Showing papers on "Linear elasticity published in 1977"

Shear behaviour of elastohydrodynamic oil films

Abstract: A simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. The model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain. The nonlinear viscous function can take any convenient form. It has been found that an Eyring 'sinh law' fits the measurements on five different fluids, although the higher viscosity fluids at high pressure are well described by the elastic/perfectly plastic equations of Prandtl-Reuss. The proposed equation covers the complete range of isothermal behaviour: linear and nonlinear viscous, linear viscoelastic, nonlinear viscoelastic and elastic/plastic under any strain history. Experiments in support of the equations are described. The nonlinear Maxwell constitutive equation is expressed in terms of three independent fluid parameters: the shear modulus $G$, the zero-rate viscosity $\eta $ and a reference stress $\tau _{0}$. The variations of these parameters with pressure and temperature, deduced from the experiments, are found to be in broad agreement with the Eyring theory of fluid flow.

Conservation laws in elasticity of the J-integral type

Abstract: Conservation laws which are expressible as functionals linear in the strain energy and its derivatives are laws of the same type as theJ-integral. For finite elastic deformations of homogeneous bodies, relations between the conservation laws are shown through the use of inverse deformation results. Completeness of the laws are established for homogeneous materials and for materials whose strain energies satisfy objectivity, isotropy, or are homogeneous functions. Laws for a class of membranes inflated by pressure are derived and applied to a cylindrical membrane. For infinitesimal deformations of linear elastic bodies, new laws which relate two independent equilibrium states are presented and applied to the problem of a line crack in a plate under mixed-mode loading conditions. A relation is shown to exist between theJ-integral and the reciprocal work theroem of Betti.

Asphalt pavement design--the shell method

Abstract: The method is based on a model in which the pavement structure is regarded as a linear elastic multi-layered system in which the materials are characterised by their modulus of elasticity and Poisson's ratio. The computer program BISAR is used to compute all stresses, strains and displacements at any point in the system under any number of vertical and/or horizontal surface loads. In this way, the primary design criteria have been established, i.e. the compressive strain at the top of the subgrade and the horizontal tensile strain in the asphalt. Secondary criteria such as permissible stresses in cementitious base layers, permanent deformation of the asphalt, etc., are also included. The permissible value for compressive subgrade strain has been derived from analysis of AASHO Road Test sections and structures conforming to CBR design. The permissible asphalt strain was determined from extensive laboratory measurements for various mix types at different stiffness moduli of the asphalt. In the application of the asphalt fatigue criterion allowance is made for the influence of the transverse distribution of wheel loads and for effects of healing and intermittent loading. The traffic data are converted into an equivalent number of standard design axle load applications. To introduce the influence of the ambient temperature a procedure has been developed to relate the mean annual or monthly air temperature to an effective asphalt temperature, depending on the thickness of the asphalt. The moduli of subgrade and unbound base layers should be determined at appropriate stress levels whereby the latter modulus is a function of the subgrade modulus. It is demonstrated that the modulus of a given mix, relevant for the structural design, can also be derived with sufficient accuracy using a nomograph to provide a practical system for road engineers, sets of design charts have been prepared from which combinations of thicknesses of the asphalt and unbound base layers can be derived for various mean annual temperatures, for a number of typical mixes and for various sugbrade moduli. Special attention is paid to a method of predicting the permanent deformation (rut depth) of the asphalt layers during the expected service life of the pavement. The application of the design method to the design of pavements for aircraft with multiple wheels is also dealt with the various laboratory tests and full scale road trials carried out to investigate the validity of the design method have been summarised. The practical use of the design method is illustrated by means of some examples. /Author/

The finite deformation of an inhomogeneity in two-dimensional slow viscous incompressible flow

Abstract: The theory of the elastic fields round ellipsoidal inclusions and inhomogeneities together with the well-known analogy between linear elasticity and slow incompressible viscous flow are used to develop the governing equations for the finite deformation of a viscous ellipsoidal inhomogeneity in a viscous matrix undergoing a general linear time-dependent flow at infinity. The governing equations are then solved for an inhomogeneity in the form of an elliptic cylinder in a linear two-dimensional flow whose stream lines at infinity are steady. The behaviour of the inhomogeneity under pure shear and simple shear is considered in detail and it is shown that the boundaries of certain deforming inhomogeneities remain unchanged during simple shear. These steady inhomogeneities can appear also in general linear two-dimensional applied flows. In such flows the behaviour is influenced both by the initial shape and orientation of the inhomogeneity and by its viscosity. Inhomogeneities which are rather viscous or subject to an applied flow with high vorticity deform periodically, while most others elongate indefinitely. The patterns of behaviour may be described in terms of a number of regimes which can be classified by considering the singularities of the differential equations governing the variations of shape and orientation of the inhomogeneity, or, equivalently, by studying the invariants of the corresponding one-parameter Lie groups. Finally, some obvious extensions of the treatment are indicated. These make it possible to consider inhomogeneities (such as holes) whose volume does not remain constant, and which have constitutive relations more general than those of a linear viscous material. In this paper we discuss the slow finite deformation of a viscous ellipsoidal inhomogeneity in a matrix of different viscosity. The problem of the deforming inhomogeneity in viscous flow has been treated by a number of workers, but usually with the main interest either in the phenomenon of the ultimate bursting of a drop or in the calculation of the properties of a suspension of such inhomogeneities; for a recent brief review see Hinch (1975). The theories have thus not been concerned primarily with the progressive finite deformation of the inhomogeneity in nonsteady flow, but have dealt with an inhomogeneity which undergoes small or limited

Matched asymptotic expansions in nonlinear fracture mechanics—III. In-plane loading of an elastic perfectly-plastic symmetric specimen

Abstract: T he matched expansion method, introduced by the authors in two earlier papers (1976) devoted to mode III loading, is applied to the practically important case of mode I loading of a symmetric specimen. The method allows the linear elastic far-field to be considered separately from the elasto-plastic near-tip field, except for coupling through a set of parameters that are determined explicitly in the matching. The effects of the plasticity are thus found, once and for all, from the solution of a set of standard elasto-plastic problems for a semi-infinite crack in an infinite body, whose properties may be tabulated. The solution for any particular specimen geometry and loading then follows from a small set of linear elastic solutions for the specimen, which define, through coefficients γij appearing in their near-tip expansions, all the parameters in the “inner” and “outer” solutions. The effects of plasticity appear in these parameters only through a set of constants Cti that define the far-field expansions of the “inner” (near-tip) solutions: they are material constants, depending upon the constitutive relation for the material, but not upon specimen geometry and loading. The J-integral, being obtainable from the far-field, is expressed as an explicit asymptotic series in the loading parameter e, whose coefficients are given as functions of the “elastic” parameters γij and the material constants Ci. It is demonstrated that a plastic-zone correction term, ry, can be chosen to yield a two-term asymptotic expansion for J; the value of ry depends upon the yielding model only through the constant C1. The Dugdale (1960) model of yielding is treated, as a simple example for which all calculations can be performed analytically, and for which exact solutions are available for comparison. Finally, the near-tip solutions are constructed for a material obeying the Mises yield criterion and associated flow-rule, using a specially developed finite element program. The first eight of the constants Ci are tabulated, which suffice to define the J-integral up to terms of order e6 (where e is a loading parameter) and some representative near-tip features are displayed graphically. The computed value of C1 shows that the conventionally adopted value for the plastic-zone correction ry is too large by a factor of roughly 2.8, if it is to yield a genuine asymptotic estimate for J. As an example, the “elastic” parameters γij are found, from a boundary collocation program, for a centre-cracked square plate subjected to tensile loading; and a plot of J versus load, and the plastic-zone shape at a particular load level, are displayed.

Crack propagation, resulting from a monotonic increasing applied stress, in a linear viscoelastic material

Abstract: A theory of crack propagation, resulting from the application of a monotonic increasing applied stress, in a linear viscoelastic material, is derived based upon an energy balance fracture criterion. It is shown that for a Maxwell solid the crack growth law can be derived either from a global energy balance taking full account of the energy dissipation resulting from viscoelastic flow, or from a local energy balance taking account of the dissipation in the failure zones. The local energy balance method allows the derivation of the crack growth law for more general linear viscoelastic solids. The theory predicts the well known Griffith condition for fracture when the material is simply linear elastic. For a crack having failure zones in a linear viscoelastic solid the growth law for a constant applied stress is where c(t) is the time dependent half-crack length, 641-1 is the yield or crazing stress in the failure zone, K(t) is the time dependent stress intensity factor, Γ is the fracture energy, ν is Poisson's ratio and J(t) is the uniaxial creep function of the viscoelastic material. This growth law is valid if either J(t)≡0 for all times t>0 (i.e. a Maxwell solid) or if 641-1 641-2 641-3

An integral associated with the state of a crack tip in a non-elastic material

Abstract: An integral has been proposed for a non-elastic material whose value is determined by the magnitude of the singularities at the tip of a crack but which may be evaluated mainly far away from the crack, where the state of deformation may be determined numerically with higher accuracy than near the crack. The integral is intended for situations when plasticity is too great for linear elastic fracture mechanics to be appropriate, but may be related to the stress intensity factors in the linear elastic case. Its value has been calculated for a central or edge crack in a uniformly loaded and unloaded plate with a non-work-hardening elastic plastic material when the loading is either tension or longitudinal shear. It has also been calculated for a non-work-hardening material for a central sloping crack under tension and for a central crack under a quadratic temperature gradient for which previously suggested contour integrals are no longer path independent even in the linear elastic case.

Dynamic Analysis of Horizontally Curved I-Girder Bridges

Abstract: The finite element displacement method is used with annular plate, thin-walled curved beam, straight prismatic beam, and frame-type diaphragm elements for the dynamic analysis of horizontally curved I-girder highway bridges. The stiffness and inertia properties of the finite elements are obtained within the bounds of linear elasticity and small displacement theories. Warping of the girder cross section due to torsion is included in the analysis. The moving vehicle on the bridge is idealized as a sprung mass supported on two unsprung masses. The centrifugal forces arising due to the motion of the vehicle in a circular path are included in the analysis. The inclusion of the vehicle inertia in the bridge-vehicle interaction results in a set of differential equations, the coefficients of which depend on the speed of the vehicle and thus are not constant. These equations of motion are solved by the linear-acceleration method of numerical integration.

Failure of granite cylinders under impact loading

Abstract: The temporal variation of axial and radial particle velocity components on the surface of granite cylinders was measured when the latter were struck on end by like cylinders of equal diameter. The experimental results were compared with two-dimensional axisymmetric finite difference calculations in which several different constitutive models were evaluated. Fairly good agreement with experiment was obtained with a simple intuitive representation of the brittle fracture process in which locally linear elastic behavior prevailed until such time as any principal stress component exceeded in tension some small prescribed value. The subsequent failure mechanism admits the opening of cracks, multiple cracking to the point of pulverization and possible reclosure under polyaxial compression. The fracture zone development is strongly influenced by a feedback effect from the interaction of the elastic field in the virgin material with signals propagating forward from the existing fracture zone.

Basic Equations of the Plane Theory of Elasticity

Abstract: The equations of the plane theory of elasticity apply to two cases of equilibrium of elastic bodies which are of considerable interest in practice, namely: to the case of plane strain and to the case of the deformation of a thin plate under forces applied to its boundary and acting in its plane *. These two cases will be discussed in detail in the following two sections.

Determination of pavement layer moduli from surface deflection data for pavement performance evaluation

Abstract: The method of obtaining the layer moduli reported in this paper relies on the use of layered elastic theory computer programs. It is necessary that the thickness and Poisson's ratio be known for each layer, although the results are not highly sensitive to the value assumed for Poisson's ratio. In the method, points on a two-dimensional surface deflection basin are fitted to the field data. Iteration is required to align the measured and computed points by adjusting the assumed values for the layer moduli. Presently the method relies on trial and error iteration using the BISTRO computer program. It is suggested that a new program be developed which, would enable a more direct determination of the moduli. A case study is presented to demonstrate the application of the method to the evaluation of a test pavement for a heavy duty haul road to support 400,000-pound gross vehicle weight lignite coal trucks. Deflections of the test pavements were obtained using the Dynaflect apparatus and a fully loaded haul vehicle. The major difference in the magnitude of surface loading served to illustrate that the assumption of linear elasticity is not valid over such a range. The evaluation of one of the eight test pavements is illustrated by example, and it is predicted that the pavement will not be satisfactory for the anticipated loading and traffic. The prediction was borne out after several years of use. /Author/

Infinitesimal discontinuities in initially stressed relativistic elastic solids

Abstract: In this paper we examine the influence of a general initial state of stress upon the propagation of infinitesimal (or weak) discontinuities in nonlinear relativistic elasticity. This influence, which materializes in alterations in the wave speeds, the general nonseparability in longitudinal and transverse waves, and the growth of the amplitude of infinitesimal discontinuities so as to form shock fronts, is first studied on the basis of a model of relativistic elasticity called hypoelasticity of zeroth order. The analytical treatment, however, is manageable only for the case of principal wave fronts for which the spatial direction of propagation coincides with a principal direction of the initial state of stress and, consequently, the wave fronts separate into longitudinal and transverse ones. Such notions as those of apparent elasticity moduli appear naturally in the analysis. Then a model of thermodynamical relativistic elasticity, referred to as neo-Hookean elasticity, is shown to be representable, insofar as wave-front propagation is concerned, by a special model of hypoelasticity of first order. The qualitative results obtained before concerning the influence of initial stresses are shown to apply equally to this description.

Surface stress and the equilibrium shape of an elastic crystal

Abstract: We consider the deformation, due to surface stress, of an elastic crystal in the shape of a square. We reduce the problem to one of linear elasticity and obtain a solution using the finite‐element method.

Long-Life Notch Strength Reduction Due to Local Biaxial State of Stress

Abstract: An explanation for the observed difference between the values of the long-life, fatigue-notch strength-reduction factor (Kf ) and the elastic stress concentration factor based on the influence of biaxial-notch root stresses is presented for a limited class of problems for which the local stresses are elastic. Results of three-dimensional linear elasticity solutions for notch stresses are reviewed and then used in conjunction with a biaxial fatigue criterion suitable for elastic straining to predict values of Kf for a variety of notched components for which values have been experimentally determined. Comparison of actual and predicted values of Kf indicates a close correspondence suggesting that the influence of local biaxial stresses on the life to crack initiation of notched components is responsible for the reduced notch severity observed in many of the experimental investigations of long-life (fatigue) notch sensitivity.

Atomic simulation of the a/2 (111) (110) edge dislocation in alpha-iron

Abstract: Computer simulation is used to obtain the plane strain and non-plane strain equilibrium core structures for the a/2 (111) (110) edge dislocation in alpha-iron as modelled by the Johnson potential. The methods of the integral formulation of anisotropic linear elasticity theory are employed in the analysis. Flexible core boundary methods reveal that positive volume expansions of 0.57b2 and 0.62b2 result from introduction of the dislocation (plane and non-plane strain cases respectively) into the perfect lattice. The non-plane strain relaxation reveals significant displacement parallel to the dislocation line in the compressive near-core region. This effect arises as a consequence of the strong nonlinearity of the core region elastic field.

Dynamic expansion of a compressible hyperelastic spherical shell

Abstract: This paper is concerned with the finite spherically symmetric motion of a compressible hyperelastic spherical shell, subjected to a spatially uniform step funtion application of pressure at its inner surface. A method, given in a previous paper [1], for the determination of the field of characteristics, for expansion of a spherical cavity in an unbounded solid is adapted to consider the spherical shell problem. Results are presented graphically for a particular strain energy function and are compared with results obtained for an incompressible material and from linear elasticity theory.

Fully monitored motorway trials in the netherlands corroborate linear elastic design theory

Abstract: This study investigates the applicability of theoretical design models by means of comprehensive experiments on fully monitored motorway trial sections. In two types of construction--one with full depth asphalt and one with a sand-cement sub-base--strains at different levels in the structures, surface deflections and soil pressures were measured under a variety of conditions of temperature and of load, speed and lateral position of the loading vehicle. Both test sections were constructed in stages. The paper gives representative results of measurements for two stages of construction of the pavements. Material properties of subgrade, sand cement and asphalt and the thicknesses of the layers were measured both in situ and in the laboratory. These data were then introduced into the BISAR computer program, based on linear elastic multi-layer theory, to calculate values of strains, deflections and soil pressures corresponding to the measured quantities and the results have been compared with one another. In addition, BISAR computations have been made to investigate the influence of a variation in test parameters (moduli, Poisson's ratios, shape and stress distribution of wheel contact area, etc.). The widths (durations) of recorded regular strain signals were used to characterize the loading time. Measured and computed quantities are mostly presented as a function of the stiffness modulus of the asphalt (incorporating the effect of temeprature and loading time). Generally, the results showed increasing scatter at lower asphalt moduli. Computations showed that variations in test conditions also have their strongest influence at low asphalt stiffness. In some cases a stronger difference between longitudinal and transverse strain was found as compared with the computed values. On the whole, measured values of strains, deflections and soil pressures showed good agreement with the corresponding computed values, particularly with higher asphalt moduli. /Author/