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Material strength: stress and strain

Fig.: .

stress, $\sigma$, in $\mathrm{N\,m^{-2}}$
strain, $\epsilon=\frac{\Delta l}{l_0}$, in %
elastic deformation
Hooke's law, $\Delta l\propto F$
$$\sigma=E\epsilon$$
Young's modulus, $E$
shear stress
$$\tau=G\gamma$$ bulk modulus, $K$
$$p_{\textrm{hyd}}=K\frac{\Delta V}{V}$$

Fig.: .
Fig.: .

tensile test
necking
elongation, $\epsilon_{\parallel}=\frac{\Delta l}{l_0}$
contraction, $\epsilon_{\perp}=-\frac{\Delta d}{d_0}$
Poisson's ratio, $\nu=-\frac{\epsilon_{\parallel}}{\epsilon_{\perp}}$
isotropic medium
$$E=2G(1+\nu)=3K(1-2\nu)$$

Plastic deformation and dislocations

Fig.: .
Source: Anderson, Leaver; Materials Science.

block slip

Fig.: .

slip system:
slip plane and slip direction
principal slip system
most densely packed lattice direction
most densely packed plane containing the slip direction

Fig.: .

single crystals
dislocations
glide
yield stress
work hardening

Fig.: .
Source: Anderson, Leaver; Materials Science.
Fig.: .
Source: Anderson, Leaver; Materials Science.

edge dislocation
screw dislocation
mixed dislocation

Fig.: .
Source: Anderson, Leaver; Materials Science.
Fig.: .
Source: Anderson, Leaver; Materials Science.

Burgers circuit
Burgers vector, $\vec{b}$,
dislocation type
glide plane contains dislocation and its Burgers vector

Fig.: .
Source: Anderson, Leaver; Materials Science.

dislocation width

Fig.: .
Source: Anderson, Leaver; Materials Science.

mobility
cross slip

Fig.: .
Source: Anderson, Leaver; Materials Science.

vacancies
climb

Stress tensor

Fig.: .
Modified from image by Jose A Sorrentino via Wikimedia under this CC licence.

elastic strain $$\epsilon=\frac{1}{E}\sigma$$
$$\epsilon_i=\frac{1}{E}\sigma_i\qquad\textrm{where}\qquad i=x,y,z\quad\textrm{(or 1,2,3)}$$ necking $$\epsilon_j=-\frac{\nu}{E}\sigma_i\qquad\textrm{where}\qquad j\neq i=x,y,z\quad\textrm{(or 1,2,3)}$$ combined
$$\epsilon_i=\frac{1}{E}(\sigma_i-\nu\sigma_j-\nu\sigma_k)\qquad(i,j,k~\textrm{cyclic})$$ $$\epsilon_{ii}=\frac{1}{E}(\sigma_{ii}-\nu\sigma_{jj}-\nu\sigma_{kk})\qquad(i,j,k~\textrm{cyclic})$$ shear strain
$$\gamma_{ij}=\frac{1}{G}\tau_{ij}$$ stress vector, strain vector and stiffness matrix
$$\vec{\epsilon}=\mathbf{\Phi_e}\vec{\sigma}= \left(\begin{array}{c}\epsilon_{11}\\\epsilon_{22}\\\epsilon_{33}\\\gamma_{12}\\\gamma_{23}\\\gamma_{31}\\\end{array}\right)= \left(\begin{array}{cccccc} 1/E&-\nu/E&-\nu/E&0&0&0\\ -\nu/E&1/E&-\nu/E&0&0&0\\ -\nu/E&-\nu/E&1/E&0&0&0\\ 0&0&0&1/G&0&0\\ 0&0&0&0&1/G&0\\ 0&0&0&0&0&1/G \end{array}\right) \left(\begin{array}{c}\sigma_{11}\\\sigma_{22}\\\sigma_{33}\\\sigma_{12}\\\sigma_{23}\\\sigma_{31}\\\end{array}\right)$$

Plastic strain

dislocations
no volume change
$$\frac{\Delta V}{V}=\sum_i\epsilon_i=\frac{1-2\nu}{E}\sum_i\sigma_i$$ Therefore, $\nu=\frac{1}{2}$ for plastic deformation.
incremental strain
$${\rm d}\epsilon^e_{11}=\frac{1}{E}({\rm d}\sigma_{11}-\nu{\rm d}\sigma_{22}-\nu{\rm d}\sigma_{33})\qquad\textrm{for elastic strains}$$ $${\rm d}\epsilon^p_{11}={\rm d}\lambda\left(\sigma_{11}-\frac{1}{2}\sigma_{22}-\frac{1}{2}\sigma_{33}\right)\qquad\textrm{for plastic strains}$$ plastic stiffness, ${\rm d}\lambda$
orientation of stress cube
principal stresses, $\sigma_i$ (as opposed to $\sigma_{ii}$)
effective stress, $\overline{\sigma}$, and effective strain, $\overline{\epsilon}$
$$\overline{\sigma}=\sqrt{\frac{\sum_{ij}(\sigma_i-\sigma_j)^2}{2}} \qquad\textrm{and}\qquad \overline{\epsilon}=\sqrt{\frac{\sum_{ij}(\epsilon_i-\epsilon_j)^2}{2}}$$ $${\rm d}\lambda=\frac{\overline{\epsilon}}{\overline{\sigma}}$$

Anelastic deformation

Fig.: .

anelastic strain
damping capacity or internal friction
relaxation

measuring strain by diffraction