Strain gradient plasticity

Determination of the size of the representative volume element for random composites: statistical and numerical approach

Modeling of the evolution of distributed damage such as microcracking, void formation, and softening frictional slip necessitates strain-softening constitutive models. The nonlocal continuum concept has emerged as an effective means for regularizing the boundary value problems with strain softening, capturing the size effects and avoiding spurious localization that gives rise to pathological mesh sensitivity in numerical computations. A great variety of nonlocal models have appeared during the last two decades. This paper reviews the progress in the nonlocal models of integral type, and discusses their physical justifications, advantages, and numerical applications.

Nonlocal integral formulations of plasticity and damage: Survey of progress

2.4. Block Copolymer Containing Hybrids 151 3. Block Copolymer Ordering in Thin Films 153 3.1. General Process Steps 153 3.2. Morphology of Thin Films 154 3.3. Thickness-Dependent Nanopatterning 154 3.3.1. Ultrathin Films: Monomolecular Films 155 3.3.2. Sub-L0 Thick Films 155 3.3.3. Thick Films 157 3.4. Placement Control: Directed Self-Assembly (DSA) 162 3.4.1. Topographic Guiding Patterns: Graphoepitaxy 163

Block copolymer based nanostructures: materials, processes, and applications to electronics.

Stimulus-responsive shape memory materials: A review

A variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term proportional to the second gradient of the ensemble average of strain. This nonlocal constitutive equation is derived in explicit closed form for isotropic material in the one case in which there exists a well-founded physical and mathematical basis for describing the material's statistics: a matrix reinforced (or weakened) by a random dispersion of nonoverlapping identical spheres. By assessing, when the applied loading is spatially-varying, the magnitude of the nonlocal term in this constitutive equation compared to the portion of the equation that relates ensemble average stresses and strains through a constant “overall” modulus tensor, we derive quantitative estimates for the minimum representative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous “effective modulus” constitutive models for composites can be expected to apply. Remarkably, for a maximum error of 5% of the constant “overall” modulus term, we show that the minimum RVE size is at most twice the reinforcement diameter for any reinforcement concentration level, for several sets of matrix and reinforcement moduli characterizing large classes of important structural materials. Such estimates seem essential for determining the minimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mechanics theory for composites. Finally, we relate our nonlocal constitutive equation explicitly to the ensemble average strain energy, and show how it is consistent with the stationary energy principle.

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

Composite materials of extremely high stiffness can be produced by employing one phase of negative stiffness. Negative stiffness entails a reversal of the usual codirectional relationship between force and displacement in deformed objects. Negative stiffness structures and materials are possible, but unstable by themselves. We argue here that composites made with a small volume fraction of negative stiffness inclusions can be stable and can have overall stiffness far higher than that of either constituent. This high composite stiffness is demonstrated via several exact solutions within linearized and also fully nonlinear elasticity, and via the overall modulus tensor estimate of a variational principle valid in this case. We provide an initial discussion of stability, and adduce experimental results which show extreme composite behavior in selected viscoelastic systems under sub-resonant sinusoidal load. Viscoelasticity is known to expand the space of stability in some cases. We have not yet proved that purely elastic composite materials of the types proposed and analyzed in this paper will be stable under static load. The concept of negative stiffness inclusions is buttressed by recent experimental studies illustrating related phenomena within the elasticity and viscoelasticity contexts.

Dramatically stiffer elastic composite materials due to a negative stiffness phase

Why do experiments detect Cosserat-elastic effects for porous, but not for stiff-particlereinforced, materials? Does homogenization of a heterogeneous Cauchy-elastic material lead to micropolar (Cosserat) effects, and if so, is this true for every type of heterogeneity? Can homogenization determine micropolar elastic constants? If so, is the homogeneous (effective) Cosserat material determined in this way a more accurate representation of composite material response than the usual effective Cauchy material? Direct answers to these questions are provided in this paper for both two- (2D) and three-dimensional (3D) deformations, wherein we derive closed-form formulae for Cosserat moduli via homogenization of a dilute suspension of elastic spherical inclusions in 3D (and circular cylindrical inclusions in 2D) embedded in an isotropic elastic matrix. It is shown that the characteristic length for a homogeneous Cosserat material that best mimics the heterogeneous Cauchy material can be derived (resulting in surprisingly simple formulae) when the inclusions are less stiff than the matrix, but when these are equal to or stiffer than the matrix, Cosserat effects are shown to be excluded. These analytical results explain published experimental findings, correct, resolve and extend prior contradictory theoretical (mainly numerical and limited to two-dimensional deformations) investigations, and provide both a general methodology and specific results for determination of simple higher-order homogeneous effective materials that more accurately represent heterogeneous material response under general loading conditions. In particular, it is shown that no standard (Cauchy) homogenized material can accurately represent the response of a heterogeneous material subjected to a uniform plus linearly varying applied traction, while a homogenized Cosserat material can do so (when inclusions are less stiff than the matrix).

/pdf/analytical-derivation-of-cosserat-moduli-via-homogenization-4v3i1jiy13.pdf

Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials

An exact asymptotic analysis is presented of the stress and deformation fields near the tip of a quasistatically advancing plane strain tensile crack in an elastic-ideally plastic solid. In contrast to previous approximate analyses, no assumptions which reduce the yield condition, a priori, to the form of constant in-plane principal shear stress near the crack tip are made, and the analysis is valid for general Poisson ratio ν. Specific results are given for ν = 0.3 and 0.5, the latter duplicating solutions in previous work by L.I. Slepyan, Y.-C. Gao and the present authors. The crack tip field is shown to divide into five angular sectors of four different types ; in the order in which these sweep across a point in the vicinity of the advancing crack, they are : two plastic sectors which can be described asymptotically (i.e., as r → 0, where r is distance from the crack tip) in slip-line terminology as ‘constant stress’ and ‘centered fan’ sectors, respectively ; a plastic sector of non-constant stress which cannot be described asymptotically in terms of slip lines; an elastic unloading sector; and a trailing plastic sector of the same type as that directly preceding the elastic sector. Further, these four different sector types constitute the full set of asymptotically possible solutions at the crack tip. As is known from prior work, the plastic strain accumulated by a material point passing through such a moving ‘centered fan’ sector is O(ln r) as r → 0 ; it is proved in the present work that the plastic strain accumulated by a material point passing through the ‘constant stress’ sector ahead of a growing crack must be less singular than In r as r → 0. As suggested also in earlier studies, the rate of increase of opening gap δ at a point currently at a distance r behind, but very near, the crack tip is given for crack advance under contained yielding by 
δ = αJσ0+β(σ0E)a ln(Rr)
where a is crack length, σ0 is tensile yield strength, E is Young's modulus, J is the value of the J-integral taken in surrounding elastic material, and the parameters α and R are undetermined by the asymptotic analysis. The exact solution for ν = 0.3 gives β = 5.462, which agrees very closely with estimates obtained from finite element solutions. An approximate analysis based on use of slip line representations in all plastic sectors is outlined in the Appendix.

http://esag.harvard.edu/rice/103_DruganRiSh_CrackElPl_JMPS82-83.pdf

Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids

https://bertoldi.seas.harvard.edu/files/bertoldi/files/nacre-2008.pdf

W. J. Drugan

Papers

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

Dramatically stiffer elastic composite materials due to a negative stiffness phase

Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials

Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids

Nacre: An orthotropic and bimodular elastic material