The analysis of cyclically loaded creeping structures for short cycle times

In a recent paper [1] it was shown that the evaluation of certain bounding solutions for a structure subjected to cyclic loading was equivalent to assuming that the cycle time Δt was short compared with a stress redistribution time. Comparisons between values which are likely to occur in creep design situations indicated that Δt may often be assumed to be small and the bounding solution may be expected to closely approximate the actual stress history. In this paper the solution for the limiting case when Δt → 0 is evaluated for a class of constitutive relationships which may be expressed in terms of a finite number of state variables. Strain-hardening viscous, visco-elastic and Bailey-Orowan equations are discussed and particular solutions for which the residual stresses remain constant in time are derived. The solution for a non-linear visco-elastic model indicates that, for the stationary cyclic state, the constitutive equation need only predict the creep strain over a discrete number of cycles and need not predict the strains during a cycle. This observation should considerably simplify creep analysis.The solution of a simple example demonstrates the similarity between the predicting of the various constitutive relationships for isothermal problems. In fact they provide virtually identical solutions when expressed in terms of reference stress histories. The finite element solution of a plate containing a hole and subjected to variable edge loading is also presented for a viscous material. The solutions show behaviour which is similar to that of the two bar structure.     

Estimates of the rupture life of structural components subjected to proportional cyclic loading

The process of creep in metals is associated with physical mechanisms which cause internal damage. This damage weakens the material; so that for a given stress level, the strain-rate increases with time. As a result of this behaviour, stress redistribution occurs which can greatly influence the ruptuie life of structural components. By use of the appropriate constitutive relations which model the tertiary portion of the creep curve, it is possible to estimate the life of a structural component by use of finite element methods. Unfortunately, the procedure is demanding on computer time; and, as a result, considerable attention has been given to the possibility of the use of bounding techniques which ease the computing problem and which are particularly useful at the early stage of design. Techniques have already been developed for bounding the rupture life of structural components subjected to constant loading and these are found to be useful. In this paper, bounding procedures are developed for structural components subjected to proportional cyclic loading. The results may be expressed in terms of a representative rupture stress, so that structural-component performance can readily be related to material behaviour.     

Analytical elasto-creep model of interfacial thermal stresses and strains in trilayer assemblies

A two-dimensional model has been developed for thermal stresses, elastic strains, creep strains, and creep energy density at the interfaces of short and long trilayer assemblies under both plane stress and plane strain conditions. Both linear (viscous) and non-linear creep constitutive behavior under static and cyclic thermal loading can be modeled for all layers. Interfacial stresses and strains are approximated using a combination of exact elasticity solutions and elementary strength of materials theories. Partial differential equations are linearized through a simple finite difference discretization procedure. The approach is mathematically straightforward and can be extended to include plastic behavior and problems involving external loads and a variety of geometries. The model can provide input data for thermal fatigue life prediction in solder or adhesive joints. For a typical solder joint, it is demonstrated that the predicted cyclic stress–strain hysteresis shows shakedown and a rapid stabilization of the creep energy dissipation per cycle in agreement with the predictions of finite element analysis.     

Multiaxial creep and cyclic plasticity in nickel-base superalloy C263

Physically-based constitutive equations for uniaxial creep deformation in nickel alloy C263 [Acta Mater. 50 (2002) 2917] have been generalised for multiaxial stress states using conventional von Mises type assumptions. A range of biaxial creep tests have been carried out on nickel alloy C263 in order to investigate the stress state sensitivity of creep damage evolution. The sensitivity has been quantified in C263 and embodied within the creep constitutive equations for this material. The equations have been implemented into finite element code. The resulting computed creep behaviour for a range of stress state compares well with experimental results. Creep tests have been carried out on double notched bar specimens over a range of nominal stress. The effect of the notches is to introduce multiaxial stress states local to the notches which influences creep damage evolution. Finite element models of the double notch bar specimens have been developed and used to test the ability of the model to predict correctly, or otherwise, the creep rupture lifetimes of components in which multiaxial stress states exist. Reasonable comparisons with experimental results are achieved. The γ^′ solvus temperature of C263 is about 925 °C, so that thermo-mechanical fatigue (TMF) loading in which the temperature exceeds the solvus leads to the dissolution of the γ^′ precipitate, and a resulting solution treated material. The cyclic plasticity and creep behaviour of the solution treated material is quite different to that of the material with standard heat treatment. A time-independent cyclic plasticity model with kinematic and isotropic hardening has been developed for solution treated and standard heat treated nickel-base superalloy C263. It has been combined with the physically-based creep model to provide constitutive equations for TMF in C263 over the temperature range 20–950 °C, capable of predicting deformation and life in creep cavitation-dominated TMF failure.     

Variation of crack-opening stresses in three-dimensions: finite thickness plate

Crack growth and closure behavior of a center cracked finite thickness plate subjected to constant amplitude cyclic load is investigated by means of a three-dimensional elastic-plastic finite-element analysis. Results are obtained for initial half crack length c_i to half plate thickness t ratios of c_i/t = 3.891 and 1.465 which shall be referred to, respectively, as thin and thick plate. A constant amplitude load with R = S_min/S_max = 0.1 and S_max/σ₀ = 0.25 is applied, where S stands for the stress amplitude and σ₀ the effective yield stress. Crack closure for the thinner plate is found to be largest at and near the free plate surface and to decrease toward the interior during the unloading portion of cyclic loading. The closure pattern stabilizes at the interior and exterior regions, respectively, for c_i/t = 3.981 at 0.34S_max and 0.56S_max and for c_i/t = 1.465 at 0.26S_max and 0.46S_max.A load-reduced displacement technique was used to determine crack-opening stresses at specified locations in the plate from the displacements calculated after 7th cycle (using unloading and reloading portions of cyclic loading). All locations were on the plate exterior surface and were located behind the crack tip and at the centerline of the crack. The opening stresses at the specified points as certain percentage of the maximum stress amplitude were obtained.     

Second-order strain accumulation in workhardening media

Summary Second-order or cross effects are the result of quadratic tensor terms in the constitutive equations of isotropic elastic, viscous and visco-elastic media, which are required by the condition of tensor invariance of those relations. These effects are most pronounced when they are clearly separable from the first-order deformation, as in the case of second-order elongation and volume change of an elastic cylinder subject to a twisting moment (Poynting effect, dilatancy) or of second-order normal stress in the case of shear flow of polymeric liquids (Weissenberg effect).An accumulating second-order effect (Ronay effect) has been discovered in experiments on strain-hardening metal specimens in reversed torsion. While thePoynting effect vanishes at zero strain in elastic solids and theWeissenberg effect at zero velocity in polymeric fluids, the second-order strain increments accumulate in strainhardening media with the number of repeated torsion cycles. Hence their observation is simple and does not require the elaborate procedures necessary for the observation od second-order effects in elastic solids, viscous fluids and visco-elastic substances.It can be shown that the observed second-order strain accumulation (Ronay effect) is implied by thePrager-Hill stress strain-increment relation for strain-hardening media, combined with theKadshevich-Novozhilov formulation of kinematic hardening, provided that the arbitrary condition that strain-increment and stress change sign simultaneously is not imposed.Paper read at the Annual Meeting of the German Rheologists, Berlin-Dahlem June 7–10, 1966.     

A method for the evaluation of design limits for structural materials in a cyclic state of creep

The purpose of the present paper is to demonstrate how the minimum theorems proposed in an accompanying paper (Ponter and Boulbibane, 2002) can be utilised in the prediction of the deformation and life assessment of structures subjected to cyclic mechanical and thermal loadings. The developed method, which is based upon bounding theorems and an associate programming method, the Linear Matching method, takes into account the changes in residual stress field occurring within a cycle. Although the solution provided a bound on the inelastic work, it also appears that generally the displacements predicted by this solution are smaller than those that would be predicted by the rapid cycle solution. By way of illustration a simple non-linear viscous model is adopted and a number of solutions are presented involving a Bree plate problem subjected to cyclic histories of load and temperature. An elastic follow-up factor is identified as a key design parameter for high temperature dwell periods.     

Creep analysis with a stress range dependent constitutive model

Many materials exhibit the stress range dependent creep behavior. The power law creep observed for a certain stress range changes to the viscous type creep as the stress value decreases. Recently published experimental data for advanced heat resistant steels indicate that the high creep exponent (in the range 7–12) may decrease to the low value of approximately unity within the stress range relevant for engineering structures. The aim of this paper is to analyze the influence of the stress range dependent power-law-viscous creep transition on the behavior of structures at elevated temperature. A constitutive model for the minimum creep rate is introduced to describe both the linear and the power law creep depending upon the stress level. To demonstrate basic features of the stress range dependent creep modeling, several elementary examples from structural mechanics are presented. They include a stress relaxation problem, a beam subjected to pure bending and a pressurized thick-walled cylinder. Based on the uni-axial transition stress the transition value of the external load is estimated such that above this value the power law can be applied. For the loading levels below this value the character of the stress distribution as well as the stress values are essentially influenced by the viscous creep.     

Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model

A simple model consisting of the Upper Convected Maxwell constitutive equation and a kinetic equation for destruction and construction of structure, first proposed by Fredrickson in 1970, is used here to reproduce the complex rheological behavior of viscoelastic systems that also exhibit thixotropy and rheopexy under shear flow. The model requires five parameters that have physical significance and that can be estimated from rheological measurements. Several steady and unsteady flow situations were analyzed with the model. The model predicts creep behavior, stress relaxation and the presence of thixotropic loops when the sample is subjected to transient stress cycles. Such behavior has been observed with surfactant-based solutions and dispersions. The role of the characteristic time for structure built up, λ, in the extent and shape of the thixotropic loops is demonstrated.     

Arrest characteristics of an impacted crack dissipating plastic energy

When a structural member is accidentally struck, an initial defect may grow and then arrest. An estimate of its size increase is made by considering the geometry of a centrally cracked panel subjected to a step function load in time. Two load amplitudes differing by a factor of five are considered. Under impact, the crack accelerates and then decelerates prior to arrest. The dynamic characteristics depend on the interaction of the elastic-plastic stress waves intervening with the physical boundaries. This effect is assessed quantitatively by computing for the energy stored in a unit volume of material and by incorporating sliding nodes in the finite element method. The energy dissipated by plastic deformation must be accounted for as it is no longer available for creating new macrocrack surface. Obtained are the near tip normal stresses that are found to change from compression to tension. Their magnitude is considerably larger than the corresponding static values. Increase in crack length changes from 0.87% to 20.6% when the magnitude of the impact load is raised five times. The rate of change of the strain energy density factor ΔS with crack growth Δa is found to be governed by the condition ΔS/Δa = const. during loading while dynamic relaxation corresponded to a nonlinear behavior. The physical implication of this remains to be clarified in view of the fact that plasticity theory may not adequately explain the near tip crack behavior.     

Influence of load amplitude and uniaxial tensile properties on fatigue crack growth

The rate at which energy is accumulated within a unit volume of material in fatigue is assumed to depend not only on load-time history but also on the specimen size and geometry in addition to material type. A threshold level for the hysteresis strain energy density function accumulated in the material is used for predicting macrocrack growth. This is accomplished by application of the incremental theory of plasticity for each increment of crack growth. The accumulated hysteresis strain energy density factor ΔS to crack growth increment Δa ratio is found to be constant for fixed specimen size and loading, i.e., . Results are obtained for the cylindrical bar specimens with a penny-shaped defect at the center subjected to a constant amplitude and frequency loading. The resistance curves in the ΔS versus Δa plot are parallel lines as specimen size is altered. This information provides a rational means for predicting the influence of specimen size on fatigue lifetime.The results are also compared with those found for geometrically similar plate specimens with line cracks. Cylinder bar specimens of the same material are found to sustain more load cycles prior to catastrophic failure.     

Mechanics of adhesive contact on a power-law graded elastic half-space

We consider adhesive contact between a rigid sphere of radius R and a graded elastic half-space with Young's modulus varying with depth according to a power law E=E₀(z/c₀)^k (0<k<1) while Poisson's ratio ν remaining a constant. Closed-form analytical solutions are established for the critical force, the critical radius of contact area and the critical interfacial stress at pull-off. We highlight that the pull-off force has a simple solution of P_cr=−(k+3)πRΔγ/2 where Δγ is the work of adhesion and make further discussions with respect to three interesting limits: the classical JKR solution when k=0, the Gibson solid when k→1 and ν=0.5, and the strength limit in which the interfacial stress reaches the theoretical strength of adhesion at pull-off.     

Stress intensity factor range and propagation mode of surface cracks under rolling–sliding contact

Finite element analyses were conducted in order to evaluate the mode I and mode II stress intensity factors for inclined edge cracks under cyclic contact load under rolling and rolling–sliding condition. The SIF range depends on crack orientation, crack length to Hertzian contact zone half-width ratio, friction between the crack faces and friction on the contact surface. The results were combined in two compact functions that determine the ΔK_I and ΔK_II values. The crack propagation mode and direction were investigated using both the maximum stress criterion and the minimum strain energy density criterion. The results are displayed in graph form, which allows a fast evaluation of the crack growth condition.     

A viscoplastic constitutive model for nickel-base superalloy,part 2: modeling under anisothermal conditions

In Part 2 of this study, extensive deformation tests were carried out on the nickel-base polycrystalline superalloy IN738LC under isothermal and anisothermal conditions between 450 and 950 °C. Under the isothermal conditions, the material showed almost no rate/time-dependency below 700 °C, while it showed distinct rate/time-dependency above 800 °C. Regarding the cyclic deformation, slight cyclic hardening behavior was observed when the temperature was below 700 °C and the imposed strain rate was fast, whereas in the case of the temperature above 800 °C or under slower strain rate conditions, the cyclic hardening behavior was scarcely observed. Unique inelastic behavior was observed under in-phase and out-of-phase anisothermal conditions: with an increase in the number of cycles, the stress at higher temperatures became smaller and the stress at lower temperatures became larger in absolute value although the stress range was approximately constant during the cyclic loading. In other words, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at lower temperatures. Based on the experimental results, it was assumed that evolution of the variable Y that had been incorporated into a kinematic hardening rule in Part 1 of this study is active under higher temperatures and is negligible under lower temperatures. The material constants used in the constitutive equations were determined with the isothermal data, and were expressed as functions of temperature empirically. The extended viscoplastic constitutive equations were applied to the anisothermal cyclic loading as well as the monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present viscoplastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions, owing to the consideration of the variable Y.     

Influence of mean stress on the fatigue crack growth characreristics of CrlMo steel tube at elevated temperature

The fatigue crack growth characteristics of CrlMo steel have been investigated at 861 K over the R-ratio range 0.1–0.7 utilising a dwell time of 10 min. at maximum load. All tests were conducted under load control in a laboratory air environment. It was established that the R-ratio significantly affected the fatigue crack extension behaviour inasmuch that with increasing R-ratio, the critical ΔK level for the onset of creep fatigue interactive growth, ΔK^IG, decreased from 20 to 7 MPa√m and the threshold stress intensity, ΔK_th, decreased from 9 to about 3 MPa√m. At intermediate ΔK levels, i.e. between ΔK_th and ΔK^IG, the fatigue crack extension rates, for all R-ratio values, resided on or slightly below the CTOD line, which represents the upper bound for contrnuum controlled fatigue crack growth. Creep fatigue interactive growth was typified by crack extension rates that reside above the CTOD line with a ΔK^IG dependence; the attainment of some critical creep condition or crack linkage condition which causes the abrupt change in crack extension behaviour at ΔK^IG; and crack extension occurs almost exclusively in an intergranular manner. The R-ratio and ΔK^IG followed a linear relation. A literature review concerning the effect of temperature on the threshold fatigue crack growth characteristics of low alloy ferritic steels demonstrated powerful effects of temperature; the magnitude of these effects, however, were dependent upon the testing temperature regime and R-ratio level. The effect of R-ratio on ΔK_th was greatest at temperatures >400°C, significant at ambient temperatures and least in the temperature range 90°C to <300°C. The relationship between temperature and ΔK_th, at a given R-ratio, exhibited a through and a minimum ΔK_th value was observed in the temperature range 200–250°C. The magnitude of the temperature effects on ΔK_th decreased with increasing R-ratio. Such effects of temperature and R-ratio on ΔK_th was reasonably explained in terms of crack closure effects. Finally, the present elevated temperature fatigue crack growth data exhibited massive crack extension enhancement values when compared to ambient near-threshold fatigue crack growth data for CrlMo steel. Such large enhancement values were the combined effects of temperature (environment) and frequency.     

Investigations into cyclic creep of materials (review)

Conclusions In this review, we analyzed the studies known to the author on cyclic creep of structural materials published since 1936. Some of the studies could not be examined but references to these studies and a brief analysis of the results can be found in [3, 54–56, 115]. The classification of these studies and interpretation of the results presented by the author and also the terminology used do not pretend to be unambiguous and universal. Slghtly different approaches to solving this problem were proposed in, for example, [7, 55, 66].It is important to continue investigations in the cyclic creep area in the following directions:generalization of the unidimensional models of cyclic creep for the multiaxial stress states;experimental examination and construction of the theory of cyclic creep for nonstationary loading conditions;examination of the development of cyclic creep in heterogeneous and anisotropic materials;development of the method of calculating the stress-strain state and endurance of structural members in the creep conditions in cyclic loading.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev, Translated from Prikladnaya Mekhanika, Vol. 23, No. 12, pp. 3–19, December, 1987.     

岩石拉压蠕变试验仪研制及应用

为了探索岩石在周期性恒定拉伸、压缩荷载作用下的蠕变行为,结合杠杆原理,设计研制了一种岩石杠杆式拉伸、压缩蠕变试验仪。该仪器具有挂重质量小、可方便切换拉、压荷载等特点。首先,通过标定好的数显式拉压荷重传感器对该试验仪拉伸、压缩荷载进行了校正,得出试验仪压缩、拉伸荷载杠杆扩力比分别为81.29倍与59.46倍,挂重质量与施加在岩样荷载呈线性关系(压缩作用下线性关系相关系数R2=0.99975,拉伸作用下R2=0.9991),荷载施加稳定。最后,采用该试验仪对红砂岩进行了单轴恒定拉、压循环荷载下的蠕变试验,探讨了受荷岩样拉压蠕变规律。上述成果丰富了岩石蠕变测试与研究内容,有助于岩石力学试验测试的发展。     

Transient analysis of a three-dimensional plate by the ray grouping technique

The propagation of transient waves in an elastic plate excited by an arbitrary loading is investigated. Exact three-dimensional transform solutions are derived for a plate suddenly loaded on one of its bounding surfaces, with the inversion effected by Cagniard's technique. The solution is based on three canonical problems for finding the n-th reflected waves from only the information on the (n − 1)-th reflected waves. The technique automatically groups rays which arrive simultaneously at one point, thus simplifying the computations needed by the ray tracing technique. A nonaxially symmetric sample loading is selected for demonstrating the technique, and numerical results are presented. The Ray Grouping Technique can be extended to the layered medium case.     

Modeling unidimensional creep by the method of isochrones

Conclusions We have constructed a unidimensional creep model which describes all characteristic stages of the process and considers cyclicity of loading. The base creep experiment can be limited to the steady-state stage and does not entail determination of a damage function.It was shown that the character of creep can be determined from the type of instantaneous strain curve. It was established in particular that only transient creep will occur for media which exhibit linear strain-hardening, all stages will be realized in media exhibiting exponential strain-hardening, and an ideally plastic medium will behave over time as a linearly viscous medium.Creep is accelerated in media in which a cyclic load reduces the ductility indices. If strain-hardening occurs, then a cyclic load retards creep.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 12, pp. 71–79, December, 1988.     

Plasticity of a high strength steel alloy

Experimentally determined plastic constitutive equations of the parabolic form (σ − σ_y = β(ε − ε_y)1/2) are presented for a high strength alloy steel. Two deformation moduli β were required to describe the quasistatic compression data, both of which, as well as the point of change, were predicted by a mode index and transition strain structure of a general theory of plasticity. Dynamic strain, duration of impact, and final strain distribution were measured on specimen rods subjected to axial, symmetric, and constant velocity impact. The dynamic yield stress was 16% higher than the quasistatic value. The dynamic response function, deduced from a simple wave propagation theory, was also parabolic and a single deformation modulus, equal to the initial quasistatic value, applied. Thus, it was shown that the form of the quasistatic response function was preserved in dynamic loading, and that the increase of the dynamic stress was due to the increase of the yield stress.