In a laser microwelding process, a pulsed Nd laser is used to weld two thin titanium sheets,

each with a thickness of $100$ m$.$ The laser operates with pulse shaping, where the power

increases linearly during the first half of the pulse and decreases linearly during the second

half. The total pulse duration is $10$ ms$.$

The peak power of the laser pulse is $1000$ W$,$ and the base power $($at the start and end of the

pulse$)$ is $100$ W$.$ The thermal conductivity of titanium is $22$ W$/$m$$K$,$ and its specific heat

capacity is $520$ J$/$kg$$K$.$ The density of titanium is $4500$ kg$/$m$3.$

Assume the energy absorption efficiency of the laser is $30\%,$ and the weld area has a circular

cross$-$section with a diameter of $0\mathrm{.}5$ mm$.$

$1.$ Calculate the total energy delivered to the weld during one laser pulse.

$2.$ Estimate the temperature rise in the titanium sheets during one pulse, assuming all the

absorbed energy is used to raise the temperature.