Protein Crystals: Molecular to Continuum Level Models Based on Crystal Plasticity Theory - Multiscale Modeling in Biomechanics and Mechanobiology - page 16

as temperature. It has been shown that for most materials the probability function
satisfies a power type expression

2 n − 1

˄ ʱ
˄ y

f p ˄ ʱ , ˇ = ʻ

(˄ ʱ ) sgn (2.7)

˄ y where ʻ and n are material constants and is the critical resolved shear stress of

the slip-system ʱ . The rate of shear strain on a slip-system ʱ is

ʳ ʱ = ˕ ʱ ˁ ʱ b ʱ ¯ v ʱ (2.8)

ˁ ʱ is the dislocation density, and b ʱ is the Burgers
vector. Combining Eqs. ( 2.6 )-( 2.8 ) one obtains:

˕ ʱ is a material parameter, where

2 n − 1

˄ ʱ
˄ y

ʳ ʱ = ʻ (˄ ʱ )

sgn (2.9)

˄ ʱ on the slip-system can be related to the Cauchy

where the resolved shear stress

stress tensor ˃ in the fixed coordinate system as:

˄ ʱ = ˃ : P ʱ (2.10)

Using Eq. ( 2.9 ), the overall accumulated slip ʳ in a the crystal can be obtained by

t

N

ʳ ʱ dt

ʳ = (2.11)

ʱ = 1

0

The accumulated slip can be used as a good measure for evaluation of the defor-

mation propensity of a crystal having specific orientation with respect to the external
load.
According to the normality rule in plasticity, a yield function f

ʳ

(˃, ˇ)

could be

defined such that

= ʻ ∂

f ( ˃ , ˇ )

∂ ˃

D P (2.12)

where is a positive parameter which depends on the type of dislocation barriers.

Comparing Eqs. ( 2.4 ) and ( 2.5 ) and solving the differential Equation ( 2.12 ), a yield
surface for protein crystals can be defined as:

ʻ

⊛ ⊞

2 n

N

P ʱ
˄ y

1
2 n

˃ :

⊝ ⊠

f ( ˃ , ˇ ) = − 1 (2.13)

ʱ = 1