Do modern jet engines monitor and record exposure to particles of dust or sand?
As of now, there are no commercially available engine particle ingestion sensors, though a few are under development. One of the companies involved in development lays out the reasons for the absence of any such sensors:


  As of today, no commercial off-the-shelf particulate sensor is in use because of the special design and difficult requirements for operation and performance in this harsh environment, according to Dr. Hai Lin of Hal Technology in Fontana, California.  


There are multiple sensors under development like the Dual Optical Embedded Dust Sensor which is used to determine the particle ingestion by the AGT1500 engine in M1 Abrams, where dust and sand is a bigger problem, if anything.

All of these sensors work on a similar principle- they measure the scattering of light by the particles using a optical sensor.


  The sensor detects light scattering from moving particles as optical pulsed signals whose amplitudes are proportional to size and whose number is proportional to concentration. This sensor technology can determine particle size, distribution, concentration and further be extended for composition. 


In future most of these systems are expected to available commercially, as 


  While continuing to advance this technology for particulate composition determination, Hal Technology is pursuing a licensing agreement or partnership with sensor or engine manufacturers to have the sensor integrated into engine heath management systems of the future.