# Stress and strain

## Introduction to glacier stress and strain

Glacier on the Antarctic Peninsula. Glaciers always flow downslope. From Wikimedia Creative Commons.

In Part I (Deformation and Sliding), we learned that glaciers flow downslope in response to their driving stresses, which are a function of the weight of the ice and gravity. ‘Strain’ is the deformation of glacier ice in response to this stress. The gravitational driving stress and the ability of ice to deform control a glacier’s velocity, which is an important parameter that we must measure if we are to understand glacier behaviour and the dynamic response of glaciers to climate change.

Note that we are referring to ice surface slope; glaciers can flow up hills as flow towards the glacier terminus is driven by glacier surface slope, not bed topography.

This webpage investigates glacier stress and strain in more detail, in order to better understand why glaciers deform and flow.

## Newton’s Three Laws

Newtown’s Three Laws of Motion

Before we begin, we must understand stress and strain and how that applies to glaciers. We must first recall Newton’s Three Laws:

1.  Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

2. The relationship between an object’s mass, m, its acceleration a and the applied force is:

F = ma

Where F is the force, m is the mass of an object (in kilograms, kg), and a is the acceleration of an object (metres per second, or m s-2). Acceleration and force are vectors, and the direction of the force vector is the same as the direction of the acceleration vector. Force is measured in Newtons (1 N = 1 kg m s-2).

Gravity (g) is a force, and the weight of an object is a result of its mass and its acceleration due to gravity (which is a constant, 9.81 m s-2):

W = mg

For ice, the mass of ice is a function of the density of ice (ρi) and the thickness of the ice (h). So for a glacier, the weight is:

W =  ρigh

3. For every action, there is an equal and opposite reaction.

## Types of stress and strain

### Introduction to Stress

‘Stress’ is a measure of how hard a material is pushed or pulled as a result of external forces. ‘Strain’, on the other hand, measures the amount of deformation that occurs as a result of the stress. There are various kinds of stress. A Force is something that changes the motion of a mass (from stationary to moving, or from uniform movement to another movement). For a glacier, the acceleration due to gravity (g) and the mass of a block of ice exert of downwards force on the bed.  A Force is always a push, a pull or a twist that changes the motion or shape of an object (Benn & Evans 1998).

When a force is applied to an object, the intensity depends on the area over which the force is distributed. My friend Jo weighs the same (more or less) when she is wearing stiletto heels as she does without. However, if she stands on my toes in her stiletto heels, it hurts a lot more than if she stands on my toes in bare feet! The stress is much greater, because the force is distributed over a much smaller cross-sectional area. Stress is measured in Pascals (1 Pa = 1 N m-2).

Glacier stress and Strain. Modified from Benn and Evans, 1998.

Stress can act at right angles to a surface (normal stress) or parallel to a surface (shear stress). Shear stress means that two tractions are acting parallel to the surface; in the case of a glacier, the gravitational driving stresses are resisted by basal drag and lateral drag, controlled by friction.

### Basal Shear Stress

At the base of a glacier with minimal slope, the normal stress (σ) acting on the bed is mainly a result of the weight of a glacier.

σρigh

Stress therefore increases with ice thickness, and normal stresses therefore increase from 0 at the ice surface to their maximum at the ice-bed interface. In small, simple glaciers, basal shear stress is also controlled by slope (α). So, if τb represents basal shear stress, then:

τb = ρigh sin αS

Table 1. Parameters used in equations

 Symbol Parameter Unit σ Normal stress sigma Pa τb Basal shear stress tau Pa ρi Density of ice (900 kg per metre cubed) roe Kg m3 g Gravitational acceleration (9.81 m s-2) m s-2 h Ice thickness m αS Ice surface slope alpha Degrees (°)

Glaciers flow because ice deforms as a result of basal shear stress. In fact, basal shear stress varies across the glacier bed because glaciers do not flow on hard, smooth surfaces. Over a rough bed, some areas will have higher stresses, and some areas will have lower stresses.

Glacier driving stresses are typically around 105 Pa at the base (Patterson, 1994). This uniform driving stress is due to the rheology of ice; ice deforms through non-linear viscous flow, with an effective viscosity that allows ice to support a basal shear stress (τd) of approximately 105 Pa. Glaciers thicken until they reach stresses of this magnitude, and further thickening or steepening leads to increased flow and thinning (Marshall et al., 2011).

Basal driving stresses do of course deviate substantially from 105 Pa, due to longitudinal and horizontal stresses, valley shape, and the effects of basal sliding or subglacial sediment deformation. Ice fields, with a low surface slope, will probably have a lower basal shear stress of less than 105 Pa, while larger glaciers and steep valley glaciers may have driving stresses well in excess of 105 Pa (Marshall et al., 2011).

Longitudinal stress modifies basal shear stress and is the pushing or pulling effect of the ice. Accelerating ice imposes tensile stresses, and decelerating ice imposes compressional stresses. This may result in brittle fracture of the ice, and can be seen in crevasses as the ice flows over a steeper part of the bed, for example.

Other stresses resisting ice flow include transverse stresses, such as lateral drag against the valley walls for a valley glacier, or with slower-moving ice for an ice stream. Vertical shear stress gradients occur when there is extensive shearing at the bed and limited shearing at the ice surface. Vertical resistive stresses may be important where the flow regime changes, such as near the ice divide or ice margin (Pattyn, 2003), but vertical resistive stress is much smaller than the other normal or shear stress components.

Driving and resisting stresses operating on a block of ice on an inclined slope.

### Strain

Strain is the change in shape of an object following the application of stress. Different materials respond differently to the same stress. In our case, ice, water, rocks and air all have different properties that define their strain responses (Benn & Evans 1998).

Strain may be elastic or permanent. Elastic strain is a temporary change in a material, and when the stress is removed, the material reverts to its original shape. The change in shape stores strain energy. At a certain level of stress, the strain energy is released, resulting in failure or permanent deformation. The level of stress at which failure occurs is the yield stress. Deformation may be brittle, where the material fractures, or ductile, where flow or creep occurs.

## Glacier flow

Glacier flow is a combination of the deformation of the ice and the bed, and sliding of ice over its bed. Glacier velocity therefore equals deformation plus basal sliding.

### Glacier Deformation

Glaciers flow because permanent deformation occurs as a result of strain in response to stress. Strain may include deformation of the ice or the sediments at the ice-bed interface, or sliding at the ice-bed interface. Resistance to strain depends on ice temperature, crystal structure, bed roughness, debris content, water pressure and other factors.

Creep is the deformation of ice crystals. Movement can occur between or within ice crystals (Cuffey & Paterson 2010). The relationship between creep and stress can be given by Glen’s Flow Law:

ε = Aτn

Where ε is the strain rate, A and n are constants, and τ is the basal shear stress. The constant n = 3, and the value of A is dependent on ice temperature, crystal orientation, debris content and other factors. Glacier ice may form beautiful folds or structures in response to creep.

Crevasses open up when the glacier is stretched (under tensile stress)

Brittle deformation occurs when ice cannot creep fast enough to accommodate the driving stresses. Crevasses are obvious examples of brittle failure in a glacier.

Compression at the margins of the glacier may result in thrusting. A thrust is a fault in the ice as a result of brittle deformation.

Deformation of the bed is dependent on the porewater pressure. If subglacial sediments are poorly drained and are saturated with water, then high porewater pressures develop. If porewater pressures are similar to the ice overburden pressure, then they support the normal stresses acting between the grains. The normal stress is reduced.

### Basal Sliding

Basal sliding occurs only where a glacier is at pressure melting point at its base. Most of the fast ice flow associated with ice streams comes about because of basal sliding. Wet glacier ice on a smooth surface is slippery. The sliding at the ice-bed interface is controlled by freezing to the bed, bed roughness, the quantity of water at the bed, and the amount of rock debris in the basal glacier ice.

Glacier beds are rough, not smooth. Bumps in the surface of the glacier bed cause melting on the upstream side, and re-freezing on the downstream side. This is called regelation, and it occurs because pressures mount up behind obstacles to ice flow. Ice melts under pressure, and this lubricates the bed of the glacier.

Meltwater at the ice-bed interface reduces the adhesion of the glacier to its bed, making it more slippery and enhancing sliding. If a glacier is flowing over a rock bed, a water film may enhance sliding and submerge minor obstacles, making the bed smoother.

## Summary

Understanding how and why glaciers flow means that we must understand glacier stress and strain. Glacier stress occurs because glaciers flow downslope under their own weight, and the strain rate is the ice response to this stress. The response of glacier ice to this stress is to deform and creep. Glacier ice also melts under pressure, and meltwater at the ice-bed interface encourages glacier sliding. The total velocity of a glacier is a function of ice deformation, bed deformation and glacier sliding.

## References

Benn, D. I., and Evans, D. J. A. (1998). “Glaciers and Glaciation.” Arnold, London, 734 pp.

Cuffey, K. M., and Paterson, W. S. B. (2010). “The Physics of Glaciers, 4th edition.” Academic Press, 704 pp.

Marshall, S.J., White, E.C., Demuth, M.N., Bolch, T., Wheate, R., Menounos, B., Beedle, M.J., Shea, J.M., 2011. Glacier Water Resources on the Eastern Slopes of the Canadian Rocky Mountains. Canadian Water Resources Journal / Revue canadienne des ressources hydriques 36, 109-134.

Paterson, W., 1994. The physics of glaciers, 480 pp. Pergamon, New York.

Pattyn, F., 2003. A new three-dimensional higher-order thermomechanical ice sheet model: Basic sensitivity, ice stream development, and ice flow across subglacial lakes. Journal of Geophysical Research: Solid Earth 108, 2382.

## 7 thoughts on “Stress and strain”

1. Hey, I am beginner of Glacier study. Your blog is very helpful. Just want to thank you.

Panshu

2. I’m surprised there is no mention or consideration of the amount of heat from the land mass at the interface listed as reason for glaciers movement down slope and that stress is considered the main factor. Is it possible that areas in the ground supporting the glacier can have varied temperatures? I notice that so many glaciers cover areas previously volcanic. Since heat rises, it would seem to be a cause of glacial movement worth considering.
Also, I am puzzled by the lack of rocks and “debris” in the photos of glaciers. I’ve looked extensively and carefully, even the picture at the top of this article, and don’t see the evidence. There are dirty areas or layers, as volcanic ash accumulates from the atmosphere between snow accumulation, but shouldn’t we be able to see parts of these valleys that have been” carved” by this ice within the glaciers as the calving process cross sections these massive ice forms?
Thank you for considering my questions.
Jdc

• Hello,

Geothermal heat flux is actually included in the full mass balance equation. This is a page that I am intending to write soon! Geothermal heat flux can contribute to basal melt and so facilitate sliding at the base of the glacier. However, ice is a poor conductor so, unless ice is very thin or the geothermal heat flux very high, tends not to influence melt at the ice surface.

In terms of debris cover, this is an important control on ablation. A thin layer of debris will enhance melt as dark surfaces absorb more energy, but a thick layer of debris will insulate the ice and reduce surface melt. This relationship is controlled by the Østrem curve. Again, another post about this is clearly required!

3. I really appreciate your explanation of pressure as a person in high heels. Most physics explanations are about men and “men’s things” such as sports and manual labor. I am happy to see a “feminine” example added to the mix. Very important to show that physics and science are welcoming to people who are traditionally feminine.

4. Hello Bethan, in ‘Basal Sheer Stress’ Section, you also mention a term ‘Basal driving stress’. So what the difference between them? thanks

5. Hi Bethan&co, thanks so much for your fabulous work. These are just so systematically and so easy to understand. You even write the glossary and put some useful links here which are really sweet. I feel very happy to read all the pages, you words make the study of glaciology quite interesting. Thanks again for all the job you have done.

6. Shear stress is the force applied by flowing liquid to its boundary. In this case, the liquid is storm water and the boundary is the channel surface. Shear stress is also occasionally referred to as the “tractive force.” Put simply, shear stress describes the force of water that is trying to drag the channel surface downstream with it….. http://shearstress.net/