Using this neighborhood radius ensures that theadjacent explosive nodes are not missed and that the detonation does not jump across a node that doesnot contain an explosive material. For any node X0in the neighborhood of node X, the detonation timefor node X is the minimum of its detonation time and the detonation time of a node X0plus the time ittakes the detonation to propagate from X0to X. This process continues until detonation times have beencalculated for all explosive nodes. The Huygen’s construction ensures that detonations propagate aroundobstacles and isolated regions of explosive material do not detonate.When detonation occurs and the detonation products are ideal gases, the pressure of the gas is set to onehalf the Chapman–Jouguet pressure (PC J ), which is given by (35) or specified by user input. To obtainthe correct energy in the ideal-gas reaction products from the detonating explosive, the initial pressuremust be set to one half the Chapman–Jouguet pressure. Once detonated, the gas expands adiabaticallyfrom this pressure. From (34), the pressure P at a time when the expansion is X is given byP = 12PC J V0V γ= 12PC J X−γ. (38)With P given by (38), the PFF between gas nodes for an expansion X is given by (31).The ideal gas law and (38) imply that the temperature T at a time when the expansion is X isT = TC J V0V γ−1= TC J X−γ+1, (39)where TC J is given by (37) or specified by user input. The ratio of molar specific heats in (38) and (39)is given by (36) or specified by user input.8. Examples of explosive loading of concrete structuresIn this section, we illustrate the capability of EMU to model explosive loading by discussing two exam-ples of explosive material detonating within concrete structures.For the first example, consider a 0.05-m thick spherical concrete shell with an inside radius 0.25m.The interior is filled with an explosive whose density is 1780 kg/m3 and detonation speed is 8590m/s.The explosive is detonated at time zero at the center of the sphere. Although this structure is not large,this problem was used to debug the detonation model.We simulated about 200µs of the explosion scenario. Using 2 processors, the simulation requiredabout 0.76 h. The grid spacing is 0.01m, and there are 133,185 nodes in the computational model.Figure 13 is a cross section view of the materials at time zero and the calculated detonation times inthe explosive. In the left graphic in this figure, the explosive is orange and the concrete is purple. Theconcrete is not reinforced. The detonation times in the right graphic vary from purple (less than 0.4µs)to orange (greater than 40µs). The calculated detonation times vary from 0 to about 35µs. The orangeapplies to the concrete, which is not an explosive material.Since peridynamic theory does not use stresses in its formulation, stress and pressure are not outputduring a simulation. Although temperature plays no role in the current models in EMU, a temperaturefield was included in the data model for future developments. The temperature field can be used toobserve the propagation of the detonation, since upon detonation, the temperature is given by (37) or user input and subsequently decreases during the adiabatic expansion as given by (39). Figure 14 showstemperatures calculated during the simulation of the explosion loading of the concrete shell.Figure 14 shows the detonation propagating from shortly after initiation (1.4µs) to 35µs when all theexplosive has detonated. The temperatures are in K.At detonation, the temperature of the detonation products is about 4400K. As the gas expands adia-batically, its temperature decreases as shown in the figure. The initial temperature of the concrete shell is293K. The temperature of the concrete will always be 293K since there is no heat transfer mechanism in the current version of EMU. No distortion of the concrete shell is evident during the times shown inFigure 14. Later times must be shown to observe expansion and damage to the shell.Figure 15 shows materials during the simulation. To observe voids and damage as the detonationpropagates, this figure contains only a 2-cm slice about the center of the sphere. This figure showsmaterials and indicates the progression of damage to the shell during the simulation. The graphics at13µs, 23µs, and 35µs show voids produced as the gas expands.The shell has not expanded much by the time the explosive is completely detonated at 35µs. However,by 43µs there is some expansion of the shell, more at 65µs, and still more at 65µs. At 43µs somegas has escaped through small cracks in the concrete. More fractures through the structure are evident at65µs and the concrete is completely fragmented by 86µs. Although the simulation extends to 200µs,well after the latest time shown in Figure 15, we do not show results from later times since these fragmentsjust continue to expand outward.For the second example, we consider a much larger structure. 极限荷载近场动力学建模英文文献和中文翻译(8):http://www.751com.cn/fanyi/lunwen_34840.html