Homogenisation of General Continua: S. Toll
Homogenisation of General Continua: S. Toll
Homogenisation of General Continua: S. Toll
S. Toll
Chalmers University of Technology, Department of Applied Mechanics, Gteborg, Sweden
ABSTRACT: This paper raises some fundamental issues about the formulation of continuum theories and the
transition between different scales. Emphasis is put on generality with respect to extended continua and multi-
physics as well as lack of scale separation. The generic continuum model is established as an approximation of
conserved extensive quantities by volume integrals of continuous density fields. Conditions for homogenisation
are derived by introducing alternative sets of fields of different resolution and requiring those to approximate the
same extensive quantities on a given scale. This leads to general equivalence conditions which must be satisfied
by any homogenisation procedure. Specific results are given for the standard thermomechanical continuum and
a more general electro-thermomechanical micropolar continuum. The usual field averages are obtained only
under severe restrictions, such as static conditions and widely separated scales.
107
Copyright 2005 Taylor & Francis Group plc, London, UK
Any change of
within must be accompanied leading to the conditions
by an equal and opposite change outside. In continuum
theory one assumes that such exchange can happen in
either of three ways: by flux across a surface , by
external supply through some long-range interaction
or by internal production. Thus the change of is
108
Copyright 2005 Taylor & Francis Group plc, London, UK
3.1 Scales not widely separated Table 1. Field variables of the standard continuum.
To obtain the effective field locally ((x) rather than
s p j
(x)), we consider the Taylor expansion of the
effective field, mass
mom g
ang r r g ( ) r
ener u + 12 2 g h q (u + 12 2 )
4.1 Homogenisation
Equation (16) also provides the appropriate extrapola- The equivalence equations for the standard continuum
tion to evaluate the effective fields near macroscopic become
boundaries.
109
Copyright 2005 Taylor & Francis Group plc, London, UK
and combining equations (27) and (28) yields the Hill-
Mandel condition, e.g., Nemat-Nasser & Hori (1993),
Equations (29), (30) and (31) simply require that 4.4 Nonlinear kinematics
the fluctuation fields , and
,
are uncorrelated with space. This is thus In order to obtain simple average relations for the
a necessary condition for the existence of macrofields kinematic fields, one must make the rather severe
in the standard continuum. It is not very restrictive, assumption that is uncorrelated with ,
however, and does not require widely separated scales.
4.2 Widely separated scales Let and be different mappings from one and
the same material configuration X , into two spatial
In the case where the scales sufficiently separated that configurations and and (which are different in
the macrofields may be taken to be uniform, or at least general): x = (X , t) (t), and x = (X , t) (t)
linear, on the scale , equations (24)(28) reduce to subject to (X , t0 ) = (X , t0 ) = X X . We may
then define the velocities
110
Copyright 2005 Taylor & Francis Group plc, London, UK
Table 2. Field variables of the electro-thermomechanical micropolar continuum.
s p j
mass
charge J
momentum + P g T
angular momentum a + r ( + P) r g ( T ) r + a
energy u + G + 12 ( 2 + a w) g h + w 12 ( 2 + a w) u q S
111
Copyright 2005 Taylor & Francis Group plc, London, UK
along with the condition r = r, which states A number of general conditions for homogenisa-
that the density fluctuation must be uncorrelated with tion have emerged. (i) Supply fields (such as body
space. These are 39 equivalence equations in 39 force) and production fields (such as an internal heat
variables. source field) must be scale invariant. Thus any inter-
actions that are specific to a microscale must be
described as a flux of a conserved quantity, not as
5.2 Widely separated scales
action-at-a-distance. (ii) Averages must be taken over
Assuming there are no micropolar fields a, w or couple a large enough volume that they are unique, within
stress on the microscale, these reduce to required accuracy. (iii) Depending on the continuum
model assumed (whether standard or extended to some
degree) certain fluctuation fields may have to be
spatially uncorrelated.
Effective fields may well exist even when scales
are not widely separated. Such effective fields may
be expressed as a series expansion in gradients of the
average, where the first term is the average itself. They
are also defined right up to a macroscopic boundary.
An admissible macrofield will automatically satisfy
the same balance equations as the microfields. The
number of integral conditions should be the same as the
number of field variables on either scale (whichever is
greatest).
The standard continuum model requires the lack
of spatial correlation of certain fluctuation fields, this
requirement may be relaxed by introducing micropolar
degrees of freedom. The redundant Hill-Mandel con-
dition arises only when thermal effects are neglected.
The standard average relations for the nonlinear kine-
matic fields, e.g. the deformation gradient tensor, are
only obtained after assuming that velocity and mass
density are statistically uncorrelated.
The electro-thermomechanical continuum is only
briefly outlined here. The objective was to point to
the possibility of truly multiphysical homogenisation,
and to show how some non-standard fields such as
couple stress should be averaged. It is certainly not
straightforward to apply those results unless they are
considerably simplified.
REFERENCES
6 CONCLUDING REMARKS
112
Copyright 2005 Taylor & Francis Group plc, London, UK