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3. THE CONCEPT OF STRESS

Here a deformable body during a finite motion is considered. Since the current configuration of

many problems is not known, it is not convenient to work with stress tensors which are expressed

in terms of spatial coordinates. For some cases it is more convenient to work with stress tensors

that are referred to the reference configuration or an intermediate configuration.

3.1 Traction vectors and stress tensors

Here surface tractions which are related to either the current or the reference configuration are

introduced. These surface tractions are expressed in terms of unique stress fields acting on a

normal vector to a plane surface.

Surface tractions

Consider a deformable body occupying an arbitrary region of physical space with

boundary surface at time .

B

t

It is postulated that arbitrary forces act on parts or the whole of the boundary surface (called

external forces), and on an (imaginary) surface within the interior of that body (called internal

forces) in some distributed manner.

Let the body now be cut by a plane surface which passes any given point with spatial

coordinates

x e

ix at time . Since interaction of the two portions are considered, forces are

transmitted across the internal plane surface. The infinitesimal resultant (actual) forceacting

on a surface element is denoted by . Note the distributed so-called resultant couples not

occurring in the classical formulation of continuum mechanics are omitted.

t

df

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For every surface element

d ds d = =f t T S

t

, (3.1)

( , , ), ( , , ).t= =t t x n T T X N (3.2)

Here,

t is the Cauchy (or true) traction vector(force measured per unit surface area defined in the

currentconfiguration), acting on with outward normalds n .

T is the first Piola-Kirchhoff (or nominal) traction vector (force measured per unit surface

area defined in the referenceconfiguration), and points in the samedirection as the Cauchy

traction vector t . The (pseudo) traction vector T does not describe the actual intensity. It

acts on the region and is, in contrast to the Cauchy traction vector t , a function of the

referential position X and the outward normal to the boundary surface . (Hence,

the dashed line for in the figure.)

N0

T

The vectors t and that act across the surface elementsT ds and with respective normalsdS n

and are referred to as surface tractionsor as contact forces, stress vectorsor just loads.

Typical surface tractions are contact and friction forces or are caused by liquids or gases.

N

Cauchy's stress theorem

There exists unique second-order tensor fields and so that, P

( , , ) ( , ) ,i ij jt t t n= =t x n x n s

( , , ) ( , ) ,i iR Rt t T P=T X N X NP

(3.3.1)

N=

Cauchy's postulate

Cauchy's stress theorem(3.3.2)

(Proofis omitted.)

where,

is a symmetricspatial tensor field called the Cauchy (or true) stress tensor,

P is a tensor field called the first Piola-Kirchhoff (or nominal) stress tensor, (sometimes

simply called the Piola stress).

The index notation in eqn (3.3.2) indicates that , is a two-point tensor in which one indexdescribes spatial coordinates

Pix , and the other material coordinates RX . In section 4.3, pg 147

(GAH) it is established that is symmetric under the assumption that resultant couples are

neglected.

Equation (3.3) which relates the surface traction with the stress tensor, is one of the most

important axioms in continuum mechanics and is known as Cauchy's stress theorem (or as

Cauchy's law). Basically, it states that if traction vectors such as t or T depend on the outward

unit normals n or , then the traction vectors must be linear inN n or , respectively.N

An immediate consequence of eqn (3.3) are the following relationships,

( , , ) ( , , ), ( , , ) ( , , )t t t t = - - = - -t x n t x n T X N T X N . (3.4)

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Equation (3.3.1) in convenient matrix notation (useful for computational purposes) is,

{ } [ ]{ }=t n , (3.5)

{ } [ ] { }1 11 12 13

2 21 22 23

3 31 32 33

, ,

t n

t n

t n

= =

t n

s s s

s s s

s s s

1

2

3

= , (3.6)

where [ ] is the Cauchy stress matrix.

Relation between the Cauchy stress tensor and first Piola-Kirchhoff stress tensor P .

From eqns (3.1) and (3.2): .( , , ) ( , , )t ds t dS =t x n T X N

Using eqn (3.3),

( , ) ( , )t ds t dS =x n X N P . (3.7)

Using Nanson's formula n (eqn (2.2.18) RWO),ds J S= NG d

S

G

dJ S d=N N G P .

i.e., . (3.8)iR ij jRJ P J=P = G, s

The passage from to and back is known as the Piola transformation (cf. definition on pg.

83 GAH). Strictly speaking, in order to obtain the two-point tensor

P

P , with componentsiR

ij

P , we

have performed a Piola transformation on the second index of tensor , with components s .

From eqn (3.8),

T T1 1,ij iR jR ji

P FJ J

= = = = PF s s , . (3.9)

which implies,

T T .=PF FP (3.10)

Hence, P is in general not symmetric.

Example 3.1 A deformation of a body is described by,

1 2 2 1 3

1 16 , ,

2 33.x X x X x= - = = X . (3.11)

The Cauchy stress tensor for a certain point of a body is given by its matrix representation as,

[ ] 20 0 0

0 50 0 kN/cm

0 0 0

=

. (3.12)

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Determine the Cauchy traction vector t and the first Piola-Kirchhoff traction vector T acting on

a plane, which is characterized by the outward unit normal in the current configuration.2=n e

Solution: From eqn (3.11), calculate ,iiRR

xF

X

=

[ ] -11 12 613

0 6 0 0 2 0

0 0 , 0 0

0 0 0 0 3

,

-

= = -

F F . (3.13)

and Components of the first Piola-Kirchhoff stress tensor from eqn (3.8)

,

det 1.J= =F

ij jRP J GsiR =

[ ] [ ][ ]

16

2

0 0 0 0 0 0 0 0

0 50 0 2 0 0 100 0 0 kN/cm0 0 0 0 0 3 0 0 0

J

-

= = =

P

G . (3.14)

Determine : Nanson's formula N d ds J S=n NG T1

d dS sJ

=N nF ,

1 12 2

112

13

0 0 0

d 6 0 0 1 d 0 d d

0 0 0 0

S s s s

= - = =

N e . (3.15) 1=N e

e

Finally using Cauchy's stress theorem,

{ } [ ]{ } 20 0 0 0 0

0 50 0 1 50 kN/cm

0 0 0 0 0

= = =

t n ,

{ } [ ]{ } 20 0 0 1 0

100 0 0 0 100 kN/cm

0 0 0 0 0

= = =

T NP .

i.e., and ; and have the same direction (as expected by eqn (3.1)). The

magnitude of is twice that of

250=t 2100=T e t T

T t , because from eqn (3.15), the deformed area is twice the

undeformed area.

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1 nle 3.1tra vecc & str ten-11jan2013

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