On the variationally consistent computational homogenization of elasticity in the incompressible limit

Background Computational homogenization is a well-established approach in material modeling with the purpose to account for strong micro-heterogeneity in an approximate fashion without excessive computational cost. However, the case of macroscopically incompressible response is still unresolved. Methods The computational framework for Variationally Consistent Homogenization (VCH) of (near) incompressible solids is discussed. A canonical formulation of the subscale problem, pertinent to a Representative Volume Element (RVE), is established, whereby complete macroscale incompressibility is obtained as the limit situation when all constituents are incompressible. Results Numerical results for single RVEs demonstrate the seamless character of the computational algorithm at the fully incompressible limit. Conclusions The suggested framework can seamlessly handle the transition from (macroscopically) compressible to incompressible response. The framework allows for the classical boundary conditions on the RVE as well as the generalized situation of weakly periodic boundary conditions

Variational formulation of generalized interfaces for finite deformation elasticity

The objective of this contribution is to formulate generalized interfaces in a variationally consistent manner within a finite deformation continuum mechanics setting. The general interface model is a zero-thickness model that represents the finite thickness “interphase” between different constituents in a heterogeneous material. The interphase may be the transition zone between inclusion and matrix in composites or the grain boundaries in polycrystalline solids. The term “general” indicates that the interface model here accounts for both jumps of the deformation as well as the traction across the interface. Both the cohesive zone model and elastic interface model can be understood as two limits of the current interface model. Furthermore, some aspects of material modeling of generalized interfaces are elaborated and a consistent model is proposed. Finally, the proposed theory is elucidated via a series of numerical examples. © 2017, The Author(s) 2017

Mode II dominant fracture of layered composite beams and wide-plates: a homogenized structural approach

Brittle delamination fracture under Mode II dominant conditions in unidirectional composite and layered beams and wide-plates is studied using a homogenized structural model based on a zigzag approach. The model captures the unstable propagation of cracks, snap-back and -through instabilities, the effects of the interaction of multiple cracks on the macrostructural response and of the layered structure on the energy release rate. The layered structure and the delaminations are described by introducing local enrichments, in the form of zigzag functions and cohesive interfaces, to a classical first-order shear deformation plate theory. The model applies to layers with principal material directions parallel to the geometrical axes, depends on only three displacement variables and the solution of specific problems requires only in-plane discretization, for any numbers of layers and delaminations. Closed form solutions are derived for the energy release rate in bi-material beams and applications are presented to homogeneous, bi-material and layered, simply supported and cantilever, bend-beams, with one and two delaminations

Mini-Workshop: Mathematics of Magnetoelastic Materials

The unifying theme of the workshop was the mathematical modeling, analysis, and numerical simulation of materials which involve magnetic and elastic interactions. During the workshop we identified several open problems from the calculus of variations, partial differential equations and modeling which appear to be essential in the understanding of the behavior of magnetoelastic materials

Mechanics of Materials: Towards Predictive Methods for Kinetics in Plasticity, Fracture, and Damage

The workshop dealt with current advances of computational methods, mathematics and continuum mechanics directed at thermodynamically consistent forms of constitutive equations for complex evolutionary phenomena in modern materials such as plasticity, fracture and damage. The main aspects addressed in presentations and discussions were multiphysical description of new materials, (visco)plasticity, fracture, damage, structural mechanics, mechanics of materials and dislocation dynamics

Multiscale Modelling of Reinforced Concrete

Since concrete cracks at relatively low tensile stresses, the durability of reinforced concrete structures is highly influenced by its brittle nature. Cracks open up for ingress of harmful substances, e.g. chlorides, which in turn cause corrosion of the reinforcement. Crack widths are thus limited in the design codes, and accurate prediction methods are needed. For structures of more complex shapes, current computational methods for crack width predictions lack precision. Hence, the development of new simulation tools is of interest. In order to properly describe the crack growth in detail, cracking of concrete, constitutive behaviour of steel, and the bond between them must be accounted for. These physical phenomena take place at length scales smaller than the dimensions of large reinforced concrete structures. Thus, multiscale modelling methods can be employed to reinforced concrete. This thesis concerns multiscale modelling of reinforced concrete. More specifically, a two-scale model, based on Variationally Consistent Homogenisation (VCH), is developed. At the large-scale, homogenised (effective) reinforced concrete is considered, whereas the underlying subscale comprises plain concrete, resolved reinforcement bars, and the bond between the two. Each point at the large-scale is associated with a Representative Volume Element (RVE) defining the effective response through a pertinent boundary value problem. In a numerical framework, the procedure pertains to a so-called FE 2 (Finite Element squared) algorithm, where each integration point in the discretised large-scale problem inherits its response from an underlying RVE problem. In order to properly account for the concrete–reinforcement bond action, the large-scale problem is formulated in terms of a novel effective reinforcement slip variable in addition to homogenised displacements. In a series of FE 2 analyses of a plane problem pertaining to a reinforced concrete deep beam with distributed reinforcement layout, the influence of boundary conditions on the RVE, as well as the sizes of the RVE and the large-scale mesh, are studied. The results of the two-scale analyses with and without incorporation of the effective reinforcement slip are compared to fully-resolved (single-scale) analysis. A good agreement with the single-scale results in terms of structural behaviour, in particular load-deflection relation and average strain, is observed. Depending on the sub-scale boundary conditions, approximate upper and lower bounds on structural stiffness are obtained. The effective strain field gains a localised character upon incorporation of the effective reinforcement slip in the model, and the predictions of crack widths are improved. The two-scale model can thus describe the structural behaviour well, and shows potential in saving computational time in comparison to single-scale analyses

Describing the macroscopic behavior of surfaces based on atomistic models

Diese Arbeit beschreibt die Modellierung von Grenz- und Oberflächen. Ein neuartiger Multiskalen-Modellierungsrahmen, basierend auf molekulardynamischen Simulationen, dient zur Bestimmung von Materialeigenschaften freier Oberflächen auf der Makroskala. Ein wesentlicher Aspekt dafür ist die Entwicklung eines thermodynamisch konsistenten Homogenisierungsansatzes auf Basis des Prinzips der Energieminimierung. Dabei werden die Parameter, welche die Energie auf der Makroskala beschreiben, aus dem Vergleich mit der Energie auf der atomistischen Skala ermittelt. Dieser Homogenisierungsansatz wird zudem auch auf thermoelastische Materialien erweitert. Danach liegt der Fokus auf einer kontinuumsmechanischen Modellierung von Oberflächen und einer starken Kopplung zwischen der Physik des Bulk- und Oberflächenmaterials. Dazu wird die Oberflächenverzerrung direkt aus der grundlegenden, dreidimensionalen Verzerrung des Bulkmaterials bestimmt. Dieses Vorgehen wird anhand zweier Ansätze implementiert. Schließlich wird für beide Ansätze eine numerische Implementierung in die Finite-Elemente Methode hergeleitet.This work describes the modeling of interface and surface materials. A novel multiscale framework determines continuum material properties of free surfaces on the macroscale based on molecular dynamics simulations. A key aspect is the derivation of a thermodynamically consistent homogenization approach by the principle of energy minimization. The parameters describing the energy on the macroscale are determined from the comparison with the corresponding energy on the atomistic scale. In addition, this Ritz-homogenization approach is enlarged to thermoelastic materials. Afterwards, the work focuses on the continuum mechanically surface modeling and a strong coupling between bulk and surface physics. To be more precise, the surface deformation is directly determined from the underlying three-dimensional bulk deformation. Therefore, two approaches of the surface deformation are implemented. Finally, a numerical implementation of both approaches in a finite-element framework is derived

Computational homogenization of microfractured continua using weakly periodic boundary conditions

Abstract Computational homogenization of elastic media with stationary cracks is considered, whereby the macroscale stress is obtained by solving a boundary value problem on a Statistical Volume Element (SVE) and the cracks are represented by means of the eXtended Finite Element Method (XFEM). With the presence of cracks on the microscale, conventional BCs (Dirichlet, Neumann, strong periodic) perform poorly, in particular when cracks intersect the SVE boundary. As a remedy, we herein propose to use a mixed variational format to impose periodic boundary conditions in a weak sense on the SVE. Within this framework, we develop a novel traction approximation that is suitable when cracks intersect the SVE boundary. Our main result is the proposition of a stable traction approximation that is piecewise constant between crack-boundary intersections. In particular, we prove analytically that the proposed approximation is stable in terms of the LBB (inf-sup) condition and illustrate the stability properties with a numerical example. We emphasize that the stability analysis is carried out within the setting of weakly periodic boundary conditions, but it also applies to other mixed problems with similar structure, e.g. contact problems. The numerical ex- * Corresponding author. Email address: [email protected] (Erik Svenning) Preprint submitted to Comput. Methods Appl. Mech. Engrg. November 18, 2015 amples show that the proposed traction approximation is more efficient than conventional boundary conditions (Dirichlet, Neumann, strong periodic) in terms of convergence with increasing SVE size

A unified computational framework for process modeling and performance modeling of multi-constituent materials

This thesis presents new theoretical and computational developments and an integrated approach for interface and interphase mechanics in the process and performance modeling of fibrous composite materials. A new class of stabilized finite element methods is developed for the coupled-field problems that arise due to curing and chemical reactions at the bi-material interfaces at the time of the manufacturing of the fiber-matrix systems. An accurate modeling of the degree of curing, because of its effects on the evolving properties of the interphase material, is critical to determining the coupled chemo-mechanical interphase stresses that influence the structural integrity of the composite and its fatigue life. A thermodynamically consistent theory of mixtures for multi-constituent materials is adopted to model curing and interphase evolution during the processing of the composites. The mixture theory model combines the composite constituent behaviors in an effective medium, thereby reducing the computational cost of modeling chemically reacting multi-constituent mixtures, while retaining information involving the kinematic and kinetic responses of the individual constituents. The effective medium and individual constituent behaviors are each constrained to mutually satisfy the balance principles of mechanics. Even though each constituent is governed by its own balance laws and constitutive equations, interactive forces between constituents that emanate from maximization of entropy production inequality provide the coupling between constituent specific balance laws and constitutive models. The mixture model is cast in a finite strain finite element framework that finds roots in the Variational Multiscale (VMS) method. The deformation of multi-constituent mixtures at the Neumann boundaries requires imposing constraint conditions such that the constituents deform in a self-consistent fashion. A set of boundary conditions is presented that accounts for the non-zero applied tractions, and a variationally consistent method is developed to enforce inter constituent constraints at Neumann boundaries in the finite deformation context. The new method finds roots in a local multiscale decomposition of the deformation map at the Neumann boundary. Locally satisfying the Lagrange multiplier field and subsequent modeling of the fine scales via edge bubble functions results in closed-form expressions for a generalized penalty tensor and a weighted numerical flux that are free from tunable parameters. The key novelty is that the consistently derived constituent coupling parameters evolve with material and geometric nonlinearity, thereby resulting in optimal enforcement of inter-constituent constraints. A class of coupled field problems for process modeling and for performance molding of fibrous composites is presented that provides insight into the theoretical models and multiscale stabilized formulations for computational modeling of multi-constituent materials