Homogenization of the 1D Peri-static/dynamic Bar with Triangular Micromodulus

Eriksson, Kjell; Stenström, Christer

doi:10.1007/s42102-020-00042-x

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…respectively, where V ( H x ) is the indicator function of H x . The peridynamic solution of equation (6) with C described by equation (12) is investigated in detail by both numerical and analytical methods in 1D case [10,28,31,35] and 2D case [37]. If we assume a linear microelastic material, then in general, the stiffness tensor for the linear microelastic materials can be shown to read as [7,33]…”

Section: Preliminariesmentioning

confidence: 99%

“…It is assumed that f ^ ( x ^ , x ) is Riemann-integrable; that does not imply the bounding of f ^ ( x ^ , x ) as | x ^ − x | → 0 . In particular, for 1D case, the “peridynamic stress” σ ( x ) [29,31] is defined as…”

Section: Preliminariesmentioning

confidence: 99%

“…Numerical results are obtained for one-dimensional (1D) case. A simplified 1D structure are analyzed because it is possible to dispense with some assumptions (widely used in two-dimensional (2D) and three-dimensional (3D) cases) and to focus our attention to the direct use of a large body of analytical and numerical results obtained for a 1D homogeneous and inhomogeneous peridynamic bar of deterministic structure [29][30][31][32][33][34][35]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems

Buryachenko¹

2022

Mathematics and Mechanics of Solids

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We consider a static problem for statistically homogeneous matrix linear bond-based peridynamic composite materials (CMs) subjected to body force with compact support. Estimation of the effective displacements is performed by the exploitation of the most popular tools and concepts used in conventional local elasticity of composite materials (CMs) with their adaptation to peridynamics. The method is based on estimation of a perturbator introduced by one inclusion inside the infinite peristatic matrix subjected to the body force. The statistical averages of both the displacements and stresses are estimated by summation of these perturbators for all possible locations of inclusions. It leads to obtaining of a new general integral equation (GIE) where a renormalizing procedure is not required for the involved absolutely convergent integrals. It allows us to estimate not only the effective nonlocal constitutive equation but also to evaluate the statistical field averages (inhomogeneous and nonlocal) inside the phases at the fine scale that is critically important for advanced modeling (e.g. for any nonlinear phenomena potentially considered in the future). In particular, in the generalized effective field method (EFM) proposed, the effective field is evaluated from self-consistent estimations using closing of a corresponding integral equation in the framework of the quasi-crystalline approximation. In so doing, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. The method proposed is a promising ingredient at the construction of provably robust data-driven surrogate equations describing both the effective behavior and effective displacement concentration operator inside inclusions. Some opportunities for generalizations of the method proposed are considered. Numerical results for the estimation of effective deformations are obtained for one-dimensional (1D) statistically homogeneous peridynamic composite bar with the prescribed self-equilibrated body forces.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems

Buryachenko¹

2022

Mathematics and Mechanics of Solids

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The J-area integral applied in peridynamics

Stenström

Eriksson

2021

Int J Fract

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The J-integral is in its original formulation expressed as a contour integral. The contour formulation was, however, found cumbersome early on to apply in the finite element analysis, for which method the more directly applicable J-area integral formulation was later developed. In a previous study, we expressed the J-contour integral as a function of displacements only, to make the integral directly applicable in peridynamics (Stenström and Eriksson in Int J Fract 216:173–183, 2019). In this article we extend the work to include the J-area integral by deriving it as a function of displacements only, to obtain the alternative method of calculating the J-integral in peridynamics as well. The properties of the area formulation are then compared with those of the contour formulation, using an exact analytical solution for an infinite plate with a central crack in Mode I loading. The results show that the J-area integral is less sensitive to local disturbances compared to the contour counterpart. However, peridynamic implementation is straightforward and of similar scope for both formulations. In addition, discretization, effects of boundaries, both crack surfaces and other boundaries, and integration contour corners in peridynamics are considered.

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The essential work of fracture in peridynamics

et al. 2023

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In this work, the essential work of fracture (EWF) method is introduced for a peridynamic (PD) material model to characterize fracture toughness of ductile materials. First, an analytical derivation for the path-independence of the PD J-integral is provided. Thereafter, the classical J-integral and PD J-integral are computed on a number of analytical crack problems, for subsequent investigation on how it performs under large scale yielding of thin sheets. To represent a highly nonlinear elastic behavior, a new adaptive bond stiffness calibration and a modified bond-damage model with gradual softening are proposed. The model is employed for two different materials: a lower-ductility bainitic-martensitic steel and a higher-ductility bainitic steel. Up to the start of the softening phase, the PD model recovers the experimentally obtained stress–strain response of both materials. Due to the high failure sensitivity on the presence of defects for the lower-ductility material, the PD model could not recover the experimentally obtained EWF values. For the higher-ductility bainitic material, the PD model was able to match very well the experimentally obtained EWF values. Moreover, the J-integral value obtained from the PD model, at the absolute maximum specimen load, matched the corresponding EWF value.

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Homogenization of the 1D Peri-static/dynamic Bar with Triangular Micromodulus

Cited by 6 publications

References 21 publications

Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems

Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems

The J-area integral applied in peridynamics

The essential work of fracture in peridynamics

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