Measurement of the E-modulus without the use of the extensometer is only possible by measuring the force and deflection on the circular plate of this material
Use strain gages long one on four sides, or use nondestructive testing (Impact Echo) to find the dynamics modmulish then use the static elevator modlua to be 0.8 of it, or maybe use fiber optic sensors...just some thoughts 😊
You can replace the extensometer by a CCD camera, if you have the equipment. You get the strain value and you can correlate it to the string to determine the young modulus