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Table of contents

High strain rate and impact experiments. Chapter 33 in Handbook of Experimental Solid Mechanics. A detailed consideration of the state of the art for each of these types of experiments, based on examples from the literature, follows. Most of the inelastic and particularly the plastic deformations due to impacts at rapid velocities occur at high strain rates. The deformations may lead to large strains and high temperatures. Some experimental techniques have been developed to measure material properties at high strain rates.

Here, the committee considers experimental techniques that develop controlled high rates of deformation in the bulk of the specimen rather than techniques that develop high strain rates just behind a propagating wave front. The main experimental techniques for measuring the rate-dependent properties of various materials are described in Figure the stress states developed by the various techniques may not necessarily be identical. One outstanding recent review of these methods is that of Field et al.

The now-classical experimental technique in the high-strain-rate domain is the Kolsky bar, or split-Hopkinson pressure bar, experiment 30 for determining the mechanical properties of various materials e. This technique is now in use throughout the world. Since the fundamental concept involved in this technique was developed by Kolsky, 34 the term Kolsky bar will be used here.

Kolsky bar experiments may include compression, tension, torsion, or combinations of all of these. FIGURE Experimental techniques used for the development of controlled high-strain-rate deformations in materials. The Kolsky bar consists of two long bars, called the input and output bars, that are designed to remain elastic throughout the test. These bars sandwich a small specimen, usually cylindrical, which is expected to develop inelastic deformations.

The bars are typically made of high-strength steels, such as maraging steel, with a very high yield strength and substantial toughness. Other bar materials that have been used include T6 aluminum, magnesium alloys and poly methyl methacrylate for testing very soft materials , and tungsten carbide for testing ceramics. One end of the input bar is impacted by a projectile made of a material identical to that of the bars; the resulting compressive pulse propagates down the input bar to the specimen.

Several reverberations of the loading wave occur within the specimen; a transmitted pulse is sent into the output bar and a reflected pulse is sent back into the input bar. The strain gage signals are typically measured using high-speed digital oscilloscopes with at least bit accuracy and preferably with differential inputs to reduce noise. Many extensions and modifications to the traditional Kolsky bar system have been developed over the last five.

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Review of experimental techniques for high rate deformation and shock studies. International Journal of Impact Engineering 30 7: Material characterization at high strain-rates. Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta-W alloys. Acta Materialia 45 3: Evaluation of ceramic specimen geometries used in a split Hopkinson pressure bar.

Strain rate sensitivity of polymers in compression from low to high strain rates. An investigation of the mechanical properties of materials at very high rates of loading. Proceedings of the Physical Society: Section B 62 Classic split-Hopkinson pressure bar testing. Mechanical Testing and Evaluation. FIGURE High-strain-rate behavior of T6 aluminum determined through servohydraulic testing, compression and torsional Kolsky bars, and high-strain-rate pressure-shear plate impact. Most of these are listed in Table I of a review 36 by Field et al.

Both computational and experimental results have shown that this extended capability can be attained not only without violating the requirements for valid high-rate testing but also while improving the precision and accuracy of the experimental results. The experiment involves the impact of plates that are flat and parallel but inclined relative to their direction of approach.

The specimen is a very thin say, m , very flat plate of the material being investigated. The target is positioned in a special fixture, known as the target holder, within an evacuated chamber. The flyer and the anvil plates are aligned before impact using an optical technique. Rotation of the projectile is prevented by a key in the projectile, which glides within a matching keyway machined in the barrel. At impact, plane longitudinal compressive and transverse shear waves are generated in the specimen and the target plate propagating at the longitudinal wave speed c l and the shear wave speed c s.

These waves reverberate within the specimen, causing the normal stress and the shear stress to. A rigorous assessment of the benefits of miniaturization in the Kolsky bar system. Experimental Mechanics 44 5: Pressure-shear plate impact testing. The specimen thickness is greatly exaggerated for clarity.

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As Yadev notes, information on the stress levels sustained by the specimen material is carried by the normal and transverse waves propagating into the target plate. The whole experiment is over before any unloading waves from the periphery of the plates reach the point of observation. Thus only plane waves are involved, and a one-dimensional analysis is not only sufficient but also rigorously correct. Like most plate impact experiments, this is a uniaxial strain experiment in that no transverse normal strains can occur during the time of interest.

Measurements of particle velocities at the free surface of the target plate are made using laser interferometry off a diffraction grating that is photodeposited onto the rear surface. The superimposed hydrostatic pressures that can be exerted during the high-strain-rate, pressure-shear plate impact experiment may be as high as 10 GPa, depending on the impedances of the flyer and target plates and the projectile velocity.

The superimposed hydrostatic pressures must always be remembered when comparing high-strain-rate, pressure-shear plate impact data with data obtained using the other techniques shown in Figure , since all of the other techniques can generate essentially uniaxial stress states, typically corresponding to low hydrostatic pressures. In particular, while the effect of pressure on the flow stress of most metals is negligible in comparison with the effect of strain rate, the effect of pressure on the strength of polymers, ceramics, glasses, and amorphous materials may be substantial, even in comparison with the effect of strain rate.

Experiments designed to study the propagation of large-amplitude stress waves within materials constitute a very broad class of impact experiments. The interest here is in experiments that examine the interactions of waves with materials, particularly exciting inelastic modes such as plasticity, cracking, or other kinds of damage. In contrast to the experiments in the preceding section, the experiments in this section all generate strain rates and stress states that vary in both space and time, and the wave propagation is fundamentally dispersive i.

High-temperature pressure-shear plate impact experiments on ofhc copper. Journal of the Mechanics and Physics of Solids 46 A new experimental technique. International Journal of Solids and Structures 37 The plate impact experiments are far more common since they can explore a wider range of the phenomena that arise in impact events. In the timescales associated with ultrahigh-strain-rate experiments, uniaxial strain conditions are sampled.

Such results are difficult to compare with results obtained at high and very high strain rates typically obtained with uniaxial stress experiments , particularly if the material has pressure-dependent properties. The strain rates developed in large-amplitude wave propagation experiments, where shocks are developed, can be on the order of 10 6 to 10 8 s —1 , but they only exist for a short time behind a propagating wave front, and because of inelastic dissipation, as well as reflections from surfaces, the strain rates will vary with position in the impact plate.

The temperatures behind the wave front may be substantial and must be accounted for as well. Comparisons of material properties estimated using wave propagation experiments and high-strain-rate experiments the distinction made in this chapter can therefore require careful parsing of experimental conditions. A shock wave generated during a plate-impact experiment propagates at a shock speed U S that varies with the particle velocity u p , and it is commonly observed that these two variables are related linearly or nearly so: Large numbers of experiments have been performed to determine these parameters in various materials.

A summary of such data is presented in Meyers 45 ; another useful reference is Gray. Experimental details are often emphasized in these conference proceedings. The main experimental issues associated with shock wave plate impact experiments are 1 the development of gun-launching facilities at the appropriate velocities; 2 the accurate measurement of projectile velocity; 3 the measurement of the stress state within the specimen, typically through the use of stress gauges; and 4 the measurement of the particle velocities in the targets, typically through the use of interferometers such as the velocity interferometry system for any reflector VISAR.

The gun-launch facilities associated with shock physics experiments are typically extremely specialized facilities run by a small number of companies and the national laboratories, and extraordinary precautions must be taken to ensure safety. Most of these facilities offer gas guns, light gas guns, or powder guns; for the higher velocities, multistage guns are typically required.

Since kinetic energy increases with the square of the velocity, reaching higher velocities typically requires the use of lower-mass sabots and flyers. The typical results obtained from shock experiments include the determination of the Hugoniot elastic limit this is essentially a uniaxial strain version of the traditional elastic limit , the determination of the Hugoniot curve itself, and depending on the instrumentation the determination of the hydrostat and possibly of the shear strength.

Note that there are independent methods such as the diamond anvil cell for measuring hydrostat response. Another problem with shock experiments is that it is quite difficult, although not impossible, to do recovery experiments. Additionally, it is difficult to separate strength and failure from overall thermodynamic response lateral gages can be introduced but have their own attendant issues , because diagnostics typically provide only the longitudinal stress, which is a combination of the spherical and deviatoric stresses.

A sophisticated capability exists within the national laboratories for analyzing the results of shock wave experiments in fine detail, and much of the current understanding of the high-pressure behavior of materials comes from such experiments. These methods are generally incapable of determining the rate-dependent shear strength of materials, except under very special conditions. This leads to one of the primary difficulties in understanding material behavior under the extreme conditions developed in the armor problem: The experimental techniques available are generally either most sensitive to material behavior under high pressure or under high strain rate, but they rarely provide accurate information under combined high pressure and ultrahigh strain rate.

As the shock is passing a material point, the stress increases very. Bar impact tests on alumina AD AIP Conference Processing volume Dynamic Behavior of Materials. Shock wave testing of ductile materials, Pp. Mechanics of energy transfer and failure of ductile microscale beams subjected to dynamic loading. Journal of Mechanics and Physics of Solids 58 8: Once the shock has passed, the material is under a state of high pressure but at relatively low strain rates since there is no relative particle motion.

Upon release, however, the material again experiences high strain rates, but now starting at high pressures. Further, under uniaxial strain conditions, self-confinement conditions prevail. Efforts to bridge this gap are recommended. A great deal can be learned about material behavior the constitutive law rather than the failure process by analyzing the results of a suite of experiments that include all of those discussed above, combined with a microscopic analysis of the deformation mechanisms active within the material before loading and postmortem.

This capability would be significantly enhanced by in situ determination of the active mechanisms, an essential requirement for ceramics and polymers. There are individual exceptions where multiscale, multiphysics methods are being developed, but this is not a concerted effort that will furnish the tools needed to address the future of protection material systems. There have been significant advances in the models and algorithms addressing the limitations of existing armor codes, which must be integrated into existing or completely new codes to achieve the next level of understanding of armor material response.

Investigating Dynamic Failure Processes. Turning now to the experimental study of dynamic failure processes, many are active mechanisms within the impact events, as discussed earlier. In a broad sense, failure processes consist of brittle fracture at various length scales, void growth associated with ductile fracture, void collapse, the development of adiabatic shear bands, and a variety of structural instabilities such as necking.

For the case of the dynamic brittle fracture process in ceramics, an example of the state of the art in the understanding of the dynamic failure process is that provided in Paliwal et al. High-speed photographs were correlated in time with stress measurements in the specimen Figure In the fully transparent samples, dynamic activation, growth, and coalescence of cracks and resulting damage zones from spatially separated internal defects were directly observed and correlated with the macroscopic loss of load-carrying capacity and the ultimate catastrophic failure of this material.

Identical experiments on the coated material showed only the dynamic progressive failure on the specimen surface, not the origin of the failure images at the internal defects. Therefore, the actual failure mode differs from what is suggested by the photographs of the opaque ceramic undergoing dynamic compression. By means of high-speed photographs on transparent AlON, these authors obtained real-time data on the damage kinetics, which suggest that the cause of the final failure for AlON under dynamic loading was the formation of a damage zone that propagates unstably, not splitting parallel to the loading axis.

This is an example of the value of using a model material—the transparency of the AlON allows determination of internal failure processes that would be otherwise inaccessible—and demonstrates that modeling approaches that only generate axial splitting modes do not properly describe the dynamic failure processes, even though they may capture surface features.

Nor would the postmortem examination of fragments in such experiments provide this critical information on dynamic failure processes in the material. An excellent example of an experimental technique that provides critical information on the dynamic failure process of void growth is provided by the work of Chhabildas and co-workers, who use a line VISAR to examine the process of spallation. This spall strength is used in a number of armor design approaches as a measure of when the material will fail under hydrostatic tension. Their results show Figure that there is a stochastic character to the spall process, and that the spall strength of the material is not a single number but a result of the specific microstructural defect distribution within the target.

This latter idea was articulated in Wright and Molinari 51 and Wright and Ramesh 52 in terms of a model of. Direct observation of the dynamic compressive failure of a transparent polycrystalline ceramic AlON. Journal of the American Ceramic Society 89 7: Determination and interpretation of statistics of spatially resolved waveforms in spalled tantalum from 7 to 13 GPa. International Journal of Plasticity 25 4: A physical model for nucleation and early growth of voids in ductile materials under dynamic loading. Journal of the Mechanics and Physics of Solids 53 7: Dynamic void nucleation and growth in solids: A self-consistent statistical theory.

Journal of the Mechanics and Physics of Solids56 2: The stress-time and damage-time curve at the bottom corresponds to the photographs at the top the times at which each photograph is taken are shown through the matched numbers on the stress-time plot. Note how the damage begins at internal flaws in the material; subsequent damage interactions lead to cooperative growth of a damage front.

The experiment on the projectile impact of an aluminum plate described earlier in this chapter is an excellent example of an experiment designed to investigate impact phenomenology. Such experiments are highly instrumented and highly controlled versions of the real impact and are valuable for determining the sometimes unexpected couplings that can occur between material properties, failure processes, and system behavior.

Such experiments, if they are designed to promote a detailed and specific understanding of the impact phenomenology, are particularly useful when performed on model material systems. A large fraction of the impact phenomenology experiments described in the literature has the different objective of providing a broad and generalized evaluation of the performance of the material system under a specific threat.

While these performance evaluation experiments have a critical role to play in the evaluation of armor systems, it is difficult to use them to extract guidelines for the design of improved armor systems. The combination of a very experienced investigator and a large database of experimental data can be a powerful tool in armor development, but this should not be the primary approach to armor development.

Several recent developments in experimental methods hold great promise for addressing complex protection materials problems. These include improved temporal and spatial resolution, the development of high-speed cameras and associated triggering electronics, and the coupling of com-. The critical recent development for two failure processes is the in situ real-time observation of the active mechanism. Postmortem evaluations require the assumption of a mechanism and then the use of circumstantial evidence to verify the assumed mechanism, which can lead to erroneous conclusions in impact problems.

A variety of very sophisticated experimental techniques has been developed for addressing major parts of the problem. However, some important technical gaps remain. An example is the characterization of the high-strain-rate response of brittle armor materials such as ceramics and glasses under combinations of high pressure and shear representative of ballistic penetration. Currently available experimental techniques are generally most sensitive to material behavior under either high pressure or high strain rate, but they rarely provide accurate information under combined high pressure and ultrahigh strain rate.

However, some important technical gaps remain, including the following:. Such has been the case in protection materials technology, where, as discussed in this chapter, significant effort has been devoted in the last 50 years to developing the basic science, algorithms, simulation software, and hardware infrastructure to meet this goal. The committee suggests directions in which these technologies could be further developed so as to have a bigger impact on protection materials. A path forward is recommended to advance the simulation-aided design of the next generation of protection materials and systems.

Background and State of the Art. The fundamental macroscopic properties of materials influencing armor performance—high strength, elastic stiffness, and ductility—are well known even if the manner in which they combine to create the most effective performance is still far from certain. A clear avenue for improving armor performance is thus to continue the quest for ever lighter, stronger, and stiffer yet more malleable materials Chapter 5.

The correlation of material properties with armor performance is usually difficult to establish exactly, as performance depends on specific details of the threat and the armor system and on their complex interactions. For example, for a given threat and a certain mass of protective material with predetermined macroscopic properties, armor performance is found to be strongly dependent on the layout of the material in the armor system—in other words, the material system.

The foregoing suggests that details of the mechanical response of the material system such as wave propagation, localized plastic deformation and fracture, crack propagation, and others play an important role in determining armor performance because they affect the ability to erode projectiles, diffuse or divert the load, distribute damage away from the impact location, dissipate energy at sufficiently fast rates, and delay failure.

A key unanswered question in armor applications is this: There is a universe of possibilities among which the answer will be found. A science-based approach with the ability to quantitatively describe the details of the physical event constitutes a sine qua non to achieve this goal. It can assess the role of operative, possibly competing, mechanisms that influence macroscopic behavior.

On the one hand, they provide the input data on material properties and behavior for the simulations. On the other hand, lab-scale and field tests with adequate instrumentation provide quantitative data that can be compared with simulation results to validate the models. Two key components missing in this picture are 1 multiscale, multiphyscis material models and algorithms incorporating information about the subscale or microstructural response, especially for improving the description of material damage and failure; and 2 the quantification of uncertainty for the overall problem analysis, including simulations and experiments.

Decades of research and development have given us mathematical formulations, computational algorithms, and computer software which, to varying degrees, depending on the method and the problem, possess many of the desired attributes sought in a computational framework for numerically solving the fundamental continuum equations governing the response of protection materials.

These responses include versatility, robustness, efficiency, and scalability. That history may be summarized as follows.


The vast majority of the codes in use for the analysis of protective materials employ explicit second-order accuracy for time integration and first-order spatial accuracy. The success of low-order explicit methods can be explained by their simplicity, robustness, and scalability. However, low-order methods pose a key limitation for the proper description of some features of material response where important opportunities for improvements in protection performance may be found—for example, multiscale structured materials, including composites, fabrics, phononic band-gap topological materials, and highly nonlinear granular chains.

New classes of high-order accuracy implicit or semi-implicit algorithms have emerged from academic research that could be incorporated in existing or new codes to gain new levels of physical detail in protection material simulations. The finite-element method has been the traditional Lagrangian approach. Its main advantage is that the description of material state and history, as well as the evolution of material boundaries and interfaces, is a natural outcome of the simulation.

Its main disadvantage is the distortion of the mesh elements induced by large deformations, which invalidates or breaks the numerical method. Particle-based Lagrangian discretizations avoid this problem as the neighborhood of interacting particles is allowed to evolve freely. However, this introduces discontinuous jumps in the continuum notion of the gradient fields strains, for example associated with the particles, which result in convergence problems. Lagrangian meshless methods 54 have emerged as a way to combine the advantages of particle methods and finite elements.

The recent peridynamic formulation of the continuum problem 55 , 56 is also a promising Lagrangian approach that results in a nonlocal particle method with a rigorous mathematical framework and a natural introduction of a characteristic length, as required for modeling material damage. The main advantage of Eulerian formulations, by contrast, is that the computational grid does not distort and thus allows for unconstrained deformations. These formulations originally found their main application in fluid dynamics. Since the governing equations involve the material-time derivative of the stress tensor, the constitutive models need to be formulated in rate form in terms of frame-indifferent stress-rate measures.

In addition, the material state and history, including the elastic response, need to be convected with the flow, which introduces an additional source of complexity and errors. This approach includes the tracking of free boundaries and material interfaces, all of which require special treatment, in contrast to the Lagrangian approach. Owing to the low order of the advection algorithms used, the associated dispersion errors grow over time and the. Computational methods in Lagrangian and Eulerian hydrocodes. Computer Methods in Applied Mechanics and Engineering 99 Meshfree and Particle Methods.

John Wiley and Sons. International Journal for Numerical Methods in Engineering 81 Crack nucleation in a peridynamic solid. International Journal of Fracture Another disadvantage of Eulerian codes is that the fixed grid must cover the entire region of interest. The arbitrary Lagrangian-Eulerian formulation, 57 a combination of both approaches, attempts to exploit the advantages of each approach by allowing high distortions to be represented in a Lagrangian framework.

Combinations of finite-element and meshless-particle methods have been developed, as have combinations of finite-element and Eulerian methods. The rest of this section summarizes the history and evolution of the codes developed and utilized for armor applications. It was a significant new computational capability at that time. During the same time frame implicit finite-element methods were introduced.

However, finite-element methods for analyzing fast dynamic response only became practical when explicit methods for integrating time were introduced. Three different codes emerged in the s implementing this approach: Because of some of the limitations associated with Eulerian codes, Lagrangian approaches for severe distortions continued to be developed.

In , two three-dimensional erosion algorithms were published 67 , 68 that allowed highly distorted elements to be discarded eroded and the interfaces to be automatically updated as the solution progressed. Although this approach introduces some inaccuracies, it allows problems with very severe distortions to be simulated in a Lagrangian framework. One well-known limitation of element erosion is that it gives the wrong energy-release rate when a crack propagates at an angle to the mesh. However, it has been recently shown that a local averaging of the energy in the computation of the energy-release rate eliminates the mesh bias and results in convergent approximations.

Of the particle methods, the smoothed particle hydrodynamics approach, which included material strength, was introduced in Libersky and Petschek. An arbitrary Lagrangian-Eulerian computing method for all flow speeds. Journal of Computational Physics 14 3: Calculation of elastic-plastic flow. Fundamental Methods in Hydrodynamics. Analysis of elastic-plastic impact involving severe distortions.

Journal of Applied Mechanics 43 Ser E 3: A program for transient analysis of structures and continua. University Press of Virginia. Defense Advanced Research Projects Agency. A three-dimensional shockwave physics code. International Journal of Impact Engineering 10 Eroding interface and improved tetrahedral element algorithms for high-velocity impact computations in three dimensions. International Journal of Impact Engineering 5 A three-dimensional impact-penetration algorithm with erosion. Convergence analysis for a smeared crack approach in brittle fracture.

Interfaces and Free Boundaries 9 3: An eigendeformation approach to variational fracture. Multiscale Modeling and Simulation 7 3: Smooth particle hydrodynamics with strength of materials. FIGURE Optimal transportation mesh-free simulation of a steel plate perforated by a steel projectile striking at various angles. For the past 20 years there has been a strong emphasis on the development of a wide range of meshless-particle algorithms for a wide range of applications.

A promising new direction combines elements of optimal transportation theory with meshfree max-ent interpolation of the fields, material-point sampling of material states, and provably convergent erosion schemes to account for fracture see Figure The optimal transportation mesh-free method is an example of this approach. Compared to other particle methods, optimal transportation exhibits strong and provable convergence properties and eliminates tension instabilities that afflict traditional particle methods. An approach based on the conversion of elements into particles has shown to effectively deal with the problem of element distortion.

All of the element variables are transferred to the particle, and the particle is attached to the face of an adjacent element if one exists. With this approach, most of the problem is represented by accurate and efficient elements, with only the highly distorted regions represented by particles. Very severe distortions can be represented in a Lagrangian framework. An alternative approach to alleviate deformation-induced mesh distortion in Lagrangian finite-element algorithms is to adaptively and continuously regenerate the mesh during the simulation. An additional advantage of adaptive remeshing methods is the ability to optimally refine the mesh for maximum accuracy.

This idea was applied successfully to penetration mechanics problems in axisymmetric conditions. Recent advances in computational geometry and mesh optimization have enabled the extension of this idea to three dimensions. Figure shows its application to simulating the oblique impact of a spherical-nosed steel penetrator on an aluminum target. One of the issues with adaptive remeshing approaches is the error introduced in the transfer of field variables from the old to the new mesh, which tends to produce artificial diffusion.

This problem can be somewhat alleviated by adapting the mesh locally instead of completely regenerating it. Another important issue is scalability in parallel calculations. It is well established in computational geometry that algorithms involving general topological changes in the data structures are very hard to implement in parallel and are usually inherently nonscalable because they require the propagation communication of unstructured and evolving data among processors.

As a result, parallel calculations are efficient only up to a few tens of processors at best. Another significant concern is that different codes produce different answers for the same problem, a telling indication of the lack of convergence in the solution. This is illustrated in Figure , where five different computa-. Conversion of 3D distorted elements into meshless particles during dynamic deformation.

International Journal of Impact Engineering 28 9: FIGURE Example of a Lagrangian finite-element simulation that uses adaptive re-meshing and refinement to eliminate element distortion and to optimize the mesh. The five approaches use 1 a finite-volume Eulerian algorithm, 2 an erosion algorithm that discards highly distorted elements, 3 a generalized particle algorithm where the entire problem is represented by particles, 4 a conversion algorithm that converts distorted elements into particles, and 5 a hybrid algorithm where the pressures are computed with particles and strength is computed with elements.

On the one hand it is encouraging that several different approaches can provide good general agreement with the experimental results. On the other hand, there are noticeable differences in the size of the fragments and the response of the tungsten projectile. Comparison of numerical methods in the simulation of hypervelocity impact. Although much progress has been made in developing computational frameworks for the analysis of protection materials, all of the algorithms have strengths and weaknesses. Additional Challenges in Computational Framework Another important issue for future threats that may subject protective materials to conditions well in the nonlinear shock physics regime has to do with the numerical treatment of shock-type discontinuities.

It has been widely established that special computational methods are needed to address the jump discontinuities associated with shocks that arise in materials under extreme compressive loadings. Although this method has proven to be a robust and simple approach for capturing shock, it introduces errors and problems. Concomitant with errors in the energy are errors in the density and the shock speed.

Emerging high-order implicit or semi-implicit methods developed by the fluid mechanics community for compressible turbulence, in which both the compressibility shock and viscous effects are important, have significant potential for solid materials as well. A critical missing component in the protective material simulation tools in use today is the ability to represent material damage and failure explicitly. A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method.

Computer Methods in Applied Mechanics and Engineering At these surfaces of discontinuity, fracture processes can be described by cohesive zone models CZMs of fracture 75 , 76 via a phenomenological traction-separation law. The key advantage of CZMs is their ability to encode in the calculation well-established laws of fracture mechanics governing the nucleation, propagation, branching, and coalescence of cracks. Camacho and Ortiz 79 presented the first formulation of this method for impact problems involving extensive fracture and fragmentation. They demonstrated that the extrinsic CZM was successful at capturing conical crack patterns in ceramic plate impact as long as finely resolved meshes were employed in the calculations.

Cohesive elements provide a notable alternative to erosion and are one of the key innovations brought to ballistic calculations in the s. This development finally enabled the robust and reliable tracking of sharp cracks and complex fracture and fragmentation properties. Cohesive-element calculations have proven highly predictive and have been extensively validated in a number of areas of application by, for example, Bjerke and Lambros 80 and Chalivendra et al.

A full three-dimensional description of crack patterns in ceramic plate impact has recently been enabled by a new extension of CZM based on a discontinuous Galerkin reformulation of the continuum problem. Figure shows the ability of the method to capture conical as well as radial and lateral cracks in ceramic plate impact.

One of the issues commonly attributed to CZM based on interface elements is that the set of available paths for crack propagation is constrained by the mesh, which is a form of mesh dependency. A variety of methods have been put forth to enable arbitrary crack paths in simulations for the purpose of reducing mesh dependency. Essentially, these approaches allow surfaces of discontinuity to propagate through the interior of volumetric elements see, for example, the extended finite-element method, 83 , 84 the embedded localization line method, 85 , 86 , 87 and the cohesive segments method.

Another issue is the possibility of describing crack branching, especially in three dimensions. So far, these types of methods have only been implemented for single-processor computations. Their scalability is marred by the same problem of propagating topological changes across processors alluded to in the discussion of parallel adaptive remeshing. Efforts are currently under way to include methods of the extended finite-element type in existing codes: A critical missing component of protective material simulation tools in use today is the ability to represent material damage and failure explicitly.

In addition to having numerical algorithms it is essential to have computational models to accurately represent the response of the materials. Numerous computational material models have been developed during the past 20 years, but. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 8 2: The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics 7 C: Statistical properties of residual stresses and intergranular fracture in ceramic materials. Numerical simulation of fast crack growth in brittle solids.

Journal of the Mechanics and Physics of Solids 42 9: Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures 33 Theoretical development and experimental validation of a thermally dissipative cohesive zone model for dynamic fracture of amorphous polymers. Journal of the Mechanics and Physics of Solids 51 6: Experimental validation of large-scale simulations of dynamic fracture along weak planes. International Journal of Impact Engineering 36 7: Elastic crack growth in finite elements with minimal remeshing.

International Journal for Numerical Methods in Engineering 45 5: A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46 1: Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. International Journal for Numerical Methods in Engineering 30 3: An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Computational Mechanics 12 5: Numerical simulation of dynamic fracture using finite elements with embedded discontinuities.

International Journal of Fracture 2: A cohesive segments method for the simulation of crack growth. These models have ranged from those for simple dynamic flow stress and dynamic failure strain to very complex models that include microstructural details. Currently, some models for materials are advanced enough to provide helpful and meaningful results, such as those illustrated in the first section of this chapter, but details of failure are not sufficiently robust to allow the predictive design of material systems to protect against specific threats.

For projectile-target interaction computations, the materials are usually modeled using phenomenological models that compute strength and failure as a function of strain, strain rate, temperature, and pressure. For metals the most commonly used strength models are the Johnson-Cook model, 91 the Zerilli-Armstrong models, 92 the SteinbergGuinan-Lund models, 93 the Bodner-Partom models, 94 and.

A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. Last accessed April 7, Dislocation-mechanics-based constitutive relations for material dynamics calculations. Journal of Applied Physics 61 5: A constitutive model for metals applicable at high-strain rate.

Journal of Applied Physics 51 3: Constitutive equations for elastic-viscoplastic strain-hardening materials. Generally these models require a characterization of a specific material, and it is not possible to predict the strength from a microstructural description of the material.

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There are fewer failure models available, with the Johnson-Cook failure model 96 being the most widely used. Although this is primarily a phenomenological model, it includes some physically based features of the ductile fracture mechanism. However, they are still to be incorporated and widely adopted in production codes for ballistic analyses.

Recent contributions have proposed improvements to the characterization of failure in the Johnson-Cook model and the Gurson model. For ceramics there are fewer models available. A unique feature of ceramics, compared to other materials such as metals , is that they have such high compressive strengths that they cannot be tested with typical laboratory stress-strain tests. This characteristic has made it difficult to directly obtain failure data under high compressive pressures and to obtain the shear strength of failed ceramic under high pressures. The JHB phenomenological model used in the illustrative example in the beginning of the chapter has an intact strength, a failed strength, a failure component based on plastic strain and pressure, and bulking.

Recently Deshpande and Evans proposed a mechanism-based model to compute damage and failure in ceramics based on microstructural parameters such as fracture toughness, crack growth rates, flaw size, and densities. In these approaches, damage is considered a state variable of the material, whose history evolves according to prescribed phenomenological laws. These laws describe the effect of the operative driving forces and mechanisms such as stress intensity and triaxiality, which depend on the material type brittle or ductile and characteristics such as defect size and porosity.

Damage models require additional parameters that must be calibrated to experiments or that sometimes have physical meaning, such as initial porosity, defect size and distribution, toughness, and so forth. This is always accompanied by a localization of the deformation in narrow regions, which is a precursor to failure. There is a fundamental mathematical problem with continuum damage models and any other model describing weakening material response—for example, the models of de Borst and Sluys and Sluys et al.

For elliptic equations, waves cannot propagate as their speeds become imaginary, and the softening region collapses to a vanishing width. This, in turn, implies that no energy is dissipated by the softening material, which is far from the real material response. What happens in reality is that there is always a physical process that limits the localization process and introduces a characteristic length scale in the problem, which is not considered in the classical continuum equations.

In the presence of softening, the numerical solution of the conventional continuum problem provides an erroneous resolution of the physical phenomenon. The element or grid size effectively sets the length scale necessary to regularize the problem as it imposes a lower bound for the localization zone width. However, this is just an illusion, because the solution does not converge as the mesh is refined.

A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Metallurgica 36 1: Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics 21 1: On the mechanism of ductile failure in high-strength steels subjected to multi-axial stress states. Journal of the Mechanics and Physics of Solids 24 A criterion for ductile fracture by the growth of holes.

Journal of Applied Mechanics 35 2: Continuum theory of ductile rupture by void nucleation and growth: Part I, yield criteria and flow rules for porous ductile media. On fracture locus in the equivalent strain and stress triaxiality space. International Journal of Mechanical Sciences 46 1: Modification of the Gurson Model for shear failure. Numerical algorithms and material models for high-velocity impact computations. International Journal of Impact Engineering 38 6: Inelastic deformation and energy dissipation in ceramics: A mechanism-based constitutive model.

Journal of the Mechanics and Physics of Solids 56 Localisation in a Cosserat continuum under static and dynamic loading conditions. Computer Methods in Applied Mechanics and Engineering 90 Wave propagation, localization and dispersion in a gradient-dependent medium. International Journal of Solids and Structures 30 9: In other words, not only the model parameters but also the mesh size are tied to a specific application.

The illustrative example in the introduction to this chapter for the projectile penetrating the aluminum plate was approached in this manner. This is clearly a significant limitation. A proper mathematical treatment of softening material response necessarily involves the modification of the classical governing equations in a way that the physically relevant length scale is introduced.

A number of generalizations of the classical formulation have been proposed to this end. They involve either the introduction of higher-order derivatives in the constitutive model gradient models, as, for example, Aifantis and Fleck and Hutchinson or the spatial averaging of strains nonlocal models such as Bazant et al. Both generalizations reflect the fact that micromechanical processes in the localization zone have an inherently nonlocal character. In the particular case of gradient-type softening or damage models, it can be shown that an internal length scale exists and that the resulting set of governing equations is well posed, having wave speeds that remain real in the softening regime.

The immediate computational consequence of this reformulation is that softening-induced mesh dependence is eliminated. These models have not permeated production computational frameworks, primarily for two reasons: Multiscale modeling might be one way to address this issue. The ability to consistently incorporate the effect of micromechanical features on material response would enable rational microstructure design. There is therefore a critical need to develop descriptions of material behavior directly rooted in the first principles of micromechanics, a long-standing aspiration of solid mechanics.

This requires new mathematical frameworks; multiscale, multiphysics constitutive models; and numerical algorithms. Multiscale modeling is a rational and systematic way to construct hierarchical models for the behavior of complex material with the least amount of empiricism and uncertainty. In this approach, the pertinent unit processes at every length scale in the hierarchy of material behavior are identified.

The processes at any scale are the average of the unit processes taking place at the length scale just below. This inductive process ceases at the atomic scale, at which point the fundamental theories describing atomic bonds take over.

For instance, as part of the Caltech advanced simulation and computing program, a full multiscale model of material response was developed for tantalum. The foundational theory on which the hierarchy rests is quantum mechanics and, in particular, the electronic structure of metals.

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