-
AN APPROACH TO MODELING EXTREME LOADING OF STRUCTURES USINGPERIDYNAMICSWe address extreme loading of structures using peridynamics. The peridynamic model is a theory of con-tinuum mechanics that is formulated in terms of integro-differential equations without spatial derivatives.It is a nonlocal theory whose equations remain valid regardless of fractures or other discontinuities thatmay emerge in a body due to loading. We review peridynamic theory and its implementation in the EMUcomputer code. We consider extreme loadings on reinforced concrete structures by impacts from massiveobjects. Peridynamic theory has been extended to model composite materials, fluids, and explosives. 36410
- 上一篇：汽车行业引入先进的实时技术英文文献和中文翻译
- 下一篇：PLC6层立体车库设计英文文献和中文翻译
Wediscuss recent developments in peridynamic theory, including modeling gases as peridynamic materialsand the detonation model in EMU. We then consider explosive loading of concrete structures. Thiswork supports the conclusion that peridynamic theory is a physically reasonable and viable approach tomodeling extreme loading of structures.1. IntroductionIn this paper, we address the extreme impact and explosive loading of structures using peridynamic theory,or peridynamics. The present paper is the first publication of results and developments in peridynamictheory to model extreme loading of large structures. Peridynamics is a theory of continuum mechanicsthat is formulated in terms of integro-differential equations without spatial derivatives [Silling 2000].Conventional continuum-mechanics theory is formulated in terms of partial differential equations withspatial derivatives. However, these derivatives do not exist at discontinuities, and the conventional theorybreaks down as a spatial discontinuity develops. Peridynamics replaces the spatial derivatives of conven-tional continuum-mechanics theory with integrals and assumes that particles in a continuum interactacross a finite distance as in molecular dynamics. Therefore, an attractive feature of peridynamics is thatits equations remain valid regardless of any fractures or discontinuities that may emerge in a structuredue to loading. Implementation of peridynamics does not use stress intensity factors and does not requirea separate law that tells cracks when and where to grow. Cracks emerge spontaneously as a result of theequations of motion and material model. They grow in whatever direction is energetically favorable forgrowth.Although peridynamic theory is relatively new compared to conventional continuum-mechanics theory,its development is continuing and it has been applied to solve a number of problems. The method hasorigins in the work of Rogula [1982] and Kunin [1982] on nonlocal behavior in crystals. Silling [2000] expanded and developed the model as a general way to formulate initial-boundaryvalue problems in which the spontaneous occurrence of discontinuities is important. He proposed theterm “peridynamic” from the Greek roots for “near” and “force”. He demonstrated that the reformulatedapproach permits the solution of fracture problems using the same equations either on or off the cracksurface or crack tip without knowledge of the initial location of the crack. Since particles separated bya finite distance can interact with each other as in molecular dynamics, the theory is nonlocal. Althoughthere are other nonlocal theories of continuum mechanics, they do not attempt to eliminate the spatialderivatives.Silling [2003] described a numerical method for solving initial-value problems within peridynamictheory. Accuracy and numerical stability were addressed in [Silling and Askari 2005] This work alsoshowed how to include nonhomogeneous materials, such as fiber-reinforced composite materials, asperidynamic materials.Silling et al. [2003] applied peridynamics to study deformation of a peridynamic bar.
Their solutionexhibits features that are not present in the classical result but converges with the classical result in thelimit of short-range forces. Gerstle and Sau [2004] and Gerstle et al. [2005] applied peridynamic theory tothe quasistatic deformation of concrete. They illustrated the deformation and fracture of small plain con-crete samples and the effect of using rebar reinforcement. Silling and Bobaru [2005] applied peridynamictheory to study stretching and tearing of membranes. They also studied string-like structures, similar tolong molecules, that sustain tensile loads while interacting with each other through intermolecular andcontact forces. Their work is an early effort to apply peridynamic theory at the nanoscale.For several years, we have used peridynamic theory to investigate damage resulting from aircraft im-pacting buildings and other large structures made of reinforced concrete. In the present paper, we discussthis approach to modeling extreme loading of large structures. We also present some recent developmentsin peridynamics, which provide an approach to modeling loading from explosive detonations.Since peridynamic theory is relatively new, we first review peridynamic theory. In Section 2, we statethe fundamental integro-differential equation of peridynamic theory and introduce the pairwise forcefunction (PFF). The PFF is the force per unit volume squared between two particles. This interactionis called a bond. Constitutive properties of a material are given by specifying the PFF. Thus, in peridy-namics, material response, damage, and failure are determined at the bond level. Bond properties canbe derived from material properties including elastic modulus, yield properties, and fracture toughness[Silling 2000].After discussing some properties of the PFF, we discuss the material models that we use for extreme-loading analyses. These materials are called proportional, microelastic or microplastic materials. Forthese materials, failure of a bond occurs when the stretch exceeds a value called the critical stretch.In Section 3, we discuss the numerical method used to solve the integro-differential equations ofperidynamics in the EMU computer code. In this section, we also review accuracy and stability forexplicit time integration. EMU is the first computer code that is based on the peridynamic theory ofcontinuum mechanics.Section 4 contains a discussion of the EMU computer code. EMU is mesh free and Lagrangian. Ituses explicit time integration to advance the solution in time and executes on parallel computers.1 In Section 5, we consider impacts of aircraft into reinforced concrete structures. We first discuss asimulation of an experiment that was performed at Sandia National Laboratories. In the test, an F4-Phantom impacted a massive, essentially rigid reinforced concrete wall [Sugano et al. 1993]. Becauseof the strength of the target, there was no perforation of the target. In the second example, there isperforation. An aircraft impacts a cylindrical structure made of reinforced concrete. In this example, wedemonstrate the ability of peridynamics to model deformation and fracture leading to perforation. Theability to model fracture in perforation problems is important since a target starts to weaken long beforea penetrator gets through and the fracture growth process determines fragment properties.We next focus on explosive loading. Before introducing the EMU detonation model, we describemodeling gases as peridynamic materials in Section 6 since the detonation products are gases. How tomodel fluids as peridynamic materials is particularly important since the pairwise interactions throughthe linearized PFF imply a Poisson’s ratio of 1/4 [Silling 2000]. We describe the EMU detonation modelin Section 7.In Section 8, we consider explosive loading of a spherical shell and a large reinforced concrete structurecontaining water.Section 9 contains a summary and conclusions of this work. There, we also discuss current workand future directions in the development and applications of peridynamic theory and the EMU computercode.2. Peridynamic theory and material modelingConsider a peridynamic body that occupies a domain R as shown in Figure 1.In the peridynamic theory, the force density on a particle at point x and time t is assumed to be givenbyρ(x)d2dt2u(x, t) =ZZZ KRf u(x0, t)−u(x, t), x0− x dV0+ b(x, t), (1)where ρ(x) is the density at x, x and x0are points in the reference configuration, t is the time, u isthe displacement vector, R is the domain of the body, f is the pairwise force function, and b is the body-force density. (In Equation (1) and elsewhere, bold quantities are vectors unless stated otherwise.)The integral in Equation (1) is taken over the volume occupied by R. All functions are assumed to besufficiently well behaved that the integral exists.Equation (1) is the fundamental integro-differential equation of peridynamic theory. It is based onNewton’s second law for all points within the domain of analysis.