Averaged Strain Energy Density Research Articles

Geometrically nonlinear stress–strain behavior of hyperelastic solid foams

The present study is concerned with an effective stress analysis of cellular solids in the finite strain regime. The homogenization of the microstructure is performed by means of a strain energy based RVE-procedure which assumes macroscopic equivalence of a representative volume element for the given microstructure and a similar volume element consisting of the effective medium, if the average strain energy density in both volume elements is equal provided that the deformation gradient with respect to both elements is equal in a volume average sense. Disordered microstructures are considered by means of a randomized periodic model in conjunction with a stochastic approach. The model is applied to an analysis of the effective stress–strain behavior of two-dimensional model foams with periodic and disordered microstructure. Special interest is directed to effects of the geometric nonlinearity.

A New Thermal-Fatigue Life Prediction Model for Wafer Level Chip Scale Package (WLCSP) Solder Joints

A new empirical equation for predicting the thermal-fatigue life of wafer level chip scale package (WLCSP) solder joints on printed circuit board (PCB) is presented. The solder joints are subjected to thermal cycling and their crack lengths at different thermal cycles are measured. Also, the average strain energy density around the crack tip of different crack lengths in the corner solder joint is determined by a time-dependent nonlinear fracture mechanics with finite element method. The solder is assumed to be a temperature-dependent elastic-plastic and a time-dependent creep material.

Strain relaxation in AlGaN under tensile plane stress

Relaxation of tensile strain in AlxGa1−xN layers of different compositions epitaxially grown on GaN/sapphire is investigated. Extended crack channels along 〈211¯0〉 directions are formed if the aluminum content exceeds a critical value, which decreases with increasing layer thickness. This process is found to limit the average strain energy density to a maximum value of 4 J/m2. By calculating the stress distribution between cracks and the strain energy release rate for crack propagation, the relaxed strain as measured by x-ray diffraction is correlated to the crack density, and the onsets of crack channeling and layer decohesion are fitted to a fracture toughness of 9 J/m2. Moreover, the crack opening at the surface is found to linearly increase with the stress. Annealing of samples above the growth temperature introduces additional tensile stress due to the mismatch in thermal expansion coefficients between the layer and substrate. This stress is shown to relieve not only by the formation of additional cracks but also by the extension of cracks into the GaN layer and a thermal activated change in the defect structure.

Solder Joint Reliability of Wafer Level Chip Scale Packages (WLCSP): A Time-Temperature-Dependent Creep Analysis

A novel and reliable wafer level chip scale package (WLCSP) is investigated in this paper. It consists of a copper conductor layer and two low cost dielectric layers. The bump geometry consists of the eutectic solder, the copper core, and the under bump metallurgy. Nonlinear time-temperature-dependent finite element analyses are performed to determine the shear stress, shear creep strain, shear stress and shear creep strain hysteresis loops, and creep strain energy density of the corner solder joint. The thermal-fatigue life of the corner solder joint is then predicted by the averaged creep strain energy density range per cycle and a linear fatigue crack growth rate theory. The WLCSP solder bumps are also subjected to shear test. Finally, the WLCSP solder joints are subjected to both mechanical shear and thermal cycling tests. [S1043-7398(00)01004-5]

A VELOCITY METHOD FOR ESTIMATING DYNAMIC STRAIN AND STRESS IN PIPES

A velocity method for estimating dynamic strain and stress in pipe structures is investigated. With this method, predicted or measured spatial average vibration velocity and theoretically derived strain factors are used to estimate maximum strain at the ends of pipes. Theoretical investigation shows that the strain at a point is limited by an expression proportional to the square root of the strain energy density, which in turn is related to its cross-sectional average. For a reverberant field or for an infinite pipe, the average strain energy density is proportional to the mean square velocity. Upon this basis, the non-dimensional strain factor is defined as the maximum strain times the ratio of the sound velocity to the spatial root mean square vibration velocity. Measurements are made confirming that this is a descriptive non-dimensional number. Using a spectral finite element method, numerical experiments are made varying the pipe parameters and considering all 16 homogeneous boundary conditions. While indicating possible limitations of the method when equipment is mounted on pipes, the experiments verify the theoretical results. The velocity method may become useful in engineering practice for assessments of fatigue life.

Geometrical, Mechanical, and Structural Adaptation of Mouse Femora Exposed to Different Loadings

Juvenile mice were utilized as a model system to study the influence of different activity regimens on the change in bone morphology and load-bearing capacity of their femora. Two treatment groups were employed: one exposed to 4g hypergravity (HG) and the other to chronic digging in high ground corn cob litter (HL). A voluntary normal exercise group (NE) served as control. One femur from each animal was used to determine the geometrical properties of the bone; the other femur was used in a load-to-failure experiment. Electron micrographs of femur cross sections were digitized to yield geometrical properties of the bone. Young's modulus and the stress and strain at failure for the bone were determined via a modified Euler beam model. Strain energy was evaluated directly from load-deflection curves obtained during the load-to-failure experiments. Average strain-energy density was then derived from these results. Cross-sectional geometrical properties (diameters, wall thicknesses, area, moments of inertia, a...

Surface remodeling of trabecular bone using a tissue level model.

We propose a two-dimensional mathematical model of trabecular bone remodeling that simulates the surface-based addition and removal of material in the actual physiological process. The model is based on a finite element representation of individual trabecular struts in which the material properties of the subtrabecular elements are constant. The remodeling stimulus is strain energy density, sensed and communicated through the osteocytic network as proposed by Mullender et al. We propose a modified osteocyte communication scheme that incorporates bone-lining cells and examines the implications of set point locations in one or the other of these two cell types. This model produces trabecular struts that align with its general loading direction. Placing the set point in the bone-lining cells rather than in the osteocytes makes the model more sensitive to changes in the other biological parameters. Introduction of a dead zone causes the model to reach a less oscillatory equilibrium in fewer iterations and produces better in-filling of trabecular strut intersections. The model gravitates to equilibrium states in which the average strain energy density is inversely proportional to the bone volume fraction to the 3.2 power.

Rayleigh-Ritz optimal design of orthotropic plates for buckling

This paper is concerned with the structural optimization problem of maximizing the compressive buckling load of orthotropic rectangular plates for a given volume of material. The optimality condition is first derived via variational calculus. It states that the thickness distribution is proportional to the strain energy density contrary to popular claims of constant strain energy density at the optimum. An engineers physical meaning of the optimality condition would be to make the average strain energy density with respect to the depth a constant. A double cosine thickness varying plate and a double sine thickness varying plate are then fine tuned in a one parameter optimization using the Rayleigh-Ritz method of analysis. Results for simply supported square plates indicate an increase of 89% in capacity for an orthotropic plate having 100% of its fibers in <TEX>$0^{\circ}$</TEX> direction.

Reduced effective misfit in laterally limited structures such as epitaxial islands

Numerical finite element calculations have been reported to determine a correction function Φ that describes the reduction of the misfit that occurs when laterally limited structures such as faceted islands or mesa structures are grown on a substrate. The reduction of the average strain energy density is calculated in these three-dimensional islands and compared to the constant strain energy density in a continuous layer. Ratios Φ are obtained from the calculation of different island geometries, i.e., different facet angles γ and different aspect ratios island width l to island height h. These discrete values are fitted by a function which can easily be applied to the full range of aspect ratios (l/h≳0) and facet angles (0°&amp;lt;γ&amp;lt;90°). Faceted Ge(Si) islands on Si(001) substrate, grown from the solution in the Stranski–Krastanov growth mode, serve as an example for the calculation. Experimental and theoretical values for the critical thickness of these islands agree well. This result demonstrates the drastic influence of islanding on misfit strain distribution in island and substrate as well and, consequently, on the strong increase of the critical thickness as determined by the mechanical equilibrium theory of Matthews and Blakeslee.

A Simple Analysis of Average Mechanical Behavior and Strain Energy Density of Misoriented Grains in a Textured Film

AbstractDifferences in strain energy density between grains of different orientation within a thin film may drive morphological changes such as abnormal grain growth, hillocking, and sunken grains. Models of these morphological changes generally assume that the behavior of each grain is the same as that of a blanket film of corresponding orientation. In order to investigate this assumption as it applies to the behavior of a (100)-oriented grain within a {111}-fiber textured copper film, simple continuum analytical and finite element models are applied to the case of a film on a rigid substrate subjected to temperature changes. The simple models indicate that the behavior of a misoriented grain does not simply duplicate that of a blanket film of the same orientation; the behavior of the surrounding grains must also be considered. The strain energy density difference between the grain and the surrounding film is less than that predicted by blanket film behavior. An elasticity equation is presented which describes the average stress-temperature behavior of widely dispersed misoriented grains within a textured film.

Stress singularities along a cycloid rough surface

The stress concentration along a rough surface is of importance for understanding the nucleation of misfit dislocations and cracks in heteroepitaxial thin films and for general flaw initiation at material surfaces exposed to environmental corrosion. In order to seek a basic understanding on this issue, a cycloid wavy surface subject to a uniform bulk stress is adopted as a model problem. The elastic stress and displacement fields are determined using Muskhelishvili's conformal mapping method. It is shown that a cusped cycloid surface generates a crack-like singular stress field within a thin surface layer; under uniform tension this singularity is found to have identical strength to a row of periodic parallel cracks. The path-independent J-integral requires that the average surface strain energy density be identical to that in the bulk, implying that any stress magnification along a rough surface must be compensated by unloading along some complementary portions of the surface. Along the cusped cycloid surface, the strain energy distribution becomes Dirac singular at the cusp tips, the rest of the surface being completely stress-free. Thus the effect of cycloid cusps is to redistribute and concentrate all the surface strain energy per wavelength at a single (cusp) point, and because of that we can claim that the cycloid surface is the most efficient stress concentrator at a fixed wavelength. Even at a moderate bulk stress level this concentration may be sufficient to cause failure or nucleation of defects.The full evolution of a rough surface under stress and other corrosion mechanisms must be solved by numerical methods. Our analytic solutions for a cycloid surface are of significant value for guiding numerical computations and gaining insights into essential features of the evolution process. From a global point of view, we show that a cusped cycloid surface becomes energetically favorable once the surface wavelength exceeds a critical value determined from the competition between the strain energy and the surface energy. From a local point of view, analysis of surface diffusion behaviors along an almost cusped cycloid surface indicates that the cusps are stable once they develop. The critical condition for formation of a cusped cycloid corresponds to the Griffith energy balance being exactly satisfied at the cusp tips while the chemical potential remains nearly constant along the rest of the surface. This implies spontaneous Griffith brittle fracture at tension cusps if plastic relaxation is not present to relieve the stress singularity.

Survey on additives for polymers

Nowadays, hyperelastic materials such as rubbers and elastomers have become increasingly important in research and industry. However, the existence of any geometric discontinuities like a crack or notch will affect the behavior of rubbers. Thus, the prediction of rupture condition in these materials is of paramount importance. One of the energy-based criteria, which has attracted many attentions due to its convenience in use and high precision, is the averaged strain energy density (ASED) criterion. Based on this criterion, a limited amount of energy in a specified volume around the stress concentration region controls the fracture of components. This criterion, which was already utilized extensively for brittle materials, has been recently extended for being used in cracked components having non-linear behavior like rubbers. On the other hand, by examining the previous studies, it can be concluded that despite the importance of rounded V-shaped notch, no research has been carried out in the presence of this type of stress concentration in hyperelastic materials. As a consequence, this study is devoted to investigate the rupture of rubbers containing a round-tip V-shape notch using the ASED criterion. To this end, first, some experiments are performed for rounded-tip V-notched rubber samples. Afterwards and in order to utilize the ASED criterion for predicting the rupture loads of tested samples, some finite element modeling considering both geometric and material non-linearities are performed. The comparison between the estimations of the ASED criterion and the corresponding experimental data reveals the high efficacy of this criterion in hyperelastic materials weakened by round-tip V-shape notches.

Continuum characterization of jointed rock masses: Part I—The constitutive equations

Hill has proved that the average strain energy density in any region of an elastic and inhomogeneous material can be calculated from the average values of the stresses and strains within that region. This concept has been used to derive the general constitutive equations of a rock mass containing an orthogonal set of discontinuous joints intersecting an anisotropic rock material. The constants required for the continuum characterization of the jointed mass are the ‘joint stress concentration factors’ B N1 and B T1 (see Fig. 1 of this paper). These are defined as the ratio of stresses along the joint to the overall stresses in the rock. Exact expressions for the stress concentration factors have been obtained for the case of a rigid rock (i.e. the intact material between the joints) containing staggered compliant joints, and a correction is proposed for the case where the rock is (elastically) deformable. These results suggest that interlocking betweem blocks of rock may become significant even for a slight offset along the joints, such as may occur during shear deformation of a rock mass containing an initially continuous orthogonal joint set. Stress concentration factors computed independently from the results of a finite element program for a jointed mass compare excellently with the above-mentioned theoretical results. It is further shown that tensile stresses are developed inside a rock with staggered joints, and may be as high as twice the overall shear stresses or the overall compressive stresses. Typical stress distributions within a block are given in Fig 6(a) and 6(b) of this paper. It is concluded that a rock mass is rendered anisotropic by any joint set having a preferred orientation. Expressions from which the elastic moduli of the equivalent continuum anisotropic rock mass may be obtained are presented in Table 1.