Dai Okumura

Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation

In this paper, we establish a homogenization framework to analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The resulting conditions are verified by analyzing numerically the in-plane biaxial buckling of an elastic hexagonal honeycomb. It is thus shown that three kinds of experimentally observed buckling modes of honeycombs i.e., uniaxial, biaxial and flower-like modes, are attained and classified as microscopic symmetric bifurcation. It is also shown that the multiplicity of bifurcation gives rise to the complex cell-patterns in the biaxial and flower-like modes.

Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression

Abstract In this paper, employing the homogenization theory and the microscopic bifurcation condition established by the authors, we discuss which microscopic buckling mode grows in elastic honeycombs subject to in-plane biaxial compression. First, we focus on equi-biaxial compression, under which uniaxial, biaxial and flower-like modes may develop as a result of triple bifurcation. By forcing each of the three modes to develop, and by comparing the internal energies, we show that the flower-like mode grows steadily if macroscopic strain is controlled, while either the uniaxial or biaxial mode develops if macroscopic stress is controlled. Second, by analyzing several cases other than equi-biaxial compression, it is shown that a second bifurcation from either the uniaxial or biaxial mode to the flower-like mode, which is distorted, occurs under biaxial compression in a certain range of biaxial ratio under macroscopic strain control. Finally, the possibility of macroscopic instability under biaxial compression is discussed.

Swelling-Induced Buckling Patterns in Gel Films with a Square Lattice of Holes Subjected to In-Plane Uniaxial and Biaxial Pretensions

In this study, we investigate swelling-induced buckling patterns in gel films containing a square lattice of holes subjected to in-plane pretensions. In accord with experiments, we simulate poly(dimethylsiloxane) (PDMS) films being prestrained and then swelled using toluene. Films are subjected to uniaxial and bi-axial pretensions before swelling to investigate the potential ability of this system to generate complex buckling patterns. Finite element analysis is performed using an inhomogeneous field theory for polymeric gels. The resulting patterns are found to be highly diverse and depend sensitively on the type and magnitude of pretensions. The patterns arise from either transformation into diamond plate patterns (DPPs) or no pattern transformation. Diagrams of pattern transformation contain three regions of DPPs, transitional patterns, and monotonous patterns. Pretensions both distort the initial arrangement of the square lattice of holes and delay the onset of transformation into DPPs.

Long-Wave In-Plane Buckling of Elastic Square Honeycombs

Abstract In this study, microscopic buckling of elastic square honeycombs subject to in-plane compression is analyzed using a two-scale theory of the up-dated Lagrangian type. The theory allows us to analyze microscopic bifurcation and post-bifurcation behavior of periodic cellular solids. Cell aggregates are taken to be periodic units so that we can discuss the dependence of buckling stress on periodic length. Then, it is shown that microscopic buckling occurs at a lower compressive load as periodic length increases, and that long-wave buckling occurs just after the onset of macroscopic instability if the periodic length is sufficiently long. It is further shown that the macroscopic instability is of the shear type, leading to a simple formula to evaluate the lowest in-plane buckling stress of elastic square honeycombs.

Microscopic Interlaminar Stress Analysis of CFRP Cross-Ply Laminate Using a Homogenization Theory

Microscopic stress distributions at an interlaminar area in a CFRP cross-ply laminate are analyzed three-dimensionally using a homogenization theory in order to investigate microscopic interaction between 0°- and 90°-plies. It is first shown that a cross-ply laminate has a point-symmetric internal structure on the assumption that each ply in the laminate has a square array of long fibers. Next, the point-symmetry is utilized to reduce the domain of homogenization analysis by half. Moreover, the substructure method is combined with the homogenization theory for reducing consumption of computational resources. The present method is then employed for analyzing stress distributions at an interlaminar area in a carbon fiber/epoxy cross-ply laminate under in-plane off-axis tensile loading. It is thus shown that microscopic shear stress significantly occurs at the interface between 0°- and 90°-plies. It is also shown that the microscopic interaction between two plies is observed only in the vicinity of the interface.

Warpage Variation Analysis of Si/Solder/Cu Layered Plates Subjected to Cyclic Thermal Loading

In cyclic thermal tests of Si/solder/OFHC-Cu (silicon/solder/oxygen-free high conductivity copper)-layered plates, the authors observed either the cyclic growth or cyclic recovery of warpage to occur depending on the heat treatment of the copper before soldering. In this study, the test results are numerically analyzed by assuming three material models for the solder and two material models for the copper. It is shown that the test results are reproduced well if proper material models are used in finite element analysis. It is revealed that the so-called multiaxial ratcheting was induced in the solder, while the uniaxial type of ratcheting or cyclic strain recovery occurred in the copper. As a result, the Armstrong and Frederick model is suggested to be valid for the multiaxial ratcheting in the solder at such low strain rates as in the cyclic thermal tests, whereas the Ohno and Wang model is shown to be appropriate for the copper. To confirm this unexpected result for the solder, the Armstrong and Frederick model is applied to the multiaxial ratcheting of another solder at three strain rates.

Effects of Two Scaling Exponents on Swelling-Induced Softening of Elastomers under Equibiaxial and Planar Extensions

We study the effects of two scaling exponents on the mechanical properties of swollen elastomers under equibiaxial and planar extensions. Two scaling exponents are introduced to extend the Flory-Rehner free energy function, and are adjusted based on the previous study. Results show that swelling-induced strain softening is apt to occur under equibiaxial extension compared to uniaxial extension. The additional tensile stress in a lateral direction enables it to occur in relatively poor solvents, and accelerates the onset point. Planar extension shows more complicated responses because the stress in the constrained direction changes dramatically depending on the combination of two scaling exponents and the Flory-Huggins interaction parameter.

Effect of Strain Hardening on Monotonic and Cyclic Loading Behavior of Plate-Fin Structures with Two Pore Pressures

We perform finite element homogenization (FEH) analysis to investigate the effect of strain hardening on the monotonic and cyclic loading behavior of plate-fin structures with two pore pressures. As a typical base metal of plate-fin structures, 316 stainless steel is considered and assumed to be the viscoplastic material that obeys the Ohno-Wang kinematic hardening rule. The plate-fin structures are assumed to be periodic and subjected to uniaxial monotonic and cyclic loadings in the stacking direction. A periodic unit cell is used for FEH analysis. Results are compared with those based on three special cases derived from Hill’s macrohomogeneity equation. It is found that the mean pore pressure entirely affect the homogenized viscoplastic behavior. It is further found that the differential pore pressure causes the remarkable accumulation of ratcheting strain in the periodic unit cell, although this internal ratcheting gives no effect on macroscopic relations, resulting in providing a closed hysteresis loop for the plate-fin structures.

Anisotropy in Buckling Behavior of Kelvin Open-Cell Foams Subject to Uniaxial Compression

This paper describes buckling modes and stresses of elastic Kelvin open-cell foams subjected to [001], [O11] and [111] uniaxial compressions. Cubic unit cells and cell aggregates in model foams are analyzed using a homogenization theory of the updated Lagrangian type. The analysis is performed on the assumption that the struts in foams have a non-uniform distribution of cross-sectional areas as observed experimentally. The relative density is changed to range from 0.005 to 0.05. It is thus found that long wavelength buckling and macroscopic instability primarily occur under [001] and [011] compressions, with only short wavelength buckling under [111] compression. The primary buckling stresses under the three compressions are fairly close to one another and almost satisfy the Gibson-Ashby relation established to fit experiments. By also performing the analysis based on the uniformity of strut cross-sectional areas, it is shown that the non-uniformity of cross-sectional areas is an important factor for the buckling behavior of open-cell foams.

Grain Size Dependence of Yield Stress Due to Strain Energy of Geometrically Necessary Dislocations

The self strain energy of geometrically necessary dislocations (GNDs) in single crystals is considered to inevitably introduce the higher-order stress work-conjugate to slip gradient. It is shown that this higher-order stress changes stepwise in response to in-plane slip gradient, and thus significantly influences the initial yielding of polycrystals. The self strain energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin (2002). The theory developed is applied to 2D and 3D model crystal grains of diameter D, leading to a D -1 -dependent term with a coefficient determined by grain shape. This size effect term is verified using published experimental data of several polycrystalline metals. It is thus found that the D -1 -dependent term is successful for predicting not only the grain size dependence of initial yield stress but also the dislocation cell size dependence of flow stress in the submicron to several micron range of D. Introduction As is well known, polycrystals exhibit the dependence of yield stress on grain size. The Hall-Petch relation is said to be established for this dependence in the conventional range of grain size. The Hall-Petch plot, however, usually has nonlinearity, as grain size is reduced from the conventional range. It is noted that the nonlinearity can occur at even around one to several microns of grain size, leading to a stronger, grain size dependence of yield stress than that by the Hall-Petch relation, as was observed in [1-3]. It is also noted that such fine grained polycrystals tend to clearly show initial yielding points at stresses markedly depending on grain size [1-6]. It is of interest to note further the following observation in several experiments [7]: in stage II where secondary slip systems become active, dislocation cells of submicron to several microns in size usually develop to cause the inverse proportionality between flow stress and dislocation cell size, which is stronger than that by the Hall-Petch relation. It is thus worthwhile to analyze the grain size dependence of yield stress in the range of submicron to several microns. The size effects mentioned above can be analyzed using the so-called strain gradient plasticity theories, which have been proposed in several studies so far [e.g., 8-15]. The theories are classified into two groups, i.e., first-order and high-order theories. Gurtin [14] developed a higher-order theory, in which a higher-order stress was introduced as the work-conjugate to slip gradient in single crystals. His theory seems promising for analyzing the size dependence of yield stress, because the constraint of slip at grain boundaries is explicitly represented as the boundary condition of slip. Okumura et al. [16] thus implemented his theory in a homogenization method to analyze the yield behavior of a 2D model polycrystal; however, no marked dependence on grain size was obtained with respect to initial yield stress. In this study, the self strain energy of geometrically necessary dislocations (GNDs) is considered in order to analyze the grain size dependence of yield stress. First, by discussing the self strain energy of GNDs in single crystals at small strains, the higher-order stress work conjugate to slip gradient is inevitably introduced. It is pointed out that this higher-order stress changes stepwise in response to Key Engineering Materials Online: 2007-08-15 ISSN: 1662-9795, Vols. 345-346, pp 3-8 doi:10.4028/www.scientific.net/KEM.345-346.3