Issue 50

A. Kakaliagos et alii, Frattura ed Integrità Strutturale, 50 (2019) 481-496; DOI: 10.3221/IGF-ESIS.50.40 490 Bombard Stress Powder Chamber Connection to Breech No Yes Bronze Material Strength Analytical FEM Yield Tensile Von Mises Stress [MPa] 180 211 219 220 240 Fraction to Yield 0.81 0.96 0.99 1.0 1.09 Table 6: Stresses in the gunpowder chamber. The computer model revealed that the maximum von Mises stress at gunpowder chamber connection to cannon breech was at 219 MPa (Fig.5b, Table 6). Moving towards the gun muzzle, distortions of bore internal diameter were progressively increasing, reaching the maximum value of 169 microns at 900 mm from cannon breech. This value corresponds to the maximum dilatation of the inner cylinder surface for a cylinder without connection to cannon breech (Fig.5c). It was concluded that the maximum von Mises stress at the connection of gunpowder chamber internal surface to cannon breech was close to gun material yield strength. Herein, ancient bronze material yield strength was considered at 220 MPa with 240 MPa tensile strength, reflecting bronze made up of copper, tin and small amounts of lead. Bronze elastic modulus was set at 100,000 MPa and Poisson ratio at 0.3. Realizing that under internal blast pressure with R set at 1,000 the von Mises combined stress is close to bronze yield limit, it was concluded that the rise of atmospheric pressure in the gun powder chamber due to powder ignition could be set at this value, producing ultimately a muzzle velocity at 216 m/sec (Eq.(1)). This result is in agreement to the overall firing capacity of smooth barrel guns, as they can deliver a muzzle velocity in the range of 0.3 to 2.0 Mach (Collins [9]). Powder chamber nonlinear response under internal pressure Aiming to approach nonlinear effects which potentially could imply a premature failure of the Gun, the powder chamber was considered as a thick cylinder under internal pressure, whereby, at the inner portion of the cylinder a plastic zone was formed (Fig.6). The plastic-elastic interface is defined by a cylindrical surface of radius r p , which effectively divides the cylinder cross sectional area into two separate adjacent zones, hence, cylindrical rings. In the inner ring, the material is in plastic state and at the outer cylindrical ring, the gun material behaves elastically (Liu [15]). The internal blast pressure q p which brings the plastic boundary interface at a certain plastic radius r p is defined with Eq.(11). Correspondingly, the radial displacement δ p at the boundary interface and the associated radial displacement Δ at the inner surface of the plastic ring are evaluated with Eqs.(12,13), respectively. The resulting radial displacement at any radial distance r from the elastic-plastic boundary zone till gunpowder chamber outer surface is defined with Eq.(14). In Eqs.(11-14), E is the bronze elastic modulus, σ y the corresponding yield stress, Poisson’s ratio equals 0.3, whereby, b is the internal powder chamber radius and the external radius respectively (Vullo [14]). The elastoplastic push over curve for the unrestrained powder chamber cylinder, hence, without connection to gun breech was evaluated using E=100,000 MPa, σ y =220 MPa, with a=772 mm and b=124mm (Curve A - Fig.7). It was identified that the yield pressure for this curve was at 123.67 MPa with corresponding radial displacement Δ=206 microns. In general, the stresses in the plastic zone, hence, the circumferential stress σ 1 and the radial stress σ 2 together with the axial stress σ 3 were defined with Eqs.(15-17) (Vullo [14]). As identified previously in Figs.4,5, the von Mises combined stress due to gunpowder chamber connection to gun breech was close to yield limit (Table 6). To capture this effect a point on the elastic branch of curve A (Fig.7) with coordinates q p =101,325 MPa and Δ=169 microns adequate to reflect yielding of gunpowder chamber connection to gun breech was identified (Fig.7). Realizing the fact that a premature local yielding occurs when the powder chamber is connected to gun breech (Figs.4,5), it was decided to produce a new curve, hence, curve B, by scaling down the “original” push over curve A by a load factor of 1.22 (Fig.7). This load factor reflects the ratio of yielding pressures for the powder chamber with and without connection to the breech, hence, 123,670/101,325=1.22. The developed curve B would use the same elastic path as curve A and enter yield phase at Δ=169 microns (Fig.7). This procedure was considered adequate for the purposes of present analysis, in order to derive first hand data for the nonlinear powder chamber response with connection to gun breech subjected to blast internal pressure. Considering the input from Fig.7 it was assumed that the radial displacement Δ occurs at a location approximately 900 mm from gun breech, as derived previously, for the elastic case, in Fig.5c. Push over curve B shall be used subsequently in order to assess the combined action of internal pressure and induced temperature due to powder ignition inside the chamber.