u (x) = y (x) - x

, and its gradient is

D u = Dy - I

. Linear elasticity expands about an undeformed stress-free reference state, taking

y = x + ϵ v

with

∣ D u ∣ ≪ 1

and external loads of the same small order. A rigid translation need not itself be small; it is the deformation gradient that controls the infinitesimal strain tensor.

For

W \in C^{2}

with

D W (I) = 0

, a Taylor series gives

W (I + H) = W (I) + \frac{1}{2} A_{i A j B} H_{i A} H_{j B} + o (∣ H ∣^{2}), A_{i A j B} = \frac{\partial ^{2} W}{\partial F _{i A} \partial F _{j B}} (I) .

(1)

The constant

W (I)

can be set to zero. The tensor

A

is the elasticity tensor. Commutation of continuous mixed partial derivatives proves its major symmetry:

A_{i A j B} = A_{j B i A} .

(2)

Material frame indifference at a stress-free reference also makes the quadratic energy depend only on the infinitesimal strain tensor

e = (H + H^{T}) /2

, leading to the usual minor symmetries after spatial and reference indices are identified.

In this scaling, an external force of order

ϵ

may be evaluated on the reference geometry: its change due to the order-

ϵ

displacement contributes only at order

ϵ^{2}

. Thus the leading forces are fixed dead loads. Likewise

F = I + O (ϵ)

J = 1 + O (ϵ)

and

S = O (ϵ)

, so the Piola transformation gives

T = J S F^{- T} = S + O (ϵ^{2})

. Hence Cauchy and first Piola stresses agree to linear order. This reasoning uses the usual small-load, stress-free linearization; an order-one prestress or leading follower-load effect requires a different expansion.