The radius of the liner is 18.288m. There is a 1.524m annulusbetween the concrete and the steel liner.Figure 10 shows results from the EMU simulation of this scenario. In this figure, the concrete is yellowand green and the rebar is orange, and the remaining colors are various parts of the aircraft including thejet fuel. The time simulated is about 0.64 s. The simulation took about 26.5 h using 64 processors, andthe time steps varied from 13µs to 20µs. The grid spacing is 0.229m, and there are 580,624 nodes inthe computational model.Figure 10 shows that before 0.17 s, the concrete wall is breached and the steel liner is deformedconsiderably. The breach in the concrete wall occurred soon after the hard parts of the aircraft, frontlanding gear, and center fuel tank (color green) impact the outer wall. By 0.64 s most of the aircraft isinside the structure. Only the tail end, parts of the wings, and some fuel from the wing tanks are notinside the structure. The center fuel tank remains intact.Figure 11 shows the progression of internal damage to the structure starting at 0.10 s. This figure showsonly the front half of the structure so that a breach will not be obscured by the back of the structure. At0.10 s, fracturing and cracks in the concrete are evident and over 99% of the bonds are broken near theimpact location. Much failure occurs along the rebar. However, there is cracking in the concrete alongdiagonals beginning where rebars intersect. Damage appears to occur first in the rebar, and the rebarseems to fail completely before the concrete fails.Although there is much damage at 0.10 s, the concrete and liner are not penetrated. There are a fewsmall holes in the structure near the impact location. Breech of the concrete wall and liner occur before0.18 s as the concrete continues to fracture and the liner is torn. After 0.34 s, the breach gets wider as parts of the wing penetrate and enter the structure. At the end of the simulation, over half of the frontpart of the structure is damaged.The results in Figure 11 show the power of the peridynamics method in problems where deformationand fracture are expected. This figure shows the dynamic evolution of fracture with cracks emergingspontaneously as a result of the equations of motion and material model, and growing in whateverdirection is energetically favorable for growth.6. Gases as peridynamic materialsSince the detonation products in an explosion are gases, we must determine how to model gases as bond-based, peridynamic materials. In this section, we develop a general expression for the PFF of a gas. Thisexpression depends on how the internal energy per unit volume of the gas changes with the expansionof the gas, X, whereX = vv0= ρ0ρ. (19)In (19), v and v0 are the deformed and reference specific volumes of the gas, respectively, and ρ and ρ0are the deformed and reference densities, respectively. Implementation of a gas model requires approxi-mating the expansion at a node from the undeformed and deformed bond lengths between this node andall gas nodes within its horizon. We provide the expression for the expansion that is presently used inEMU and complete this discussion of gases as peridynamic materials by developing the PFF for a gas.To derive a general expression for the PFF of a gas, consider (9) and (10). From (9), the micropotential,w, may be written asw(x, t) =η(x,t) Zη0f (η, ξ ) • dη, (20)where η0 is some fiducial state of stretch for evaluation of the micropotential. Since the integral (10)vanishes outside the horizon of x, H(x), (10) may be written using (20) asWu(x, t) = 12ZZZ KH(x)" η(x0,t) Zη0f (η, ξ ) • dη#dVξ . (21)Let all the bonds be held fixed except for bond k at a given value of ξ . Then, for an incremental stretchdpk in bond k, (21) becomes dW, wheredW = 12f • dη1Vξ = 12f • d(η+ξ )1Vξ = 12f kdpk1Vk . (22)In (22), fk is the magnitude of the bond force per unit volume squared in bond k due to this incrementalstretch dpk in this bond, and 1Vξ and 1Vk are volume elements associated with bond k. The later twoequalities follow since ξ is fixed under the stretch and f is parallel to η+ξ as stated in (6).The quantity fk in (22) is the PFF for a gas. This quantity can be expressed in terms of the expansionX as follows. Changes in the energy per unit volume, dW, result from the stretch dpk . Therefore, using the definitionof dW and the chain rule, (22) may be written asdW = ∂W∂pkdpk = dWd X∂ X∂pkdpk (23)since W depends on pk through its dependence on X.Equating (22) and (23) and solving for fk yields a general expression for the PFF of a gasfk = 21VkdWd X∂ X∂pk. (24)The PFF can be obtained from (24) once the energy per unit volume, W, is known as a function of theexpansion, X, and X is known as a function of the incremental stretches for all gas bonds within thehorizon of a node.There are many ways to approximate the expansion X. 极限荷载近场动力学建模英文文献和中文翻译(6):http://www.751com.cn/fanyi/lunwen_34840.html