Introduction to the Invariant Formulation of Anisotropic Constitutive Equations
The aim of this Chapter is to present some elementary notion for non-specialists in the invariant formulation of anisotropic constitutive equations. Much of this Chapter is taken from [1].
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INTRODUCTION TO THEINVARIANT FORMULATION OF ANISOTROPie CONSTITUTIVE EQUATIONS
J.P. Boehler University of Grenoble, France
1. INTRODUCTION The aim of this Chapter is to present some elementary notion for non-specialists in the invariant formulation of anisotropic constitutive equations. Much of this Chapter is taken from [1]. We consider the very simple case of a two-dimensional anisotropic material of constitutive equation specifying a symmetric 2nd order tensor T, which is a function of two symmetric second order tensors D and T
In (1),
~
F(D,
~:
0
( 1)
is the structural tensor, taking into account the symmetries of
the material internal structure, D is the mechanical agency and T
the
mechanical response of the material. We use the absolute notation, not to be related to coordinates at this point. For definiteness, T is the stress tensor, D is the strain tensor in the case of elastic behavior or the
J. P. Boehler (ed.), Applications of Tensor Functions in Solid Mechanics © Springer-Verlag Wien 1987
14
J.P. Boehler
rate of strain tensor in the case of plastic behavior. In genera1, the principal directions (e ' e ) of the response T do - 1 - 2 principal directions (E ' E ) of the agency ~· -1 - 2 The constitutive equation
not coincide with the
(1) is sketched in the Figure below,
where the material, in its initial configuration, is represented by a circle and the attached orthonormal frame (v , v ); the tensorsTand D -1
-2
-
are represented by their associated ellipses. Similar sketches will be used throughout this Chapter.
F Initial configuration of the material
Agency
F
...
D
depends on
Response
~
T
the material the behavior
Constitutive relation 2. PRINCIPLE OF ISOTROPY OF SPACE 2.1 Physical concept Constitutive equations are subjected to the invariance requirements of the Principle of Isotropy of Space
(or Principle of Material Indif-
ference [2]). We shall apply it here to the simple situation involved in the equation (1).
15
Introduction to the Invariant Formulation
A consequence of this Principle is that an arbitrary transformation Q of the orthogonal group 0 and applied to both the material and the agency D, results in the same orthogonal transformation of the material response T. A naive statement of this is that the orientation of the material in the space has no effect on its constitutive relation. This is sketched in the Figure below.
T
D
} =
ara'
Consequence of the Principle of Isotropy of Space 2.2 Mathematical concept
The transformation of the body
D =>
and the transformation of the agency
result in the same transformation of the response: T => Finally we obtain: VQ
or:
E
~(9~9t' 9~9t) = 9~(~,
0 VQ
E
0
F(D,
Relation (2) indicates the invariance
pgt
~) = !T
Definition of an isotropic material A direct consequence of this definition is that the principal directions , e ) of the E , E ) of the agency -D and the principal directions (e -2 -1 -2 response T are the same. To see that, we apply the reflection S with res-
-1
pect to the direction
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