# Showing papers in "Journal of The Mechanics and Physics of Solids in 1965"

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TL;DR: In this article, the elastic moduli of two-phase composites are estimated by a method that takes account of the inhomogeneity of stress and strain in a way similar to the Hershey-Kroner theory of crystalline aggregates.

Abstract: T he macroscopic elastic moduli of two-phase composites are estimated by a method that takes account of the inhomogeneity of stress and strain in a way similar to the Hershey-Kroner theory of crystalline aggregates. The phases may be arbitrarily aeolotropic and in any concentrations, but are required to have the character of a matrix and effectively ellipsoidal inclusions. Detailed results arc given for an isotropic dispersion of spheres.

3,289 citations

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TL;DR: In this paper, the breaking strength of tungsten or molybdenum wires, uniaxially aligned in a copper matrix, was found to be a linear function of the wire content.

Abstract: T ensile tests at a variety of temperatures have been carried out on composites consisting of tungsten or molybdenum wires, uniaxially aligned in a copper matrix. Both continuous and discontinuous wires have been used, and both brittle and ductile tungsten wires. It is found that the breaking strength is a linear function of the wire content. A simple theory to explain this is developed and auxiliary experiments to check the theory are described. Some simple predictions about the behaviour of fibre reinforced metals are made from the results.

2,122 citations

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TL;DR: In this paper, a heuristic analysis is given for the determination of the elastic moduli of a composite material, the several constituents of which are each isotropic and elastic, and the results are intended to apply to heterogeneous materials composed of contiguous, more-or-less spherical grains of each of the phases.

Abstract: A heuristic analysis is given for the determination of the elastic moduli of a composite material, the several constituents of which are each isotropic and elastic. The results are intended to apply to heterogeneous materials composed of contiguous, more-or-less spherical grains of each of the phases.

1,571 citations

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TL;DR: In this article, the internal inhomogeneities of stress and strain in an arbitrarily deformed aggregate of elasto-plastic crystals are evaluated theoretically using a tensor constitutive law of a general kind.

Abstract: T he internal inhomogeneities of stress and strain in an arbitrarily deformed aggregate of elasto-plastic crystals are evaluated theoretically. A tensor constitutive law of a general kind is assumed for the individual crystals. The implied mechanical properties of the aggregate as a whole are estimated by means of a self-consistent model akin to one used by H ershey (1954), K roner (1958, 1961) and B udiansky and W u (1962), but differing in significant respects.

1,347 citations

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TL;DR: In this paper, the plane strain bulk modulus and the two shear moduli of multiphase transversely isotropic fiber reinforced materials of arbitrary transverse phase geometry, are bounded from above and below in terms of phase moduli and phase volume fractions.

Abstract: T he Plane strain bulk modulus and the two shear moduli of multiphase transversely isotropic fibre reinforced materials of arbitrary transverse phase geometry, are bounded from above and below in terms of phase moduli and phase volume fractions. Particular attention is given to the important special case of two-phase fibre reinforced materials.

484 citations

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TL;DR: In this article, the overall moduli of an arbitrary fiber composite with transversely isotropic phases are estimated by a method previously used for dispersion composites and polycrystals.

Abstract: T he overall moduli of an arbitrary fibre composite with transversely isotropic phases are estimated by a method previously used for dispersion composites and polycrystals. In all cases the estimates lie between the best bounds available at present. For practical purposes the method also appears to be sufficiently reliable in detail, except when either the concentration or rigidity of the fibres is extreme.

484 citations

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TL;DR: In this article, it was shown that for a work-hardened material the hardness increases with increasing cone angle for cone angles above 105° and that the deformation process appears to be a radial compression similar to that described by S amuels and M ulhearn.

Abstract: Q uasi-static indentations with diamond cones have been made in specimens of copper and mild steel work-hardened by varying amounts. The relationship between the hardness so found, the cone angle, and the yield stress is given. The results agree well with the indentation theory of L ockett (1968a) based on classical slip line analysis. This shows that for a work-hardened material the hardness increases with increasing cone angle for cone angles above 105°. Lockett's theory is degenerate below 105° and the experiments here also show a change in deformation mode in this region. Although this agreement with theory appears satisfactory, the deformation process appears to be substantially different from that implied in Lockett's analysis. For large angle cones the deformation resembles a ‘radial’ compression similar to that described by S amuels and M ulhearn (1957): for small cone angles the deformation resembles more clearly the classical slip-line form given by Lockett, though this is precisely the region where the theory becomes degenerate. The effective strain produced by each cone during the indentation process itself is determined for each cone and it is shown that it is possible to construct the stress-strain curve of a given material from a series of cone indentations.

315 citations

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TL;DR: In this paper, the equilibrium positions of an edge dislocation inside a circular inclusion were investigated using elasticity solutions, and the conditions under which the centre of the inclusion is stable equilibrium position were given.

Abstract: Equilibrium positions of an edge dislocation inside a circular inclusion are investigated using elasticity solutions. The conditions under which the centre of the inclusion is stable equilibrium position are given, and it is shown that the equilibrium positions away from the centre are unstable. The results indicate that the Poisson's ratios play an important role in characterizing the different types of behaviour.

93 citations

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TL;DR: The static perturbation technique is the basic tool in a wide class of stability investigations in solid mechanics as discussed by the authors, and perturbations are in terms of a parameter representing progress along any prospective equilibrium path.

Abstract: A ttention is directed to the fact that the static perturbation technique is the basic tool in a wide class of stability investigations in solid mechanics This seems often to have been overlooked, sometimes with disadvantage The perturbations are in terms of a parameter representing progress along any prospective equilibrium path The method is illustrated by applying it to general conservative (and some non-conservative) finite-dimensional systems The leading perturbation problems are worked out in detail for a model of the elastic/plastic column, and a new exact finite deflection solution of the latter is obtained for comparison

79 citations

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TL;DR: In this article, the loss of stability of a general conservative structural system described by n generalized coordinates, a loading parameter, and an imperfection parameter, is studied and explicit expressions for the derivatives of the post-buckling path of the perfect system and for derivatives relating the critical load of an imperfect system to the magnitude of the imperfection parameters are derived intrinsically without resort to power-series expansions.

Abstract: The loss of stability of a general conservative structural system described by n generalized coordinates, a loading parameter, and an imperfection parameter, is studied. Attention is restricted to discrete critical points of the ‘perfect’ system and three branching points which might be described as asymmetric, stable-symmetric and unstable-symmetric are examined in detail. For each point explicit expressions for the derivatives of the post-buckling path of the ‘perfect’ system and for the derivatives relating the critical load of an ‘imperfect’ system to the magnitude of the imperfection parameter are derived intrinsically, without resort to power-series expansions. The general theory is finally applied to three ‘models’ designed to illustrate the salient features of the three branching points under consideration.

71 citations

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TL;DR: In this paper, a new method of limit analysis of structures is introduced whereby a continuous structure made of a perfectly plastic material is replaced by a discrete mathematical model and the stress analysis problem is then interpreted as a linear programming problem.

Abstract: A new method of limit analysis of structures is introduced whereby a continuous structure made of a perfectly plastic material is replaced by a discrete mathematical model. The stress analysis problem is then interpreted as a linear programming problem. By means of this method a lower bound is found for the plastic collapse load of a square plate, both clamped and simply supported at the edge, under uniform hydrostatic pressure.

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TL;DR: In this article, the authors used the theory of the thermally activated deformation of metals, and calculated activation volumes for the single crystals and polycrystalline specimens of high purity aluminium were tested in compression at strain rates to 500 sec−1 using the split Hopkinson pressure bar method.

Abstract: S ingle crystal and polycrystalline specimens of high purity aluminium were tested in compression at strain rates to 500 sec−1 using the split Hopkinson pressure bar method. Using the theory of the thermally activated deformation of metals, activation volumes for the single crystals were calculated and found to be dependent upon both orientation of the crystal with respect to the loading axis and the average strain. For the same strain, activation volume for the polycrystalline material is near the average of the variously oriented single crystals, indicating that the same rate controlling mechanism is operative. The addition of impurities in aluminium leads to a reduction in rate sensitivity as evidenced by an increase in activation volume. Finally, the application of thermal activation theory to high strain-rate testing is discussed.

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TL;DR: In this article, a comparison between Eshelby's formulations of the static inclusion poblem in terms of point-forces and dislocations is made, and the difference between the physical processes to which they correspond is demonstrated by considering the stress waves arising from the sudden creation of an inclusion by either process.

Abstract: A comparison is made between Eshelby's formulations of the static inclusion poblem in terms of point-forces and in terms of dislocations. Although the two formulations are mathematically equivalent, the difference between the physical processes to which they correspond is demonstrated by considering the stress waves arising from the sudden creation of an inclusion by either process. Associated energy effects are also examined. In the course of the analysis, general relations for time-dependent spontaneous deformations and dislocations are discussed, and it is shown how relations for the latter may be deduced from relations for the former.

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TL;DR: In this article, the creep and recovery behaviour of polyproylene fibres of different structure was analysed using the approach proposed previously (W ard and O nat 1963) and used to define the nature of the hereditary functionals which describe the non-linear viscoelastic behaviour.

Abstract: I n this paper the creep and recovery behaviour of several polyproylene fibres of different structure are described. The results were analysed using the approach proposed previously (W ard and O nat 1963) and the creep and recovery used to define the nature of the hereditary functionals which describe the non-linear viscoelastic behaviour. It was found that higher order functionals become increasingly important with increasing stress level and with increasing time of creep or recovery. Although no detailed attempts have been made to relate the viscoelastic behaviour to the structure of the fibres, it appears that the creep and recovery behaviour arc systematically influenced by the degree of over-all molecular orientation, by the crystalline structure and by the fibre molecular weight.

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TL;DR: In this article, the theoretical yield surfaces under combined incremental stress are computed from these stress fields, which satisfy conditions of equilibrium, continuity and the single crystal stress-strain relation throughout the aggregate.

Abstract: H eterogeneous slip and stress fields of a fine-grained polycrystalline aggregate of differently oriented crystals of long square cylinders were calculated by L in (1964) under uniaxial loadings from single crystal slip characteristics. These stress fields satisfy conditions of equilibrium, continuity and the single crystal stress-strain relation throughout the aggregate. From these stress fields the theoretical yield surfaces under combined incremental stress are here computed. Aggregates of crystals with the same isotropic elastic constants have been found to have a plastic potential coinciding with the yield surface. The initial yield surface during loading beyond the elastic range does not expand uniformly as given by isotropic hardening theories nor shift as a whole as predicted by the kinematic hardening, but changes both in shape and size as in linear piecewise plasticity. Theoretically, a vertex exists on the yield surface. At the vertex, the direction of the incremental plastic strain vector is not uniquely determined, but lies within the cone of normals to the tangent planes at the vertex and depends on the direction of the incremental stress vector. Incremental plastic strain vectors are theoretically calculated for different incremental stress ratios. Experimentally observed yield surface is actually the surface in stress space giving finite measurable incremental plastic strain. These surfaces theoretically have no vertex but a rounded nose of large curvature at the loaded point as observed in a number of experimental observations.

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TL;DR: In this paper, the effect of small imperfections on the buckling behavior of elastic structural systems with n degrees of freedom is discussed using the energy principles of mechanics, and the concept of generalized co-ordinates.

Abstract: Using the energy principles of mechanics, and the concept of generalized co-ordinates, the effect of small imperfections on the buckling behaviour of elastic structural systems with n degrees of freedom is discussed. Attention is restricted to conservative systems that buckle at a point of bifurcation when no imperfections are present. Two different classes of imperfections, namely major and minor imperfections, and their effect on the buckling behaviour are established. Expressions relating these imperfections with the critical load and critical deflection of the system are derived. The results of a few experiments on a model structure are discussed and give some qualitative verification of the theory.

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TL;DR: In this article, the dynamic Young's modulus of a composite of tungsten containing twenty per cent of silver by volume was evaluated up to 1020°C and it was shown that the temperature dependence of the modulus as well as the absolute value of modulus is closely approximated by the upper bound theory of H ashin and S htrikman.

Abstract: T he dynamic Young's modulus of a composite of tungsten containing twenty per cent of silver by volume was evaluated up to 1020°C. In the temperature range 25°C–800°C it was shown that the temperature dependence of the modulus as well as the absolute value of the modulus is closely approximated by the upper bound theory of H ashin and S htrikman . Above 800°C the modulus decreases much more rapidly than predicted by the theory. This rapid drop in modulus is believed to be associated with interphase boundary shearing between the tungsten and silver phases. A large discontinuity in modulus (over 10% change) is observed at the melting point of silver. Above the melting point of silver the modulus of the composite is about equal to the modulus of a tungsten sample containing an equivalent percentage of pores suggesting that there is no strengthening of tungsten from the presence of molten silver.

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TL;DR: In this paper, the application and limitations of a one-dimensional theory of bar impact, particularly as it relates to a specific experimental configuration known as the thin wafer or split Hopkinson pressure-bar technique, were discussed.

Abstract: Many different impact test arrangements have been utilized to investigate the dynamic properties of metals. This paper discusses the applications and limitations of a one-dimensional theory of bar impact, particularly as it relates to a specific experimental configuration known as the ‘thin wafer’ or ‘split Hopkinson pressure-bar’ technique. An analysis of this experiment, which utilized a bilinear approximation for the stress-strain curve of the aluminium being tested, led previous investigators to demand the incorporation of strain rate effects in the constitutive relation of metals. However, the present nonlinear analysis of the same test demonstrates that the experimental data for this particular metal can be predicted from a strain rate independent theory. This new analysis indicates the influence of assumptions which neglect certain stress wave propagation phenomena. These mechanical effects are shown to be an important contribution to dynamic observations which had been attributed only to intrinsic material properties , such as strain rate dependence.

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TL;DR: In this article, the Coulomb-Mohr yield condition in plastic flow is considered, and the velocity equations used are those proposed by S pencer (1964) based on the assumptions of material incompressibility and the postulate that deformation occurs by shear along the characteristic curves of the stress equations.

Abstract: Steady radial flow of an isotropic, cohesive and frictional, non-hardening rigid/plastic solid through a rough wedge-shaped channel under conditions of plane strain is considered; the weight of the material is neglected. The stresses are assumed to satisfy the Coulomb-Mohr yield condition in plastic flow, and the velocity equations used are those proposed by S pencer (1964) based on the assumptions of material incompressibility and the postulate that deformation occurs by shear along the characteristic curves of the stress equations. General stress and velocity solutions are presented. It is shown that the assumption of a Coulomb law of friction at the walls of the channel is incompatible with conditions for radial flow unless the material is cohesionless. Some stress and velocity distributions for cohesionless materials are calculated.

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Brown University

^{1}TL;DR: In this paper, the change in shape of spherically divergent stress pulses in three viscoelastic olids, namely polyethylene, polymethylmethacrylate and polystyrene, have been made.

Abstract: M easurements of the change in shape of spherically divergent stress pulses in three viscoelastic olids, namely polyethylene, polymethylmethacrylate and polystyrene, have been made. The stress pulses were produced by the detonation of small explosive charges on blocks of these materials and the pulses were detected with condenser microphones. The stress amplitudes were sufficiently small for the theory of linear viscoelasticity to apply and the attenuation and dispersion of the pulses enabled values to be obtained for the real and imaginary parts of the bulk modulus. It was found that for both polyethylene and polymethylmethacrylate the ‘bulk loss’ was about one fifth of the mechanical loss in shear, while for polystyrene the bulk losses were too small for an accurate quantitative estimate to be made. It was shown, however, that for this material, also, the assumption that the bulk loss was a constant fraction of the loss in shear gave good agreement with the observed results. It is suggested that, since the bulk loss appears to follow the shear loss, the same microscopic dissipative processes may be responsible for both types of loss, since, on a molecular scale, density changes of long chain polymers must be accompanied by local shear.

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TL;DR: In this article, the effect of nonlinearity at an acceleration wave of arbitrary initial form is discussed on the assumption that the material can be modelled by a hypo-elastic equation of state.

Abstract: The effect of non-linearity at an acceleration wave of arbitrary initial form is discussed on the assumption that the material can be modelled by a hypo-elastic equation of state. Closed form expressions arc obtained for the variation in strengths of isochoric and irrotational waves in the special case when the wave propagates into a uniform region. For an irrotational wave it is shown that the variation in strength predicted by linear theory is in error. Either shocks form or the deformation ‘forgets’ the details of the disturbance at any finite time. Linear theory becomes progressively worse with increasing time. The technique developed can be extended to more complicated wave phenomena.

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TL;DR: In this paper, the oblique shadow moire method was employed to measure the deflexions of plates and the equipment used was simple, reducing considerably the scatter in the results.

Abstract: An initial and three subsequent yield surfaces were investigated for aluminium 6061-T 651. The study was approached using theory of plates rather than thin-walled tubes because it was desired to investigate the entire two-dimensional space. The oblique shadow moire method was employed to measure the deflexions of plates and the equipment used was simple, reducing considerably the scatter in the results. Since the plate thickness was a variable as far as strength criteria were concerned the study was limited to the relative shape of yield surfaces.
Combinations of various ratios of bending moments were obtained in the second and fourth quadrants of the two-dimensional bending moment space by using rhomboid plates loaded anticlastically. The first and third quadrants were investigated using a superposition of a square plate supported at the four corners and loaded at the centre, and a square plate supported at two diagonally opposite corners and loaded at the centre. This approach produced combinations of various ratios of bending moments in plates loaded synclastically.
To establish the initial yield surface for comparing it to subsequent yield surfaces, a series of specimens were loaded anticlastically, synclastically or subjected to uniaxial bending. The principal bending moments M1 and M2 at yielding were plotted in the two dimensional bending moment space. The shape of the initial locus fell between the Tresca and Mises loci.
A first subsequent yield surface was established by pre-loading a series of plates anticlastically into the plastic region with M1 = − M2. A second subsequent yield surface was defined in the same manner as the first subsequent surface except that the direction for pre-bending into the plastic range was such that M1 ≈ − 12M2. A third partial subsequent yield surface was found in the fourth quadrant only. The pre-loading path in this case was the same as in the first subsequent surface (M1 = − M2), the only difference being that the pre-loading was advanced further in the plastic region and it was followed by a recovery of the plates for three months at room temperature.
The initial and the three subsequent yield surfaces were compared and important features of the behaviour of metals subjected to any type of biaxial loading and to any straining history were disclosed.

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TL;DR: In this article, the authors developed general equations governing the lateral and in-plane buckling of a gridwork composed of a network of intersecting beams and showed that for certain boundary conditions simple and theoretically exact solutions for these equations may be derived.

Abstract: A gridwork (or grillage) composed of a network of intersecting beams can buckle if axial loads are applied to the members. General equations governing the lateral and in-plane buckling of such systems are developed. It is shown that for certain boundary conditions simple and theoretically exact solutions for these equations may be derived.

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TL;DR: In this paper, the principal (stress, strain rate) lines, characteristics of the velocity solution of any kinematically determined axially-symmetric plastic deformation problem which is governed by Tresca's yield condition and the associated flow rule, are shown.

Abstract: The principal (stress, strain rate) lines, characteristics of the velocity solution of any kinematically determined axially-symmetric plastic deformation problem which is governed by Tresca's yield condition and the associated flow rule, are shown to be also characteristics of the stress solution, even if arbitrary volume forces (e.g. inertia) are allowed for. The characteristic stress equations are deduced and applied to two illustrative examples, already dealt with in a former paper.

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TL;DR: In this article, a sensitive etching technique was used to detect the earliest stages of plastic yielding from the observation of individual slip dislocations, and a comparison showed that there is no significant elastic superstressing effect; slip begins in the notch at the load predicted from the elastic stress concentration factor and from the tensile yield stress.

Abstract: N otch bend tests have been made on 3 % Si steel at 295°K and 77°K, and compared with plain tensile tests on the same material. A sensitive etching technique has been used to detect the earliest stages of plastic yielding from the observation of individual slip dislocations. The comparison shows that there is no significant elastic superstressing effect; slip begins in the notch at the load predicted from the elastic stress concentration factor and from the tensile yield stress. Elastic constraint keeps the plastic strain in the notch very small until the general yield load is approached. The fracture strength is high, at 77°K, because cleavage cracking is preceded by twinning, in the root of the notch, and twinning is preceded by slip.

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TL;DR: In this paper, the effect of lateral inertia on the propagation of an elastic-plastic pulse along a bar is investigated. And the authors conclude that some observations set forth as proof of the existence of a strain-rate effect might equally be explained, at least qualitatively, on the basis of a lateral inertia effect.

Abstract: I t is reasonable to expect that under rapid dynamic loading lateral inertia effects would play a significant role in the propagation of an elastic-plastic pulse along a bar. The one-dimensional approximate equations of elasticity are modified to include the effect of lateral inertia. These equations are solved analytically for step function loading. The equations are then differenced and the stability of the resulting difference equations is discussed. Numerical solutions of the difference equations are compared with the analytic solutions. The equations are then modified to allow for plastic deformation and hysteresis in addition to the lateral inertia effect and are solved numerically. These solutions are compared with the results of strain-rate dependent and strainrate independent theories. It is concluded that some observations set forth as proof of the existence of a strain-rate effect might equally be explained, at least qualitatively, on the basis of a lateral inertia effect.

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TL;DR: In this article, the propagation of stress waves from a unit step impact pulse concentrated obliquely at a point on the upper surface of an infinite elastic plate whose thickness is of the order of magnitude of a wave length is analyzed.

Abstract: A n analysis is presented of the propagation of stress waves from a unit step impact pulse concentrated obliquely at a point on the upper surface of an infinite elastic plate whose thickness is of the order of magnitude of a wave length. An expansion method of Cagniard is employed to separate successively reflected waves of increasing time delay. The problem of performing a Laplace transform and its inversion is solved by use of Cagniard's artifice. Stress expressions and sets of numerical principal stress values are obtained for the stress state at points along the z -axis, for oblique angles of incidence of 30° and 60°. Stress surfaces are drawn to depict the principal stress state variation with time and the jump discontinuity wave fronts signalling respective wave arrivals.

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TL;DR: In this article, the dynamic Young's modulus-composition relation in eutectic alloy systems, without terminal solid solubility, follows either a linear relation or a less than linear relation, where the moduli differences between the two phases are small or when the microstructure of the alloy exhibits a strong fibred texture such as is obtained in a cold drawn material.

Abstract: T he dynamic Young's modulus-composition relation in eutectic alloy systems, without terminal solid solubility, follows either a linear relation or a less than linear relation. Essentially linear relations are obtained for systems where the moduli differences between the two phases are small or when the microstructure of the alloy exhibits a strong fibred texture such as is obtained in a cold drawn material. Eutectic systems containing a random distribution of phases with large moduli differences between the phases exhibit less than linear relations between modulus and composition. Experimental data on modulus of cast Ag-Pb alloys are in close agreement with the upper bound theory of Hashin and Shtrikman.

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TL;DR: In this article, a comparison between the deflections and internal loads induced in a plate and in the equivalent gridwork is made, based on which the accuracy of a gridwork analysis can be estimated.

Abstract: T he rectangular mesh gridwork analogy for plates is formally established by combining finite difference calculus and differential calculus methods. A comparison is made between the deflections and internal loads induced in a plate and in the equivalent gridwork. From this comparison the accuracy of an equivalent gridwork analysis can be estimated.

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TL;DR: In this paper, a slab of material of infinite extent and of a given thickness is subjected to a pressure step travelling along one face of it at a uniform speed greater than the wave propagation speeds in the material.

Abstract: A slab of material of infinite extent and of a given thickness is subjected to a pressure step travelling along one face of it at a uniform speed greater than the wave propagation speeds in the material. The response of various material media to such high-velocity loadings forms the subject of this study. The present paper is the first of a series concerning this problem and deals with the case of a classical linear elastic material and of a general linear viscoelastic material. The deformation and stress fields in these materials are computed and compared. In particular, the effects due to stress relaxation, the reflection characteristics of the stress waves at the free surface, the decay of the wave strength, etc. are discussed in detail.