Part 3 · Kinematics

3.7 Infinitesimal strain and rotation

Source: Favata’s notes, Kinematics §2.1–2.2, p. 23–26 (15 Oct instalment)

Objectives

Say when a deformation is small, and what is thrown away when it is.
Split a small displacement gradient into a strain and a rotation, and read each one off.
Recipe: $E = sym H$ , $W = skw H$ , $w$ from $[W]$ .
Pictures: a square under $I + E$ and $I + W$ ; a point that rotates along the tangent instead of the circle.
Theorem: $R \approx I + W$ .
Vocabulary: smallness parameter, infinitesimal theory, infinitesimal strain tensor, infinitesimal rotation tensor, infinitesimal rigid displacement, axial vector, local decomposition.

Primary goal

Replace the exact, nonlinear measures of the last sections ( $C$ , $U$ , $R$ , $E_{G}$ ) by linear ones in $H$ , valid when $F$ is close to $I$ . Multiplication $F = RU$ becomes addition $H = E + W$ .

How small is small

By Theorem 3.3.1, $d ℓ / d ℓ_{0} = ∣ Ft ∣$ : if $F$ is close to $I$ , lengths change little.

Definition 3.7.1Smallness parameter

$ε := x \in Ω sup ∣ F (x) - I ∣ = x \in Ω sup ∣ H (x) ∣,$ the greatest distance between $F$ and the identity; the second form uses $H = F - I$ (Definition 3.2.5). The infinitesimal theory neglects infinitesimals of order higher than the first in $ε$ .

Why a small H means small length changesbeyond the notes

For a unit $t$ : $∣ t + Ht ∣ - ∣ t ∣ \leq ∣ Ht ∣ \leq ∣ H ∣$ . The last step is Cauchy–Schwarz on each row: $∣ Ht ∣^{2} = \sum_{i} (\sum_{j} H_{ij} t_{j})^{2} \leq \sum_{i, j} H_{ij}^{2} = ∣ H ∣^{2}$ .

Warning

Small strain is not enough. A rigid rotation by $90°$ has no strain at all ( $C = I$ ), yet $∣ H ∣ = ∣ R - I ∣$ is large. The infinitesimal theory needs small strains and small rotations (Exercise 3.7.2).

The infinitesimal strain tensor

Write $F = I + ε (x) H$ with $ε (x) := ∣ H (x) ∣ \leq ε$ and $∣ H ∣ = 1$ : all the size is in one scalar, so terms of order $ε^{2} (x)$ can be dropped everywhere at once.

Proposition 3.7.2C to first order

$C = F^{⊤} F = I + H^{⊤} + H + H^{⊤} H \approx I + H^{⊤} + H,$ since $H^{⊤} H = ε^{2} (x) H^{⊤} H$ is second order. (Equation 2.2.)

Warning

The notes print the quadratic term of (2.2) as $\frac{1}{2} H^{⊤} H$ ; multiplying out $(I + H)^{⊤} (I + H)$ gives it without the $\frac{1}{2}$ . It is dropped anyway.

From here on, terms superlinear in $H$ are neglected. The Green–Saint-Venant tensor becomes $E_{G} = \frac{1}{2} (C - I) \approx \frac{1}{2} (H + H^{⊤})$ : this is what its $\frac{1}{2}$ was for.

Definition 3.7.3Infinitesimal strain tensorEssential

$E := sym H .$ It characterises the metric changes completely in the infinitesimal theory.

Key observation 3.7.4What E measuresbeyond the notes

For a unit fibre $t$ : $(d ℓ / d ℓ_{0})^{2} = Ct \cdot t \approx 1 + 2 Et \cdot t$ , so $\frac{d ℓ - d ℓ _{0}}{d ℓ _{0}} \approx Et \cdot t$ : the relative elongation. $E_{11}$ is the one along $e_{1}$ .
Only the symmetric part counts: for skew $W$ , $Wt \cdot t = 0$ , so $Ht \cdot t = Et \cdot t$ .

The infinitesimal rotation tensor

Like any tensor, $H = sym H + skw H$ (Theorem 2.3.2).

Definition 3.7.5Infinitesimal rotation tensor

$W := skw H .$

Theorem 3.7.6The rotation to first order

Neglecting terms superlinear in $H$ , the rotation of the polar decomposition is $R \approx I + W .$

Proof

$U^{2} = C \approx I + 2 E$ by Proposition 3.7.2. And $(I + E)^{2} = I + 2 E + E^{2} \approx I + 2 E$ , so $U \approx I + E$ (the square root is unique, Theorem 2.7.10; the dropped $E^{2}$ is not written in the notes).
$(I + E) (I - E) = I - E^{2} \approx I$ , so $U^{- 1} \approx I - E$ .
$R = F U^{- 1} \approx (I + H) (I - E) \approx I + H - E = I + W$ .

So a rotation about $x_{0}$ moves $x$ by $W (x - x_{0})$ . In a rigid displacement $\nabla u = R - I = W$ is constant, and integrating from $x_{0}$ :

Definition 3.7.7Infinitesimal rigid displacement

$u_{rig} (x) = u (x_{0}) + W (x - x_{0}) = u (x_{0}) + w \times (x - x_{0}),$ with $w$ the axial vector of $W$ : $w /∣ w ∣$ is the axis, $∣ w ∣$ the (infinitesimal) angle. (Equations 2.3 and 2.4; the notes’ “(2.4) can be rewritten” means (2.3).)

Example 3.7.8Remark 2.1

$[W] = 0 γ - β - γ 0 α β - α 0$ has axial vector $w = α e_{1} + β e_{2} + γ e_{3}$ . Check one column: $W e_{1} = γ e_{2} - β e_{3} = w \times e_{1}$ .

Non-example 3.7.9I + W is not a rotationbeyond the notes

$(I + W)^{⊤} (I + W) = I - W^{2} \neq = I$ , and $∣ (I + W) a ∣^{2} = ∣ a ∣^{2} + ∣ Wa ∣^{2}$ : a linearised rotation lengthens everything a little. It is a rotation only to first order.

The split H = E + W

Key observation 3.7.10The product becomes a sumbeyond the notes

$(I + W) (I + E) = I + E + W + WE \approx I + H = F$ . So $F = RU$ becomes $H = E + W$ : a strain plus a rotation. The other order, $(I + E) (I + W)$ , gives the same, so to first order the left decomposition coincides with the right one.

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Figure 3.7.1 Drag the tips of

F e_{1}

and

F e_{2}

. The orange square is

I + E

, stretched along its principal directions; the violet one is

I + W

, turned. The teal square, the exact

F

, does both. Make

H

symmetric or skew (it snaps) and one of the two parts disappears. Then make

H

large and watch the first-order stretch of

e_{1}

drift away from the exact one.

Local decomposition

Around any $x_{0}$ , $f (x) = f (x_{0}) + F (x_{0}) (x - x_{0}) + o (∣ x - x_{0} ∣)$ : near a point every deformation behaves as if it were homogeneous. So the constant tensors above become fields:

Definition 3.7.11Local decompositions

$F (x) = R (x) U (x)$ : $R (x)$ is the local rotation tensor, the rigid rotation of points close to $x$ ; $U (x)$ the local stretch tensor, the deformation around $x$ .
In the infinitesimal theory, $H (x) = E (x) + W (x)$ : the local infinitesimal deformation and local infinitesimal rotation tensors.

Warning

The notes call $U (x)$ the “local Cauchy–Green tensor”; it is the stretch tensor. The Cauchy–Green tensor is $C = U^{2}$ .

Planar rotations: exact and infinitesimal

If the displacement lies in the plane normal to $e_{3}$ ( $u \cdot e_{3} = 0$ ), an infinitesimal rigid displacement is $u_{rig} (x) = u (x_{0}) + φ e_{3} \times (x - x_{0}), u (x_{0}) \cdot e_{3} = 0, (x - x_{0}) \cdot e_{3} = 0,$ with $φ$ the angle about $e_{3}$ : $w = φ e_{3}$ .

Let $x$ rotate by $φ$ about a fixed $x_{0}$ , with $x - x_{0}$ along $e_{1}$ . Exactly, $u_{rig} (x) = ∣ x - x_{0} ∣ ((cos φ - 1) e_{1} + sin φ e_{2}),$ and to first order in $φ$ ( $cos φ - 1 \approx 0$ , $sin φ \approx φ$ ) $u_{rig} (x) \approx ∣ x - x_{0} ∣ φ e_{2} = φ e_{3} \times (x - x_{0}) .$

Key observation 3.7.12Along the tangent, not the circle

Under an infinitesimal rotation, $x$ moves along the line through $x$ perpendicular to $x - x_{0}$ : the tangent to its circle. The error is about $∣ x - x_{0} ∣ φ^{2} /2$ , second order, and the approximation improves as $φ$ decreases.

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Figure 3.7.2 Drag the point round the circle. The teal arrow is the exact displacement; the violet one is the infinitesimal rotation, which runs up the tangent. Bring the angle down: halving it shrinks the red gap to a quarter, so the gap vanishes faster than the displacement itself.

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Figure 3.7.3 How far can you rotate before the linearised displacement is off by 1 % of the radius? Angles snap to whole degrees; the slider under Angle steps one degree at a time.

Exercises

Exercise 3.7.1beyond the notes

$[H] = 1 - 2 0 430 00 - 2 \times 1 0^{- 3}$ . Find $E$ , $W$ , $w$ , and the relative elongations along $e_{1}$ , $e_{2}$ and $(e_{1} + e_{2}) / 2$ .

Solution

$[E] = 110130 00 - 2 \times 1 0^{- 3}$ , $[W] = 0 - 3 0 300000 \times 1 0^{- 3}$ , $w = (W_{32}, W_{13}, W_{21}) = (0, 0, - 3) \times 1 0^{- 3}$ : $0.003$ rad clockwise about $e_{3}$ . Elongations $Et \cdot t$ : $1 \times 1 0^{- 3}$ , $3 \times 1 0^{- 3}$ , and $\frac{1}{2} (E_{11} + 2 E_{12} + E_{22}) = 3 \times 1 0^{- 3}$ . The exact values $∣ Ft ∣ - 1$ are $1.0020$ , $3.0080$ , $3.0020$ ( $\times 1 0^{- 3}$ ): off by order $∣ H ∣^{2}$ .

Exercise 3.7.2beyond the notes

For a rigid rotation by $θ$ about $e_{3}$ (Example 2.6.5), find $E = sym H$ and evaluate $E_{11}$ at $1°$ and $10°$ . What does it say about the infinitesimal theory?

Solution

$H = R - I$ , so $E_{11} = E_{22} = cos θ - 1$ , the rest zero; $W$ has $W_{21} = sin θ \approx θ$ . $E_{11} = - 1.5 \times 1 0^{- 4}$ at $1°$ , $- 1.5 \times 1 0^{- 2}$ at $10°$ : a pure rotation of $10°$ shows a 1.5 % “shortening”. The exact $E_{G}$ is zero for every $θ$ . The infinitesimal theory needs small rotations, not only small strains.

Exercise 3.7.3beyond the notes

Show that $(I + E) (I + W)$ and $(I + W) (I + E)$ differ by a second-order term, and say when they are equal.

Solution

Their difference is $EW - WE$ , a product of two first-order tensors. It vanishes when $E$ and $W$ commute, e.g. for a pure dilatation $E = e I$ .