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Applying the uniform-approximation technique, consistent plate theories of different orders are derived from the basic equations of the three-dimensional linear theory of elasticity. The zeroth-order approximation allows only for rigid-body motions of the plate. The first-order approximation is identical to the classical Poisson-Kirchhoff plate theory, whereas the second-order approximation leads to a Reissner-type theory. The proposed analysis does not require any a priori assumptions regarding the distribution of either displacements or stresses in thickness direction. IntroductionPlates are important elements in structural engineering and have been widely studied by engineers during the last century. A comprehensive bibliography can be found in [14,26,32]. Plates are characterized by the fact that their extension in one direction, e.g. in the thickness (characteristic length h), is considerably less than that in the other two directions (characteristic in-plane length a). The surface that bisects the plate continuum transversely is assumed to be plane and is called for short, mid-plane. Loads are applied perpendicular to the mid-plane or through moments acting around the in-plane axes. Plate theories attempt to describe the three-dimensional state of stresses and displacements in a plate continuum by two-dimensional quantities defined on a surface. Therefore, plate theories are inherently approximative.Several possibilities exist to derive plate theories. One uses polynominal expansions in the thickness direction both for the displacements and for the stresses. These series expansions have to be truncated at a specific order, and often they are complemented by a set of a priori assumptions, mostly motivated by engineering intuition, [16,22].The asymptotic method (cf. e.g.[7]) for the derivation of governing plate equations develops two sets of differential equations, one for the ''interior'' of the plate and the other one for the ''boundary layer''. Especially for dynamic problems, where the characteristic in-plane dimension, i.e. the wave length k, is much smaller than the planar extension a, this method supplies accurate results and allows for reliable error estimations; it requires, however, advanced mathematical techniques.An alternative approach ''lives'' completely on the mid-surface. Translations, rotations, or, generally, directors are attached to each point of the material surface leading to Cosserat-or director-type theories, [20,33]. A historical review of plate theories may be found in [1,30]; the latest achievements in plate and shell theories have been discussed in [29].Any two-dimensional approximation of the governing equations of three-dimensional continuum mechanics yields errors and, possibly, contradictions. In order to assess the validity of an approximation and to indicate its range of applicability, various features have been

By Fourier‐series expansion in thickness direction of the plate with respect to a basis of scaled Legendre polynomials, several equivalent (and therefore exact) two‐dimensional formulations of the three‐dimensional boundary‐value problem of linear elasticity in weak formulation for a plate with constant thickness are derived. These formulations are sets of countably many PDEs, which are power series in the squared plate parameter. For the special case of a homogeneous monoclinic material, we obtain an approximative plate theory in finitely many PDEs and unknown variables by the truncation approach of the uniform‐approximation technique. The PDE system is reduced to a scalar PDE expressed in the mid‐plane displacement. The resulting second‐order theory, considered as a first‐order theory, is equivalent to the classical Kirchhoff theory for the special case of an isotropic material and equivalent to Huber's classical theory for an actual monoclinic material. However it remains shear‐rigid as a second‐order theory. Therefore, it is modified by an a‐priori assumption to a theory for monoclinic materials, that presumes the former equivalences, considered as a first‐order theory, but is in addition equivalent to Kienzler's theory as a second‐order theory for the special case of isotropy, which implies further equivalences to established shear‐deformable theories, especially the Reissner‐Mindlin theory and Zhilin's plate theory. The presented new second‐order plate theory for monoclinic materials is finally a system of two coupled PDEs of differentiation order six in two variables.

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A B S T R A C T The endurance limit and the mechanisms of fatigue crack initiation in the high-cycle regime were investigated using round specimens of the bearing steel SAE 52100 in a bainitic condition under longitudinal forces, torsional moments and combinations of these loads. Three specimen types were examined: smooth specimens and specimens with circumferential notches with radii of 1.0 and 0.2 mm. The surfaces of the specimens including the notches were ground resulting in compressive residual stresses in the nearsurface region. The influence of mean and multiaxial stresses on the endurance limit can be understood by consideration of crack initiation mechanisms and micromechanics. Crack initiation occurred at oxides, carbonitrides and at the surface. The oxides had little adhesion to the bainitic matrix and acted like pores. The carbonitrides were well bonded to the matrix and caused stress concentrations due to their higher elastic modulus when compared to that of the matrix. The mechanisms of crack initiation could be related to the load type: loads with rotating principal stresses cause more damage for nitrides than for oxides. Increasing maximum stresses are more dangerous for nitrides than for oxides, and damage the surface more than the nitrides. Normal stresses produce more damage for oxides than shear stresses. The endurance limits were calculated by means of an extended weakest-link model which combines volume and surface crack initiation with individual fatigue criteria. For volume crack initiation, the criterion of Dang Van was used. For the correct description of the surface crack initiation, a criterion proposed by Bomas, Mayr and Linkewitz was applied. With this concept, a prediction of the endurance limit is possible. The influence of the notch geometry on the endurance limit is well characterized. A 0 = reference surface d = net diameter of the specimens E = elastic modulus f = load frequency F = longitudinal load, distribution function K t,σ = concentration factor for normal stress K t,τ = concentration factor for shear stress m = exponent of the Weibull distributionCorrespondence: H. Bomas.

The ductile fracture behavior of different specimens is analyzed by continuum damage-mechanics techniques. A model introduced by Gurson and modified by Needleman and Tvergaard has been implemented in the finite element program package, ADINA. The damage parameters of the model are measured and calculated from smooth tension tests, and the characteristic material distance is estimated from compact tension experiments. A steel, ASTM A710, and a weld metal for the steel, ASTM A508, are investigated. The damage parameters determined from the smooth bars are used to predict the deformation and fracture behavior of notched round bars and of sidegrooved compact specimens. For the weld metal, a side-grooved WOL-X-specimen is also simulated. In every case, a satisfactory agreement of prediction and experiment is observed. In order to investigate the influence of the stress state (constraint) in cracked specimens, a series of numerical computations of different specimen geometries and loading situations is performed utilizing the same set of parameters of the ASTM A710 steel. The slopes of the predicted J-resistance curves increase with increasing ratio of tension versus bending load and with decreasing relative crack length.

Material conservation and balance laws of elementary beam theory have been derived. The application to beams with discontinuities in the stiffness results in a surprisingly simple formula to calculate stress intensity factors of cracked beams. (IWM

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