Bridging between continuous and
discrete dislocations
Marcelo Epstein
Calgary, Canada
deLeónfest 2013
Dec. 16-19, ICMAT, Madrid
Singular vs. smooth
Singular
defects
Smooth
defects
Physical view
Continuous dislocations
• Two classical approaches:
– Heuristic elevation of the discrete picture: the lattice is
replaced by a smooth frame field (repère mobile)
(Burgers’ vector
torsion of a connection)
– Constitutively based, without any allusion to the
underlying atomic structure
• Another possibility:
– A purely geometric approach that will encompass the
continuous and the discrete cases
The purely constitutive approach
• Pioneers: Kondo (1950s), Noll and Wang (1960s)
• Later developments:
- The material groupoid
- Integrability conditions
- Higher-order materials
- Functionally graded materials
Continuum kinematics
The deformation gradient F at a point is a linear map that transforms an
infinitesimal cube into a parallelepiped.
Constitutive response
• Different materials respond differently to their
deformation (elasticity, viscoelasticity, memory,
nonlocality, …)
• In a simple elastic body the response can be
encapsulated in a single scalar function:
W = W (F; X)
• Physically, this function represents stored potential
energy (per unit mass), just like a spring in a box.
Material isomorphism
• We want to compare the material response at two different points X1
and X2 to check if perhaps they are made of the same material (i.e.,
if they have, so to speak, the same ‘chemical identity’).
X1
X2
body B
Material isomorphism
• For the whole body we write:
W  W (F; X)
• Two points are materially isomorphic
if there exists a linear map P12
between their tangent spaces such
that:
W (FP12 ; X1 )  W (F; X2 )
identically for all (non-singular) F.
• Surgical analogy: a perfect graft.
Material symmetry
• A material symmetry G at X is a material
automorphism, i.e.:
W (FG; X)  W (F; X)
F GL(3, R)
• The collection of material symmetries at a point X
form a subgroup GX of the special linear group (or
unimodular group):
G X  SL(3, R)
(Example: isotropy)
Material symmetry
F
G
X
W(FG; X) = W(F; X)
A game of arrows
Given a body, we draw an arrow for every material isomorphism (including
automorphisms). The arrow has its tail at the source point and its tip at the target point. The
set of arrows will not be empty, since the identity is always available at each point.
As a direct consequence of the fact that material isomorphism is an equivalence relation, we
note that:
1. If the tip of an arrow meets the tail of another, the arrow starting at the tail of the first
and ending at the tip of the second represents a material isomorphism, and so it must also
be a member of the arrow set.
2. Since material isomorphisms are invertible, if an arrow is a member of the set so must be
the opposite arrow.
The material groupoid
• The construction just described can be formalized more
rigorously as a groupoid, called the material groupoid
associated with the given constitutive law.
• A groupoid is transitive if there exists at least one arrow
connecting any two points. A groupoid is a Lie groupoid if
certain smoothness conditions are satisfied.
• A body is said to be smoothly uniform if its material groupoid
(which always exists) happens to be a transitive Lie groupoid.
• Physically, uniformity means that all the points of the body are
made of the same material. The presence of dislocations can be
then associated with the flatness of any of the G-structures that
can be derived from the material groupoid.
The purely geometric approach: Tools
• ALGEBRA
– Covectors
– Wedge product
– Exterior algebra
– p-covectors
• DIFF. GEOMETRY
– Differential 1-forms
– Exterior derivative
– Closed forms
– Differential p-forms
ANALYSIS
de Rham`s currents are to differential forms what
Schwartz`s distributions are to real functions
Geometric interpretation of a covector
Geometric interpretation of a 1-form
Physical interpretation and
integrability
Smooth vs. singular
Currents
Exterior derivatives and boundaries
Dislocations
An edge dislocation
Proof of Frank’s first rule
Possibilities
The singular counterpart?