Multiscale finite element method for a locally nonperiodic heterogeneous medium

Fish, Jacob; Wagiman, Amir

doi:10.1007/bf00371991

Cited by 111 publications

(52 citation statements)

References 6 publications

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“…Among the noteworthy SGEMs are the s-version of the finite element method [19,20,21,22] with application to strong [23,24] and weak [25,26,27,28] discontinuities, various multigrid-like scale bridging methods [29,30,31,32], the Extended Finite Element Method (XFEM) [33,34,35] and the Generalized Finite Element Method (GFEM) [36,37] both based on the Partition of Unity (PU) framework [38,39] and the Discontinuous Galerkin (DG) [40,41] method. Multiscale methods based on the concurrent resolution of multiple scales are often called as embedded, concurrent, integrated or hand-shaking multiscale methods.…”

Section: Introductionmentioning

confidence: 99%

Multiscale enrichment based on partition of unity

Fish¹,

Yuan²

2004

Numerical Meth Engineering

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139

110

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A new Multiscale Enrichment method based on the Partition of Unity (MEPU) method is presented. It is a synthesis of mathematical homogenization theory and the Partition of Unity method. Its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible. MEPU is perfectly suited for enriching the coarse scale continuum descriptions (PDEs) with fine scale features and the quasi-continuum formulations with relevant atomistic data. Numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi-continuum formulations.

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Section: Introductionmentioning

confidence: 99%

Multiscale enrichment based on partition of unity

Fish¹,

Yuan²

2004

Numerical Meth Engineering

Self Cite

139

110

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“…(35) to the stress, strain, and plastic strain tensors, substituting the resulting terms into Eq. (15), and gathering the terms of equal order…”

Section: Multiple Temporal Scale Analysismentioning

confidence: 99%

“…This is in contrast to the classical (spatial) mathematical homogenization where the microscale displacement field is taken as a perturbation of the macroscale field. Spatial homogenization in the presence of non-periodic conditions have been previously investigated using stochastic [11,12,13,14] and deterministic [15,16,17,18,19] methods. Mathematical analysis of the spatial homogenization theory with almost-and non-periodic fields have been conducted in Refs.…”

Section: Introductionmentioning

confidence: 99%

A Nonlocal Multiscale Fatigue Model

Fish

Oskay

2005

Mechanics of Advanced Materials and Structures

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A nonlocal multiscale model in time domain is developed for fatigue life predictions. The method is based on the mathematical homogenization theory with almost periodic fields. The almost periodicity reflects the effects of irreversible deformations in time domain in the form of accumulation of damage. Multiple temporal scales are introduced to decompose the original boundary value problem into micro-chronological (temporal unit cell) and macrochronological (homogenized) problems. A nonlocal Gurson type constitutive law is revisited for cyclic loading, calibrated and validated against fatigue crack propagation experiments on 316L austenitic stainless steel specimens.

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“…By integrating (10) over and making use of (13), we have (18) where and (19) The constitutive equating for the phase average field can be expressed as (20) where is the phase average stress, and the overall homogenized stress field turns into (21) where is the volume fractions for phase in RVE satisfying . The phase free energy density corresponding to the nonlocal constitutive equation (20) is given as (22) and the corresponding phase damage energy release rate and the energy dissipation inequality [9][39] applied to the phase average field can be expressed as (23) (24) It should be noted that the nonlocal character of the phase average damage and the constitutive equation (20) has been proved in [20].…”

Section: Figure 1 Macroscopic and Microscopic Structuresmentioning

confidence: 99%

“…Compute the principal components of by (18) and the damage equivalent strain by (37) in terms of and .…”

Section: Stress Update Proceduresmentioning

confidence: 99%

Computational mechanics of fatigue and life predictions for composite materials and structures

Fish

2002

Computer Methods in Applied Mechanics and Engineering

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A multiscale fatigue analysis model is developed for brittle composite materials. The mathematical homogenization theory is generalized to account for multiscale damage effects in heterogeneous media and a closed form expression relating nonlocal microphase fields to the overall strains and damage is derived. The evolution of fatigue damage is approximated by the first order initial value problem with respect to the number of load cycles. An efficient integrator is developed for the numerical integration of the continuum damage based fatigue cumulative law. The accuracy and computational efficiency of the proposed model for both low-cycle and high-cycle fatigue are investigated by numerical experimentation. IntroductionIn recent years several fatigue models have been developed within the framework of continuum damage mechanics (CDM) [10][12] [29][35] [36]. Within this framework, internal state variables are introduced to model the fatigue damage. The degradation of material response under cyclic loading is simulated using constitutive equations which couple damage cumulation and mechanical responses. The microcrack initiation and growth are lumped together in the form of the evolution of damage variables from zero to some critical value. Most of the existing CDM based fatigue damage models are based on the classical (local) continuum damage theory even though it is well known that the accumulation of damage leads to strain softening and loss of ellipticity in quasi-static problems and hyperbolicity in dynamic problems (see, for example, [45] revealed the intrinsic links between the nonlocal CDM theory and fracture mechanics providing a possibility for building a unified framework to simulate crack initiation, propagation and overall structural failure under cyclic loading.When applying the CDM based fatigue model to life prediction of engineering systems, the coupling between mechanical response and damage cumulation poses a major computational challenge. This is because the number of cycles to failure, especially for high-cycle fatigue, is usually as high as tens of millions or more, and therefore, it is practically not feasible to carry out a direct cycle-by-cycle simulation for the fully coupled models even with today's powerful computers. An efficient integrator, or the so-called

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Multiscale finite element method for a locally nonperiodic heterogeneous medium

Cited by 111 publications

References 6 publications

Multiscale enrichment based on partition of unity

Multiscale enrichment based on partition of unity

A Nonlocal Multiscale Fatigue Model

Computational mechanics of fatigue and life predictions for composite materials and structures

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