Macroscopic Loading Research Articles

Studies on Micro- to Macroscopic Mechanical Behavior of Porous Polymer under Compaction

The 2D extended homogenization model based on molecular chain network theory is employed to investigate the micro-to macroscopic mechanical behavior of polymer with randomly distributed voids under macroscopic compaction. A parametric study is performed to quantify the effect of the volume fraction of voids and the macroscopic stress triaxiality of loading conditions on the compaction behavior of porous polymer. The results suggest that the onset of localized shear band at the ligament between voids leads to the macroscopic yield of porous polymer. Furthermore, the microscopic localized shear deformation behavior is promoted in the polymer with high-volume fraction of voids under high macroscopic stress triaxiality loading condition, which results in the early appearance of the macroscopic yield. After the macroscopic yield, a remarkable strain hardening is shown in the macroscopic response of porous polymer under high macroscopic stress triaxiality loading condition, which is due to the onset and propagation of a large number of shear bands at the ligaments between voids. On the other hand, microscopic buckling develops at the narrowest ligament in the polymer with high-volume fraction of voids under high macroscopic stress triaxiality loading condition, which leads to the relative low macroscopic deformation resistance. Furthermore, to develop the macroscopic constitutive model of polymer with high-volume fraction of voids under high macroscopic stress triaxiality loading condition, a modified Gurson model is proposed which takes account of microscopic buckling and gives a good agreement with the unit cell computational results.

Numerical particle-scale study of swelling pressure in clays

The particle level responses to different external loadings of a montmorillonitic clay soil are investigated numerically. The soil is saturated by a solution of monovalent counterions, of varying concentrations. We use finite element micromechanical models (based on the Poisson-Nernst-Planck equations) to estimate counterion and electrical potential distributions around individual clay particles at various distances from one another, since analytical solutions are not possible for these complex arrangements of particles. Disjoining pressures are then estimated using the Van’t Hoff relation and Maxwell stress tensor. As the distance between the clay particles decreases and double-layers overlap, the concentration of counterions in the micropores between clay particles increases. This increase lowers the chemical potential of the pore fluid and creates a chemical potential gradient in the solvent that generates the so-called “disjoining” or “osmotic” pressure. Because of this disjoining pressure, it is clear that particles need not contact one another in order to carry an “effective stress”. This work may lead towards theoretical predictions of the macroscopic load deformation response of montmorillonitic soils based on micro-electro-chemo-mechanical modelling of particles.

Nonlinear hierarchical multiscale modeling of cortical bone considering its nanoscale microstructure

We have used a hierarchical multiscale modeling scheme for the analysis of cortical bone considering it as a nanocomposite. This scheme consists of definition of two boundary value problems, one for macroscale, and another for microscale. The coupling between these scales is done by using the homogenization technique. At every material point in which the constitutive model is needed, a microscale boundary value problem is defined using a macroscopic kinematical quantity and solved. Using the described scheme, we have studied elastic properties of cortical bone considering its nanoscale microstructural constituents with various mineral volume fractions. Since the microstructure of bone consists of mineral platelet with nanometer size embedded in a protein matrix, it is similar to the microstructure of soft matrix nanocomposites reinforced with hard nanostructures. Considering a representative volume element (RVE) of the microstructure of bone as the microscale problem in our hierarchical multiscale modeling scheme, the global behavior of bone is obtained under various macroscopic loading conditions. This scheme may be suitable for modeling arbitrary bone geometries subjected to a variety of loading conditions. Using the presented method, mechanical properties of cortical bone including elastic moduli and Poisson's ratios in two major directions and shear modulus is obtained for different mineral volume fractions.

Instability during cohesive zone growth

Tensile microcracking of quasi-brittle materials is studied by means of micromechanics, based on (i) an elasto-damaging cohesive zone model accounting for cohesive softening and (ii) a dilute distribution of non-interacting microcracks of uniform orientation and size. Considering virgin microcracks (initially without cohesive zones), macroscopic tensile load increase results in growth of cohesive zones ahead of stationary (non-propagating) cracks and, subsequently, in crack propagation which, notably, will be encountered before the cohesive zones are fully developed i.e. onset of instable cohesive zone growth will be encountered at a load level (i) at which tractions are still transmitted across the inner edges of the cohesive zones and (ii) at which the separation at the inner edges of the cohesive zones is smaller than its critical value. Focusing on onset of instable cohesive zone growth, the chosen approach allows for accessing quantities characterizing the stability limit (e.g., the intensity of the macroscopic loading and the opening at the inner edges of the cohesive zones), without raising the need for non-linear Finite Element analyses. It is shown that the tensile macrostrength of materials containing virgin microcracks is larger than the one related to cracks with already initially fully developed cohesive zones, and related strength differences are quantified for a wide class of cohesive softening behavior. The proposed model is validated by comparing model predictions with an exact solution (available for the special case of constant cohesive tractions) and with results from reliable Finite Element analyses. The paper will be of interest for engineers involved in testing and/or in modeling of quasi-brittle media including cementitious materials and rock.

Numerical homogenization of foam‐like structures based on the FE2‐approach

AbstractThe computation of foam–like structures is still a topic of research. There are two basic approaches: the microscopic model where the foam–like structure is entirely resolved by a discretization (e.g. with Timoshenko beams) on a micro level, and the macroscopic approach which is based on a higher order continuum theory. A combination of both of them is the FE2‐approach where the mechanical parameters of the macroscopic scale are obtained by solving a Dirichlet boundary value problem for a representative microstructure at each integration point.In this contribution, we present a two–dimensional geometrically nonlinear FE2‐framework of first order (classical continuum theories on both scales) where the microstructures are discretized by continuum finite elements based on the p‐version.The p‐version elements have turned out to be highly efficient for many problems in structural mechanics. Further, a continuum–based approach affords two additional advantages: the formulation of geometrical and material nonlinearities is easier, and there is no problem when dealing with thicker beam–like structures.In our numerical example we will investigate a simple macroscopic shear test. Both the macroscopic load displacement behavior and the evolving anisotropy of the microstructures will be discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Dynamic Mechanical Environment of the Chondrocyte: A Biphasic Finite Element Model of Cell-Matrix Interactions Under Cyclic Compressive Loading

Cyclic mechanical loading of articular cartilage results in a complex biomechanical environment at the scale of the chondrocytes that strongly affects cellular metabolic activity. Under dynamic loading conditions, the quantitative relationships between macroscopic loading characteristics and solid and fluid mechanical variables in the local cellular environment are not well understood. In this study, an axisymmetric multiscale model of linear biphasic cell-matrix interactions in articular cartilage was developed to investigate the cellular microenvironment in an explant subjected to cyclic confined compressive loading. The model was based on the displacement-velocity-pressure (u-v-p) mixed-penalty weighted residual formulation of linear biphasic theory that was implemented in the COMSOL MULTIPHYSICS software package. The microscale cartilage environment was represented as a three-zone biphasic region consisting of a spherical chondrocyte with encapsulating pericellular matrix (PCM) that was embedded in a cylindrical extracellular matrix (ECM) subjected to cyclic confined compressive loading boundary conditions. Biphasic material properties for the chondrocyte and the PCM were chosen based on previous in vitro micropipette aspiration studies of cells or chondrons isolated from normal or osteoarthritic cartilage. Simulations performed at four loading frequencies in the range 0.01-1.0 Hz supported the hypothesized dual role of the PCM as both a protective layer for the cell and a mechanical transducer of strain. Time varying biphasic variables at the cellular scale were strongly dependent on relative magnitudes of the loading period, and the characteristic gel diffusion times for the ECM, the PCM, and the chondrocyte. The multiscale simulations also indicated that axial strain was significantly amplified in the range 0.01-1.0 Hz, with a decrease in amplification factor and frequency insensitivity at the higher frequencies. Simulations of matrix degradation due to osteoarthritis indicated that strain amplification factors were more significantly altered when loss of matrix stiffness was exclusive to the PCM. The findings of this study demonstrate the complex dependence of dynamic mechanics in the local cellular environment of cartilage on macroscopic loading features and material properties of the ECM and the chondron.

Mesoscopic analysis of the utilization of hardening model for a description of softening behavior based on disturbed state concept theory

Mesoscopic characteristics of a clayey soil specimen subjected to macroscopic loading are examined using a medical-use computerized tomography (CT) instrument. Disturbed state concept (DSC) theory is based on the utilization of the hardening model. DSC indirectly describes material behavior by claiming that the actual response of the material is expressed in terms of the relative intact (RI) response and the fully adjusted (FA) response. The occurrence of mesoscopic structural changes of material has similarities with the occurrence of a macroscopic response of the material under loadings. In general, the relative changing value of a softening material is three to five times more than that of a hardening material. Whether special zones exist or not in a specimen cross section does not affect the following conclusion: hardening material and softening material show mechanical differences with CT statistical indices values prominently changing, and the change is related to the superposing of a disturbance factor. A new disturbance factor evolution function is proposed. Thus, mesoscopic statistical indices are introduced to describe macroscopic behavior through the new evolution function. An application of the new evolution function proves the effectiveness of the amalgamation of a macroscopic and a mesoscopic experimental phenomenon measurement methods.

Spherical and acicular representation of hydrates in a micromechanical model for cement paste: prediction of early-age elasticity and strength

Early-age stiffness and strength evolution of cement paste is studied in the framework of continuum micromechanics. Based on the self-consistent scheme, elastic and strength properties are upscaled from the scale of several micrometers up to the scale of several hundreds or thousands of micrometers. Four material phases are considered: clinker, hydration products, water and air. We assign a spherical geometry to clinker grains and pores, while we investigate both spherical and acicular (needle-type) shapes as geometrical representation of the micrometer-sized hydration products. As regards macroscopic poromechanical boundary conditions, two extreme cases are considered: drained conditions and sealed conditions, respectively. These choices allow for studying the influence of (i) the morphological representation of hydrates, and of (ii) the bulk stiffness of water, on the micromechanical prediction of early-age behavior of cement paste, including setting and the hydration-dependent evolutions of both elastic stiffness and uniaxial compressive strength. The newly proposed strength model is based on a von Mises-type elastic limit criterion for individual hydrates. Corresponding deviatoric stress peaks within hydrates are estimated through quadratic stress averages. In this way, the micromechanical strength criterion is formulated in terms of macroscopic loading (stresses or strains, respectively). Model-predicted elasticity and strength evolutions are compared with data from experimental testing of cement pastes with water–cement ratios ranging from 0.35 to 0.60. Satisfactory agreement between model predictions and experiments allows for two conclusions: the morphology of hydrates significantly influences micromechanics-based elastic stiffness estimates of cement paste particularly at very early ages, whereas elastic properties of mature cement paste can be estimated reliably on the basis of both spherical or acicular shaped hydrates. The development of a reliable strength model, however, requires consideration of hydrates as non-spherical particles, no matter what age of cement paste is considered.

Effect of Size-Dependent Cavitation on Micro- to Macroscopic Mechanical Behavior of Rubber-Blended Polymer

In rubber-blended polymer, the onset of cavitation in the particles relaxes the high triaxiality stress state and suppresses the onset of crazing in the polymer. As a result, large plastic deformation is substantially promoted compared with single-phase polymer. On the other hand, it is also well known that the onset of cavitation depends on the size of particle. To investigate the size dependence of cavitation behavior in the particle, a theoretical analysis is done employing a void model under plane strain condition, which takes into account the surface tension and the limiting stretch of the void. Continuously, to study the effect of the size-dependent cavitation on the micro- to macroscopic mechanical behavior of the blend, a computational model is proposed for the blend consisting of irregularly distributed heterogeneous particles containing the void with surface force. The results indicate that when the size of the particle decreases to a critical value that depends on both the initial shear modulus of particle and the surface tension on the surface of void, the increase of the critical stress for the onset of cavitation becomes remarkable and consequently, the onset of cavitation is eliminated. When the particle is embedded in polymer, the relation between average normal stress, which is acting on the interface of particle and matrix, and volumetric strain of particle shows dependence on the size of particle but no dependence on the triaxiality of macroscopic loading condition. For the blend consisting of particles smaller than the critical value, the onset of cavitation is eliminated in particles and as a result, the conformation of the shape of particle to the localized shear band in matrix becomes difficult and the shear deformation behavior tends to occur all over the matrix. Furthermore, in this case, the area of the maximum mean stress is confined to the area adjacent to the particle and the value of it increases almost linearly throughout the whole deformation process, which would lead to the onset of crazing in matrix. On the other hand, it is clarified that the onset of cavitation is predominant in the localized microscopic region containing heterogeneous particles and therefore, the plastic deformation is promoted in this region.

Constitutive Modeling of the Stress-Stretch Behavior of Two-Dimensional Triangulated Macromolecular Networks Containing Folded Domains

The mechanical behavior of the red blood cell membrane is governed by the lipid bilayer which resists changes in surface area and the underlying spectrin network which resists changes in shape. The constituent spectrin chains of the network consist of a series of domains along the chain, which exhibit noncovalent interactions. Upon sufficient extension of a chain, each folded domain undergoes mechanically-induced unfolding after reaching a chain force between 10 and 35pN. Individual spectrin chains within the network experience their first unfolding event at different levels of macroscopic strain depending on the macroscopic loading conditions and the orientation of each constituent chain with respect to the macroscopic loading. A microstructurally-informed continuum level constitutive model is developed which tracks individual chain deformation behavior as well as the overall macroscopic network stress-strain behavior. Using the introduced continuum approach and statistical mechanics based models of the chain force-extension behavior together with a transition state model of domain unfolding; a constitutive model for the membrane stress-stretch behavior is constructed. Uniaxial tension and simple shear behaviors of the membrane are simulated incorporating the unfolding of the individual chains. A Taylor averaging approach is used as a first approximation to account for the irregularities in the spectrin network which result in a near plateau-like force behavior with increasing stretch.

A Method of Numerical Material Testing in Nonlinear Multiscale Material Analyses

A method of numerical material testing is developed for characterizing equivalent or macroscopic mechanical and/or other physical properties of composite materials with heterogeneous microstructures. The proposed method is prepared for testing on a variety of heterogeneous microstructures and nonlinear material behavior of constituents and their interfaces under various patterns of macroscopic loading. The numerical material testing in this method is intended to be carried out by means of the finite element method with a quasi-static implicit solution scheme in conjunction with the homogenization theory for periodic microstructures, i.e., unit cells.

多数誘導機群の波及電圧崩壊と機器保護が系統負荷特性に与える影響

As is well known, two of the fundamental processes which give rise to voltage collapse in power systems are the on-load tap changers of transformers and the dynamic characteristics of loads such as induction machines. It is well established that, of these two, the former makes for a slower collapse while the latter makes for a faster collapse. However, in realistic simulations, the load levels of the induction machines are not uniform and it may be expected that some of the loads will collapse first, followed by the collapse of the loads which did not go into instability during the preceding collapses. In such situations, the overall equivalent collapse behavior viewed from the bulk transmission level becomes somewhat different from the simple collapse driven by one aggregated induction machine. This paper studies the process of cascaded voltage collapse among many induction machines by time simulation, in which the load distribution on a feeder line is modeled by several hundred induction machines and static impedance loads. It is shown that in some cases voltage collapse actually cascades among induction machines, with the macroscopic load dynamics viewed from the upper level resulting in a slower collapse than expected from the aggregated load model. We also show the effects of machine protection of induction machines, which results in slower collapse. © 2009 Wiley Periodicals, Inc. Electr Eng Jpn, 170(2): 19–27, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20894

Mechanical Properties of two novel planar lattice structures

Two novel statically indeterminate planar lattice materials are designed: a new Kagome cell (N-Kagome) and a statically indeterminate square cell (SI-square). Their in-plane mechanical properties, such as stiffness, yielding, buckling and collapse mechanisms are investigated by analytical methods. The analytical stiffness is also verified by means of finite element (FE) simulations. In the case of uniaxial loading, effective modulus, yield strength, buckling strength and critical relative density are compared for various lattice structures. At a critical relative density, the collapse mode will change from buckling to yielding. Elastic buckling under macroscopic shear loading is found to have significant influence on failure of lattice structures, especially at low relative densities. Comparison of the analytical bulk and shear moduli with the Hashin–Shtrikman bounds indicates that the mechanical properties of the SI-square honeycomb are relatively close to being optimal. It is found that compared with the other existing stretching-dominated 2D lattice structures, the N-Kagome cell possesses the largest continuous cavities for fixed relative densities and wall thicknesses, which is convenient for oil storage, disposal of heat exchanger, battery deploying and for other functions. And the initial yield strength of the N-Kagome cell is slightly lower than that of the Kagome cell. The SI-square cell has similar high stiffness and strength as the mixed cell while its buckling resistance is about twice than that of the mixed cell.

Probing the internal geometry of a woven composite during deformation using an x-ray microdiffraction imaging technique

X-ray diffraction is an important tool for studying multiphase materials because it can resolve parameters from each phase independently. When coupled with a high-flux, microfocussed x-ray beam, scanning microdiffraction experiments are possible. This letter reports on the use of this technique to image a fiber reinforced composite with a complex woven lamina geometry. These systems are difficult to study with other experimental techniques because the fibers are inaccessible and the matrix is often opaque. However, microfocused x-ray diffraction reveals how macroscopic load affects the weave microgeometry by reorienting the embedded fibers.

A multiscale approach of fatigue and shakedown for notched structures

The aim of this paper is to analyse the fatigue phenomena in the presence of stress gradients. It is well-known that most fatigue criteria fail to predict the lifetime of components in the presence of high stress concentrations or stress gradients, as it is the case in the neighbourhood of cracks, holes notches and encountered for example in riveted or threaded structures. Proposed is a numerical approach in the framework of the high cycle fatigue domain in order to give a qualitative answer. The work starts from the numerical computation of macroscopic loading corresponding to some fatigue experiments on specimens with an inclusion of metallic grains embedded in a macroscopic matrix. The computed fields are then analysed in terms of the HCF (high cycle fatigue) criterion [1], which is based on the estimation of the shakedown limit at the grain scale. The infinite lifetime prediction is based on the assumption that fatigue occurs if at least one grain fails, i.e. reaches plastic shakedown. The predictions at mesoscopic and macroscopic scales are close if the macroscopic stress distribution is homogeneous. However in the case of the stress gradient, lifetime predicted at the macroscopic scale is underestimated when compared to the predictions made at the mesoscopic scale. Another result is that the gap between microscopic and macroscopic predictions obtained from these numerical computations can roughly be estimated by a diminution of stress of the same order of magnitude as found in the experiments and phenomenological observations.

Void growth by dislocation-loop emission

Experimental results from spall tests on aluminum reveal the presence of a dense dislocation structure in an annulus around a void that grew under the tensile pulse when a shock wave was reflected at the free surface of the specimen. The proposition is that dislocation emission from the void surface under load is a viable mechanism for void growth. To understand void growth in the absence of diffusive effects, the interstitial-loop emission mechanism under tensile hydrostatic stress is investigated. First, the micromechanics of pile-up formation when interstitial loops are emitted from a void under applied macroscopic loading is reviewed. Demand for surface energy expenditure upon void-surface change is taken into consideration. It is demonstrated that in face-centered cubic metals loop emission from voids with a radius of ∼10 nm is indeed energetically possible in the hydrostatic stress environment generated by shock loading. On the other hand, the levels of hydrostatic stress prevalent in common structural applications are not sufficient to drive loops at equilibrium positions above a ∼10 nm void. However, for voids larger than about 100 nm, the energetics of loop emission are easily met as a necessary condition even under the low stress environment prevalent in structural applications.

Microscopic and macroscopic instabilities in finitely strained porous elastomers

The present work is an in-depth study of the connections between microstructural instabilities and their macroscopic manifestations—as captured through the effective properties—in finitely strained porous elastomers. The powerful second-order homogenization (SOH) technique initially developed for random media, is used for the first time here to study the onset of failure in periodic porous elastomers and the results are compared to more accurate finite element method (FEM) calculations. The influence of different microgeometries (random and periodic), initial porosity, matrix constitutive law and macroscopic load orientation on the microscopic buckling (for periodic microgeometries) and macroscopic loss of ellipticity (for all microgeometries) is investigated in detail. In addition to the above-described stability-based onset-of-failure mechanisms, constraints on the principal solution are also addressed, thus giving a complete picture of the different possible failure mechanisms present in finitely strained porous elastomers.

Crack opening conditions at ‘corner nodes’ in FE analysis with cracking along mesh lines

On the basis of a recent work proposed by the authors on a double-minimization method for evaluating inter-element forces and stresses transmitted across mesh lines, the crack opening conditions at a corner node of the FE mesh, from where several lines (potential cracks) emanate, is examined in this paper. The study is developed locally as a post-processing step of a standard displacement-based FE calculation, in terms of an always-increasing external (macroscopic) load factor μ. The cracking laws for each potential crack line are assumed rigid-plastic with hyperbolic failure criterion in terms of normal and shear components of the stress traction at that point. It is observed that, as μ increases, in general such point may undergo up to four phases of evolution until a crack can effectively open through it. First, while stress tractions across mesh lines at the point are all below cracking criterion, forces may be evaluated with the double minimization method recently proposed. Second, cracking criterion is reached for one of the lines only. Stress evaluation requires a modified minimization method with one (hyperbolic) constraint; however, crack still does not open at the node because of the lack of kinematic continuity. Third, cracking criterion is satisfied for a second of the lines converging at the nodal point. Stress tractions may then be calculated with a system of equations involving the two hyperbolic constraints alone and no minimization is needed. But in general the through crack cannot open yet at this stage because of non-coincident flow rules, until either (i) a third line reaches the cracking criterion, or (ii) these get reoriented to exhibit parallel directions in the global reference system. Two simple examples of application are provided which illustrates the development of the various cracking stages and shows different situations that may take place.

Selective Epitaxial Growth of GaAs on Ge Substrates with a SiO2 Pattern

We have selectively grown thin epitaxial GaAs films on Ge substrates with the aid of a 200 nm thin SiO2 mask layer. The selectively grown structures have lateral sizes ranging from 1 μm width up to large areas of 1 by 1 mm2. The growth is fully selective, thanks to an optimized growth procedure, consisting of a 13 nm thin nucleation layer grown at high pressure, followed by low pressure growth of GaAs. This growth procedure inhibits the nucleation of GaAs on the mask area and is a good compromise between reduction of loading effects and inhibition of anti phase domain growth in the GaAs. Nevertheless, both microscopic and macroscopic loading effects can still be observed on x-section SEM images and profilometer measurements. X-ray diffraction and low temperature photoluminescence measurements demonstrate the good microscopic characteristics of the selectively grown GaAs.

Simultaneous occurrence of double micro/macro stress singularities for multiscale crack model

It is rare to find two distinct stress singularities with different order at the same crack tip while the local displacements remained regular. Such a discovery was made when developing a dual scale micro/macro crack model and validated from a closed form asymptotic analytical solution for a micro/macro crack subjected to mixed boundary conditions. The two different orders of the stress singularities are 0.75 and 0.25 for an incompressible material. The existence of multi-singularities with different orders at the crack point is an unexplored area of research. For a sharp notch wedge, multi-stress singularities at the wedge point can prevail and they depend on the notch angle and Poisson’s ratio. Physical implications of such solutions when viewed at the microscopic scale are particularly significant as they tend to suggest that the multiple combinations of geometries and materials can yield so many different combinations of distortional and dilatational effects. No wonder microscopic irregularities seem to be the rule. Discontinuous numerically solutions, however, may or may not correspond to singularities. In view of the recent research emphases on multiscaling, the multiple singularity solution at a point seems to signify the inherent characteristics of micro-defects. They are irregular and defy self-similarity. The double singularity alone implies that a microcrack can extend in two different directions from the same location, known as branching in the very rapid propagation of a macrocrack. At the microscopic scale, double singularity implies the possibility that branching can occur under macroscopic static load. Such a behavior of static branching of microcracks associated with double singularity is being established. A fundamental aspect of this model is the inherent existence of restraining stress from the material that is essential for the microcrack and only sometimes for the macrocrack. As a rule, the microcrack being smaller would react to the material restraint more readily. Such a feature is necessary in the model in addition to the effects of micro/macro material interaction and the macrocrack characteristic length with reference to the macrocrack length. In normalized form, the minimum of three parameters, say μ ∗, σ ∗ and d ∗, would be necessary for the formulation of the dual scale micro/macro crack model. The combined effects of μ ∗ (micro/macro material), σ ∗ (load and restraint) and d ∗ (characteristic length) cannot be over emphasized in multiscale modeling. In the spirit of classical fracture mechanics, a micro/macro crack intensification factor K micro macro is defined and obtained in closed form. It contains three basic parameters μ ∗, d ∗ and σ ∗ being unique to all microcracks while two parameters ϕ 1 and ϕ 2 can be used to describe additional features of the microstructure. More precisely, the specific material microstructure can be modeled by two or more parameters associated with the eigenvalues selected rather than using the constitutive relations in the traditional manner. The asymptotic form of K micro macro is exact such that the case specific nature of microcracking can be distinguished by two constants. The contribution of this work is to establish the potential use of the dual singularity solution. The basic concept of the method applies to other multiscale problems in mechanics and physics.