1.9 The Stress Tensor

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In the previous section, it was explained that tk(x) is the stress on an infinitesimal surface

at x perpendicular to gk. The components σ kl of t k are defined by

tk = σ klgl . (1.245)

32 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

Now, t (n) = tknk can be rewritten as

t (n) = σ klnkgl (1.246)

or, since nk = gk

· n,

t (n) = σ klgl

· (gk

· n) = (σ klgl

⊗ gk) · n (1.247)

which shows that σ kl is a second-order contravariant tensor (the so-called Cauchy stress

tensor) and that the stress vector on an infinitesimal surface perpendicular to n can be

obtained by the scalar product of the transpose of the stress tensor at that point with n, in

component notation:

t l

(n)

= σ klnk. (1.248)

The Cauchy stress is also called the true stress since it is defined in the spatial state of

reference. It is the stress the deformed state truly experiences.

An important property of σ kl follows from Equation (1.240) in component notation:

eij lg

j

k σ kl = 0 (1.249)

where eij l is the alternating symbol. Since g

j

k

= δ

j

k one finds

σ kl = σ lk (1.250)

that is, the stress tensor is symmetric. Letting

σ := σ klgk

⊗ gl (1.251)

Equation (1.250) is equivalent to

σ = σ T (1.252)

and t (n) = σT · n, Equation (1.247), is transformed into t (n) = σ · n.

For the special case of t k, Equation (1.247) reduces to

t k = σT · gk. (1.253)

The term v ,k · t k in the energy balance, Equation (1.243), becomes (see also Equation (1.163))

v ,k · t k =gk

· (∇ ⊗v) · σT · gk

= (v ⊗∇) : σ T = (v ⊗∇) : σ (1.254)

yielding for the complete energy equation

ρ

Dε

Dt

= (v ⊗∇) : σ −∇ · q + ρh. (1.255)

Using the definition in Equation (1.166)

ρ

Dε

Dt

= l : σ −∇ · q + ρh (1.256)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 33

or

ρ

Dε

Dt

= d : σ −∇ · q + ρh. (1.257)

Since σ is symmetric, all its eigenvalues are real. The meaning of the eigenvalues can

be clarified by looking for the maximum normal stress in a point. Since t (n) = n · σ, the

normal stress σ on an infinitesimal surface with normal n is given by

σ = n · σ · n. (1.258)

Maximizing σ with the constraint _n_ = n · g_ · n = 1 yields

∂

∂n

_n · σ · n − λ(n · g_ · n)_ = 0. (1.259)

g_ is the contravariant metric tensor whose components gkl satisfy

gkl = gk · gl . (1.260)

Equation (1.259) leads to the eigenvalue problem

(σ − λg_) · n = 0. (1.261)

Similar to Equation (1.121), one can write

σ =

3

_

i=1

λiσ (ni ⊗ ni ) (1.262)

where ni are the complementary basis vectors to the eigen one-forms of σ . However, contrary

to C the tensor σ is not positive definite, since σ in Equation (1.258) can be negative

(pressure). In general, the stress eigenvectors do not coincide with the strain eigenvectors.

Consequently, ni in Equation (1.262) is usually distinct from ni in Equation (1.136).

The force on an infinitesimal area da can be written as

dF = t (n) da = σ · n da

= σ · da

= σ · JF

−T · dA

= J σ · (F

−T · N) dA

=: T (N) dA (1.263)

where Equation (1.65) was used. The vector T (N) represents an equivalent stress vector on

the surface in the reference configuration and satisfies

T (N) = J σ · F

−T · N. (1.264)

Defining the Piola–Kirchhoff tensor of the first kind by an expression similar to Equation

(1.247):

T (N) := PT · N (1.265)

34 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

one finds

P = JF

−1 · σ . (1.266)

Notice that P is a two-point tensor, in component notation:

PKk = JXK

,lσ lk . (1.267)

The tensor P is not symmetric. Indeed, σ = σ T is equivalent to

F · P = PT · FT. (1.268)

To remediate this, a Piola–Kirchhoff stress tensor of the second kind, S, is defined by

S := P · F

−T = JF

−1 · σ · F

−T. (1.269)

This tensor is symmetric and satisfies

S = SKLGK ⊗ GL. (1.270)

One also defines the Kirchhoff stress τ by

τ := J σ . (1.271)

Equation (1.257) can now also be written as

ρ0

Dε

Dt

= d : τ − J∇ · q + ρ0h. (1.272)

In the balance equations in the previous section, a couple of quantities were defined on

surfaces in the spatial configuration such as the heat vector q. Similar to the derivation in

Equation (1.263), an equivalent quantity in the reference configuration is defined by

q · da = Q· dA (1.273)

yielding

Q = J q · F

−T. (1.274)

Analogously, one defines

S = J s · F

−T (1.275)

for the entropy flux. Do not confuse the infinitesimal length dS, Equation (1.8), with the

Piola–Kirchhoff stress tensor of the second kind S, Equation (1.269), and the entropy

vector S, Equation (1.275). The context should clarify what is meant.

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