Plasticity for Structural Engineers

Notation

Stresses and Strains

? 1, ? 2, ? 3

principal stresses, tensile stress positive

? ij

stress tensor

S ij

stress deviator tensor

?

normal stress

?

shear stress

p = ? I 1

hydrostatic pressure or spherical stress

? oct = ? I 1

octahedral normal stress

? oct = ?? J 2

octahedral shear stress

? m = ? oct

mean normal stress

? m = ?? J 2

mean shear stress

S 1, S 2, S 3

principal stress deviators

? 1, ? 2, ? 3

principal strains, tensile strain positive

? ij

strain tensor

e ij

strain deviator tensor

?

normal strain

?

engineering shear strain

? ? = I ? 1

volumetric strain

? oct = ? I ? 1

octahedral normal strain

? oct = 2 ?? J ? 2

octahedral engineering shear strain

e 1, e 2, e 3

principal strain deviators

Invariants

I 1

= ? 1 + ? 2 + ? 3 = ? ii

= first invariant of stress tensor

J 2

=

Sij Sij

=

? [ ? x ? ? y) 2 + ( ? y ? ? z) 2 + ( ? z ? ? x) 2] + ?

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