Continuum mechanics Ian Hewitt, University of Oxford - - PowerPoint PPT Presentation

Continuum mechanics

Ian Hewitt, University of Oxford hewitt@maths.ox.ac.uk

Continuum mechanics

A continuum approximation treats a material as having a continuous distribution of

• mass. It applies on scales much larger than inter-molecular distances.

Each ‘point’ of the continuum can be ascribed properties, such as density, temperature, velocity, pressure, etc. Continuum mechanics provides a mathematical framework to describe how these properties vary in space and time.

ρ(x, t) ) T(x, t) ) u(x, t) ) p(x, t)

Continuum mechanics can be used to describe both ‘fluids’ and ‘solids’ - we focus on fluids.

Kinematics

• Strain rate
• Coordinate systems / derivatives

Dynamics

• Stress tensor

Conservation laws

• Conservation of momentum
• Constitutive laws
• Navier-Stokes equations
• Conservation of mass
• Conservation of energy

Boundary conditions Depth-integrated models

Kinematics

Coordinate systems

Eulerian description Spatial coordinates, fixed in space

(x, t) x

Lagrangian description (X, t) Spatial coordinates, fixed in material

X

We usually choose these as the coordinates of a reference configuration at t = 0

x = (x, y, z) = (x1, x2, x3) x(X, t)

Material paths are governed by

Dx Dt = u x u x|t=0 = X

where is the time rate of change for fixed

D Dt = X

(i.e. the derivative ‘following the fluid’) velocity u = (u, v, w) = (u1, u2, u3)

!

Coordinate systems

A stake drilled into the ice tracks the ice motion in a Lagrangian system. A weather station on the ice surface measures atmospheric properties in a (roughly) Eulerian framework. Fluid models are usually written in an Eulerian coordinate system.

Material derivative

T = f(x, t)

Given some function of Eulerian coordinates (e.g. temperature) we can calculate the material derivative using the chain rule (recall )

x = x(X, t) DT Dt = ∂T ∂t + u · rT

rate of change with respect to time at fixed x

∂T/∂t

rate of change with respect to x rate of change of with respect to time at fixed

= X x r u rT

local term advective term The material derivative is also called the ‘convective’ derivative or ‘total’ derivative. In components,

DT Dt = ∂T ∂t + ui ∂T ∂xi = ∂T ∂t + u∂T ∂x + v∂T ∂y + w∂T ∂z

We use the summation convention (repeated indices imply a sum):

ui ∂T ∂xi =

3

X

i=1

ui ∂T ∂xi

Material derivative

Example The rate of change of temperature as measured by a skier has components due to:

• the temperature decreasing through the evening
• the temperature increasing as they travel downhill

DT Dt = ∂T ∂t + u · rT

Strain rate

Strain is a measure of deformation. The strain rate is a measure of how fast strain is changing. One dimension Consider the rate of change of length of a small fluid element Time stretching rate

Dt ∂x 1 dx D Dt(dx) = ∂u ∂x X D Dt(dx) = du = ∂u ∂xdx

• dx

6 ˙ εij = 1 2 ✓ ∂ui ∂xj + ∂uj ∂xi ◆

Three dimensions Time

dx = ˆ s ds dx = ˆ s ds 1 ds D Dt(ds) = 1 2ˆ sT (ru + ruT )ˆ s = ˆ si ˙ εijˆ sj

where the strain rate tensor is The strain rate is now described by a rank-two tensor (a matrix)

x y x y x y z

Strain rate

Examples

q u = @ x z 1 A @

• A

˙ εij = @ 1 1 1 A u = @ z 1 A @ A ˙ εij = @

1 2 1 2

1 A

Dynamics

We define the Cauchy stress tensor as the force per unit area in the i direction on the face with normal in the j direction.

Stress tensor

Stress is force per unit area. The stress state is described by a rank-two tensor. At each point in the material, consider a small cube.

σ = σij = σij

ijnj

q σ = σij = @ σxx σxy σxz σyx σyy σyz σzx σzy σzz 1 A

Due to conservation of angular momentum, the stress tensor must be symmetric. The stress acting on a general surface with unit normal is

si = σijnj

• r, in index notation,

n n s s = σ · n

We define the pressure by

s = σ · n p = 1 3σii

and the deviatoric stress tensor by

+ τ σ = pδ + τ

• r

✓ ◆ σij = pδij + τij

Constitutive law

The constitutive law describes a relationship between stress and strain rates - it characterises the rheology of the material For a Newtonian fluid (e.g. water)

τij = 2η ˙ εij η is the viscosity

For ice, it is common to use Glen’s flow law

τ = q

1 2τijτij

n ≈ 3 ≈ − A ≈ 2.4 × 1024 Pa3 s1 × 0 C ˙ εij = A(T)τ n−1τij

at This can be written in the form of a Newtonian fluid but with an effective (non-constant) viscosity

τij = 2η ˙ εij η = 1 2Aτ n−1

ε τ ε τ n = 1 n = 3 n = ∞ ˙ ε τ

Conservation laws

Conservation of mass

Time

dM dt = Q Z

sources - sinks A major concern at Karthaus …

Conservation of mass

ace

a d

bed

ab dM dt = Z

surface

a dS + Z

bed

ab dS Qc Z Z Qc M

Conservation of mass

Conservation of mass applies to each arbitrary (Eulerian) volume in the ice.

V

V

∂V u · n

If the material is incompressible, , we obtain

r · u = 0 Dρ Dt = 0

Use divergence theorem

Z Z Z

V

∂ρ ∂t dV = Z

V

r · (ρu) dV

Since this is true for any V

Z ∂ρ ∂t + r · (ρu) = 0

‘sources and sinks’ here are due to material flowing through the boundary

Z Z d dt Z

V

ρ dV = Z

∂V

ρu · n dS Z Z

Conservation of mass

An alternative derivation is to consider arbitrary material (Lagrangian) volumes

∂V u · n

Use Reynolds transport theorem Since this is true for any V

Z ∂ρ ∂t + r · (ρu) = 0

since the boundary is now a material surface, no mass crosses it.

V (t) d dt Z

V (t)

ρ dV = 0 Z Z Z

V (t)

∂ρ ∂t + r · (ρu) dV = 0 V (t)

Conservation of momentum

We apply a similar argument to conserve momentum for each volume V Momentum conservation is equivalent to Newton’s second law:

V

∂V u · n

flux of momentum through boundary surface forces body force (gravity)

d dt Z

V

ρui dV = Z

∂V

ρuiujnj dS + Z

∂V

σijnj dS + Z

V

ρgi dV

Apply divergence theorem Use that volume is arbitrary

Z Z ∂ ∂t(ρui) + ∂ ∂xj (ρuiuj) = ∂σij ∂xj + ρgi ✓ ◆ Z Z Z Z Z

V

∂ ∂t(ρui) dV = Z

V

∂ ∂xj (ρuiuj) + ∂σij ∂xj + ρgi dV

Use conservation of mass

✓ ◆ ρ ✓∂u ∂t + u · ru ◆ = r · σ + ρg

(in vector form) Rate of change of momentum is equal to the forces acting

Navier-Stokes equations

We have derived mass and momentum equations for an incompressible fluid

r · u = 0 ✓ ◆ ρ ✓∂u ∂t + u · ru ◆ = rp + r · τ + ρg

Combining with the Newtonian rheology gives the Navier-Stokes equations

τij = 2η ˙ εij r · u = 0 ρ ✓∂u ∂t + u · ru ◆ = rp + ηr2u + ρg ✓ ◆

constant viscosity is used here this term is non linear!

Reynolds number

✓ ◆ ∂u ∂t + u · ru = 1 ρr · σ + g u ⇠ U ⇡ 100 m y−1 ⇠ ⇡ u · ru ⇠ 10−14 m s−2 r ⇠ σ ⇠ ρgz ⇠ g ⇠ g ⇡ 9.8 m s−2 ⇠ ⇡ x ⇠ L ⇡ 1000 m

Estimate the size of terms in the momentum equation for an ice sheet The inertial terms on the left are much much smaller than those on the right. More generally, the relative size of these terms is measured by the Reynolds number

Re = ρUL η

• this is a measure of how ‘fast’ the flow is.

For small Reynolds number (‘slow flow’) we have the Stokes equations

r · u = 0 0 = rp + r · τ + ρg ˙ εij = A(T)τ n−1τij

High Reynolds number flows

✓ ◆ ∂u ∂t + u · ru = 1 ρr · σ + g

For flows with high Reynolds number (e.g. most atmosphere and ocean processes) we can usually ignore the viscous terms. However, such flows are often turbulent, and there are Reynolds stresses (due to fluctuations in the velocity field) that have to be parameterised

∂u ∂t + u · ru = 1 ρrp r · ⌦ u0u0↵ + g ⌦

0↵

Reynolds stresses When inertia is important we may also have to worry about the effects of Earth’s rotation

Du Dt + 2Ω ∧ u + Ω ∧ (Ω ∧ x) Du Dt

becomes

Conservation of energy

The same methods work to derive an energy equation. Rate of change of energy is equal to the work done by forces and net conductive heat transfer

V

∂V u · n

flux of energy through boundary work done against surface forces work done against gravity

d dt Z

V

ρ(e+ 1

2|u|2) dV =

Z

∂V

ρ(e+ 1

2|u|2)u·n dS+

Z

∂V

krT·n dS+ Z

∂V

u·σ·n dS+ Z

V

ρu·g dV

conductive transfer Applying the usual arguments …

ρcp ✓∂T ∂t + u · rT ◆ = r · (krT) + τij ˙ εij De Dt = cp DT Dt ◆

Boundary conditions

Kinematic boundary conditions

At a rigid boundary (e.g. the glacier bed* in absence of melting/ freezing), we must usually have no normal flow

u · n = 0

For a viscous fluid we also usually have no slip However, glaciers often slide at the base, for which we adopt a sliding law relating basal speed and basal shear stress

τ b = f(|ub|) ub |ub| | | τ b = σ · n (n · σ · n)n

At a material boundary (e.g. the glacier surface in absence of accumulation or melting) we insist particles must stay on the surface

D Dt (z s(x, y, t)) = 0 ∂s ∂s ∂s ∂s ∂t + u ∂s ∂x + v ∂s ∂y = w · ub = u (u · n)n = 0

If there is mass transfer (e.g. accumulation) at a boundary, conservation

• f material demands a modification of the kinematic condition

∂s ∂t + u ∂s ∂x + v ∂s ∂y = w + a

Dynamic boundary conditions

At free boundaries we apply atmospheric stress conditions

σ · n = pan n · σ · n = pa τ s = σ · n (n · σ · n)n = 0

This is often broken into normal and shear components (atmospheric pressure is often chosen as the gauge pressure and set to zero)

Stokes equations + boundary conditions

∂u ∂x + ∂v ∂y + ∂w ∂z = 0 0 = ∂p ∂x + ∂τxx ∂x + ∂τxy ∂y + ∂τxz ∂z 0 = ∂p ∂y + ∂τyx ∂x + ∂τyy ∂y + ∂τyz ∂z 0 = ∂p ∂z + ∂τzx ∂x + ∂τzy ∂y + ∂τzz ∂z ρg ∂s ∂t + u ∂s ∂x + v ∂s ∂y = w + a σ · n = 0 z = s(x, y, t) z = b(x, y) u ∂b ∂x + v ∂b ∂y = w τ b = f(|ub|) ub |ub| ˙ εij = A(T)τ n−1τij

Shallow approximation (lubrication theory)

0 = ∂p ∂x + ∂τxz ∂z 0 = ∂p ∂z ρg p = τxz = 0 z = b(x) τxz = f(ub) z = s(x, t) ∂s ∂t + u ∂s ∂x = w + a u ∂b ∂x = w ∂u ∂x + ∂w ∂z = 0 1 2 ∂u ∂z = A|τxz|n−1τxz

(1) (4) (2) (3) (5) (6) (7) (8) Integrate (1) with (7) and (8)

∂h ∂t + ∂q ∂x = a q = Z s

b

u dz Z h = s − b

Integrate (2)-(4) with (5) and (6)

− q = −2A(ρg)n n + 2 hn+2

• ∂s

∂x

• n−1 ∂s

∂x

Depth-integrated equations directly

ace

a d ) = q(x, t) −

• − q(x + dx, t) +

∂h ∂t + ∂q ∂x = a

• ∂

∂t(h dx) = q(x, t) − q(x + dx, t) + a dx ∂h ∂t + q(x + dx, t) − q(x, t) dx = a dx → 0

Depth-integrated mass conservation Rearrange Take limit

• dx

Summary

Continuum variables can be described in terms of Eulerian or Lagrangian coordinates. The material derivative is the derivative following fluid particles. The principles of mass and momentum conservation lead to coupled PDEs for velocity, pressure and deviatoric stress. Together with a constitutive law these lead to the Navier-Stokes or Stokes equations. Various boundary conditions are applicable for different types of bounding surfaces. Stress and strain rate tensors describe the forces and the rates of deformations in the material.

Continuum mechanics Coordinate systems

• Treats fluid as a continuous distribution, with properties (e.g. temperature) assigned to each

point.

• Eulerian (coordinate is fixed in space, material points moves through the coordinates)

x

• Lagrangian (coordinate moves with material, material points defined by initial position)

X T = f(x, t)

e.g. satellite footprint e.g. GPS stake

| Dx Dt = u x|t=0 = X

Rates of change

• Eulerian derivative
• Material derivative

T = f(x(X, t), t) = F(X, t) ∂T ∂t = ∂f ∂t

• x fixed
• DT

Dt = ∂F ∂t

• X fixed

DT Dt = ∂T ∂t + u · rT

• using chain rule

(rate of change ‘following the fluid’) (also called ‘convective’ or ‘total’ derivative)

Continuum mechanics

Notation

• x = (x, y, z) = (x1, x2, x3)

u = (u, v, w) = (u1, u2, u3)

• Eulerian coordinates
• Velocities
• Summation convention

(repeated indices are summed over) Strain rate tensor

✓ ◆ ˙ εij = 1 2 ✓ ∂ui ∂xj + ∂uj ∂xi ◆

Stress tensor

✓ ◆ σij = pδij + τij

• Describes local rates of deformation
• Describes force per unit area in i direction on a surface with normal in j direction
• Pressure
• Deviatoric stress tensor

r · u = ∂ui ∂xi =

3

X

i=1

∂ui ∂xi ! ✓ ◆ p = 1 3σii + τij

• si = σijnj
• Kronecker delta

X δij = ( 1 i = j i 6= j ✓ ◆

Constitutive law

• Relates stress to strain rate - general form

3 τij = cijkl ˙ εkl τij = 2η ˙ εij

• Newtonian fluid
• Glen’s law for ice

τ = q

1 2τijτij

n ≈ 3 ≈ − A ≈ 2.4 × 1024 Pa3 s1 η viscosity

Conservation of mass

q ∂ρ ∂t + r · (ρu) = 0

• Consider rate of change of mass in a control volume
• If incompressible , then

r · Dρ Dt = 0 Dt r · u = 0

• using divergence theorem, and since volume is arbitrary,

× 0 C ˙ εij = A(T)τ n−1τij

at

Z Z d dt Z

V

ρ dV = Z

∂V

ρujnj dS

Conservation of momentum

• Apply Newton’s second law to a control volume
• manipulating, and since volume is arbitrary,

ρDu Dt = r · σ + ρg

• e.g. for incompressible Newtonian fluid, these are the Navier-Stokes equations:

Dt r · u = 0 Dt r r ρ ✓∂u ∂t + u · ru ◆ = rp + ηr2u + ρg

Reynolds number

• measures the importance of the inertia terms (LHS of momentum eqn).

Re = ρUL η

• typically very small for ice flow, so approximate as ‘Stokes flow’,

Z Z Z Z d dt Z

V

ρui dV = Z

∂V

ρuiujnj dS + Z

∂V

σijnj dS + Z

V

ρgi dV 0 = rp + r · τ + ρg

Boundary conditions (general)

• rigid boundaries

u · n = 0

(no normal flow) (no slip) friction law to account for slip

· u (u · n)n = 0 τ b = f(|ub|) ub |ub|

• ub = ui (uknk)ni

| | τ b = σijnj (nkσkjnj)ni

• glacier surface

w = ∂f ∂t + u∂f ∂x + v∂f ∂y z = f(x, y, t)

Glacier boundary conditions

z = s(x, y, t) σijnj = paδij

• free surfaces

(kinematic condition)

σijnj = paδij

(stress continuity)

∂s ∂t + u ∂s ∂x + v ∂s ∂y w = a

• glacier bed

accumulation/ablation atmospheric pressure

= a pa u ∂b ∂x + v ∂b ∂y w = ab ( ⇡ 0 ) = ab basal accumulation z = b(x, y)

Shallow-layer models

• Exploit small aspect ratio to reduce complexity of the model.
• Derive by systematically approximating and integrating the governing equations, or by using

conservation laws for depth averaged quantities.

• Depth-averaged conservation of mass

∂h ∂t + r · q = a h = s b Z

• Depth-averaged conservation of momentum

Z 0 = ρghrs τ b + r · (hτ) ✓ ◆

• r

r · τ = ✓ 2τ xx + τ yy τ xy τ xy τ xx + 2τ yy ◆

ice depth ice flux membrane stress

h = s b q = hu = Z s

b

u dx