204 R. Segev
Since in the sequel we consider only the smooth case, we will use “variational stresses” to refer to the densities.
5. The Cauchy stress induced by a variational stress
In [4] we defined a canonical mapping p
σ
: L J
1
W ,
m
T
∗
→ L W,
m−1
T
∗
, that assigns to a variational stress density S a Cauchy stress σ satisfying the following relation.
At every x ∈
we suppress the evaluation at x in the notation φ
∧ σ w = S j
φ⊗w
. Here, j
φ⊗w
is roughly the jet at x of a section whose value is 0 ∈ W
x
and its derivative is φ
⊗ w. More precisely, if u : → W is the section whose first jet at x is j
φ⊗w
, then, u satisfies the following conditions: ux
= 0; denoting the zero section of W by 0, T
x
u −
T
x
∈ L T
x
, T
0x
W
x
induces the linear mapping φ ⊗w through the isomorphism of T
0x
W
x
with W
x
. The local representative of p
σ
is as follows. If σ = p
σ
S, then, using the local representatives of σ and S as in the previous sections,
σ
β 1...ˆı...m
= −1
i−1
S
+i β
1...m
, no sum over i .
The mapping p
σ
is clearly linear and surjective.
6. The divergence of a variational stress
Given a variational stress density S its generalized divergence Div S is the section of the bundle L W,
V
m
T
∗
defined by DivSw
= d p
σ
Sw − S J
1
w .
The local expression for Div Sw is S
i α
1...m,i
− S
α 1...m
w
α
d x
1
∧ . . . ∧dx
m
, which shows that Div S depends only on the values of w and not its derivative. With these
definitions one obtains for the case where F
✂
w =
Z
✂
S j
1
w that
F
✂
w =
Z
✂
b
✂
w +
Z
∂
✂
t
✂
w
where t
✂
w = ι
∗
✂
σ w and Div S
+ b
✂
= 0. We conclude that every variational stress induces a unique force system
{b
✂
, t
✂
} through the Cauchy stress it induces and its diver- gence. Actually, we obtained a decomposition of S j
1
w into an exact differential and a term
Notes on stresses for manifolds 205
that is linear in the values of w. The converse is also true. If we have a force system that sat- isfies Cauchy’s postulates, then, the induced Cauchy stress enables us to define a section S of
L J
1
W , V
m−1
T
∗
by S j
1
w = bw + dσ w. Clearly, writing the local expression
for S, it is linear in the jet of w. Hence, F
✂
w =
Z
✂
bw
+ Z
✂
dσ w =
Z
✂
S j
1
w .
If for a given variational stress Div S = 0, then S j
1
w = dσ w, for σ = p
σ
◦ S.
References
[1] S
EGEV
R., Forces and the existence of stresses in invariant continuum mechanics, J. of
Math. Phys. 27 1986, 163–170.
[2] S
EGEV
R., The geometry of Cauchy’s fluxes, Archive for Rat. Mech. and Anal. 154 2000,
183–198. [3] S
EGEV
R.
AND
R
ODNAY
G., Cauchy’s theorem on manifolds, Journal of Elasticity 56
2000, 129–144. [4] S
EGEV
R.
AND
R
ODNAY
G., Divergences of stresses and the principle of virtual work on
manifolds, Technische Mechanik 20 2000, 129–136. AMS Subject Classification: 73A05, 58A05.
Reuven SEGEV Department of Mechanical Engineering
Ben-Gurion University P. O. Box 653
Beer-Sheva 84105, ISRAEL e-mail:
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206 R. Segev
Rend. Sem. Mat. Univ. Pol. Torino Vol. 58, 2 2000
Geom., Cont. and Micros., II
B. Svendsen
∗
NON-LOCAL CONTINUUM THERMODYNAMIC EXTENSIONS OF CRYSTAL PLASTICITY TO INCLUDE
THE EFFECTS OF GEOMETRICALLY-NECESSARY DISLOCATIONS ON THE MATERIAL BEHAVIOUR
Abstract. The purpose of this work is the formulation of constitutive models for the inelastic material behaviour of single crystals and polycrystals in which
geometrically-necessary dislocations GNDs may develop and influence this be- haviour. To this end, we focus on the dependence of the development of such dis-
locations on the inhomogeneity of the inelastic deformation in the material. More precisely, in the crystal plasticity context, this is a relation between the density of
GNDs and the inhomogeneity of inelastic deformation in glide systems. In this work, two models for GND density and its evolution, i.e., a glide-system-based
model, and a continuum model, are formulated and investigated. As it turns out, the former of these is consistent with the original two-dimensional GND model of
Ashby 1970, and the latter with the more recent model of Dai and Parks 1997. Since both models involve a dependence of the inelastic state of a material point on
the history of the inhomogeneity of the glide-system inelastic deformation, their incorporation into crystal plasticity modeling necessarily implies a correspond-
ing non-local generalization of this modeling. As it turns out, a natural quantity on which to base such a non-local continuum thermodynamic generalization, i.e.,
in the context of crystal plasticity, is the glide-system scalar slip deformation. In particular, this is accomplished here by treating each such slip deformation as
either 1, a generalized “gradient” internal variable, or 2, as a scalar internal degree-of-freedom. Both of these approaches yield a corresponding generalized
Ginzburg-Landau- or Cahn-Allen-type field relation for this scalar deformation determined in part by the dependence of the free energy on the dislocation state in
the material. In the last part of the work, attention is focused on specific models for the free energy and its dependence on this state. After summarising and briefly dis-
cussing the initial-boundary-value problem resulting from the current approach as well as its algorithmic form suitable for numerical implementation, the work ends
with a discussion of additional aspects of the formulation, and in particular the connection of the approach to GND modeling taken here with other approaches.
∗
I thank Paolo Cermelli for helpful discussions and for drawing my attention to his work and that of Morton Gurtin on gradient plasticity and GNDs.
207
208 B. Svendsen
1. Introduction
