Kinematic Constraints, Constitutive Equations and Failure Rules for Anisotropic Materials
It is common in many branches of continuum mechanics to treat material as though it is incompressible. Although no material is truly incompressible, there are many materials in which the ability to resist volume changes greatly exceeds the ability to resi
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KINEMATIC CONSTRAINTS, CONSTITUTIVE EQUATIONS AND FAlLURE RULES FOR ANISOTROPIC MATERIALS
A.J.M. Spencer
The University of Nottingham, England
1. KINEI-1ATIC CONSTRAINTS
It is common in many branches of continuum mechanics to treat material as though it is incompressible.
Although no material is truly
incompressible, there are many materials in which the ability to resist volume changes greatly exceeds the ability to resist shearing deformations; examples are liquids with low viscosity, like water, and some natural and artificial rubbers.
For such materials, the assumption
of incompressibility is a good approximation in many circumstances, and often greatly simplifies the solution of specific problems.
It should
be noted, though, that there are occasions when even a small degree of compressibility may produce a major effect; an example is the propagation of sound waves in water.
Incompressibility is an example of a
kinematic constraint; it
restricts the range of admissible deformations.
Another example is the
constraint of inextensibility in specified directions.
J. P. Boehler (ed.), Applications of Tensor Functions in Solid Mechanics © Springer-Verlag Wien 1987
Some highly
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A.J.M. Spencer
anisotropic materials exhibit strong resistance to extension in particular directions, compared to their shear resistance and resistance to extension in other directions.
Obvious examples are fibre composite materials
composed of strong stiff aligned fibres reinforcing a relatively soft matrix.
Materials of this kind may, approximately, be treated as
inextensible in the fibre direction, and analysis of their behaviour is often greatly simplified by making this approximation.
As in the case of
incompressibility, some caution is needed, because slight inextensibility can produce large effects.
However, the approximation is often useful,
and results derived from i t can be used as a basis on which to construct more accurate solutions.
The mechanics of these ideal fibre-reinforced
materials is described in
[1].
We refer quantities to a fixed reetangular coordinate system. typical particle has position vector X and coordinates X configuration at time t
R
= 0.
A
in its reference
At a subsequent time t the same particle
occupies the position x with coordinates x.. l
The deformation is
described by the dependence of x on X and t, thus X=
x(X,t),
or
X. l
The deformation gradient tensor F has components FiR' where F.
1R
=
ax. ;ax 1
R
.
We employ the finite strain tensors C and B, with components CRS and B ..
l]
respectively, where
c
T F F,
B
FFT
dX. dX.
CRS
FiRFiS
dX. dX.
l l ---
B ..
axR axs
l]
F. F. lR JR
___2:. _ 2
axR axR
and also the infinitesimal strain tensor E, with components E .. , where l]
E
E,.
lJ
ax.
Clx
.J
.!. ( __ l +_2 -eS 2
ax.J ax.l
ij ·
The velocity v is regarded as a function of x and t.
The rate-of-
Kinernarie Constraints and Constitutive Equations
189
deformation tensor has components Do 0' where ~J
avo
Do~J 0
av OJ
- l. (-~___2
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