Unloading of thick-walled cylinders that have been plastically deformed

Two analytical solutions are used to predict load-strain relations for unloading of thick-walled cylinders. The solutions assume that the material is an isotropic-hardening material that obeys the von Mises yield condition. The loading function for the material for the unloading of the cylinders was obtained from tension-compression specimens that were unloaded and reverse loaded from several points along the tension stress-strain diagram. Good agreement is indicated between the unloading load-strain curves obtained from two thick-walled cylinders made of SAE 1045 steel and the curves predicted by the analytical solutions. The analytical solutions predict that the beneficial circumferential compressive residual stresses at the inside of the cylinders decrease by about 50 percent during the unloading.     

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.     

Elevated-temperature creep and relaxation of torsion-tension members

Based on the assumption that each material satisfies the condition for isotropic hardening for a von Mises material, an incremental solution is developed to predict axial-strain creep curves and maximum shearing-strain creep and relaxation curves for solid circular torsion-tension members subjected to proportionate and nonproportionate stepped loading including creep in tension and relaxation in torsion. Test data are obtained from torsion-tension members made either of annealed OFHC copper at 800°F (427°C) or hot-rolled SAE 1045 steel at 950°F (510°C). The loading histories include either four stepped proportionate load changes, four stepped nonproportionate load changes, or torsion-tension loading in which the axial load remains constant and the torsional load is allowed to relax during two loading periods of the test. Each test duration is about 100 h. Good agreement is indicated between the predicted and measured creep and relaxation curves.     

Time-dependent ratchetting experiments of SS304 stainless steel

The time-dependent strain cyclic characteristics and ratchetting behaviours of SS304 stainless steel were investigated by uniaxial/multiaxial cyclic loading tests at room and elevated temperatures (350 and 700 °C). The effects of loading rate, peak/valley strain or stress holds, ambient temperature and non-proportional loading path on the cyclic softening/hardening and ratchetting behaviours of the material were discussed. It is shown that: the cyclic deformation of the material presents remarkable time-dependence at room temperature and 700 °C; the cyclic hardening feature and ratchetting strain depend significantly on straining or stressing rate, hold-time, ambient temperature and the non-proportionality of loading path; the time-dependent ratchetting is resulted from the slight opening of hysteresis loop and visco-plasticity together, and the viscosity is a dominating factor at 700 °C; at 350 °C, abnormal rate-dependence and quick shakedown of ratchetting are observed due to the dynamic strain aging of the material at this temperature. Some significant conclusions are obtained, which are useful to construct a constitutive model to describe the time-dependent ratchetting behaviour of the material. It is also stated that the unified visco-plastic constitutive model discussed here cannot provide reasonable simulation to the time-dependent ratchetting at 700 °C, especially to that with certain peak/valley stress hold, since the effect of the high viscosity on time-dependent ratchetting cannot be properly described by using a unified visco-plastic flow rule.     

Evaluation of multiaxial theories for room-temperature plasticity and elevated-temperature creep and relaxation of several metals

Based on the assumption that the material satisfies the condition of isotropic hardening for either a von Mises or a Tresca material, finite-strain theories are derived for solid circular torsion members for the conditions that the inelastic deformations are either time independent or time dependent. In the latter case, both creep and relaxation theories are derived. At room temperature the theories are evaluated for each of eight metals using finite-strain data from tension, compression and torsion members. Of the six metals that are found to satisfy the condition required for the isotropic-hardening model, two are von Mises, one is Tresca, and the other three are between von Mises and Tresca. At elevated temperatures, the theories are evaluated for each of five of the latter six metals, using data from tension and torsion members. Material properties obtained from the tension specimens are used to predict creep and relaxation curves for the torsion members. Contrary to the results at room temperature, creep curves for the torsion members do not all fall within the region bounded by von Mises and Tresca theories. In the case of relaxation, either excellent agreement is obtained between the von Mises strain-hardening theory and experimental data or the theory is 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.     

Experimental creep buckling of aluminum cylinders in axial compression

Creep-buckling tests were conducted on aluminum alloy 2024-0 circular cylinders in axial compression at 500° F having nominalR/t values of 90 and 50. Creep-buckling times for a variety of applied creep-stress values were compared with theoretical predictions of Gerard's unified theory of creep buckling of columns, plates and shells. In this theory, creep-buckling solutions are analogous to plastic-buckling solutions, provided that the material parameters used in the theoretical relation are developed from constant-strain-rate stress-strain data derived by a graphical process from compressive-creep data. The theoretical data were evaluated using appropriate classical plastic-buckling theory and previously obtained creep data on the 2024-0 aluminum material at 500° F. End shortening of the cylinders was autographically recorded during the tests and creep-buckling times were obtained from an analysis of the end-shortening record. A comparison of theory and test data indicated that the theory was somewhat conservative in predicting creep-buckling times. The discrepancy may have been due, in part, to the uncertainty in determining the precise time at which the experimental cylinders buckled. The cylinders withR/t～90 buckled in the axisymmetric mode for the lower creep stresses while, forR/t～50, all buckling occurred in the axisymmetric mode.     

Compression-impact testing of aluminum at elevated temperatures

This paper presents and experimental technique for determining compressive stress-strain curves well into the plastic range of relatively soft metals at strain rates from 300 to 2000 sec^?1 at six temperatures from 30 to 550° C. More than 100 curves were obtained on annealed 1100° F aluminum. The strain-rate dependence in these tests could be fitted quite well either by a power function (log-log plot) or by a semilogarithmic plot, but the power function gave a better correlation of the present data with that obtained at lower strain rates by Alder and Phillips.¹     

Dynamic Equibiaxial Flexural Strength of Borosilicate Glass at High Temperatures

A novel high temperature ring-on-ring Kolsky bar technique was employed to investigate the dynamic equibiaxial flexural strength of borosilicate glass at temperatures ranging from room temperature up to 750°C. This technique provided non-contact heating of the glass specimen and prevented thermal shocks in the specimen. Experimental results at the loading rate of 22.5 MN/s showed significant temperature dependence on the flexural strength. To explore the mechanisms of this temperature effect, controlled surface cracks were introduced on the tensile surface of the glass specimens using a Vickers indentation technique. These surface cracks were then heat treated under the same thermal histories as those tested in the high temperature dynamic experiments. The evolution of crack morphologies at 200°C, 550°C and 650°C were examined. The results indicate that residual stress relaxation may play an important role in the strengthening below 200°C, while crack healing and blunting may account for the strengthening above 500°C.     

Analysis of biaxial-fatigue damage at elevated temperatures

Two-level cumulative-damage fatigue tests were conducted on tubular 304 stainless-steel specimen under biaxial-strain conditions at elevated temperatures. Effects of temperature, biaxiality and sequence of straining were investigated. The experimental results are forwarted with a new approach that utilizes the Miner cumulative-damage rule. This approach has shown that fatigue damage at elevated temperatures of 538°C (1000°F) and 649°C (1200°F) accelerates and decelerates as a result of time of exposure to a given loading sequence. The effect of biaxiality is shown through the behavior of the material under axial and shear-strain components. The axial (tensile) strain component has shown to be the severest detrimental damaging component when compared to a shear-strain component. A damage mechanism emerges from the interaction of temperature and loading sequence. Its significance can be observed only when a certain life ratio has been exhausted.     

Multiaxial-stress studies on rigid polyurethane foam

Thin-walled tubes of rigid polyurethane foam, at 5 lb/cu ft and 4 lb/cu ft nominal density, were subjected to combined stresses. A universal-test machine applied the axial tensile or compressive stress; an independent pressurization system supplied the internal pressure to the specimen. Changes in diameter, length and wall thickness were measured at room temperature and at ?50°F. Failure envelopes in 2-dimensional stress space were obtained for the two densities at room temperature and for the greater density at ?50°F. For the wall-thickness range investigated (0.15≤t≤0.25 in.), no effect on the failure envelope was observed. Material constants were obtained in terms of orthotropic elastic theory (at 5 lb/cu ft and room temperature) to describe the material in terms of structure.     

Temperature dependent bulge test for elastomers

The bulge test is a particularly convenient testing method for characterizing elastomers under biaxial loading. In addition, it is convenient to utilize this test for validating material models in simulation due to the heterogeneous strain field induced during inflation. During the bulge test the strain field for elastomers covers uniaxial tension at the border to pure shear and equibiaxial tension at the pole. Elastomeric materials exhibit a hyperelastic material behavior, with a dependency on temperature and loading rate. The temperature effect on the mechanical behavior during biaxial loading is considered in the present study. A bulge test setup combined with a temperature chamber is developed in order to characterize this effect, and an exemplary temperature dependent characterization of a poly(norbornene) elastomer is performed with this setup. The equibiaxial stress–strain curves measured at 60 °C, 20 °C and −20 °C are presented.     

Large strain creep analysis of thick-walled cylinders

The finite-strain theory has been used to study the creep behaviour of a thick-walled cylinder under large strains. The analysis is divided into two parts. In part 1 the creep deformation of a thick-walled cylinder of an anisotropic material subjected to internal pressure has been discussed. The effect of the anisotropy has been depicted graphically. It is found that the anisotropy of the material has a significant effect on the axial stress, strain and strain rate. Part 2 of the paper deals with the creep analysis of cylinders of either isotropic or anisotropic materials subjected to combined internal and external pressures. The effect of the anisotropy is found to be similar to that found in part 1. It is seen, however, that the introduction of external pressure results in decreasing the strain rate and thus increasing the life of the cylinder.     

Creep of concentrically loaded circular plates

Using a strain-energy approach and the Rayleigh-Ritz procedure, a nonlinear compressible theory was developed to predict the creep deflections of circular plates laterally loaded by a point force acting at the center of the plates. Both clamped and simply supported boundary conditions were considered. The theory applies to transient as well as to steady-state creep. The stress-strain-time relations for the material were represented by a family of isochronous stress-strain curves. For analytical purpose, each curve was approximated by an arc hyperbolic sine function. Experimental data were obtained from eight plates made of high-density polyethylene tested in a controlled-atmosphere room. Material properties were obtained from tension and compression specimens. All test members were subjected to the same load history. An initial load was held constant for a specified time and then increased by 10 percent of its initial value at four equal intervals of time. Good agreement was found between theory and experiment.     

Linear stability analysis for a thermoviscoplastic material under cyclic axial loading

Some mechanical properties exhibit a very strong dependence upon temperature; these evolutions can be properly analyzed by the steady state response in cyclic loading. To relate experimental conditions to thermomechanical characteristics, the existence and the stability of steady state solutions are studied for cylinders submitted to cyclic compression. The material, considered as rigid viscoplastic, is modeled by a non-Newtonian temperature dependent viscous law. Closed form solutions are obtained in the framework of a large deformation theory by neglecting thermal expansion and inertia effects. Steady state regime is analyzed. The stress versus strain rate response and the temperature distribution are established as functions of the geometry of the cylinder, the loading characteristics and the material parameters. The stability of steady state solutions is analyzed with use of a linear perturbation scheme.Received: 4 July 2002, Accepted: 5 August 2004, Published online: 24 February 2005PACS: 46.15.Ff, 83.60.St Correspondence to: F. Dinzart     

The Effect of Temperature on Anisotropy Properties of an Aluminium Alloy

The temperature influence on the mechanical behaviour during plastic deformation of an AA5754-O aluminium alloy has been investigated by several experimental tests. First, monotonous tensile tests were carried out from room temperature up to 200°C with a classical tensile machine and with a less conventional testing apparatus involving the heating of the sample by Joule effect. With this second testing apparatus, the strain fields and tensile curves were obtained in function of temperature by means of a non-contacting optical 3D deformation measuring system. Moreover, shear tests were performed in the same temperature range. It is shown that the anisotropy coefficients are rather constant within this temperature range, with a relative variation less than 8%. For both tensile and shear tests, the stress levels are similar at the beginning of straining at room temperature and 150°C, except that the Portevin?CLe Chatelier (PLC) phenomenon disappears at elevated temperature, and then evolves differently. At 200°C, the stress level is clearly below whatever the deformation. In the framework of drawing process, the formability of this alloy at temperatures higher than 150°C seems to be improved.     

Testing the space-shuttle thermal-protection coating

A light-weight insulation material and its protective glassy coating will protect thespace shuttle from temperatures as high as 1250°C (2300°F). The critical performance characteristics of the brittle coating are investigated using testing techniques developed to accommodate these extreme environments and the delicate material. These include an ultimate-strain test-specimen geometry which circumvents problems created by flawed edges, as well as a tension specimen preparation and loading system with which premature failures due to excessive bending moment are avoided. Additionally, an elevated-temperature mechanical strain transducer—useable at more than 870°C (1600°F)—is described. Potential alterations to this sensor are discussed which would make it functional at up to 1600°C (3000°F).     

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．     

Remarks on the Instability of an Incompressible and Isotropic Hyperelastic, Thick-Walled Cylindrical Tube

The problem of instability of a hyperelastic, thick-walled cylindrical tube was first studied by Wilkes [1] in 1955. The solution was formulated within the framework of the theory of small deformations superimposed on large homogeneous deformations for the general class of incompressible, isotropic materials; and results for axially symmetrical buckling were obtained for the neo-Hookean material. The solution involves a certain quadratic equation whose characteristic roots depend on the material response functions. For the neo-Hookean material these roots always are positive. In fact, here we show for the more general Mooney–Rivlin material that these roots always are positive, provided the empirical inequalities hold. In a recent study [2] of this problem for a class of internally constrained compressible materials, it is observed that these characteristic roots may be real-valued, pure imaginary, or complex-valued. The similarity of the analytical structure of the two problems, however, is most striking; and this similarity leads one to question possible complex-valued solutions for the incompressible case. Some remarks on this issue will be presented and some new results will be reported, including additional results for both the neo-Hookean and Mooney–Rivlin materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Three dimensional analytical solution for finite circular cylinders subjected to indirect tensile test

This paper derives a new three-dimensional (3-D) analytical solution for the indirect tensile tests standardized by ISRM (International Society for Rock Mechanics) for testing rocks, and by ASTM (American Society for Testing and Materials) for testing concretes. The present solution for solid circular cylinders of finite length can be considered as a 3-D counterpart of the classical two dimensional (2-D) solutions by Hertz in 1883 and by Hondros in 1959. The contacts between the two steel diametral loading platens and the curved surfaces of a cylindrical specimen of length H and diameter D are modeled as circular-to-circular Hertz contact and straight-to-circular Hertz contact for ISRM and ASTM standards respectively. The equilibrium equations of the linear elastic circular cylinder of finite length are first uncoupled by using displacement functions, which are then expressed in infinite series of some combinations of Bessel functions, hyperbolic functions, and trigonometric functions. The applied tractions are expanded in Fourier–Bessel series and boundary conditions are used to yield a system of simultaneous equations. For typical rock cylinders of 54 mm diameter subjected to ISRM indirect tensile tests, the contact width is in the order of 2 mm (or a contact angle of 4°) whereas for typical asphalt cylinders of 101.6 mm diameter subjected to ASTM indirect tensile tests the contact width is about 10 mm (or a contact angle of 12°). For such contact conditions, 50 terms in both Fourier and Fourier–Bessel series expansions are found sufficient in yielding converged solutions. The maximum hoop stress is always observed within the central portion on a circular section close to the flat end surfaces. The difference in the maximum hoop stress between the 2-D Hondros solution and the present 3-D solution increases with the aspect ratio H/D as well as Poisson’s ratio ν. When contact friction is neglected, the effect of loading platen stiffness on tensile stress in cylinders is found negligible. For the aspect ratio of H/D = 0.5 recommended by ISRM and ASTM, the error in tensile strength may be up to 15% for both typical rocks and asphalts, whereas for longer cylinders with H/D up to 2 the error ranges from 15% for highly compressible materials, and to 60% for nearly incompressible materials. The difference in compressive radial stress between the 2-D Hertz solution or 2-D Hondros solution and the present 3-D solution also increases with Poisson’s ratio and aspect ratio H/D. In summary, the 2-D solution, in general, underestimates the maximum tensile stress and cannot predict the location of the maximum hoop stress which typically locates close to the end surfaces of the cylinder.