A Macroscopic-Level Hybrid Lattice Particle Modeling of Mode-I Crack Propagation in Ductile and Brittle Materials G. Wang 1 , A. Al-Ostaz, A. H.-D. Cheng Department of Civil Engineering, University of Mississippi P.R. Mantena Department of Mechanical Engineering, University of Mississippi Abstract This paper presents a numerical method, known as Hybrid Lattice Particle Modeling (HLPM), for the study of mode I crack formation and propagation in twodimensional geometry subject to a fixed-grip condition. The HLPM combines the strength of two numerical techniques, a Particle Model (PM) with a Lattice Model (LM), for the simulation of solid subject to dynamic loading, resulting in large deformation and possible fragmentation. A Lennard-Jones-type potential is employed to describe the nonlinear dynamic interaction among macroscopic-size particles. Crack initiation and propagation is investigated for materials with different Young’s modulus and tensile strength. It is demonstrated that crack patterns and propagation closely match the anticipated physical behavior of ductile and brittle materials. The outcome of this investigation is applied to the study of an infrastructure material retrofitted using a ductile coating to restrict the crack propagation.
Keywords:
fracture mechanics, cracks, lattice model, particle model, constitutive
relations. PACS: 68.45.kg, 61.43.Bn, 40.30.My, 46.30.Nz, 83.20._d, 83.80.Nb
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Corresponding author: Ge Wang, Email:
[email protected] Tel: + 662 915 5369, Fax: + 662 915 5523
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1. Introduction Dynamic fracture is a complex and multi-scale physical phenomenon. From the microscopic point of view, fracturing is a process that material becomes separated due to the successive breakage of atomic bonds. Since the intrinsic strength properties at atomic structure level are available, molecular dynamics (MD) analysis has been attempted at the nano scale. However, although MD simulation has benefitted from the rapid development of computing power and is becoming increasingly popular, the present state of computer technology is still far from being able to meet the demands of the macroscopic tasks. For example, we currently still cannot simulate a 1 × 1 × 1 cm 3 cubic copper body at atomic level because the body consists of 10 24 copper atoms, a number so large that no computer in the world can handle it. The second difficulty is its incapability in reaching the practical time scales. For instance, the laboratory dynamic fracture experiments generally last in microseconds (1 microsecond = 10−6 second), while the MD model time steps are typically in the nano ( 10−9 ) or pico ( 10−12 ) second range. As such, MD is limited to a
narrow range of solving nano- to micrometer scale problems. For this reason, a numerical tool for the modeling of dynamic fracture at macroscopic level is needed. Particle modeling (PM) is one of many branches of discrete element approaches. PM was originally proposed by Greenspan [1997]. In essence, PM can be regarded as an upscale MD, but applied to large length scale and time scale problems. However, Greenspan [1997] developed such model with in mind more fluid modeling than solid modeling. As a consequence, there was no direct linkage to the solid material properties, making PM an empirical model without demonstrated validation with real engineering problems. Lattice model, on the other hand, has a long history of success in modeling micromechanics solid problems [Askar, 1985; Noor, 1988; Ostoja-Starzewski, 2002, 2007]. Lattice model, however, does not have the flexibility of particle models, in which the particles can be subjected to very large deformation (displacement) and even fragmentation. Wang and Ostoja-Starzewski [2005] developed a modified PM, which is now renamed a hybrid lattice particle model (HLPM) [Wang, et al., 2008] in which the particle’s Lennard-Jones potential can be correctly mapped into lattice model spring
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constants, which are in turn matched with the material’s continuum-level elastic moduli and strength. Successful PM simulations have been achieved in predicting the fracture pattern of an epoxy plate with randomly distributed holes in tension [Ostoja-Starzewski & Wang, 2006], and predicting the dynamic failure of a polymeric material (nylon-6,6) due to the impact of an indenter [Wang, et al., 2008]. Recently there is another successful HLPM simulation accomplished in predicting the dynamic failure of vinyl ester with the same empirical conditions as in nylon-6,6 test [Wang, et al., 2008]. The comparison of HLPM result with the according experimental observation is illustrated in Figure 1(a, b). From Figure 1(a, b), it is seen that measured load peak happens around t ≈ 1.38 ms, and the measured deflection at the load peak is 2.60 mm, with the total impact energy equal to 0.7 J. The corresponding HLPM simulated result shows that the load peak happens around t ≈ 1.42 ms, and the deflection at load peak is 2.66 mm, and the total impact energy calculated is 0.76 J. Similarly to the HLPM simulation of nylon-6,6 [Wang, et al., 2008], although the simulated load profile is not exactly the same as the experiment, we observe similar characteristics, including the fluctuating profile with roughly the same period. The simulated peak load is also reasonable close to the experimental value. Hence we conclude that the HLPM simulation compares favorably with the experimental measurements. After gaining the confidence of HLPM from the above-mentioned fracture study cases, in this paper, we investigate the modeling capability of the HLPM on the initiation and propagation of mode I fracture in ductile and brittle materials employing a fixed-grip condition. As the mechanism for fracture formation and propagation in the lattice and the particle model is very different from that of the continuum mechanics based fracture mechanics model, it is not clear that the physical phenomenon of stable and unstable fracture growth can be correctly predicted. In short, the continuum model uses the stress intensity factor and energy release rate concepts for fracture creation and propagation; while the discrete HLPM uses the tensile/compression strength between bonds and the first principle based dynamic interaction among the particles. To have confidence in these models for simulating dynamic fracture problems, both numerical models need to be tested and validated. 3
In what follows, we first briefly introduce the HLPM algorithm. It is then applied to several two-dimensional dynamic fracture problems. Particularly, the stable and unstable fracture growth corresponding to the ductile and brittle materials can be faithfully reproduced, using only the physically interpreted Lennard-Jones potential constants. Finally, the HLPM is applied to the investigation of a functionally designed composite material of an infrastructure material coated with a ductile layer for the protection of fracture propagation. The ultimate application is aimed at the retrofitting of failing infrastructure.
2. Model Description The hybrid lattice particle model (HLPM)—also called lattice particle simulation, discrete modeling, or quasi-molecular modeling—is a dynamic simulation model that typically uses a relatively small number of particles of macroscopic sizes, representing solid and/or fluid mass. The particles’ location and velocity evolves according to the laws of Newtonian mechanics. The force interaction between particles is modeled after Wang & Ostoja-Starzewski [2005] (reason: conventional LM only works for linear considerations), which is matched up with the Young’s modulus and tensile strength of the material as well as energy and mass. Hence the HLPM has the flexibility of both the PM and the LM. The simplicity of the theory of HLPM has some advantage over other discrete element approaches. Ease of numerical implementation is the second advantage of HLPM. Since the physical size of each particle is ignored other than its equivalent mass, the algorithm of coding a HLPM computation is fairly easy. In principle, the distance of particle spacing can decrease to a few angstroms; in that case we recover a molecular dynamics like model. Hence the HLPM is fairly flexible in modeling physical phenomena of all sizes, limited only by the number of particles needed in the modeling (computational power). The theoretical derivation of non-thermal-based PM can be briefly reviewed as follows. In HLPM, the interaction force between neighboring (quasi-)particles, F , takes the same form as in MD:
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F =−
G H + r p rq
(1)
Here G, H, p and q are positive constants that can be determined by the equivalent mass, energy, elastic moduli, and tensile strengths of the quasi-particle system. The condition q > p ≥ 1 is required to ensure that a much stronger repulsive force than attractive force
when r → 0 , where r is the distance between two particles. Ashby & Jones (1980) presented a simple method for evaluating continuum-type Young’s modulus E (GPa) and tensile stress σ (r ) (MPa) of the material from the force field F (r ) , namely
E=
S0 r0
(2)
and
σ ( r ) = NF (r )
(3)
where S 0 = ( dF / dr ) r = r0 , and r0 is the equilibrium spacing between contiguous particles,
N is the number of bonds per unit area, equal to 1 / r02 . Tensile strength, σ TS , is given at the necking position rd , where dF (rd ) / dr = 0 , such that
σ TS = NF (rd )
(4)
Eqs. (2) and (3) imply that material’s properties, in terms of Young’s modulus E and tensile stress σ (r ) , are dependant upon the choice of (p, q) under a fixed r0 . Wang & Ostoja-Starzewski (2005) has concluded that the larger the (p, q), the larger the E and
σ TS (r ) , leading to a more brittle material. Just as in MD, the non-linear dynamical equation of motion for each particle Pi of the PM system is given by
r r ⎡⎛ G H ⎞ r ji ⎤ d 2 ri i i mi 2 = ∑ ⎢⎜ − p + q ⎟ ⎥ , dt rij ⎟⎠ rij ⎥⎦ ⎢⎣⎜⎝ rij
i≠ j
(5)
The leapfrog method, with second-order accuracy, is employed in the PM simulations. The safe time step is after the result by Hockney & Eastwood (1999):
Ω Δt
