A. Cylindrical coordinates

Many problems are such that it is advantageous to use cylindrical (r, O, z) instead of Cartesian (x, y, z) coordinates. Cylindrical basis vectors (er, eo, e z )

are expressed in the Cartesian basis (e x, ey, e z ) as follows:

e r = cos(O)ex + sin(O)e y , eo = - sin(O)ex + cos(O)ey (A.l)

See Fig. 8.1 for an illustration. Both cylindrical and Cartesian bases are orthonormal. The position vector of a point M(r,O,z) W.r.t. the frame (O,ex,ey,e z ) is:

~

x -=OM= rer + zez (A.2)

Differentiation of (A.l) gives:

de r = (dO)eo,dee = -(dO)er (A.3)

Let v be a vector field defined in the cylindrical basis as follows:

v(r, O, z) = F(r, O, z)er + G(r, O, z)eo + H(r, O, z)ez (A.4)

The gradient (V' v) of v is defined by:

dv = (V'v) . (dx) (A.5)

Differentiating (A.4) and using (A.3), we obtain:

( âF âF âF) dv = âr dr + âO dO + âz dz e r + F(dO)eo

( âG âG âG) + âr dr + âO dO + âz dz eo - G(dO)er

( âH âH âH) + a:;:dr + ao dO + azdz e z (A.6)

Differentiation of (A.2) gives:

dx = (dr)e r + (rdO)ee + (dz)e z (A.7)

546 A. Cylindrical coordinates

Using (A.6) and (A.7), Eq. (A.5) can be written in the following matrix form:

[ 8F dr + !(8F - G)rd() + 8F dz 1 8r r 88 8z 8C dr + !(8C + F)rd() + 8C dz 8r r 88 8z

8H dr + ! 8H (rd(}) + 8H dz 8r r 88 8z

(A.8)

, .. dX

where the 3 x 3 matrix (V'v) is given in the cylindrical basis as follows:

[

8F 8r

V'v = ~~ 8H ar

The divergence of v is given by:

!(8F -G) r 88 !(8C +F) r 88

18H r7fi[

8F 1 8z 8C 8z 8H {fi

. 8F l 8G 8H dlV v == tr(V'v) = 8r +;:-( 8(} + F) + 7);'

(A.9)

(A.lO)

using (A.9). Now let g(r,(},z) be a scalar field. The gradient (V'g) of 9 is defined by:

dg = (V'g) . (dx), (A.l1)

where the 3 x l array (V' g) is given in the cylindrical basis as follows:

V'g = [ ~~ l, 8z

(A.12)

using (A.7). The gradient (V'V'g) of (V'g) is found from (A.9) and (A.12) as follows:

[ !t..s. !(h _ !!!s.) 8r2 r 8r8 r 88

V'V'g = .!l..(!!!s.) !(!~ + !!s.) 8r 5 88 r r 88 8r b.. !h 8r8z r 888z

(A.13)

The Laplacian (Llg) of 9 is defined by:

_ 8 2g l 182g 8g 8 2g Llg = tr (V'V'g) = 8r2 + ;:-(;:- 8(}2 + 8r) + 8z2 (A.14)

This can be rewritten as follows:

l 8 8g l 82g 82g Llg = ;:- 8r (r 8r) + r2 8(}2 + 8z2 (A.15)

Consequently, the following result follows:

A. Cylindrical coordinates 547

Consider a second-order symmetric tensor a (e.g., stress (1" or strain 1:) and a vector u. In Cartesian coordinates, the following result is easily established:

(A.17)

This can be written in the following intrinsic form which is valid in cylindrical coordinates for instance

div(a . u) = (diva) . u + a: (\lu), (A.18)

where div designates the divergence operator. We now apply the result to the cylindrical basis by taking u to be equal to er , e() and ez, successively,

(diva) . er div(a· e r ) - a: (\le r )

oarr l (oaer ) oazr l ar + ~ 7iiJ + arr + fu - ~aee,

(diva) . ee div(a· e()) - a: (\lee) oar() l oa()() oaze l ar + ~(7iiJ + are) + fu + ~ar(),

(diva)· e z div(a· e z ) - a: (\le z )

oarz l (oaez ) oazz ar + ~ 7iiJ +arz + fu' (A.19)

For the projection along e r ) we used (A.9) with v = e r , Le. F = l and G = H = O, and Eq. (A.lO) with

Le. F = arn G = aer and H = azr. The projections along e() and ez are obtained in a similar fashion. As an application of results (A.19), equilibrium equations (div (1" + f = O) are obtained in cylindrical coordinates by setting a = (1", i.e.

oarr ~ oar() oazr arr - ae() f - O· ar + r of) + oz + r + r - ,

oare loa()() oaze 2 1, O -- + --- + -- + -are + () = ar r of) oz r

oarz loa()z oazz arz f O --+---+--+-+ z= OZ r of) OZ r

(A.20)

The infinitesimal strain tensor is defined by: 1: = (\lu + ,;r u)/2, where u is the displacement. Using (A.9), the components in the cylindrical basis are given as follows:

548 A. Cylindrical coordinates

âUr Err

âr ,

Eee = 1 (âUO ) ;: â() + Ur ,

EOr ~ (~âUr _ Ue + âUo) = ErO, 2râC r âr âuz

Ezz âz ,

Erz ~ (âur âUz )_ 2 âz + âr - Ezr

Eez ~ (âue + ~ âUz ) = Eze 2 âz r â(}

(A.21)

We can go from Cartesian to cylindrical coordinates via the following 3 x 3 tmnsformation matrix:

[ cos(} sin(} O 1

[P] = - sin() cos() O O O 1

(A.22)

The rows correspond to the components of e r , eo and e z in the Cartesian hasis, respectively. Matrix [P] has the following properties:

[p][p]T = [pf[p] = [8], det[P] = 1,

where [8] is the 3 x 3 identity matrix. A vector v transforms from one or­thonormal hasis (eI, e2, e3) to another (ei, e2' e3) according to:

{v*} = [P]{v}, {v} = [Pf{v*} (A.23)

An application to cylindrical versus Cartesian coordinates gives:

vr = (cos ())v", + (sin ())vy, Vo = - (sin ())v", + (cos ())vy (A.24)

A second-order tensor a transforms from one orthonormal hasis (ei) to an­other (ei) according to:

[a*]" = [P][a][Pf, [a] = [p]T[a*][P] (A.25)

If a is symmetric (e.g., stress U or strain f.), then applying results (A.25) to cylindrical versus Cartesian hases, we ohtain:

arr a",,,, cos2 () + ayy sin2 () + 2a",y sin(fJ) cos(())

aee = a",,,, sin2 () + ayy cos2 () - 2a",y sin(()) cos(())

arO (a yy - a",,,,) sin () cos () + a",y (cos2 () - sin 2 9) = aOr

arz = a",z cos () + ayz sin () = azr

aez = -a",z sin() + ayz cos() = aze (A.26)

A. Cylindrical coordinates 549

In order to obtain the expressions of the Cartesian components in terms of the cylindrical ones, we need not to develop (A.25)b, it suffices to substitute 8 with (-8) in Eqs. (A.26). Note also that arr +aoo + azz = axx +ayy + azz ,

'-"'" '-"'" because (tr a) is an invariant.

B. Cardan's formulae

Consider the following cubic equation where A, B and Care given parameters:

x3 + Ax2 + Bx + C = O (B.1)

A closed-form solution exists and is named after Cardan (1501-1576). A typ­ical example of (B.l) is the search for the eigenvalues (TJ1, TJ2, TJ3) of a second­order symmetric tensor 7] (e.g., stress U or strain €). lndeed, as mentioned in Sec. 1.4, those eigenvalues satisfy the following equation:

where I J (7]) , J = 1, 2, 3, are the principal invariants of 7],

Actually, it is better to first compute the eigenvalues r;J of the deviatoric part of 7] defined by:

Il (7]) dev 7] = 7] - -3-1,

because we have h(dev 7]) = O and thus A = O in (B.l). Once the r;/s are found, the eigenvalues of 7] are computed as follows:

Another example where the solution of Eq. (B.1) is needed is the strain localization condition in 2D (Sec. 19.11.1).

There are various ways of presenting Cardan's formulae. We shall give hereafter an implementat ion which proved to be computationally robust but has one problem which will be explained at the end of the appendix.

The following change of variable:

A X=x+-

3

transforms (B.l) into the following equation:

(B.2)

552 B. Cardan's formulae

X 3 +pX +q = O, (B.3)

where p and q are defined as follows:

(B.4)

We only consider the case when p < O (see comments at the end). We intro­duce the following notation:

(B.5)

If it is found numerically that C</> > 1, then it is reset to 1. Likewise, if C</> < -1, then it is reset to (-1). Next, we compute the following angle:

1 cP3 == :3 arccos CrjJ (B.6)

Finally, the solutions of the original equation (B.I) are the following:

A -3 + 2P3 COScP3,

-~ - P3[COScP3 + (sin cP3)v'3J,

A . - 3 - P3 [cos cP3 - (SIn cP3) v'3] (B.7)

The solution procedure given here is robust but there is one problem: In theory, if (4p3 + 27q2 > O) then there is one real solution and two complex ones. This case is not detected with our algorithm, which always returns three real solutions. A workaround is to always check whether those solutions are physically acceptable or not.

c. Matrices for the representation of second­and fourth-order tensors

C.l Storage

Let a,b be second-order symmetric tensors (e.g., stress and strain tensors), and e, D, E fourth-order tensors (e.g., Hooke's operator in linear elastic­ity, tangent operators in elasto-plasticity, etc.) For computations, a or bare stored in 6 x 1 arrays as follows:

{a} = [ an (C.l)

This is not the traditional way of storing stress and strain tensors. That method distinguishes between stress and strain: shear components of strain are multiplied by a factor of 2 while the shear components of stress are kept as they are. 1 However, we shall show hereafter that definit ion (C.l) has some nice properties which the traditional method does not possess.

The inner product of a and b is given by the scalar:

(C.2)

The traditional storage does not lead to such expressions if both tensors are stresses or both of them are strains.

If e relates a and b in the following way (e.g., linear elasticity or incre­mental elasto-plasticity, etc.)

(C.3)

then it must have the following symmetries (because a and bare symmetric),

(C.4)

and it is stored in a 6 x 6 matrix according to:

{t} [1011 1022 1033 21012 21023 21031] T

{O"} [0"11 0"22 0"33 0"12 0"23 0"31]T

554 C. Matrices for the representation of second- and fourth-order tensors

al1 Cl111 C1l22 C1l33 Cll12 V2 C1l23V2 a22 C2211 C2222 C2233 C2212 V2 C2223 V2 a33 C3311 C3322 C3333 C3312 V2 C3323 V2

a12V2 C12l1 V2 C1222 V2 C1233 V2 2C1212 2C1223

a23V2 C2311 V2 C2322 V2 C2333 V2 2C2312 2C2323

a31V2 C3111 V2 C3122 V2 C3I33 V2 2C3112 2C3123

Cl131 V2 bl1

C2231 V2 b22

C3331 V2 b33 (C.5)

2C1231 b12V2

2C2331 b23V2

2C3131 b31 V2

If C has the additional symmetries (e.g., Hooke's operator in linear elasticity):

(C.6)

then Eq. (C.5) shows that the 6 x 6 matrix [C] is symmetric. The fourth-order tensor E = C : D, defined in component form by:

(C.7)

is stored as a 6 x 6 matrix [E] which is the product of the 6 x 6 matrices [C] and [D], i.e.

[E] = [C][D] (C.8)

The re ader can check for example that:

[E]ll = E l111 , [Eh4 = E1l12V2, [E]55 = 2E2323 , etc.

The fourth-order identity tensor 1, defined in component form by:

1 I ijk1 = 2(c5ik c5j1 + c5il c5jk ) (C.9)

is stored as the 6 x 6 matrix [1]:

1 O O O O O O 1 O O O O O O 1 O O O O O O 1 O O

(C.lO)

O O O O 1 O O O O O O 1

i.e., the 6 x 6 identity matrix (if one stores the components Iijkl as they are, one finds the last 3 terms in the diagonal equal to 1/2 instead of 1).

C.I Storage 555

The isotropic elasticity tensor C defined by (A and ţi being Lame's coef-ficients) :

C = 2fJ,l + Al 01, i.e. Cijkl = J1.(rSik rSjl + rSi/rSjk ) + ArSijrSkl (C.lI)

is stored as the 6 x 6 matrix [C] computed by:

[C] = 2J1.[I] + A{I}{I}T, i.e. (C.12)

l O O O O O l O l O O O O l

[C] 2J1. O O l O O O

+A l

[ l l l O O O ] O O O l O O O O O O O l O O O O O O O l O

A + 2J1. A A O O O A A+2J1. A O O O A A A + 2J1. O O O

(C.13) O O O 2J1. O O O O O O 2J1. O O O O O O 2J1.

This is consistent with storing the stress u and the strain € as the 6 x l arrays defined in Eq. (C.I); one can also check that the more general 6 x 6 matrix defined in Eq. (C.5) reduces to that of Eq. (C.13) in the isotropic case defined by Eq. (C.lI). (Note: with a traditional storage, the last three terms in the diagonal are simply J1.).

If a fourth-order tensor C is the the tensor product of two symmetric second-order tensors a and b,

(C.14)

then it is stored as a 6 x 6 matrix [C] (where {a} and {b} are 6 x l arrays defined in (C.I)):

[C] = {a}{b}T, i.e. (C.15)

al1 bl1 allbn au b33 allb12 V2 allb23 V2 all b31 V2 a22 bll a22 bn a22 b33 a22 b12V2 a22 b23V2 a22 b31V2

[C]= a:n b11 a33 b22 a33 b33 a33 b12V2 a33 bz3V2 a33 b31 V2 a1Z bll V2 a12 b22V2 a1Z b33 V2 2a12b12 2a12b23 2a12b31

aZ3 b11 V2 a23 b22V2 a23 b33V2 2a23b12 2a23b23 2a23b31

a31 bl1 V2 a31 b22 V2 a31 b33 V2 2a31 b12 2a31 b23 2a31 b31 (C.16)

556 C. Matrices for the representation of second- and fourth-order tensors

This is consistent with the matrix storage of fourth-order tensors as defined in Eq. (C.5), and unlike the traditional storage, there is no problem and no distinction to be made when both a and bare stress tensors or both are strain tensors.

C.2 Change of coordinates

Consider two orthonormal bases (ei) and (ei), i = 1,2,3. The change of coordinates from the first basis to the second is determined by a 3 x 3 matrix [P] defined by:

Since: Oij = e; . ej = (Pikek) . (Pjlel) = PikPjlOkl,

it appears that the matrix [P] verifies:

(C.17)

(C.18)

Let v be a vector represented by 3 x 1 arrays {v} in the first basis and {v*} in the second. We have:

Therefore, the following transformation rules hold:

{v*} = [P]{v}, {v} = [p]T{v*} (C.19)

A second-order tensor acan be represented by 3 x 3 matrices [a] in the first basis and [a*] in the second. We have:

a:j = (a· ej) . e; = PjkPil(a· ek) . el = PjkPilalk

Consequently, the following transformation rules hold:

[a*] = [p][a][p]T, [a] = [p]T[a*][P] (C.20)

We now assume that a is symmetric. In order to write the transformat ion rules in component form, one can use Eq. (C.20), or use a procedure which is explained hereafter (and which will be useful for the transformation of fourth-order tensors).

In component form, Eq. (C.20)a gives:

a;j PikaklPjl

PilPjlall + Pi2Pj2a22 + Pi3Pj3a33

+ (Pi1 Pj2 + Pi2 Pj1 )a12 + (Pi2 Pj3 + Pi3 Pj2 )a23 + (Pi3 Pj1 + Pi1 Pj3 )a31

C.2 Change of coordinates 557

(Note that we do have aij = aii)' Equations (C.20) can then be written in the following matrix forms:

{a*} = [q){a}, {a} = [qf {a*}

where {a*} and {a} are 6 x 1 arrays defined as in (C.1),

{a*} = {a} [ all

a~2 a 33 ai2V2 ahV2 a 31 V2 ( T

a22 a33 a12V2 a23V2 a31 V2 ]

and [q] is a 6 x 6 matrix given by:

PllPll P12 P12 P13 P13 Pll PI2 V2

P21 P21 P22 P22 P23 P23 P21 P22V2

[q] P31 P31 P32 P32 P33 P33 P31 P32 V2

Pll P21 V2 P12 P22V2 P13 P23V2 PU P22 + P12P21

P21 P31 V2 P22 P32V2 P23 P33V2 P21 P32 + P22 P31

Pl1 P31 V2 P12 P32 V2 P13 P33 V2 Pl1 P32 + P12 P31

P12P13V2 P11 P13V2

P22 P23 V2 P21 P23V2

P32 P33 V2 P31 P33 V2

P12P23 + P13P22 Pu P23 + P13 P21

P22 P33 + P23 P32 P21 P33 + P23 P31

P12P33 + PI3 P32 PU P33 + P13P31

(C.21)

(C.22)

Note that unlike the traditional storage, the expres sion of [q] is unchanged, whether we transform stress or strain tensors. The reader can count the number of operations (especially the multiplications) to see which one of transformations (C.20) or (C.21) is computationally cheaper.

From Eqs. (C.21) it is easy to check that:

[q][q]T = [qf[q] = [1] (C.23)

In linear elasticity, the relation (T = C : f: is derived from a potential w (strain energy density):

âw (T= -,

âf:

. 1 wlth w = -f: : C : f:

2 (C.24)

We wish to find a matrix transformation rule for the fourth-order tensor C. We follow the method of Lekhnitskii (1981). In the basis (ei), we can write Eq. (C.24)b as:

(C.25)

558 C. Matrices for the representation of second- and fourth-order tensors

where {E} is a 6 x 1 array defined as in Eq. (C.l), and [C] a 6 x 6 matrix defined as in Eq. (C.5).

In the basis (ei), we can write Eq. (C.24)b as:

(C.26)

where {f*} is a 6 xl array defined as in Eq. (C.l), and [C*] a 6 x 6 matrix defined as in (C.5).

Equating the expressions of w in Eqs. (C.25) and (C.26) and using the transformation rule (C.2l)a, we obtain:

(C.27)

Since this relation musţ hold for any {f}, we deduce that:

[C] = [qf[C*][q] (C.28)

Using the transformation rule (C.2l)b and equating the expressions of w in Eqs. (C.25) and (C.26), we obtain:

[C*] = [q][C][q]T (C.29)

Note that transformation rules (C.28) or (C.29) are also valid if C is not Hooke's operator but has the same symmetries as that tensor. Finally, no transformation is needed if C is isotropic since [C*) = [C).

D. Zero-stress constraints

In the various stress update algorithms that were presented in Chaps. 12, 13, 15, 16, 17 and 18 it was assumed that alt strain or deformation gradient components are known. There are cases however where the assumption is not valid. If we have a (local) plane stress state (as for plates or shells) the out-of­plane component of strain is unknown. AIso, for a beam, only the axial strain component is known. This appendix shows how to deal with those cases for some important classes of material models.

D.I Small-strain J2 elasto-plasticity

We keep the notations of Sec. 12.10.2. We show how the return mapping algorithm is modified in plane stress when the out-of-plane strain component E33 is unknown. A trial stress is defined as follows:

(D.l)

where all components of €tr are equal to those of € except 4'3 which is com­puted such that (Ţ~3 = o. Using a well-known technique (see Chap. 7), the following value is found:

(D.2)

The stress-elastic strain relation reads:

(D.3)

The reader can check that this can be rewritten as in (Doghri, 1995):

(D.4)

where:

560 D. Zero-stress constraints

8ij = It~3 = c5i3 c5j3 - ~c5ij, O 1

-3 O 1

-3 1 2

"6 = dev = 3 O O

(D.5)

O O O O

using the 6 x 1 array notat ion (Appendix C). The volumetric and deviatoric parts of the stress-strain relations are:

tr (7 tr (7tr + 3K(t33 - t~~);

S = str - 2GLlep + 2G(t33 - t~~)"6 (D.6)

Combining the first equation with the incremental plastic flow relation

3 s LleP = --Llp, (D.7)

20"eq

we find:

(D.8)

Taking the inner product of each side of the equation and using the yield condition:

O"eq = O"y + R(p),

the following scalar equation is found:

(O"eq + 3GLlp)2 = (0";;)2 + 4G2(t33 - t~~)2 + 6G(t33 - t~~)8~~,

which can be rewritten as follows:

(D.9)

k1 (p, t33) = 3GLlp + O"y + R(p) - {(0";;)2 + 2G(t33 - t~~)[38~~

+2G(t33 - t~~)]} 1/2 = O (D.lO)

The extra-unknown t33 is such that 0"33 = O, Le.

1 833 + "3tr (7 = O

This can be rewritten as follows, using (D.6)a and (D.8),

k2 (p,t33) = (K + 4~) (t33 - t~~)[O"y + R(p)]

+G [tr (7tr + 3K(t33 - t~~)] Llp = O , ' .. 3Ktr e

(D.11)

(D.12)

D.2 General small-strain models 561

In summary, the problem is reduced to finding two scalar unknowns: p and t:33

which satisfy two scalar nonlinear equations (D.lO) and (D.12). This system can be solved iteratively using Newton's method:

(âk1 )(it)[p(iHl) _ (it)] + (âk1 )(it)[ (it+l) _ (it)] _ -k (it) (it)) â p â f 33 f 33 - 1 P , f 33 P f33

(ââk2 )(it)[P(it+l) _ p(it)] + (ââk2 )(it)[f~~+l) _ f~~)] = -k2(p(it),f~~)) (D.13) P f33

The reader can check that the four partial derivatives are given as follows:

âk1 dR âp 3G + dp'

âk1 G[3sM; + 4G(f33 - 4'3)] ât:33 k1 - 3GL1p - [O'y + R(p)]

âk2 (4G) tT dR = K + 3 (f33 - t:33 ) dp + 3GKtr t: âp

âk2 (K + 4~) [O'y +R(p)] +3GKL1p (D.14)

âf33

The algorithm presented here is the same as in (Aravas, 1987). Another, projection-based algorithm is proposed in (Simo and Taylor, 1986).

D.2 General small-strain models

As we have seen in the previous section, enforcing a plane stress condition for h elasto-plasticity, which is perhaps the simplest nonlinear material model, is rather involved. For more sophisticated models, the algorithm can become very tedious. That is why it may be advantageous to use the following pro­cedure which is both very simple and general.

The plane stress condition is considered as a nonlinear equation where the unknown is f33:

(D.15)

This equation is solved iteratively with Newton's method:

(it) + (â0'33 )(it) [ (it+l) _ (it)]_ O 0'33 â f 33 f 33 -

t:33 (D.16)

------(it ) c3333

So, at each iteration (it), the constitutive routine is called with aU strain

components given, including f~~). The routine computes the stress (T and the material (consistent or algorithmic) tangent c at tn+l' If the plane stress

562 D. Zero-stress constraints

condition (D.15) is not satisfied, then we iterate again by computing E~~+1) from (D.16).

For a beam with axis 1, only Ell is known. Components E22 and 1:33 are found from the conditions:

(D.17)

Similarly to the plane stress case -(D.16)- these two nonlinear equations are solved iteratively using Newton's method.

D.3 General finite-strain models

In this section, we extend the algorithm of the previous section to the finite­strain regime. In plane stress, component F33 of the deformation gradient is unknown and should be computed such that the 33-component of the Cauchy stress u is nil. As we shall see hereafter, it is easier to write the condition as follows:

(D.18)

where T = (det F)u is the Kirchhoff stress. This equation is solved iteratively with Newton's method:

(it) + ( 8733 )(it) [F,(it+1) _ F,(it)j = O 7 33 8F33 33 33 (D.19)

The problem is to compute (8733/8F33) knowing the spatial tangent operator c. The reader can assume hyperelasticity for now, but the results we shall find are also valid for elasto-plasticity.

The following reiat ion between Kirchhoff and first Piola-Kirchhoff stresses hold:

(D.20)

Taking the partial derivative w.r.t. FpQ and making extensive use of results found in Sec. 15.1.3, we obtain the following successive equalities:

8PiK 8FpQ F jK + PiK8jp8KQ

AKiQpFjK + PiQ8jp

= (SKQ 8ip + FiMFpNCKMQN)FjK + P iQ 8jp

(PjQ8ip + P iQ8jp ) + FjKFiMFrRFpNCKMRN(F-l)Qr

(PjQ8ip + P iQ8jp ) + Cjirp(F-l)Qr, (D.21)

where c is the spatial tangent operator. Now, using the relation between T

and P again,

D.3 General finite-strain models 563

(D.22)

we can recast (D.21) in the following simple format:

(D.23)

where asum over r is assumed. The Jacobian needed in (D.19) is found as an application: (i, j) = (3,3) and (p, Q) = (3,3),

(D.24)

In summary, for plane stress, at each iteration (it), the constitutive routine is called with aU deformation gradient components given, including Fi;t). The routine computes the Kirchhoff stress T and the spatial tangent operator c at tn+l. If the plane stress condition (D.18) is not satisfied, then we iterate again by computing Fi;t+l) from (D.19) and (D.24)

For a beam with axis 1, only Fll is known. Components F22 and F33 are found from the conditions:

(D.25)

These two nonlinear equations are solved iteratively using Newton's method. The four partial derivatives which are needed are given directly by (D.23).

Exercise: Consider a finite-strain elasto-plastic model based on a multi­plicative deformat ion gradient decomposition and hyperelasticity and show that result (D.23) still holds. Hint: use Eq. (16.35) and review the above calculations carefully.

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Index

acceleration, 338 accumulated plastic strain, 302 acoustic tensor, 473 additive decomposition, 321, 413 Airy stress function, 167, 196 algorithmic tangent operator, see

consistent tangent operator Almansi-Euler strain, 346 - algorithm, 416 Armstrong-Frederick model, see

kinematic hardening axisymmetric problem - 2D, 198 - plate, 146 - thermo-elasticity, 239

back stress, see kinematic hardening stress

backward Euler integration, 316 Baushinger effect, 302 beam, 49 beam-column, 268 Beltrami-Mitchell compatibility

equations, 24 bending moment - beam, 58 - plate, 130, 143, 148 bending problem - beam, 51 - plate, 131 bending stiffness, 130, 297 bi-harmonie, 25, 167 bifurcation - continuous, 475 - discontinuous, 475 - elastic damage, 471 Biot's strain, see nominal strain body force - 2D, 187 boundary condition, 9, 388 - beam, 60

- plate, 133 boundary-value problem, see local or

weak formulations , 313 brittle material, 16, 203, 241 buckling, 249 bulk modulus, 20, 377

Cardan's formulae, 551 Castigliano's theorem, 44 - beam, 68, 85 Cauchy stress, 351, 371 Cauchy-Green strain - left, 340, 375, 399 - right, 340, 370, 400 Cayley-Hamilton theorem, 376, 418 centroid, 50 Chaboche-Marquis model, see

kinematic hardening change of coordinates - cylindrical, 545 - matriees for, 556 - polar, 193 characteristie volume, 498 circular arch, 214 circular section - torsion, 113, 114 circular tube - torsion, 121 Clausius-Duhem inequality, 321, 367,

399 - ductile damage, 445 - kinematic hardening, 428 - nonlocal model, 503 Codazzi-Gauss compatibility condi-

tions, 280 coil winding, 206 compatibility equations, 12 complementary energy, 38 composite material, 26, 205, 242 compression of a disk, 230 conie surface, 283

conjugate function, 98, 105 conjugate variables, see dual variables conservat ion of linear momentum, 350 conservat ion of mass, 349 consistent tangent operator, 318 - ductile damage, 459 - kinematic hardening, 433 - viscoplasticity, 333 continuity equations, 11 - beam, 60 contour lines - torsion, 100 Cosserat mechanics, 496 creep, 329 critical buckling load, 252 critical buckling stress, 254 critical damage, 441 cross section, 49 current configuration, 337 curvature - beam, 54 - plate, 129, 145 - shell, 289 - yield surface, 320 curved beam, 209 curvilinear coordinates, 273 cyclic plasticity, 303, 423, 435 cylindrical coordinates, see change of

coordinates - torsion, 109

damage mechanics, 439 damage model - ductile, 440, 442 - elastic, 469, 506, 509 damage potential, 444 damage threshold, 441 damage variable, 439 dead load, 388 deformat ion gradient, 339, 416 deviatoric, 6 direct method - stability, 252 directional derivative, 385, 503 Dirichlet problem, 99 discontinuity - beam, 63 - plate, 151 displacement - shell, 285, 287 dissipation, 322 - elastic damage, 506, 510 - macroscopic, 502, 517

- mechanical, 324, 399 distortion energy, 22 divergence operator

Index 575

- cylindrical coordinates, 546, 547 Drucker's stability criterion, 312 dual variables, 33, 69, 322, 356 ductile material, 203, 226, 241

eccentric force, 267 effective stress, 440, 443 eigenvalues, see principal values eigenvectors, see principal directions Einstein's summation convention, 1 elastic predictor, 315, 404 elastic unloading, 309 elasticity domain, 304, 307, 325, 399 - ductile damage, 443 - kinematic hardening, 426 elasticity tensor - material, 371, 375 - spatial, 372, 376, 387, 395, 407 elasto-plasticity, 305 elliptic section - torsion, 110 ellipticity - rate problem, 473 energy method - stability, 252, 256, 269 entropy, 366 equation of state, 321, 400 - ductile damage, 443 - kinematic hardening, 426 equilibrium equations, 8 - 2D, 163, 164, 195 - beam, 58 - cylindrical coordinates, 547 - plate, 126, 144 - shell, 292 Eshelby's results, 524 Euler's method - buckling, 252 Eulerian description, 338, 351 exponential algorithm, 404, 419

fatigue, 462 fictitious cut - beam, 58 - plate, 147 finite element method, 47, 313 first moment, 50 Flamant's solution, 224 flow rule, 307, 322, 403 - viscoplasticity, 331

576 Index

Fourier series - 2D, 169, 182, 200 - plate, 136, 157 Fourier's heat con duct ion law, 235 fracture Mechanics, 465 free energy, 321, 367, 399, 410 - duc ti le damage, 442 - elastic damage, 506, 510 - kinematic hardening, 426 - macroscopic, 500, 515 fundamental form - first, 274 - second, 278

Gâteaux derivative, 385 Galerkin's method, 45 generalized standard material, 323 - nonlocal model, 503 geometric stiffness, 389 Gibbs vector, 420 gradient operator - curvilinear coordinates, 283 - cylindrical coordinates, 545, 546 gradient plasticity, 498 Green-Lagrange strain, 344, 345 - algorithm, 416 Green-Naghdi-McInnis rate, 364, 417

hardening modulus, 311, 327 - ductile damage, 446 - kinematic hardening, 429 hardening stress, 302, 399, 423 harmonic, 96 heat equation, 234, 324 heat flux, 235, 366 Helmholtz's free energy, see free energy Hill's criterion, 306 Hill's linear comparison solid, 473 Hill's maximum dissipation principle,

307, 324, 402 hollow elliptic section - torsion, 118, 119 homogenization, 521 - nonlocal model, 498 Hooke's operator, 17, 305 hydrostatic pressure, 19, 373 hydrostatic stress, 24 hyperelasticity, 369, 401 hypoelasticity, 369, 372, 413

identity tensor, 2 incompressibility, 20, 307, 340, 359,

373,392 indicator function, 323

influence function - plate, 140 influence line, 69 initial yield stress, 302 internal energy, 366 internal force, 3 - plate, 130 internal length, 496, 502 internal load, see stress resultant internal variable, 321, 399 - ductile damage, 442 - kinematic hardening, 426 isochoric deformat ion, 6, 350, 374, 382,

403 isotropic material, 18, 374, 379, 399

J 2 flow theory, 305, 411 Jaumann rate, 363, 372 - algorithm, 420

kinematic hardening, 425, 442 - stress, 423 kinematically admissible, 31 kinetic energy, 353, 366 Kirchhoff stress, 356, 370, 399, 413 Kirchhoff-Love theory - plate, 127, 135 - shell, 287 Kirsh's solution, 223 Kronecker's symbol, 2 Kuhn-Tucker conditions, 308

Levy's method - plate, 136 Lagrange multiplier, 373, 390, 392 Lagrangian description, 338, 354, 355,

388 Lagrangian multiplier, 308 Lame's problem, 201, 205, 207, 222 Lame's coefficients, 18 Laplacian operator - cylindrical coordinates, 546 laws of thermodynamics, 366 - first, 366 - second, 366 Lee's multiplicative decomposition, 397 Leibnitz's formula, 220 Lejeune-Dirichlet theorem, 252 Lemaitre-Chaboche model, 441, 442 length variation, 343 Lie derivative, 364, 371, 400 linear elasticity, 16, 33, 305 linearization, 385 - of constitutive equations, 387

- of deformation, 386 - of elasto-statics, 387 - of pressure, 393 - of weak formulation, 389 local formulation, 9, 29, 351, 356 local state method, 320 localization mode, 473 localization surface, 473 localization theorem, 349 logarithmic strain, 344, 348 logarithmic stretch, 396, 401

macro-crack, 439, 490 macroscopic approach, 519 material description, see Lagrangian

description material time derivative, 338 Maxwell's geometric compatibility, 475 Maxwell-Betti theorem, 42, 47 - beam, 68, 80 - plate, 140 mechanical problem, 237 membrane analogy, 102 membrane problem, 245 - plate, 130 membrane theory, 298 micro/macro approach, 519 mid-point - rotation, 420 - rule, 415, 420 - spin tensor, 420 - strain, 416 mid-surface - shell, 273 middle fiber, 49 mixed formulation, 390 mixed stress, 391 Mohr's stress circles, 7 moment of inertia, 50 Mooney-Rivlin model, 378 Mori-Tanaka model, 526 motion,338 multi-connected, 12 - torsion, 104

Nanson's formula, 354 Navier equations, 23 Navier-Bernoulli assumption, 59 necking, 359 neo-Hookean model, 377 nominal strain, 344, 346 nominal stress, 354, 370 non-associative plasticity, 325

Index 577

- ductile damage, 442 - kinematic hardening, 425 non-conservative load, 258 nonlocal model, 496 normal force - beam, 57 normal stress - beam, 64 normality rule, see flow rule , 323 - generalized, 325, 427, 443 Norton's power law, 331, 332

objective stress rate, 363, 413 objectivity, 359, 370 - incremental, 422 octahedral shear stress, 14 Ogden's model, 380

penalty, 378, 380, 392 Piola identity, 387, 393 Piola-Kirchhoff stress - first, 354 - second, 357, 370, 375, 379 plane strain, 163, 195 - generalized, 28, 201, 240, 243 plane stress, 129, 164, 195, 296 - h elasto-plasticity, 559 - algorithms, 559 - general finite-strain models, 562 - general small-strain models, 561 - generalized, 131, 170 plastic buckling, 255 plastic corrector, 316 plastic loading, 309 plastic multiplier, 326 - ductile damage, 445 - kinematic hardening, 428 - viscoplasticity, 332 plastic potential, 325 - ductile damage, 443 - kinematic hardening, 427 plastic power, 308 plastic strain, 302, 408 plastic work - ductile damage, 449 Poisson's ratio, 19 polar coordinates, see change of

coordinates polar decomposition, 340 - algorithm, 417 potential energy, 35, 389, 390 - plate, 139, 148 - torsion, 108

578 Index

Poynting effect, 383 Prandtl's stress function, 98 Prandtl-Reuss equation, 305 principal axes of inertia, 51 principal curvature lines, 273 principal directions, 340, 379, 401, 417 principal invariants, 6, 374, 418 principal stretches, 340, 374, 379, 418 principal values, 6 proper-orthogonal, 2, 419 pull-back, 403 push-forward, 372, 375

radial return algorithm, 317, 411 radius of gyration, 254 rate of deformation, 347, 370, 399, 413 rate-dependent plasticity, see

viscoplasticity reciprocity theorem, see Maxwell-Betti

theorem rectangular section - torsion, 116 reference configuration, 337 reference temperature, 233 Reissner-Mindlin theory - plate, 135 representative volume element (RVE),

500, 519 return mapping algorithm, 315, 406 - ductile damage, 450 - kinematic hardening, 430 - viscoplasticity, 333 Reuss model, 524 rigid body motion, 35, 200, 360 Ritz's method, 45 - plate, 160 - torsion, 117 Rodrigues formula, 420 rotation matrix or tensor, 340, 413 - algorithm, 417, 419 rotation of a disk, 215, 218

Saint-Venant's principle, 25, 53, 134 - 2D, 176, 180, 211 - thermo-elasticity, 240, 246 - torsion, 97 self-consistent model, 524 self-equilibrium, 37 semi-crystalline polymer, 527 - amorphous phase, 531 - inclusion, 528 - intermediate phase, 536 - micro-domain, 533

- polydomain, 533 - slip system, 528 semi-infinite plate, 224, 228 shape variation, 22 shear force - beam, 57 - plate, 143, 148 shear modulus, 19, 377 shear reduced area, 67, 70 shear stress, 7 - beam, 64 Simo's algorithm, 404, 406, 408, 410 simple shear, 313, 381 simply connected, 12 slenderness ratio, 254 small-perturbation hypothesis, 10 spatial description, see Eulerian

description specific heat, 234 specific heat supply, 234 spectral decomposition, 341, 406, 418 spin tensor, 347 - algorithm, 420 square section - torsion, 118 stability, 250 state varia bie, 320 statically admissible, 37 statically determinate problem, 66, 71 statically indeterminate problem, 66,

77 steady state, 236 stored energy, see strain energy strain, 5 - average, 521 - cylindrical coordinates, 547 - plate, 129 - shell, 286, 288 strain energy, 21, 40, 369 - beam, 67 - plate, 139, 148 strain energy rele ase rate, 441, 443 strain hardening, 310, 311 strain increment, 415 strain localizat ion - algorithm, 513 - analytical results, 477 - ductile damage, 481, 483, 490 - Rice's presentation, 474 strain softening, 310, 311 - ductile damage, 450, 483 - elastic damage, 469 strength criteria, 12

stress, 3 - average, 522 stress concentration, 115, 218, 224 stress power, 353 stress resultant, 57 - beam, 62 - plate, 123 - shell, 290 stress triaxiality, see triaxiality ratio stretch tensor - left, 340 - right, 340 strong formulation, see local formula-

tion structural heating, 324 superposition principle, 25, 42 - 2D, 182, 183, 230 - plate, 151, 153, 155 surface of revolution, 280

tangent operator, 310 - ductile damage, 445 - kinematic hardening, 429 tensor, 1 - matrices for change of coordinates,

556 - matrices for storage, 553 thermal conductivity, 235 thermal expansion, 233 thermal problem, 236 thermal strain, 233 thermal stress, 237 thermo-mechanical problem, 235 thermodynamic force, 321 - ductile damage, 442 - kinematic hardening, 426 - nonlocal model, 502 thin walled section - torsion, 121 torsional rigidity, 99, 106 transformat ion rule, 1, see change of

coordinates - cylindrical coordinates, 548 transient, 245 transport formula, 348 Tresca's criterion, 15 triangular section - torsion, 113 triaxiality ratio, 441, 447 true stress, see Cauchy stress Truesdell rate, 364 two-scale approach, see micro/macro

approach

Index 579

uniaxial stress, 312, 358, 384 - macro. scale, 542 - strain localization, 480 uniqueness, 34, 35, 38 - rate problem, 473

variational formulation, 33, 37, 390 velocity, 338 velocity gradient, 347 virtual power theorem, 353 virtual work theorem, 32, 388 - beam, 90 viscoelasticity, 535 viscoplasticity - function, 329, 332 - Perzyna's formulation, 331 viscosity, 330, 534 viscous stress, 330 Voigt model, 523 volume variation, 5, 22, 48 von Mises equivalent stress, 13, 306

warping function, 95 weak formulation, 32, 353, 355, 388 well posed problem, 10 - rate problem, 473

yield condition or criterion, 302, 399 - kinematic hardening, 425 yield function, 304 - ductile damage, 443 - kinematic hardening, 426 - viscoplasticity, 331 yield surface, 304 Young's modulus, 17