弹性力学问题的一阶有限元解法

求解弹性力学问题的应力时,如果采用常规的位移有限元法,需要先求得单元的节点位移,再经过求导运算得到。为了解决这种求解方式引起的应力精度下降的问题,提出了弹性力学问题的一阶多变量形式,使得应力与位移精度同阶,并推导了弱形式。采用有限元方法,对弹性力学问题给出了一阶解法的二维、三维数值算例,并且将一阶解法的结果与常规位移有限元法的解进行了比较。数值计算结果表明,一阶解法有效提高了应力的精度,并且应力的误差和节点位移的误差具有相同的收敛阶,验证了本文方法的有效性,为提高有限元法的应力精度提供了新的思路。     

Normal surface displacement around a circular hole by reflection holographic interferometry

In a previous paper¹, a fringe-compensation technique was developed to improve the possibilities of stress analysis by real-time holographic interferometry. The technique is specially well suited for the measurement of small displacements in the direction of viewing. As an application of this method, the surface displacements caused by strains in the thickness direction are measured around a circular hole in a plate loaded in tension in its plane. Independent prior knowledge of the in-plane displacement is required, however, in data processing. An analytical solution to the problem is used for that purpose. The experimental results are compared to those obtained theoretically from the classical two-dimensional analysis, and from a three-dimensional analysis. The two-dimensional theory assumes a state of ‘generalized plane stress’. The three-dimensional theory, made by Alblas², takes into account the existence of stresses in the thickness direction, and the variation of the in-plane stresses through the thickness. Both theories give the same results away from the hole. They differ significantly, however, when the hole boundary is approached, where the proximity of the hole induces three-dimensional effects. The experimentally measured displacement is found to be in good agreement with both theories away from the hole. Close to the hole, a large departure from the two-dimensional results is observed. The experimental results here are close to those of three-dimensional results. The experiment is thus in good agreement with the three-dimensional theory over the whole field. But the two-dimensional theory is valid only at large distances from the hole.     

15万立米气柜残余应力及工作应力测试分析

报导了１５万立米大型煤气气柜挂圈走台的残余应力及气柜升降时工作应力的测试结果。该气柜服役十余年后发生煤气泄漏，为分析其原因，进行了残余应力及气柜修复前后的工作应力测试，结果表明：修复前残余应力及工作应力均达到很高水平，气柜升起时伴有严重侧偏现象，修复后应力状态有明显好转。     

Forward roll coating flows of viscoelastic liquids

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the two-dimensional film splitting flow that occurs in forward roll coating is unstable; a three-dimensional steady flow sets in, resulting in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the two-dimensional flow is determined by the competition of capillary and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are non-Newtonian polymeric solutions and dispersions, most of the theoretical analyses of film splitting flows relied on the Newtonian model. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case.Forward roll coating flow is analyzed here with two differential constitutive models, the Oldroyd-B and the FENE-P equations. The results show that the elastic stresses change the flow near the film splitting meniscus by reducing and eventually eliminating the recirculation present at low capillary number. When the recirculation disappears, the difference of the tangential and normal stresses (i.e., the hoop stress) at the free surface becomes positive and grows dramatically with fluid elasticity, which explains how viscoelasticity destabilizes the flow in terms of the analysis of Graham [M.D. Graham, Interfacial hoop stress and instability of viscoelastic free surface flows, Phys. Fluids 15 (2003) 1702–1710].     

变宽度板在拉伸和弯曲作用下的三维应力集中

根据一种修正的余能原理，建立了具有一个无外力圆柱面的三维杂交应力元，元内假定应力场满足三维柱坐标表示的平衡方程及无外力圆柱面上的外力边界条件；当元退化为二维时，也满足协调条件。单元位移场选择与相邻单元协调。数值算例表明这种特殊杂交应力元在相当粗的网格下，能十分有效地分析变宽度薄／厚板在拉伸与弯曲作用下的三维(及二维)应力集中。     

On consistent first approximations in the general linear theory of thin elastic shells

Summary This paper, as a number of earlier ones, is concerned with the rational establishment of twodimensional differential equations for the approximate analysis of stress and strain in elastic layers with spacecurved middle surface. It has been known for some time that the principal difficulty of this problem is to establish rational two-dimensional constitutive equations which correspond to a given system of constitutive equations for the layer treated as a three-dimensional continuum. — In an earlier publication [18] the point had been made that since two-dimensional shell theory was concerned with stress resultants and stress couples, it ought to be advantageous to derive such a theory from a three-dimensional theory in which force stresses as well as moment stresses were incorporated, even for media which, actually, were incapable of supporting moment stresses. — The earlier work [18] had indicated that, mathematically, the advantages of approaching the derivation of two-dimensional shell theory from three-dimensional moment stress elastically theory had to do with the form of the compatibility equations for strain in such a three-dimensional theory. Briefly, with these three-dimensional compatibility equations it becomes possible to concentrate all three-dimensional aspects of the shell problem in a three-dimensional system of integro-differential constitutive equations, and the task of deriving rational two-dimensional constitutive equations becomes nothing but the task of establishing suitable asymptotic expansions for the solutions of these three-dimensional integro-differential equations. In the work in [18] this task had not actually been carried out. The present paper establishes a significant rearrangement of the system of integro-differential equations, in such a way that the nature of the necessary asymptotic expansions is made evident. — With this, explicit results are obtained which include the system of two-dimensional constitutive equations of Koiter and Sanders for an iotropic homogeneous medium, as well as a system of constitutive equations for a class of shells for which the normals to the middle surface are not directions of elastic symmetry, as well as a system of constitutive equations for shells which are sufficiently soft in transverse shear to make transverse shear deformation a first-order effect.

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Übersicht In dieser Veröffentlichung wird die rationelle Aufstellung der zweidimensionalen Differentialgleichungen für die näherungsweise Bestimmung von Spannungen und Verformungen in elastischen Schichten mit räumlich gekrümmter Mittelfläche behandelt. Es ist bekannt, daß die Hauptschwierigkeit dabei im Aufstellen von zweidimensionalen Stoffgleichungen besteht, die einem gegebenen System von Stoffgleichungen für eine als dreidimensionales Kontinuum behandelten Schicht entsprechen. In einer früheren Veröffentlichung [18] wurde darauf hingewiesen, daß es vorteilhaft sein könnte, eine solche Theorie aus einer dreidimensionalen Theorie abzuleiten, in der sowohl Momentspannungen als auch Kraftspannungen berücksichtigt werden. Das gilt auch für solche Stoffe, die in Wirklichkeit nicht in der Lage sind, Momentenspannungen aufzunehmen. — Es wurde seinerzeit gezeigt, daß die Vorteile einer Ableitung der genäherten zweidimensionalen Schalentheorie aus der dreidimensionalen Elastizitätstheorie mit der Form der Verträglichkeitsbedingungen für die Verformungen in dieser dreidimensionalen Theorie zusammenhängen. Mit Hilfe dieser dreidimensionalen Verträglichkeitsbedingungen wird es möglich, alle dreidimensionalen Aspekte des Schalenproblems in einem dreidimensionalen System von Integro-Differentialgleichungen für das Stoffverhalten zu konzentrieren, so daß die Ableitung zweidimensionaler Stoffgleichungen nichts anderes ist, als das Aufstellen geeigneter asymptotischer Reihenentwicklungen für die Lösungen dieser dreidimensionalen Integro-Differentialgleichungen. Das wurde jedoch in [18] noch nicht ausgeführt. In der vorliegenden Veröffentlichung wird das System der Integro-Diffe-rentialgleichungen so umgeformt, daß die Art der notwendigen asymptotischen Entwicklungen deutlich wird. Auf diese Weise werden explizite Ergebnisse erhalten, die das System der zweidimensionalen Stoffgleichungen von Koiter und Sanders für ein isotropes homogenes Medium einschließen. Desgleichen sind darin enthalten die Stoffgleichungen für eine Klasse von Schalen, für die die Normalen zur Mittelfläche nicht mit den Richtungen der elastischen Symmetrie übereinstimmen, sowie auch die Stoffgleichungen für Schalen, die hinreichend weich gegenüber Querschub sind, so daß Querschubdeformationen als Effekte erster Ordnung auftreten.

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A report on work supported by the Office of Naval Research, Washington, D.C.     

A prediction of in vivo mechanical stresses in blood vessels using thermal expansion method and its application to hypertension and vascular stenosis

Mechanical stimuli play critical roles in cardiovascular diseases, in which in vivo stresses in blood vessels present a great challenge to predict. Based on the structural–thermal coupled finite element method, we propose a thermal expansion method to estimate stresses in multi-layer blood vessels under healthy and pathological conditions. The proposed method provides a relatively simple and convenient means to predict reliable in vivo mechanical stresses with accurate residual stress. The method is first verified with the opening-up process and the pressure-radius responses for single and multi-layer vessel models. It is then applied to study the stress variation in a human carotid artery at different hypertension stages and in a plaque of vascular stenosis. Our results show that specific or optimal residual stresses exist for different blood pressures, which helps form a homogeneous stress distribution across vessel walls. High elastic shear stress is identified on the shoulder of the plaque, which contributes to the tearing effect in plaque rupture. The present study indicates that the proposed numerical method is a capable and efficient in vivo stress evaluation of patient-specific blood vessels for clinical purposes.     

Inversion of speckle interferometer fringes for hole-drilling residual stress determinations

Speckle interferometric fringe patterns record stress-relief displacements induced by the drilling of blind-holes into prestressed objects. The quantitative determination of residual stress state from such stress patterns is difficult because of the ambiguity in the order of the observed fringes. The plane stress magnitudes are provided directly from selected fringe positions using a stochastic, iterative least squares minimization approach. The inversion requires prior knowledge of the experimental geometry and an appropriate uniaxial stress-relief displacement basis function derived from three-dimensional finite element calculations. Superpositioning of the rotated and scaled displacement basis functions allows the stress-relief relaxation for any biaxial state of stress to be determined. In this paper, fringe patterns were forward modeled from a large ensemble of calculated biaxial stress-relief displacement fields. Inversion of these noise-free fringe patterns reproduced the biaxial stresses with negligible error. Analysis of more realistic fringe patterns that include speckle noise gave stress magnitude errors that diminished rapidly with the number of selected points to better than 3 percent for 100 points. Sensitivity of the optical method is influenced by a number of factors, but the ensemble of model fringe patterns studied indicates that the stress magnitudes (nomalized with respect to the material's Young's modulus) from 3×10^–4 to 10^–2 can accurately be determined with visible laser radiation. The method is amenable to automation and can easily be extended to study near surface gradients in the residual stresses or applied to other optical recording techniques such as moiré and phase-shifting interferometry.     

Experimental stress analysis of a nuclear-reactor pressure vessel

The stresses from internal pressure in a nuclear-reactor pressure vessel were investigated using both strain-gaged and photoelastic models. The methods used in obtaining complete stress distributions along the nozzles from strain-gage data are described, and comparisons are made between the distributions obtained from the photoelastic model and those derived from strain-gage data. The effects of nonradial attachment of a nozzle to a spherical shell upon the stresses in the nozzle and in the shell are shown. Finally, comprisons are made between the experimental results and those for simplified analytical models.     

Use of electrical-resistance strain elements in three-dimensional stress analysis

A method for constructing resistance-wire strain gages which may be imbedded in a plastic model without materially altering the stress pattern in the model is presented. The methods used to calibrate the gages and the photoelastic tests made to investigate the effect of the gages on the stress pattern are described. The application of this new three-dimensional technique to evaluate the stress distribution in a thick-walled cylindrical pressure vessel is discussed. Correlation between experimental data and calculated values is given.     

Experimental study of large-diameter thin-wall pressure vessels

An experimental stress analysis of three cylindrical pressure vessels with radius/thickness ratios ranging from 100 to 238 and different head closures is described. Brittle coatings and electrical strain gages were employed to determine stress distributions over the entire outer surface of the vessels. Electrical strain gages alone were used to determine stresses on the inside surface of the vessels. Particular emphasis was placed on determining stress concentrations and on nonlinear effects produced by geometric imperfections. An attempt was also made to correlate the failure, which started in the cylindrical portion of the three vessels, with the elastic-stress distribution. It was found that the imperfections in the cylinder were not significant if the vessel was fabricated from a ductile steel. However, if the vessel was constructed from a high-strength but brittle steel, the imperfections significantly lower the bursting strength of the vessel.     

含人工脱粘层固体装药结构应力应变场分析

针对某翼柱型固体装药结构,选取1/16子结构进行了有限元分析。通过将其分割成若干具有单一拓扑型式的组合体,获得了精度较高的结构化网格。在三维线性粘弹性理论和边界非线性理论基础上,基于接触算法模拟了人工脱粘层的边界条件;研究了固化降温过程含人工脱粘层固体装药结构的变形特征和应变场,并与无人工脱粘层的计算结果进行了对比,结果表明:人工脱粘层可以显著地改善固体装药结构端部的应变水平,最大的等效VonMises应变从无人工脱粘的30.1%下降为23.0%。此外,探讨了人工脱粘层前缘的应力水平以及工程上界面脱粘的快速近似计算方法。     

A multilayer-reflection technique for three-dimensional photoelastic studies of perforated plates in bending

A photoelastic investigation was conducted to determine the stress-concentration factors around a large, symmetrically reinforced central hole in a square plate under 1∶1 and 2∶1 biaxial bending. Tapered-edge rings served as the reinforcement, and a major objective was to determine the ring proportions such that the maximum stress at the hole would be equal to the value which would be present in an unperforated plate under the same nominal stress. Because the stress distribution at the periphery of a hole in such a plate structure varies in the radial, tangential and thickness direction, it was necessary to employ a three-dimensional photoelastic technique. There were a number of serious disadvantages in the use of any of the standard procedures and a new three-dimensional technique for room-temperature use was developed which is particularly suitable for the determination of boundary stresses around holes in bending experiments. With the technique in its present state of development, the three-dimensional isochromatic distribution in the plate can be determined from a single model and, from this, the boundary value of stress. The new technique utilized a laminated-plate model. Selective aluminizing of the laminations allowed for the determination of fringe-order distributions in the thickness direction as well as in the radial and circumferential directions at the boundary of the hole in flat models. Uniaxial maximum fringe orders were determined and, from these, the biaxial values were obtained by superposition.     

Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness

This paper studies the stress and displacement distributions of functionally graded beam with continuously varying thickness, which is simply supported at two ends. The Young’s modulus is graded through the thickness following the exponential-law and the Poisson’s ratio keeps constant. On the basis of two-dimensional elasticity theory, the general expressions for the displacements and stresses of the beam under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at two ends, are analytically derived out. The unknown coefficients in the solutions are approximately determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beams. The effect of Young’s modulus varying rules on the displacements and stresses of functionally graded beams is investigated in detail. The two-dimensional elasticity solution obtained can be used to assess the validity of various approximate solutions and numerical methods for the aforementioned functionally graded beams.     

Theory of elastic and photoelastic isodynes. Samples of application in composite structures

The theoretical foundations of a system of new methods which the authors call isodyne photoelasticity are presented. It is shown that the plane-elastic isodynes related to two-dimensional stress fields can be obtained experimentally using particular scattered-light techniques. The experimentally obtained photoelastic isodynes allow non-destructive analysis of stresses in structures with local three-dimensional stress states.Isodyne photoelasticity is used to determine stresses in composite structures, in particular the stresses at internal discontinuities and the delaminating stresses. The results are used to assess the reliability of some analytical solutions. The theories of isodynes and of the developed experimental techniques of isodyne photoelasticity are presented in a manner compatible with contemporary concepts of the models of reality—physical models, mathematical models, and experimental models constructed of matter and energy.Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1985.     

On attainable lower boundary of the three-dimensional Coulomb-Tresca invariant

An attainable lower boundary of the three-dimensional Coulomb-Tresca invariant is constructed to facilitate the search of plasticity conditions for isotropic bodies which, just as the Tresca-Saint Venant plasticity condition, ensure the hyperbolic analytic type of three-dimensional equations of the mathematical theories of plasticity based on the generalized associate flow law. The construction is performed by using the system of three “two-dimensional” tangential stresses related to the given three-dimensional stress state. It is proved that the Coulomb-Tresca invariant attains its lower bound in any plane strain state where the out-of-plane principal normal stress is intermediate (or median).     

三向应力Mohr圆的真实构成及剪应力作用方向的确定

三向应力Mohr圆的构成在传统上是借助公式推证而得，并以平面图形来表示，缺乏三维的真实感和直观性。在应力应变分析中，对于平面应力问题，可以通过平面应力摩尔圆确定过一点不同斜面上的应力分量及其作用方向。对于三维问题，利用摩尔圆图解法可以确定某一斜面上的正应力和剪应力的数值，但不能表示剪应力的作用方向。剪应力的作用方向需要通过另外的图解方法来确定。本文分别从坐标系旋转和数值计算的角度解释了三向应力Mohr圆的构成过程，形象地说明了Mohr圆的物理本质。针对三向应力Mohr圆不能表示剪应力作用方向的问题，通过矢量运算，给出了剪应力作用方向的确定公式。     

Residual Stress Analysis on Thick Film Systems by the Incremental Hole-Drilling Method – Simulation and Experimental Results

The residual stress state in thick film systems, as for example thermal spray coatings, is crucial for many of the component’s properties and for the evaluation of the integrity of the coating under thermal and/or mechanical loading. Therefore it is necessary to be able to determine the local residual stress distribution in the coating, at the interface and in the substrate. The incremental hole-drilling method is a widely used method for measuring residual stress depth profiles, which was already applied for thermally sprayed coatings. But so far no reliable hole-drilling evaluation method exists for layered materials having a stress gradient in depth. The objective was to investigate, how far existing evaluation methods of the incremental hole-drilling method that are only valid for residual stress analysis of homogenous material states can be applied to thick film systems with coating thicknesses between 50 μm and 1000 μm and to point out the application limits for these already existing methods. A systematic Finite Element (FE) study was carried out for coating systems with an axisymmetric residual stress state σ₁?=?σ₂. It is shown that conventional evaluation methods developed for homogeneous, non-layered material states can be successfully applied for a stress evaluation in the substrate and the coating for small and for sufficiently large coating thicknesses, respectively, regardless of the type of evaluation algorithm used i.e. the differential or the integral method. The same accounts for material combinations that have a Young’s modulus ratio close to one, between 0.8 and 1.2. The studies indicated that outside the given ranges case specific calibration must be applied to calculate reliable results. Further, calibration data were determined case specifically for a selected model coating system. The accuracy of a residual stress determination using these case specific calibration data was examined and the sensitivity of the evaluation with respect to an accurate knowledge of the boundary conditions of the coating system i.e. the coating thickness and the Young modulus was studied systematically. Finally, the calibration data were applied on a thermally sprayed aluminium coating on a steel substrate analysis and the results for using the incremental hole drilling method were compared to results from X-ray stress analysis.     

The gamma-ray-irradiation method applied to three-dimensional thermal photoelasticity

It has recently been shown that a gamma-ray irradiation fixes the birefringence in photoelastic models of Araldite B. The method has hitherto only been used to fix the birefringence in specimens subjected to constant forces. This work applies the irradiation method to determine the thermal stresses in a thick-walled cylinder with a temperature gradient along the radius. The model behaved in the following way during the experiment. The material relaxed when the temperature gradient was applied, and the model was irradiated when it was stress free. Real stresses, which were equal to the thermal stresses but with opposite sign, appeared when the temperature gradient was removed. As the problem is mathematically two-dimensional, it is possible to determine the birefringence by means of the slicing technique. Good agreement was obtained between the stress determined experimentally and those calculated analytically. The irradiation sensitivity of a number of materials has been determined, and some new materials have been used that are superior in two important ways to that used earlier in that they require a much lower irradiation dose and their mechanical properties are considerably better.