Mechanics of Materials with Periodic Truss or Frame Micro-Structures

Martinsson, Per-Gunnar; Babuška, Ivo

doi:10.1007/s00205-006-0044-2

Cited by 20 publications

(21 citation statements)

References 23 publications

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“…For lattices that can be modeled as trusses there exists a d × d matrix σ (0) (ξ) such that an identity analogous to (3.11) holds (Theorem 6.5 of Ref. 13). The matrix σ (0) (ξ) can be computed from σ (us) (ξ) through a simple formula.…”

Section: Elastostatic Equilibrium Of Mechanical Trussesmentioning

confidence: 95%

“…All its entries are second-order polynomials in ξ and moreover, all its eigenvalues satisfy c|ξ| 2 ≤ λ i ≤ C|ξ| 2 for some c > 0 and C < ∞ (Lemma 4.1 of Ref. 13). Since σ (ε) (ξ) = ε −2 σ (us) (εξ) we find that each one of the q 2 blocks of S (0) (ξ) equals σ (0) (ξ) −1 .…”

Section: Elastostatic Equilibrium Of Mechanical Trussesmentioning

confidence: 97%

“…It follows from Lemma 4.1 in Ref. 13 that for any connected mono-atomic lattice there exists a positive definite matrix M such that this series takes the form…”

Section: Homogenization Of Mono-atomic Latticesmentioning

confidence: 98%

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Homogenization of Materials With Periodic Truss or Frame Micro-Structures

Martinsson

Babuška

2007

Math. Models Methods Appl. Sci.

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The equations that govern elastostatic equilibrium between a prescribed force field and an unknown displacement field for materials with periodic skeletal micro-structures are studied. It is shown that as the size of the micro-structure tends to zero, the displacement field will converge to the solution of a constant coefficient partial differential equation. This equation is shown to be either a classical or a micro-polar continuum elasticity equation, depending on the micro-structural geometry and the nature of the external load field. Convergence is proved for representative model problems in Sobolev energy norms and in the maximum norm. In addition, it is shown that by considering pseudodifferential homogenized equations, any order of convergence can be achieved.

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Section: Elastostatic Equilibrium Of Mechanical Trussesmentioning

confidence: 95%

Section: Elastostatic Equilibrium Of Mechanical Trussesmentioning

confidence: 97%

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Homogenization of Materials With Periodic Truss or Frame Micro-Structures

Martinsson

Babuška

2007

Math. Models Methods Appl. Sci.

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“…Throughout this section, we assume that the lattice of the discrete strain model is connected [19]. We begin this section by briefly reviewing the discrete elastic model introduced in [21].…”

Section: The Abcs For Discrete Elasticitymentioning

confidence: 99%

Exact Artificial Boundary Conditions for Continuum and Discrete Elasticity

Lee¹,

Caflisch²,

Lee³

2006

SIAM J. Appl. Math.

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For the continuum and discrete elastic equations, we derive exact artificial boundary conditions (ABCs), often referred to as transparent boundary conditions, that can be applied at a planar interface below which there are no forces. Solution of the elasticity equations can then be performed using this interface as an artificial boundary, often with greatly reduced computational effort, but without loss of accuracy. A general solvability requirement is presented for the existence of an artificial boundary operator for discrete systems (such as discrete elasticity) on an unbounded (semi-infinite) domain. The solvability requirement is validated by introducing a sum-of-exponentials ansatz for the solution below the artificial boundary. We also derive a new expression for the total energy for the system, involving only the region above the artificial boundary. Numerical examples are provided to confirm and illustrate the accuracy and effectiveness of the results. Introduction.Many of the boundary value problems arising in applied mathematics are formulated on unbounded domains. It is in general a nontrivial task to solve such problems numerically [6], since the numerical solution naturally requires boundary conditions at a finite depth in the body.The main motivation of the present work comes from the numerical simulation of strain fields in semi-infinite domains. For the strain equations, the use of a physical boundary condition, such as the zero displacement field at a certain depth, has been a common practice [21]. On the other hand, due to the long range of elastic interactions, the zero boundary condition must be imposed at considerable depth in order to accurately compute the strain field [4], which entails large computational cost.The purpose of this paper is to derive exact artificial boundary conditions (ABCs) such that the solution on the (bounded) computational domain coincides with the exact solution on the unbounded domain. Such exact artificial boundary conditions are oftentimes referred to as transparent boundary conditions (TBCs) [6].There have been various works on ABCs for a wide range of problems. For example, certain ABCs for the Poisson and Helmholtz equations on infinite domains are investigated in [1] using domain decomposition and Fourier techniques. For general elliptic problems, approximate ABCs and error estimates are performed within the finite element framework in [3]. Boundary element methods for homogeneous elasto-

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“…Stretch dominated lattice materials then resemble periodic trusses, albeit on a smaller length scale than typical structural trusses (Hutchinson and Fleck, 2006;Martinsson and Babuška, 2007). Experimental and computational studies of discrete truss structures reveal complicated dynamic behavior, including the existence of band gaps (Howard and Pao, 1998;Signorelli and von Flotow, 1988) and microjetting under impact loading (Winter et al, 2014).…”

Section: Introductionmentioning

confidence: 97%

Wave propagation in equivalent continuums representing truss lattice materials

Messner

Barham

Kumar

et al. 2015

International Journal of Solids and Structures

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Stiffness scales linearly with density in stretch-dominated lattice meta-materials offering the possibility of very light yet very stiff structures. Current additive manufacturing techniques can assemble structures from lattice materials, but the design of such structures will require accurate, efficient simulation techniques. Equivalent continuum models have several advantages over discrete truss models of stretch dominated lattices, including computational efficiency and ease of model construction. However, the development an equivalent model suitable for representing the dynamic response of a periodic truss in the small deformation regime is complicated by microinertial effects. This paper derives a dynamic equivalent continuum model for periodic truss structures suitable for representing long-wavelength wave propagation and verifies it against the full Bloch wave theory and detailed finite element simulations. The model must incorporate microinertial effects to accurately reproduce long wavelength characteristics of the response such as anisotropic elastic soundspeeds. The formulation presented here also improves upon previous work by preserving equilibrium at truss joints for simple lattices and by improving numerical stability by eliminating vertices in the effective yield surface.

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Mechanics of Materials with Periodic Truss or Frame Micro-Structures

Cited by 20 publications

References 23 publications

Homogenization of Materials With Periodic Truss or Frame Micro-Structures

Homogenization of Materials With Periodic Truss or Frame Micro-Structures

Exact Artificial Boundary Conditions for Continuum and Discrete Elasticity

Wave propagation in equivalent continuums representing truss lattice materials

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