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Modeling of One Inclusion in the Infinite Peristatic Matrix Subjected to Homogeneous Remote Loading

  • Valeriy A. BuryachenkoEmail author
Original Articles
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Abstract

A basic problem of micromechanics is analysis of one inclusion in the infinite matrix subjected to a homogeneous remote loading. A heterogeneous medium with the bond-based peridynamic properties (see Silling, J Mech Phys Solids 48:175–209, 2000) of constituents is considered. At first, volumetric boundary conditions are set up at the external boundary of a final domain obtained from the original infinite domain by truncation. One also considers a periodization approach (for a dilute concentration of inclusion). At last, a group of the iteration methods is considered where the displacement field is decomposed as linear displacement corresponding to the homogeneous loading of the infinite homogeneous medium and a perturbation field introduced by one inclusion. This perturbation field is found by the iteration method for entirely infinite sample with an initial approximation given by a driving term which has a compact support. The form of the mentioned solutions is adapted for subsequent incorporation into one or another micromechanical approach for peristatic composite materials. The methods are demonstrated by numerical examples for 1D case. A convergence of numerical results for the peristatic composite bar to the corresponding exact evaluation for the local elastic theory is shown.

Keywords

Inclusion Inhomogeneous materia Peristatics Non-local methods 

1 Introduction

The prediction of the behavior of composite materials (CM) in terms of the mechanical properties of constituents and their microstructure is a central problem of micromechanics. In its turn, the stress fields estimations in the constituents are based on a substitution into the one or another micromechanical scheme of a solution (called basic problem) for one inclusion inside the infinite matrix subjected to some homogeneous effective field. In the case of locally elastic properties of constituents, for both the ellipsoidal homogeneous inclusion and homogeneous effective field, this solution has an analytical representation in the form of Eshelby [24] tensor (see for references [44, 65]). For the general inclusion shape, the various numerical methods have been developed. So, the finite element analysis (FEA) in the truncation method is used for the modeling of the infinite medium with one inclusion by an increased size sample (see for references [8]). Immediate considerations of the infinite medium were performed by either the boundary integral equation methods (see, e.g., [17, 34]) or the volume integral equation ones (see for references, e.g., [8]).

Proposal of peridynamics by Silling [52] (see also [5, 36, 38, 56, 57]), 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 on 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 over finite distance (called a 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 (see [55, 57]) 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 horizons. The horizon can encompass discontinuities or different materials. A simplified version derived from this approach is the so-called bond-based approach, in which interactions only occur between pairs of material points within a horizon. As it 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 in three dimensions or ν = 1/3 in two dimensions [52]. 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, 55]). 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 [42] to differentiate the static problems considered in the current paper from the dynamic problems.

Buryachenko [10, 13] established similarity of the general integral equations for both locally elastic CMs and peristatic ones that opens the opportunities to straightforward generalization of their solutions for locally elastic CMs to the peristatic ones. These models contain as a necessary ingredient the solution for one inclusion in the infinite matrix subjected to the homogeneous effective field (called basic problem). At first, the scheme by the FEA for the locally elastic basic problem is generalized to the peristatic sample of increasing size subjected to the volumetric homogeneous displacement loading. While numerically solving a problem initially formulated on the unbounded domain, one typically truncates this domain. The artificial volumetric boundary conditions (ABC) are set up at the external boundary of a final domain obtained from the original infinite domain by truncation (see [60]). In a simplest case of a homogeneous remote loading at infinity, the corresponding artificial boundary conditions are also assumed to be homogeneous. A necessary requirement of ABC is that the solution found inside the truncated domain is close to the corresponding solution in the origin infinite domain. The infinite medium is modeled by an increased size sample with controlled stabilization of the solution for this sample.

The theory of multi-scale analysis of locally elastic composites of periodic structure is well developed (see for references, e.g., [3, 25, 40, 47]). It is usually implied that a ratio of the representative size of microstructure and the overall dimension of the structure is negligibly small. An alternative sort of truncation method in peristatics of CM is periodization method when a unite cell (UC, see [14, 15, 16, 37]) size is increased while the inclusion size is fixed or, in other words, dilute concentration of inclusion is analyzed, i.e., the infinite medium with one inclusion is modeled as a periodic structure CM with dilute concentrations of inclusions. For the peristatic periodic structure CM, one introduces new volumetric periodic boundary conditions (PBC) at the interaction boundary of a representative unit cell (UC). Introduction of the volumetric periodic boundary conditions allows us to generalize classical computational homogenization approach to their peristatic version.

Immediate consideration of the infinite medium can be performed by the Green function (see [61, 63]) technique which is reduced to the volume integral equation method (VIEM) well developed in the theory of locally elastic CMs (see for references [8]). Introduction of the micropolarization tensor by Buryachenko [10, 13] (generalizing the notion of polarization tensor in local elasticity, see, e.g., [8]) provides a great scope for generalization of the VIEM to its peristatic counterpart that can be easily performed as a straightforward exploitation of the method [9] proposed for solution of a corresponding basic problem with nonlocally elastic (in the sense of Eringen [23]) properties of constituents. However, this version of the VIEM is not considered in the current publication in more details.

In the next approach, the displacement field is decomposed as linear displacement corresponding to the homogeneous loading of the infinite homogeneous medium and a perturbation field introduced by one inclusion. This perturbation field is found by the iteration method for entirely infinite sample analogously to a solution for the nonlocally elastic (in the sense of Eringen [23]) basic problem proposed by Buryachenko [9]. The discretization of the nonlocal linear equations of peristatics are reduced to the linear matrix equations. The direct method of triangulation of the corresponding N × N matrix A becomes prohibitive in terms of computer time and storage if the matrix is quite large for either (or both) 2D and 3D dimensions or comparable scale of nonlocal effects (defined by the horizon) and the field and material scales. Furthermore, most large problems are sparse, and the sparsity is lost to a considerable extent during the triangulation procedure. For dealing with these very large matrix with many nonzero entries, it is advisable to use a class of iterative methods which never alter the matrix A and require the storage of a few vectors of length N at a time.

In many physics and engineering areas, nature may be best represented or modeled by unbounded or semi-unbounded domains. While the existence and uniqueness of solutions of finite systems have been completely resolved, much remains unknown for infinite systems corresponding to the problems for one inclusion in the infinite matrix. An excellent detailed account of the colorful history of infinite matrices is given by Bernkopf [4]). Establishment of a sufficient conditions guarantying the existence and uniqueness of infinite matrix equations are usually restricted by the matrixes of special structures (see [51]]) like diagonal dominance, tridiagonal, and sign distributions, which are frequently nonsingular. The purpose of this paper is to propose a method for solution of infinite matrix equations corresponding to the problems for one inclusion in the infinite peristaltic matrix. Establishment of the right hand side of this equation as a discretization of a fictitious body force with compact support has a beneficial effect on development of an iteration solution method which falls out as a bonus.

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. The scheme of the truncation method is presented in Section 3. Periodization method is considered in Section 4. The iterative method for one inclusion in the infinite matrix is considered in Section 5. The method proposed is in fact the Jacoby method generalized to the band iteration matrix of the infinite order. General representations of Sections 35 are immediately analyzed for 1D cases in Section 6. Close solutions of the truncation method and iterative one are demonstrated. A convergence of numerical results for the peristatic composite bar to the corresponding exact evaluation for the local elastic theory are shown.

2 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 heterogeneity v with indicator functions V and bounded by the closed smooth surfaces Γ defined by the relations Γ(x) = 0 (x ∈Γ), Γ(x) > 0 (xv), and Γ(x) < 0 (xv). Initially, no restrictions are imposed on the elastic symmetry of the phases or on the geometry of the heterogeneity.

We first consider the local basic equations of thermoelastostatics of composites

$$ \begin{array}{@{}rcl@{}} \nabla\boldsymbol{\sigma}(\textbf{x})&=&-\textbf{b}(\textbf{x}), \end{array} $$
(1)
$$ \begin{array}{@{}rcl@{}} \boldsymbol{\sigma}(\textbf{x})&=&\textbf{L}(\textbf{x})\boldsymbol{\varepsilon}(\textbf{x}), \ \ \text{or}\ \ \ \boldsymbol{\varepsilon}(\textbf{x})=\textbf{M}(\textbf{x})\boldsymbol{\sigma}(\textbf{x}), \end{array} $$
(2)
$$ \begin{array}{@{}rcl@{}} \boldsymbol{\varepsilon}(\textbf{x})&=&[\nabla {\otimes}\textbf{u}+(\nabla{\otimes}\textbf{u})^{\top}]/2, \ \ \nabla\times\boldsymbol{\varepsilon}(\textbf{x})\times\nabla=\textbf{0}, \end{array} $$
(3)
where ⊗ and 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):
$$ \textbf{L}(\textbf{x})=(dk,2\mu)\equiv dk(\textbf{x})\textbf{N}_1 + 2\mu(\textbf{x})\textbf{N}_2, $$
(4)
N1 = δδ/d, N2 = I-N1 (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,b) the notation g1(x) ≡g(x) −g(0) is used.

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

$$ {}^L\hat{\mathcal{{\boldmath{L}}}}({}^L\textbf{u})(\textbf{x})+\textbf{b}(\textbf{x})=\textbf{0},\ \ \ {}^L\hat{\mathcal{{\boldmath{L}}}}({}^L\textbf{u})(\textbf{x}):=\nabla[\textbf{L}\nabla^L\textbf{u}(\textbf{x})], $$
(5)
where \(^{L}\hat {\mathcal {{\boldmath {L}}}}({}^{L}\textbf {u})(\textbf {x})\) is an elliptic differential operator of the second order.

In this section, we also summarize the linear peristatics model introduced by Silling [52] (see also [32, 54]). 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 the second kind

$$ \hat{\mathcal{{\boldmath{L}}}}(\textbf{u})(\textbf{x})+\textbf{b}(\textbf{x}) = \textbf{0}, \ \ \ \hat{\mathcal{{\boldmath{L}}}}(\textbf{u})(\textbf{x})\!:=\!\!\int \textbf{f}(\textbf{u}(\hat {\textbf{x}})-\textbf{u}(\textbf{x}),\hat{\textbf{x}}-\textbf{x},\textbf{x})d\hat {\textbf{x}} , $$
(6)
where f is a pairwise force function (called also a bond force) whose value is the force vector that the point located at \(\hat {\textbf {x}}\) (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 \( \textbf {f}(\textbf {u}(\hat {\textbf {x}})-\textbf {u}(\textbf {x}),\hat {\textbf {x}}-\textbf {x},\textbf {x})=\textbf {f}(\textbf {u}(\hat {\textbf {x}})-\textbf {u}(\textbf {x}),\hat {\textbf {x}}-\textbf {x})\). 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). Because of this, the superscripts L(⋅) will be correspond hereafter to the local case. The body force density function b(x) is assumed to be self-equilibrated
$$ \int \textbf{b}(\textbf{x})d\textbf{x}= 0 $$
(7)
and vanished outside some loading region: b(x) = 0 for |x| > aδ.

The relative position of two points in the reference configuration is denoted as ξ

$$ \boldsymbol{\xi}=\hat{\textbf{x}}-\textbf{x} $$
(8)
and their relative displacement as η
$$ \boldsymbol{\eta}=\textbf{u}(\hat {\textbf{x}})-\textbf{u}(\textbf{x}). $$
(9)
All deformations are assumed small and the reference and deformed configurations are taken to be the same. The further points are apart the weaker their interaction. Only points \(\hat {\textbf {x}}\) inside some neighborhood \({\mathcal {H}}_{\textbf {x}}\) of x interact with x:
$$ \textbf{f}(\boldsymbol{\eta},\boldsymbol{\xi},\textbf{x})\equiv \textbf{0}\ \ \ \forall \hat{\textbf{x}}\not \in {\mathcal{H}}_\textbf{x}, $$
(10)
which means that the point x can not “see” outside \({\mathcal {H}}_{\textbf {x}}\). The vector \(\boldsymbol {\xi }=\hat {\textbf {x}}-\textbf {x}\) (\(\hat {\textbf {x}}\in {\mathcal {H}}_{\textbf {x}}\)) is called a bond to x, and the collection of all bonds to x form the domain \({\mathcal {H}}_{\textbf {x}}\). Without loss of generality, it is assumed that a shape of \({\mathcal {H}}_{\textbf {x}}\) is spherical: \({\mathcal {H}}_{\textbf {x}}=\{\hat {\textbf {x}}:\ |\hat {\textbf {x}}-\textbf {x}|\leq l_{\delta }\}\) and a positive number lδ, called the horizon, does not depend on x. The pairwise force function f is required to have the following properties (for ∀η,ξ and ∀xw):
$$ \textbf{f}(-\boldsymbol{\eta},-\boldsymbol{\xi},\textbf{x}+\boldsymbol{\xi})\equiv -\textbf{f}(\boldsymbol{\eta},\boldsymbol{\xi},\textbf{x}) \ \ \ \forall \boldsymbol{\eta},\boldsymbol{\xi} $$
(11)
meaning assurance of conservation of linear momentum, and
$$ (\boldsymbol{\eta}+\boldsymbol{\xi})\times\textbf{f}(\boldsymbol{\eta},\boldsymbol{\xi},\textbf{x})\equiv \textbf{0} \ \ \ \forall \boldsymbol{\eta},\boldsymbol{\xi}, $$
(12)
which assures conservation of angular momentum and means that the force vector between these particles \(\hat {\textbf {x}}\) 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

$$ \textbf{f}(\boldsymbol{\eta},\boldsymbol{\xi},\textbf{x})=\textbf{C}(\boldsymbol{\xi},\textbf{x})\boldsymbol{\eta} \ \ \ \forall \boldsymbol{\eta},\boldsymbol{\xi}. $$
(13)
The corresponding micropotential given by
$$ \widetilde w (\boldsymbol{\eta},\boldsymbol{\xi},\textbf{x})={1\over 2}\boldsymbol{\eta}^{\top}\cdot\textbf{C}(\boldsymbol{\xi},\textbf{x})\boldsymbol{\eta} $$
(14)
defines the local strain energy density (the energy per unit volume in the body at a given point)
$$ \mathcal{{W}}(\textbf{x})={1\over 2}{\int}_{{\mathcal{H}}_\textbf{x}}\widetilde w (\boldsymbol{\eta},\boldsymbol{\xi},\textbf{x}) d\hat {\textbf{x}}, $$
(15)
where the factor of 1/2 appears because each endpoint of a bond “owns” only half the energy in the bond and the forces in the prestressed reference configuration are assumed to be vanished since otherwise these forces can be incorporated into the body-force b. Here, the material’s micromodulus function C contains all constitutive information and its value is a second-order tensor given by
$$ \textbf{C}(\boldsymbol{\xi},\textbf{x})={\partial \textbf{f}(\textbf{0},\boldsymbol{\xi},\textbf{x})\over \partial \boldsymbol{\eta}} \ \ \ \forall \boldsymbol{\xi}. $$
(16)

For consistency with Newton third law (also following the requirement from Eq. 11) 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

$$ \textbf{C}(-\boldsymbol{\xi})=\textbf{C}(\boldsymbol{\xi})=\textbf{C}^{\top}(\boldsymbol{\xi}) \ \ \ \forall \boldsymbol{\xi}, $$
(17)
where the properties of C are discussed in detail in Silling [52]. For example, for the micromodulus functions with the step-function and triangular profiles
$$ \textbf{C}(\boldsymbol{\xi})= \textbf{C} V({\mathcal{H}}_\textbf{x}), \ \ \textbf{C}(\boldsymbol{\xi})=\textbf{C}(1-|\boldsymbol{\xi}|/l_{\delta})V({\mathcal{H}}_\textbf{x}), $$
(18)
respectively, where \(V({\mathcal {H}}_{\textbf {x}})\) is the indicator function of \({\mathcal {H}}_{\textbf {x}}\). The peristatic solution of Eq. 6 with C described by Eqs. (18) is in detail investigated by both numerical and analytical methods in 1D case (see [5, 7, 42, 62]) and 2D case (see [29, 30]). If we assume a linear microelastic material then, in general, the stiffness tensor for a linear microelastic materials can be shown to read as (see [20, 21, 52])
$$ \textbf{C}(\boldsymbol{\xi})= \lambda (\boldsymbol{\xi})\boldsymbol{\xi}\otimes\boldsymbol{\xi}, $$
(19)
where a scalar function λ(ξ) depends on the specific material model and for a linear isotropic material, λ depends on ξ only trough ξ = |ξ|. For the special case of a linear microelastic materials
$$ \textbf{C}(\boldsymbol{\xi})=c{\boldsymbol{\xi}\otimes\boldsymbol{\xi}\over |\boldsymbol{\xi}|^3}V({\mathcal{H}}_\textbf{x}), \ \ \ \text{i.e} \ \ \ C_{ij}(\boldsymbol{\xi})=c{\xi_i\xi_j\over (\xi_k\xi_k)^{3/2}}V({\mathcal{H}}_\textbf{x}), $$
(20)
where a constant c depends again on the material but also on the dimension d of the problem. The proportionality factor c (20) (or C (18)) 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]). A direct consequence of the assumption (19) 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 [52].

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

$$ \begin{array}{@{}rcl@{}} \textbf{C}(\textbf{x},\hat{\textbf{x}}) = \left\{\begin{array}{ll} \textbf{C}^{(1)}(\textbf{x},\hat{\textbf{x}}), & \text{for} \ \textbf{x},\hat{\textbf{x}}\in v,\\ \textbf{C}^{(0)}(\textbf{x},\hat{\textbf{x}}), & \text{for} \ \textbf{x},\hat{\textbf{x}}\in v^{0},\\ \textbf{C}^{i}(\textbf{x},\hat{\textbf{x}}), & \text{for} \ \textbf{x}\!\in\! v,\hat{\textbf{x}}\!\in\! v^{0} \text{or}\ \ \textbf{x}\!\in\! v^{0},\hat{\textbf{x}}\!\in\! v,\\ 0, & \text{for} \ |\textbf{x}-\hat{\textbf{x}}|>l_{\delta}, \end{array}\right. \end{array} $$
(21)
which can also be presented in the form
$$ \begin{array}{@{}rcl@{}} \textbf{C}(\textbf{x},\hat{\textbf{x}}) \!&=&\! \textbf{C}^{(1)}(\textbf{x},\hat{\textbf{x}})V^{(1)}(\textbf{x})V^{(1)}(\hat {\textbf{x}})\\ &&+\textbf{C}^{(0)}(\textbf{x},\hat{\textbf{x}})V^{(0)}(\textbf{x})V^{(0)}(\hat {\textbf{x}}) \\ \!&&+\! \textbf{C}^{i}(\textbf{x},\hat{\textbf{x}})[V^{(1)}(\textbf{x})V^{(0)}(\hat {\textbf{x}})\\ &&+ V^{(0)}(\textbf{x})V^{(1)}(\hat {\textbf{x}})]. \end{array} $$
(22)
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 that \(\textbf {C}^{(1)}(\textbf {x},\hat {\textbf {x}})\geq \textbf {C}^{i}(\textbf {x},\hat {\textbf {x}})\geq \textbf {C}^{(0)}(\textbf {x},\hat {\textbf {x}})\), or
$$ \begin{array}{@{}rcl@{}} \textbf{C}^i(x,\hat x)\!&=&\!(\textbf{C}^{(0)}(\textbf{x},\hat {\textbf{x}})+\textbf{C}^{(1)}(\textbf{x},\hat {\textbf{x}}))/2, \\ \textbf{C}^i(\textbf{x},\hat {\textbf{x}})\!&=&\! \min[\textbf{C}^{(0)}(\textbf{x},\hat {\textbf{x}}),\textbf{C}^{(1)}(\textbf{x},\hat {\textbf{x}})], \end{array} $$
(23)
(see [2, 54]) where \(\textbf {x},\hat {\textbf {x}}\in v_{\Gamma }\) is called the interaction interface (see for details 48]). Adaptive grid refinement technique was proposed in [6] for analysis of peristatic problems in the visinity of interfase that involves a variable horizon size.

Substitution of Eq. 13 into Eq. 11 leads to the following property of C(ξ,x):

$$ \textbf{C}(\boldsymbol{\xi},\textbf{x})=\textbf{C}(-\boldsymbol{\xi},\textbf{x}+\boldsymbol{\xi}). $$
(24)

The peridynamic theory is traditionally based on the using 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 \(\hat {\textbf {f}}:\ R^{d}\times R^{d}\to R^{d}\) by

$$ \begin{array}{@{}rcl@{}} \hat{\textbf{f}}(\textbf{p},\textbf{q})=\left\{\begin{array}{lll} \textbf{f}(\textbf{u}(\textbf{p})-\textbf{u}(\textbf{q}),\textbf{p}-\textbf{q},\textbf{q}), & \text{if} \ \textbf{p},\textbf{q}\in w,\\ \textbf{0}, & \text{otherwise}. \end{array} \right. \end{array} $$
(25)
It is assumed that \(\hat {\textbf {f}}(\hat {\textbf {x}},\textbf {x})\) is Riemann-integrable that does not imply the bounding of \(\hat {\textbf {f}}(\hat {\textbf {x}},\textbf {x})\) as \(|\hat {\textbf {x}}-\textbf {x}|\to 0\). Then by adapting Cauchy’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 \(\hat {\textbf {x}}\) to the right of z exert on all material particles to its left (see, e.g., [58, 62]). So, for dD (d = 1,2,3) cases
$$ \begin{array}{@{}rcl@{}} \boldsymbol{\sigma}(\textbf{x})\!&=&\!\mathcal{\boldmath{L}}^{\sigma}(\textbf{u}), \\ \mathcal{\boldmath{L}}^{\sigma}(\textbf{u})\!&:=&\! {1\over 2} {\int}_S{\int}_0^{\infty}{\int}_0^{\infty} (y+z)^{d-1}\hat{\textbf{f}}(\textbf{x}+y\textbf{m},\textbf{x}-z\textbf{m})\\ &&\otimes\textbf{m} dzdyd{\Omega}_\textbf{m}, \end{array} $$
(26)
were S stands for the unit sphere and dΩm denotes a differential solid angle on S in the direction of any unite vector m. Lehoucq and Silling [32] proved (26) for 3D case while the case d = 2 [13] can be justified in a similar manner. Equation 26 at d = 1 can also be reduced to the representations [58, 62] by the variable exchange xzr, x + ys. Indeed, the origin-centered unit 1D “sphere” is the set {-1 , 1}, which has a measure of 2. Then the integrand of Eq. 26 is reduced to integrand of the corresponding equation for 1D case [58, 62] by the use of the equality (11).

3 Truncation Method

Due to nonlocality, the equilibrium equation (6) should be accompanied by a “boundary” condition, imposed as a volumetric constraint in so-called the interaction domain wΓ (in opposite to the local elasticity case where the boundary conditions are imposed precisely at the bounding surface Γ(0), see for details [31, 52]); 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, 41]) \(\overline {\overline {w}}=w\cup w_{\Gamma }\) stands the nonlocal closure of w. The most popular shape for wΓ with prescribed the forces and displacements is a boundary layer of thickness given by the horizon lδ (see [35]); \(w_{\Gamma }=\{w\oplus {\mathcal {H}}_{\textbf {0}}\}\backslash w\), where \(w\oplus {\mathcal {H}}_{\textbf {0}}\) is a Minkovski outer parallel (delation) of w (\({\mathcal {A}}\oplus {\mathcal {B}}:= \cup _{\textbf {x}\in {\mathcal {A}},\textbf {y}\in {\mathcal {B}}}\{\textbf {x}+\textbf {y}\}\)); in such a case w is the internal region of \(\overline {\overline {w}}\) (see [52]). 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, the nonlocal Dirichlet volume-constrained problem is given as (see [19, 41])

$$ \begin{array}{@{}rcl@{}} \left\{ \begin{array}{ll} -\widehat{\mathcal{\boldmath{L}}} (\textbf{u})=\textbf{b}& \text{on} \ w,\\ \textbf{u}=\textbf{h}, & \text{on}\ w_{{\Gamma} u}=w_{\Gamma}. \end{array}\right. \end{array} $$
(27)
The conditions (27) are called displacement loading. It is assumed that \(\overline {\overline {w}}\) is obtained by cutting out of a domain from periodic heterogeneous medium filling the entire space Rd. The interaction domain wΓ is covered by the surfaces Γr := {y : dist(y,w) = r} (0 ≤r ≤ lδ) so that for ∀ywΓ (y ∈Γr) one can attach a normal vector n(y)⊥Γr; i.e. Γ0 and \({\Gamma }_{l_{\delta }}\) are the boundaries of w and wΓ, respectively. In such a case, the Dirichlet (272) volumetric boundary conditions are called homogeneous loading conditions if there exist some symmetric constant tensors \(\boldsymbol {\varepsilon }^{w_{\Gamma }}\) such that
$$ \begin{array}{@{}rcl@{}} \textbf{h}(\textbf{y})&=& \boldsymbol{\varepsilon}^{w_{\Gamma}}\textbf{y}, \ \textbf{y}\in w_{\Gamma}. \end{array} $$
(28)
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).

Peridynamics is a continuum theory where each infinitesimal volume interacts with an infinite numbers of other volumes within the horizon. However, in numerical implementation described by Silling and Askari [54], 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 ∈ [1,Nmax] of nodes covering the total macrovolume w. The spatially discretized form of the equilibrium (6) and (13) replaces the integral by the finite sum for each node p

$$ \sum\limits_{q\in {\mathcal{H}}_p}\textbf{C}(\textbf{x}_p,\textbf{x}_q)(\textbf{u}_q-\textbf{u}_p)\bar V_{pq}=-\textbf{b}_p, $$
(29)
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 cutoff 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 [43, 49]. In modified algorithm by Hu et al. [28] (with an accuracy estimations [49]), 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 Γ separating the phases v(0) and v(1), then the locally elastic model require 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 [48, 50]). It was showed that nonlocal interface problems converge to their classical local counterparts in the limit of the horizon going to zero lδ/a → 0. 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 is described by Eq. 6 with the constant horizon and micromodulus (21) determined as an average value of the micromoduli in the matrix and inclusion (231), see [2] (and also [37] where one proposed a variation of the extended interface micromodulus \(\textbf {C}^{i}(\textbf {x},\hat {\textbf {x}})\) in a spirit of the functional graded materials theory described in, e.g., [8]).

The equilibrium (29) can be presented in a standard matrix form (xpw)

$$ \begin{array}{@{}rcl@{}} \sum\limits_q {\mathbb{K}}_{pq}\textbf{u}_q\! &=&\! \textbf{b}_p, \end{array} $$
(30)
$$ \begin{array}{@{}rcl@{}} {\mathbb{K}}_{pq}\! &=&\! \textbf{C}(\textbf{x}_p,\textbf{x}_q)\bar V_{pq} - \delta_{pq}\sum\limits_{r\in {\mathcal{H}}_p}\textbf{C}(\textbf{x}_p,\textbf{x}_r)\bar V_{pr}, \end{array} $$
(31)
where \({\mathbb {K}}_{pq}\) is a rectangular matrix (dNmax × d(Nmax + NwΓ)) while a displacement uq is presented by a vector (d(Nmax + NwΓ) × 1). Equation (30) should be solved in an accompany with the corresponding homogeneous volumetric boundary conditions [e.g., Dirichlet-type volumetric boundary condition (28)]. It means inversion of a global square stiffness matrix \(\hat {\mathbb {K}}\) (dNmax × dNmax) for all nodes p ∈ [1,dNmax] in Eq. 32 where a displacement u and an fictitious body force vector \(\hat {\textbf {b}}\) (defined by the volumetric boundary condition (28)) are presented by the vectors (dNmax × 1)
$$ \begin{array}{@{}rcl@{}} \sum\limits_q \hat{\mathbb{K}}_{pq}\textbf{u}_q\! &=&\! \hat{\textbf{b}}_p, \end{array} $$
(32)
$$ \begin{array}{@{}rcl@{}} \hat {\mathbb{K}}_{pq}\! &=&\! {\mathbb{K}}_{pq}W(\textbf{x}_q), \end{array} $$
(33)
$$ \begin{array}{@{}rcl@{}} \hat{\textbf{b}}_p\! &=&\! -\sum\limits_{r\in {\mathcal{H}}_p}\textbf{C}(\textbf{x}_p,\textbf{x}_r)\boldsymbol{\varepsilon}^{w_{\Gamma}}\textbf{x}_r \bar V_{pr} W^{w{\Gamma}}(\textbf{x}_r) \end{array} $$
(34)
where \(\textbf {h}\equiv \boldsymbol {\varepsilon }^{w_{\Gamma }}\textbf {x}\) (\(\boldsymbol {\varepsilon }^{w_{\Gamma }}\equiv \)const., xwΓ) (28), and the indicator function \(W^{\overline {\overline {w}}}\) of the nonlocal closure \(\overline {\overline {w}}=w\cup w_{\Gamma }\) is splitted as
$$ \begin{array}{@{}rcl@{}} W^{\overline{\overline{w}}}(\textbf{x})=W(\textbf{x})+W^{w{\Gamma}}(\textbf{x}). \end{array} $$
(35)

Thus, the solution u(x) (xw) based on inverting a truncated matrix \(\hat {\mathbb {K}}_{pq}\) (p,q = 1,…,dNmax) is assumed to be found. In truncation domain method, an infinite integration domain Rd is replaced by corresponding finite domain wRd with control of the error. For estimation of the size of the truncation domain, let us the geometrical boundary Γw of the domain w in 2D is prescribed in the polar coordinate system Γw(x) = {x : x1 = r(ϕ)cos(ϕ),x2 = r(ϕ)sin(ϕ)}. The positions of the points y ∈Γρ of the surface Γρ is expressed through the points x ∈Γw by a homothety transformation:

$$ \textbf{y}=\textbf{x}_i+\rho(\textbf{x}-\textbf{x}_i), $$
(36)
where xi = 0 is a homothetic center coinciding with the particle center. Once the parameter 1 < ρ is chosen, the distribution of the points y ∈Γρ is determined. For a sequence ρk = k we estimate a sequence uρ in the domain wρ with the boundary Γρ and analyze a convergence of uρ by estimation of tolerance defined by the relative difference of uρ and uρ+ 1
$$ \begin{array}{@{}rcl@{}} {\Delta}\!&=&\!||\textbf{u}^{\rho+ 1}-\textbf{u}^{\rho}||_{L_2} /||\textbf{u}^{\rho+ 1}||_{L_2},\ \ \ {\Delta}\leq 0.1\% \end{array} $$
(37)
$$ \begin{array}{@{}rcl@{}} ||(\cdot)||_{L_2}\!&=&\! \left( (\text{mes} v_{\Gamma})^{-1} {\int}_{v_{\Gamma}} (\cdot)^2(\textbf{y})d\textbf{y}\right)^{1/2}, \end{array} $$
(38)
where 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 vl := vvΓ because according to [10, 13] the effective moduli depend on the strain distribution u(x) only in the domain xvl.

The found solution u(x) (xw) allows us to finalize the solution of the basic problem by the following representations for the displacements and strains (xqvl)

$$ \begin{array}{@{}rcl@{}} \textbf{u}(\textbf{x}_q)\! &=&\! \textbf{A}^u(\textbf{x}_q)\boldsymbol{\varepsilon}^{w {\Gamma}}, \end{array} $$
(39)
$$ \begin{array}{@{}rcl@{}} \boldsymbol{\varepsilon}(\textbf{x}_q)\! &=&\! \textbf{A}^{\epsilon}(\textbf{x}_q)\boldsymbol{\varepsilon}^{w {\Gamma}}, \end{array} $$
(40)
respectively, where Au(xq) is the displacement influence function while the strain influence function A𝜖(xq) is obtained from Au(xq) by numerical differentiation. Hereafter, we introduced the notion of the field (displacement, strain, and stress) influence functions widely used in micromechanics (see, e.g., [8, 25]) [no confusion with the term “influence (or weighting) function” exploited in peridynamics, see, e.g, [50, 57]]. It should be mentioned that the strain influence function A𝜖(xq) averaged over the inclusion v is used in the different methods of micromechanics of locally elastic CMs (see for references [8]). However, the peridynamic theory is traditionally based on the using of the displacement fields u(x) rather than the strain ε(x) fields which are not conceptually necessary. Moreover, the fictitious body forces (34) with a discontinuity at x ∈Γ also produce a discontinuity of u(x) at x ∈Γ because in general the displacement field u(x) has the same smoothness as the body force field (see Silling et al. [58]). For elimination of this difficulties, Buryachenko [17] (see also [16]) proposed a micromechanical model of peristatic CMs based on the averaging of weighted displacement influence function Au(s) over the surface svl. However, just for better visualisation (e.g., for comparison with a locally elastic solution) of comparative numerical 1D analyses of different methods (see Section 6), we will estimate the strain influence function A𝜖(xq).

Of course, a classical problem for a finite peristatic sample (with or without inclusions) was intensively analyzed in many publications even for more general constitutive laws (such as nonlinear and state-based approaches [43, 49, 50, 54, 57]) and more advanced numerical methods. So, in mesh adaptivity [6, 7], a given quadrature model is refined where additional resolution is needed (e.g., in the vicinity of interface). In coarsening approach by Silling [53], the detailed geometry of a body is replaced by a succession of different geometries providing essentially the same result as in the original detailed problem. Macek and Silling [35] (see also [36]) implemented the peridynamics approach into the ABAQUS commercial FE code by the use of truss elements as mesh with appropriate stiffness properties which represents the peridynamic bonds.

Thus, there is a room for improvement of the quadrature method considered (30)–(35) and its generalization to more general constitutive laws. The goal of the current presentation is a formulation of a systematic scheme for obtaining of the basic problem solution (38) and (39) having control of the error (37). The form of this solution is adapted for subsequent incorporation into one or another micromechanical method (such as, e.g., generalized Mori-Tanaka approach, see for details [18]) of random structure peristatic CMs. In general, the displacement (38) and strain (39) influence functions depend on the size of the domain w. However, the satisfaction of the tolerance requirements (37) provides a stabilization of the solution Au(x) and A𝜖(x) in the extended inclusion xvl. It should be mentioned that the estimation of the tolerance (37) in the same domain xvΓ for two solutions uρ+ 1 and uρ with the different domains of definition is more convenience than control of the limit

$$ a/L \to 0. $$
(41)

4 Periodization Method

In this section, we extract some results by Buryachenko [16] in the form adopted for subsequent presentation.

An alternative sort of truncation method is periodization method where the problem for one inclusion inside the infinite matrix is replaced by a periodic structure with an increasing ratio lΩ/a of a unite cell (UC) size lΩ and the inclusion size a. So, the domain w is constructed from a periodic field of the cubic unit cells (UC) Ωα: w = ∪Ωα, where each UC with the side length 2lΩ is labeled by the multiplet of integer numbers α = (α1,…,αd) ∈ Z+, here αi = 0,± 1 (i = 1,…,d). The centers of UCs are presented as \(\textbf {x}^{\Omega }_{\boldsymbol {\alpha }}= 2l^{\Omega }\alpha _{1}\textbf {e}_{1}+\ldots + 2l^{\Omega }\alpha _{d}\textbf {e}_{d}\) in the Cartesian coordinate frame e1,…,ed. Each identical unit cell contains a homogeneous matrix v(0) and inclusion v with an indicator functions V and bounded by the closed smooth surfaces Γ defined by the relations Γ(x) = 0 (x ∈Γ), Γ(x) > 0 (xv), and Γ(x) < 0 (xv). In the context of the principle of separation of scales assumed, the microscopic length scale |Ωα| is much smaller than the characteristic size of the macroscopic sample |w|.

For simplicity of notations we will consider 2D case w = ∪Ωij (i,j = 0,± 1,± 2,…) with the square unit cells Ωij. Let a representative unit cell Ω00 with the corner points \(\textbf {x}^{c}_{kl}\) (k,l = ± 1) has the boundary \({\Gamma }^{0}=\cup {\Gamma }^{0}_{ij}\) where the boundary partition \(\textbf {x}_{ij}^{0}\in {\Gamma }^{0}_{ij}\) separates the UCs Ω00 and Ωij (i = 0,± 1, j = ±(1 −|i|)). The position vectors \(\textbf {x}^{0}_{\alpha }:=\textbf {x}^{0}_{ij}\in {\Gamma }^{0}_{ij}\) or \(\textbf {x}^{0}_{\beta }:=\textbf {x}^{0}_{kl}\in {\Gamma }^{0}_{kl}\) of the opposite sides of Ω00 (α = −β) are expressed through the position vectors of the corner points \(\textbf {x}^{c}_{mn}\) (i = 0,± 1, j = ±(1 −|i|), k = −i, l = −j, m,n = ± 1)

$$ \begin{array}{@{}rcl@{}} \textbf{x}^0_{\alpha}&=&\textbf{x}^0_{\beta}+\textbf{x}^c_{1,-1}-\textbf{x}^c_{-1,-1}, \end{array} $$
(42)
$$ \begin{array}{@{}rcl@{}} \textbf{x}^0_{\alpha}&=&\textbf{x}^0_{\beta}+\textbf{x}^c_{-1,1}-\textbf{x}^c_{-1,-1} \end{array} $$
(43)
for the i = 1, j = 0 and i = 0, j = 1, respectively, of the boundary Γ0.

For visualization of the corresponding periodic boundary conditions (PBC) in locally elastic UC, a toroidal edge correction (see [45]) is often used. In such a case, each 2D square box Ω00 can be regarded as a torus, so that points on opposite edges are considered to be closed (a 3D case is analyzed in a similar manner). In peristatics, if the source point xp + ξ ∈Ωα (x ∈Ω00, HCode \(\boldsymbol {\xi }\in {\mathcal {H}}_{p}\)) then the peristatic counterparts of the locally elastic PBC called new volumetric periodic boundary conditions (PBC) represent periodic displacements and antiperiodic tractions

$$ \begin{array}{@{}rcl@{}} \textbf{u}(\textbf{x}_p+\boldsymbol{\xi})\! &=&\! \textbf{u}(\textbf{x}_p+\boldsymbol{\xi}-2\boldsymbol{\alpha}^l) + 2\boldsymbol{\varepsilon}^{w_{\Gamma}}\boldsymbol{\alpha}^l, \end{array} $$
(44)
$$ \begin{array}{@{}rcl@{}} \textbf{t}(\textbf{x}^0_{\bf\alpha})\! &=&\! -\textbf{t}(\textbf{x}^0_{\bf\beta}), \end{array} $$
(45)
respectively, where β = −α and \(\textbf {x}^{0}_{\bf \alpha }\) are defined by Eq. 41 while antiperiodic traction \(\textbf {t}(\textbf {x}_{\bf \alpha })=\boldsymbol {\sigma }(\textbf {x}^{0}_{\bf \alpha })\textbf {n} (\textbf {x}^{0}_{\bf \alpha })\) (26) is defined on the boundaries \(\textbf {x}^{0}_{\bf \alpha }\in {\Gamma }^{0}_{\bf \alpha }\) [α = (ij), i = 0,± 1,±(1 −|i|)] with the outward normal unit vectors \(\textbf {n} (\textbf {x}^{0}_{\bf \alpha })=-\textbf {n} (\textbf {x}^{0}_{\bf \beta })\).

It was showed (see for details [16]) that due to the volumetric PBC, the governing equation depends only on the displacements inside the UC Ω00 rather than on the displacement inside the extended UC \({\Omega }_{00{\Gamma }}:= {\Omega }_{00} \oplus {\mathcal {H}}_{\textbf {0}}\). In fact, the volumetric PBC (42) incorporating into the equilibrium Eq. 6 produces the new closed integral equation which does not require any additional interface conditions at the geometrical boundary Γ0. The discretization of this equation in the form (32) acts as a macro-to-micro transition of the deformation-driven type, where the overall deformation εwΓ is controlled. Determination of the microstructural displacements u(x) (x ∈Ω00) in an accompany with the volumetric PBC (42) at the dilute concentration of inclusions [a/lΩ → 0, compare with Eq. 40] allows to finalize the solution of the basic problem with estimation of the displacement Au(xq) (38) and strain A𝜖(xq) (39) influence functions. Control of accuracy is executed by the tolerance requirement (37). It should be mentioned that the theory of computational homogenization of periodic structure peristatic CMs was developed in [17]. However, a dilute approximation (a/lΩ ≪ 1) with prescribed tolerance requirement (37) was not considered before.

5 Iterative Methods

5.1 Basic Iterative Methods

One of the main feature of linear peristatics (6) is that each unknown up appears in only a few equations 2nl + 1 (where 2lδ = 2nlh) of the linear system in a “neighborhood” of the p th equation. In such a case for the full space w = Rd, the infinite coefficient matrix \(\mathbb {K}\) is banded with some bandwidth kb (so that \(\mathbb {K}_{pq}= 0\) at |pq| > kb) defined by both the numeration of nodes p and nl. In particular, the bandwidth for 1D problem equals nl.

Let us consider the governing equation for an infinite Rd (d = 1,2,3) homogeneous peristatic medium (6)

$$ \widehat{\mathcal{\boldmath{L}}}^{(0)}(\textbf{u}^{0})(\textbf{x})+\textbf{b}(\textbf{x})=\textbf{0} $$
(46)
with the micromodulus C(0). The self-equilibrated body-force density b(x) (44) produces a displacement corresponding to the homogeneous strain
$$ \textbf{u}^{0}(\textbf{x})\equiv \boldsymbol{\varepsilon}^{0}\textbf{x}, \ \ \ \boldsymbol{\varepsilon}^{0}=\text{const}. $$
(47)
We are coming now to consideration of a single fixed inclusion v(0) with the micromodulus C(1)(ξ) in the infinite homogeneous medium with the properties C(0)(ξ). Then the solution u(x) can be split into two ones as
$$ \textbf{u}(\textbf{x})=\textbf{u}^0(\textbf{x})+\textbf{u}_1(\textbf{x}), $$
(48)
where
$$ \begin{array}{@{}rcl@{}} \widehat{\mathcal{\boldmath{L}}}^{(0)}(\textbf{u}^{0}) (\textbf{x})+\textbf{b}(\textbf{x})\! &=&\! \textbf{0}, \end{array} $$
(49)
$$ \begin{array}{@{}rcl@{}} \widehat{\mathcal{\boldmath{L}}}(\textbf{u}_1) (\textbf{x})+\widehat{\mathcal{\boldmath{L}}}_1(\textbf{u}^{0}) (\textbf{x})\! &=&\! \textbf{0}, \end{array} $$
(50)
and the operator \(\widehat {\mathcal {\boldmath {L}}}\) is decomposed as \(\widehat {\mathcal {\boldmath {L}}}\equiv \widehat {\mathcal {\boldmath {L}}}^{(0)}+\widehat {\mathcal {\boldmath {L}}}_{1}\) (44), where \(\widehat {\mathcal {\boldmath {L}}}_{1}\) is defined by Eq. 44 with the replacements C(0)C1. In so doing, Eq. 48 is more preferable to solve than Eqs. 6 and 13 (see, e.g., [64]) because a fictitious body force density of Eq. 48 has a compact support \(\mathcal {\boldmath {L}}_{1}(\textbf {u}^{0})(\textbf {x})\equiv \textbf {0}\) (xvl, vl = {x : |x|≤ a + lδ}). Hence, u1(x) passes to the constant u1(x) →const. at the removal of x from vl: |xy|→ (\(\forall \textbf {y}\in {v^{l}_{i}}\)) and, therefore, for numerical solution of Eq. 48 only some finite domain xvl0 := {x : |x| < al0} should be discretized. However, the main advantage of the decompositions (47) and (48) (48) is elimination in Eq. 48 of the nonlocal volumetric boundary conditions prescribed over a nonzero volume (see, e.g., Eq. 28 and [19, 31, 35, 52]) of Eq. 48 while the solution of Eq. 47 is assumed to be known.

Indeed, Eq. 48 can be recast in the form (30) (p,q = …,− 2,− 1,0,1,2,…)

$$ \begin{array}{@{}rcl@{}} \sum\limits_q {\mathbb{K}}_{pq}\textbf{u}_{1|q}\! &=&\! \tilde{\textbf{b}}_p, \end{array} $$
(51)
where the infinite banded matrix \({\mathbb {K}}_{pq}\) is defined by Eq. 31 while a fictitious body force has the following representation
$$ \tilde{\textbf{b}}_p=-\sum\limits_{r\in {\mathcal{H}}_p}\textbf{C}_1(\textbf{x}_p,\textbf{x}_r)\boldsymbol{\varepsilon}^0(\textbf{x}_r-\textbf{x}_p) \bar V_{pr}. $$
(52)
It should be mentioned that the indices p,q = …,− 2,− 1,0,1,2,… in Eq. 49 are renumbered with respect to the indices p,q = (1,dNmax) (32). In general, the matrix \({\mathbb {K}}\) (49) is an infinite matrix; however, in our current practice, the matrix \({\mathbb {K}}\) (49) is, in fact, a truncated submatrix \(({\mathbb {K}}_{pq})\) (49) (p,q = [Nfinite,Nfinite] with the size 2Nfinite × 2Nfinite described below. The vector \(\tilde {\textbf {b}}_{p}\) in the right hand side of Eq. 49 has a compact support (\(\tilde {\textbf {b}}_{p}\equiv \textbf {0},\ \forall \textbf {x}_{p}\in R^{d}\setminus v^{l}\)) because C1(xp,xq) vanishes in the domain Rdvl. It means that nonzero elements of the vector \(\tilde {\textbf {b}}_{p}\) is limited by the indices |p|≤ d(Nv + NvΓ) where Nv and NvΓ are the numbers of nodes in the inclusion v and the external part of interaction interface vlv, respectively.

The last established feature of the right hand side of Eq. 49 allows us to exploit the iterative methods which are especially effective for analyses of infinite banded matrices. The popular iterative methods passes from one iteration to the next by modifying one or a few components of an approximate vector solution at a time. Improvement of iterations is controlled by the residual vector \(\tilde {\textbf {b}}-\mathbb {K}\textbf {u}\). Namely, Eq. 49 can be written in the form (k = 1,2,…)

$$ \mathbb {D}\textbf{u}_1^{(k)}=\mathbb{F}\textbf{u}_1^{(k-1)}+\tilde{\textbf{b}}_p, $$
(53)
where \(\mathbb {K} =\mathbb {D}-\mathbb {F}\) is a splitting and \(\textbf {u}_{1}^{(0)}\) is a starting vector. For the iteration to be practical, it must be easy to solve the system (51). In the simplest case, this is the Jacobi method where \(\mathbb {D}= \text {diag}(\ldots , \mathbb {K}_{-1 -1},\mathbb {K}_{00}, \mathbb {K}_{1 1},\ldots \)) is a diagonal matrix. The diagonal entries of \(\mathbb {D}\) are all nonzero that enables the inverse infinite matrix \(\mathbb {D}^{-1}= \text {diag}(\ldots , \mathbb {K}^{-1}_{-1 -1},\mathbb {K}^{-1}_{00}, \mathbb {K}^{-1}_{1 1},\ldots \)) to be found. Inversion \(\mathbb {D}^{-1}\) of the diagonal matrix makes it possible to reduce (51) to an explicit recursive relation
$$ \textbf{u}_1^{(k)}=\mathbb{B}\textbf{u}_1^{(k-1)}+\tilde {\tilde{\textbf{b}}},\ \ \mathbb{B}:=\mathbb {D}^{-1}\mathbb{F}, \ \ \tilde {\tilde{\textbf{b}}}:=\mathbb {D}^{-1}\tilde{\textbf{b}} $$
(54)
with the driving term usually chosen as an initial approximation
$$ \textbf{u}_1^{(0)}=\tilde{ \tilde{\textbf{b}}}. $$
(55)
It is also possible a choice of the initial approximation \(\textbf {u}_{1}^{(0)}\) in the form of the locally elastic solution (5)
$$ \textbf{u}_1^{(0)}=^L\textbf{u}_1. $$
(56)
It should be mentioned that using of the initial approximation (54) implies, in fact, the correction of the \(\tilde {\textbf {b}}(\textbf {x})\) at xwΓ analogously to Eq. 34 with replacement \(\boldsymbol {\varepsilon }^{w_{\Gamma }}\to ^{L}\!\boldsymbol {\varepsilon }_{1}\).

Thus, the indices p of nonzero elements of the vectors \( \tilde {\textbf {b}}_{p}\) and \(\textbf {u}_{1|p}^{(0)}\) are bounded by the inequality |p|≤ d(Nv + NvΓ). The iteration matrix \(\mathbb {B}\) (completely determining the convergence (or not) of an iterative method (51)) is the infinite banded matrix (as \(\mathbb {K}\)) with the same bandwidth kb. Then multiplication of the matrix \(\mathbb {B}\) and the vector \(\textbf {u}_{1|p}^{(0)}\) leads to the vector \(\textbf {u}_{1|p}^{(1)}\) with the indices of nonzero elements bounded by an inequality |p|≤ d(Nv + NvΓ + kb). In a similar manner, the k’s iteration \(\textbf {u}_{1|p}^{(k)}\) has a compact support with the indices of nonzero elements |p|≤ N(k) := d(Nv + NvΓ + kb × k) that defines the size of the domain \(v_{i}^{l0}\). It should mentioned that the matrix \({\mathbb {B}}\) (521) is an infinite matrix. However, owing to the mentioned compact support of \(\textbf {u}^{(k-1)}_{1}\), only the submatrix \(({\mathbb {B}})_{pq}\), p,q = −[d(Nv + NvΓ + kb × (k − 1)),d(Nv + NvΓ + kb × (k − 1))], is used. It means that in the iteration scheme (521), one exploits the finite matrix \(({\mathbb {B}})_{pq}\), p,q = −[d(Nv + NvΓ + kb × (k − 1)),d(Nv + NvΓ + kb × (k − 1))], with the size depending on the iteration number k.

In the Jacoby iteration, each component of the next iterate \(\textbf {u}_{1}^{(k + 1)}\) is expressed through all components of the previous step \(\textbf {u}_{1}^{(k)}\). However, in the Gauss-Seidel iteration, the newly computed component \(u_{1|p}^{(k)}\) is changed within a working vector that is redefined at each step (|p|≤ N(k))

$$ u_{1|p}^{(k + 1)}=(\mathbb{D}_{pp})^{-1}\left[ \sum\limits_{q=-N^{(k)}}^{p-1}\mathbb{F}_{pq}u_{1|q}^{(k + 1)}+ \sum\limits^{q=N^{(k)}}_{p + 1}\mathbb{F}_{pq}u_{1|q}^{(k)}+\tilde b_p\right]. $$
(57)

The successive overrelaxation method (SOR) corrects the p th component following the relaxation sequence (0 < ω ≤ 1)

$$ u_{1|p}^{(k + 1)}=\omega u_{1|p}^{GS}+(1-\omega)u_{1|p}^{(k)}, $$
(58)
in which \(u_{1|p}^{GS}\) is defined by the expression on the right-hand side of Eq. 56; note that if ω = 1 then this is just the Gauss-Seidel method.

The sequence \(\textbf {u}_{1}^{(0)}, \textbf {u}_{1}^{(1)}, \textbf {u}_{1}^{(2)},\ldots \) converges to a unique solution u = u1 if the norm of the iteration matrix \(\mathbb {B}\) turns out to be small “enough” (less than 1) \(|\!|\mathbb {B}|\!|<1\) where \(|\!|\mathbb {B}|\!|\) denotes the Frobenius norm of \(\mathbb {B}\). A convergence of u(k) is controlled by estimation of tolerance defined by the relative difference of u(k+ 1) and u(k) and the residual vector

$$ \begin{array}{@{}rcl@{}} {\Delta}_2&=&||\textbf{u}^{(k + 1)}-\textbf{u}^{(k)}||_{L_2} /||\textbf{u}^{(k + 1)}||_{L_2}, \end{array} $$
(59)
$$ \begin{array}{@{}rcl@{}} {\Delta}_{3}&=&||\tilde{\textbf{b}}-\mathbb{K}\textbf{u}^{(k + 1)}||_{L_2} /||\textbf{u}^{(k + 1)}||_{L_2}, \end{array} $$
(60)
where L2 norm is evaluated over the domain vl (as in Eq. 372). The simulation is stopped when the tolerance max(Δ23) reaches 0.1%.

5.2 An Iterative Tikhonov Regularization Method

It may appear that the ill-conditioning issue (see e.g., [7]), that occurs with the use of some micromodulus function, prevents computing the numerical solution to the static problem (32) or (49). A measure of the ill-posedness of the mentioned equations can be performed by using the condition number \(\text {cond}\mathbb {K}= |\!|\mathbb {K}|\!||\!|\mathbb {K}^{-1}|\!|\). If \(\mathbb {K}\) is highly ill-conditioned, one can use a regularization method to solve this system. So, the known Tikhonov (see Tikhonov and Arsenin [59]) regularization equation is

$$ \tilde{\mathbb{K}}\textbf{u}_1=\textbf{b}^K, \ \ \tilde{\mathbb{K}}:=(\mathbb{K}^{\top}\mathbb{K}+\lambda\textbf{I}), $$
(61)
where \(\textbf {b}^{K}:=\mathbb {K}^{\top }\tilde {\textbf {b}}\), I is the unit infinite matrix, and the superscript signifies the transpose. The optimal value λ is chosen according to the method described in e.g., [27, 33, 39]).

The methods for inversion of the nonsingular finite matrix \(\tilde {\mathbb {K}}\) (32) are well developed (see, e.g., [26, 46]). We modify the Tikhonov regularization method for the infinite matrix \(\tilde {\mathbb {K}}\) by using the preconditioner and iteration for solving an ill-posed linear infinite system (49). So, we begin with the following preconditioner

$$ \textbf{P}=\mathbb{K}^{\top}+\lambda \mathbb{K}^{-1}. $$
(62)
Applying the operator P (59)-(49), we obtain the following equation
$$ (\mathbb{K}^{\top}+\lambda \mathbb{K}^{-1})\mathbb{K}\textbf{u}= (\mathbb{K}^{\top}+\lambda \mathbb{K}^{-1})\tilde{\textbf{b}}, $$
(63)
which is reduced to
$$ \tilde{\mathbb{K}}\textbf{u}=\textbf{b}^K+\lambda\textbf{u}. $$
(64)
It is interesting that Eq. 61 is similar to the Tikhonov regularization (58) with additional term λu at the right-hand side in Eq. 61. It gives an opportunity to form an iterative process to find the solution of u by
$$ \tilde{\mathbb{K}}\textbf{u}_1^{[n + 1]}=\textbf{b}^K+\lambda\textbf{u}_1^{[n]}. $$
(65)
Starting from \(\textbf {u}_{1}^{[n]}=\textbf {0}\) (n = 0), we can apply the SOR to solve the above linear system and generate a sequence of vectors \(\textbf {u}_{1}^{(k)[n]}\) (k = 1,...). In each inner iteration, we apply the SOR to solve the linear system (58) under a convergence criterion specified in Section 5.1. When the following convergence criterion is satisfied, we also stop the outer iterative sequence, and then the solution of u1 is obtained.

Thus, the basic problem is assumed to be solved and the displacement Au(xq) (38) and strain A𝜖(xq) (39) influence functions are found with the required tolerance (57). It should be mentioned that the solution of the basic problem by Buryachenko [11] is in fact a simplified first order approximation of the iteration method (51) proposed before at the intuitional level of rigorousness for entirely 1D case without any tolerance requirements.

6 Numerical Results 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 v = [−a,a] and the volumetric boundary wΓ: \(\bar {\bar w}=w_{\Gamma }\cup w\), wΓ = [−Llδ,−L] ∪ [L,L + lδ]. The midpoint quadrature scheme (similar to [11, 12]) is realized for the uniform grid xi = (0,± 1,± 2,…,±(Nmax + NwΓ)) with the numbers of material points 2Nmax + 1 and 2NwΓ + 1 inside both w and the interaction boundary wΓ, respectively, and a constant step size h = L/Nmax.

We consider a heterogeneous peristatic bar with the step-function profiles (181) (k = 0,1, E(1)/E(0) = 5)

$$ \begin{array}{@{}rcl@{}} C(\xi)=\left\{\begin{array}{ll} C^{(k)}, & \text{for} \ |\xi|<l_{\delta},\\ 0, & \text{for} \ |\xi|>l_{\delta}. \end{array}\right.,\ \ C^{(k)}= 3E^{(k)}/l_{\delta}^3; \end{array} $$
(66)
the triangle profile (182) [and the micromodulus (19) and (20)] can be considered in a similar manner.
For the periodic structure CMs, the rule of the interaction boundary wΓ is performed by the interaction UC’s boundary ΩΓ. In Fig. 1, one presents the estimation of the relative strains \(\tilde \varepsilon (x):=\varepsilon (x)/\varepsilon ^{0}\) and \(\tilde \varepsilon ^{\Omega } (x):=\varepsilon (x)/\langle \varepsilon \rangle \) in the peristatic composite bar by the truncation method (curve 3) and periodization one (curve 1), respectively, at a/L = a/lΩ = 0.06 and Na = 200. In fact, the relative strains \(\tilde \varepsilon (x)\) and \(\tilde \varepsilon ^{\Omega } (x)\) are the strain influence functions (39) (see the discussion at the end of Section 3). The curves of strain influence functions vs x/a behave according to the general features of peristatic solutions for a homogeneous peristatic bar with a fictitious body force considered in [7, 58]. Indeed, the fictitious body forces (34) with a discontinuity at x = ±a also produce a discontinuity of u(x) at x = ±a (it is not demonstrated in Fig. 1) because in general the displacement field u(x) has the same smoothness as the body force field (see Silling et al. [58]). In addition, the discontinuity in the micromodulus C(ξ) (63) 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) [it is well observed for the curves 1 and 3 in Fig. 3 in the form of peaks of ε(x) (k = 0) at x = a ± lδ, lδ/a = 0.25]. Locally elastic solution for the periodic bar \(^{L}\!\tilde \varepsilon ^{\Omega } (x)\) is shown by the curve 2. The solution by the truncation method (curve 3) is close to the corresponding locally elastic solution \(\tilde \varepsilon (x)=[1 + 2H(|x|/a-1)]/3\) at |x|− a > 2lδ, where H(x) is the Heaviside function.
Fig. 1

Relative strains \(\tilde \varepsilon (x)\) (3) and \(\tilde \varepsilon ^{\Omega } (x)\) (1) vs x/a. Locally elastic solution \(^{L}\!\tilde \varepsilon ^{\Omega } (x)\) (2)

For the iterative method with the initial approximation (54), we estimate influence of the ratio Na/Nmax with the fixed Nmax = 2500, lδ/a = 0.25, and the iteration number k = 100. In Fig. 2, one presents the curves ε(x) ∼ x/a with the different rations Na/Nmax = 1/25,1/10,1/2 corresponding to maxxε(x)/ε0ε(1.25)/ε0 = 1.333, 1.347, 1.357, respectively. As can be seen, a change of the ratio Na/Nmax in 12 times yields a negligible distinction of the curves 1 and 3 that indicates on effectiveness and robustness of the iterative method proposed (so Δ2 = 3.4E − 4, Δ3 = 6.3E − 9) in 1D case.
Fig. 2

Relative strain \(\tilde {\varepsilon }(x)\) vs x/a for \(N^{a}/N^{\max }=\) 1/2 (1), 1/10 (2), 1/25 (3)

We analyze the influence of iteration number on distribution of ε(x)/ε0x/a for the fixed Na = 600 and lδ/a = 0.25. As can be seen in Fig. 3, increasing of the iteration number k from 3 to 10 and 100 (see the curves 1, 2, and 3, respectively, in Fig. 3) lead to variation of the pick of strains maxε(x)/ε0 = maxε(1.25)/ε0 from 1.381 to 1.379 and 1.348, respectively. It means that 10 iteration provides sufficient accuracy. The curve 4 in Fig. 3 is estimated by the truncation method (as in Fig. 1) has the pick maxε(x)/ε0 = ε(1.25)/ε0 = 1.31. In so doing, the compute times spent for estimation of curves 2 and 4 differ by two decimal order.
Fig. 3

Relative strains \(\tilde {\varepsilon }(x)\) vs x/a estimated by the iterative method k = 3(1), 10(2), and 100(3) and truncation method (4)

In Fig. 4, the fixed Na = 600 is considered with the different scale rations lδ/a = 0.05, 0.25, 0.5, 1.0 (curves 1–4, respectively) corresponding to the maxxε(x) = 1.344, 1.356, 1.357, 1.364 and minxε(x) = 0.297, 0.296, 0.300, 0.333, respectively. As it is expected ε(x) →Lε(x) in the limit of the horizon going to zero lδ/a → 0 at some distance of x from the inclusion boundary x = a. The peristatic curves 1–4 estimated for Na = 600 and Na = 200 (only the results for Na = 600 are depicted in Fig. 4), respectively, are pretty close to each other with the exception of the point x = a where the curve undergoes unexpectedly large break. Due to this reason, the estimations are performed for Na = 600.
Fig. 4

Relative strain vs x/a for lδ/a = 0.05, 0.25, 0.5, 1.0 (curves 1–4, respectively)

7 Conclusion

One presented three different methods for the modeling of the basic problem for one inclusion inside the infinite matrix subjected to some homogeneous effective remote field. Each method has advantages and disadvantages and it is crucial for the analyst to be aware of their range of applications. In particular, both the truncation and periodization methods supported by the FEA are well developed in the locally elastic problems. The FEA is especially effective for the constant fields ε0 ≡ const. From other side, the VIE method in local elasticity enables one to restrict discretization to the extended inclusion \({v^{l}_{i}}\) only, and an inhomogeneous structure of both the inclusion and remote field ε0≢ const. presents no problem in the framework of the same numerical scheme. The VIE method has well developed routines for the solution of integral equations (such as, e.g., the quadrature schemes) and allows to analyze arbitrary inhomogeneous fields ε0≢ const. However, the VIE method is quite time-consuming and no optimized commercial softwares exist for its application. Because of this, the truncation method combined with commercial softwares FEA (such as, e.g., ABAQUS) is apparently most popular in engineering practice for locally elastic media. The mentioned features of the different methods are analyzed in [8] in the context of locally elastic problems. However, in the case of peridynamics, computational advantage of the truncation method realized by the FEA [35] is leveled off (with resect to the VIE method) due to nonlocality of the constitutive law. Nevertheless, universality of commercial software FEA provides a long-term advantage of the truncation method in engineering practice. In general, for dealing with the infinite banded matrices (12), it is advisable to use a class of iterative methods proposed which never alter the infinite matrix \(\mathbb {K}\) and require the storage of a few vectors of finite length at a time. One demonstrated effectiveness and robustness of the iterative method proposed in 1D case.

Finally, the found displacement influence functions Au(x) (38) (having control of the error such as Eqs. 37 or 57) can be incorporated into one or another method of micromechanics of peristatic random structure CMs (see, e.g., a generalized Mori-Tanaka approach [18]) that will by considered in subsequent publications.

Notes

Acknowledgements

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

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Authors and Affiliations

  1. 1.Micromechanics and Composites LLCDaytonUSA

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