For $1< p<\infty$ we prove an $L^{p}$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$. More precisely, let $\Omega \subset \mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that

\[ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \]

$P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$

$P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$

$P\times \nu =0$

$\partial \Omega$

$\nu$

$\partial \Omega$

$\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$

$P$

\begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*}

$P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$

$\Gamma \subseteq \partial \Omega$

The high-frequency vibration technology was introduced to relieve the quenched residual stress in the Cr12MoV steel based on the high-frequency vibration system that mainly consisted of an electromagnetic vibrator and an amplitude boost unit. The high-frequency vibratory stress relief (VSR) experiments were conducted to study the effectiveness of the high-frequency vibration technology. In addition, the high-frequency vibration plasticity model was developed based on the thermal activation theory to reveal the mechanism of the high-frequency VSR. The results show that the high-frequency VSR has good effects on eliminating residual stress, while the surface hardness for the Cr12MoV steel remains almost the same. Moreover, there are no changes in the grain size of the Cr12MoV steel during the high-frequency VSR, while the dislocation density for the Cr12MoV steel during the high-frequency VSR decreases by 27.21%. The decrease of dislocation density in the Cr12MoV steel is the essence of residual stress relaxation. The findings confirm the significant effects of high-frequency vibration on metal plasticity and provide a basis to understand the underlying mechanism of the high-frequency VSR.

A model has been developed to describe the rheology of austenite at high temperatures, characteristic from hot strip rolling of low-alloyed steels. It is based on the evolution of a single dislocation density through storage and strain rate-dependent dynamic recovery. The model gives good predictions of the stress-strain curves in a wide range of temperatures, strain rates and chemical compositions. It is now applied on industrial hot strip mills to predict the rolling loads of a wide range of industrial steels.

The peak profile shape analysis has been preferentially used in the evaluation of X-ray and synchrotron powder diffraction pattern. However, neutron diffraction facilities of new generation frequently offer the instrumental resolution high enough to efficiently study the effects of broadening of neutron diffraction profiles. The present paper describes the procedure for a detailed evaluation of Bragg peak shape based on the method of transformed model fitting (TMF) which has been recently developed particularly for the treatment of neutron diffraction profiles. Microstructure modeling is performed in the reciprocal space and the convolution of the model with the instrumental resolution curve is fitted to the profiles recorded in the diffraction experiment.

The modified Williamson–Hall and Warren–Averbach methods were used successfully for analyzing experimentally observed anisotropic X-ray diffraction line broadening and for determining reliable values of crystallite size and dislocation density in cerium oxide. The modified Williamson–Hall plot gives 22.3(2) nm for volume-weighted crystallite size, while the modified Warren–Averbach produces 18.0(2) nm for area-weighted grain size. The dislocation density and effective outer cut-off radius of dislocations obtained from the modified Warren–Averbach method are 1.8(3)×1015 m−2 and 15.5(1) nm, respectively.

The growth of precipitates in a deformed Cu–Ni–Si alloy with an aging treatment and the rearrangement of dislocations were investigated using small-angle X-ray scattering method and XRD line-profile analysis. The small-angle X-ray scattering method was used for characterizing the growth behavior of the precipitates. The results showed that the precipitates grew gradually to a few nanometers in radius when aged under the condition that the alloy exhibited a maximum of the hardness due to precipitation hardening. The growth rate rose from the onset of the overaging, where the hardness started to decrease. The line-profile analysis of copper-based alloy diffraction peaks using modified Williamson–Hall and modified Warren–Averbach procedures yielded a variation in the dislocation densities of the alloy as a function of the aging time. The dislocation density of the alloy before the aging treatment was estimated to be 1.7×1015 m−2 and its high value was held up to the peak-aging time. With the onset of the overaging, however, the dislocation density distinctly decreased by about 1 order of magnitude indicating that a large amount of the dislocations rearranged to release the alloy from the high dislocation-density state. The results suggest that the massive rearrangement of dislocations was accompanied with coarsening of the precipitates.

Let $F^{\rm p} \in {\rm GL}(3)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form ${\rm Curl}[{F^{\rm p}}]\cdot (F^{\rm p})^T$ applied to rotations controls the gradient in the sense that pointwise $ \forall R \in C^1(\mathbb{R}^3, {\rm SO}(3)): \Arrowvert {\rm Curl}[R] \cdot R^T \Arrowvert_{\mathbb{M}^{3\times3}}^2 \ge \frac{1}{2} \Arrowvert{\rm D}R\Arrowvert_{\mathbb{R}^{27}}^2$ .This result complements rigidity results[Friesecke, James and Müller, Comme Pure Appl. Math.55 (2002) 1461–1506; John, Comme Pure Appl. Math.14 (1961) 391–413; Reshetnyak, Siberian Math. J.8 (1967) 631–653)] as well as an associated linearized theorem saying that $ \forall A \in C^1(\mathbb{R}^3, \mathfrak{so}(3)): \Arrowvert {\rm Curl}[A]\Arrowvert_{\mathbb{M}^{3\times3}}^2 \ge \frac{1}{2} \Arrowvert{\rm D}A\Arrowvert_{\mathbb{R}^{27}}^2 = \Arrowvert\nabla{\rm axl}[A]\Arrowvert_{\mathbb{R}^9}^2$ .