Nonlinear Continua Part 5 pot

3.8 wlm = wmlwe get wt1 · wn2 = wt2 · wn1= The above result, a direct consequence of the Cauchy Theorem, is known as the projection theorem or reciprocal theorem of Cauchy Malvern 1969.J

3.2 The Cauchy stress tensor 71

Fig 3.3 Tangent surfaces at P have the same traction vector

3.2.1 Symmetry of the Cauchy stress tensor (Cauchy Theorem)

In Sect 4.4.2 we will prove that from the equilibrium equations of a nonpolar continuum we get

that is to say, the Cauchy stress tensor is symmetric

In the following figure, at the pointS , inside the w-configuration of a contin-uum body, the stress tensor isw

Cutting the continuum with the surface wV1, we get atS the traction vector

wt1 = wn1 · w >

if we cut with wV2, the traction vector atS is:

wt = wn · w =

Secant surfaces at a point inside a continuum

In an arbitrary coordinate system {w{l} we can write

wt1 · wn2 = wq1 l

wlm wq2 m >

wt2 · wn1 = wq2l wlm wq1m >

and since from Eq (3.8) wlm = wmlwe get

wt1 · wn2 = wt2 · wn1= The above result, a direct consequence of the Cauchy Theorem, is known as the projection theorem or reciprocal theorem of Cauchy (Malvern 1969).JJJJJ

3.3 Conjugate stress/strain rate measures

Let us assume, at an instant (load level)w, a continuum body B in equilibrium under the action of external body forceswband external surface forceswt Assuming a velocity field wv(wx) on B, the power provided by the external forces is:

wSh{w =

Z

w Y

wwb · wvwdY +

Z

w V

wt · wvwdV = (3.9a)

Using Eq (3.7) we can rewrite Eq.(3.9a) as

3.3 Conjugate stress/strain rate measures 73

wSh{w =

Z

w Y

wwb · wv wdY +

Z

w V

¡w

n · w¢

· wvwdV = (3.9b) From the Divergence Theorem (Hildebrand 1976),

wSh{w =

Z

w Y

£ w

wb · wv + u ·¡w

 · wv¢ ¤w

introducing Eq (2.110a-2.110c) and after some algebra,

wSh{w =

Z

w Y

£w

 : wl + ¡w

wb + u · w ¢

· wv ¤ w

dY = (3.9d) Since the w-configuration is an equilibrium configuration, the following equation (to be proved in Chap 4, Eq.(4.27b)) holds

wwb + u · w = w G

wv

and an obvious result that we also need is:

Gwv

Gw ·wv = G

Gw

 1 2

wv · wv

¸

The kinetic energy of the body B, at the instant w, is defined as

wN =

Z

w Y

w 2

wv · wvwdY =

Z

p

1 2

wv · wvdp = (3.9g)

In the second integral of the above equation we integrate over the mass of the body B Since the mass of the body is invariant,

GwN

Z

p

Gwv

Finally, using the decomposition of the velocity gradient tensor in Eq (2.112a) and considering that since (w) is a symmetric tensor and (w$) is a skew-symmetric one,

we get

wSh{w = GwN

Z

w Y

We define

wS =

Z

w Y

as the stresses power ObviouslywSis the fraction ofwSh{wthat is not trans-formed into kinetic energy and that is either stored in the body material or dissipated by the body material, depending on its properties (see Chapter 5) From Eq (3.10) we define the spatial tensors w and wd to be energy conjugate (Atluri 1984)

In what follows we will define other pairs of energy conjugate stress/strain rate measures

3.3.1 The Kirchho stress tensor

From Eqs (3.10) and (2.34d) and the mass-conservation principle (to be dis-cussed in Chapter 4, Eq.(4.20d)) we get

wS =

Z

w Y

w : wdwdY =

Z

 Y



w

w : wddY = (3.11)

The Kirchho stress tensor is defined as

w =



w

where,: density in the reference configuration andY : volume of the refer-ence configuration

It is important to note that although the Kirchho stress tensor was in-troduced by calculatingwSvia an integral defined over the reference volume,

Eq (3.12) clearly shows that w is defined in the same space where w is defined: the spatial configuration Hence, using in thew-configuration an arbi-trary curvilinear coordinate system {w{d} with covariant base vectorswgdwe obtain

w = wde wg

d

wg

and

wde =



w

3.3.2 The first Piola-Kirchho stress tensor

From Eqs (3.11) and (3.9i) we can write

wS =

Z

 Y

In Chap 2 we learned how to derive representations in the reference config-uration of tensors defined in the spatial configconfig-uration via pull-back operations

We will now obtain a representation of the Kirchho stress tensor in the form

of a two-point tensor

In the reference configuration we define an arbitrary coordinate system {{D} with covariant base vectors g

D; and in the spatial configuration a system {w{d} with covariant base vectors wg

d We also define a convected system {l} with covariant base vectors in the reference configuration egl and covariant base vectors in the spatial configurationwegl

In the spatial configuration we can write the Kirchho stress tensor as

3.3 Conjugate stress/strain rate measures 75

w = w˜lm wegl wegm (3.15a)

a pull-back of the above tensor to the reference configuration is:

wT` = w˜lm egl egm= h

wT`iLM

gL gM = (3.15b)

We define a two-point representation of w as

w

P = w˜lm egl wegm = £w

P¤Lm 

g

L

wg

After some algebra,

w

SLm = w˜op C

{L Co

Cw{m

Cp =

"

wst C

o

Cw{s

Cp

Cw{t

#

C{L Co

Cw{m

Cp

= wsm ¡w

[1¢L

The second-order two-point tensor wP is the first Piola-Kirchho stress tensor It is apparent from Eq (3.15d) that it is a non-symmetric tensor

We can write, due to the symmetry of the Kirchho stress tensor:

S =

Z

 Y

wde wodedY =

Z

 Y

wed wodedY = (3.16a)

Hence, using Eq.(3.15d):

S = Z

 Y

w

SEd w[eE wode dY = (3.16b) Using Eq (2.111a) we have

S =

Z

 Y

w

SEd w[˙dE dY =

Z

 Y

w

P · · wX˙ dY = (3.17)

We can also write the above as (Malvern 1969):

S = Z

 Y

w

The above equation defines the two-point tensors w

PW and w

X˙ as energy conjugates

We will not try to force a so-called “physical interpretation” of the first Piola-Kirchho stress tensor; instead we will regard it only as a useful math-ematical tool

3.3.3 The second Piola-Kirchho stress tensor

The pull-back configuration ofw to the reference configuration is:

wT` = h

wlm âw

[1đL l

âw

[1đM m

i

gL gM = (3.19)

The tensor wT`, defined by the above equation, is the second Piola-Kirchho stress tensor and it is a symmetric tensor

Using Bathe’s notation (Bathe 1996), we identify the second Piola-Kirchho stress tensor, corresponding to thew-configuration and referred to the config-uration inw = 0 asw

S

From Eqs (3.15d) and (3.19) we get

w

VLM = âw

[1đM m

w

SLm that is to say,

w

VLM = êw

!âw

SLmđôLM

=

JJJJJ From Example 2.16:

w

%˙DE = êw

!âw

gdeđô

and since from Eq (3.19),

w

VDE = êw

!âw

deđôDE

(3.20b) using Eqs (3.11) and (2.88), we get

wS =

Z

 Y

w

S : w%˙ dY = (3.20c)

From Eq (3.20c) we define the tensors wS and w% to be energy conjugate˙ (Atluri 1984)

Here, we will also not try to force a “physical interpretation” of the second Piola-Kirchho stress tensor

An important point to be analyzed is the transformation ofw

Sunder rigid-body rotations

• Let us first consider the w-configuration of a certain body B At an arbitrary pointS the Cauchy stress tensor isw

3.3 Conjugate stress/strain rate measures 77

• Let us now assume that we evolve from the w-configuration to a (w + w)-configuration imposing on B and on its external loads a rigid body rotation

At the pointS we can write:

w+w

and therefore,

 For the external loads,

 For a velocity vector,

 For the external normal vector,

For an arbitrary force vector wf (it can be a force per unit surface, per unit volume, etc.) and considering the evolution described above,

w+wf · w+wv = w+ww R · wf ·w+wv

= wf ·w+ww RW · w+wv

= wf ·w+ww RW · w+ww R · wv

and since the rotation tensor is orthogonal,

w+wf · w+wv = wf · wv= The above equation states the intuitive notion that a rigid-body rotation cannot aect the value of the deformation power performed by the external

Atw we can write

and at (w + w),

Introducing Eq (3.22d) in the above,

w+wt = ¡w+w

w R · wn¢

And with Eq (2.28a) and (3.22a), we finally have

wt = wn · hw+w

w RW · w+w · w+ww Ri

For deriving the above equation, we used thatw+ww R · wt = wt · w+ww RW = Hence,

w+w = w+ww R · w · w+ww RW = (3.23e) The above equation indicates that the Cauchy stress tensor fulfills the cri-terion for objectivity under isometric transformations, established for Eulerian tensors in Sect 2.12.2

We define an arbitrary system {w{d 0

} in the w-configuration and a system {w+w{d} in the (w + w)-configuration Hence, from Eq (3.23e),

w+wd

e = w+ww Ud

f 0 wf 0

g 0

¡w+w

w UW¢g 0

and using Eq (2.28c), we get

w+wd

e = w+ww Ud

f 0 wf 0

g 0 w+w

w Uo

p 0 w+wjoe wjp 0 g 0

(3.24b) therefore,

w+wdo = wf0p0 w+ww Udf0 w+w

It is easy to show that for the Kirchho stress tensor we can also write

w+w = w+ww R · w · w+ww RW = (3.25) From Eq (3.19), we obtain

w+w

 VLM = w+wlm ¡w+w

 [1¢L

l

¡w+w

 [1¢M

but since,

w+w

 [d

D = w+ww Ud

0 w

[d 0

¡w+w

 [1¢D

d = ¡w

[1¢D

d 0

¡w+w

w UW¢d 0

using Eqs (3.24c) and (3.26c) in Eq (3.26a), we get

3.3 Conjugate stress/strain rate measures 79 w+w

therefore,

w+w

The above equation indicates that the second Piola-Kirchho stress ten-sor fulfills the criterion for objectivity under isometric transformations, estab-lished for Lagrangian tensors in Sect 2.12.2

3.3.4 A stress tensor energy conjugate to the time derivative of the Hencky strain tensor

In Sect 2.8.5 we defined the logarithmic or Hencky strain tensor

Let us now define, via a pull-back operation, the following stress tensor:

With the notationwU(·) we define the pull-back of the components of the tensor (·) using the tensorwR (Simo & Marsden 1984), that is to say, w is

an unrotated representation ofw

From the symmetry of w , the above definition implies the symmetry of

w =

We will now demonstrate, following (Atluri 1984), that for an isotropic material w and w

H˙ are energy conjugate.

We can write Eq (3.14) as

wS =

Z

 Y

wDE êw

Uâw

gdeđô

DE

therefore,

wS =

Z

 Y

wDE wUdD wgdewUeE dY = (3.29b)

From Eq (2.28c),

w

UdD = âw

UWđO o

and using the above, the integrand in Eq (3.29b) is:

wDE âw

UWđO o

wjdo jDO wgde w

Ue

It is also easy to show that

w

RW · wd · wR = hâw

UWđO o

wjdo jDO wgde wUeEi

gD gE = (3.29e)

Hence, using Eq (2.118b),

wS =

Z

 Y

1 2

w : Ể

w

U˙ · wU1 + wU1 · wU˙Ễ

dY = (3.29f)

In order to simplify the algebra, in what follows we will work in a Cartesian system; [D] will be the matrix formed with the Cartesian components of a second-order tensor A

From Eqs (2.122a-2.122d), we get

[wX ] [˙ w

X]1 = [wUO] [w] [˙ w]1 [wUO]W + [w O] (3.30a)

 [wUO] [w] [wUO]W [w O] [wUO] [w]1 [wUO]W and

[wX ]1 [wX] = [˙ w

UO] [w]1 [w] [˙ w

+ [wUO] [w]1 [wUO]W [w O] [wUO] [w] [wUO]W  [w O]

it follows from the above two equations that

1

2

n

[wX ] [˙ w

X]1 + [wX]1 [wX ]˙ o

= [wUO] [w]1 [w] [˙ w

UO]W (3.30c)

+ 1

2 [

w

UO] [w]1 [wUO]W [w O] [wUO] [w] [wUO]W

 1

2 [

w

UO] [w] [wUO]W [w O] [wUO] [w]1 [wUO]W >

and using once again Eqs (2.122a-2.122d), we get

1

2

n

[wX ] [˙ wX]1 + [wX]1 [wX ]˙ o

= [wUO] [w]1 [w] [˙ wUO]W (3.30d)

+ 1

2 [

w

X]1 [w O] [wX]  1

2 [

w

X] [w O] [wX]1 = Using the result in Example 2.17, we can write

1

2

n

[wX ] [˙ wX]1 + [wX]1 [wX ]˙ o

= [wK]  [˙ w O] [oqwX] (3.30e) + [oqw

X ] [w

 O] +1

2 [

w

X]1 [w O] [wX ]  1

2 [

w

X ] [w

 O] [wX ]1 = Using the above in Eq (3.29f) and working with the matrix components,

wS =

Z

 Y

[w ] n

[wK]˙   [w O] [oqwX] (3.31a) + [oqwX] [w O]+ 1

2 [

w

X1] [w O] [wX ]

 1

2 [

w

X ] [w O] [wX1]

¾

dY =

Since [w ] and [w

X ] are symmetric and [w

O] is skew-symmetric, we can rewrite the above equation as

3.4 Objective stress rates 81

wS =

Z

 Y

[w ] [wK]˙  dY



Z

 Y

n £ [w ] [w

K]¤

  £

[wK] [w ]¤



o [w O] dY

+1

2

Z

 Y

n £

[wX ]1 [w ] [wX ]¤

  £

[wX ] [w ] [wX]1¤



o [w O] dY =

(3.31b)

We will show in Chap 5 that for isotropic materials the Eulerian tensors

w (w ) andw

Vhave coincident eigenvectors (they are coaxial)

Taking into account that

w

and the definition ofw in Eq (3.28) we conclude that the Lagrangian tensors

w , w

U and wH are also coaxial for isotropic materials

Obviously, the coaxiality of w and w

H implies that [w ] [wK]  [wK] [w ] = [0] (3.33a) and the coaxiality of w and w

U implies that [wX]1 [w ] [w

X]  [w

X ] [w ] [w

X ]1 = [0]= (3.33b) Finally, for isotropic materials,

wSLvrw=P dw= =

Z

 Y

and therefore in this casew and w

H˙ are energy conjugates.

3.4 Objective stress rates

In Sect 2.14.1 we show that the adequate tool for deriving objective rates of Eulerian tensors is the Lie derivative

The Lie derivative of the Cauchy stress tensor is:

³

w ´de

Ow v(w)¤de

= w˙de  wfe wodf  wdf woef (3.35) the above stress rate is known as Oldroyd stress rate (Marsden & Hughes 1983)

Example 3.4 JJJJJ

To derive the expression of Oldroyd’s stress rate we start from Eq (2.129),

¡

Ow v w¢de

= Cwde

Cwde

Cw{s

wys  C

wyd

C{S

¡w

[1¢S s

wse

wye

C{S

¡w

[1¢S s

wds=

From Eq (2.109a)

w

X˙ =



Cwyd

C{D + w[sDwd

so wyo

¸

wgdgD >

and Eq (2.111b)

w

X˙ = wl · wX>

we get,

Cwyd

C{D = wodpw[pD  wuvd wyv w[uD = Then, from

w = wde wg

d

wg

e >

we get,

w =˙



Cwde

Cwde

Cw{o

wyo + wpe wpod wyo + wdp wpoe wyo

¸

wgdwge =

Replacing in the first equation, after some algebra, we finally get

¡

Ow v w¢de

= w˙de  wods wse  woes wds =

JJJJJ

• The Lie derivative of the Kirchho stress tensor is:

h

w ide

=£

Ow v(w )¤de

= w˙de  wfe wodf  wdf woef (3.36) the above rate is known as the Truesdell stress rate (Marsden & Hughes 1983)

From Eq (3.19),

wlm = wVLM w[lL w[mM >

hence,

3.4 Objective stress rates 83

Cwlm

Cwlm

Cw{s

wys = wV˙LM w[lL w[Mm + wVLM C

wyl

C{L

w

[mM +wVLM w[lL C

wym

C{M

w! (wV˙LM)ilm

= wV˙LM w[lL w[Mm = Therefore,

h

w! (wV˙LM)ilm

= Cwlm

Cwlm

Cw{s

wys

wop (w[1)Lo(w[1)Mp Cwyl

C{L

w

[mM

 wop(w[1)Lo(w[1)Mp Cwym

C{M w[l

L =

Using algebra along the lines of Example 3.4 and Eq.(3.36) we finally get,

w lm = h

w! (wV˙LM)ilm

=

JJJJJ

The above example shows that the necessary and su!cient condition for the second Piola-Kirchho stress tensor to remain constant is that the Trues-dell stress rate is zero (Eringen 1967)

We can also perform pull-back and push-forward operations using the rota-tion tensorwR(Simo & Marsden 1984) Let us consider an arbitrary Eulerian stress tensor wt( e.g.w or w ), and perform on its components a w



U(wt)¤DE

= wwde(wUW)Dd(wUW)Ee = (3.37a)

In order to simplify our calculations we will now work in a Cartesian

U(wt)¤

= [wU]W [ww] [w

Taking into account that d

dw

© [U]Wª

= ©d

dw[U]ªW

> we get d

dw

£w

U(wt)¤

= [wU]W [w˙w] [w

U] + [w

U]˙ W [ww] [w

U] + [w

U]W [ww] [w

U] (3.37c)˙ and using Eqs (2.115a) and (2.116a) we get

d

dw

£w

U(wt)¤

= [wU]W [w˙w] [w

U]  [wU]W [w U] [ww] [wU]

+ [wU]W [ww] [w U] [wU] = (3.37d)

h

Ow

 R(wt)i

= [wU] d

dw

£w

U(wt)¤

[wU]W (3.37e)

we finally arrive at

h

Ow

 R(wt)i

= [w˙w]  [w

U] [ww] + [ww] [w U] (3.37f) that in an arbitrary spatial coordinate system leads to

h

Ow

 R(wt)ide

= w˙wde  w U fd wwfe + wwdf w U fe = (3.38) The above Lie derivative is the well-known Green-Naghdi stress rate (Di-enes 1979, Marsden & Hughes 1983, Pinsky, Ortiz & Pister 1983, Simo & Pister 1984, Cheng & Tsui 1990) From its derivation it is apparent that the Green-Naghdi stress rate is objective under isometric transformations, there-fore it is known as a corotational stress rate

In the case of the Kirchho stress tensor, its Green-Naghdi rate is the

w

U-push-forward of the rate ofw

If as a reference configuration we use thew-configuration (Dienes 1979), we will fulfill Eq (2.118d), that is to say,

t$ = w

Using the above in Eq (3.38) we obtain the Jaumann stress rate (Truesdell

& Noll 1965)

It is important to realize that (Dienes 1979):

• The Jaumann stress rate is only coincident with the corotational stress rate whenwU  g

• When formulating a corotational constitutive relation, we will only be able

to use the Jaumann stress rate when the spatial and reference configura-tions are coincident