Biomedical Engineering Reference
In-Depth Information
Fig. 3.21 The principle of
determinism and the principle
of local action imply that the
state X is dependent on the
states of all other material
points Y of a body K , from
the remote past to the present
time
quantities. The desired relation between stress and strain measures will be derived
based on appropriate strain energy functions w. Hence, the generation of consti-
tutive equations is treated extensively, based on rational mechanics.
3.2.6.1 Strain Energy Functions
w is referred to as a strain energy function (abbreviated with sef in the following)
describing purely mechanical processes. Such a function is considered as a con-
stitutive equation and must thus satisfy the principles of rational mechanics, i.e.
causality, determinism, local action and objectivity.
Principle of Causality. This principle determines the choice of dependent and
independent variables in the context of material model generation. Restriction to
pure mechanical processes, a relation between w and the motion v is constituted,
where w is identified as the dependent and v as the independent variable.
Principle of Determinism. The current state of a material point of the con-
tinuum is determined through the current motion and the entire motion-history of
all other material points of the continuum body. This means: w at the material
point X at (the present) time t with the position vector X is determined through
the motion history v of all material points Y of the body K from the remote past
s =-? to the present time s = t. Hence, using ( 3.47 ), the most general material
equation of w (in a mechanical process) takes the following form:
t
t
w ð X ; t Þ¼ f
h y ; X ; t f
h v ð Y ; s Þ; X ; t i:
ð 3 : 151 Þ
s ¼1
X ; Y 2 K
s ¼1
X ; Y 2 K
Notes:In( 3.151 ), f is a scalar-valued ''functional'' prescription (such as an
integral, a differential or a function). The use of angle brackets '' hi '' emphasizes the
distinction of a functional and a function which usually is parenthesized. According
to ( 3.151 ), w can also be an explicit function of position X and time t thus comprising
inhomogeneous and aging materials. Inhomogeneous materials may exhibit different
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