A slit web material has a curvilinear structure that is substantially longer than it is wide, with specially shaped slits, “tilde-slits”, which permit relatively easy expansion upon deployment. The cuts are arranged in continuous rows of tilde-slits. In any given row, the tilde-slits follow one-after-the-other in a curvilinear direction. Adjacent rows of slits are parallel to each other, but are offset from one another such that a line drawn between adjacent tilde-slits in adjacent rows is not perpendicular to the direction of the rows. The invention contemplates that the material dispenses from a continuous roll. If the slits are arranged in the longitudinal direction, then the web material expands in the width direction only upon deployment. However, if the slits are arranged such that the row direction is at some angle to the longitudinal direction, then the web material expands in both directions upon deployment. In this case, a special dispenser is not required, and the material expands in both directions as it is pulled off the roll prior to cutting a desired length of material from the roll.