Theories of Creep in Ceramics

Mathematical models that have been proposed for creep in ceramics are described. Emphasis is on models involving grain boundary motion (sliding or flow). In Lifshitz models the crystalline grains elongate with strain; the elongation results from diffusion, slip, or solution and precipitation. In Rachinger models the grains do not elongate during creep. The sliding strain can be accommodated by viscous flow of a glassy phase at the grain boundaries, or if there is no boundary glass by diffusion or slip in superplastic models. Sliding of a glass-free boundary can result in cavitation, cracking, or formation of boundary dislocations or triple point folds.

Most models of ceramic creep at high temperatures predict a steady state (stage II) creep rate that depends on the applied stress, grain size, and temperature. A general equation for the creep rate as a function of these factors, as well as the elastic modulus and a diffusion coefficient, is used to compare models. The models give different exponents for the functional dependence of creep rate on grain size and strain and different temperature dependencies. These differences are compared in tables, and the main mechanistic features of the models are described in the text.

The purpose of this review is to describe creep models rather than to compare them with experimental results or to select the most applicable models. There are few critical experimental tests that allow selection of the most accurate models; such experiments are suggested as the next step in choosing between the models for specific experimental results.     

铁基块体非晶合金在纳米压痕过程中的蠕变行为研究

利用纳米压痕技术研究了{［（Fe_0.6Co_0.4）_0.75B_0.2Si_0.05］_0.96Nb_0.04}₉₆Cr₄铁基块体非晶合金的室温蠕变行为及不同的加载速率对该块体非晶合金蠕变变形的影响.{［（Fe_0.6Co_0.4）_0.75B_{0.2< 关键词： 块体非晶合金 蠕变 EVEV模型 蠕变速率敏感指数}     

Creep in oriented thermoplastics

Sheets of oriented low-density polyethylene possessing transverse symmetry were prepared by simple tensile drawing at room temperature. The degree of anisotropy was varied by varying the draw ratio. Classic elasticity theory shows that five constants are necessary to characterize the deformation behavior of an elastic material possessing such symmetry. The time-dependent equivalents of these constants have been determined from the simultaneous measurement of longitudinal and lateral strain during tensile creep of specimens cut at 0, 45, and 90 deg to the draw direction. Special creep apparatus, able to handle small, flexible specimens, was developed for this study. The shear compliance obtained from the tensile creep studies was in good agreement with the value obtained from torsional creep measurements on a 0-deg specimen, for the entire range of draw ratios. The contraction measurements have been used to provide additional evidence for deformation mechanisms suggested by the longitudinal strain measurements and, when combined with tensile measurements, have allowed the volume changes occurring during tensile creep to be calculated.     

Creep duration analysis in terms of the field theory of defects

From the equations of motion within the field theory of defects, creep curves are derived and a relationship between the applied stress and the time to rupture under different deformation conditions is obtained. The creep duration as a function of the applied stress and the initial strain rate, as well as the ultimate strain, specifying the material rupture, are found.     

Relaxation and Creep of Dry Bulk Solids

In former investigations it has been shown that creep (constant stress, altering strain) and relaxation (constant strain, decreasing stress) can be observed with dry bulk solids. Both effects are covered when investigating the time dependent behaviour of bulk solids where time dependence can also mean an increase of the deformation resistance with increasing deformation rate. In this paper the investigated time dependent effects do not include time consolidation. The effects of creep and relaxation are often neglected for bulk solids because in many applications the influence of these time dependent behaviours is of minor importance. A deeper insight into the bulk solids flow characteristics and mechanisms can only be obtained when time dependence is taken into consideration.     

Creep of multidirectionally fibre-reinforced composites

A model for the creep of metal matrix composites multidirectionally reinforced by short fibres is proposed. The reinforcement is described by the effective stiffness tensor of a multidirectional arrangement of continuous fibres and the internal damage of the composite during creep due to fibre fragmentation is introduced by assigning a heuristic nonlinear stress–strain relationship to the fibres. Based on the model, the load partitioning between matrix and fibres is computed. The macroscopic creep behaviour is simulated for composites exhibiting different fibre orientation distributions and different heuristic nonlinear stress–strain functions. The computational results rationalize the creep behaviour of multidirectional fibre-reinforced composites. For a two-dimensional random orientation distribution, a good qualitative match between simulation and experimental results is obtained for compressive loading and for in-plane tensile loading. For loading normal to the reinforcement plane, the model overestimates the creep resistance. In this case, the formation and growth of cavities seems to govern the creep deformation of the composite.     

Research on nonlinear constitutive relationship of permanent deformation in asphalt pavements

To predict correctly the rut depths in asphalt pavements, a new nonlinear viscoelastic-elastoplastic constitutive model of permanent deformation in asphalt pavements is presented. The model combines a generalized Maxwell model with an elastoplastic one. Then from the creep theory, the linear and nonlinear constitutive equations of the generalized Maxwell model are obtained. From the nonlinear finite element method for the rutting of the asphalt pavement, the rut depths of 4 asphalt-aggregate mixtures are obtained. And the results are compared with the ones from the finite element method by SHRP and the experiments by SWK/UN. The results in this paper are better than the ones by SHRP, and agree with the ones of the experiment by SWK/UN. This shows that the nonlinear viscoelastic-elastoplastic constitutive model, which is presented in this paper for the rutting of the asphalt pavement, is effective. The properties, such as nonlinear elasticity, plasticity, viscoelasticity and nonlinear viscoelasticity, which affect the rutting of an asphalt pavement, can be shown in the model. And the characteristics of the permanent deformation of the asphalt pavement can be presented entirely in the model.     

Molecular-Dynamics Simulation of Grain-Boundary Diffusion Creep

Molecular-dynamics (MD) simulations are used, for the first time, to study grain-boundary diffusion creep of a model polycrystalline silicon microstructure. Our fully dense model microstructures, with a grain size of up to 7.5 nm, were grown by MD simulations of a melt into which small, randomly oriented crystalline seeds were inserted. In order to prevent grain growth and thus to enable steady-state diffusion creep to be observed on a time scale accessible to MD simulations (of typically 10^-9s), our input microstructures were tailored to (i) have a uniform grain shape and a uniform grain size of nm dimensions and (ii) contain only high-energy grain boundaries which are known to exhibit rather fast, liquid-like self-diffusion. Our simulations reveal that under relatively high tensile stresses these microstructures, indeed, exhibit steady-state diffusion creep that is homogenous (i.e., involving no grain sliding), with a strain rate that agrees quantitatively with that given by the Coble-creep formula.     

Influence of a viscoelastic foundation on the stability of Beck's column: An exact analysis

The stability of Beck's column supported by three different viscoelastic foundations, viz., the standard linear solid, the Maxwell and the Kelvin-Voigt, is investigated. Closed form stability criteria are obtained for the entire range of system parameters through an exact dynamic analysis for each foundation model. The results for the Kelvin-Voigt model show that for a given stiffness parameter of the foundation the critical load increases with an increase in damping and reaches a limiting value for large damping. Unlike the case of conservative loading, the Maxwell foundation is shown to have a positive influence on the stability of this non-conservative problem. Furthermore, for this model, an optimum combination of foundation parameters exists to yield the maximum flutter load. The standard linear solid foundation combines the characteristics of Maxwell and Kelvin-Voigt models, as expected.     

Effect of High-Temperature Structure and Diffusion on Grain-Boundary Diffusion Creep in fcc Metals

Molecular dynamics simulations of high-energy twist and tilt bicrystals of fcc palladium reveal a universal, liquid-like, isotropic high-temperature diffusion mechanism, characterized by a rather low self-diffusion activation energy that is independent of the boundary type or misorientation. Medium-energy grain boundaries exhibit the same behavior at the highest temperatures; however, at lower temperatures the diffusion mechanism becomes anisotropic, with a higher, misorientation-dependent activation energy. Our simulations demonstrate that the lower activation energy at elevated temperatures is caused by a structural transition, from a solid boundary structure at low temperatures to a liquid-like structure at high temperatures. We demonstrate that the existence of such a transition has important consequences for diffusion creep in nanocrystalline fcc metals. In particular, our simulations reveal that in the absence of grain growth, nanocrystalline microstructures containing only high-energy grain boundaries exhibit steady-state diffusion creep with a creep rate that agrees quantitatively with that given by the Coble-creep formula. Remarkably, the activation energy for the high-temperature creep rate is the same as that characterizing the universal high-temperature diffusion in high-energy energy bicrystalline grain boundaries.     

Experimental study of room-temperature indentation viscoplastic ‘creep’ in zirconium

In this paper we have studied the mechanisms of so-called ‘indentation creep’ in a zirconium alloy. Nanoindentation was used to obtain strain rate data as the sample was indented at room temperature, at a homologous temperature below that for which creep behaviour would be expected for this material. A high value of strain rate was obtained, consistent with previous work on indentation creep. In order to elucidate the mechanism of time-dependent deformation, a load relaxation experiment was performed by uniaxial loading of a sample of the same alloy. By allowing relaxation of the sample from a peak load in the tensile test machine, a similar stress exponent was obtained to that seen in the nanoindentation creep test. We conclude that for metals, at temperatures below that at which conventional creep will occur, nanoindentation ‘creep’ proceeds through deformation on active slip systems that were initiated by prior loading beyond the plastic limit. It is therefore more appropriate to describe it as a viscoplastic process, and not as creep deformation.     

Climb via vacancy diffusion of edge dislocations in 2D dislocation microstructures

The prevention of strength degradation of components is one of the great challenges in solid mechanics. In particular, at high temperatures material may deform even at low stresses, a deformation mode known as deformation creep. One of the microstructural mechanisms that governs deformation creep is dislocation motion due to the absorption or emission of vacancies, which results in motion perpendicular to the glide plane, called dislocation climb. However, the importance of the dislocation network for the deformation creep remains far from being understood. In this study, a climb model that accounts for the dislocation network is developed, by solving the diffusion equation for vacancies in a region with a general dislocation distribution. The definition of the sink strength is extended, to account for the contributions of neighbouring dislocations to the climb rate. The model is then applied to dislocation dipoles and dislocation pile-ups, which are dense dislocation structures and it is found that the sink strength of dislocations in a pile-up is reduced since the vacancy field is distributed between the dislocations. Finally, the importance of the results for modelling deformation creep is discussed.     

Constitutive description of primary and steady-state creep deformation behaviour of tempered martensitic 9Cr–1Mo steel

The model based on the coupled sine hyperbolic creep rate relation with the evolution of internal stress as a function of strain provides better understanding of primary and secondary creep behaviour of tempered martensitic 9Cr–1Mo steel. The predicted evolution of internal stress as an increase in the internal stress value (or, decrease in effective stress) with strain/time appropriately described the observed decrease in creep rate during primary creep in the steel. The applicability of the model has been demonstrated by comparing experimental and predicted creep strain–time and creep rate–strain/time data of 9Cr–1Mo steel at 793 and 873 K for quenched and tempered and simulated post-weld heat treatment conditions. Irrespective of prior heat treatment and test temperature, the optimised parameters associated with the internal stress values exhibited linear variations with applied stress. The influence of prior heat treatment on primary and secondary creep characteristics of the steel is reflected on the rate constant values associated with the model. At all temperatures and heat treatment conditions, good agreement between the experimental and predicted steady-state creep rates demonstrate the further applicability of the model.     

Effect of thermomechanical processing on the creep behaviour of Udimet alloy 188

Udimet alloy 188 was subjected to grain-boundary engineering involving thermomechanical processing in an attempt to improve the creep performance and determine the effects on creep deformation processes. The as-received sheet was cold-rolled to either 10, 25 or 35% reduction per pass followed by a solution treatment at 1191°C for 1 h plus air cooling. This sequence was repeated four times and the resultant microstructure and grain-boundary character distribution were described using electron backscatter diffraction. The fraction of general high-angle grain boundaries tended to increase with increased cold rolling. The 10 and 25% cold-rolled materials exhibited lower creep rates than the 35% cold-rolled material. The measured creep stress exponents and activation energies suggested that dislocation creep with lattice self-diffusion was dominant at 760°C for stresses ranging between 100 and 220 MPa. A transition in the creep exponent below the applied stresses of 100 MPa indicated that a different secondary creep mechanism was rate-controlling at low stresses. A significant amount of grain-boundary cracking was observed both on the surface and subsurface of deformed samples, but surface cracks were greater in number and size than those within the bulk. The cracking behaviour was similar in both vacuum and air environments, indicating that grain-boundary cracking was not caused by environment. To assess the mechanisms of crack nucleation, in situ scanning electron microscopy was performed during elevated-temperature (T ≤ 760°C) tensile-creep deformation. Sequential secondary electron imaging and electron backscatter diffraction orientation mapping were performed in situ to allow the evolution of crack nucleation and linkage to be followed. Cracking occurred preferentially along general high-angle grain boundaries and less than 15% of the cracks were found on low-angle grain boundaries and coincident site lattice boundaries. A fracture initiation parameter analysis was performed to identify the role of slip system interactions at the boundaries and their impact on crack nucleation. The parameter was successful in separating the population of intact and cracked general high-angle boundaries at lower levels of strain, but not after crack coalescence dominated the fracture process. The findings of this work have significant implications regarding grain-boundary engineering of this alloy and potentially for other alloy systems.     

Molecular dynamics simulations of the surface tension of n-hexane,n-decane and n-hexadecane

Molecular dynamics simulations have been used to compute the surface tension of linear alkanes. The OPLS force field has been compared with the SKS force field for alkanes (n-hexane, n-decane and n-hexadecane) over two ranges of temperature: high temperatures where no experimental data are available for surface tension and lower temperatures where comparisons may be made with experiments. At high temperatures, for a given coexistence density, these two models predict a similar surface tension. For a given temperature the two models yield different surface tensions. However, these deviations can be attributed to differences in the prediction of the coexistence curves. For the SKS model the computed coexistence properties have been compared with experimental data. The simulation data are in reasonable agreement with the experimental data.     

Homogenization and two-scale models for liquid phase epitaxy

We consider models for liquid phase epitaxy without and with elasticity. The models are based on continuum models for fluid flow and transport of adatoms in the liquid solution and a BCF–model for the growth of the solid phase. Using homogenization by formal asymptotic expansion, we obtain two–scale models that are appropriate to describe the evolution of microstructures in the solid phase for processes of technically relevant macroscopic length scales. The two–scale models consist of macroscopic equations for fluid flow and solute transport in the liquid and microscopic cell problems for the growth and elastic deformation of the solid. For the case without elasticity and a phase field approximation of the BCF–model, an estimate of the model error is presented.     

Structural micromechanics of oriented polymers

To clarify whether the interfibrillar slippage occurs on plastic deformation of oriented polymers, flow creep of ultrahigh molecular weight polyethylene (UHMW PE) samples with various connectedness of microfibrils has been studied in a dead load mode at room temperature. The flow creep rate of melt-crystallized and gel-cast UHMW PE films drawn to various draw ratios, as well as of modified gel-crystallized samples (cross-linked/grafted or washed free of low molecular weight fraction) has been measured with the help of a unique laser interferometric technique (Doppler creep rate meter). The technique allows one to measure creep rates for deformation increments as small as 0.3 μ within an accuracy 1%. The interferometric technique enabled us to observe an extremely high variability of flow creep rate. It was recognized that the creep process accelerates or slows from time to time. A length of a loaded sample increased by multiple consecutive deformation jumps (or steps). The size distribution of the steps appeared to be controlled by the structure of interfibrillar regions. The influence of the latter on the variability of creep rate confirms a hypothesis that suggests a contribution of interfibrillar slippage to plastic deformation of oriented polymers. The observed phenomenon has been attributed to stick-slip motion of microfibrils and their aggregates sliding on each other under the action of applied stress. It was found that the creep rate decreases with increasing interfibrillar interaction.     

Creep strain recovery of Fe-Ni-B amorphous metallic ribbon

Creep strain recovery and structural relaxation of the amorphous metallic glass Fe₄₀Ni₄₁B₁₉ after longtime loading at different annealing temperatures below the glass transition temperature have been studied using anisothermal differential scanning calorimetry (DSC) and dilatometry (TMA). It has been demonstrated that structural relaxation effects depend on the stress-annealing temperature of the amorphous ribbon. The structural relaxation states of the amorphous ribbon annealed at different temperatures under and without applied stress have been compared. The activation energy spectra were calculated from the anisothermal dilatometric measurements using the modern method based on the Fourier transformation technique. The influence of the annealing temperature on the shape of creep strain recovery spectra has been analyzed.