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Journal articles on the topic 'Hill elastic bounds'
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1
Torquato, S., and F. Lado. "Improved Bounds on the Effective Elastic Moduli of Random Arrays of Cylinders." Journal of Applied Mechanics 59, no. 1 (March 1, 1992): 1–6. http://dx.doi.org/10.1115/1.2899429.
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Improved rigorous bounds on the effective elastic moduli of a transversely isotropic fiber-reinforced material composed of aligned, infinitely long, equisized, circular cylinders distributed throughout a matrix are evaluated for cylinder volume fractions up to 70 percent. The bounds are generally shown to provide significant improvement over the Hill-Hashin bounds which incorporate only volume-fraction information. For cases in which the cylinders are stiffer than the matrix, the improved lower bounds provide relatively accurate estimates of the elastic moduli, even when the upper bound diverges from it (i.e., when the cylinders are substantially stiffer than the matrix). This last statement is supported by accurate, recently obtained Monte Carlo computer-simulation data of the true effective axial shear modulus.
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Murshed, Muhammad Ridwan, and Shivakumar I. Ranganathan. "Hill–Mandel condition and bounds on lower symmetry elastic crystals." Mechanics Research Communications 81 (April 2017): 7–10. http://dx.doi.org/10.1016/j.mechrescom.2017.01.005.
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Qiu, Y. P., and G. J. Weng. "Elastic Moduli of Thickly Coated Particle and Fiber-Reinforced Composites." Journal of Applied Mechanics 58, no. 2 (June 1, 1991): 388–98. http://dx.doi.org/10.1115/1.2897198.
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Based on the models of Hashin (1962) and Hashin and Rosen (1964), the effective elastic moduli of thickly coated particle and fiber-reinforced composites are derived. The microgeometry of the composite is that of a progressively filled composite sphere or cylinder element model. The exact solutions of the effective bulk modulus κ of the particle-reinforced composite and those of the plain-strain bulk modulus κ23, axial shear modulus μ12, longitudinal Young’s modulus E11, major Poisson ratio ν12, of the fiber-reinforced one are derived by the replacement method. The bounds for the effective shear modulus μ and the effective transverse shear modulus μ23 of these two kinds of composite, respectively, are solved with the aid of Christensen and Lo’s (1979) formulations. By considering the six possible geometrical arrangements of the three constituent phases, the values of κ, and of κ23, μ12, E11, and ν12 are found to always lie within the Hashin-Shtrikman (1963) bounds, and the Hashin (1965), Hill (1964), and Walpole (1969) bounds, respectively, but unlike the two-phase composites, none coincides with their bounds. The bounds of μ and μ23 derived here are consistently tighter than their bounds but, as for the two-phase composites, one of the bounds sometimes may fall slightly below or above theirs and therefore it is suggested that these two sets of bounds be used in combination, always choosing the higher for the lower bound and the lower for the upper one.
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Li, Yingzhe, and Jay D. Bass. "Single Crystal Elastic Properties of Hemimorphite, a Novel Hydrous Silicate." Minerals 10, no. 5 (May 10, 2020): 425. http://dx.doi.org/10.3390/min10050425.
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Hemimorphite, with the chemical formula Zn4Si2O7(OH)2·H2O, contains two different types of structurally bound hydrogen: molecular water and hydroxyl. The elastic properties of single-crystal hemimorphite have been determined by Brillouin spectroscopy at ambient conditions, yielding tight constraints on all nine single-crystal elastic moduli (Cij). The Voigt–Reuss–Hill (VRH) averaged isotropic aggregate elastic moduli are KS (VRH) = 74(3) GPa and μ (VRH) = 27(2) GPa, for the adiabatic bulk modulus and shear modulus, respectively. The average of the Hashin–Shtrickman (HS) bounds are Ks (HS) = 74.2(7) GPa and and μ (HS) = 26.5(6) GPa. Hemimorphite displays a high degree of velocity anisotropy. As a result, differences between upper and lower bounds on aggregate properties are large and the main source of uncertainty in Ks and μ. The HS average P wave velocity is VP = 5.61(4) km/s, and the HS S-wave velocity is VS = 2.77(3) km/s. The high degree of elastic anisotropy among the on-diagonal longitudinal and pure shear moduli of hemimorphite are largely explained by its distinctive crystal structure.
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Zhao, Y. H., and G. J. Weng. "Effective Elastic Moduli of Ribbon-Reinforced Composites." Journal of Applied Mechanics 57, no. 1 (March 1, 1990): 158–67. http://dx.doi.org/10.1115/1.2888297.
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Based on the Eshelby-Mori-Tanaka theory the nine effective elastic constants of an orthotropic composite reinforced with monotonically aligned elliptic cylinders, and the five elastic moduli of a transversely isotropic composite reinforced with two-dimensional randomly-oriented elliptic cylinders, are derived. These moduli are given in terms of the cross-sectional aspect ratio and the volume fraction of the elliptic cylinders. When the aspect ratio approaches zero, the elliptic cylinders exist as thin ribbons, and these moduli are given in very simple, explicit forms as a function of volume fraction. It turns out that, in the transversely isotropic case, the effective elastic moduli of the composite coincide with Hill’s and Hashin’s upper bounds if ribbons are harder than the matrix, and coincide with their lower bounds if ribbons are softer. These results are in direct contrast to those of circular fibers. Since the width of the Hill-Hashin bounds can be very wide when the constituents have high modular ratios, this analysis suggests that the ribbon reinforcement is far more effective than the traditional fiber reinforcement.
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Torquato, S., and F. Lado. "Bounds on the Effective Transport and Elastic Properties of a Random Array of Cylindrical Fibers in a Matrix." Journal of Applied Mechanics 55, no. 2 (June 1, 1988): 347–54. http://dx.doi.org/10.1115/1.3173681.
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This paper studies the determination of rigorous upper and lower bounds on the effective transport and elastic moduli of a transversely isotropic fiber-reinforced composite derived by Silnutzer and by Milton. The third-order Silnutzer bounds on the transverse conductivity σe, the transverse bulk modulus ke, and the axial shear modulus μe, depend upon the microstructure through a three-point correlation function of the medium. The fourth-order Milton bounds on σe and μe depend not only upon three-point information but upon the next level of information, i.e., a four-point correlation function. The aforementioned microstructure-sensitive bounds are computed, using methods and results of statistical mechanics, for the model of aligned, infinitely long, equisized, circular cylinders which are randomly distributed throughout a matrix, for fiber volume fractions up to 65 percent. For a wide range of volume fractions and phase property values, the Silnutzer bounds significantly improve upon corresponding second-order bounds due to Hill and to Hashin; the Milton bounds, moreover, are narrower than the third-order Silnutzer bounds. When the cylinders are perfectly conducting or perfectly rigid, it is shown that Milton’s lower bound on σe or μe provides an excellent estimate of these effective parameters for the wide range of volume fractions studied here. This conclusion is supported by computer-simulation results for σe and by experimental data for a graphite-plastic composite.
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Park, N. J., H. J. Bunge, H. Kiewel, and L. Fritsche. "Calculation of Effective Elastic Moduli of Textured Materials." Textures and Microstructures 23, no. 1 (January 1, 1995): 43–59. http://dx.doi.org/10.1155/tsm.23.43.
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Effective elastic constants of polycrystalline materials were determined with a recently developed method. This method bases on the modelation of the actual material by a cluster of 100 to 500 single crystals. In the present version of the scheme parallelepipeds are used. The ODF was calculated with the series expansion method. The transformation of this ODF into a finite sum of single orientations permits to assign any grain an individual orientation.Reliable results for the effective elastic moduli of textured materials are reported. They lie always within the bounds of Voigt and Reuss. The very high anisotropic substances, e.g. shape-memory-alloys, show a significant deviation from the Hill values.
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Ku, A. P. D., and R. P. Nordgren. "On Plastic Collapse of Media With Random Yield Strength." Journal of Applied Mechanics 68, no. 5 (February 6, 2001): 715–24. http://dx.doi.org/10.1115/1.1388011.
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This paper concerns the plastic collapse of an elastic/perfectly plastic medium with randomly variable yield strength under a fixed load. The yield strength is represented by a Gaussian random field of known statistical properties. Using the theorems of limit analysis and the methods of reliability theory, algorithms are developed for the computation of upper and lower bounds on the probability of plastic collapse. By varying the magnitude of the fixed load, bounds on the probability distribution function for the collapse load can be computed. Results are given for uniform pressure applied to a rectangular region of the surface of an elastic/plastic half-space. For the corresponding plane problem, results for the classical Hill and Prandtl failure mechanisms are compared. Three-dimensional results are found to differ significantly from those of the plane problem. Comparison is made with results of a previous approximate method for three-dimensional problems.
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Zhang, Jun, and Martin Ostoja-Starzewski. "Frequency-dependent scaling from mesoscale to macroscale in viscoelastic random composites." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2188 (April 2016): 20150801. http://dx.doi.org/10.1098/rspa.2015.0801.
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This paper investigates the scaling from a statistical volume element (SVE; i.e. mesoscale level) to representative volume element (RVE; i.e. macroscale level) of spatially random linear viscoelastic materials, focusing on the quasi-static properties in the frequency domain. Requiring the material statistics to be spatially homogeneous and ergodic, the mesoscale bounds on the RVE response are developed from the Hill–Mandel homogenization condition adapted to viscoelastic materials. The bounds are obtained from two stochastic initial-boundary value problems set up, respectively, under uniform kinematic and traction boundary conditions. The frequency and scale dependencies of mesoscale bounds are obtained through computational mechanics for composites with planar random chessboard microstructures. In general, the frequency-dependent scaling to RVE can be described through a complex-valued scaling function, which generalizes the concept originally developed for linear elastic random composites. This scaling function is shown to apply for all different phase combinations on random chessboards and, essentially, is only a function of the microstructure and mesoscale.
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Vermeulen, Arnold C., Christopher M. Kube, and Nicholas Norberg. "Implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry." Powder Diffraction 34, no. 2 (April 30, 2019): 103–9. http://dx.doi.org/10.1017/s088571561900037x.
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In this paper, we will report about the implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for general, triclinic crystal symmetry. With applying appropriate symmetry relations, the point groups of higher crystal symmetries are covered as well. This simplifies the implementation effort to cover the calculations for any crystal symmetry. In the literature, several models can be found to estimate the polycrystalline elastic properties from single crystal elastic constants. In general, this is an intermediate step toward the calculation of the polycrystalline response to different techniques using X-rays, neutrons, or ultrasonic waves. In the case of X-ray residual stress analysis, the final goal is the calculation of X-ray Elastic constants. Contrary to the models of Reuss, Voigt, and Hill, the Kröner–Eshelby model has the benefit that, because of the implementation of the Eshelby inclusion model, it can be expanded to cover more complicated systems that exhibit multiple phases, inclusions or pores and that these can be optionally combined with a polycrystalline matrix that is anisotropic, i.e., contains texture. We will discuss a recent theoretical development where the approaches of calculating bounds of Reuss and Voigt, the tighter bounds of Hashin–Shtrikman and Dederichs–Zeller are brought together in one unifying model that converges to the self-consistent solution of Kröner–Eshelby. For the implementation of the Kröner–Eshelby model the well-known Voigt notation is adopted. The 4-rank tensor operations have been rewritten into 2-rank matrix operations. The practical difficulties of the Voigt notation, as usually concealed in the scientific literature, will be discussed. Last, we will show a practical X-ray example in which the various models are applied and compared.
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McMahon, B. T., S. K. Haigh, and M. D. Bolton. "Bearing capacity and settlement of circular shallow foundations using a nonlinear constitutive relationship." Canadian Geotechnical Journal 51, no. 9 (September 2014): 995–1003. http://dx.doi.org/10.1139/cgj-2013-0275.
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The design of shallow foundations is dominated by issues of settlement rather than bearing capacity per se. The ability to predict the settlement of foundations at a given factor of safety is hence of key importance in design. In this paper, the energy method for a linear-elastic, perfectly plastic method utilizing the von Mises’ yield criterion with associated flow developed and reported by McMahon et al. in 2013 is extended to consider the nonlinear behaviour of soil. The energy method is used to investigate the load–settlement behaviour of shallow foundations by utilizing an ellipsoidal cavity-expansion mechanism and deformation fields within the boundaries of the classical Hill and Prandtl mechanisms. An elastic mechanism obtained from an analysis in ABAQUS was also investigated using this energy method. The upper-bound approach demonstrates that the cavity-expansion mechanism produces a better solution at small values of settlement, whereas at greater settlements the Prandtl mechanism is shown to produce a more optimal upper-bound solution.
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Sandercock, Thomas G., and C. J. Heckman. "Force From Cat Soleus Muscle During Imposed Locomotor-Like Movements: Experimental Data Versus Hill-Type Model Predictions." Journal of Neurophysiology 77, no. 3 (March 1, 1997): 1538–52. http://dx.doi.org/10.1152/jn.1997.77.3.1538.
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Sandercock, Thomas G. and C. J. Heckman. Force from cat soleus muscle during imposed locomotor-like movements: experimental data versus Hill-type model predictions. J. Neurophysiol. 77: 1538–1552, 1997. Muscle is usually studied under nonphysiological conditions, such as tetanic stimulation or isovelocity movements, conditions selected to isolate specific properties or mechanisms in muscle. The purpose of this study was to measure the function of cat soleus muscle during physiological conditions, specifically a simulation of a single speed of slow walking, to determine whether the resulting force could be accurately represented by a Hill-type model. Because Hill-type models do not include history-dependent muscle properties or interactions among properties, the magnitudes of errors in predicted forces were expected to reveal whether these phenomena play important roles in the physiological conditions of this locomotor pattern. The natural locomotor length pattern during slow walking, and the action potential train for a low-threshold motor unit during slow walking, were obtained from the literature. The whole soleus muscle was synchronously stimulated with the locomotor pulse train while a muscle puller imposed the locomotor movement. The experimental results were similar to force measured via buckle transducer in freely walking animals. A Hill-type model was used to simulate the locomotor force. In a separate set of experiments, the parameters needed for a Hill-type model (force-velocity, length-tension, and stiffness of the series elastic element) were measured from the same muscle. Activation was determined by inverse computation of an isometric contraction with the use of the same locomotor stimulus pattern. During the stimulus train, the Hill-type model fit the locomotor data fairly well, with errors <10% of maximal tetanic tension. A substantial error occurred during the relaxation phase. The model overestimated force by ∼30% of maximal tetanic tension. A nonlinear series elastic element had little influence on the force predicted by a Hill model, yet dramatically altered the predicted muscle fiber lengths. Further experiments and modeling were performed to determine the source of errors in the Hill-type model. Isovelocity ramps were constructed to pass through a selected point in the locomotor movement with the same velocity and muscle length. The muscle was stimulated with the same locomotor pulse train. The largest errors again occurred during the relaxation phase following completion of the stimulus. Stretch during stimulation caused the Hill model to underestimate the relaxation force. Shortening movements during stimulation caused the Hill model to overestimate the relaxation force. These errors may be attributed to the effects of movement on crossbridge persistence, and/or the changing affinity of troponin for calcium between bound and unbound crossbridges, neither of which is well represented in a Hill model. Other sources of error are discussed. The model presented represents the limit of accuracy of a basic Hill-type model applied to cat soleus. The model had every advantage: the parameters were measured from the same muscle for which the locomotion was simulated and errors that could arise in the estimation of activation dynamics were avoided by inverse calculation. The accuracy might be improved by compensating for the apparent effects of velocity and length on activation. Further studies are required to determine to what degree these conclusions can be generalized to other movements and muscles.
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"The overall constitutive behaviour of nonlinearly elastic composites." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 422, no. 1862 (March 8, 1989): 147–71. http://dx.doi.org/10.1098/rspa.1989.0023.
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This work uses the general framework of Hill for the definition of overall mechanical properties of composite materials undergoing large deformations, together with Ball’s constitutive assumption of polyconvexity, and ideas based on a Hashin–Shtrikman variational structure developed recently by Talbot & Willis for nonlinear problems, to obtain rigorous first-order (depending only on the initial volume fractions of the phases) and second-order (depending also on the overall isotropy of the composite) bounds on the overall properties of composite materials with nonlinearly elastic phases. Although the proposed first-order upper bound, or Voigt bound, agrees with a previous result, the corresponding lower bound, or Reuss bound, is significantly tighter than the previously available result. The proposed second-order bounds are completely new, and are of course tighter than the first-order bounds. Additionally, a ‘self-consistent’ estimate is given for the case when the phases are neo-hookean.
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Hanh, Vuong Thi My, Pham Duc Chinh, Vu Lam Dong, and Le Hoai Chau. "Estimates for the elastic moduli of 2D hexagonal-shape orthorhombic crystals with in-plane random crystalline orientations." Vietnam Journal of Mechanics, June 13, 2019. http://dx.doi.org/10.15625/0866-7136/13183.
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Numerical finite element simulations on the homogenization problem for large random-aggregate samples of a particular 2D hexagonal-shape-geometry random polycrystals from the base crystals of orthorhombic symmetry have been performed. At sufficiently large random-aggregate samples, the scatter intervals converge toward the Voigt-Reuss-Hill bounds, and then our recently constructed bounds, which have been specified for the aggregates.
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Chen, Haofeng. "Lower and Upper Bound Shakedown Analysis of Structures With Temperature-Dependent Yield Stress." Journal of Pressure Vessel Technology 132, no. 1 (December 7, 2009). http://dx.doi.org/10.1115/1.4000369.
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Based upon the kinematic theorem of Koiter (1960, “General Theorems for Elastic Plastic Solids,” in Progress in Solid Mechanics 1, J. N. Sneddon and R. Hill, eds., North-Holland, Amsterdam, pp. 167–221) the linear matching method (LMM) procedure has been proved to produce very accurate upper bound shakedown limits. This paper presents a recently developed LMM lower bound procedure for shakedown analysis of structures with temperature-dependent yield stress, which is implemented into ABAQUS using the same procedure as for upper bounds. According to the Melan’s theorem (1936, “Theorie statisch unbestimmter Systeme aus ideal-plastichem Baustoff,” Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A, 145, pp. 195–210), a direct algorithm has been carried out to determine the lower bound of shakedown limit using the best residual stress field calculated during the LMM upper bound procedure with displacement-based finite elements. By checking the yield condition at every integration point, the lower bound is calculated by the obtained static field at each iteration, with the upper bound given by the obtained kinematic field. A number of numerical examples confirm the applicability of this procedure and ensure that the upper and lower bounds are expected to converge to the theoretical solution after a number of iterations.
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Gad, A. I., and X. L. Gao. "Two Versions of the Extended Hill’s Lemma for Non-Cauchy Continua Based on the Couple Stress Theory." Mathematics and Mechanics of Solids, September 9, 2020, 108128652094857. http://dx.doi.org/10.1177/1081286520948576.
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Two versions of the extended Hill’s lemma for non-Cauchy continua satisfying the couple stress theory are proposed. Each version can be used to determine two effective elasticity (stiffness) tensors: one classical and the other higher order. The classical elasticity tensor relates the symmetric part of the force stress to the symmetric strain, whereas the higher-order elasticity tensor links the deviatoric part of the couple stress to the non-symmetric curvature. Four sets of boundary conditions (BCs) are identified using Version I, and three sets of BCs are obtained from Version II of the extended Hill’s lemma, which can all satisfy the Hill–Mandel condition. For each BC set selected, admissibility and average field requirements are checked. Furthermore, the equilibrium is examined for the cases with the kinetic BCs, and the compatibility is checked for the cases with the kinematic BCs. To illustrate the two newly proposed versions of the extended Hill’s lemma, a homogenization analysis is conducted for a two-phase composite using a meshfree radial point interpolation method. The effective elastic constants obtained in this analysis are compared with those predicted by the Voigt and Reuss bounds and computed through a finite element model constructed using COMSOL, which verifies and supports the current method.