# Mathematical modelling of deformation mechanisms in ice: Dissertation

Kari Santaoja

Research output: ThesisDissertationMonograph

1 Citation (Scopus)

### Abstract

Mathematical modelling of deformation mechanisms in polycrystalline isotropic ice over wide temperature and strain rate ranges was investigated in this thesis. The proposed 3-dimensional constitutive equations describe both time-dependent ductile deformation mechanisms and microcracking associated with Hookean deformation showing a brittle nature. A model to describe the high strain rates operative mechanisms, which form the microcrack nucleus, was introduced. It was assumed that the required four independent deformation systems were: basal glide (2 systems) and twinning (2 systems). At very high strain rates, when the dislocation motion on the basal plane is not fast enough, the total permanent response of the icebody causing microcracking may be resulting from twinning. The prominent microcracking (damage) at high strain rates is described by a fourth-order tensor. The damage evolution law is based on the potential energy theorem. The observed micromechanisms in ice are modelled for both the condition for microcrack formation and the effects of microcracks on the mechanical properties of ice. The proposed approach connects continuum damage mechanics with fracture mechanics. The proposed theory reveals the Hall-Petch type of relation, although the frictional stress vanishes. Computed examples satisfactorily gave the measured compressive strength of ice and predicted the ratio of the peak compressive stress to the peak tensile stress as 1.85: 1, which is much lower than the measured one - 8.6 : 1. At elevated temperatures when the strain rate is low, the ductile response of ice is dominant. The following mechanisms are modelled: Hookean deformation, delayed elasticity associated with grain boundary sliding and viscoplaticity associated with dislocation movement. The uniaxial material model is based on the equation proposed by Sinha. However, a more effective approximate time integration method was applied in computing the delayed elastic strain. This work gives a micromechanical representation for the (viscoclastic) delayed elasticity. This was applied during the derivation of the 3-dimensional constitutive SADS-equation, which coincides in uniaxial tension with the Ashby and Duval modification of Sinha's model. A Schapery's nonlinear viscoelasticity theory based time integration method, which is a modification to the procedure applied in the uniaxial case, was proposed for the SADS-equation. Although this method gives realistic predictions during loading it does not allow the material to relax correctly.
Original language English Doctor Degree Helsinki University of Technology Määttänen, Mauri, Supervisor, External person 19 Apr 1990 Espoo VTT Technical Research Centre of Finland 951-38-3617-7 Published - 1990 G4 Doctoral dissertation (monograph)

### Fingerprint

deformation mechanism
strain rate
microcrack
ice
twinning
constitutive equation
modeling
dislocation
elasticity
continuum damage mechanics
viscoelasticity
damage
fracture mechanics
tensile stress
ductile deformation
grain boundary
potential energy
compressive strength
sliding
mechanical property

### Keywords

• ice
• deformation
• mechanical properties
• mechanics
• microstructure
• mathematical models
• fractures (materials)
• cracking (fracturing)

### Cite this

Santaoja, K. (1990). Mathematical modelling of deformation mechanisms in ice: Dissertation. Espoo: VTT Technical Research Centre of Finland.
Santaoja, Kari. / Mathematical modelling of deformation mechanisms in ice : Dissertation. Espoo : VTT Technical Research Centre of Finland, 1990. 236 p.
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title = "Mathematical modelling of deformation mechanisms in ice: Dissertation",
abstract = "Mathematical modelling of deformation mechanisms in polycrystalline isotropic ice over wide temperature and strain rate ranges was investigated in this thesis. The proposed 3-dimensional constitutive equations describe both time-dependent ductile deformation mechanisms and microcracking associated with Hookean deformation showing a brittle nature. A model to describe the high strain rates operative mechanisms, which form the microcrack nucleus, was introduced. It was assumed that the required four independent deformation systems were: basal glide (2 systems) and twinning (2 systems). At very high strain rates, when the dislocation motion on the basal plane is not fast enough, the total permanent response of the icebody causing microcracking may be resulting from twinning. The prominent microcracking (damage) at high strain rates is described by a fourth-order tensor. The damage evolution law is based on the potential energy theorem. The observed micromechanisms in ice are modelled for both the condition for microcrack formation and the effects of microcracks on the mechanical properties of ice. The proposed approach connects continuum damage mechanics with fracture mechanics. The proposed theory reveals the Hall-Petch type of relation, although the frictional stress vanishes. Computed examples satisfactorily gave the measured compressive strength of ice and predicted the ratio of the peak compressive stress to the peak tensile stress as 1.85: 1, which is much lower than the measured one - 8.6 : 1. At elevated temperatures when the strain rate is low, the ductile response of ice is dominant. The following mechanisms are modelled: Hookean deformation, delayed elasticity associated with grain boundary sliding and viscoplaticity associated with dislocation movement. The uniaxial material model is based on the equation proposed by Sinha. However, a more effective approximate time integration method was applied in computing the delayed elastic strain. This work gives a micromechanical representation for the (viscoclastic) delayed elasticity. This was applied during the derivation of the 3-dimensional constitutive SADS-equation, which coincides in uniaxial tension with the Ashby and Duval modification of Sinha's model. A Schapery's nonlinear viscoelasticity theory based time integration method, which is a modification to the procedure applied in the uniaxial case, was proposed for the SADS-equation. Although this method gives realistic predictions during loading it does not allow the material to relax correctly.",
keywords = "ice, deformation, mechanical properties, mechanics, microstructure, mathematical models, fractures (materials), cracking (fracturing)",
author = "Kari Santaoja",
year = "1990",
language = "English",
isbn = "951-38-3617-7",
series = "Valtion teknillinen tutkimuskeskus. Tutkimuksia - Research Reports",
publisher = "VTT Technical Research Centre of Finland",
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school = "Helsinki University of Technology",

}

Santaoja, K 1990, 'Mathematical modelling of deformation mechanisms in ice: Dissertation', Doctor Degree, Helsinki University of Technology, Espoo.
Espoo : VTT Technical Research Centre of Finland, 1990. 236 p.

Research output: ThesisDissertationMonograph

TY - THES

T1 - Mathematical modelling of deformation mechanisms in ice

T2 - Dissertation

AU - Santaoja, Kari

PY - 1990

Y1 - 1990

N2 - Mathematical modelling of deformation mechanisms in polycrystalline isotropic ice over wide temperature and strain rate ranges was investigated in this thesis. The proposed 3-dimensional constitutive equations describe both time-dependent ductile deformation mechanisms and microcracking associated with Hookean deformation showing a brittle nature. A model to describe the high strain rates operative mechanisms, which form the microcrack nucleus, was introduced. It was assumed that the required four independent deformation systems were: basal glide (2 systems) and twinning (2 systems). At very high strain rates, when the dislocation motion on the basal plane is not fast enough, the total permanent response of the icebody causing microcracking may be resulting from twinning. The prominent microcracking (damage) at high strain rates is described by a fourth-order tensor. The damage evolution law is based on the potential energy theorem. The observed micromechanisms in ice are modelled for both the condition for microcrack formation and the effects of microcracks on the mechanical properties of ice. The proposed approach connects continuum damage mechanics with fracture mechanics. The proposed theory reveals the Hall-Petch type of relation, although the frictional stress vanishes. Computed examples satisfactorily gave the measured compressive strength of ice and predicted the ratio of the peak compressive stress to the peak tensile stress as 1.85: 1, which is much lower than the measured one - 8.6 : 1. At elevated temperatures when the strain rate is low, the ductile response of ice is dominant. The following mechanisms are modelled: Hookean deformation, delayed elasticity associated with grain boundary sliding and viscoplaticity associated with dislocation movement. The uniaxial material model is based on the equation proposed by Sinha. However, a more effective approximate time integration method was applied in computing the delayed elastic strain. This work gives a micromechanical representation for the (viscoclastic) delayed elasticity. This was applied during the derivation of the 3-dimensional constitutive SADS-equation, which coincides in uniaxial tension with the Ashby and Duval modification of Sinha's model. A Schapery's nonlinear viscoelasticity theory based time integration method, which is a modification to the procedure applied in the uniaxial case, was proposed for the SADS-equation. Although this method gives realistic predictions during loading it does not allow the material to relax correctly.

AB - Mathematical modelling of deformation mechanisms in polycrystalline isotropic ice over wide temperature and strain rate ranges was investigated in this thesis. The proposed 3-dimensional constitutive equations describe both time-dependent ductile deformation mechanisms and microcracking associated with Hookean deformation showing a brittle nature. A model to describe the high strain rates operative mechanisms, which form the microcrack nucleus, was introduced. It was assumed that the required four independent deformation systems were: basal glide (2 systems) and twinning (2 systems). At very high strain rates, when the dislocation motion on the basal plane is not fast enough, the total permanent response of the icebody causing microcracking may be resulting from twinning. The prominent microcracking (damage) at high strain rates is described by a fourth-order tensor. The damage evolution law is based on the potential energy theorem. The observed micromechanisms in ice are modelled for both the condition for microcrack formation and the effects of microcracks on the mechanical properties of ice. The proposed approach connects continuum damage mechanics with fracture mechanics. The proposed theory reveals the Hall-Petch type of relation, although the frictional stress vanishes. Computed examples satisfactorily gave the measured compressive strength of ice and predicted the ratio of the peak compressive stress to the peak tensile stress as 1.85: 1, which is much lower than the measured one - 8.6 : 1. At elevated temperatures when the strain rate is low, the ductile response of ice is dominant. The following mechanisms are modelled: Hookean deformation, delayed elasticity associated with grain boundary sliding and viscoplaticity associated with dislocation movement. The uniaxial material model is based on the equation proposed by Sinha. However, a more effective approximate time integration method was applied in computing the delayed elastic strain. This work gives a micromechanical representation for the (viscoclastic) delayed elasticity. This was applied during the derivation of the 3-dimensional constitutive SADS-equation, which coincides in uniaxial tension with the Ashby and Duval modification of Sinha's model. A Schapery's nonlinear viscoelasticity theory based time integration method, which is a modification to the procedure applied in the uniaxial case, was proposed for the SADS-equation. Although this method gives realistic predictions during loading it does not allow the material to relax correctly.

KW - ice

KW - deformation

KW - mechanical properties

KW - mechanics

KW - microstructure

KW - mathematical models

KW - fractures (materials)

KW - cracking (fracturing)

M3 - Dissertation

SN - 951-38-3617-7

T3 - Valtion teknillinen tutkimuskeskus. Tutkimuksia - Research Reports

PB - VTT Technical Research Centre of Finland

CY - Espoo

ER -

Santaoja K. Mathematical modelling of deformation mechanisms in ice: Dissertation. Espoo: VTT Technical Research Centre of Finland, 1990. 236 p.