Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
16
2
2018
VARIATIONAL INEQUALITIES FOR HETEROGENEOUS MICROSTRUCTURES BASED ON COUPLE-STRESS THEORY
101-119
Sourish
Chakravarty
Department of Mechanical and Aerospace Engineering, University at Buffalo, The State
University of New York, Buffalo, New York, 14260, USA
Sonjoy
Das
Department of Mechanical and Aerospace Engineering, University at Buffalo, The State
University of New York, Buffalo, New York, 14260, USA
Ali R.
Hadjesfandiari
Department of Mechanical and Aerospace Engineering, University at Buffalo, The State
University of New York, Buffalo, New York, 14260, USA
Gary F.
Dargush
Department of Mechanical and Aerospace Engineering, University at Buffalo, The State
University of New York, Buffalo, New York, 14260, USA
In this work, we view mesoscopic material volume elements consisting of heterogeneous microstructures as couple-stress
continua to account for underlying length-scale dependence. We use a recently established self-consistent version
of couple-stress theory that results in a skew-symmetric couple-stress tensor, along with the energy-conjugate
mean-curvature tensor. Using this new theoretical framework, we establish a generalized Hill energetic equivalence relationship that leads to a homogeneous material representation at the macroscale point associated with the mesoscopic volume element. We identify the necessary and sufficiency conditions that enable the extension of the couple-stress continuum framework and its application to incorporate the mesoscale features into the macroscale continuum description. We establish the concept of a micromechanically consistent macroscopic elastic constitutive tensor within this paradigm and also propose special kinematically and statically uniform boundary conditions, analogous to previous work in classical elasticity. This then leads to determination of two suitable matrices that bound the matrix representation of the macroscopic elastic constitutive tensor in the positive definite sense. Similar bounds based on classical mechanics are found to be critical quantities in several aspects of multiscale material modeling. We envisage that the theoretical work presented here will be useful in analyzing coarse-grained heterogeneous microstructures with inherent characteristic length-scale features contained within the mesoscopic material volume element.
HIGHER-ORDER EFFECTIVE MODEL DESCRIBING A NON-ISOTHERMAL THIN-FILM FLOW
121-130
Eduard
Marušic-Paloka
Department of Mathematics, Faculty of Science, University of Zagreb Bijenicka 30, Zagreb,
10000, Croatia
Igor
Pažanin
Department of Mathematics, Faculty of Science, University of Zagreb Bijenicka 30, Zagreb,
10000, Croatia
We study the flow and heat transfer inside a thin layer of lubricant film between two surfaces. We start from the Stokes
equation coupled with the heat equation including the viscous dissipation term. A new second-order asymptotic model
is proposed, correcting the non-isothermal Reynolds system. Rigorous justification by error estimate of the formally
derived effective model is provided.
RANDOM WALK SIMULATION MODEL OF DIFFUSION IN CIRCULAR AND ELLIPTICAL PARTICULATE COMPOSITES
131-142
Jian
Qiu
Department of Mechanical and Materials Engineering, University of Denver, Colorado 80208
Yun-Bo
Yi
Department of Mechanical and Materials Engineering, University of Denver, Denver, Colorado 80208, USA
We developed a random walk model for solving the diffusion problem arising from composite materials of particulate
inclusions with different shapes. Our two-phase material system contains circular or elliptical impermeable inclusions that are randomly embedded in a matrix material. A computational algorithm was developed for random walk simulation of molecules and to estimate the effective diffusion coefficient of the composites. The random walk model has been validated by solutions from finite element analysis and effective medium theories. Our computational results show that the density of random distribution and the volume fraction of inclusions have significant effects on effective diffusion coefficients. Moreover, the aspect ratio of inclusions can significantly reduce diffusion speed when the volume fraction of inclusions is greater than? 30%.
A MULTISCALE MESH-FREE APPROACH TO MODELING DAMAGE OF AN ULTRA-HIGH-PERFORMANCE CONCRETE
143-161
Jesse A.
Sherburn
U.S. Army Engineer Research and Development Center, Vicksburg, Mississippi, USA
William F.
Heard
U.S. Army Engineer Research and Development Center, Vicksburg, Mississippi, USA
Brett A.
Williams
U.S. Army Engineer Research and Development Center, Vicksburg, Mississippi, USA
Paul A.
Sparks
U.S. Army Engineer Research and Development Center, Vicksburg, Mississippi, USA
A multiscale mesh-free formulation based on the reproducing kernel particle method (RKPM) is used to develop a
damage model for ultra-high-performance concrete (UHPC). The damage evolution law is derived from the Helmholtz
free energy, where the energy released from the cracked microstructure is equated to the homogenized macroscale response at the continuum level. In order to perform the microscale calculations for UHPC, the fracture energy of the
UHPC must be determined. In this study, the fracture energy of a UHPC is experimentally determined by performing
single-edge notched three-point beam tests. The fracture energy is required in order to perform physically based RKPM
microscale calculations. The damage evolution law is then determined from the microscale calculations and applied
to a quasi-static macroscale RKPM calculation. The multiscale framework is compared to a typical phenomenological damage model to show its ability to accurately reproduce softening behavior that exists under loading of a UHPC.
FREE VIBRATION PROPERTY ANALYSIS OF COMPOSITE LAMINATED MICROPLATES BASED ON DIFFERENT HYPOTHESES IN COUPLE STRESS CONSTITUTIVE EQUATIONS
163-186
Shengqi
Yang
State Key Laboratory of Structural Analysis for Industrial Equipment, Department of
Engineering Mechanics, Dalian University of Technology, Dalian, 116024, China
Shutian
Liu
State Key Laboratory of Structural Analysis for Industrial Equipment, Department of
Engineering Mechanics, Dalian University of Technology, Dalian, 116024, China
The free vibration property of composite laminated microplates is analyzed analytically and numerically based on two
kinds of couple stress plate models. The laminated microplates are composed of orthotropic plies with different ply angles
which are modeled as orthotropic couple stress media. The two kinds of plates models are called as the Simplifiedmodel
which is based on simplification hypotheses on the rotation and curvatures (ωz = 0, χxz =0, χyz = 0) in couple stress
constitutive equations, and the Complete model without any hypotheses. Hamilton principle is employed to derive
the governing equations of free vibration based on the hypotheses on the rotation and/or the curvatures in the couple
stress theory and the assumption on the displacements in the plate theory. A kind of finite element is constructed for
couple stress microplates, and its convergence and precision are verified through typical examples. Solutions for typical
examples with different boundary conditions are obtained in closed form using Navier’s technique and by solving the
eigenvalue equation, and/or by the finite element method. The application scope of the Simplified model is investigated
through comparing the results with the Complete model’s. It is noted that, although the simplified model can give
accurate free vibration modes for Kirchhoff and Mindlin plates, but for Reddy plates, large errors may occur. Thus, it
is needed to use the complete model for free vibration analysis of high order theory of microplates.
A VARIANT OF THE S-VERSION OF THE FINITE ELEMENT METHOD FOR CONCURRENT MULTISCALE COUPLING
187-207
Wei
Sun
State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084,
China
Jacob
Fish
Civil Engineering and Engineering Mechanics, Columbia University, New York, New York
10027, USA
Hachmi
Ben Dhia
Laboratoire MSSMat, CentraleSupélec, Université Paris-Saclay, UMR CNRS 8579, France
A variant of the s-version of the finite element method (hereafter coined the s-method) for concurrent multiscale coupling is developed. The proposed method is inspired by a combination of the s-version of the finite element method and the Arlequin method. It features a superposition of a local (fine) mesh, which partly overlaps a global (coarse) mesh, and appropriate homogeneous boundary conditions on both meshes that enforce solution continuity. Its performance in terms of accuracy and computational efficiency in solving a class of multiscale continuum mechanics problems is evaluated by virtue of comparison to the fine reference single mesh and the Arlequin method. Numerical studies are conducted for one-, two-, and three-dimensional problems. For select local and global meshes, the cause of accuracy gains in comparison to the Arlequin method, while having almost the same gain in CPU time, with respect to the discrete single fine mesh for both approaches, is explained.