Stress-strain relationships in yarns subjected to rapid impact loading: 2. Breaking velocities, strain energies, and theory neglecting wave propagation

Abstract

The behavior of a yarn specimen fastened at one end t o a h ead mass and at t he o ther end t o a small tail mass is anal yzed for longitudinal impact of t he specimen at t he head. Th e a nalys is leads to a basic formula for " limi ting breaking velocity," which is a cha racteristic property of t he ma terial a nd independent of t he dim ensions of t he sp ecimen. A simple pTo cedure is described for ob taining its value. The values for cotto n, ny lo n, and undra wn ny lon yarns tes ted at room temperat ure are fou nd to be 130, 228, a nd 550 m eters per seco nd, respective ly. The practical app li catio n of t he limit ing break ing ve loci ty to safety enginee ri ng is shown by a n exampl e on safety lines. FonTIul as aTe d eriv ed for co mput ing energy to a n.v stra in und er impact loadin g, a nd t he co mpu ted values are found to a gr ee wi t h t hose ob tain ed fr om t he a rea under the im pact load-extens ion curves.

Stress-strain relationships in yarns subjected to rapid impact loading: 5. Wave propagation in long textile yarns impacted transversely

“…This is the case considered by von Karman, and to which his solution, eq (1), (2), and (3), is applicable. A combination of olutions for the preceding special cases gives the solution to the problem in which a long filament initially at rest along an axis in the x' direction is impacted transversely at x= o with velocity V. The boundary conditions that must be satisfied for the x>O half of the filament are ~=O and 1] = Vt at x=O, and ~= 1] = 0 at X= co.…”

“…The innermost, called the plastic wave by von Karman, propagates at velocity (2) In the region between the elastic-and the pIa tic-wave fronts the train increases to Ep , the maximum value obtained a a result of the impact. The strain remain constant at this value in the wake of the plastic-wave front.…”

“…According to von Karman, the strain incr eases gradually in th e region b etween two wave fronts, an "elasLic wave" fron t 2 propagatin g at velocity (1) and a " plastic wave" fron t 2 propagating more slowly at velocity (2) In eq (1) and (2), M is the mass p er unit length of the unstrained filament material, and dTjdf. is the slope of the tension-strain curve, evaluated at zero strain in eq (1) and at strain f. p in eq (2). In the wake of th e plastic-wave front the strain in th e filament has the constant valu e €1J" The value of f. p depends upon the speed at which the filament is impacted longitudinally.…”

“…The velocity of each wavelet is proportional to the square root of the slope of the tension-strain curve, hence in order that the wavelets do not overtake each other, the slope of the tension-strain curve must decrease , as e increases (be concave downward ). The final wavelet in the train, or plastic wave, propagates with velocity Gp given by eq (2). The strain in the wake of the plastic-wave front has the constant value ep • Filament material in the wake of the plastic-wave front flows inwarsl at the constant velocity W given by eq (3).…”

“…T he b ehavior of textile yarns hold in clamps 40 to 60 cm apart and impacted transversely was demon tra ted and di scu sed in th e fom th pap er of Lhi s serie [1,2,3,4). 1 The pr esent paper deals wi th the theor:r of transver se-wave propagation in :ram s of infini te length wher e ther e are no clamps to r eflect t he waves.…”