Finite-strain pressure-tension loading of thick-walled cylinders including temperature effects

Experimental data were obtained from thick-walled cylinders made of hot-rolled SAE 1045 steel at room temperature, annealed OFHC copper at room temperature and at 500°F (260°C), and annealed aluminum alloy 1100 at room temperature and at 305° F (152° C). Experimental pressure-strain curves were compared with curves predicted by two different analytical solutions. One solution is a finite-strain, incremental, compressible analytical solution. The second solution is a finite-strain, total-strain, incompressible analytical solution which was corrected to make it applicable for compressible materials. With both solutions, the material is assumed to be an isotropic-hardening material that obeys the von Mises yield condition. The loading function for the material was obtained from tension specimens tested at the some temperature and loading rate as the thick-walled cylinders. Good agreement was found between each solution and experimental data.     

Inelastic behavior of thick-walled cylinders subjected to nonproportionate loading

Inelastic behavior of thick-walled cylinders subjected to nonproportionate loading was studied by the testing of specimens made of C1045 steel and of annealed copper. Several theories were reviewed. A closed-form solution proposed by Mendelson¹² was used to predict external strains for open-end and closed-end thick-walled cylinders. An incremental theory proposed by Chu¹³ was used to provide incremental solutions for open-end thick-walled cylinders, and for cylinders subjected to nonproportionate loading. Test data for open-end and closed-end thick-walled cylinders made of C1045 steel and of annealed copper were in excellent agreement with the incremental theory. Larger values were predicted by use of the closed-form solution for circumferential strains than actual test data for open-end thick-walled cylinders at large depth of yielding. For cylinders subjected to nonproportionate loading, excellent agreement was indicated between the incremental theory and the experiments for the plot of axial load vs. circumferential strain for specimens made of both metals. Agreement between the incremental theory prediction of axial strains for the specimens made of annealed copper and test data is quite satisfactory. Larger values were predicted by the incremental theory for axial strain than experimental data for specimens made of C1045 steel. The error was conservative.     

Bursting pressure of thick-walled cylinders subjected to internal and external pressures,axial load and torsion

A finite-total-strain, incompressible, analytical solution is presented to predict load-deformation relations for loads from zero to failure for thick-walled cylinders subjected to internal pressure, external pressure, axial load and torsion. The solution assumes that the material is an isotropic hardening material that obeys the von Mises yield condition. The flow law incorporates the prandtl-Reuss stressstrain relations and a loading function represented by the tension true-stress vs. true-strain diagram. Poisson's ratio is assumed to be equal to one-half for both elastic and plastic strains. The difference between the strains given by the incompressible solution and the correct strains are calculated for one set of elastic loads; the strains given by the incompressible solution are then corrected based on the assumption that each correction is proportional to the increase in the given component of load. Good agreement is indicated between the corrected incompressible solution and data obtained from cylinders made of either SAE 1045 steel, OFHC copper, or aluminum alloy 1100. Paper was presented at 1975 SESA Spring Meeting held in Chicago, IL on May 11–16. This investigation was funded by Rock Island Arsenal and was conducted at Research Directorate, GEN Thomas J. Rodman Laboratory, under the Laboratory Research Cooperative Program of ARO-D.     

Creep of thick-walled cylinders subjected to internal pressure and axial load

A creep theory is presented to predict deformations at any specified time for a thick-walled cylinder subjected to internal pressure and axial load. The theory is based on the usual assumptions that the deformations are infinitestimal, that the material is incompressible and that the total strain theory is valid. The stress-strain-time relation for the material is assumed to be represented by an isochronous stress-strain diagram which is approximated by an arc hyperbolic sine function. The experimental part of the investigation included tests of thick-walled cylinders made of high-density poly-ethylene whose ratio of outside to inside radii were either 1.5 or 2.0. The test cylinders were either tested as closed-ented cylinders with internal pressure or subjected to a combination of internal pressure and axial load. Also, the application of the theory for varying load conditions was studied. Good agreement was found between theory and experiment.     

Collapse of thick-walled cylinders under external pressure

Hydrostatic collapse tests performed on thick-walled capped cylinders are described. Finite-element predictions which incluce the effects of end-cap stiffening, cross-section ovalities and material strain hardening are compared to experimental results. The analyses correctly predict the sequence of events leading to collapse, but experimental failure pressures are significantly below predictions. It is concluded that the von Mises yield criterion used in the analysis did not accurately represent the yield behavior of the 1018 steel tubing material of the test-cylinders for the triaxial-stress conditions of interest.     

Exact shell solutions for conical springs

In this article, the equations of equilibrium of conical disk springs of thin and moderate thickness are obtained through the variational principles for thin-walled and thick-walled conical shells. The closed form analytical solutions based on the common deformation hypotheses for the equations of thin- and thick-walled truncated conical shells were achieved. The results of calculations of reaction forces, based on analytical formulae, were compared with the results of finite element analysis, demonstrating the good accuracy of the derived formulae. The theory is extended to incorporate the anisotropy of the material. The problem for optimal anisotropy is solved. The minimal stiffness of the spring is achieved, if the upmost modulus of the orthotropic material is in the meridional direction. Analogously, the highest stiffness is attained, if the maximal elastic modulus circumferentially oriented. Engineering applications of the current theory potentially include Bellville springs and slotted disk springs with moderate flatness for automotive and industrial applications.     

An experimental study on spontaneous adiabatic shear band formation in electro-magnetically collapsing cylinders

The formation of shear bands in collapsing thick-walled cylinders (TWC) occurs in a spontaneous manner. The advantage of studying spontaneous, as opposed to forced, shear localization, is that it highlights the inherent susceptibility of the material to adiabatic shear banding without prescribed geometrical constraints. In the case of spontaneous shear localization, the role of microstructure (grain size and grain boundaries) on localization, is still unresolved. Using an electro-magnetic set-up, for the collapse of thick-walled cylinders, we examined the shear band formation and evolution in seven metallic alloys, with a wide range of strength and failure properties. To assess microstructural effects, we conducted systematic tests on copper and Ti6Al4V with different grain sizes. Our results match quite well with previously reported data on much larger specimens, showing the absence of a size effect, on adiabatic shearing. However, the measured shear band spacings, in this study, do not match the predictions of, existing analytical models, indicating that the physics of the problem needs to be better modeled.     

不同拉压特性的厚壁圆筒极限内压统一解

厚壁圆筒在实际工程领域中应用广泛,若能精确计算出极限内压,对预防事故发生,降低风险有重要意义.工程中存在许多材料,其拉压强度和拉压模量均存在差异,这些差异对极限内压的大小有显著影响.以往研究表明,仅考虑拉压强度与拉压模量的一个方面,计算结果与实际情况存在一定的误差.本文基于双剪统一强度理论,综合考虑中间主应力效应及材料拉压强度和拉压模量的不同,推导了内压作用下厚壁圆筒的弹、塑性状态的应力分布及弹性极限内压、塑性极限内压与安定极限内压的统一解,通过与其他文献对比分析验证了本文计算结果的正确性,分析了半径比、统一强度理论参数、拉压强度比与拉压模量系数对弹性极限内压、塑性极限内压及安定极限内压的影响.结果表明:统一解均随半径比和统一强度理论参数的增大而增大,随拉压强度比的增大而减小,弹性极限内压随材料拉压模量系数的增大而减小,当壁厚增加到一定值后,安定极限内压随材料拉压模量系数的增大而减小;材料的拉压模量不同、拉压强度差异对厚壁圆筒的安定性影响显著,考虑中间主应力效应可使材料的潜能得到更充分发挥,极限内压随半径比的变化规律可为选择合理壁厚提供参考,该结论可为厚壁圆筒的工程应用提供理论依据.     

Chirality in the Torsion of Cylinders with Trigonal Symmetry

Solutions are obtained for a class of torsion problems for cylinders of a material with trigonal material symmetry. In particular, the solutions for elliptical, circular and equilateral triangular cross-sections are presented. These solutions show that the stress distributions are non-chiral and the same as they would be if the material were isotropic; however the in-plane displacements are chiral and different from the isotropic case. The results show that there will be transverse, in-plane stress interactions between the layers of a composite cylinder composed of concentric cylinders of different trigonal materials in torsional loading. Such composite cylinders are structural designs used by nature and by man. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the radial oscillations of incompressible, isotropic, elastic and limited elastic thick-walled tubes

The finite amplitude, free radial oscillations of a thick-walled circular cylindrical tube are studied for an arbitrary incompressible, isotropic and homogeneous rubber-like material having limiting molecular chain extensibility. First, based on classical results for hyperelastic tubes, some results for thick-walled Mooney-Rivlin tubes are described graphically in the phase plane. Then the periodicity of the finite amplitude, free oscillations of a general limited elastic, thick-walled tube is studied; and some analytical results for the Gent model are illustrated in several numerical examples. Results for thick-walled Gent tubes are compared with those for corresponding Mooney-Rivlin tubes; and the motion of thin-walled Gent tubes is illustrated in the phase plane. Physical conclusions are presented. The period of small amplitude oscillations of an arbitrary elastic or limited elastic tube is derived from relations obtained by a linearization of a general class of equations of which the tube problem is a special case. Classical results of the linear theory are thereby recovered and compared with results for Mooney-Rivlin and Gent tubes.     

软化材料厚壁筒的解析解及其稳定性分析

将弹塑性材料的应力应变全过程曲线简化为三线性模型(弹性-线性软化-残余理想塑性),并假设材料服从Tresca屈服准则和关联流动法则,推导出受内压厚壁筒的解析解.在这个解析解的基础上,讨论了厚壁筒的平衡稳定性问题,内压达到临界载荷时,厚壁筒丧失稳定性,其临界载荷就是软化塑性材料厚壁筒的承载能力.     

Double-layered piezo-thermoelastic hollow cylinder under some coupled loadings

In the present paper, a long thick-walled piezo-thermoelastic hollow cylinder with double layers is studied. The effects of temperature on the performance of the cylinder are obtained. Based on the theory of elasticity, the exact solutions of the cylinder under some coupled loadings are found. In the present paper, differences of a piezoelectric parameter between the two layers is taken into account. For comparison, numerical results have been carried out for both double-layered and graded cylinders. At the end of the present paper some discussions are addressed.     

竖井开挖的不稳定性和岩爆的力学机制

基于稳定性理论和弹塑性厚壁筒的解析解,研究了竖井开挖过程的平衡路径,并将屈服后的井壁压力--位移平衡路径曲线应用到围岩的稳定性分析.研究表明,竖井围岩的不稳定及其岩爆的发生不仅取决于岩石材料应力应变全过程曲线的峰后特性,还取决于竖井开挖过程对应的平衡路径曲线的类型.平衡路径曲线能够描述围岩结构加载屈服后的力学行为.平衡路径的曲线形状只可能有两种类型,一种曲线上的点都处于稳定的平衡状态,虽有塑性变形的累积,但不会失稳崩落;另一种曲线则发生极值点型井壁失稳并伴有位移突跳.竖井开挖过程的平衡路径曲线是判定围岩岩爆发生的关键因素.极值点型失稳和位移突跳是均质围岩中岩爆可能发生的一种力学机制.      

INSTABILITY OF SHAFT EXCAVATION AND MECHANISM OF ROCKBURST

基于稳定性理论和弹塑性厚壁筒的解析解,研究了竖井开挖过程的平衡路径,并将屈服后的井壁压力--位移平衡路径曲线应用到围岩的稳定性分析.研究表明,竖井围岩的不稳定及其岩爆的发生不仅取决于岩石材料应力应变全过程曲线的峰后特性,还取决于竖井开挖过程对应的平衡路径曲线的类型.平衡路径曲线能够描述围岩结构加载屈服后的力学行为.平衡路径的曲线形状只可能有两种类型,一种曲线上的点都处于稳定的平衡状态,虽有塑性变形的累积,但不会失稳崩落;另一种曲线则发生极值点型井壁失稳并伴有位移突跳.竖井开挖过程的平衡路径曲线是判定围岩岩爆发生的关键因素.极值点型失稳和位移突跳是均质围岩中岩爆可能发生的一种力学机制.     

The symmetrical adhesive contact problem for circular elastic cylinders

The rolling contact problem involving circular cylinders is at the heart of numerous industrial processes, and critical to any elastohydrodynamic lubrication analysis is an accurate knowledge of the associated contact pressure for the static dry problem. In a recent article [1] the authors have obtained new horizontal pressure distributions, both exact and approximate for various problems involving the symmetrical contact of circular elastic cylinders. In [1] it is assumed that only the circumferential horizontal displacement is prescribed in the contact region while the vertical circumferential displacement is left arbitrary and is assumed to take on whatever value is predicted by the deformation. The advantage of this assumption is that the problem reduces to a single singular integral equation which by transformations can be simplified to an integral equation involving the standard finite Hilbert transform. Here we consider the more general displacement boundary value problem within the contact region, and to be specific we examine the problem with zero vertical circumferential displacement and prescribed horizontal circumferential displacement. The solution of this problem involves two coupled singular integral equations for the horizontal and vertical pressure distributions. Basic equations and some approximate analytical solutions are obtained for symmetrical contact of circular elastic cylinders by both parallel plates and circular cylinders which are either rigid or elastic. Numerical results for the approximate analytical solutions are given for contact by rigid parallel plates and rigid circular cylinders.     

he analytical solutions of thick-walled cylinder of softening material and its stability

将弹塑性材料的应力应变全过程曲线简化为三线性模型(弹性-线性软化-残余理想塑性), 并 假设材料服从Tresca屈服准则和关联流动法则,推导出受内压厚壁筒的解析解. 在这个解析解 的基础上,讨论了厚壁筒的平衡稳定性问题,内压达到临界载荷时,厚壁筒丧失稳定性,其 临界载荷就是软化塑性材料厚壁筒的承载能力.     

考虑拉压强度差效应的厚壁圆筒承载能力分析

应用双剪统一强度理论，考虑材料的拉压异性和同性，推导了在内压力和轴力联合作用下的厚壁圆筒的塑性极限载荷表达式．在该表达式中，当反映中间主应力效应的系数取不同的值时，就能得到按Tresca屈服准则、线性逼近的Mises屈服准则和双剪应力屈服准则的计算结果，并且绘制了在相应准则下的极限应力线图．从而可知：在三维应力状态下，应用该理论，可以获得极限载荷分析的精确解；极限载荷线图与三种屈服准则的屈服曲线是相吻合的；计算的结果可以用于拉压异性和同性的材料，为工程应用提供了理论依据．     

Rate dependent finite strain constitutive model of polyurea

Continuous loading and unloading experiments are performed at different strain rates to characterize the large deformation behavior of polyurea under compressive loading. In addition, uniaxial compression tests are carried out with multistep strain history profiles. The analysis of the experimental data shows that the concept of equilibrium path may not be applied to polyurea. This finding implies that viscoelastic constitutive models of the Zener type are no suitable for the modeling of the rate dependent behavior of polyurea. A new constitutive model is developed based on a rheological model composed of two Maxwell elements. The soft rubbery response is represented by a Gent spring while nonlinear viscous evolution equations are proposed to describe the time-dependent material response. The eight material model parameters are identified for polyurea and used to predict the experimentally-measured stress-strain curves for various loading and unloading histories. The model provides a good prediction of the response under monotonic loading over wide range of strain rates, while it overestimates the stiffness during unloading. Furthermore, the model predictions of the material relaxation and viscous dissipation during a loading-unloading cycle agree well with the experiments.     

非线性Mohr－Coulomb体边坡的极限荷载

本文根据拟线性偏微分方程组的特征线理论，求得无重双曲、抛物Ｍｏｈｒ－Ｃｏｕｌｏｍｂ体边坡极限荷载的解析解．用两参数的双曲线和抛物线可以较好拟合由实验提供的破坏曲线．由其解发现屈服准则和坡角对极限荷载影响很大，理想塑性体的极限荷载也非最小．     

Edge impact of an elastic-plastic semi-infinite plate

The edge impact of an elastic-plastic semiinfinite plate subject to conditions of plane strain is investigated analytically and experimentally. The theoretical analysis is based on the strain-rate-independent theory of plastic-wave propagation. The plate is initially unstressed; the boundary condition for the edge of the plate corresponds to constant-velocity longitudinal impact except that the step in velocity has a finite rise time. Calculations are carried out according to both the approximate one-dimensional theory and the two-dimensional theory for an elastic-plastic isotropic work-hardening material. The rise time for both solutions is chosen so as to optimize the agreement between theoretical and experimental strain-time profiles. A numerical solution of the two-dimensional equations is obtained by using a difference method developed by Clifton;¹ the onedimensional approximation is solved by the well-known method of characteristics. The problem was approximated experimentally by the axial collision of 4-in.-diam annealed aluminum thick-walled cylinders with a diameter-to-wall-thickness ratio of ten. For impact velocities of 90, 130 and 160 ips (corresponding to maximum strains of 0.12, 0.22 and 0.36 percent, respectively), dispersive characteristics and maximum strain amplitudes of the strain wave are found to be in good agreement with the theoretical predictions of both solutions. However, the two-dimensional solution indicates that the stresses, strains and velocities in regions of high strain-rate are highly nonuniform across the plate thickness.