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Damage Tensor

Estimations of microcracking in ceramics can be made using the young's modulus in the damaged material relative to the undamaged material. A damage variable, $D$, may be defined by

$D=1-{\frac {E}{E_{0}}}$

Damage tensor

(A course on Damage Mechanics)

Compression spring mechanics

Max shear stress

$\tau _{max}={\frac {Tr}{J}}+{\frac {F}{A}}={\frac {8FD}{\pi d^{3}}}+{\frac {4F}{\pi d^{2}}}$

Spring index

$C={\frac {D}{d}}$

High stresses make springs with a spring index below 4 difficult to manufacture, while springs with an index above 12 are more likely to buckle.

Critical buckling curves

Direct shear factor

$K_{s}=1+{\frac {1}{2C}}$

Wahl factor

The Wahl factor incorporates shear stress as well as the stress concentration factor due to spring curvature

$K_{w}={\frac {4C-1}{4C-4}}+{\frac {0.615}{C}}$

Torsion spring mechanics

Angular deflection of helical torsion spring

$\theta ={\frac {ML}{EI}}$

Angular deflection in revolutions

$\theta '={\frac {10.2MDN_{a}}{d^{4}E}}$

Spring rate of torsional spring

$k={\frac {M}{\theta '}}\approxeq {\frac {d^{4}E}{10.2DN}}$

Number of active turns in helical torsion spring

$N_{a}=N_{b}+N_{e}$

$N_{e}={\frac {L_{1}+L_{2}}{3\pi D_{m}}}$

Diameter reductions

$D_{m}={\frac {D}{XXXXX}}$

Body length increase

$L=d(N_{b}+1+\theta )$

Stress

$\sigma ={\frac {32T}{\pi d^{3}}}K_{B}$

where $K_{B}$ is a bending stress correction factor

$K_{B,ID}={\frac {4C^{2}-C-1}{4C(C-1)}}$

$K_{B,ID}\approx {\frac {4C-1}{4C-4}}$

$K_{B,OD}\approx {\frac {4C+1}{4C+4}}$

http://springipedia.com/torsion-design-theory.asp

Approximation from modulus for many metals and ceramics (ashby)

$T_{m}\approx {\frac {E\Omega }{100k}}$

where $E$ is the modulus, $k$ is the Botlzman's constant, and $\Omega$ is the volume-per-atom in the structure.

An even more approximate estimate of the melting temperature from the modulus has been observed as the following for metals and ceramics:

$T_{m}\approx {\frac {E}{0.12}}$

where $E$ is the modulus in GPa and $T_{m}$ is the melting temperature in kelvin

Adhesive strength tests

A variety of tests are used to measure the strength of adhesive bonds under different loading conditions. Adhesives joints are often heavily influenced by temperature, moisture, and strain rate and therefore should be controlled/ measured during the tests. Adhesive failures can occur in a variety of different ways depending on the loading conditions and the strength of the bonds of the substrate. For example, when testing the tensile strength of an adhesive, the adhesive may debond from the substrate or fail within the adhesive itself if the bond between the adhesive and substrate are strong. The loading conditions throughout the sample are often not identical, making quantitative results difficult to achieve.

Tensile test

Adhesive tensile testing attempts to measure the tensile strength of the adhesive bond. The tensile strength of adhesives is often difficult to measure because the failure modes typically are not solely tensile (peeling often occurs at the edges of the samples). Results are therefore often qualitative and comparative. ASTM D897 is a common standard for adhesive tensile tests.

Lap shear test

The lap shear test is used to assess the strength of an adhesive bond under shear loading conditions. The output of the test is the stress required for failure of the adhesive bond. Lap shear tests require an adhesive bond between flat interfaces of the desired material. ASTM D2919 is a common standard for the lap shear test.

Adhesives joints are often heavily influenced by temperature and moisture and therefore should be controlled during lap shear tests.

Mechanical loss coefficient

Expand this and alter language based on multiple sources

The mechanical loss coefficient is the degree to which a material dissipates vibrational energy. It is a measure of the fractional energy dissipated for a load/unload cycle. Elastic energy is stored in a material subject to a stress/strain and dissipates when the stress/strain is relieved.

Stored elastic energy

$U=\int _{\sigma }^{\sigma _{max}}\sigma \,d\varepsilon ={\frac {1}{2}}{\frac {\sigma ^{2}}{E}}$

Dissipated energy

$\Delta U=\oint \sigma \,d\varepsilon$

Loss coefficient

$\eta ={\frac {\Delta U}{2\pi U}}$

Relationship to resonance factor, $Q$, for $\eta <0.01$

$\eta ={\frac {D}{2\pi }}={\frac {\Delta }{\pi }}=tan\,\delta ={\frac {1}{Q}}$

where

• $D=\Delta U/U$ = specific damping capacity
• $\Delta$ = log decrement
• $\delta$ = phase-lag between stress and strain

Material Damping

$tan\ \delta ={\frac {E''}{E'}}$

Crack growth measurement methods

Optical

Microscope or telescope is used to track crack tip. Surface of sample must be highly polished. This often prevents accurate measurements in corrosive environments The measurements are typically 2-dimensional and of a single despite cracks being a 3D objects which vary through the thickness of the material.

Ultrasonic

Measures the ultrasonic reflections from the crack. The reflectivity varies with surface roughness of the crack. Crack tip plasticity also affects the ultrasonic pulse attenuation and velocity. Ultrasonic probes often cannot be used in aggressive environments and is not very sensitive to small crack growths.

Compliance

Crack growth increases the specimen compliance. Calibration is required for many sample geometries which do not already have theoretical calibrations.

Potential drop

As the crack growths, the electrical resistance of the sample increases by reducing the size of the cross-sectional area. High current is typically needed in order to improve sensitivity. The system must be calibrated and usually uses a calibration polynomial of the form

${\frac {a}{w}}=A_{0}+A_{1}\left({\frac {V}{V_{0}}}\right)+A_{2}\left({\frac {V}{V_{0}}}\right)^{2}+A_{n}\left({\frac {V}{V_{0}}}\right)^{n}$

Doyleite

Doyleite, Al(OH)_3, is a form of aluminum hydroxide.

Doyleite has a triclinic structure with a space group of ${\bar {P}}1$ or P1.

Ductile phase crack bridging

To predict toughening provided by large scale cracking, weighted functions for the tractions across bridged crack faces have been developed. In this case, the crack shielding as a result of large-scale crack bridging can be defined by:

$\Delta K_{lsb}=\int _{L}\alpha \sigma (x)h(a,x)\ dx$

where $\sigma (x)$ is the traction function along the bridge zone and $h(a,x)$ is weight function for the traction. A weight function defined by Fett and Munz is shown below.

Relative robustness of materials

The robustness of a material to thermal shock, or thermal shock resistance is characterized by the thermal shock parameter:

$R_{\mathrm {T} }=k\,{\frac {\sigma _{\mathrm {adm} }}{\alpha \,E/(1-\nu )}}\,$,

where the further parameters is:

• $\nu$, the Poisson ratio

This formula can be simplified by introducing the brittle θ parameter:

$R_{\mathrm {T} }=(1-\nu )\,k\,\theta _{brittle}$,

This formula can be simplified as:

$R_{\mathrm {T} }=\,k\,\theta$,

where the theta parameter is:

$\theta ={\frac {\sigma _{\mathrm {adm} }}{E/(1-\nu )}}$,

notably, this means that in this case the elasticity modulus is not the Young modulus, but rather:

${\frac {E}{1-\nu }}$,

Thermal shock parameter in the physics of solid-state lasers

The laser gain medium generates heat. This heat is drained through the heat sink. The transfer of heat occurs at a certain temperature gradient. The non-uniform thermal expansion of a bulk material causes the stress and tension, which may break the device even at a slow change of temperature. (for example, continuous-wave operation). This phenomenon is also called thermal shock. The robustness of a laser material to the thermal shock is characterized by the thermal shock parameter. (see above)

Roughly, at the efficient operation of laser, the power $P_{\mathrm {h} }$ of heat generated in the gain medium is proportional to the output power $P_{\mathrm {s} }$ of the laser, and the coefficient $q$ of proportionality can be interpreted as heat generation parameter; then, $P_{\mathrm {h} }=qP_{\mathrm {s} }.$ The heat generation parameter is basically determined by the quantum defect of the laser action, and one can estimate $q=1-\omega _{\mathrm {s} }/\omega _{\mathrm {p} }$, where $\omega _{\mathrm {p} }$ and $\omega _{\mathrm {s} }$ are frequency of the pump and that of the lasing.

Then, for the layer of the gain medium placed at the heat sink, the maximal power can be estimated as

$P_{\mathrm {s,max} }=3{\frac {R_{\mathrm {T} }}{q}}{\frac {L^{2}}{h}},\,$

where $h$ is thickness of the layer and $L$ is the transversal size. This estimate assumes the unilateral heat drain, as it takes place in the active mirrors. For the double-side sink, the coefficient 4 should be applied.

Thermal loading

The estimate above is not the only parameter which determines the limit of overheating of a gain medium. The maximal raise $\Delta T$ of temperature, at which the medium still can efficiently lase, is also the important property of the laser material. This overheating limits the maximal power with an estimate

$P_{\mathrm {s,max} }=2{\frac {k\Delta T}{q}}{\frac {L^{2}}{h}}\,$

Combination of the two estimates above of the maximal power gives the estimate

$P_{\mathrm {s,max} }=R{\frac {L^{2}}{h}}\,$

where

$R={\textrm {min}}\left\{{\begin{array}{c}3R_{\mathrm {T} }/q\\2k\Delta T/q\end{array}}\right.$

Estimates of maximal value of loss $\beta$, at which desirable output power $P$ is still available in a single disk laser, versus normalized power $s={\frac {\omega _{\rm {p}}}{\omega _{\rm {s}}}}{\frac {PQ}{R^{2}}}$, and experimental data (circles)

is thermal loading; parameter, which is important property of the laser material. The thermal loading, saturation intensity $Q$ and the loss $\beta$ determine the limit of power scaling of the disk lasers . Roughly, the maximal power at the optimised sizes $L$ and $h$, is of order of $P={\frac {R^{2}}{Q\beta ^{3}}}$. This estimate is very sensitive to the loss $\beta$. However, the same expression can be interpreted as a robust estimate of the upper bound of the loss $~\beta ~$ required for the desired output power $P$:

$~\beta _{\mathrm {max} }=\left({\frac {R^{2}}{PQ}}\right)^{\frac {1}{3}}.$

All the disk lasers reported work at the round-trip loss below this estimate. The thermal shock parameter and the loading depend on the temperature of the heat sink. Certain hopes are related with a laser, operating at cryogenic temperatures. The corresponding Increase of the thermal shock parameter would allow to soften requirements for the round-trip loss of the disk laser at the power scaling.

Crack Tip Stress Field

The crack tip stress field in plane stress or plane strain conditions in a homogeneous, isotropic elastic solid, can be represented as:

$\sigma _{ij}={\frac {K_{I}}{\sqrt {2\pi r}}}\sigma _{ij}^{I}(\theta )+{\frac {K_{II}}{\sqrt {2\pi r}}}\sigma _{ij}^{II}(\theta )+T\delta _{i1}\delta _{j1}$

where $\delta _{ij}$ is the Kronecker delta and $r$ and $\theta$ are polar coordinated centered on the crack tip.