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Plane constrained shear of single crystals

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Abstract

This paper studies the plane constrained shear problem for single crystals having one active slip system and subjected to loading in both directions within the small strain thermodynamic dislocation theory proposed by Le (J Mech Phys Solids 111:157–169, 2018). The numerical solution of the boundary value problem shows the combined isotropic and kinematic work hardening, the sensitivity of the stress–strain curves to temperature and strain rate, the Bauschinger effect, and the size effect.

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Abbreviations

\(\beta \) :

Plastic slip

\(\varvec{\sigma }\) :

Stress tensor

\(\varvec{\varepsilon }\) :

Total strain tensor

\(\varvec{\varepsilon }^\text {e}\) :

Elastic strain tensor

\(\varvec{\varepsilon }^\text {p}\) :

Plastic strain tensor

\(\chi \) :

Configurational temperature

\(\chi _0\) :

Steady-state configurational temperature

\(\gamma (t)\) :

Shear amount (as control parameter)

\(\gamma _D\) :

Energy of one dislocation per unit length

\(\mu \) :

Shear modulus

\(\nu \) :

Poisson’s ratio

\(\rho \) :

Total density of dislocations

\(\rho ^\text {g}\) :

Density of non-redundant dislocations

\(\rho ^\text {r}\) :

Density of redundant dislocations

\(\tau \) :

Resolved shear stress (Schmid stress)

\(\tau _\text {B}\) :

Back stress

\(\tau _\text {T}\) :

Taylor stress

\(\tau _\text {Y}\) :

Flow stress

\(\varphi \) :

Angle between slip direction and \(x_1\)-axis

\(\mathbf {m}\) :

Unit vector normal to the slip plane

\(\mathbf {s}\) :

Unit vector showing the slip direction

\(\mathbf {u}\) :

Displacement vector

\(a^2\) :

The minimally possible area occupied by one dislocation

b :

Magnitude of Burgers’ vector

h, w, L :

Height, width, and depth of the slab

q :

Dimensionless plastic strain rate

\(q_0\) :

Dimensionless total strain rate

T :

Kinetic-vibrational temperature

\(t_0\) :

Time characterizing the depinning rate

\(T_P\) :

Energy barrier expressed in the temperature unit

Dot over quantities:

Time rates

Bar over quantities:

Quantities averaged over the thickness

Tilde over quantities:

Rescaled (dimensionless) quantities

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Authors and Affiliations

  1. Lehrstuhl für Mechanik - Materialtheorie, Ruhr-Universität Bochum, 44780, Bochum, Germany

    F. Günther

  2. Materials Mechanics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    K. C. Le

  3. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    K. C. Le

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  1. F. Günther
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Günther, F., Le, K.C. Plane constrained shear of single crystals. Arch Appl Mech 91, 2109–2126 (2021). https://doi.org/10.1007/s00419-020-01872-3

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