The mobility of soft tissue can cause inaccurate needle insertions. Particularly in steering applications that employ thin and flexible needles, large deviations can occur between pre-operative images of the patient, from which a procedure is planned, and the intra-operative scene, where a procedure is executed. Although many approaches for reducing tissue motion focus on external constraining or manipulation, little attention has been paid to the way the needle is inserted and actuated within soft tissue. Using our biologically inspired steerable needle, we present a method of reducing the disruptiveness of insertions by mimicking the burrowing mechanism of ovipositing wasps. Internal displacements and strains in three dimensions within a soft tissue phantom are measured at the needle interface, using a scanning laser-based image correlation technique. Compared to a conventional insertion method with an equally sized needle, overall displacements and strains in the needle vicinity are reduced by 30% and 41%, respectively. The results show that, for a given net speed, needle insertion can be made significantly less disruptive with respect to its surroundings by employing our biologically inspired solution. This will have significant impact on both the safety and targeting accuracy of percutaneous interventions along both straight and curved trajectories.