Elastic Strain Components Research Articles

Temperature dependence of strains and stresses in undercritical cubic superlattices and heterojunctions.

Strained superlattices and heterojunctions subject to variable temperature exhibit changes in their elastic and/or thermal strain and stress components, relative to their values at room temperature. We consider systems grown in arbitrary directions, with thicknesses smaller than the critical value (undercritical systems). In lowest order, the changes are linear with the temperature. The dependence on temperature of the thermal expansion coefficients is taken into account and shown to improve agreement with data. Criteria are established for predicting the form of such changes in any combination of material constituents. Specific applications are treated in detail and comparison is made with existing data from the literature. The effective linear thermal expansion coefficients of the structure, parallel and perpendicular to the direction of growth, are formulated explicitly. The present results are transcribed to the parallel problem of a hydrostatic pressure in the most general case; this extends previously published work, which refers to material constituents with a lattice misfit smaller than the bulk modulus misfit. The latter assumption is valid for most material combinations but not all. \textcopyright{} 1996 The American Physical Society.

A study of stationary crack-tip deformation fields in thin sheets by computer vision

The in-plane deformation fields near a stationary crack tip for thin, single edge-notched (SEN) specimens, made from Plexiglas, 3003 aluminum alloy and 304 stainless steel, have been successfully obtained by using computer vision. Results from the study indicate that (a) in-plane deformations ranging from elastic to fully plastic can be obtained accurately by the method, (b) for U, $$\varepsilon _{xx} $$ and $$\varepsilon _{xy} $$ , the size of the HRR dominant zone is much smaller than forV and $$\varepsilon _{yy} $$ , respectively. Since these results are in agreement with recent analytical work, suggesting that higher order terms will be needed to accurately predict trends in the data, it is clear that the region where the first term in the asymptotic solution is dominant is dependent on the component of the deformation field being studied, (c) the HRR solution can be used to quantity $$\varepsilon _{yy} $$ only in regions where theplastic strains strongly dominate the elastic strain components (i.e., when $$\left| {\varepsilon _{yy}^{el} /\left. {\varepsilon _{yy}^{pl} } \right|} \right.< .03 - .05$$ ); forV, the HRR zone appears to extend somewhat beyond this region, (d) the displacement componentU does not have the HRR singularity anywhere within the measurement region for either 3003 aluminum or 304 SS. However, the displacement componentV agrees with the HRR slope up to the plastic-zone boundary in 3003 aluminum ( $$5{\text{ }}J/\sigma _0< r< 2{\text{5 }}J/\sigma _0 $$ ) and over most of the region where measurements were obtained ( $$J/\sigma _0< r< 8{\text{ }}J/\sigma _0 $$ ) in 304 SS and (e) the effects of end conditions must be included in any finite-element model of typical SEN specimen geometries to accurately calculate theJ integral and the crack-tip fields.

Calculation of elasto-plastic rigidities using the exact ilyushin yield surface

The analysis of failure loads and mechanisms of plated structures requires knowledge of the elasto-plastic rigidities when elements of the structure are fully plastic. Previous attempts to calculate these rigidities from a single-layer analysis have given results which differ from those obtained by multi-layer analysis. It is shown that by using an exact representation of the Ilyushin yield surface, and by considering the normal direction to that surface, it is possible to obtain a unique stress distribution through the thickness of the plate, which allows the correct value of the rigidities to be found. The reasons for the discrepancy are discussed, and it is concluded that it arises from incompatible assumptions regarding the relative magnitudes of the elastic and plastic strain components. A numerical example is given to illustrate the phenomenon.

Special algorithm for computer calculation of the point group invariants

A computer procedure for the calculation of point group invariants was developed with the use of the reductive projection algorithm. Corresponding symmetrical degrees of the relevant representation were projected onto a unit irreducible representation in order to obtain the invariant linear combinations of the basis functions. A complete set of the icosahedral invariants of the elastic strain components was obtained by means of this procedure for the elastic 6D representation and the elastic constants up to the fourth-order were found.

Pressure tuning of strains and piezoelectric fields in cubic superlattices and heterostructures: Linear effects.

Strained superlattices and heterostructures under hydrostatic pressure exhibit changes in the elastic strain and stress components which, in the lowest order, are linear in the applied pressure. An arbitrary direction of growth is assumed. Criteria are established for predicting the form of such changes in any combination of material constituents. The pressure dependence of the tetragonal distortion is shown to affect the built-in piezoelectric fields, wherever present. Expressions are derived for the effective linear compressibilities parallel and perpendicular to the direction of growth, and for the effective bulk compressibilities and bulk moduli of the system.

Constitutive Model for Ice

A uniaxial constitutive model for ice is developed in which the total strain is taken to be the sum of instantaneous elastic, delayed elastic, and viscous strain components. Nonlinear behavior of selected terms in the model allow the representation of softening due to internal damage. The predictions of the uniaxial model are compared with published test results for constant strain‐rate, constant stress, and constant stress‐rate tests. The model is generalized for plane stress, and predictions are compared with published data for biaxial tests.

Use of Cantilever Specimens to Determine Microcreep Properties

Abstract Cantilever beams were employed to determine the microyield strength (MYS) and the microstress relaxation (MSR) behavior of composites for stable member applications. Cantilever beams were designed and the compliance determined using resistance strain gages. Two different composites were tested: 2124-T6/30% SiC and Al-2.5Mg/20% SiC. A 2124-T6/30%SiC composite, previously tested under axial compression, was used to compare its data with the results of this study using bend specimens. The latter material was chosen because of its lower yield strength and modulus. The cantilever beam geometry is a significantly easier configuration to test and obtain over one per million resolution in strain than equivalent axial compression specimens. This allows cost-effective development of microcreep data using readily available standard mechanical test equipment. Additional test development work may permit this technique to be developed into an ASTM standard. It also offers the opportunity to measure the elastic, plastic, and anelastic strain components to develop engineering materials with improved MYS and MSR properties.

A transversely isotropic inelastic constitutive model of blood vessels.

The present paper is concerned with the formulation of an anisotropic constitutive model of blood vessels typically observed in peripheral arteries. The hysteresis loops under cyclic loadings are assumed to be caused by the loading path dependence and the loading rate dependence. In view of this, the total deformation is divided into the sum of the elastic part and the inelastic part. The elastic part is established by postulating a strain energy density function of an exponential type whose exponent is the quadratic form of the transversely isotropic type with respect to the elastic strain components. The inelastic part, on the other hand, is formulated by modifying the viscoplastic model proposed in the previous paper, that is, the transverse isotropy is incorporated into the kinematic hardening rule. Comparisons of the simulation results with the literature show that the present model can describe the anisotropic behavior appearing in the cyclic hysteresis loop and in the stress relaxation.

Raman intensities near second-order transitions: RP5O14 ferroelastics (where R is a lanthanide).

This paper provides a theory of Raman intensities for soft optic modes at second-order structural phase transitions. It combines the general theory of Fleury [Comments Solid State Phys. 4, 167 (1972)], which gave expressions for the total light scattering cross section (including Raman, Brillouin, and central-mode components) with the theory of Errandonea [Phys. Rev. B 21, 5221 (1980)], which described a specific form of coupling between the optic-mode order parameter Q and certain elastic strain components ${e}_{j}$ for the R${\mathrm{P}}_{5}$${\mathrm{O}}_{14}$ family of ferroelastics. The result of combining these two theories is a set of explicit expressions for the Raman intensities of the soft optic mode alone as a function of temperature, in both phases, and for all Raman tensor components. The predictions are compared with experimental data for both ZZ, YY, XX, and ZX polarizability components above and below ${T}_{c}$ for both ${\mathrm{LaP}}_{5}$${\mathrm{O}}_{14}$ and ${\mathrm{TbP}}_{5}$${\mathrm{O}}_{14}$; good agreement between theory and experiment is obtained.

RHEOLOGICAL CHARACTERIZATION OF APPLE CORTEX

The incremental creep and recovery theory was used to develop a constitutive relationship for apple cortex without assuming the nature and the characteristics of the material. Successive loading and unloading cycles of increasing duration were applied on the cylindrically shaped specimens of apple cortex. The elastic strain component was linear with stress. The viscoelastic and permanent strain components were time dependent and nonlinear functions of stress.Three‐step creep tests and stress‐strain tests at various deformation rates were also conducted on cylindrically shaped specimens of apple cortex. Prediction of strains under these loading histories by using the constitutive equation showed a close agreement between the theoretical and experimental strains. An equivalence between the critical creep stress (found from the incremental creep and recovery tests) and the ultimate stress, obtained from the stress‐strain tests at various deformation rates, were also established.

A theory for the creep of steel fibre reinforced cement matrices under compression

The paper presents a theoretical model to predict the creep of cement matrices reinforced with randomly oriented discrete steel fibres. The theory considers the composite to be represented by an aligned steel fibre which is surrounded by a thick cylinder of the cement matrix. The fibre provides restraint to the flow component of creep of the matrix through the fibre-matrix interfacial bond strength. The delayed elastic strain component of creep is unaffected by the fibre. The fibre-matrix interfacial bond strength,τ, is shown to be primarily a function of the shrinkage of the cement matrix and the radial deformation caused by the sustained axial stress. In addition, the state of stress in the matrix at the interface is suggested to influence greatly the bond strength,τ. The validity of the theory is established by means of experimental data on concrete and mortar matrices reinforced with melt extract and hooked steel fibres, at sustained stress-strength ratios of 0.3 and 0.55. Finally an empirical expression is derived to determine the creep of steel fibre reinforced concrete, based on a knowledge of the creep in unreinforced matrices and fibre size and volume fraction.

A Comprehensive Viscoelasto-Plastic Characterization of Sand-Asphalt Compressive and Tensile Cyclic Loading

Abstract A comprehensive constitutive law for a sand-asphalt mixture subjected to either compressive or tensile cyclic loading is developed. The elastic, plastic, viscoelastic, and viscoplastic strain components are incorporated into the model as they are found to be simultaneously present during the loading process. The various parameters of the model are extracted from a series of repeated uniaxial creep and creep-recovery experiments performed under either compressive or tensile constant stress. The experiments are conducted at constant temperature and at various stress levels, time periods, and number of cycles. The material law presentation is followed by a parametric study that is intended to illustrate the relative quantitative influence of various parameters on the resilient and residual strain components.

Parkes revisited: On rigid-plastic and elastic-plastic dynamic structural analysis

In this paper we re-examine a classical problem of rigid-plastic structural dynamics solved by Parkes [1], namely that of finding the deformations of a beam carrying a mass at its tip which is subjected to a short pulse loading. A considerable body of “elementary rigid-plastic theory” has been developed based on the neglect of elastic strain components, the idealization of perfectly plastic behavior (absence of strain hardening or strain rate sensitivity), and the assumption of linear field equations ignoring effects of geometry changes. This theory provides a valuable conceptual framework for problems of dynamic plastic structural response, although corrections are required for many practical applications; for example, Parkes introduced a correction for effects of high strain rates shown to be needed by his experiments on steel beams. The neglect of elastic strains — elastic moduli taken infinite — is a crucial assumption, and its validity is the subject of the present paper. We study it by making comparison between “exact” numerical solutions furnished by an advanced finite element computer program for an elastic-plastic beam, and the rigid-plastic solution slightly modified to allow for large deflections. Further insight is gained by comparisons also with a simplified elastic-plastic approach based on treatment of elastic and plastic action in artificially separated stages. The main conclusions are that the initial travelling-hinge phase of the rigid-plastic solution is essentially a fiction, but that the subsequent mode form phase is a prominent and significant feature of the late-time part of the “actual” elastic-plastic response.

Strength and deformability of clayey soils in tension

1. The failure of soil in the investigated range of initial states (density and moisture content) and loading paths is determined by the mechanism of breaking (brittle fracture), and the elastic component of limit strain in tension e 3 (e)* is constant.

Kinematics of deformation processes in materials subjected to finite elastic-plastic strains

Some aspects of the analysis of elastic-plastic deformation processes with both elastic and plastic finite strain components are studied. The discussion is intentionally restricted to the geometric description of such situation. As a starting point the intermediate configuration concept is accepted but some important generalizations to it appear to be necessary. Basing upon this concept the properly invariant forms of strain and rate of strain measures are defined being referred to the the reference, intermediate and actual configurations, respectively. The applicability of the different strain measures to the constitutive equations is discussed.

Inelastic plane-wave propagation in anisotropic rocks

The equations for combined elastic and inelastic plane-wave propagation in anisotropic rocks are developed in terms of familiar laws of plasticity. The total strain is the sum of the elastic and plastic contributions. The elastic component of strain is related to the stress according to Hooke's law, and the plastic strain is obtained from the yield function and an associated flow rule. The general equations apply to materials of any symmetry, but specific calculations are performed for materials of transverse isotropy related to parallel planes of weakness, i.e., bedding planes. Wave speeds, stress amplitudes, etc., depend on the direction of wave propagation with respect to the bedding planes. Velocities of the inelastic waves differ in loading and unloading. Analytical and numerical solutions to the problem of a finite-duration step load applied over a planar surface are presented. It is shown that wave attenuation is also dependent on wave propagation direction.

The effect of frequency on the cyclic strain and fatigue behavior of cast rené at 1600° F

Controlled strain fatigue tests have been carried out for two heat treatments of cast Rene 80 at 1600°F at several frequencies and strain ranges. Results were fitted to frequency-modified fatigue equations for elastic and plastic strains using regression analysis. Save for a more pronounced cyclic softening, little difference in fatigue behavior was found in the two heat treatments. This material exhibits a normal frequency behavior for the elastic strain component of the total strain, and an inverse frequency effect for the plastic strain component, resulting in a negligible frequency effect on total strain in the high strain, short life regime. This behavior is attributed to the persistence of a transgranular mode of crack propagation in preference to the more usual intergranular mode in all tests including those at frequencies as low as 0.03 cpm. On the other hand, crack nucleation was more characteristic of cast nickel-base superalloys at 1600°F, that is, by oxide ridging at surface grain boundaries. A low transition fatigue life was found, such that for fatigue lives greater than 1000 cycles, elastic behavior essentially prevails.

The mechanical properties of leather

The many uses of leather have largely relied on the range of mechanical properties which it can provide, according to the raw material employed in its manufacture and the manufacturing processes themselves. The contrasting behaviour of a stiff sole leather and of a fine gloving leather exemplify this point. The last twenty years have seen intensive investigation of many mechanical properties of leather and the design of test methods now accepted internationally. At ambient temperatures and humidities most types of leather show mainly elastic behaviour, although delayed elastic effects may give the semblance of plasticity. The stress relaxation-time relation for constant linear strain shows the stress decaying linearly with log (time). The stress decay becomes discontinuous after sufficient time. The stress-strain relation for extension of leather strips is often markedly non-linear even at low strains (<2%). Two dimensional extension of leather has been analysed using an instrument allowing independent extension in two perpendicular directions. To a first approximation each stress component is linearly related to the two elastic strain components in the perpendicular directions. As with other materials of biological origin, the mechanical behaviour of leather varies from place to place in the skin, not only over its area, but also through its thickness. The extent of variation is briefly discussed and related to the underlying fibre structure. Leather which has been strained and then subjected to either heat alone or heat and moisture, shows much more extensive plasticity than occurs at lower temperatures. This behaviour has been used to enable leather to be given appropriate shapes, as in the heat setting of upper leathers. Quantitative studies of heat setting are reported and the influence of such variables as temperature, moisture content of the applied air stream, the air stream velocity and the duration of treatment are discussed. The plastic deformation obtained in this way is contrasted with “run” in gloving leather.

Elastic-Plastic Deformation at Finite Strains

In some circumstances, elastic-plastic deformation occurs in which both components of strain are finite. Such situations fall outside the scope of classical plasticity theory which assumes either infinitesimal strains or plastic-rigid theory for large strains. The present theory modifies the kinematics to include finite elastic and plastic strain components. For situations requiring this generalization, dilatational influences are usually significant including thermomechanical coupling. This is introduced through the consideration of two coupled thermodynamic systems: one comprising thermoelasticity at finite strain and the other the irreversible process of dissipation and absorption of plastic work. The present paper generalizes a previous theory to permit arbitrary deformation histories.

Finite-Strain Elastic—Plastic Theory with Application to Plane-Wave Analysis

In the generation of plane waves in metal plates by detonation of contact explosives, the pressures reached can be sufficiently high to produce finite elastic and plastic strain components, and appreciable changes in temperature due to thermomechanical coupling effects. A theory of elastic—plastic deformation is developed to include these conditions. Finite elastic and plastic deformations are represented in terms of the initial and final configurations of the body, and an unstressed configuration subject only to plastic flow. In general, this configuration cannot be represented by a continuous displacement function. The analysis of the kinematics of these three states is carried out for general deformation history. The discussion of the application of finite deformation thermoelastic theory is limited to the conditions in a plane wave with fixed principal directions. Plasticity theory with the consideration of variable temperature is also restricted to the plane-wave situation. The influence of the thermodynamic irreversibility of plastic flow is included in an entropy production term which modifies the temperature variation. Application to the plane-wave propagation problem is discussed.