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Approximate kinematical relations in plasticity
APPROXIMATE
KINEMATICAL PLASTICITY
RELATIONS
IN
J. CASEY Department of Mechanical Engineering. University of Houston. University Park. Houston. TX 77004. U.S.A. Abstract-Within the framework of finite plasticity, it is shown how a number of kinematical approximations can be systematically derived from the multiplicative decomposition of the deformation gradient. Particular attention is devoted to two types of approximate theories: (a) those in which elastic deformations are of a different order of magnitude than plastic deformations; and (b) those in which rotations are of a different order of magnitude than strains.
I. INTRODUCTION
In the classical theory of plasticity, in which deformations are assumed to be infinitesimal, the strain and rotation tensors can be additively decomposed into elastic and plastic parts. Correspondingly, in finite plasticity, the deformation gradient can be muitiplicatively decomposed into elastic and plastic parts[ I-31. These, in turn, can be written as products of stretches and rotations by means of the polar decomposition theorem. The strain tensors associated with the finite stretch tensors differ from the strain tensors of infinitesimal plasticity by nonlinear terms, and, moreover, do not obey an additive law of decomposition. The present paper is concerned with situations which are intermediate between those envisaged by infinitesimal plasticity on the one hand, and by finite plasticity on the other. Two types of deformations are of special interest:_(a) those in which elastic deformations are of a different order of magnitude than plastic deformations; and (b) those in which rotations are of a different order of magnitude than strains. For such deformations, the following kinds of questions arise: (i) How are the measures of strain and rotation related to those of infinitesimal plasticity? (ii) Is an additive decomposition of strain possible, even if the appropriate measures of strain are nonlinear? Questions of this type will be considered for a variety of approximate theories. The method utilized herein for the construction of intermediate approximate theories is suggested by recent work of Naghdi and Vongsampigoon(41 and Casey and Naghdi[S]. Recall that, in constructing a theory of infinitesimal deformation, one usually adopts as a measure of smallness the magnitude of the displacement gradient (see e.g. Subsection 3.2 of [6]). Both strains and rotations are then necessarily small. However, if only the strains are assumed to be small, then the rotations need not be small, and vice versa. The basic idea introduced by Naghdi and Vongsarnpigoon[4] is that the magnitude of rotation can be measured in terms of the magnitude of strain. This leads to a theory of moderate rotation, in which the strains are of order e, but the rotations are of order E”*. Casey and Naghdi[S] considered the complementary theory in which rotations are of order E while strains are of order e In. For an elastic material, the latter theory of moderute strain was identified in 151as representing physically nonlinear elasticity.
For elastic-plastic materials, one can also consider theories of moderate rotation and of moderate strain: we do this in Section 6 below. But, even more interestingly, in plasticity theory one can suppose that plastic deformations are moderate while elastic deformations remain small, and vice versa. Such developments are discussed in Section 5. In constructing the approximate theories of Sections 5 and 6, we start out with the exact expressions contained in Section 2 and proceed to the second-order theory detailed in Section 4. Infinitesimal plasticity is discussed in Section 3 for purposes of comparison. Lagrangian kinematical measures are used in the explicit developments 671
J. C\\I
672
>
of Sections 2 through 6, and corresponding developments based on Eulerian measures are indicated in Section 7. Throughout the paper, direct tensor notation is employed. The transpose, determinant, and norm of a second-order tensor A are denoted by A’. det A, and /I A //, respectively, and the inverse of A, if it exists, is denoted by A-‘. I stands for the identity
tensor.
Additional
mathematical
2. EXACT
background
material can be found in 14-61.
EXPRESSIONS
Consider an elastic-plastic body 93 moving in three-dimensional space. Choose an arbitrary fixed reference configuration K() of 3, and let K be the present configuration of 53 at time 1. Denote the deformation gradient relative to the reference configuration by F (det F > 0). Next, introduce an intermediate stress-free configuration gi, thereby obtaining the multiplicative decomposition F = F,F,, ,
(2.
I)
with det F, > 0, det Fp > 0. Locally, F,, maps the reference configuration into a stressfree configuration iz, and F,. maps K into the present configuration. The factors F,. and F,, are interpreted as elastic and plastic parts of F, respectively. A multiplicative decomposition was utilized by Backmanll], Lee and Liu[Z], and LeeI3l.t Issues regarding the existence and uniqueness of the decomposition (2. I), and the matter of appropriate invariance requirements for F,. and F,, have been addressed in [7-IO]. The possibility of accommodating the decomposition (2. I) within the framework of the general theory of plasticity of Green and Naghdi[ I I, 121was established in [7]. Performing right polar decompositions on the three tensors in (2.1), we obtain F = RM,
F, = R,.M,,
F,, = R,M,,,
(2.2)
in which R, R,, and RI, are proper orthogonal tensors, representing rotations, and M, M,, and Mp are symmetric positive definite tensors, representing stretches. We also define total, elastic, and plastic Lagrangian strain tensors by E = 3(F”F - I), E, = f(F.!F, - I), E,, = j(F;F,, - I).
(2.3)
From (2.1) and (2.3),.~.~it follows that E - Ep = F7’E P “FP’
(2.4)
Introduce now the displacement gradient H, and analogous tensors associated with F, and Fp: H = F - I,
H, = F, -
I,
H,, = F,, -
I.
(2.5)
It is clear from (2.1) and (2.9i.2.3 that H = H, + H, + H,H,, .
(2.6)
Each of the three tensors H, H,, and HP can be additively decomposed into its symmetric and skew-symmetric parts as follows: H = e + w, H, = e, + w,,. H, = e, + w,.
(2.7)
t Backman[ I] wrole (2. I) in inverse form, for the purpose of calculating Eulerian strains, but he then proceeded to consider Lagrangian strains. Lee and Liu[?] added the condition that the intermediate configuration be locally stress-free.
with e = h(H
+ H’)
= e’,
w = $(H - H’)
= -w’I
e,. = i(H,. + Hz.)= e,?,H',. = ;(H,. - H,!)= -w
et,= 4(H,, + H$ = e,!, Using all the relations
(2.8)
w,, = i(H - l-l/,) = -w;,
in (2.3). (2.5), (2.7) and (2.8). WC obtain
E = e + $H”H = e + i(e’ + ew - we - w’), E,.= e,. + JHZ’H,. = e, + i(e: + e,.w,. - w,.e,. - wf).
(1.9)
E,, = e,, + ;tHb’H,, = e,, + $(e,‘, + e,,w,, - w,,e,, - H’S). Furthermore,
with the aid of (2.6). all the equations
in (2.8). and (2.7)2.~. WC find that
e = e, + e, + f{H,.H,, + H,!‘H?} = e,. + e, + J{e,.e,, + e,.w,, + w,.e,, + w,,w,, + e,,e,. - w,pef.- ellwl. + w~~w~~J
(2. IO)
and
w = w, + w,, + 4{H,H,, - H; Hf.} = w,, + w,, + ${e,.e,, + e,.w, + w,.e,, + w,.w,, - e,,e,. + w,,e,. + e,,w,. - w,tw,,l.
(2.11)
The tensors e and w are strain and rotation measures appropriate for infinitesimal deformations. However, even for finite deformations e and w are unambiguously defined by (2.8)I.z. The exact expression in (2.9) which relates e and w to the finite strain tensor E is especially useful in constructing approximate theories in which strain and rotation are allowed to have different orders of magnitude. In this connection, see Section 3 of IS].
3. INFlNlTESlMAL
PLASTICITY
Let a measure E of smallness be defined by E = max{sup II I-I,. II, i?
sup II I+,, II}
(3.1)
Ko
(in which sup stands for supremum). If Z is any tensor-valued function of (H,., H,) defined in a neighborhood of (0, 0) and satisfying the condition that there exists a nonnegative real constant K such that
11Z (I < Kr” as E + 0, where n is a positive
number,
(3.2)
then we write
2 = O(C). In the light of (2.8)~~.~
(3.3)
and (3.1),
e, = O(E),
w,. = O(r),
e,, = Ok).
w,, = Ok).
(3.4)
J. C,\\I \I
07-l It
follows
from (2.2)2.2q (2.5)2.1. (2.S)3.4. (3.1). and (3.4),.3 that
M,. = (F;!F,.)“’
= 1 + e,. f O(E’).
M,, = (F,!F,,)“’
= I + e,, + O(E‘).
(3.0
and
M,: ’ = 1 - e,. + O(E). Likewise, by virtue of (2.2h, R, = F,.M,’
(2.5h,
M,; ’ = I - e,, + O(E’).
(3.6),.,.
= I + w,. + Ok’),
(3.6)
(2.7)z.2, and (3.4),.3.3.Jq
R,, = F,,M,’
= 1 + w,, + O(E’).
(3.7)
Also, in view of (2.21~3, (3.Qr.2, (3.7),~, (2.8L~~, (2.7),.J. and (3.4)1.2.J.4.
F,’
= M, ‘R: = I - H,. + O(E’),
It is clear from (2.9)~,
F,’
= M, ‘R;
= 1 -
H,
+ O($).
(3.8)
(3.1). and (3.4),,3 that
E,. = e,. + O(E’) = O(r).
E,, = e,, + O(E’) = O(r).
(3.9)
The relations (3.9),.? are the justification for calling e,. and e,, infinitesimal elastic strain and infinitesimal plastic strain, respectively. The tensors w,. and w,, are referred to as infinitesimal elastic and plastic rotations, respectively. and are related to R,. and R,, through (3.7),.?. It follows from (2.6) and (3.1) that H = H,. + H,, + O(E’)
= O(E).
(3.10)
Hence, in view of (2.5),, (2.2),, and (2.8)1.2, M
=
(FTF)“2
= 1 + e + O(E’),
R = I + w + Ok’), Finally,
it is evident
= I - e + O(E?),
(3.11)
= I - H + O($).
from (2.10),, (2.1 I),, (3.1). and (3.4)I.J that
e = e, + e,, + O(G) In the infinitesimal
F-’
M-’
theory,
= O(E),
w = w,. +
W,’
+
O(G) = O(E).
(3.12)
all terms of O(eZ) are neglected.
4. SECOND-ORDER
RELATIONS
In this section, we examine the form which the various kinematical relations take when terms up to O(r’) are explicitly retained. It is obvious that (3.4),.~.~.4 still hold, and that all of the terms in (2.6), (2.9)1.?.3, (2. 1O),.2, and (2.1 1)1.2 must be retained. However, we observe that now
H = O(E),
e = Ok),
w = O(E),
E = O(E),
E,. = O(t).
E,, = O(E).
(4.1)
Following
the procedure
of Casey and Naghdi[S],
we write M in the form
M = I + cxoe + ale’ + azew + cr,we + a4wz + 0(c3), where the coefficients
a(), (x1, a?, a3, and a4 are to be determined.
(4.2)
To do this, we note
Approximak
kincm;hc;d
rchionh
in plablici(y
675
from (2.2)1, (2.5)). and (2.7), that M’ = I -t 2e + eL!+ ew - we - w’.
(4.3)
Then, substituting (4.2) in (4.3) and solving for the coefficients. we deduce that M = I + e + d(ew - we - w’) + O($).
(4.4)
M,. = I + e,. + j(e,.w,. - w,.e,. - w:) + O($),
(4.5)
Similarly,
M,, = I + e,, + B(e,,w,,- w,,e,, - wt) + O(e’). Using the same procedure, we also obtain M--l = I - e + i(2e’ - ew + we + w2) + O(E’), M, ’ = 1 - e, + t(2ef - e,.w,. + w,.ec + w:) + O(G),
(4.6)
M/Y’ = I - e,, + 1(2cz - e,,w,, + wPe,, + w;) + O(G). It follows from (4.6)1.2.3,(2.2)1.2.~,(2.5)1.2.3,and (2.7),.*~ that R=I+w
- &ew + we - w’) + O(G),
R, = I + w,. - 4(et.wc, + w,.e,. - w:) + O(d),
(4.7)
R, = I + w,, - j(e,,w,, + w,,e,, - w$) + O(E~).
Furthcrmorc, proceeding as in (4.2). WCfind that F-’ = I - H + H’ + O(rJ), F, ’ = I - H,. + H,?.+ O(e’),
(4.8)
F, ’ = I - H,, + H; + O(E’). The latter set of expressions may also be deduced from (4.6)1.2.3.and (4.7),.2.J in the same manner as (3.8),.* were established. Finally, with the use of (2.4), (2.%, (2.9)?, (3.1). (3.4)l.t.3.4, and (4.1)>, we obtain E - E, = E, + Hi’e,. + e,H,, + O(E’)
= E,, + (e,>- w,.)e,. + e,.(e,, +
(4.9) W,,)
+
O(E’)
=
O(E)-
In the second-order theory, all terms of O(e”) are neglected. The expressions (4.4), (4.6), , (4.7), and (4.8), were given by Casey and Naghdi[S]. The relationship between E, E,, E,,, e,., e,,, w,., and w, indicated in (4.9) is similar to one obtained by Backmant. However, Backman retains some terms of O(e’) in his expression, while neglecting others of the same order of magnitude. 5. ELASTIC
DEFORMATIONS
OF DIFFERENT
PLASTIC
ORDER
OF MAGNITUDE
THAN
DEFORMATIONS
In this section, we shall allow H,. and H,) to have different orders of magnitude. Two complementary cases will be considered: first, the case of small plastic deformations accompanied by moderate elastic deformations, and second, the case of small elastic deformations accompanied by moderate plastic deformations. t See equation
(16) of
1Il.
(5.1)
We say that F,. corresponds
to a modo~rrc c>ltr.stic,d~~fimnrrrio~~with respect to E if (5.2)
H,. = O(E’ ‘). It is obvious from (2.8)3.4.5.h. (5. I ), and (5.2) that now elj = O(E). Furthermore,
w,, = O(E).
e,. = O(E’.iz). w,. = OG”‘).
(5.3)
it follows from (2.6). (2.8),.?, (S. ). and (5.2) that e = O(z”‘),
H = H, + H,, + O(E”‘) = O(E’.“),
w = O(E’i’).
(5.4)
all terms must be ret&Lined. but in (2.9)J, only e,, is of order greater than In W41.2, OG”‘). Thus, ew - we - w’) = O(Z"')
E = e + b(e’ +
E,. = e,. + i(ez + e,.w,. - w,.e,. - wf) = OG”‘)
(5.5)
E, = e,, + O(E’) = e,, + O(E”‘) = O(Z). where use has been made of (5.4)~~ and (5.3)‘.2.3.5. With the aid of (5.311.2.3.4. the relations (2. IO) and (2. I I) reduce to e
e,,
=
+
e,,
o(p)
+
=
o(p),
w
=
w,.
+
w,, +
O(E3’2)= (C”‘).
(5.6)
The following expressions for the stretch tensors may be deduced from (4.4) and (4.5)‘.? with the use of (5.4)~.~, (S.~),.ZJ.~. (3. I), and (5. I): M
=
1 +
e
+
f(ew
-
we -
w’) + O(Z3") = I + O(Z”‘)
M,. = I + e,. + $(e,.w,. - w,.e,. - w;, + O(Cj”) = I + O(P),
(5.7)
M,, = 1 + e,, + O(Z3”) = I + O(E). Similarly, from (4.6)‘~~ one obtains M-
’
= I - e + &2e’ - ew + we + w’) + O(ZJ”) = I + O(Z”‘),
MC:’ = I - e, + f(2ez - e,.w,. + w,.e,. + wf) + O(E”‘) = 1 + O(E’.C), M,; ’ = I - e, - O(2”) Likewise,
(5.8)
= 1 + O(T).
(4.7)l.2,3 lead to the following expressions R = I + w - i(ew + we - w’) + O(P)
for the rotation tensors: = I f O(Z”‘)
R,. = I + w,. - d(e,.w,, + w,.e,. - w’) (, + O(p) R,, = I + w,, + O(C”‘) = I + O(E).
= I + O(p)
(5.9)
Appro\imale
Also, by virtue
of (5.4)‘.
rclalionh
in plasticit)
(5.2). (5. I), and (3. I). it is evident
F,:
’ = I - H,. + H; + O(E=) = I + O(E”‘),
H
+
H,, + O(Z”‘)
with the aid of (5.I),
=
from (4.Lo1.2.~ that
’ =
1 -
H2 + O@‘*)
677
F-
F,; ’ = 1 Moreover,
kincrnalic;d
()(E”*).
1 +
(5. IO)
= I + O(Z).
(5.3)1. (5.5)J and (3.1). we deduce from (4.9) that
E = E,. + E,, + O(Z’“) = E “ + eI’ + O(:-“‘)
(5.1 I) = O(7”‘).
An approximate theory may be based on the above results by neglecting all terms of O(Z3’*). In this th eory, additive decompositions hold in the forms (5.6)‘.2 and (5.1 I). Also, as can be seen from (5.5)? and (5.9)1, e,, and w, are appropriate measures of plastic strain and plastic rotation. 5.2
Stnrrll eh.srk~ d~‘formaiion.s, moduwrc plastic Instead of (5.1). in this subsection we let
dcforrnutions
E = sup 11H,. (I.
(5.12)
iz We say that F,, corresponds (5.12) if
pltrstic
to a ~mh~rrc
d&wt~rrion
with
respect to Z in
H ,I = O(E”:)
(5.13)
Expressions complementary to those of Subsection 5.1 may now be readily deduced by the same type of arguments as were used in that subsection. In place of (5.3), we now have
e,. = O(Z). w,. = O(Z).
e,, = OG”‘),
w,, = O(E”*).
(5.14)
The expressions (5.4)‘.,., hold in the same form (with ? being now given by (5.12)). The expression for E is of the same form as in (5.5)‘. while E,. = e,. + O(E”” ) = O(g), E,, = e,, + ice,‘, + e,,w,, -
(5.15) w,,e,, -
wf)
= O(Z”‘).
The form of the results (5.6)‘~ remains unchanged. The expressions for M, M- ‘, R. F- ’ have the same form as in (5.7)‘, (5.8)‘. (5.9)‘. and (5. IO)‘, respectively. while M,. = I + e,. + O(Z”‘) M,
’ = I -
= I + O(Z).
e,. + O(Z”‘)
= I + O(Z).
M,, = I + e,, + &e,,w,, M;’
= I -
e, + &2e,‘, -
R,, = I + w,. + O(Z-“‘) R,, = I + w,, -
w,,e,, -
= 1 -
H,. + O(Z’“)
F,;’
= I -
H,, + H;
+ O(P)
= 1 + O(P),
e,,w,, + w,,e,, + WE) + O(Z”“)
=. I + OG”‘),
= I + O(Z),
b(e,,w,, + w,,e,, -
F,’
w;,
wf,
+ O(P)
= I + O(Z).
+ O($“)
= I + O(Z”‘).
= I + o(z”2),
(5.16)
h7X
J. C‘,\>I \’
Also, paralleling
(5.1 I), it now follows
from (3.9) that
E = E,. + E,, + O@ ‘)
(5.17)
= e,. + E,, + O(?“)
= O(?“‘).
All terms of O(?“‘) arc omitted in the aproximatc theory. In this theory, additive dccompositions hold in the form (5.6),.> and (5.17). The tensors e,. and w, are appropriate measures of elastic strain and elastic rotation.
6. KOTATIONS
OF DIFFERENT
ORDER
OF MAGNITUDE
THAN
STRAINS
In this section, we consider another pair of complementary cases, namely the case of small strains accompanied by moderate rotations and the case of small rotations accompanied by moderate strains. The possibility of measuring rotation in terms of the magnitude of strain was first recognized by Naghdi and Vongsarnpigoon[4]. These authors gave a precise definition of moderate rotation and also identified sufficiency conditions for the rotation to be moderate (see Definition 4.2 and Theorem 4.4 of [4]). An alternative but equivalent definition of moderate rotation was proposed by Casey and Naghdi[S]. The latter definition will be used in Subsection 6.1. In Subsection 6.2, again following Casey and Naghdi[S], we consider small rotations accompanied by moderate strains. In the case of an elastic material, this leads to a theory of physically nonlinear elasticity (see Subsection 3.4 of [5]).
6. I Small strains, modertrre rotations In the present subsection, we set G = maxIsup
sup I( e,, II}.
II e,. (I,
E
We say that w,. and w,, correspond
to modcwrtc~ rottrtions
WC = o(p),
It is clear from (2.7)~,
(2.6), (6.l), H
w,,
=
with respect to E if
o(p).
(6.2)
and (6.2) that
()(z”‘)
= <’
(6.1)
Ku
H
I’
=
o($“),
(6.3)
H = H,. + H,, + HpH,, = o(?“‘).
It follows
from the second equations
in (2. IO) and (2. I I), (6. I), and (6.2),.2 that
e = e,. + e,, + d(w,.W,,+ w,,w,.) w = w,, + w,, + i(w,.w,,
-
+
O(P)
= O(E),
w,,w,.) + O(P)
(6.4)
= O(P).
With the aid of (6.4)1.~, (6.2)1.~, and (6. I), the finite strain tensors in (2.9),.~.~ become E
=
jw’
e -
+- O(Z-“‘) = O(Z),
E,, = e,. -
$w: + O(E"") = O(E),
E,,
$wf + O(Z”‘)
=
e,,
-
= O(E).
(6.5)
Approximate
In view ol’(6.l), be written as
kinematical
(6.311.2, (6.411.2, ml
M = I +
(3. I).
e -
relations
h7Y
in plasticit!
the stretch tensors in (4.4) and
$w’ + O(2”)
(4.5),.?
may
= 1 + O(Z),
M,. = 1 + e,. - 4~: + O(?“) = 1 + O(Z).
(6.6)
M,, = I + e,, - 4~3 + 0(Z3”) = I + O(Z). Similarly, (4.6)‘.-,.Xand (4.7)‘.~.~furnish the expressions M- ’ = I - e + $w’ + O(?)
= I + O(T),
MC:’ = I - e,. + bw? + 0(‘E3”) = I + O(Z),
(6.7)
M,;’ = I - e,, + $wf + O(Z”‘) = I + O(Z), and R = I + w + R,. = I
fw2 + Cl@“) = 1 + o(-E”2).
aw,t+ (I(?“)
+ w,. +
= I + Cl@‘“),
(6.8)
R,, = 1 + w,, + ;w; + O(E”‘) = I + 0(Z”2). With the help of (2.7)l.1.3, (6.l), (6.2),.,, (6.4),.?, and (3.1), we deduce from (4.8)‘~ that
w? +
o(p)
F
’ =
F,
1= 1 -
e,,
-
w,.
f
w: + 0(P2)
F,;
1 = 1 -
e,,
-
w,,
+
wf +
1
-
e
-
w
+
= 1 + o(p), = I + O(E"2)
(6.9)
O(Z”“) = 1 + O(E”‘).
It follows from (4.9), (6.1), (6.2),.?. (3.1) and (6.5)2 that E - Ef = E,. + O(Z”‘) = O(E),
(6.10)
and hence, by virtue of (6.5)‘.~.~, e = e,. + e,, +
I(w’ -
w;-
wf) + O(P)
= O(Z).
(6.1 I)
If (6.4)~ is substituted in (6. I I), (6.4)’ will result. Once terms of order Z3’2are neglected, an additive decomposition is obeyed by the tensors E, E,. and E,,. as can be seen from (6.10). Also, by (6.5)‘.2.3, each of these tensors differs from the corresponding infinitesimal strain tensor by a quadratic term in infinitesimal rotation. 6.2
Smull
rolutions,
modcr.crte strtrins
In this subsection, we set Z = max{sup II w,. II, sup IIw,, II). K KO We then say that e,. and e,, correspond to
moderate
strains
e,. = O(Z’“), e, = 0(Z”2).
(6.12)
with respect to Z if (6.13)
J. CAL.).
hX0
Clearly, the expressions (6.3) i.z.3 again hold with G now given by (6. II). Returning to (2.10) and (2.1 I). and invoking (6.12). (6.13),.z. and (3.1), we deduce that e = e,. + e,, + d(e,.e,, + e,,e,.) + O(Ez”)
= O(E”2).
w = w,. -+ w,, + i(e,.e,, - e,e<,) + O(Z”‘) Utilizing
(6.14)
= O(Z”‘).
(2.9)i.z.~. (6.14)1.~, (6.13),.2, sand (6. I?), WC obtain E
=
e
be’ + O@“)
+
= O(z”‘),
E,. = e,. + jef + O(?)
= O(E”‘),
(6.15)
E,, = e,, + Bef + O(E-“‘) = O(Z”‘). With the use of (4.4), (4.5)i.z, (3.1), we find that
(4.6)i.~.~. (4.7),.~.3, (6.14),.?, (6.13),.?, (6.12) and
M = I + e + OW’)
= I + O(G”‘),
M,. = I + e,. + O(E”“)
= I + O(;E”‘),
M,, = I + e,, + O(E”“)
= I + O(Z”‘).
(6.16)
while ’ =
Me
1 -
e
er +
+
MC:’ = I - e,. +
O(2”)
= 1 + O(z”‘),
e;?. + O(P)
= I + O(Z”‘),
M,; ’ = I - e,, + ef + O(E”“)
(6.17)
= I + O(E”‘),
and R = I + w + O(E”‘)
From (4.g)i.~
R,. = I + w,. + O(Z”‘)
= I + O(2),
R,, = 1 + w,, + O@“)
= I + O(Z).
F-1
= 1 -
F,’
= I - e,. -
it follows
e
-
w
e,, -
(6.18)
(6.13)1.;?. (6.12) and (3. I). we obtain
(2.7)1.2.Y. (6.14),.-.
F; ’ = I Also,
= I + O(E),
+
e2 + 0(E3”)
= 1 + O(Z”‘),
w,. + e: + O(Z”‘)
= I + O(?“‘),
w,, + e,Z,+ O(7”‘)
= I + O(Z”‘).
(6.19)
froom (4.9), (6.12), (6.13)i.z and (6.15)~ that E -
E,, = E,. + e,.e,, + e,,e,. + O(P)
= OG”‘)
(6.20)
and hence, by (6.15),.~.~, that e -
e,, = e,. -
The substitution Even when from (6.20) and noting, however, tensors only by
i(e’
-
ef -
e,Z,)+ e,e, + e,,e,. + 0(E3’2) = OG”‘).
(6.21)
of (6.14), in (6.21) reduces the latter equation to (6.14),. terms of order Zw2 are neglected in the present subsection, it is clear (6.21) that additive decompositions of strain do not hold. It is worth that the infinitesimal strain tensors differ from the Lagrangian strain a quadratic term in infinitesimal strain (see (6.1S)1.z.1).
Approximate
kinematical rehions
681
in plasticity
Besides the developments described above, several other similar approximate theories may be constructed. For example, one could take e,, of 0(Z”2) and w,,, e,, w,, of smaller orders of magnitude. All other combinations are mathematically possible also. but not all of them may have obvious physical relevance. 7. EULERIAN
MEASURES
It is also of interest to construct approximate theories on the basis of finite Eulerian measures. For this purpose, we employ left polar decompositions
F = NR.
F,. = N,.R,,
F,, = N,R,,
(7.1)
in place of (2.2). Here, N, N,., and N,, are symmetric, positive definite left stretch tensors. We define Eulerian finite strain tensors by E = $(I - F- “F- I), i?,. = &I - F,” F,‘),
i?, = i(I - F;7’ F, ‘).
(7.2)
where F-’ = (F-i)“ = (F’)-1. From (2.1) and (7.2),.z.3 it follows that E - i?,. = F,“i?,F,‘.
(7.3)
Instead of (2.5),.~.~, we introduce the displacement gradient % (relative to the present configuration K) and analogous tensors associated with F,. and F,. Thus,
H = 1 - F-‘,
H,. = I - F,‘,
R,, = I _ F,‘.
(7.4)
-We also decompose H, H,., and E,, into their symmetric and skew-symmetric parts:
H=E+G, z = i(TI + ii’), F = $(B - H’),
H,. = E,, =
i!,. +
iv<,,
I,(H, + ii’),
?I, = zp + T, =
Fp,
a(& + R;),
(7.5)
K = i(H,. - HZ.,, EP = f@i,, - iI;,.
In view of (2. I), (7.4)l.z.~ and (7.%.~.~.7.~.~, --
ti = i& + i&, - H,H,. ,
(7.6)
while E = E,, + E,,
- - +{H,,H,, + -H,.7- H,,7’ 1
--= z,, + i$ - i{q,,E, + i&F,. + K,,Z,. + w,,w,. + Z,.Z, - e,w, - a&
(7.7) + -wewpIr
and if
- - B{H,,H,. - -‘r-7’ H,. Ii,, } (7.8) ${q,,e,. + E,,G‘, + G,,C/, + 3,,%‘, - Z‘Z,, + Z,i,> + i+‘.C,>- WCEPI. = V‘, + i,, =
ii‘,
+ ii,,
Using all the relations in (7.2), (7.4) and (7% we conclude that
682
J. CASE\
For the construction of infinitesimal plasticity and second-order plasticity, we would employ a measure of smallness 6 = madsup
II R. IO supII i$ II}.
K
(7.10)
K
WC observe that, by virtue of (3. I), ([email protected] and(7.4)1.2.~. ‘is= H - HZ + O(E)), a, = He - H: + 0(e3),
(7.11j
H, = H, - H; + O(e3). Also, with the aid of (7.4),.~.~and (7.10) we deduce that F = I + F,
Tz + iP + O(S3),
= I + R, +
FP =
I +
rif + O@‘),
(7.12)
HP + Ti; + o(63),
and hence H =
a + a’
+
0(a3),
H,
= ii, + i$f +
06%
H,,
= iip + ii; +
O(6").
(7.13)
The relations (7. I I) and (7.13) allow approximate theories constructed from Eulerian measures to be related to those discussed in Sections 3,4,5, and 6 of the present paper. REFERENCES 1. M. E. Backman, Form of the relation between stress and tinite elastic and plastic strains under impulsive loading. J. Appl. Phys. 35, 2524 (1964). 2. E. H. Lee and D. T. Liu, Finite-strain elastic-plastic theory with application to plane-wave analysis. J. ADD/. Ph. 38. 19 (1967). 3. E: ‘H. Lee, Elastic-plastic deformation at finite strains. J. Appl. Mcch. 36, I (1969). 4. P. M. Naghdi and L. Vongsarnpigoon. Small strain accompanied by moderate rotation. Arch. Rution. Mech. Awl. SO, 263 (1982). 5. J. Casey and P. M. Naghdi, Physically nonlinear and related approximate theories of elasticity and their invariance properties. Arch. Rorion. Mech. Anol. in press. 6. .I. Casey and P. M. Naghdi, An invariant infinitesimal theory of motions superposed on a given motion. Arch. Rotion. Mech. Anul. 76, 355 (1981). 7. A. E. Green and P. M. Naghdi, Some remarks on elastic-plastic deformation at finite strain. In!. J. Engng Sci. 9, 1219 (1971). 8. P. M. Naghdi and J. A. Trapp, On finite elastic-plastic deformation of metals. J. Appl. Mech. 41, 254 (1974). 9. J. Casey and P. M. Naghdi, A remark on the use of the decomposition F = F,.F,, in plasticity. J. Appl. Mech. 47, 672 (1980). IO. J. Casey and P. M. Naghdi. Discussion of “A correct definition of elastic and plastic deformation and its computational significance” (by V. A. Lubarda and E. H. Lee). 1. Appl. Mech. 48, 983 (1981). II. A. E. Green and P. M. Naghdi. A general theory of an elastic-plastic continuum. Arch. Rofion. Much. Anul. 18, 251 (1%5). 12. A. E. Green and P. M. Naghdi, A thermodynamic development of elastic-plastic continua. froc. fUTAM Symp. on Irreversible Aspects of Continuum Mechanics and Transfer of Physical Churucteristics in Moving Nuids (Edited by H. Parkus and L. 1. Sedov), p. 117). Springer-Verlag, Berlin (1966).
KINEMATICAL PLASTICITY
RELATIONS
IN
J. CASEY Department of Mechanical Engineering. University of Houston. University Park. Houston. TX 77004. U.S.A. Abstract-Within the framework of finite plasticity, it is shown how a number of kinematical approximations can be systematically derived from the multiplicative decomposition of the deformation gradient. Particular attention is devoted to two types of approximate theories: (a) those in which elastic deformations are of a different order of magnitude than plastic deformations; and (b) those in which rotations are of a different order of magnitude than strains.
I. INTRODUCTION
In the classical theory of plasticity, in which deformations are assumed to be infinitesimal, the strain and rotation tensors can be additively decomposed into elastic and plastic parts. Correspondingly, in finite plasticity, the deformation gradient can be muitiplicatively decomposed into elastic and plastic parts[ I-31. These, in turn, can be written as products of stretches and rotations by means of the polar decomposition theorem. The strain tensors associated with the finite stretch tensors differ from the strain tensors of infinitesimal plasticity by nonlinear terms, and, moreover, do not obey an additive law of decomposition. The present paper is concerned with situations which are intermediate between those envisaged by infinitesimal plasticity on the one hand, and by finite plasticity on the other. Two types of deformations are of special interest:_(a) those in which elastic deformations are of a different order of magnitude than plastic deformations; and (b) those in which rotations are of a different order of magnitude than strains. For such deformations, the following kinds of questions arise: (i) How are the measures of strain and rotation related to those of infinitesimal plasticity? (ii) Is an additive decomposition of strain possible, even if the appropriate measures of strain are nonlinear? Questions of this type will be considered for a variety of approximate theories. The method utilized herein for the construction of intermediate approximate theories is suggested by recent work of Naghdi and Vongsampigoon(41 and Casey and Naghdi[S]. Recall that, in constructing a theory of infinitesimal deformation, one usually adopts as a measure of smallness the magnitude of the displacement gradient (see e.g. Subsection 3.2 of [6]). Both strains and rotations are then necessarily small. However, if only the strains are assumed to be small, then the rotations need not be small, and vice versa. The basic idea introduced by Naghdi and Vongsarnpigoon[4] is that the magnitude of rotation can be measured in terms of the magnitude of strain. This leads to a theory of moderate rotation, in which the strains are of order e, but the rotations are of order E”*. Casey and Naghdi[S] considered the complementary theory in which rotations are of order E while strains are of order e In. For an elastic material, the latter theory of moderute strain was identified in 151as representing physically nonlinear elasticity.
For elastic-plastic materials, one can also consider theories of moderate rotation and of moderate strain: we do this in Section 6 below. But, even more interestingly, in plasticity theory one can suppose that plastic deformations are moderate while elastic deformations remain small, and vice versa. Such developments are discussed in Section 5. In constructing the approximate theories of Sections 5 and 6, we start out with the exact expressions contained in Section 2 and proceed to the second-order theory detailed in Section 4. Infinitesimal plasticity is discussed in Section 3 for purposes of comparison. Lagrangian kinematical measures are used in the explicit developments 671
J. C\\I
672
>
of Sections 2 through 6, and corresponding developments based on Eulerian measures are indicated in Section 7. Throughout the paper, direct tensor notation is employed. The transpose, determinant, and norm of a second-order tensor A are denoted by A’. det A, and /I A //, respectively, and the inverse of A, if it exists, is denoted by A-‘. I stands for the identity
tensor.
Additional
mathematical
2. EXACT
background
material can be found in 14-61.
EXPRESSIONS
Consider an elastic-plastic body 93 moving in three-dimensional space. Choose an arbitrary fixed reference configuration K() of 3, and let K be the present configuration of 53 at time 1. Denote the deformation gradient relative to the reference configuration by F (det F > 0). Next, introduce an intermediate stress-free configuration gi, thereby obtaining the multiplicative decomposition F = F,F,, ,
(2.
I)
with det F, > 0, det Fp > 0. Locally, F,, maps the reference configuration into a stressfree configuration iz, and F,. maps K into the present configuration. The factors F,. and F,, are interpreted as elastic and plastic parts of F, respectively. A multiplicative decomposition was utilized by Backmanll], Lee and Liu[Z], and LeeI3l.t Issues regarding the existence and uniqueness of the decomposition (2. I), and the matter of appropriate invariance requirements for F,. and F,, have been addressed in [7-IO]. The possibility of accommodating the decomposition (2. I) within the framework of the general theory of plasticity of Green and Naghdi[ I I, 121was established in [7]. Performing right polar decompositions on the three tensors in (2.1), we obtain F = RM,
F, = R,.M,,
F,, = R,M,,,
(2.2)
in which R, R,, and RI, are proper orthogonal tensors, representing rotations, and M, M,, and Mp are symmetric positive definite tensors, representing stretches. We also define total, elastic, and plastic Lagrangian strain tensors by E = 3(F”F - I), E, = f(F.!F, - I), E,, = j(F;F,, - I).
(2.3)
From (2.1) and (2.3),.~.~it follows that E - Ep = F7’E P “FP’
(2.4)
Introduce now the displacement gradient H, and analogous tensors associated with F, and Fp: H = F - I,
H, = F, -
I,
H,, = F,, -
I.
(2.5)
It is clear from (2.1) and (2.9i.2.3 that H = H, + H, + H,H,, .
(2.6)
Each of the three tensors H, H,, and HP can be additively decomposed into its symmetric and skew-symmetric parts as follows: H = e + w, H, = e, + w,,. H, = e, + w,.
(2.7)
t Backman[ I] wrole (2. I) in inverse form, for the purpose of calculating Eulerian strains, but he then proceeded to consider Lagrangian strains. Lee and Liu[?] added the condition that the intermediate configuration be locally stress-free.
with e = h(H
+ H’)
= e’,
w = $(H - H’)
= -w’I
e,. = i(H,. + Hz.)= e,?,H',. = ;(H,. - H,!)= -w
et,= 4(H,, + H$ = e,!, Using all the relations
(2.8)
w,, = i(H - l-l/,) = -w;,
in (2.3). (2.5), (2.7) and (2.8). WC obtain
E = e + $H”H = e + i(e’ + ew - we - w’), E,.= e,. + JHZ’H,. = e, + i(e: + e,.w,. - w,.e,. - wf).
(1.9)
E,, = e,, + ;tHb’H,, = e,, + $(e,‘, + e,,w,, - w,,e,, - H’S). Furthermore,
with the aid of (2.6). all the equations
in (2.8). and (2.7)2.~. WC find that
e = e, + e, + f{H,.H,, + H,!‘H?} = e,. + e, + J{e,.e,, + e,.w,, + w,.e,, + w,,w,, + e,,e,. - w,pef.- ellwl. + w~~w~~J
(2. IO)
and
w = w, + w,, + 4{H,H,, - H; Hf.} = w,, + w,, + ${e,.e,, + e,.w, + w,.e,, + w,.w,, - e,,e,. + w,,e,. + e,,w,. - w,tw,,l.
(2.11)
The tensors e and w are strain and rotation measures appropriate for infinitesimal deformations. However, even for finite deformations e and w are unambiguously defined by (2.8)I.z. The exact expression in (2.9) which relates e and w to the finite strain tensor E is especially useful in constructing approximate theories in which strain and rotation are allowed to have different orders of magnitude. In this connection, see Section 3 of IS].
3. INFlNlTESlMAL
PLASTICITY
Let a measure E of smallness be defined by E = max{sup II I-I,. II, i?
sup II I+,, II}
(3.1)
Ko
(in which sup stands for supremum). If Z is any tensor-valued function of (H,., H,) defined in a neighborhood of (0, 0) and satisfying the condition that there exists a nonnegative real constant K such that
11Z (I < Kr” as E + 0, where n is a positive
number,
(3.2)
then we write
2 = O(C). In the light of (2.8)~~.~
(3.3)
and (3.1),
e, = O(E),
w,. = O(r),
e,, = Ok).
w,, = Ok).
(3.4)
J. C,\\I \I
07-l It
follows
from (2.2)2.2q (2.5)2.1. (2.S)3.4. (3.1). and (3.4),.3 that
M,. = (F;!F,.)“’
= 1 + e,. f O(E’).
M,, = (F,!F,,)“’
= I + e,, + O(E‘).
(3.0
and
M,: ’ = 1 - e,. + O(E). Likewise, by virtue of (2.2h, R, = F,.M,’
(2.5h,
M,; ’ = I - e,, + O(E’).
(3.6),.,.
= I + w,. + Ok’),
(3.6)
(2.7)z.2, and (3.4),.3.3.Jq
R,, = F,,M,’
= 1 + w,, + O(E’).
(3.7)
Also, in view of (2.21~3, (3.Qr.2, (3.7),~, (2.8L~~, (2.7),.J. and (3.4)1.2.J.4.
F,’
= M, ‘R: = I - H,. + O(E’),
It is clear from (2.9)~,
F,’
= M, ‘R;
= 1 -
H,
+ O($).
(3.8)
(3.1). and (3.4),,3 that
E,. = e,. + O(E’) = O(r).
E,, = e,, + O(E’) = O(r).
(3.9)
The relations (3.9),.? are the justification for calling e,. and e,, infinitesimal elastic strain and infinitesimal plastic strain, respectively. The tensors w,. and w,, are referred to as infinitesimal elastic and plastic rotations, respectively. and are related to R,. and R,, through (3.7),.?. It follows from (2.6) and (3.1) that H = H,. + H,, + O(E’)
= O(E).
(3.10)
Hence, in view of (2.5),, (2.2),, and (2.8)1.2, M
=
(FTF)“2
= 1 + e + O(E’),
R = I + w + Ok’), Finally,
it is evident
= I - e + O(E?),
(3.11)
= I - H + O($).
from (2.10),, (2.1 I),, (3.1). and (3.4)I.J that
e = e, + e,, + O(G) In the infinitesimal
F-’
M-’
theory,
= O(E),
w = w,. +
W,’
+
O(G) = O(E).
(3.12)
all terms of O(eZ) are neglected.
4. SECOND-ORDER
RELATIONS
In this section, we examine the form which the various kinematical relations take when terms up to O(r’) are explicitly retained. It is obvious that (3.4),.~.~.4 still hold, and that all of the terms in (2.6), (2.9)1.?.3, (2. 1O),.2, and (2.1 1)1.2 must be retained. However, we observe that now
H = O(E),
e = Ok),
w = O(E),
E = O(E),
E,. = O(t).
E,, = O(E).
(4.1)
Following
the procedure
of Casey and Naghdi[S],
we write M in the form
M = I + cxoe + ale’ + azew + cr,we + a4wz + 0(c3), where the coefficients
a(), (x1, a?, a3, and a4 are to be determined.
(4.2)
To do this, we note
Approximak
kincm;hc;d
rchionh
in plablici(y
675
from (2.2)1, (2.5)). and (2.7), that M’ = I -t 2e + eL!+ ew - we - w’.
(4.3)
Then, substituting (4.2) in (4.3) and solving for the coefficients. we deduce that M = I + e + d(ew - we - w’) + O($).
(4.4)
M,. = I + e,. + j(e,.w,. - w,.e,. - w:) + O($),
(4.5)
Similarly,
M,, = I + e,, + B(e,,w,,- w,,e,, - wt) + O(e’). Using the same procedure, we also obtain M--l = I - e + i(2e’ - ew + we + w2) + O(E’), M, ’ = 1 - e, + t(2ef - e,.w,. + w,.ec + w:) + O(G),
(4.6)
M/Y’ = I - e,, + 1(2cz - e,,w,, + wPe,, + w;) + O(G). It follows from (4.6)1.2.3,(2.2)1.2.~,(2.5)1.2.3,and (2.7),.*~ that R=I+w
- &ew + we - w’) + O(G),
R, = I + w,. - 4(et.wc, + w,.e,. - w:) + O(d),
(4.7)
R, = I + w,, - j(e,,w,, + w,,e,, - w$) + O(E~).
Furthcrmorc, proceeding as in (4.2). WCfind that F-’ = I - H + H’ + O(rJ), F, ’ = I - H,. + H,?.+ O(e’),
(4.8)
F, ’ = I - H,, + H; + O(E’). The latter set of expressions may also be deduced from (4.6)1.2.3.and (4.7),.2.J in the same manner as (3.8),.* were established. Finally, with the use of (2.4), (2.%, (2.9)?, (3.1). (3.4)l.t.3.4, and (4.1)>, we obtain E - E, = E, + Hi’e,. + e,H,, + O(E’)
= E,, + (e,>- w,.)e,. + e,.(e,, +
(4.9) W,,)
+
O(E’)
=
O(E)-
In the second-order theory, all terms of O(e”) are neglected. The expressions (4.4), (4.6), , (4.7), and (4.8), were given by Casey and Naghdi[S]. The relationship between E, E,, E,,, e,., e,,, w,., and w, indicated in (4.9) is similar to one obtained by Backmant. However, Backman retains some terms of O(e’) in his expression, while neglecting others of the same order of magnitude. 5. ELASTIC
DEFORMATIONS
OF DIFFERENT
PLASTIC
ORDER
OF MAGNITUDE
THAN
DEFORMATIONS
In this section, we shall allow H,. and H,) to have different orders of magnitude. Two complementary cases will be considered: first, the case of small plastic deformations accompanied by moderate elastic deformations, and second, the case of small elastic deformations accompanied by moderate plastic deformations. t See equation
(16) of
1Il.
(5.1)
We say that F,. corresponds
to a modo~rrc c>ltr.stic,d~~fimnrrrio~~with respect to E if (5.2)
H,. = O(E’ ‘). It is obvious from (2.8)3.4.5.h. (5. I ), and (5.2) that now elj = O(E). Furthermore,
w,, = O(E).
e,. = O(E’.iz). w,. = OG”‘).
(5.3)
it follows from (2.6). (2.8),.?, (S. ). and (5.2) that e = O(z”‘),
H = H, + H,, + O(E”‘) = O(E’.“),
w = O(E’i’).
(5.4)
all terms must be ret&Lined. but in (2.9)J, only e,, is of order greater than In W41.2, OG”‘). Thus, ew - we - w’) = O(Z"')
E = e + b(e’ +
E,. = e,. + i(ez + e,.w,. - w,.e,. - wf) = OG”‘)
(5.5)
E, = e,, + O(E’) = e,, + O(E”‘) = O(Z). where use has been made of (5.4)~~ and (5.3)‘.2.3.5. With the aid of (5.311.2.3.4. the relations (2. IO) and (2. I I) reduce to e
e,,
=
+
e,,
o(p)
+
=
o(p),
w
=
w,.
+
w,, +
O(E3’2)= (C”‘).
(5.6)
The following expressions for the stretch tensors may be deduced from (4.4) and (4.5)‘.? with the use of (5.4)~.~, (S.~),.ZJ.~. (3. I), and (5. I): M
=
1 +
e
+
f(ew
-
we -
w’) + O(Z3") = I + O(Z”‘)
M,. = I + e,. + $(e,.w,. - w,.e,. - w;, + O(Cj”) = I + O(P),
(5.7)
M,, = 1 + e,, + O(Z3”) = I + O(E). Similarly, from (4.6)‘~~ one obtains M-
’
= I - e + &2e’ - ew + we + w’) + O(ZJ”) = I + O(Z”‘),
MC:’ = I - e, + f(2ez - e,.w,. + w,.e,. + wf) + O(E”‘) = 1 + O(E’.C), M,; ’ = I - e, - O(2”) Likewise,
(5.8)
= 1 + O(T).
(4.7)l.2,3 lead to the following expressions R = I + w - i(ew + we - w’) + O(P)
for the rotation tensors: = I f O(Z”‘)
R,. = I + w,. - d(e,.w,, + w,.e,. - w’) (, + O(p) R,, = I + w,, + O(C”‘) = I + O(E).
= I + O(p)
(5.9)
Appro\imale
Also, by virtue
of (5.4)‘.
rclalionh
in plasticit)
(5.2). (5. I), and (3. I). it is evident
F,:
’ = I - H,. + H; + O(E=) = I + O(E”‘),
H
+
H,, + O(Z”‘)
with the aid of (5.I),
=
from (4.Lo1.2.~ that
’ =
1 -
H2 + O@‘*)
677
F-
F,; ’ = 1 Moreover,
kincrnalic;d
()(E”*).
1 +
(5. IO)
= I + O(Z).
(5.3)1. (5.5)J and (3.1). we deduce from (4.9) that
E = E,. + E,, + O(Z’“) = E “ + eI’ + O(:-“‘)
(5.1 I) = O(7”‘).
An approximate theory may be based on the above results by neglecting all terms of O(Z3’*). In this th eory, additive decompositions hold in the forms (5.6)‘.2 and (5.1 I). Also, as can be seen from (5.5)? and (5.9)1, e,, and w, are appropriate measures of plastic strain and plastic rotation. 5.2
Stnrrll eh.srk~ d~‘formaiion.s, moduwrc plastic Instead of (5.1). in this subsection we let
dcforrnutions
E = sup 11H,. (I.
(5.12)
iz We say that F,, corresponds (5.12) if
pltrstic
to a ~mh~rrc
d&wt~rrion
with
respect to Z in
H ,I = O(E”:)
(5.13)
Expressions complementary to those of Subsection 5.1 may now be readily deduced by the same type of arguments as were used in that subsection. In place of (5.3), we now have
e,. = O(Z). w,. = O(Z).
e,, = OG”‘),
w,, = O(E”*).
(5.14)
The expressions (5.4)‘.,., hold in the same form (with ? being now given by (5.12)). The expression for E is of the same form as in (5.5)‘. while E,. = e,. + O(E”” ) = O(g), E,, = e,, + ice,‘, + e,,w,, -
(5.15) w,,e,, -
wf)
= O(Z”‘).
The form of the results (5.6)‘~ remains unchanged. The expressions for M, M- ‘, R. F- ’ have the same form as in (5.7)‘, (5.8)‘. (5.9)‘. and (5. IO)‘, respectively. while M,. = I + e,. + O(Z”‘) M,
’ = I -
= I + O(Z).
e,. + O(Z”‘)
= I + O(Z).
M,, = I + e,, + &e,,w,, M;’
= I -
e, + &2e,‘, -
R,, = I + w,. + O(Z-“‘) R,, = I + w,, -
w,,e,, -
= 1 -
H,. + O(Z’“)
F,;’
= I -
H,, + H;
+ O(P)
= 1 + O(P),
e,,w,, + w,,e,, + WE) + O(Z”“)
=. I + OG”‘),
= I + O(Z),
b(e,,w,, + w,,e,, -
F,’
w;,
wf,
+ O(P)
= I + O(Z).
+ O($“)
= I + O(Z”‘).
= I + o(z”2),
(5.16)
h7X
J. C‘,\>I \’
Also, paralleling
(5.1 I), it now follows
from (3.9) that
E = E,. + E,, + O@ ‘)
(5.17)
= e,. + E,, + O(?“)
= O(?“‘).
All terms of O(?“‘) arc omitted in the aproximatc theory. In this theory, additive dccompositions hold in the form (5.6),.> and (5.17). The tensors e,. and w, are appropriate measures of elastic strain and elastic rotation.
6. KOTATIONS
OF DIFFERENT
ORDER
OF MAGNITUDE
THAN
STRAINS
In this section, we consider another pair of complementary cases, namely the case of small strains accompanied by moderate rotations and the case of small rotations accompanied by moderate strains. The possibility of measuring rotation in terms of the magnitude of strain was first recognized by Naghdi and Vongsarnpigoon[4]. These authors gave a precise definition of moderate rotation and also identified sufficiency conditions for the rotation to be moderate (see Definition 4.2 and Theorem 4.4 of [4]). An alternative but equivalent definition of moderate rotation was proposed by Casey and Naghdi[S]. The latter definition will be used in Subsection 6.1. In Subsection 6.2, again following Casey and Naghdi[S], we consider small rotations accompanied by moderate strains. In the case of an elastic material, this leads to a theory of physically nonlinear elasticity (see Subsection 3.4 of [5]).
6. I Small strains, modertrre rotations In the present subsection, we set G = maxIsup
sup I( e,, II}.
II e,. (I,
E
We say that w,. and w,, correspond
to modcwrtc~ rottrtions
WC = o(p),
It is clear from (2.7)~,
(2.6), (6.l), H
w,,
=
with respect to E if
o(p).
(6.2)
and (6.2) that
()(z”‘)
= <’
(6.1)
Ku
H
I’
=
o($“),
(6.3)
H = H,. + H,, + HpH,, = o(?“‘).
It follows
from the second equations
in (2. IO) and (2. I I), (6. I), and (6.2),.2 that
e = e,. + e,, + d(w,.W,,+ w,,w,.) w = w,, + w,, + i(w,.w,,
-
+
O(P)
= O(E),
w,,w,.) + O(P)
(6.4)
= O(P).
With the aid of (6.4)1.~, (6.2)1.~, and (6. I), the finite strain tensors in (2.9),.~.~ become E
=
jw’
e -
+- O(Z-“‘) = O(Z),
E,, = e,. -
$w: + O(E"") = O(E),
E,,
$wf + O(Z”‘)
=
e,,
-
= O(E).
(6.5)
Approximate
In view ol’(6.l), be written as
kinematical
(6.311.2, (6.411.2, ml
M = I +
(3. I).
e -
relations
h7Y
in plasticit!
the stretch tensors in (4.4) and
$w’ + O(2”)
(4.5),.?
may
= 1 + O(Z),
M,. = 1 + e,. - 4~: + O(?“) = 1 + O(Z).
(6.6)
M,, = I + e,, - 4~3 + 0(Z3”) = I + O(Z). Similarly, (4.6)‘.-,.Xand (4.7)‘.~.~furnish the expressions M- ’ = I - e + $w’ + O(?)
= I + O(T),
MC:’ = I - e,. + bw? + 0(‘E3”) = I + O(Z),
(6.7)
M,;’ = I - e,, + $wf + O(Z”‘) = I + O(Z), and R = I + w + R,. = I
fw2 + Cl@“) = 1 + o(-E”2).
aw,t+ (I(?“)
+ w,. +
= I + Cl@‘“),
(6.8)
R,, = 1 + w,, + ;w; + O(E”‘) = I + 0(Z”2). With the help of (2.7)l.1.3, (6.l), (6.2),.,, (6.4),.?, and (3.1), we deduce from (4.8)‘~ that
w? +
o(p)
F
’ =
F,
1= 1 -
e,,
-
w,.
f
w: + 0(P2)
F,;
1 = 1 -
e,,
-
w,,
+
wf +
1
-
e
-
w
+
= 1 + o(p), = I + O(E"2)
(6.9)
O(Z”“) = 1 + O(E”‘).
It follows from (4.9), (6.1), (6.2),.?. (3.1) and (6.5)2 that E - Ef = E,. + O(Z”‘) = O(E),
(6.10)
and hence, by virtue of (6.5)‘.~.~, e = e,. + e,, +
I(w’ -
w;-
wf) + O(P)
= O(Z).
(6.1 I)
If (6.4)~ is substituted in (6. I I), (6.4)’ will result. Once terms of order Z3’2are neglected, an additive decomposition is obeyed by the tensors E, E,. and E,,. as can be seen from (6.10). Also, by (6.5)‘.2.3, each of these tensors differs from the corresponding infinitesimal strain tensor by a quadratic term in infinitesimal rotation. 6.2
Smull
rolutions,
modcr.crte strtrins
In this subsection, we set Z = max{sup II w,. II, sup IIw,, II). K KO We then say that e,. and e,, correspond to
moderate
strains
e,. = O(Z’“), e, = 0(Z”2).
(6.12)
with respect to Z if (6.13)
J. CAL.).
hX0
Clearly, the expressions (6.3) i.z.3 again hold with G now given by (6. II). Returning to (2.10) and (2.1 I). and invoking (6.12). (6.13),.z. and (3.1), we deduce that e = e,. + e,, + d(e,.e,, + e,,e,.) + O(Ez”)
= O(E”2).
w = w,. -+ w,, + i(e,.e,, - e,e<,) + O(Z”‘) Utilizing
(6.14)
= O(Z”‘).
(2.9)i.z.~. (6.14)1.~, (6.13),.2, sand (6. I?), WC obtain E
=
e
be’ + O@“)
+
= O(z”‘),
E,. = e,. + jef + O(?)
= O(E”‘),
(6.15)
E,, = e,, + Bef + O(E-“‘) = O(Z”‘). With the use of (4.4), (4.5)i.z, (3.1), we find that
(4.6)i.~.~. (4.7),.~.3, (6.14),.?, (6.13),.?, (6.12) and
M = I + e + OW’)
= I + O(G”‘),
M,. = I + e,. + O(E”“)
= I + O(;E”‘),
M,, = I + e,, + O(E”“)
= I + O(Z”‘).
(6.16)
while ’ =
Me
1 -
e
er +
+
MC:’ = I - e,. +
O(2”)
= 1 + O(z”‘),
e;?. + O(P)
= I + O(Z”‘),
M,; ’ = I - e,, + ef + O(E”“)
(6.17)
= I + O(E”‘),
and R = I + w + O(E”‘)
From (4.g)i.~
R,. = I + w,. + O(Z”‘)
= I + O(2),
R,, = 1 + w,, + O@“)
= I + O(Z).
F-1
= 1 -
F,’
= I - e,. -
it follows
e
-
w
e,, -
(6.18)
(6.13)1.;?. (6.12) and (3. I). we obtain
(2.7)1.2.Y. (6.14),.-.
F; ’ = I Also,
= I + O(E),
+
e2 + 0(E3”)
= 1 + O(Z”‘),
w,. + e: + O(Z”‘)
= I + O(?“‘),
w,, + e,Z,+ O(7”‘)
= I + O(Z”‘).
(6.19)
froom (4.9), (6.12), (6.13)i.z and (6.15)~ that E -
E,, = E,. + e,.e,, + e,,e,. + O(P)
= OG”‘)
(6.20)
and hence, by (6.15),.~.~, that e -
e,, = e,. -
The substitution Even when from (6.20) and noting, however, tensors only by
i(e’
-
ef -
e,Z,)+ e,e, + e,,e,. + 0(E3’2) = OG”‘).
(6.21)
of (6.14), in (6.21) reduces the latter equation to (6.14),. terms of order Zw2 are neglected in the present subsection, it is clear (6.21) that additive decompositions of strain do not hold. It is worth that the infinitesimal strain tensors differ from the Lagrangian strain a quadratic term in infinitesimal strain (see (6.1S)1.z.1).
Approximate
kinematical rehions
681
in plasticity
Besides the developments described above, several other similar approximate theories may be constructed. For example, one could take e,, of 0(Z”2) and w,,, e,, w,, of smaller orders of magnitude. All other combinations are mathematically possible also. but not all of them may have obvious physical relevance. 7. EULERIAN
MEASURES
It is also of interest to construct approximate theories on the basis of finite Eulerian measures. For this purpose, we employ left polar decompositions
F = NR.
F,. = N,.R,,
F,, = N,R,,
(7.1)
in place of (2.2). Here, N, N,., and N,, are symmetric, positive definite left stretch tensors. We define Eulerian finite strain tensors by E = $(I - F- “F- I), i?,. = &I - F,” F,‘),
i?, = i(I - F;7’ F, ‘).
(7.2)
where F-’ = (F-i)“ = (F’)-1. From (2.1) and (7.2),.z.3 it follows that E - i?,. = F,“i?,F,‘.
(7.3)
Instead of (2.5),.~.~, we introduce the displacement gradient % (relative to the present configuration K) and analogous tensors associated with F,. and F,. Thus,
H = 1 - F-‘,
H,. = I - F,‘,
R,, = I _ F,‘.
(7.4)
-We also decompose H, H,., and E,, into their symmetric and skew-symmetric parts:
H=E+G, z = i(TI + ii’), F = $(B - H’),
H,. = E,, =
i!,. +
iv<,,
I,(H, + ii’),
?I, = zp + T, =
Fp,
a(& + R;),
(7.5)
K = i(H,. - HZ.,, EP = f@i,, - iI;,.
In view of (2. I), (7.4)l.z.~ and (7.%.~.~.7.~.~, --
ti = i& + i&, - H,H,. ,
(7.6)
while E = E,, + E,,
- - +{H,,H,, + -H,.7- H,,7’ 1
--= z,, + i$ - i{q,,E, + i&F,. + K,,Z,. + w,,w,. + Z,.Z, - e,w, - a&
(7.7) + -wewpIr
and if
- - B{H,,H,. - -‘r-7’ H,. Ii,, } (7.8) ${q,,e,. + E,,G‘, + G,,C/, + 3,,%‘, - Z‘Z,, + Z,i,> + i+‘.C,>- WCEPI. = V‘, + i,, =
ii‘,
+ ii,,
Using all the relations in (7.2), (7.4) and (7% we conclude that
682
J. CASE\
For the construction of infinitesimal plasticity and second-order plasticity, we would employ a measure of smallness 6 = madsup
II R. IO supII i$ II}.
K
(7.10)
K
WC observe that, by virtue of (3. I), ([email protected] and(7.4)1.2.~. ‘is= H - HZ + O(E)), a, = He - H: + 0(e3),
(7.11j
H, = H, - H; + O(e3). Also, with the aid of (7.4),.~.~and (7.10) we deduce that F = I + F,
Tz + iP + O(S3),
= I + R, +
FP =
I +
rif + O@‘),
(7.12)
HP + Ti; + o(63),
and hence H =
a + a’
+
0(a3),
H,
= ii, + i$f +
06%
H,,
= iip + ii; +
O(6").
(7.13)
The relations (7. I I) and (7.13) allow approximate theories constructed from Eulerian measures to be related to those discussed in Sections 3,4,5, and 6 of the present paper. REFERENCES 1. M. E. Backman, Form of the relation between stress and tinite elastic and plastic strains under impulsive loading. J. Appl. Phys. 35, 2524 (1964). 2. E. H. Lee and D. T. Liu, Finite-strain elastic-plastic theory with application to plane-wave analysis. J. ADD/. Ph. 38. 19 (1967). 3. E: ‘H. Lee, Elastic-plastic deformation at finite strains. J. Appl. Mcch. 36, I (1969). 4. P. M. Naghdi and L. Vongsarnpigoon. Small strain accompanied by moderate rotation. Arch. Rution. Mech. Awl. SO, 263 (1982). 5. J. Casey and P. M. Naghdi, Physically nonlinear and related approximate theories of elasticity and their invariance properties. Arch. Rorion. Mech. Anol. in press. 6. .I. Casey and P. M. Naghdi, An invariant infinitesimal theory of motions superposed on a given motion. Arch. Rotion. Mech. Anul. 76, 355 (1981). 7. A. E. Green and P. M. Naghdi, Some remarks on elastic-plastic deformation at finite strain. In!. J. Engng Sci. 9, 1219 (1971). 8. P. M. Naghdi and J. A. Trapp, On finite elastic-plastic deformation of metals. J. Appl. Mech. 41, 254 (1974). 9. J. Casey and P. M. Naghdi, A remark on the use of the decomposition F = F,.F,, in plasticity. J. Appl. Mech. 47, 672 (1980). IO. J. Casey and P. M. Naghdi. Discussion of “A correct definition of elastic and plastic deformation and its computational significance” (by V. A. Lubarda and E. H. Lee). 1. Appl. Mech. 48, 983 (1981). II. A. E. Green and P. M. Naghdi. A general theory of an elastic-plastic continuum. Arch. Rofion. Much. Anul. 18, 251 (1%5). 12. A. E. Green and P. M. Naghdi, A thermodynamic development of elastic-plastic continua. froc. fUTAM Symp. on Irreversible Aspects of Continuum Mechanics and Transfer of Physical Churucteristics in Moving Nuids (Edited by H. Parkus and L. 1. Sedov), p. 117). Springer-Verlag, Berlin (1966).