The creep of polycrystalline ice

Constitutive equations for polycrystalline ice are reviewed. Dimensionless plots of data are developed which suggest conditions which must be met by a successful constitutive equation. Modifications to the phenomenological law of Sinha (1978a, b) are discussed to make it meet these conditions. A model is developed which contributes to the physical basis of the more complete constitutive equation proposed by Le Gac and Duval (1980). This equation is developed further, and formulated in a dimensionless form, leading to a set of master-curves describing the deformation of polycrystalline ice, up to the stage where the minimum in the strain-rate occurs.     

A temperature dependent creep damage model for polycrystalline ice

We present a continuum damage model for the temperature dependent creep response of polycrystalline ice under a multiaxial state of stress, suited for ice in polar regions. The proposed model is based on a thermo-viscoelastic constitutive law for ice creep and a local orthotropic damage accumulation law for tension, compression and shear loadings. Orthotropic damage is represented by a symmetric second-order damage tensor and its effect on creep is incorporated through the effective stress concept. The unknown model parameters are first calibrated using published experimental data from constant uniaxial stress tests and then predictions are made for constant strain rate and multiaxial loadings. The predicted results are in good agreement with both experimental and numerical results in the literature illustrating the viability of the proposed model. The model is mainly intended for studying the failure mechanisms of polar ice at low deformation rates with depth varying temperature profiles.     

Transient and steady state creep behaviour of polycrystalline MgO

Stress change experiments during compressive creep tests at high stresses on polycrystalline MgO at 1596 K have shown that the creep rate at any instant during transient and steady state creep is predicted by the ratio,r/h, wherer is the rate of recovery (=??σ/t6t) andh is the coefficient of strain hardening (=?σ/?ε). Over most of transient and steady state creep, whenh is constant and the decrease in creep rate, \(\dot \in\) , is a direct result of a decrease inr, the variation of the creep strain,ε, with time,t, is accurately described as $$ \in = \in _0 + \in _T (1 - e^{ - mt} ) + \dot \in _s t$$ whereε ₀ is the instantaneous strain on loading,ε _T the transient creep strain,m relates to the rate of exhaustion of transient creep and \(\dot \in _s\) is the steady creep rate. Deviations from this equation occur during the initial 10 to 15% of the transient creep life, whenh increases rapidly. The variations in \(\dot \in\) ,r andh during transient and steady state creep are explained in terms of a model for creep in which the rate-determining process is the diffusion controlled growth of the three-dimensional dislocation network within subgrains to form dislocation sources allowing slip to occur.     

Viscous creep deformation of polycrystalline CaTiO3 at elevated temperatures

Creep deformation of polycrystalline CaTiO₃ has been investigated in air at temperatures between 1100 and 1200° C and stresses between 4 and 13 MPa. The results indicate that the creep deformation of this material can be described by a threshold process with associated activation energy of 200 kcal mol^–1 (836.8 kJ mol^–1). It is tentatively concluded that creep deformation of polycrystalline CaTiO₃ within the ranges of experimental parameters investigated is rate-controlled by interfacial defect creation and/or annihiliation at grain boundaries.     

On modelling steady state and transient creep of polycrystalline solids

Summary A constitutive model is proposed to describe creep deformation of polycrystalline materials under complex stress and temperature histories. The concept of piecewise linear effective stress—creep strain rate relationship is utilized. A key assumption in this model is that the back stress under a given stress and temperature reaches a saturation point. This saturation point corresponds to the steady state condition.With 12 Figures     

An evaluation of the rate-controlling flow process in Newtonian creep of polycrystalline ice

Using experimental data and theoretical calculation for Newtonian creep in polycrystalline ice, it is demonstrated that unlike most other materials, in which the rate-controlling flow process is edge dislocation climb under saturated condition, the rate-controlling flow process of polycrystalline ice is dislocation glide along the basal plane under a constant dislocation density. The dislocation density during Newtonian creep of ice is determined by the initial state instead of the magnitude of the Peierls stress. The transition stress (threshold) from power-law creep to Newtonian creep is controlled by the dislocation density instead of the Peierls stress. The activation energy of the Newtonian creep is similar to that of the self-diffusion due to the requirements of the diffusion of protons during dislocation glide.     

Crack-enhanced creep in polycrystalline material: strain-rate sensitive strength and deformation of ice

A non-linear viscoelastic creep equation for polycrystalline material is presented. It incorporates the effect of cracking and is capable of describing primary, secondary and tertiary behaviour. The model predicts the formation of microcracks and thus the damage state due to the high-temperature grain-boundary embrittlement process. This paper describes its application in formulating crack-enhanced creep and material response under constant strain-rate loading conditions (theoretically the simplest case but actually the most difficult to maintain). The formulation makes it possible to define the rate effect on stress-strain response and the rate sensitivity of strength, failure time, failure strain, damage and damage rate, strain recovery, etc. Numerical correspondence between theory and experiment was observed when predictions were compared with available closed-loop, controlled, constant strain-rate strength and deformation data on pure ice. Calculations made use of material constants determined from independent constant-load creep tests.     

Tensile cracks in polycrystalline ice under transient creep: Part I — Numerical modeling

The interaction between creep deformations and a stationary or growing crack is a fundamental problem in ice mechanics. Knowledge concerning the physical mechanisms governing this interaction is necessary: (1) to establish the conditions under which linear elastic fracture mechanics can be applied in problems ranging from ice-structure interaction to fracture toughness testing; and (2) to predict the ductile-to-brittle transition in the mechanical behavior of ice and, especially, the stability and growth of cracks subjected to crack-tip blunting by creep deformations. This requires a quantitative estimate of the creep zone surrounding a crack-tip, i.e., the zone within which creep strains are greater than the elastic strains.

The prediction of the creep zone in previous ice mechanics studies is based on the theory developed by Riedel and Rice (1980) for tensile cracks in creeping solids. This theory is valid for a stationary crack embedded in an isotropic material obeying an elastic, power-law creep model of deformation and for a suddenly applied uniform far-field tension load that is held constant with time. The deformation of ice at strain-rates ahead of a crack (i.e., 10⁻⁶ to 10⁻² s⁻¹) is dominated, however, by transient (not steady power-law) creep and the loading, in general, is not instantaneous and constant.

A numerical model is developed in this paper to investigate the role of transient creep and related physical mechanisms in predicting the size, shape and time evolution of the creep zone surrounding the tip of a static crack in polycrystalline ice. The model is based on the fully consistent tangent formulation derived in closed form (Shyam Sunder et al., 1993) and used in the solution of the physically-based constitutive theory developed by Shyam Sunder and Wu (1989a, b) for the multiaxial behavior of ice undergoing transient creep. The boundary value problem involving incompressible deformations ahead of a stationary, traction-free mode I crack in a semi-infinite medium is modeled and solved by a finite element analysis using the boundary layer approach of Rice (1968). This model is verified by comparing its predictions with (i) the known theoretical solutions for the elastic and HRR asymptotic stress and strain fields in an elastic-plastic material of the Ramberg-Osgood type, and (ii) the creep zone size for an isotropic material obeying the elastic power-law creep model of deformation.     

Tensile cracks in polycrystalline ice under transient creep: Part II — Numerical simulations

This paper discusses predictions of a numerical model presented in the companion paper (Nanthikesan and Shyam Sunder, 1995) to analyze tensile cracks in polycrystalline ice undergoing transient creep. The numerical model is based on the internal state variable constitutive theory of transient creep in ice developed by Shyam Sunder and Wu (1989a,b, 1990). The finite element model uses the boundary layer approach of Rice (1968), in conjunction with a mid-point crack-tip element and reduced integration, to simulate the asymptotic stress and deformation fields in the vicinity of the crack tip, including incompressible creep deformations.

The problem of a stationary, traction-free, tensile (mode I) crack is analyzed to predict the size, shape and time evolution of the creep-dominated fracture process zone surrounding the crack-tip. The numerical simulations quantify the effects of transient creep, material strain hardening, fabric anisotropy, loading rate, temperature, and finite fracture test-specimen boundary on the development of the creep zone. A range of stress-intensity rates from 1 to 100 kPa s⁻¹ and temperatures from −5° to −25°C is considered in the simulations.

The results from a comprehensive numerical simulation study show that: (i) transient creep increases the creep zone size by more than an order of magnitude over that for a power-law creeping material, i.e., about 40 times for the isotropic, equiaxed granular ice tested by Jacka (1984); (ii) material strain hardening significantly affects the creep zone size, i.e., the creep zone for the transversely-isotropic columnar-grained ice tested by Sinha (1978), with the crack loaded in the plane of isotropy, is about 4 times smaller than that for the granular isotropic ice; (iii) fabric anisotropy increases the size of the creep zone by a factor of at least two for cracks in the transversely-isotropic, columnar-grained ice loaded in the plane of isotropy; (iv) the Riedel and Rice (1980) equation, which was derived for an isotropic power-law creeping material subjected to a suddenly applied constant stress-intensity, overestimates the creep zone size by a factor of 4.2 for a constant stress-intensity rate loading; and (v) as the crack size increases, linear elastic fracture mechanics becomes increasingly applicable at lower loading rates and higher temperatures.     

Preparation of polycrystalline ice specimens for laboratory experiments

A perennial problem in preparing fine grained ice specimens is the convenient production of low porosity material which is homogeneous, of random c-axis orientation and of controlled grain size.A review of the nature of bubble formation in ice leads directly to the two factors which must be controlled in order to minimize gas bubble formation, and thus porosity. The factors are (1) reduction of the number of bubble nucleation sites and (2) reduction of the amount of dissolved gas available at the freezing front.An ice specimen preparation technique developed at CRREL addresses both these factors. Two techniques have proven effective in reducing the number of nucleation sites; (1) a purge technique using carbon dioxide gas and (2) a vacuum technique. A newly developed flushing technique reduces the amount of dissolved gas in the pore water during freezing.The resulting material is optically clear with some very small bubbles dispersed throughout. The radial freezing technique employed occasionally results in a narrow column of fine bubbles along the central axis of the specimen. The density is 0.917 ± 0.002 Mg/m³.An overview of techniques for forming, flooding and freezing ice specimens is presented and several major preparation methods are reviewed in detail.     

Viscoelastic solid relations for the deformation of ice

A frame indifferent differential operator law relating stress, stress-rate, strain, and strain-rate is constructed to describe the qualitative features of both constant load and constant displacement rate response in uni-axial stress experiments. No lower order differential relation can describe both responses, but the two responses are not sufficient to determine the response coefficients of the relation. The jump relation is determined for a stress discontinuity applied in a general configuration. A simple initially isotropic model is proposed to investigate the effects of loading history and the anisotropy induced in a configuration following a loading-unloading cycle. Both uni-axial stress and shear cycles are treated, followed by uni-axial stress loading in different directions. The respective deformation histories are determined in the small strain approximation, and demonstrate a significant influence of the initial load cycle on response from the unloaded configuration.     

Acoustic emissions from polycrystalline ice

The acoustic emission response from fine-grained polycrystalline ice subjected to constant compressive loads was examined. A number of tests were conducted with the nominal stress ranging from 0.8 to 3.67 MPa at a temperature of ?5°C. The acoustic emission response was recorded and the data are presented with respect to time and strain. The source of acoustic emissions in ice is considered in terms of the formation of both microfractures and visible fractures that develop without catastrophic failure of the ice. A model to describe the acoustic emission response is developed.     

Governing equations for the steady state creep in orthotropic materials

Summary A general constitutive equation for creep deformation is presented based upon the concept of tensorial internal variables. The consequences of the theory of tensor functions representation are discussed with respect to the evolution equations. In a particular case of steady evolution of internal variables the governing equation for the secondary creep rate is derived in terms of a scalar inelastic potential. The material parameters required to characterize the stationary creep behaviour of the orthotropic composite are obtained from the unidirectional tension creep tests performed on a glass woven fabric xylok composite. Further check on the theory is made for the bidirectionally loaded specimens.With 4 Figures     

Crack propagation during steady state creep

A model is proposed tor the steady propagation of a creep crack under steady state creep conditions. Creep is envisaged to take place everywhere in the solid, though higher creep rates at the crick tip lead to a local concentration of the creep strain. A critical local strain criterion is used to describe the condition for crack advance. Local damage is envisaged to accumulate at the crack tip as a result of, or in parallel with the creep strain. The model correctly predicts a dependence of crack propagation rate with the nett section stress varied to the power m, where m is the exponent of stress in the creep equation, for large values of m. An approximate dependence of propagation rate on the elastic stress intensity factor is also shown. Tnese predictions are in accord with experimental work.     

Creep and creep fracture of polycrystalline copper

Normal creep curves are recorded over extended stress ranges at 686–823 K for fine-grain copper. Analyses of the curve shape variations, together with the results of stress change experiments, do not support the view that a transition from dislocation to diffusional creep mechanisms occurs with decreasing stress. Instead, the observed behaviour patterns suggest that dislocation processes are dominant at all stress levels. However, strain accumulation within the grains becomes progressively less important as deformation is increasingly confined to the grain boundary zones when the stress is reduced below the yield stress at the creep temperature. New approaches are then introduced for rationalization of creep rate and creep life measurements, which account for the data trends taken as evidence for major mechanism changes when the creep properties are described using power law relationships.