Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 120

state in which it is subjected to the stress T i , which is held constant during this leg
of the cycle. During the second loading sequence the strain is given by

E i ¼ S ij ð T ð A Þ

þ T j Þ; T j T ð B Þ

0 (5.17H)

j j

and the work done from T i to T i þ T i is

ð

B

1

S ij ð T ð A Þ

þ T j Þ d T i ¼ S ij T ð A Þ

T ð B Þ
i

2 S ij T ð B Þ

T ð B Þ
i

W AB ¼ þ (5.18H)

j j j

O

where the factor of one-half does not appear before the first term on the right hand
side because T i is held fixed during the second loading sequence in this leg of the
loading cycle. During the third loading sequence the stress

T i

is held fixed; the

strain is given by

E i ¼ S ij ð T ð B Þ

T j T ð A Þ þ T j Þ;

0 : (5.19H)

j j

The work done during the third loading sequence of the cycle, from T i þ T i
to T i

is then

O

ð

1
2 S ij T ð A Þ

S ij ð T ð B Þ
j

þ T j Þ d T i ¼S ij T ð B Þ

j

T ð A Þ
i

T ð A Þ
i

W BC ¼ : (5.20H)

j

A

The work done during the final loading sequence, from T i

to 0, is then

ð

O

1
2 S ij T ð B Þ

S ij T j d T i ¼ T ð B Þ

i

W CO ¼ : (5.21H)

j

B

The work done in the closed cycle is then the sum W OO ¼ W OA þ W AB þ W BC

þ W CO given by

W OO ¼ S ij T ð A Þ
j

T ð B Þ
i

S ij T ð B Þ
j

T ð A Þ
i

¼ðS ij S ji Þ T ð A Þ
j

T ð B Þ
i

: (5.22H)

If the cycle is traversed in reverse (Fig. 5.2(b) ), then

W OO ¼ðS ij S ji Þ T ð A Þ

T ð B Þ
i

: (5.23H)

j