The creep deformation of vibrating structures

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.     

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.     

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.     

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.     

Rheological–dynamical analogy: Prediction of damping parameters of hysteresis damper

This research aims to predict the damping parameters of hysteresis damper based on an analytical rheological–dynamical (RDA) visco-elasto-plastic solution of one-dimensional longitudinal continuous vibrations of a bar. A visco-elasto-plastic bar or damper is an energy dissipation device. An attempt is made to estimate quantitatively the influence of material physical parameters of materials on the damping ratio in both the linear visco-elastic analysis and the nonlinear visco-elasto-plastic analysis of damper subjected to external vibration forces. Two types of damping are considered: viscous damping in the case of linear analysis, defined as stiffness and/or mass proportional and, in the case of nonlinear analysis, hysteresis damping caused by inelastic deformations of damper. Owing to the visco-elastic nature of the materials of the damper and the frequency dependence of the viscous damping ratio ξ, it is useful to consider separately the situations arising when ξ is positive (the system is stable) and when it is negative. A negative damping ratio means that the complementary solution of the response would not die away (the system is unstable because of factor e^{ξ · ω · t}). In the case of nonlinear analysis, the force–displacement relation is nonlinear, so it is very difficult to predict the actual damping and stiffness coefficients, even if the force–displacement characteristic is simply perfect elasto-plastic. Using the RDA method, which takes into account the rate of release of visco-elasto-plastic energy of the dissipation devices; nonlinear behaviors are linearized, enabling to obtain the equivalent damping and stiffness coefficients and the effective period for the damper.     

An analytical and computational study of crack initiation under transient creep conditions

An analytical method has been developed to predict creep crack initiation (CCI), based on the accumulation of a critical level of damage at a critical distance. The method accounts for the re-distribution of stress from the elastic or elastic–plastic field, experienced on initial loading, to a steady state creep stress distribution, via a transient creep region. The method has been applied to predict CCI times in a fracture specimen of type 316H stainless steel at 550 °C. The failure model has been also been implemented into a finite element (FE) framework. Reasonable and conservative predictions of CCI time can be obtained from the analytical solution relative to FE solutions. Conservative predictions of experimental CCI times are obtained when stress redistribution is taking into account. However, CCI times predicted from a steady state creep model are found to be non-conservative.     

Multi-cycle deformation of silicone elastomer: observations and constitutive modeling with finite strains

Observations are reported on a medical grade of silicone elastomer in uniaxial tensile tests up to breakage of specimens, short-term relaxation tests, and cyclic tests with monotonically increasing maximum elongation ratios. Experimental data in cyclic tests demonstrate the fading memory phenomenon: stress–strain diagrams for two specimens with different deformation histories along the first n?1 cycles and coinciding loading programs for the other cycles become identical starting from the nth cycle. A constitutive model is developed in cyclic viscoplasticity of elastomers with finite strains, and its adjustable parameters are found by fitting the experimental data. Ability of the stress–strain relations to predict the mechanical response in cyclic tests with various deformation programs is confirmed by numerical simulation.     

Creep simulation of asymmetric effects by use of stress mode dependent weighting functions

The paper presents a framework for creep modeling of materials exhibiting different behaviors in different loading scenarios, such as tension, compression and shear, respectively. To this end an additive decomposition of the flow rule is assumed into a sum of weighted stress mode related quantities. The characterization of the stress modes is obtained in the octahedral plane of the deviatoric stress space in terms of a single scalar variable, such that stress mode dependent scalar weighting functions can be constructed. Furthermore the numerical implementation into a finite element program of the resulting set of constitutive equations and aspects of the sensitivity analysis for parameter identification are addressed. Verification of the constitutive equations is succeeded for an aluminum alloy AK4-1T and a superalloy René 95, respectively. In two finite element examples the proposed model is applied to investigate the relaxation behavior of a square plate with circular hole and the evolution of creep damage in a gasturbine blade subjected to centrifugal and thermal loads.     

Cycle-dependent creep under random loading

An important aspect of the response of a metal subjected to cyclic loading and a superimposed mean load is its capacity for progressive strain accumulation. In the present investigation, random loading with tensile mean loads is applied to carbon-steel specimens. The cycle-dependent creep properties are described and a phenomenological approach to predict the number of cycles to ductile failure is proposed. At lower stress values, the fatigue-type brittle failure occurs. To predict the failure lives, the familiar fatigue-damage theory is modified for the case of random loading, and the effect of mean stress is included.     

Creep constitutive modeling of an aluminum alloy under multiaxial and cyclic loading

A constitutive model is developed to characterize creep response of polycrystalline metals. The model is based on the effective stress concept and back stress is utilized as an internal variable. A memory aspect is incorporated in the model to account for the previous maximal stress as a result of dislocation related micromechanisms. The model is used to predict the creep potential of an aluminum alloy under multiaxial and cyclic loading. The predicted results are compared with the available experimental results on an aluminum alloy (2618-T61). The associated six basic model parameters are evaluated by using an optimization technique, called ‘Box Algorithm’. These model parameters are then used to predict some unoptimized experimental data sets. The influence of multiaxial and cyclic loadings is investigated in detail for the selected aluminum alloy. Overall, excellent correlations are observed.     

Compliance of actin filament networks measured by particle-tracking microrheology and diffusing wave spectroscopy

We monitor the time-dependent shear compliance of a solution of semi-flexible polymers, using diffusing wave spectroscopy (DWS) and video-enhanced single-particle-tracking (SPT) microrheology. These two techniques use the small thermally excited motion of probing microspheres to interrogate the local properties of polymer solutions. The solutions consist of networks of actin filaments which are long semi-flexible polymers. We establish a relationship between the mean square displacement (MSD) of microspheres imbedded in the solution and the time-dependent creep compliance of the solution, <Δr ²(t)>=(k _B T/πa)J(t). Here, J(t) is the creep compliance, <Δr ²(t)> is the mean-square displacement, and a is the radius of the microsphere chosen to be larger than the mesh size of the polymer network. DWS allows us to measure mean square displacements with microsecond temporal resolution and Ångström spatial resolution. At short times, the mean square displacement of a 0.96μm diameter sphere in a concentrated actin solution displays sub-diffusion. <Δr ²(t)>∝t ^{, with a characteristic exponent =0.78±0.05, which reflects the finite rigidity of actin. At long times, the MSD reaches a plateau, with a magnitude that decreases with concentration. The creep compliance is shown to be a weak function of polymer concentration and scales as J _p ∝c –1.2±0.3. This exponent is correctly described by a recent model describing the viscoelasticity of semi-flexible polymer solutions. The DWS and video-enhanced SPT measurements of the compliance plateau agree quantitatively with compliance measured independently using classical mechanical rheometry for a viscous oil and for a solution of flexible polymers. This paper extends the use of DWS and single-particle-tracking to probe the local mechanical properties of polymer networks, shows for the first time the proportionality between mean square displacement and local creep compliance, and therefore presents a new, direct way to extract the viscoelastic properties of polymer systems and complex fluids.}     

Multi-mechanism anisotropic model for granular materials

By representing the assembly by a simplified column model, a constitutive theory, referred to as sliding–rolling theory, was recently developed for a two-dimensional assembly of rods subjected to biaxial loading, and then extended to a three-dimensional assembly of spheres subjected to triaxial (equibiaxial) loading. The sliding–rolling theory provides a framework for developing a phenomenological constitutive law for granular materials, which is the objective of the present work. The sliding–rolling theory provides information concerning yield and flow directions during radial and non-radial loading. In addition, the theory provides information on the role of fabric anisotropy on the stress–strain behavior and critical state shear strength. In the present paper, a multi-axial phenomenological model is developed within the sliding–rolling framework by utilizing the concepts of critical state, classical elasto-plasticity and bounding surface. The resulting theory involves two yield surfaces and falls within the definition of the multi-mechanism models. Computational issues concerning the solution uniqueness for stress states at the corner of yield surfaces are addressed. The effect of initial and induced fabric anisotropy on the constitutive behavior is incorporated. It is shown that the model is capable of simulating the effect of anisotropy, and the behavior of loose and dense sands under drained and undrained loading.     

Nonlinear viscoelastic-degradation model for polymeric based materials

This study presents a phenomenological constitutive model for describing response of solid-like viscoelastic polymers undergoing degradation. The model is expressed in terms of recoverable and irrecoverable time-dependent parts. We use a time-integral function with a nonlinear integrand for the recoverable part and another time-integral function is used for the irrecoverable part, which is associated with the degradation evolution in the materials. Here, the degradation is attributed to the secondary and tertiary creep stages. An ‘internal clock’ concept in viscoelastic materials is used to incorporate the accelerated failure in the materials at high stress levels. We ignore the effect of heat generation due to the dissipation of energy and possible healing in predicting the long-term and failure response of the polymeric materials. Experimental data on polymer composites reported by Drozdov (2011) were used to characterize the material parameters and validate the constitutive model. The model is shown capable of predicting response of the polymer composites under various loading histories: creep, relaxation, ramp loading with a constant rate, and cyclic loadings. We observed that the failure time and number of cycles to failure during cyclic loadings are correlated to the duration of loading and magnitude of the prescribed mechanical loads. A scalar degradation variable is also introduced in order to determine the severity of the degradation in the materials, which is useful to predict the lifetime of the structures subject to various loading histories during the structural design stage.     

Experimental study of the cyclic visco-elasto-plastic behaviour of a polyamide fibre strap

Experimental tensile tests were performed on polyamide-based (PA66) woven strap samples. A strain measuring device was used to measure the strain in the middle and effective part of the woven tensile sample. The tests were performed, on the one hand under monotonous tension at different strain rates and on the other hand under sophisticated cyclic loading histories, including relaxation and creep sequences. The analysis of experimental results was made through a visco-elasto-hysteresis model, based on the superimposition of three stress components. The proposed method allows for characterizing the steady state viscous stress as a function of strain and strain rate, the time-independent irreversible behaviour and the instantaneous modulus increasing with the strain. Based on the visco-elasto-hysteresis model, an analysis enabled us to understand and predict the change in relaxation and creep orientations during complex loading histories.     

高应力高水压下砂岩三轴蠕变特性试验研究

对砂岩进行高围压高水压条件下的三轴压缩蠕变试验。试验表明,在整个加载过程中,孔隙水压力主要起到增强轴向变形和横向变形的作用;但在加载的初始阶段,孔隙水压力在一定程度上抑制了轴向变形。当应力水平高于屈服应力时,横向蠕变速率明显大于轴向蠕变速率,且横向蠕变率先进入加速蠕变阶段。本文提出一个新的非线性黏性元件,并引入一个能判断是否进入加速蠕变阶段的计时器元件,组建一个非线性黏塑性加速蠕变启动模型,将该黏塑性模型与Burgers模型串联,构建一个新的非线性黏弹塑性蠕变模型,推导了该模型在常规三轴应力状态下的本构方程。基于试验结果,通过对非线性优化算法(BFGS)的Matlab编程,实现对本文提出蠕变模型的参数识别,识别效果比较理想。对比试验曲线与拟合曲线,二者相当吻合,验证了新提出的非线性黏弹塑性蠕变模型的正确性。     

Compression of an axisymmetric layer on a rigid mandrel in creep

An approximate solution describing the compression of an axisymmetric layer ofmaterial on a rigid mandrel under the equations of the creep theory is constructed. The constitutive equation is introduced so that the equivalent stress tends to a finite value as the equivalent strain rate tends to infinity. Such a constitutive equation leads to a qualitatively different asymptotic behavior of the solution near the mandrel surface, on which the maximum friction law is satisfied, compared with the well-known solution for the creep model based on the power-law relationship between the equivalent stress and the equivalent strain rate. It is shown that the solution existence depends on the value of one of the parameters contained in the constitutive equations. If the solution exists, then the equivalent strain rate tends to infinity as the maximum friction surface is approached, and the qualitative asymptotic behavior of the solution depends on the value of the same parameter. There is a range of variation of this parameter for which the qualitative behavior of the equivalent strain rate near the maximum friction surface coincides with the behavior of the same variable in ideally rigid-plastic solutions.     

Full field analysis of an anti-plane interface crack subjected to dynamic body forces

In this study, the transient full field response of an interface crack between two different media subjected to dynamic body force at one material is investigated. For time t < 0, the bimaterial medium is stress free and at rest. At t = 0, a concentrated anti-plane dynamic point loading is applied at the medium as shown in Fig. 1. The total wave field is due to the effect of this point loading and the scattering of the incident waves by the interface crack. An alternative methodology that is different from the conventional superposition method is used to construct the reflected, refracted and diffracted wave fields. A useful fundamental solution is proposed in this study and the full field solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying an exponentially distributed traction (in the Laplace transform domain) on the interfacial crack faces. The Cagniard–de Hoop method of Laplace inversion is used to obtain the transient solution in time domain. Exact transient closed form solutions for stresses and stress intensity factors are obtained. Numerical results for the time history of stresses and stress intensity factors during the transient process are discussed in detail.     

含与不含晶界空穴的双晶体蠕变行为研究

基于晶体滑移理论,建立了各向异性镍基合金双晶体的蠕变本构模型和蠕变寿命预测模型,通过MARC用户子程序CRPLAW将上述本构模型进行了有限元实现,并对双晶体蠕变行为进行了计算分析,考虑了:(1)晶体取向的影响;(2)垂直、倾斜和平行于外载方向的三种位向晶界情况;(3)晶界处引进空间空穴的影响。结果表明,双晶体上特别是微空穴和晶界附近区域的蠕变应力应变呈现不同的变化规律,对此晶粒晶体取向和晶界位向有较大的影响;微空穴的存在削弱了双晶体的承载能力,显著地影响了双晶体蠕变持久寿命;相同条件下,垂直晶界对双晶体模型的蠕变损伤影响最为强烈,倾斜晶界次之,平行晶界最小;微空穴的生长与晶界位向和晶体取向有强烈的依赖关系,其中垂直晶界更有利于晶体滑移和微空穴生长。