \(\hat {\mathbf {f}}(\hat {\mathbf {x}},\mathbf {x})\) as \(|\hat {\mathbf {x}}-\mathbf {x}|\to 0\). Then, by adapting Cauchy\(\prime \)s notion of stress in a crystal, one can define the “peristatic stress” σ(z) at the point z to be the total force that all material points \(\hat {\mathbf {x}}\) to the right of z exert on all material particles to its left (see, e.g., [59, 62]). For 2D and 3D cases (d = 2,3),

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\sigma}(\mathbf{x}) &=& \mathcal{L}^{\sigma}(\mathbf{u}), \\ \mathcal{L}^{\sigma}(\mathbf{u}) &:=& \frac{1}{2} {\int}_S{\int}_0^{\infty}{\int}_0^{\infty} (y+z)^{d-1}\hat{\mathbf{f}}(\mathbf{x}+y\mathbf{m},\mathbf{x}-z\mathbf{m})\otimes\mathbf{m} dzdyd\varOmega_{\mathbf{m}}, \end{array} $$
(18)

where S stands for the unit sphere and dΩm denotes a differential solid angle on S in the direction of any unit vector m. Lehoucq and Silling [35] proved Eq. 18 for 3D case while the case d = 2 can be justified in a similar manner. Equation 18 at d = 1 can also be reduced to Eq. 75 (see Section 6).

Some Averages for the Field Parameters and General Representations for the Effective Elastic Moduli

Volumetric Homogeneous Displacement Loading and Some Averages for the Field Parameters

It is assumed that the representative macrodomain w contains a statistically large number of realizations α (providing validity of the standard probability technique) of heterogeneities viv(k) of the constituent v(k) (i = 1,2,…; k = 1,2,…,N). A random event α belongs to a sample space \(\mathcal {A}\), over which a probability density p(x,α) is defined (see, e.g., [64]). For any given α, any random function g(x,α) (e.g., g = V,V(k),σ,ε) is defined explicitly as one particular member, with label α, of an ensemble realization. Then, the mean, or ensemble average is defined by the angle brackets enclosing the quantity g

$$ \langle\mathbf{g}\rangle(\mathbf{x})={\int}_{\mathcal{A}}\mathbf{g}(\mathbf{x},\alpha)p(\mathbf{x},\alpha)d\alpha. $$
(19)

No confusion will arise below in notation of the random quantity g(x,α) if the label α is removed. One treats two material length scales (see, e.g, [61]): the macroscopic scale L, characterizing the extent of w, and the microscopic scale a, related with the heterogeneities vi. Moreover, one supposes that applied field varies on a characteristic length scale Λ. We consider the scale separation of both the material scales and field one is

$$ L\gg\varLambda\geq a\geq l_{\delta}. $$
(20)

All the random quantities under discussion are described by statistically homogeneous random fields.

Due to nonlocality, the equilibrium Eq. 6 should be accompanied by a “boundary” condition, imposed as a volumetric constraint in so-called the interaction domain wΓ. It is distinguished from the local elasticity case where the boundary conditions are imposed precisely at the bounding surface Γ(0) (see for details [31, 54]); i.e., the nonlocal boundary wΓ is a d-dimensional region unlike its (d − 1)-dimensional counterpart Γ0 in local problems. The interaction domain wΓ contains points y not in w interacting with points xw. A variety of choices for the domain wΓ having positive volume are possible (see for details [19, 42] \(\overline {\overline {w}}=w\cup w_{\varGamma }\) stands or the nonlocal closure of w. The most popular shape for wΓ (with prescribed forces and displacements) is a boundary layer of thickness given by the horizon lδ (see [37]) \(w_{\varGamma }=\{w\oplus {\mathcal{H}}_{\mathbf {0}}\}\backslash w\), where \(w\oplus {\mathcal{H}}_{\mathbf {0}}\) is a Minkowski outer parallel (dilation) of w (\(\mathcal {A}\oplus {\mathcal{B}}:= \cup _{\mathbf {x}\in \mathcal {A},\mathbf {y}\in {{\mathcal{B}}}} \{\mathbf {x}+\mathbf {y}\}\)); in such a case, w is the internal region of \(\overline {\overline {w}}\) (see Silling [54]). Points in w interact with points in wΓ through the pairwise force function f, and, therefore, these external forces must be supplied through the fictitious force density b. In particular, let us for definiteness consider the nonlocal “Dirichlet” volume-constrained problem (see [19, 42])

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} -\widehat{\mathcal{L}} (\mathbf{u})=\mathbf{b}& \text{on} \ w,\\ \mathbf{u}=\mathbf{h}, & \text{on}\ w_{\varGamma u}=w_{\varGamma}. \end{array}\right. \end{array} $$
(21)

called displacement loading conditions. The Dirichlet (212) volumetric boundary conditions are called homogeneous loading conditions if there exist some symmetric constant tensors \(\boldsymbol {\varepsilon }^{w_{\varGamma }}\) such that

$$ \begin{array}{@{}rcl@{}} \mathbf{h}(\mathbf{y})= \boldsymbol{\varepsilon}^{w_{\varGamma}}\mathbf{y}, \ \mathbf{y}\in w_{\varGamma u}=w_{\varGamma}. \end{array} $$
(22)

There are no specialized restrictions on the shape and smoothness of Γ0 which is defined just by convenience of representation. For example, it is convenient for visibility, although not an essential feature of the theory, to assume that Γ0 has a circle (or oval) shape (d = 2). Due to the condition (20), it is possible to perform a domain decomposition of wΓ over non-overlapping subdomains \(w_{\varGamma }^{i}\) (\(w_{\varGamma }=\cup w_{\varGamma }^{i}\), \(w_{\varGamma }^{i}\cap w_{\varGamma }^{j}=\emptyset ,\ \ \forall i\not = j\)) such that \( \text {mes} \ w_{\varGamma }^{i}/\text {mes}\ w\ll 1\) and \(\mathbf {n}(\mathbf {y})\simeq \mathbf {n} (\mathbf {y}_{i})\) for \(\forall \mathbf {y}\in w_{\varGamma }^{i}\) where yi is a center of \(w_{\varGamma }^{i}\).

In an analogy with the volumetric boundary domain vΓi, we introduce a volumetric interface boundary (called also interaction interface, see [51]) vΓi =Γ+Γ where \(\varGamma ^{+}_{i}\) and \(\varGamma ^{-}_{i}\) are the boundary layers (internal and external, respectively) separated by the geometric boundary Γi and having a thickness given by the horizon lδ. For a general form of the inclusion vi, the external volumetric interface \(\varGamma ^{-}_{i}\) can be expressed through the Minkowski dilation \(\varGamma ^{-}_{i}=\{v_{i}\oplus {\mathcal{H}}_{\mathbf {0}}\}\backslash v_{i}\). A nonlocal closure of the inclusion \({v^{l}_{i}}:=v_{i}\oplus {\mathcal{H}}_{\mathbf {0}}\) is called an extended inclusion while \(v^{l}:=\cup {v_{i}^{l}}\) (i = 1,2,…) stands for the extended inclusion phase. In so doing, vl(0) := wvlv(0) is called a truncated matrix.

All the random quantities under discussion are described by statistically homogeneous random fields. The ergodicity condition is assumed for statistically homogeneous random functions f(x) when the spatial average estimated over one sufficiently large sample and statistical mean coincide for both the whole volume w and the individual constituent v(k) (k = 0,1):

$$ \begin{array}{@{}rcl@{}} \langle\mathbf{f}\rangle &=&\{\mathbf{f}\}\equiv \lim_{w\uparrow R^d} |w|^{-1}\int _w \mathbf{f}(\mathbf{x})d\mathbf{x}, \\ \langle \mathbf{f}\rangle^{(k)} &=& \{\mathbf{f}\}^{(k)}\equiv\lim_{w\uparrow R^d} |v^{(k)}|^{-1}{\int}_{w} \mathbf{f}(\mathbf{x})V^{(k)}d\mathbf{x}, \end{array} $$
(23)

where |w| = mesw. Hereafter, n(k) is a number density of component v(k)vi and c(k) is the concentration, i.e., volume fraction, of the component viv(k): \( c^{(k)}=\langle V^{(k)}\rangle =\overline v_{i}n^{(k)}, \ \overline v_{i}=\text {mes} v_{i}\ \ (k=1,2,\ldots ,N;\ i=1,2,\ldots ),\quad c^{(0)}=1-\langle V\rangle \).

For locally elastic CM, the statistical averages 〈σ〉 and 〈ε〉 (which can be considered as the volume averages) are usually decomposed in the form the volume averages over the phases v(0) and v(1)

$$ \{\boldsymbol{\sigma}\}=c^{(0)}\{\boldsymbol{\sigma}\}^{(0)}+c^{(1)}\{\boldsymbol{\sigma}\}^{(1)},\ \ \ \{\boldsymbol{\varepsilon}\}=c^{(0)}\{\boldsymbol{\varepsilon}\}^{(0)}+c^{(1)}\{\boldsymbol{\varepsilon}\}^{(1)}. $$
(24)

The material and field decompositions usually used in locally elastic micromechanics correspond to the volume partition of a full space w = Rd over the volumes of the matrix and inclusions

$$ 1=c^{(0)}+c^{(1)},\ \ w=v^{(0)}\cup v^{(1)} , \ \ W(\mathbf{x})=V^{(0)}(\mathbf{x})+V^{(1)}(\mathbf{x}). $$
(25)

However, in peridynamics, the material decomposition corresponds to another partition of the full space

$$ 1=c^{l(0)}+c^{l(1)},\ \ w=v^{l(0)}\cup v^{l(1)}, \ \ W(\mathbf{x})=V^{l(0)}(\mathbf{x})+V^{l(1)}(\mathbf{x}). $$
(26)

over the extended inclusions \(v^{l(1)}:=v^{(1)}\oplus {\mathcal{H}}_{0}\) and the truncated matrix vl(0) := wvl(1) with the indicator functions Vl(1)(x) and Vl(0)(x), respectively. Hereafter, we use the volume averages

$$ \langle \mathbf{f}\rangle^{l(k)} = \{\mathbf{f}\}^{l(k)}\equiv\lim_{w\uparrow R^d} |v^{l(k)}|^{-1}{\int}_{w} \mathbf{f}(\mathbf{x})V^{l(k)}d\mathbf{x}, $$
(27)

over the extended inclusions vl(1) and the truncated matrix vl(0), respectively. It should be mentioned that the inclusions vi and vj cannot overlap while overlapping of corresponding external inclusions \({v^{l}_{i}}\) and \({v^{l}_{j}}\) is possible. However, for simplicity purposes, we will also assume non-overlapping of the external inclusions \({v^{l}_{i}}\) and \({v^{l}_{j}}\).

General Representation for Effective Moduli

In this section, we consider the effective elastic properties of statistically homogeneous heterogeneous macrodomain w subjected to homogeneous loading when statistical averages 〈σ〉(x) ≡const. and 〈ε〉(x) ≡const. are homogeneous and 〈u〉(x) = 〈εx (i.e., 〈u〉(x)≠〈u〉). In such a case, independently on the nature of the constitutive laws (either local or nonlocal) in the micro (or nano) points, the field macrovariables 〈σ〉 and 〈ε〉 are related by a local material effective tensor of either the elasticity or compliance:

$$ \langle\boldsymbol{\sigma}\rangle=\mathbf{L}^{*}\langle\boldsymbol{\varepsilon}\rangle, \ \ \text{or} \ \ \langle\boldsymbol{\varepsilon}\rangle=\mathbf{M}^{*}\langle\boldsymbol{\sigma}\rangle, $$
(28)

respectively. The representations for the effective moduli L (23) are obtained for both the local (conventional elasticity) and nonlocal (nonlocal elasticity and peristatics) problems. Due to the scale separation hypothesis (20), statistical homogeneities of both the material and field parameters, and homogeneity of the boundary loading, the nonlocal boundary conditions do not affect the statistical average of constitutive Eq. 28.

However, a direct practical use of Eq. 231 involves difficulties because it requires estimation of averages inside each phase of the load stresses 〈σ(k). The mentioned equations are significantly simplified for the conventional elasticity of matrix composites with the homogeneous matrix when a decomposition

$$ \mathbf{L}(\mathbf{x})=\mathbf{L}^{(0)}+\mathbf{L}_1(\mathbf{x}) $$
(29)

holds and the jumps of the moduli L1(x) vanish inside the matrix xv(0). Averaging of the constitutive equation at the fine scale (21) yields

$$ \langle\boldsymbol{\sigma}\rangle=\mathbf{L}^{(0)}\langle\boldsymbol{\varepsilon}\rangle+ \langle^L \boldsymbol{\tau}(\mathbf{x})\rangle, $$
(30)

where

$$ ^L \boldsymbol{\tau}(\mathbf{x})=\mathbf{L}_1(\mathbf{x})\boldsymbol{\varepsilon}(\mathbf{x}) $$
(31)

denotes the stress polarization tensor which is simply a notational convenience and vanishes inside the matrix: Lτ(x) ≡0 (xv(0)). Then

$$ \mathbf{L}^{*}=\mathbf{L}^{(0)}+\mathbf{R}^{*}, \ \ \ \langle^L \boldsymbol{\tau}\rangle:=\mathbf{R}^{*}\langle\boldsymbol{\varepsilon}\rangle. $$
(32)

A fundamental advantage of the representation (32) is the necessity to estimate the stress and strain fields only inside the inclusions (xv(1)) rather than inside entire composite (xw). It is found to be possible due to the fact that a constant tensor can be always taken out from the brackets 〈(⋅)〉 of statistical averages, e.g.,

$$ \langle\mathbf{L}^{(0)}\boldsymbol{\varepsilon}\rangle=\mathbf{L}^{(0)}\langle\boldsymbol{\varepsilon}\rangle. $$
(33)

We will propose the generalization of Eq. 31 to the peristatic composites by the use of generalizations of some basic auxiliary representations for micromechanics of conventional elasticity. Namely, an analog of the decomposition (29) can be presented in the form

$$ \begin{array}{@{}rcl@{}} \hat{\mathcal{L}}&=&\hat{\mathcal{L}}^{(0)}+\hat{\mathcal{L}}_1 \end{array} $$
(34)

with the operator summands \(\hat {{\mathcal{L}}}^{(0)}\) and \(\hat {{\mathcal{L}}}_{1}\) which are corresponded to the micromoduli decomposition C(0) and \(\mathbf {C}_{1}^{}\), respectively,

$$ \begin{array}{@{}rcl@{}} \mathbf{C}(\mathbf{x},\hat{\mathbf{x}})&=&\mathbf{C}^{(0)}(\mathbf{x},\hat{\mathbf{x}})+\mathbf{C}_1(\mathbf{x},\hat{\mathbf{x}}), \end{array} $$
(35)

and the jump of micromodulus \(\mathbf {C}_{1}(\mathbf {x},\hat {\mathbf {x}}):=\mathbf {C}^{(1)}(\mathbf {x},\hat {\mathbf {x}})- \mathbf {C}_{0}(\mathbf {x},\hat {\mathbf {x}})\) vanishes inside the truncated matrix \( \mathbf {x},\hat {\mathbf {x}}\in v^{l(0)}\). Both decompositions (29) and (35) are supported by a single assumption of the matrix homogeneity meaning invariantness of both tensors L(0)(x) and \(\mathbf {C}^{(0)}(\mathbf {x},\hat {\mathbf {x}})\), respectively, with respect to parallel transaction.

Furthermore, in an analogy with the micropotential \(\widetilde w\) [58], one defines the notions of the micropolarization tensors (introduced in [14, 17])

$$ \begin{array}{@{}rcl@{}} \widetilde{\boldsymbol{\tau}}(\mathbf{x},\hat{\mathbf{x}})&=&\mathbf{C}_1(\mathbf{x},\hat{\mathbf{x}})\boldsymbol{\eta}(\mathbf{x},\hat{\mathbf{x}}) , \end{array} $$
(36)
$$ \begin{array}{@{}rcl@{}} \widetilde{\boldsymbol{\tau}}^{(0)}(\mathbf{x},\hat{\mathbf{x}})&=&\mathbf{C}^{(0)} (\mathbf{x},\hat{\mathbf{x}})\boldsymbol{\eta}(\mathbf{x},\hat{\mathbf{x}}), \end{array} $$
(37)

which produce the local stress polarization tensor

$$ \boldsymbol{\tau}(\mathbf{x})=\mathcal{L}^{\sigma}(\widetilde{\boldsymbol{\tau}})(\mathbf{x}), $$
(38)

while \({\mathcal{L}}^{\sigma }(\widetilde {\boldsymbol {\tau }}^{(0)})\) constitutes an action of the operator \({\mathcal{L}}^{\sigma }\) on the medium with the material properties of the matrix \(\mathbf {C}(\mathbf {x},\hat {\mathbf {x}})\equiv \mathbf {C}^{(0)}(\mathbf {x},\hat {\mathbf {x}})\), and the displacement fields \(\mathbf {u}(\hat {\mathbf {x}})\), u(x) of the real composite material. Then the operator counterpart of Eq. 30 for the peristatic composite material can be decomposed as

$$ \boldsymbol{\sigma} (\mathbf{x})=\mathcal{L}^{\sigma}(\widetilde{\boldsymbol{\tau}}^{(0)})(\mathbf{x})+\boldsymbol{\tau}(\mathbf{x}). $$
(39)

Here, the local stress micropolarization tensor τ(x) is defined as (d = 1,2,3)

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\tau}(\mathbf{x})&=&\frac{1}{2}{\int}_S{\int}_0^{\infty}{\int}_0^{\infty} (y+z)^{(d-1)}\{\mathbf{C}_1(\mathbf{x}+y\mathbf{m}, \mathbf{x}-z\mathbf{m}) \\ &\cdot&[\mathbf{u}(\mathbf{x}+y\mathbf{m})-\mathbf{u}(\mathbf{x}-z\mathbf{m})]\}\otimes \mathbf{m} dzdyd\varOmega_{\mathbf{m}}. \end{array} $$
(40)

The main advantage of the decomposition (39) is a realizability of generalization of Eq. 33 (see for details [14])

$$ \langle\mathcal{L}^{\sigma}(\widetilde{\boldsymbol{\tau}}^{(0)})\rangle=\mathbf{L}^{(0)}\langle\boldsymbol{\varepsilon}\rangle $$
(41)

for statistically homogeneous peristatic composites subjected to the homogeneous volumetric loading. Equation 41 has a very attractive physical meaning. Namely, we formally replaced \(\mathbf {C}(\mathbf {x},\hat {\mathbf {x}})\) in the extended inclusion phase vl by the micromodulus of the matrix \(\mathbf {C}^{(0)}(\mathbf {x},\hat {\mathbf {x}})\) while a real random strongly inhomogeneous displacement field u(x) (xw) is fixed. Nevertheless, the average of the stress operator \(\langle {\mathcal{L}}^{\sigma }(\tilde {\boldsymbol {\tau }}^{(0)})\rangle (\mathbf {x})\) defined at this displacement field exactly coincide with the term L(0)ε〉 for the corresponding locally elastic CM (compare the right hand sides of Eqs. 33 and 41). It means that in the average sense, the peristatic truncated matrix (with the micromodulus \(\mathbf {C}^{(0)}(\mathbf {x},\hat {\mathbf {x}})\)) is deformed identically to the locally elastic matrix (with the modulus L(0)) deformation. In so doing, the modulus L(0) uniquely expressed through \(\mathbf {C}^{(0)}(\mathbf {x},\hat {\mathbf {x}})\) as an effective modulus (see [17]) of the peristatic CM with zero concentration of inclusions. No assumptions about \(\mathbf {C}^{(0)}(\mathbf {x},\hat {\mathbf {x}})\) (including an arbitrary shape of \({\mathcal{H}}_{\mathbf {x}}\)) was used at the derivative of Eq. 41.

Equation 41 leads to subsequent simplification of the averaged Eq. 39

$$ \langle\boldsymbol{\sigma}\rangle=\mathbf{L}^{(0)}\langle\boldsymbol{\varepsilon}\rangle+\langle\boldsymbol{\tau}\rangle, $$
(42)

that yields the representation for the effective elastic modulus (28)

$$ \mathbf{L}^{*}=\mathbf{L}^{(0)}+\mathbf{R}^{*},\ \ \ \langle\boldsymbol{\tau}\rangle=\mathbf{R}^{*}\langle\boldsymbol{\varepsilon}\rangle. $$
(43)

Formally the averaged equations for the elastic (2) and peristatic (6) composites are identical (compare Eq. (32) with Eq. 43) although the polarization tensor Lτ(x) (31) and micropolarization tensor τ(x) (40) are conceptually different. However, a common feature of τ(x) (40) and Lτ(x) (31) is that both tensors vanish inside the modified matrix (\(\mathbf {x}\in v^{l(0)}:=w\backslash v^{l},\ \ v^{l}:=v\oplus {\mathcal{H}}_{\mathbf {0}}\)) and expressed through the jumps of material properties of both the inclusions and matrix (described by the operators and tensors, respectively); for the locally elastic problem, vl(0)v(0) and \({\mathcal{H}}_{\mathbf {0}}=\emptyset \). The effective elastic moduli L (32) and (43) are defined as the proportionality factors between the statistical averages 〈σ〉 and 〈ε〉 which can be considered as the volume averages decomposed in the form of the volume averages over the phases v(0) and v(1) that allows one to estimate the effective moduli in the framework of some additional hypotheses (see Section 5). In so doing, all nonlocal effects are related with the estimation of the average 〈τl(1) over the extended inclusion phase vl(1).

Truncation Method for One Inclusion in the Matrix

We will consider in this section a macrodomain w containing one inclusion vi subjected to the prescribed body force b(x). In numerical implementation of peridynamics described by Silling and Askari [55], the region w is discretized into a set of nodes p, each with a finite known volume (called full volume) \(\bar V_{p}=h^{d}\) defined by the size h. Taken together, the nodes form a grid xp with the total number \(p\in [1,N^{{\max \limits }}]\) of nodes covering the total macrovolume w. The spatially discretized form of the equilibrium Eqs. 6 and 10 replaces the integral by the finite sum for each node p

$$ {\sum}_{q\in \mathcal{H}_p}\mathbf{C}(\mathbf{x}_p,\mathbf{x}_q)(\mathbf{u}_q-\mathbf{u}_p)\bar V_{pq}=-\mathbf{b}_p, $$
(44)

where a mid-point-type integration scheme is used with subscripts denoting the node number up = u(xp), bp = b(xp). For every interaction between the nodes q and p, only the volume fraction \(\bar V_{pq}\) of the volume \(\bar V_{q}\) of node q inside the cut-off distance lδ of node p is counted. Estimation of the partial volumes \(\bar V_{pq}\) (partial volume algorithm) for d = 1,2,3 was proposed in [49] (see also [52]). In modified algorithm by Hu et al. [28] (with an accuracy estimations in [52]), the family of the node p may include nodes q outside the horizon |xqxp| > lδ that allows for a non-vanishing force state for this pair of nodes with subsequent correction for partial volumes. The mentioned algorithms were proposed for the family points completely belonging to the same phase (either \({\mathcal{H}}_{p}\subset v^{(0)}\) or \({\mathcal{H}}_{p}\subset v^{(1)})\). If the line segment [xp,xq] intersect the geometrical interface Γi separated the phases v(0) and v(1) then the locally elastic model requires separate interface conditions defined on Γ. In contrast to local elasticity, in peridynamics, one introduces a d-dimensional set xvΓ (called extended interface or interaction interface) of thickness 2lδ (including Γ as a central d − 1 dimensional surface Γ) where a new peristatic operator is defined on displacements u(x) (see for details [51, 53]). It was showed that nonlocal interface problems convergence to their classical local counterparts outside the interface in the limit of the horizon going to zero lδ → 0 with the fixed a. Although there are no technical issues with correction of discretized peridynamic equation in the extended interface xvΓ, it is helpful to assume for simplicity only that the peristatic operator in the extended interface (fuzzy interface) is described by Eq. 6 with the constant horizon and micromodulus (14) determined as an average value of the micromoduli in the matrix and inclusion (161), see [2] (and also [39] where one proposed a variation of the extended interface micromodulus \(\mathbf {C}^{i}(\mathbf {x},\hat {\mathbf {x}})\) in a spirit of the functional graded materials theory described in, e.g., [11]).

The equilibrium Eq. 44 can be presented in a standard matrix form (xpw)

$$ \begin{array}{@{}rcl@{}} \sum\limits_q {\mathbb{K}}_{pq}\mathbf{u}_q&=&\mathbf{b}_p, \end{array} $$
(45)
$$ \begin{array}{@{}rcl@{}} {\mathbb{K}}_{pq}&=& \mathbf{C}(\mathbf{x}_p,\mathbf{x}_q)\bar V_{pq}-\delta_{pq}\sum\limits_{r\in \mathcal{H}_p}\mathbf{C}(\mathbf{x}_p,\mathbf{x}_r)\bar V_{pr}, \end{array} $$
(46)

where \({\mathbb {K}}_{pq}\) is a rectangular matrix \(\left (dN^{{\max \limits }}\times d(N^{{\max \limits }}+N^{w\varGamma })\right )\) while a displacement uq is presented by a vector \(\left (d(N^{{\max \limits }}+N^{w\varGamma })\times 1\right )\). We need to solve Eq. 45 accompanied by the corresponding homogeneous volumetric boundary conditions [e.g., Dirichlet–type volumetric boundary condition (21)]. It means inversion of a global square stiffness matrix \(\hat {\mathbb {K}}\)\((dN^{{\max \limits }}\times dN^{{\max \limits }})\) for all nodes \(p\in [1,dN^{{\max \limits }}]\) in Eq. 46 where a displacement u and an fictitious body force vector \(\hat {\mathbf {b}}\) (defined by the volumetric boundary condition (22)) are presented by the vectors \((dN^{{\max \limits }}\times 1)\)

$$ \begin{array}{@{}rcl@{}} \sum\limits_q \hat{\mathbb{K}}_{pq}\mathbf{u}_q&=&\hat{\mathbf{b}}_p, \end{array} $$
(47)
$$ \begin{array}{@{}rcl@{}} \hat {\mathbb{K}}_{pq}&=& {\mathbb{K}}_{pq}W(\mathbf{x}_q), \\ \hat{\mathbf{b}}_p&=&-\sum\limits_{r\in \mathcal{H}_p}\mathbf{C}(\mathbf{x}_p,\mathbf{x}_r)\boldsymbol{\varepsilon}^{w_{\varGamma}}\mathbf{x}_p \bar V_{pr} W^{w\varGamma}(\mathbf{x}_r) \end{array} $$
(48)

where \(\mathbf {h}\equiv \boldsymbol {\varepsilon }^{w_{\varGamma }}\mathbf {x}\) (\(\boldsymbol {\varepsilon }^{w_{\varGamma }}\equiv \)const., \(\mathbf {x}\in w_{\varGamma }\)) (22), and the indicator function \(W^{\overline {\overline {w}}}\) of the nonlocal closure \(\overline {\overline {w}}=w\cup w_{\varGamma }\) is split as

$$ \begin{array}{@{}rcl@{}} W^{\overline{\overline{w}}}(\mathbf{x})=W(\mathbf{x})+W^{w\varGamma}(\mathbf{x}). \end{array} $$
(49)

However, the coefficient matrix \(\hat {\mathbb {K}}\) (47) is often severely ill-conditioned and solution of the system of linear algebraic Eq. 47 by direct methods, such as, e.g., the least-squares method, can lead to a significant error. If \(\hat {\mathbb {K}}\) is highly ill-conditioned, one can use a regularization method to solve this system. So, the known Tikhonov [60] regularization equation is

$$ \tilde{\hat{\mathbb{K}}}\mathbf{u}=\hat{\mathbf{b}}^K, \ \ \tilde{\hat{\mathbb{K}}}:=(\hat{\mathbb{K}}^{\top}\hat{\mathbb{K}}+\lambda\mathbf{I}), $$
(50)

where \(\hat {\mathbf {b}}^{K}:=\hat {\mathbb {K}}^{\top }\hat {\mathbf {b}}\), I is the unit matrix, and the superscript signifies the transpose. The optimal value λ is chosen according to the method described in, e.g., [25].

Thus, the solution u(x) (xw) based on inverting a truncated matrix \(\hat {\mathbb {K}}_{pq}\) (\(p,q=1,\ldots , dN^{{\max \limits }})\) is assumed to be found. In the truncation domain method, an infinite integration domain Rd is replaced by corresponding finite domain wRd while having control of the error. For estimation of the size of the truncation domain, the geometrical boundary Γ0 of the domain w in 2D is prescribed in the polar coordinate system \(\varGamma ^{0}(\mathbf {x})=\{\mathbf {x}: x_{1}=r(\phi )\cos \limits (\phi ), x_{2}=r(\phi )\sin \limits (\phi )\}\). The positions of the points yΓ0ρ of the surface Γ0ρ are expressed through the points xΓ by a homothetic transformation:

$$ \mathbf{y}=\mathbf{x}_i+\rho(\mathbf{x}-\mathbf{x}_i), $$
(51)

where xi = 0 is a homothetic center coinciding with the particle center. Once the parameter 1 < ρ is chosen, the distribution of the points yΓρ is determined. For a sequence ρk = k, we estimate a sequence uk in the domain wρ with the boundary Γ0ρ and analyze a convergence of uρ by estimation of tolerance defined by the relative difference of uρ and uρ+ 1

$$ \varDelta=||\mathbf{u}^{\rho+1}-\mathbf{u}^{\rho}||_{L_2} /||\mathbf{u}^{\rho+1}||_{L_2},\ \ \ ||(\cdot)||_{L_2}= \left( (\text{mes} v_{\varGamma})^{-1} {\int}_{v_{\varGamma}} (\cdot)^2(\mathbf{y})d\mathbf{y}\right)^{1/2}, $$
(52)

where the L2 norm is evaluated over the domain vΓ. The simulation is stopped when the tolerance Δ reaches 0.1%. We control the variations of uρ only over the domain vΓ because according to [14, 17] the effective moduli depend on the displacement distribution u(x) only in the domain xvΓ.

Micromechanical Modeling

We generalize the basic hypothesis of local elasticity of CMs to their peristatic counterparts. Namely, the main hypothesis of many micromechanical methods termed as the effective field hypothesis is the following:

H1)Each inclusionviis located in the homogeneous strain field\((\mathbf {y}\in {v^{l}_{i}})\):

$$ \overline {\boldsymbol{\varepsilon}}_i(\mathbf{y}) \equiv \overline {\boldsymbol{\varepsilon}}(\mathbf{x}_i), $$
(53)

corresponding to the linear displacement\(\overline {\mathbf {u}}(\mathbf {y})= \overline {\boldsymbol {\varepsilon }}(\mathbf {x}_{i})(\mathbf {y}-\mathbf {x}_{i})\).

Thus, the solutions of Eq. 47 for the displacements and strains can be considered to be of the form

$$ \begin{array}{@{}rcl@{}} \mathbf{u}(\mathbf{x})-\overline{\mathbf{u}}(\mathbf{x})&:=&\mathcal{L}^{u u}(\mathbf{x}-\mathbf{x}_i,\overline{\mathbf{u}}), \end{array} $$
(54)
$$ \begin{array}{@{}rcl@{}} \boldsymbol{\varepsilon}(\mathbf{x})-\overline{\boldsymbol{\varepsilon}}(\mathbf{x})&:=&\mathcal{L}^{\varepsilon \varepsilon}(\mathbf{x}-\mathbf{x}_i,\overline{\boldsymbol{\varepsilon}}), \end{array} $$
(55)

respectively, where the perturbator \({\mathcal{L}}^{\varepsilon \varepsilon }(\mathbf {x}-\mathbf {x}_{i},\overline {\boldsymbol {\varepsilon }})\) is obtained from \({\mathcal{L}}^{u u}(\mathbf {x}-\mathbf {x}_{i},\overline {\mathbf {u}})\) by numerical differentiation. For the homogeneous effective strain \(\overline {\boldsymbol {\varepsilon }}\equiv \)const., the linear integral operators \({\mathcal{L}}^{u u}(\mathbf {x}-\mathbf {x}_{i},\overline {\mathbf {u}})\) and \({\mathcal{L}}^{\varepsilon \varepsilon }(\mathbf {x}-\mathbf {x}_{i},\overline {\boldsymbol {\varepsilon }})\) on the domain \(\mathbf {x}\in {v_{i}^{l}}\) are decomposed as

$$ \begin{array}{@{}rcl@{}} \mathcal{L}^{u u}(\mathbf{x}-\mathbf{x}_i,\overline{\mathbf{u}})&:=&\mathbf{L}^{u u}(\mathbf{x}-\mathbf{x}_i) \overline{\boldsymbol{\varepsilon}}, \end{array} $$
(56)
$$ \begin{array}{@{}rcl@{}} \mathcal{L}^{\varepsilon \varepsilon}(\mathbf{x}-\mathbf{x}_i,\overline{\boldsymbol{\varepsilon}})&:=& \mathbf{L}^{\varepsilon \varepsilon}(\mathbf{x}-\mathbf{x}_i)\overline{\boldsymbol{\varepsilon}}. \end{array} $$
(57)

Dilute Approximation Method (DAM)

Let the dispersion contains identical particles, of a fixed shape vi, randomly aligned throughout the matrix. We assume that the dispersion is dilute, in the sense that the volume fraction, c(1), of the particles is small and their mutual interaction can be neglected. In other words, each inhomogeneity can be treated as if it exists in a homogeneous infinite matrix without the interference by other inhomogeneities. So, each particle can be thus imagined as single, immersed into an unbounded matrix

$$ c^{l(1)}\ll 1, \ \ \overline {\boldsymbol{\varepsilon}}=\langle\boldsymbol{\varepsilon}\rangle, $$
(58)

subjected to the homogeneous volumetric displacement loading at infinity. If the interactions between inhomogeneities are completely ignored, then substitution of Eq. 58 into Eqs. 42 and 56 leads to

$$ \begin{array}{@{}rcl@{}} \mathbf{L}^{*}&=&\mathbf{L}^{(0)}+c^{l(1)}\mathbf{R}_i^l, \end{array} $$
(59)

where (\(\mathbf {x},\mathbf {y}\in {v_{i}^{l}}\))

$$ \begin{array}{@{}rcl@{}} \mathbf{R}^l_i(\mathbf{x})\langle\overline{\boldsymbol{\varepsilon}}\rangle&=& \mathcal{L}^{\sigma}(\mathbf{C}_1\boldsymbol{\eta})(\mathbf{x}), \\ \boldsymbol{\eta}(\mathbf{x},\mathbf{y})&=& \left[(\mathbf{y}-\mathbf{x})^{\top}+\mathbf{L}^{uu}_i(\mathbf{x}-\mathbf{x}_i)-\mathbf{L}^{uu}_i(\mathbf{y}-\mathbf{x}_i)\right]\overline{\boldsymbol{\varepsilon}}, \end{array} $$
(60)

and \({\mathbf {R}_{i}^{l}}:=\langle \mathbf {R}_{i}(\mathbf {x}){\rangle _{i}^{l}}\) stands for the average (27) of Ri(x) over the extended inclusion \({v_{i}^{l}}\).

It should be mentioned that the effective moduli can be estimated by the use of evaluation of average stresses in the components

$$ c^{l(1)}\ll 1, \ \ \overline {\boldsymbol{\varepsilon}}=\mathbf{M}^{(0)}\langle\boldsymbol{\sigma}\rangle, $$
(61)

that leads to the following representation

$$ \mathbf{M}^{*}=\mathbf{M}^{(0)}\left[\mathbf{I}-c^{l(1)}\mathbf{R}_i^l\mathbf{M}^{(0)}\right]. $$
(62)

In so doing, utilization of the dilute approximation method (DAM) reduces to an inequality L≠(M)− 1 although two versions (59) and (62) coincide in the asymptotic c(1) → 0. The estimations (59) and (62) are consistent only up to the first order O(c(1)) of the volume fraction of the inhomogeneities, providing however in this case an exact expression (since the particle interactions are totally neglected). Because of this, Eq. 59 will be exploited if \(\mathbf {C}(\mathbf {x},\hat {\mathbf {x}}) -\mathbf {C}^{(0)}(\mathbf {x},\hat {\mathbf {x}})\) is positive-define; otherwise, a dual scheme (62) based on the estimation of average stresses and effective compliance will be used. The dilute expressions are strictly valid only for vanishingly small inclusion volume fractions. Equations 59 and 62 were obtained under the assumption that the inclusions are dilutely dispersed in the matrix and thus do not “feel” any influence of surrounding neighbors, i.e., they are loaded by the unperturbed applied remote field. For locally elastic CM with ellipsoidal homogeneous inclusions, the representations (59) and (62) are reduced to the corresponding equations obtained by the Eshelby method. Because of this, the approach (59) and (62) are also called the generalized Eshelby method.

Generalized Mori–Tanaka Method

Modeling of overall behavior of composites with nondilute inclusion volume fraction must (explicitly or implicitly) account for the interaction between inclusions, i.e., for the effects of the surrounding inclusions on the stress and strain fields in the vicinity of a given inclusion. Now, consider a model of CM where for a typical inhomogeneity vi, the effects (or the existence) of other inhomogeneities are communicated to it through the displacement, strain and stress fields in its surrounding matrix material. When an infinite number of inhomogeneities randomly distributed in the matrix are analyzed, the average field in the matrix can be considered as good approximations of the actual fields in the matrix surrounding each inhomogeneity. As pointed out by Benveniste [7], the essential assumption in the Mori–Tanaka method (MTM [45]) for locally elastic CMs states that each inclusion vi behaves as an isolated one in the infinite matrix and subject to some effective strain field \(\overline {\boldsymbol {\varepsilon }}(\mathbf {x})\) coinciding with the average strain in the matrix:

$$ \overline{\boldsymbol{\varepsilon}}(\mathbf{x})=\langle\boldsymbol{\varepsilon}\rangle^{(0)}. $$
(63)

The assumption (63) makes it possible to determine uniquely the effective elastic properties of multicomponent composite materials. So, hypothesis (63) gives an opportunity to use the solution (601) for each inclusion vi and to find the average stress in the matrix by the use of a representation of the average strain in the whole composite over the average strain in the separate phases:

$$ \langle\boldsymbol{\varepsilon}\rangle=c^{l(0)}\langle\boldsymbol{\varepsilon}\rangle^{l(0)} + c^{l(1)}\langle\boldsymbol{\varepsilon}\rangle^{l(1)}= c^{l(0)}\langle\boldsymbol{\varepsilon}\rangle^{l(0)} + \langle\mathbf{A}_i(\mathbf{x})V_i^l(\mathbf{x})\rangle \langle\boldsymbol{\varepsilon}\rangle^{l(0)}, $$
(64)

where \(\mathbf {A}_{i}(\mathbf {x}):=\mathbf {L}^{\varepsilon \varepsilon }(\mathbf {x}-\mathbf {x}_{i})+\mathbf {I}\) (\(\mathbf {x}\in {v_{i}^{l}})\) is a strain concentration factor for a single extended inclusion in the infinite matrix while \({\mathbf {A}_{i}^{l}}=\langle \mathbf {A}_{i}(\mathbf {x}){V^{l}_{i}}(\mathbf {x}){\rangle ^{l}_{i}}\) is its average. The right hand side of Eq. 64 can be also found through the displacements at the external interface \(\varGamma _{i}^{-}\). Indeed, by the use of the Gauss-Ostrogradsky theorem, we get the average strain at the extended inclusion \({v_{i}^{l}}\)

$$ \langle {\boldsymbol{\varepsilon}}\rangle^l_i=:\mathbf{A}_i^l \langle {\boldsymbol{\varepsilon}}\rangle^{0}= \langle {\mathbf{u}}\otimes \mathbf{n}\rangle_i^{l\omega}:= |v_i^l|^{-1}{\int}_{\varGamma_i^-} {\mathbf{u}}(\mathbf{s})\otimes \mathbf{n}(\mathbf{s}) d\mathbf{s}, $$
(65)

where u(s) is a displacement on the interface boundary \(\mathbf {s}\in \varGamma _{i}^{-}\) with the outward normal unit vector n(s) on \(\varGamma _{i}^{-}\). Thus, the average strain concentration factor \({\mathbf {A}_{i}^{l}}\) is estimated through the displacement u(s) (\(\mathbf {s}\in \varGamma _{i}^{-}\)) (i.e., evaluation of the strain concentration factor Ai(x) (\(\mathbf {x}\in {v_{i}^{l}}\)) is not required). Equations 64 and 65 lead to the representations of statistical average of both the strains in the matrix and the stress polarization tensor τ in the inclusions (\(\mathbf {x}\in {v_{i}^{l}}\))

$$ \begin{array}{@{}rcl@{}} \langle\boldsymbol{\varepsilon}\rangle^{l(0)}&=& [c^{l(0)} +c^{l(1)}\mathbf{A}_i^l]^{-1} \langle\boldsymbol{\varepsilon}\rangle, \\ \boldsymbol{\tau}(\mathbf{x})&=&\mathbf{R}_i(\mathbf{x})[c^{l(0)} +c^{l(1)}\mathbf{A}_i^l]^{-1} \langle\boldsymbol{\varepsilon}\rangle. \end{array} $$
(66)

Substituting of Eq. 662 into the average constitutive equation (4.12) yields

$$ \begin{array}{@{}rcl@{}} \langle\boldsymbol{\sigma}\rangle&=& \mathbf{L}^{(0)}\langle\boldsymbol{\varepsilon}\rangle+\langle\boldsymbol{\tau} V^l\rangle= \left[\mathbf{L}^{(0)}+\langle\mathbf{R}_i V^l\rangle [c^{l(0)} +c^{l(1)}\mathbf{A}_i^l]^{-1}\right]\langle\boldsymbol{\varepsilon}\rangle. \end{array} $$
(67)

Comparison of the definition of effective moduli L with Eq. (67) leads to the representation of effective properties and the effective field:

$$ \begin{array}{@{}rcl@{}} \mathbf{L}^{*} &=& \mathbf{L}^{(0)}+ c^{l(1)}\mathbf{R}_i^l \mathbf{D}_i, \end{array} $$
(68)
$$ \begin{array}{@{}rcl@{}} \langle\overline{\boldsymbol{\varepsilon}}\rangle &=& \mathbf{D}_i\langle\boldsymbol{\varepsilon}\rangle,\ \ \ \mathbf{D}_i=[c^{l(0)} +c^{l(1)}\mathbf{A}_i^l]^{-1}. \end{array} $$
(69)

As can be seen from comparing Eqs. 582 and 63, the only difference between the dilute approximation and the Mori–Tanaka methods is the choice of the effective field \(\overline {\boldsymbol {\varepsilon }}\) representing how the effects of other inclusions are taken into account. So, the dilute approximation implies no interaction between the inhomogeneities and, therefore, the strain in the matrix coincides with the remote strain 〈ε〉. The Mori–Tanaka method also assumes the absence of all inclusions; however, the influence of these removed inclusions is described by the average strain 〈εl(0) in the truncated matrix (instead of 〈ε〉). Because of this, both the Mori–Tanaka method and dilute approximation method are invariant with respect to the possible overlapping of extended inclusions \({v_{i}^{l}}\) and \({v_{j}^{l}}\) mentioned after Eq. (27). As is expected, the Mori–Tanaka estimations (68) are reduced to the Dilute approximation (59) in the limiting case cl(1) → 0. It demonstrates a role of the dilute approximation method as a test-drive for another micromechanical methods which should be reduced to the dilute approximation approach in the limiting case cl(1) → 0.

The Mori–Tanaka method provides a better estimate of the effective modulus than the dilute approximation does for the composite with nondilute inclusion phase. In particular, we consider the classical 3D example of locally elastic CMs containing the incompressible matrix and rigid spherical inclusions. Then, we obtain representation for the effective shear muduli (see, e.g., [11])

$$ \begin{array}{@{}rcl@{}} \mu^{*}/\mu^{(0)} &=& 1+\frac{5}{2}c , \end{array} $$
(70)
$$ \begin{array}{@{}rcl@{}} \mu^{*}/\mu^{(0)} &=& 1+\frac{5}{2}\frac{c}{1-c} \end{array} $$
(71)

estimated for cl(1) = c(1) by the dilute approximation (59) and the Mori–Tanaka approach, respectively. It is well known that Eq. 71 better estimates experimental data for the effective viscosity of Newtonian suspensions than the dilute approximation (70). From the other side, if LL(0) is positive-define, there is a sense to compare the Eqs. 68 and 59 (rather than (68) and (62)) because both Eqs. 68 and 59 were obtained for the prescribed average strain 〈ε〉 while Eq. (62) was derived through the average stress 〈σ〉.

Equations 68 and 69 coincide to within notations with the corresponding MTM for the locally elastic CMs

$$ \begin{array}{@{}rcl@{}} {~}^L \mathbf{L}^{*} &=& \mathbf{L}^{(0)}+ \mathbf{R}_i \mathbf{D}_i, \end{array} $$
(72)
$$ \begin{array}{@{}rcl@{}} \langle\overline{\boldsymbol{\varepsilon}}\rangle &=& \mathbf{D}_i\langle\boldsymbol{\varepsilon}\rangle,\ \ \ \mathbf{D}_i=[c^{(0)} +c^{(1)}\mathbf{A}_i]^{-1} \end{array} $$
(73)

with conceptually different tensors Ri and Ai considered as locally elastic counterparts of the tensors \({\mathbf {R}_{i}^{l}}\to \mathbf {R}_{i}\) and \({\mathbf {A}_{i}^{l}}\to \mathbf {A}_{i}\) as lδ → 0. For locally elastic CM, the MTM as defined by Eq. 68 has been called the “direct approach” by Benveniste [7], who showed it is identical for composites with the ellipsoidal homogeneous inclusions to the “equivalent inclusion-average strain” formalism (see [45, 63]). Despite their limitations, conventional MTM provides useful accuracy for the elastic contrasts pertaining to most practically relevant composites. This combination of features makes them important tools for evaluating the effective elastic properties of inhomogeneous materials that show a matrix-inclusion topology with aligned inclusions. No assumptions about both the shape of heterogeneities and their microstructure were used at the derivative of Eqs. 5962, and 68.

Numerical Results for 1D Case

Numerical Schemes for 1D Case

At first, we consider the truncation method for 1D peristatic bar \(\bar {\bar w}\) of the length 2L + 2lδ and cross-section area A = 1 containing one inclusion vi = [−a,a] and the volumetric boundary wΓ: \(\bar {\bar w}=w_{\varGamma }\cup w\), wΓ = [−Llδ,−L] ∪ [L,L + lδ]). The midpoint quadrature scheme (similar to [13, 16]) is realized for the uniform grid \(x_{i}=(0,\pm 1,\pm 2,\ldots , \pm (N^{{\max \limits }}+N^{w\varGamma }))\) with the numbers of material points \(2 N^{{\max \limits }}+1\) and 2NwΓ + 1 inside both w and the interaction boundary wΓ, respectively, and a constant step size \( h =L/N^{{\max \limits }}\); 2Na + 1 and nl stand for the numbers of the material points in both the inclusion vi and horizon lδ, respectively.

We consider a heterogeneous peristatic bar with the step-function profiles (111) (k = 0,1)

$$ \begin{array}{@{}rcl@{}} C(\xi)=\left\{\begin{array}{ll} C^{(k)}, & \text{for} \ |\xi|<l_{\delta},\\ 0, & \text{for} \ |\xi|>l_{\delta}.,\ \ C^{(k)}=3E^{(k)}/l_{\delta}^3; \end{array}\right. \end{array} $$
(74)

the triangle profile (112) [and the micromodulus (12) and (13)] can be considered in a similar manner.

For 1D case, the peristatic stress σ(x) (18) and micropolarization tensor τ(x) (40) are represented as

$$ \begin{array}{@{}rcl@{}} \sigma(x)&=&{\int}_{\mathcal{H}^-_{x-l_{\delta}}} \left\{{\int}_{\mathcal{H}^+_{r+l_{\delta}}} C(r,s)[u(s)-u(r)]ds\right\}dr, \end{array} $$
(75)
$$ \begin{array}{@{}rcl@{}} \tau(x)&=&{\int}_{\mathcal{H}^-_{x-l_{\delta}}} \left\{{\int}_{\mathcal{H}^+_{r+l_{\delta}}} C_1(r,s)[u(s)-u(r)]ds\right\}dr, \end{array} $$
(76)

respectively, where \({\mathcal{H}}^{-}_{x-l_{\delta }}=\{r:\ x-l_{\delta }<r<x\}\) and \({\mathcal{H}}^{+}_{r+l_{\delta }}=\{s:\ x<s<r+l_{\delta }\}\). Equation 18 at d = 1 can be reduced to Eq. 75 by the variable exchange xzr, x + ys (reduction of Eqs. 40 to 76) is considered in a similar manner). Indeed, the origin-centered unit 1D “sphere” is the set [− 1,1], which has a measure of 2. Then the integrand of Eq. 18 is reduced to integrand of Eq. 75 by the use of the equality (8).

Given a node distribution xj, j = 0,± 1,± 2,…,±N over the length of \(v_{i}^{l0}\), we discretize (47) at the nodes xj using the composite midpoint integration method,

$$ {\sum}_{q=p-n_l}^{q=p+n_l}C(x_p,x_q)(\tilde u_{q}-u_{p})\bar V_{pq}= -{\sum}_{q=p-n_l}^{q=p+n_l}C(x_p,x_q)\varepsilon^{w\varGamma}x_q\bar V_{pq} W^{w\varGamma}(x_q), $$
(77)

where \(\bar V_{pj}\) is the portion of node q “volume” covered by the horizon of node p and a constant step size \(\varDelta x=l_{\delta }/n_{l}=L/N^{{\max \limits }}\) is assumed; \(\tilde u_{q}=u_{q}\) if \(|q|\leq N^{{\max \limits }}\) and \(\tilde u_{q}=0\) if \(|q|> N^{{\max \limits }}\). The error in the coordinate difference formula (77) is well known to be O(Δx2) (see [55]). Effectiveness of the quadrature scheme (77) for the homogeneous infinite bar was demonstrated by [10] who also proved uniform convergence of the peridynamic solutions to the classical solutions of static and dynamic elasticity problems in 1D case in the limit of the horizon going to zero.

After estimation of u(xp) (|xp|≤ L) by the quadrature method (77), the peristatic stress distribution (75) and the micropolarization tensor (76) can be presented in a similar form

$$ \begin{array}{@{}rcl@{}} \sigma(x_p)&=&\sum\limits_{q=j-n_l}^j\sum\limits_{r=j}^{q+n_l}C(x_q,x_r)[u(x_r)-u(x_q)]\bar V_{qp}\bar V_{rp} , \\ \tau(x_p)&=&\sum\limits_{q=j-n_l}^j\sum\limits_{r=j}^{q+n_l}C_1(x_q,x_r)[u(x_r)-u(x_q)]\bar V_{qp}\bar V_{rp}. \end{array} $$
(78)

Equation 782 allows us to estimate the average micropolarization tensor

$$ \begin{array}{@{}rcl@{}} R_i^l\varepsilon^{w\varGamma}&=& \frac{2}{{N^{\max}+N^{w\varGamma}}} \\ &\cdot& {\sum}_{p=-N^{\max}-N^{w\varGamma}} ^{N^{\max}+N^{w\varGamma}} \sum\limits_{q=j-n_l}^j\sum\limits_{r=j}^{q+n_l}C_1(x_q,x_r)[u(x_r)-u(x_q)]\bar V_{qp}\bar V_{rp}, \end{array} $$
(79)

while the average strain concentrator factor for the isolated inclusion (65) can be found from the solution (77)

$$ A^l_i\varepsilon^{w\varGamma}=(a+l_{\delta})^{-1} (u_{N^a+n^l}-u_{-N^a-n^l})/2. $$
(80)

Substitutions of Eqs. 79 and 80 into Eqs. 5962, and 77 68 lead to new representations for the effective Young moduli:

$$ \begin{array}{@{}rcl@{}} E^{*}&=& E^{(0)}+c^{l(1)}R_i^l , \end{array} $$
(81)
$$ \begin{array}{@{}rcl@{}} E^{*}&=& E^{(0)}[1-c^{l(1)}R_i^l/E^{(0)}]^{-1} , \end{array} $$
(82)
$$ \begin{array}{@{}rcl@{}} E^{*}&=& E^{(0)}+c^{l(1)}R_i^l[c^{l(0)}+c^{l(1)}A^l_i]^{-1} \end{array} $$
(83)

for the generalized dilute approximation (81) and (82), and the generalized MTM (83), respectively.

It should be mentioned that Eqs. 8183 were obtained as the straightforward particular 1D consequences of the general nD (n = 1,2,3) representations (59), (62), and (68) respectively. However, entirely 1D case for the infinite peristatic composite bar was considered by Buryachenko [13]. General integral equation (see [14]) was solved for 1D case by more cumbersome method. The final result of Buryachenko [13] in the current notation (82) has a representation

$$ E^{*}=E^{(0)}[1-c^{(1)}R_i^l/E^{(0)}]^{-1}. $$
(84)

In fact, a more accurate Eq. 82 is reduced to Eq. 84 when the volume of extended inclusions is approximated by the volume of inclusions cl(1)c(1).

Locally Elastic Inhomogeneous Bar

We will reproduce the well-known exact estimations of the effective Young modulus in the form which is the most convenient for the subsequent comparison with the peristatic heterogeneous bar. For the local elasticity theory, the constitutive law has the form

$$ \sigma(x)=E(x)\varepsilon(x),\ \varepsilon(x)=\hat E(x)\sigma(x), $$
(85)

where \(\hat E(x) :=E^{-1}(x)\). In 1D case, the stress field is a constant for any heterogeneous (even statistically inhomogeneous) bar

$$ \sigma(x)\equiv\langle\sigma\rangle= \hat E\langle\varepsilon\rangle=\text{const.} $$
(86)

while the other two parameters ε(x) and E(x) [or \(\hat E(x)\)] vary. In so doing, a deformation in any point is homogeneous (xv(k), k = 0,1, m = 1 − k)

$$ \varepsilon(x)\equiv \varepsilon^k=\langle\varepsilon\rangle^{(k)}=E^{(m)}\hat E^{(k)}\varepsilon^m.\ $$
(87)

The averaged Eq. (85) at E(x) = E(k)V(k)(x) (xv(k), k = 0,1) can be presented in the form (compare with Eqs. 30) and 31)

$$ \langle\sigma\rangle=E^{(0)}\langle\varepsilon\rangle +\langle^L \tau\rangle,\ \ \ { ^L \tau}(x):=(E(x)-E^{(0)})\varepsilon(x). $$
(88)

where 〈Lτ〉 is the strain polarization parameter averaged over \(R=(-\infty ,\infty )\). Substitution of Eq. (873) into (242) and (881) leads to the exact representation for the effective compliance \(\hat E^{*}\) presented for a two-phase bar in the form

$$ \hat E^{*}=c^{(1)}\hat E^{(1)}+c^{(0)}\hat E^{(0)}. $$
(89)

Estimation of the effective Young modulus (with exact representation (89)) can also be obtained from Eqs. 8183 in the limit of vanishing length scale lδ/a → 0. Indeed, substitutions of the local limit (\({A^{l}_{i}}\to A_{i}\) and \({R_{i}^{l}}\to R_{i}\))

$$ A_i=E^{(0)}\hat E^{(1)},\ \ \ R_i=(E^{(1)}-E^{(0)})E^{(0)}\hat E^{(1)} $$
(90)

into Eqs. 8183 yield

$$ \begin{array}{@{}rcl@{}} E^{*}&=& E^{(0)}(1+c^{(1)})- c^{(1)}(E^{(0)})^2 \hat E^{(1)}, \end{array} $$
(91)
$$ \begin{array}{@{}rcl@{}} \hat E^{*}&=& c^{(0)}\hat E^{(0)}+c^{(1)}\hat E^{(1)} , \end{array} $$
(92)
$$ \begin{array}{@{}rcl@{}} E^{*}&=& \left[c^{(0)}\hat E^{(0)}+c^{(1)}\hat E^{(1)}\right]^{-1} \end{array} $$
(93)

for the dilute approximations (91) and (92), and the MTM (93), respectively. It is interesting that the representations (92) and (93) coincide with the exact Eq. (89) and differ from Eq. 91. The mentioned coincidence is explained by a fundamental reasons. Indeed, one can conclude from Eq. 87 that the effective field acting on the inclusion is given by

$$ \overline{\varepsilon}_i=\varepsilon^{0}=\frac{\langle\sigma\rangle}{E^{(0)}}=\frac{E^{*}}{ E^{(0)}}\langle\varepsilon\rangle, $$
(94)

where the equality (941) is in fact the Mori and Tanaka [45] method (MTM) according to which the effective field \(\overline {\varepsilon }_{i}\) coincides with the volume average of the strain inside the matrix. In so doing, Eq. 942 is equivalent to the dilute approximation where each inclusion behaves as an isolated one inside the infinite homogeneous matrix. Furthermore, Eq. 87 implies that the “quasi–crystalline” approximation by Lax [34] is exactly fulfilled. Acceptance of the “quasi–crystalline” approximation for statistically homogeneous fields of identical inclusions, in turns, leads to an equivalence between the multiparticle effective field method, the method of effective field, and the MTM (see for details [11]). This seemingly unusual coincidence of results obtained by the different methods is explained by the exact conditions (861) and (871).

Numerical Examples for 1D Case

At first, the micromodulus functions with the step-function profile (74) with the same horizon lδ/a = 0.05, 0.25, and 0.5 are considered for both the inclusions and matrix with the ratios of the corresponding elastic counterparts E(1)/E(0) = 5. The midpoint quadrature scheme (75) is realized for the uniform grid with a constant step size Δx = L/4000 (corresponding to lδ/Δx = 20,100,200, respectively); an accuracy of the numerical solution for a single inclusion in the infinite homogeneous bar is analyzed by comparison of results obtained for Δx = lδ/200, lδ/100 and lδ/20.

It should be mentioned that the variation of the smoothed displacement curves \(u(x)\sim x\) (as well as the stress ones) are visually unclearly manifested. The curves u(x) vs x/a behave according to the general features of peristatic solutions for a homogeneous peristatic bar subjected to the fictitious body force mentioned by Silling et al. [59] (see also [10]). So, u1(x) also has a discontinuity at x = ±a because in general the displacement field u(x) has the same smoothness as the body force field (see [59]). In addition, the discontinuity in the micromodulus C(ξ) (41) at |ξ| = lδ, has a further effect on the smoothness of u(x) with a corresponding discontinuity of the derivatives u(k) of the order k = 1,2,… at x = ±(a + klδ). Because of this, we will pay our attention to the analyses of strain field distributions found by numerical differentiation of u(x): ε = u(x)/x (without points |x| = a). In Fig. 1, one presents the numerical results of the relative strains \(\tilde {\varepsilon }(x):= \varepsilon /\varepsilon ^{w\varGamma }\) for the peristatic solution (77) at \(N^{{\max \limits }}/N^{a}=10\) and lδ/a = 0.05, lδ/a = 0.05, 0.25, and 0.5, respectively, while the locally elastic solution is described by a step function \(^{L} \tilde {\varepsilon }(x)=[1+4H(|x|/a-1)]/5 \sim x\) (87), where H(x) is the Heaviside function. The peristatic curves estimated for \(N^{{\max \limits }}=2000\) and \(N^{{\max \limits }}=4000\), respectively, are pretty close to each other in the domain \(x\in {v^{l}_{i}}\) (exploited for a subsequent effective modulus estimation) with the exception of the point x = a where the curve estimated for undergoes large break. Due to this reason, the subsequent estimations are performed for \(N^{{\max \limits }}=4000\). The fixed Na = 400 is considered with the different scale rations lδ/a = 0.05, 0.25, 0.5 (see Fig. 1) corresponding to the \(\max \limits _{x}\tilde {\varepsilon }(x)=1.51, \ 1.54\), 1.55 and \(\min \limits _{x}\tilde {\varepsilon }(x)=0.166\), 0.164, 0.162, respectively. An inhomogeneity of \(\tilde {\varepsilon }(x)\) in comparison with locally elastic limit \(^{L} \tilde {\varepsilon }(x)\) reflects a nonlocal nature of peristatic phenomena. The inhomogeneity of \(\tilde {\varepsilon }(x)\) is localized in the vicinity of discontinuities x = ±a of displacement u(x). As is expected \(\tilde { \varepsilon }(x)\to ^{L} \tilde {\varepsilon }(x)\) in the limit of the horizon going to zero lδ/a → 0 at some distance of |xa| > 0.5a from the inclusion boundary x = ±a.

Fig. 1
figure1

Relative strain \(\tilde {\varepsilon }(x)\) vs x/a for lδ/a = 0.05(1), 0.025(2), and 0.5(3)

The classical problem of micromechanics is the estimation of effective elastic properties which are directly defined by the stress polarization parameters (78) and the displacements at the external boundary interface (80). Estimation of both the average peristatic stress polarizations (79) and displacements at the external boundary interface (80) as the functions of the homogeneous field εwΓ makes it possible to evaluate the effective Young’s modulus by Eqs. 8183, see Figs. 23, and 4 corresponding to the scales lδ/a = 0.05, 0.25, and 0.5, respectively. For the dilute approximations of the locally elastic case (lδ/a = 0), curves 1 and 2 in Figs. 23, and 4 were estimated by the exact equation (92) and Eq. 91, respectively. The dilute approximations of the peristatic composite bars are described by the curves 3 and 4 (see Figs. 23, and 4) estimated by Eqs. 83 and 82, respectively. At last, the curves 5 were plotted by the generalized Mori–Tanaka approach (84). As can be seen, the nonlocal effects have a weak action on the dilute approximation (81) (compare curves 2 and 4 in Figs. 23 and 4) while the effective moduli E estimated by the dilute approximation (82) can differ by a factor of 2 for the different scale lδ/a (see Fig. 4). In so doing, both the generalized Mori–Tanaka approach (83) and the generalized dilute approximation (82) lead to closed results (compare the curves 3 and 5 in Figs. 23 and 4) with softer results corresponding to Eq. 83.

Fig. 2
figure2

Relative effective modulus E/E(0) vs c estimated for lδ/a = 0.0 by Eqs. (92) (1), (91) (2), and for lδ/a = 0.05 (82) (3), (81) (4), and (83) (5)

Fig. 3
figure3

Relative effective modulus E/E(0) vs c estimated for lδ/a = 0.0 by Eqs. (92) (1), (91) (2), and for lδ/a = 0.25 (82) (3), (81) (4), and (83) (5)

Fig. 4
figure4

Relative effective modulus E/E(0) vs c estimated for lδ/a = 0.0 by Eqs. (92) (1), (91) (2), and for lδ/a = 0.5 (82) (3), (81) (4), and (83) (5)

In Fig. 5, we only present the estimations \(E^{*}/E^{(0)}\sim c\) obtained by the generalized Mori–Tanaka approach for the scales lδ/a = 0, 0.05, 0.25, 0.5, and 0.75, see curves 1–5, respectively.

Fig. 5
figure5

Relative effective modulus E/E(0) vs c estimated by Eqs. (83) for lδ/a = 0.0(1), 0.05(2), 0.25(3), 0.5(4), and 0.75(5)

A synergism effect between the elastic mismatch E/E(0) and the nonlocal scale lδ/a is manifested in the curves \(E^{*}/E^{(0)}\sim c\). Indeed, for vanishing elastic mismatch [\(\mathbf {C}(\mathbf {x},\hat {\mathbf {x}})\equiv \mathbf {C}^{(0)}(\mathbf {x},\hat {\mathbf {x}})\)], any nonlocality (∀lδ/a) has no effect on the effective Young modulus EE(0). The limit of vanishing length scale lδ/a accompanied with the elastic mismatch E(1)/E(0) = 5 leads to stiffening of E in comparison with E(0) in 5 times at c(1) = 1. At last, combining of both the length scale effect and elastic mismatch one tends to further stiffening of the effective modulus E (compare curves 1 and 5 in Fig. 5).

Conclusion

For the media subjected to remote homogeneous volumetric boundary loading, one proved that the effective behavior of this media is described by conventional effective constitutive equation which is intrinsic to the locally elastic theory. It was made by the most exploitation of the popular tools and concepts used in conventional elasticity of CMs and adapted to peristatics. Effective moduli are expressed through the introduced local stress polarization tensor averaged over the extended inclusion phase rather than in an entire space. One proposes a generalization of a dilute approximation method to their peristatic counterpart in the sense that the volume fraction of the particles is small and their mutual interaction can be neglected. As in a classical approach, the essential assumption in the generalized Mori–Tanaka method (MTM) states that each extended inclusion behaves as an isolated one in the infinite matrix and subject to some effective strain field coinciding with the average strain in the truncated matrix. Boss DAM and MTM are the particular implementations of the EFH (53) defining the background of the modern micromechanics of different physical natures proposed by Poisson, Faraday, Mossotti, Clausius, and Maxwell (1824–1880, see for references and details [11, 15, 41]). Homogeneity hypothesis H1 (53) does not assume any restrictions on both the material microtopology of inclusions and the field inhomogeneity of u(x),ε(x),σ(x) (xRd) (e.g., either singularity, discontinuity, or nonlocality).

The direct Mori–Tanaka method based on a single hypothesis (63) also has a specific merit in shedding new light on the weakness of the direct method that opens up outstanding possibilities to improve the method as was done in the method of effective field (MEF) generalizing its locally elastic counterpart (see for references [11]) to the peristatic analog in the subsequent publications by the author.

It should be mentioned that Buryachenko [12] analyzed the 2D composites consisting of constituents with different types of nonlocal properties introduced by Kröner [33] and Eringen [23]. It was also demonstrated that for statistically homogeneous composites subjected to the homogeneous loading, the effective properties are described by the classical local elasticity theory. Moreover, it was obtained by the generalized MEF that the nonlocal effects yield the slight stiffening of the effective properties (compare with Figs. 234, and 5) while the local fields estimated by both the nonlocal and local theories can differ by a sign. Similar effects are expected to be found for the peristatic CMs.

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Acknowledgements

Both the helpful comments of reviewers and their encouraging recommendations are gratefully acknowledged.

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Correspondence to Valeriy A. Buryachenko.

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Buryachenko, V.A. Generalized Mori–Tanaka Approach in Micromechanics of Peristatic Random Structure Composites. J Peridyn Nonlocal Model 2, 26–49 (2020). https://doi.org/10.1007/s42102-019-00023-9

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Keywords

  • Microstructures
  • Inhomogeneous material
  • Peristatics
  • Non-local methods
  • Multiscale modeling