Advances in Ma terials Physics and Che mist ry, 2012, 2, 53-55
doi:10.4236/ampc.2012.24B015 Published Online December 2012 (htt p://www.Sc iRP.org/journal/ampc)
The Minimum Energy Principle in Description of Nonlinear
Properties of Orthotropic Material
Tadeusz Wegner, Dariusz Kur pisz
Applied Mechanics Institute, Poznan University of Technology, Poznan, Poland
Received 2012
ABSTRACT
In th is paper the con ception o f theoretical d etermine the relat ions b etween material experiment al characteris tics is presen ted. On the
base of stres s-strain relatio ns for nonlin ear elasti c anisotrop ic material an d geometrical interpretation of deformation state, the gener-
al form of strain energy density function was introduced. Using this function and variational methods the relations between material
characteristics were achieved. All consi derations are illus trated by a short th eoretical example.
Keywords: Material C haracteri s tics; Mechanical Propert ies; Deformati on S tate Compon ents; Strain Energy Densit y Fu nction ;
Minimum Energy Principle; Variational Methods
1. Introduction
One of the most important nature laws is the minimum energy
principle. Thought of this principle each physical match tends
at mini mizati on of its en ergetic state. So in the case of material
the configuration of deformation state must satisfy the principle
of minimum of energy. Because the discussed law introduces
relations between deformation state components, the material
characteri stics must b e mutually coh erent. On i mpo rtant rule of
energy, as a tool to description of material mechanical proper-
ties, call attention Ogden [1], Perzyna [2,3], Petryk [4-6],
Schrod er [7], Wagner [ 8], Wegner [ 9] and other. The main aim
of this paper is construction mathematical relations between
material mechanical characteristics due to minimum of energy
principle. For affirmation of generality of considerations the
nonlinear elastic orthotropic material will be used.
2. Geometrical Interpretation of Deformation
State and Strain Energy Density Function
The deformation of material requires the work of external load.
Direct answer of material for application of external load pro-
gram is the stress. Its value is dependent on load magnitude,
deformation state and individual properties of material. So the
relations between stress and strain (there be measure of defor-
mation ) are different for d ifferent mat erials. Co nnectin g factors
of strain and stress components are material characteristics.
These relati ons for nonli near elastic orth otropic material can b e
written as
123
112213 3
112 233
12 3
221 1233
112 233
1 23
3311322
112 233
()() ,
()() ()
()() ,
() ()()
() (),
()() ()
vv
EEE
vv
EE E
vv
E EE
σσσ
ε εε
εεε
σσ σ
εε ε
εε ε
σ σσ
εε ε
ε εε

=−−



=− +−



=−− +



(1)
where
123
,, σσσ
and
123
,, εεε
are respectively principal
stress and strain components.
.
() ()
def
iiii i
E εσε ε =
for
1,2, 3i=
and
.
()
def
ij jij
v ε εε =
for
,1,2, 3ij=
and
ij≠
are experimental
material character istics obtained in u niaxial tension tests. As we
can see th e full descrip tion of materi al requ ires an experimental
assignment of nine characteristics. There is oppressive for reali-
zation. Hence it is proper to search for dependences among t hem.
2.1. Geometrical Interpretation of Deformation State
Let’s assume that the external load program is such, that the
orthotropy directions and principal deformation directions are
the same. Next separate the mat erial p iece in shape a cube, that
edges are parallel to principal orthotropy directions. The defor-
mation process of this el ement ary cube was illustrated belo w.
Every deformation state response a point on deformation
path C. On the end of this path we have a desired deformation
state
123
,,
KKK
εεε
. The motion along path C is initiated by
changeable stress components. So every displacement along
path C needs work.
Figure 1. Interpretation of def ormation state .

T. WEGNER ET AL.
54
2.2. Strain Energy Density Funct ion
The deformation work L can be expressed by the use of line
integral in the form
3
1
Ld
ii
i
C
σε =∑
∫=
(2)
Because the space of deformation components is potential
the deformation work is independent from the shape of defor-
mation path C. So we can write
11
22
33
,
,where 0,1
.
K
K
K
t
C tt
t
εε
εε
εε
=

==∈〈 〉

=

(3)
and hence by the use of (2)
13
12312 3
1
0
(, , )(,,)
KKKKKK K
ii
i
Wttt dt ε εεσεεεε
=
=
∑
∫
(4)
where
123
(, , )
i
σεε ε
is the solution of system of Equations (1).
The strain energy W is a function of deformation state compo-
nents.
3. The Principle of Minimum Energy
Because th e solution of system of Equations (1) can be write in
the form
12 32
( )(,...,)
iii i
EFv v σε = ⋅
(5)
for
1,2, 3,i=
then the equation (4) can be written as
123
13
12 232 2
1
0
(,,)
()(( ),...,( )).
KKK
K KKK
ii ii
i
W
Et Fvtvtdt
εεε
ε εεε
=
=∑
∫
(6)
The right side of equation (6) is a functional due to functions
()
ii
E ε
and
()
ij j
v ε
. Hence the detection of minimum of ener-
gy is equivalent the determination of minimum of functional (6).
Let’s put that
123 1232
3
12 232 2
1
(,,,,...,)
()(( ),...,( )).
K KKK
ii ii
i
FE EE vv
E tFvt vt ε εεε
=
=∑
(7)
If (6) has a minimum, then the following assumptions must
be satisfied
1i3
(I-III) 0
i
F
E
≤≤
∂
∀=
∂
(8)
(IV-IX) 0 for ,1,2,3, and .
ij
Fiji j
v
∂= =≠
∂
(9)
The Equations (8) and (9) introduce (on the base of minimum
of energ y), t he relatio ns between materi al mechanical characte-
ristics. So if single or twice indexed material characteristics are
known then the determination of the next characteristics is
possible.
4. An Example
Let’s assume, that we have nonlinear elastic isotropic material
under flat state of stress (in plane 1O2). In such case the ma-
terial ph ysical p rop ert ies are t he same in all d ir ection s. It means
that there are two material characteristics. Because the compo-
nent of stress state in third direction is equal zero (
30
σ =
), th e
deformation path C can be given as
11
22
:for 0,1
K
K
t
Ct
t
εε
εε
=
∈〈 〉
=

 (10)
and the system of Equation (1) reduces to
12 2
11
12
21 1
22
12
()
() ,
1 ()()
()
() .
1 ()()
KK K
K
KK
KK K
K
KK
ttv t
Et v tvt
ttv t
Et v tvt
εε ε
σε εε
εε ε
σε εε
+⋅
= ⋅
−

+⋅
= ⋅
−

(11)
The relation (6) simplifies to
112 2
12 11
012
21 1
22
12
()
( ,)[()
1 ()()
()
()]
1 ()()
KK K
KK KK
KK
KK K
KK
KK
ttv t
WEt v tvt
ttvt
E tdt
v tvt
εε ε
εε εε
εε
εε ε
εε
εε
+⋅
=−
+⋅
+−
∫
(12)
Let’s take the function
()E ε
as known. The perturbation of
functional (12) is possible due to functions
1
()
K
vt ε
and
2
()
K
vt
ε . So on the base of assumptions (8) and (9) we have
1 1222
1
12 2
2
12
0[() ()()]
()
()
0,
[1() ()]
K K KKK
K
KK K
KK
FE tvtEt
vt
ttv t
v tvt
εεε εε
ε
εε ε
εε
∂=⇔+
∂
+⋅
×=
−
(13)
1 1221
1
21 1
2
12
0[()() ()]
()
()
0,
[1() ()]
K KKKK
K
KK K
KK
FEt Etvt
vt
ttv t
v tvt
εεεε ε
ε
εε ε
εε
∂=⇔+
∂
+⋅
×=
−
(14)
Because
21
12
12
() and ()vv
εε
εε
εε
=−=−
(15)
the conditions (13) and (14) can be written as
12 21
12
()( )( )()0,
1( )()
Ev Ev
vv
εε εε
εε
−=
−
(16)
or
11
22
() ()
() ()
vE
vE
εε
εε
= (17)
For nonlinear materials the relations (15) are the definition of
transversal strain coefficient, analogous to classical Poisson
ratio definition in linear theory. The condition (17) between
transversal strain and changeable stiffness coefficients is a co-
herence condition for material characteristics of nonlinear iso-
trop ic material s, as a co nseq u ence o f the minimum energ y p rin-
ciple.
5. Conclusions
Relations between material characteristics results from mini-
mum energy principle. The strain energy density function can

T. WEGNER ET AL.
55
be treated as functional of materials characteristics. The solu-
tion of system of equations (8), (9) is not trivial in case if we
know at least one ch ar acteristi c.
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