The objective of this paper is to unveil a novel damping mechanism exhibited by 3D woven lattice materials (3DW), with emphasis on response to high-frequency excitations. Conventional bulk damping materials, such as rubber, exhibit relatively low stiffness, while stiff metals and ceramics typically have negligible damping. Here we demonstrate that high damping and structural stiffness can be simultaneously achieved in 3D woven lattice materials by brazing only select lattice joints, resulting in a load-bearing lattice frame intertwined with free, ‘floating’ lattice members to generate damping. The produced material samples are comparable to polymers in terms of damping coefficient, but are porous and have much higher maximum use temperature. We shed light on a novel damping mechanism enabled by an interplay between the forcing frequency imposed onto a load-bearing lattice frame and the motion of the embedded, free-moving lattice members. This novel class of damping metamaterials has potential use in a broad range of weight sensitive applications that require vibration attenuation at high frequencies.
Architected materials are multi-phase and/or cellular materials in which the topological distribution of the phases is carefully controlled and optimized for specific functions or properties. Nearly two decades of research has resulted in the identification of a number of topologically simple, easy to fabricate, well established structures (including honeycombs and truss lattices), which have been optimized for specific stiffness and strength, impact and blast protection, sound absorption, wave dispersion, active cooling and combinations thereof.
Over the past few years, dramatic advances in processing techniques, including polymer-based templating (e.g., stereolithography, photopolymer waveguide prototyping, two-photon polymerization) and direct single- or multi-material formation (e.g., direct laser sintering, deformed metal lattices, 3D weaving and knitting), have enabled fabrication of new architected materials with complex geometry and remarkably precise control over the geometric arrangement of solid phases and voids from the nanometer to the centimeter scale.
The ordered, topologically complex nature of these materials and the degree of precision with which their features can now be defined suggests the development of new multi-physics and multi-scale modeling tools that can enable optimal designs. The result is efficient multi-scale cellular materials with unprecedented ranges of density, stiffness, strength, energy absorption, permeability, chemical reactivity, wave/matter interaction and other multifunctional properties, which promise dramatic advances across important technology areas such as lightweight structures, functional coatings, bio-scaffolds, catalyst supports, photonic/phononic systems and other applications.
Some of the most exciting recent developments in this field are the exploration of size effects in the development of nano-architected materials with superior combinations of properties, the investigation of geometrically complex unit cell architectures that enable non-linear effective mechanical response from linear-elastic materials, novel manufacturing approaches with increased resolution and scalability, and improved design optimization tools. Here we briefly review some recent progress in these areas, and conclude with some thoughts about opportunities for future development. The collection of articles in this focus issue is a wonderful exposure to some of the latest original work in this field.
In this article, we study the computational design of multiphase architected materials comprising a stiff phase, a dissipative phase, and void space, with enhanced vibration damping characteristics under wave propagation. We develop a topology optimization framework that maximizes a figure of merit comprising of effective stiffness, density and effective damping. We also propose novel material interpolation strategies to avoid the blending of different phases at any given point in the design domain. This is achieved by carefully defining different penalization schemes for different components of the merit function. The effective stiffness of the periodic multiphase material is calculated using homogenization theory and the Bloch–Floquet theorem is used to obtain its damping capacity, allowing for the investigation of the effect of wave directionality, material microarchitecture and intrinsic material properties on the wave attenuation characteristics. It is shown that the proposed topology optimization framework allows for systematic tailoring of microstructure of the multiphase materials for wide ranges of frequencies and densities and results in the identification of optimized multiphase cellular designs with void space that are superior to fully dense topologies.
Bree Interaction Diagrams have long been one of the major visual design guides for employing and evaluating shakedown in engineering applications. These diagrams provide representations of the realms in which elastoplastic behaviors, including shakedown, are found for a material and structure under variable loads. The creation of these diagrams often relies upon some combination of upper or lower bound shakedown theorems and numerical shakedown limit determination techniques. Part of the utility of these diagrams is that, for a given structure and loading conditions, inspecting them is sufficient to determine whether shakedown will occur or not. The diagrams cannot however, give the designer insight into how the conditions for shakedown are met. This chapter presents some graphical interpretations of one of the common methods for shakedown determination: the use of Melan’s Lower Bound Theorem. The intent is to provide additional insight for designers regarding how shakedown conditions are satisfied. In this way, additional directions for modifying designs to recover shakedown behavior may also be identified. Revisiting this well-established theorem from a graphical and pedagogical approach, also provides a foundation for interdisciplinary innovation. The particular focus is on simple examples that highlight ways in which Melan’s theorem may be applied to shakedown design problems.
Viscoelastic materials are commonly used to dissipate kinetic energy in case of impact and vibrations. Unfortunately, dissipating large amounts of energy in a monolithic material requires high combinations of two intrinsic properties – Young’s modulus and loss factor, which are generally in conflict. This limitation can be overcome by designing cellular materials incorporating negative stiffness elements. Here we investigate a configuration comprising two positive stiffness elements and one negative stiffness element. This unit cell possesses an internal degree of freedom, which introduces hysteresis under a loading-unloading cycle, resulting in substantial energy dissipation, while maintaining stiffness. We demonstrate and optimize a simple implementation in a single material design that does not require external stabilization or pre-compression of buckled elements; these key features make it amenable to fabrication by virtually any additive manufacturing approach (from 3D printing to assembly and brazing) in a wide range of base materials (from polymers to metals). No additional intrinsic damping mechanism is required for the base material, which is assumed linear elastic. Furthermore, the architected material can be designed to be fully recoverable. When optimized, these architected materials exhibit extremely high combinations of Young’s modulus and damping, far superior to those of each constituent phase.
In 1903, Alexander Graham Bell developed a design principle to generate lightweight, mechanically robust lattice structures based on triangular cells; this has since found broad application in lightweight design. Over one hundred years later, the same principle is being used in the fabrication of nanolattice materials, namely lattice structures composed of nanoscale constituents. Taking advantage of the size-dependent properties typical of nanoparticles, nanowires, and thin films, nanolattices redefine the limits of the accessible material-property space throughout different disciplines. Herein, the exceptional mechanical performance of nanolattices, including their ultrahigh strength, damage tolerance, and stiffness, are reviewed, and their potential for multifunctional applications beyond mechanics is examined. The efficient integration of architecture and size-affected properties is key to further develop nanolattices. The introduction of a hierarchical architecture is an effective tool in enhancing mechanical properties, and the eventual goal of nanolattice design may be to replicate the intricate hierarchies and functionalities observed in biological materials. Additive manufacturing and self-assembly techniques enable lattice design at the nanoscale; the scaling-up of nanolattice fabrication is currently the major challenge to their widespread use in technological applications.
We report on a new class of elastic architected materials with hybrid unit cells, consisting of discrete elastic elements with non-convex strain energy and one convex (but possibly nonlinear) elastic element, to obtain a reversible multifunctional material with extreme energy dissipation. The proposed design exploits numerically optimized nonlinearities in the force–displacement response of the sub-unit-cell elements to approach the theoretical limit of specific damping capacity in any material, Psi_th=8. Specific damping capacities up to Psi=6.02 were experimentally demonstrated, which are far greater than any experimental value previously reported, including in high damping elastomers (Psi<4.5). Remarkably, this damping performance is achieved even with a single unit cell, thus avoiding the need for thick multi-cell designs. Furthermore, the proposed design offers relatively high stiffness and low transmitted stress upon compression. The proposed concept could enable the design of reversible impact-resistant structures with superior crashworthiness and energy dissipation.
In most recent earthquakes, traditional seismic design demonstrated its effectiveness in reducing casualties through ductile structural mechanisms, but allowed for extensive structural damages that accounted for tremendous economic losses. This evidence raised awareness for the need of an increased level of resiliency, mostly in low-rise buildings. Seismic isolation is a protection system that proved to be successful in prevent damages and maintaining operability. However, high costs and sever testing protocols currently discourage the extensive application of this technology to low-rise buildings. In this study, an architected periodic cellular material with unprecedented characteristic is proposed as a low-cost alternative to traditional technologies to produce seismic isolation devices for implementation in residential, retail and office buildings. Results from preliminary numerical analysis demonstrate the range of performance of this novel architected material in comparison with traditional materials. The scalability of the architected material is also addressed with the aim of investigating the feasibility of using tests on small assemblies of the constituent unit-cells instead of full scale tests to assess the structural performance of the isolators.
The development of a novel class of materials that can be used for the seismic isolation protection systems is presented. The material is an architected cellular material obtained as periodic reproduction of a unit cell in all spatial directions. The mechanical properties of the constitutive unit-cell are tailored through an optimal design of the spatial configuration of voids and solids (cellular architecture) for a given solid constituent. The presented material can be designed at different scales in order to obtain unprecedented combinations of mechanical properties, such as high stiffness and strength in the vertical direction combined with high flexibility and dissipative capability in the lateral directions. Compact-shape, light-weight isolators overcoming traditional isolators’ limitations may be obtained through optimization of the unit cell properties. A mechanistic model of the isolator is provided after calibration on randomized parametric nonlinear Finite Element analysis of the unit cell. Keywords: Seismic isolation; periodic architected material; mechanistic model.
Multistable shape-reconfigurable architected materials encompassing living hinges and enabling combinations of high strength, high volumetric change, and complex shape-morphing patterns are introduced. Analytical and numerical investigations, validated by experiments, are performed to characterize the mechanical behavior of the proposed materials. The proposed architected materials can be constructed from virtually any base material, at any length scale and dimensionality.
This article was featured in a Nature Research Highlight, Nature 535 (07/2016) 32
Designing materials with exceptional combinations of properties at low weight is a continuous goal in many industries. Cellular (i.e., porous) materials with one or more phases topologically organized in a precisely designed configuration (often denoted as architected materials, or metamaterials) are excellent candidates to reach combinations of properties that are unattainable by existing monolithic materials. Additive manufacturing techniques are perfectly suited to implement the topological complexity that is often required for optimal performance. As beneficial size effects often arise in mechanical and functional properties as dimensions are shrunk to the nanoscale, 2PP becomes an ideal platform to investigate and ultimately fabricate topologically micro-architected and nano-architected materials with truly unique properties. The chapter reviews some notable features of architected materials, surveys commonly available manufacturing approaches, and presents challenges and opportunities for 2PP fabrication.
The epidemiology of valvular heart disease has significantly changed in the past few decades with aging as one of the main contributing factors. The available options for replacement of diseased valves are currently limited to mechanical and bioprosthetic valves, while the tissue engineered ones that are under study are currently far from clinical approval. The main problem with the tissue engineered heart valves is their progressive deterioration that leads to regurgitation and/or leaflet thickening a few months after implantation. The use of bioresorbable scaffolds is speculated to be one factor affecting these valves’ failure. We have previously developed a non-degradable superelastic nitinol mesh scaffold concept that can be used for heart valve tissue engineering applications. It is hypothesized that the use of a non-degradable superelastic nitinol mesh may increase the durability of tissue engineered heart valves, avoid their shrinkage, and accordingly prevent regurgitation. The current work aims to study the effects of the design features on mechanical characteristics of this valve scaffold to attain proper function prior to in vivo implantation.
Reducing mass without sacrificing mechanical integrity and performance is a critical goal in a vast range of applications. Introducing a controlled amount of porosity in a strong and dense material (hence fabricating a cellular solid) is an obvious avenue to weight reduction. The mechanical effectiveness of this strategy, though, depends strongly on the architecture of the resulting cellular material (i.e., the topology of the introduced porosity). Recent progress in additive manufacturing enables fabrication of macro-scale cellular materials (both single-phase and hybrid) with unprecedented dimensional control on the unit-cell and sub-unit-cell features, potentially producing architectures with structural hierarchy from the nano to the macro-scale. As mechanical properties of materials often exhibit beneficial size effects at the nano-scale (e.g., strengthening of metals and toughening of ceramics), these novel manufacturing approaches provide a unique opportunity to translate these beneficial effects to the macro-scale, further improving the mechanical performance of architected materials. The enormous design space for architected materials, and the strong relationship between the topological features of the architecture and the effective physical and mechanical properties of the material at the macro-scale, present both a huge opportunity and an urgent need for the development of suitable optimal design strategies. Here we present a number of strategies for the advanced manufacturing, characterization and optimal design of a variety of lightweight architected materials with unique combinations of mechanical properties (stiffness, strength, damping coefficient…). The urgent need to form strong synergies among the fields of additive manufacturing, topology optimization and architectureproperties relations is emphasized throughout.
This paper investigates the optimal architecture of planar micro lattice materials for minimum weight under simultaneous axial and shear stiffness constraints. A well-established structural topology opti- mization approach is used, where the unit cell is composed of a network of beam elements (Timoshenko beams are used instead of truss elements to allow modeling of bending-dominated architectures); start- ing from a dense unit cell initial mesh, the algorithm progressively eliminates inefficient elements and resizes the essential load-bearing elements, finally converging to an optimal unit cell architecture. This architecture is repeated in both directions to generate the infinite lattice. Hollow circular cross-sections are assumed for all elements, although the shape of the cross-section has minimal effect on most optimal topologies under the linear elasticity assumption made throughout this work. As optimal designs identi- fied by structural topology optimization algorithms are strongly dependent on initial conditions, a careful analysis of the effect of mesh connectivity, unit cell aspect ratio and mesh density is conducted. This study identifies hierarchical lattices that are significantly more efficient than any isotropic lattice (includ- ing the widely studied triangular, hexagonal and Kagomé lattices) for a wide range of axial and shear stiff- ness combinations. As isotropy is not always a design requirement (particularly in the context of sandwich core design, where shear stiffness is generally more important than compressive stiffness), the- se optimal architectures can outperform any established topology. Extension to 3D lattices is straightforward.
The search for light yet strong materials recently benefited from novel high resolution 3D-printing technologies, which allow for fabricating lightweight porous materials with optimally designed micro-topologies. Architectural design improves mechanical properties significantly compared to stochastic porosity, as in foams. Miniaturization of the architectures offers to exploit material strengthening size-effects occurring at the nanoscale. However, these effects and their interaction with structural behavior are not yet well understood. We present tensile experiments of nanoscale alumina–polymer composite bars and cellular microarchitectures, applying 3D-printed push-to-pull mechanisms. The strength of alumina is found to strongly increase as the material thickness decreases. Below 50 nm thickness a plateau at about 5.5 GPa is reached, which is in the range of the theoretical strength. The characteristic low tensile strength of ceramics and its high variability seem not to hold at the nanoscale. Thus, when designed and fabricated appropriately, microarchitectures will facilitate carrying these size-effects beyond scales in future, allowing the use of ceramic materials far beyond what is possible to date.
3D tissue culture models are utilized to study breast cancer and other pathologies because they better capture the complexity of in vivo tissue architecture compared to 2D models. However, to mimic the in vivoenvironment, the mechanics and geometry of the ECM must also be considered. Here, we studied the mechanical environment created in two 3D models, the overlay protocol (OP) and embedded protocol (EP). Mammary epithelial acini features were compared using OP or EP under conditions known to alter acinus organization, i.e. collagen crosslinking and/or ErbB2 receptor activation. Finite element analysis and active microrheology demonstrated that OP creates a physically asymmetric environment with non-uniform mechanical stresses in radial and circumferential directions. Further contrasting with EP, acini in OP displayed cooperation between ErbB2 signalling and matrix crosslinking. These differences in acini phenotype observed between OP and EP highlight the functional impact of physical symmetry in 3D tissue culture models.
In this article, we propose a method to incorporate fabrication cost in the topology optimization of light and stiff truss structures and periodic lattices. The fabrication cost of a design is estimated by assigning a unit cost to each truss element, meant to approximate the cost of element placement and associated connections. A regularized Heaviside step function is utilized to estimate the number of elements existing in the design domain. This makes the cost function smooth and differentiable, thus enabling the application of gradient-based optimization schemes. We demonstrate the proposed method with classic examples in structural engineering and in the design of a material lattice, illustrating the effect of the fabrication unit cost on the optimal topologies. We also show that the proposed method can be efficiently used to impose an upper bound on the allowed number of elements in the optimal design of a truss system. Importantly, compared to traditional approaches in structural topology optimization, the proposed algorithm reduces the computational time and reduces the dependency on the threshold used for element removal.
This paper demonstrates that the Thermal Coefficient of resonant Frequency (TCF) of a micro glass-blown Pyrex spherical resonator can be substantially reduced by the application of a titanium (Ti) coating. Finite Elements Analysis (FEA) is used to demonstrate that the temperature dependence of the Young’s modulus of the shell material is the dominant parameter affecting the TCF of the resonator, clearly suggesting the use of a metallic compensating layer. Experimental characterization demonstrates that the TCF of a Pyrex glass-blown resonator is reduced by 70% (from 73 ppm/°C to 24 ppm/°C) by the application of a 1.33 μm thick layer of Ti. It is predicted by FEM that for a Ti layer thickness on the order of 2.5 μm the TCF will fall below 10 ppm an acceptable value for high performance resonators. This investigation is a step forward in the quest to employ the desirable properties of micro-blown resonators, such as high symmetry, manufacturing tolerances and environmental robustness.
Recent progress in micro- and nanofabrication techniques enables the creation of hierarchically architected microlattices with dimensional control over six orders of magnitude, from centimeters down to nanometers. This hierarchical control facilitates the exploration of opportunities to exploit nano-sized material effects in structural materials. In this work, we present the fabrication, characterization, and properties of hollow metallic glass NiP microlattices. The wall thicknesses, deposited by electroless plating, were varied from %60 nm up to 600 nm, resulting in relative densities spanning from 0.02 to 0.2%. Uniaxial quasi-static compression tests revealed two different regimes in deformation: (i) Structures with a wall thickness above 150 nm failed by catastrophic failure at the nodes and fracture events at the struts, with significant micro- cracking and (ii) Lattices whose wall thickness was below 150 nm failed initially via buckling followed by significant plastic deformation rather than by post-elastic catastrophic fracture. This departure in deformation mechanism from brittle to deformable exhibited by the thin-walled structures is discussed in the framework of brittle-to-ductile transition emergent in nano-sized metallic glasses.
When properly designed at ultra-low density, hollow metallic microlattices can fully recover from compressive strains in excess of 50%, while dissipating a considerable portion of the elastic strain energy. This article investigates the physical mechanisms responsible for energy loss upon compressive cycling, and attributes the most significant contribution to a unique form of structural damping, whereby elastic local buckling of individual bars releases energy upon loading. Subsequently, a simple mechanical model is presented to capture the relationship between lattice geometry and structural damping. The model is used to optimize the microlattice geometry for maximum damping performance. The conclusions show that hollow metallic microlattices exhibit exceptionally large values of the damping figure of merit, (Young’s modulus)^(1/3) (loss coefficient)/(density), but this performance requires very low relative densities (<1%), thus limiting the amount of energy that can be dissipated.
Recent progress in advanced manufacturing enables fabrication of macro-scale hollow metallic lattices with unit cells in the millimeter range and sub-unit cell features at the submicron scale. If designed to minimize mass, these metallic microlattices can be manufactured with densities lower than 1 mg/cm3, making them the lightest metallic materials ever demonstrated. Measuring the compressive stiffness of these ultralight lattices with conventional contact techniques presents a major challenge, as the lattices buckle or locally fracture immediately after contact with the loading platens is established, with associated reduction in stiffness. Non-contact resonant approaches have been successfully used in the past for modulus measurements in solid materials, at both small and large scales. In this work we demonstrate that Laser Doppler Vibrometry coupled with Finite Elements Analysis is a suitable technique for the reliable extraction of the Young’s modulus in ultralight microlattices.
This paper presents a two-degree-of-freedom analytical model for the electromechanical response of double ended tuning fork (DETF) force sensors. The model describes the mechanical interaction between the tines and allows investigation of the effect of a number of asymmetries, in tine stiffness, mass, electromechanical parameters and load sharing between the tines. These asymmetries are introduced during fabrication (e.g., as a result of undercut) and are impossible to completely eliminate in a practical design. The mechanical coupling between the tines induces a frequency separation between the in-phase and the out-of-phase resonant modes. The magnitude of this separation and the relative intensity of the two modes are affected by all the asymmetries mentioned above. Two key conclusions emerge: (i) as the external axial compressive load is increased, the in-phase mode reaches zero frequency (buckling) much faster than the out-of-phase (i.e., operational) mode, resulting in a device with a decreased load range. (ii) During the operation, balanced excitation is essential to guarantee that the out-of-phase mode remain significantly stronger than the in-phase mode, thus allowing sharp phase locked loop locking and hence robust performance. The proposed model can be used to assess the magnitude of asymmetries introduced by a given manufacturing process and accurately predict the performance of DETF force sensors. For the specific sensor characterized in this study, the proposed model can capture the full dynamic response of the DETF and accurately predict its maximum axial compressive load; by contrast, the conventional single-DOF model does not capture peak splitting and overpredicts the maximum load by ~18%. The proposed model fits the measured frequency response of the electromechanical system and its load-frequency data with coefficient of determination (R2) of 95.4% (0.954) and 99.2% (0.992), respectively.
Ordered periodic microlattices with densities from 0.5 mg/cm3 to 500 mg/cm3 are fabricated by depositing various thin film materials (Au, Cu, Ni, SiO2, poly(C8H4F4)) onto sacrificial polymer lattice templates. Young’s modulus and strength are measured in compression and the density scaling is determined. At low relative densities, recov- ery from compressive strains of 50% and higher is observed, independent of lattice material. An analytical model is shown to accurately predict the transition between recoverable “pseudo-superelastic” and irrecoverable plastic deformation for all con- stituent materials. These materials are of interest for energy storage applications, de- ployable structures, and for acoustic, shock, and vibration damping.
Molecular dynamics simulations of nanocrystalline Ni revealed that the in-plane Young’s modulus of 2.2 nm grained Ni film with 10 grains across its thickness was only 0.64% smaller than that of bulk, while it dropped to 24.1% below bulk value for ~1 grain across film. This size dependence arises from the increased number of more compliant grains adjacent to the free surface. Simulations of nanocrystalline diamond revealed that the anharmonicity of the potential curve determined the sensitivity of the Young’s modulus to variations in the sample size.
Recent advances in multiscale manufacturing enable fabrication of hollow-truss based lattices with dimensional control spanning seven orders of magnitude in length scale (from ;50 nm to ;10 cm), thus enabling the exploitation of nano-scale strengthening mechanisms in a macroscale cellular material. This article develops mechanical models for the compressive strength of hollow microlattices and validates them with a selection of experimental measurements on nickel microlattices over a wide relative density range (0.01–10%). The limitations of beam-theory-based analytical approaches for ultralight designs are emphasized, and suitable numerical (finite elements) models are presented. Subsequently, a novel computational platform is utilized to efficiently scan the entire design space and produce maps for optimally strong designs. The results indicate that a strong compressive response can be obtained by stubby lattice designs at relatively high densities (~10%) or by selectively thickening the nodes at ultra-low densities.
This article presents an innovative additive manufacturing approach for fully optimized ceramic and hybrid (ceramic/polymer) hierarchical micro-architected materials for extreme environments. The developed materials combine high-temperature capabilities, extremely low thermal conductivity, high stiffness and strength per unit weight, sufficient toughness and great resistance to oxidation. Processing involves 3D printing of a ceramic architecture, followed by bisque firing and sintering. When sufficient porosity remains after sintering (~50%), infiltration by a polymeric matrix is possible, resulting in a cellular architecture where the constituent material is a fully dense ceramic/polymer hybrid with exceptional ductility. After characterizing the microstructure and mechanical properties of the constituent material (ceramic and hybrid), we demonstrate the fabrication of a truss-core sandwich panel. Both the internal architecture and the external shape can be controlled at will in the manufacturing process. Multifunctional Thermal Protection Systems (TPS) for the next generation of high-speed aircraft (particularly hypersonics) are the prototypical, albeit not the only, application.
This paper presents a resonant double-ended tuning fork (DETF) force sensor with an experimentally demonstrated resolution of 7 nN and a compressive load range of 0.08 N, exceeding a dynamic range of 140 dB (100 parts per billion). The resonator has a scale factor of 216 kHz/N, a Q-factor exceeding 60 000 at 3-mtorr ambient pressure, and a zero-load resonant frequency of 47.6 kHz. The resonator is kept at resonance via a phase-locked loop composed of discrete elements. The sensor is implemented with a silicon-on-glass process with a 100-μm-thick 111 silicon structural layer. The sensor and the complete readout circuit are fully embedded in a compact 65 mm × 52 mm printed circuit board (PCB). The outof-plane parasitic modes of the DETF are also investigated with finite-element simulations and laser Doppler vibrometry experiments, and are verified to be outside of the device working range. The PCB is mounted on a microstage and coupled with an off-the-shelf displacement actuator to realize an economical, versatile, and robust micromechanical test frame with unprecedented combination of force and displacement resolution and range.
Plasticity theory is the mathematical formalism that describes the constitutive model of a material undergoing permanent deformation upon loading. For polycrystalline metals at low temperature and strain rate, the J2 theory is the simplest adequate model. Classic plasticity theory does not include any explicit length scale, and as a result, the constitutive behavior is independent of the sample dimensions. As the characteristic length of a sample is reduced to the micro (and nano) scale, careful experimental observations clearly reveal the presence of a size effect that is not accounted for by the classical theory. Strain gradient plasticity is a formalism devised to extend plasticity theory to these smaller scales. For most metals, strain gradient plasticity is intended to apply to objects in the range from roughly 100 nm to 100 μm. Above 100 μm, the theory converges with the classical theory and below 100 nm surface and grain boundary effects not accounted for in the theory begin to dominate the behavior. By assuming that the plastic work (or in some theories, the yield strength) depends not only on strain but also on strain gradients (a hypothesis physically grounded in dislocation theory and, in particular, in the notion of geometrically necessary dislocations (GND) associated with incompatibility due to strain gradients), an intrinsic length scale is naturally introduced, allowing the theory to capture size effects. According to most theories, the intrinsic length scale is of the order of the distance between dislocation-clipping obstacles or cellular dislocation structures (typically, submicron to tens of microns). This continuum theory is appropriate for length scales that remain large relative to the distance between dislocations. As the sample length scale is dropped below this level, dislocations must be modeled individually, and discrete dislocations simulations (DSS) are the preferred approach. At even smaller scales, molecular dynamics (MD) becomes the applicable tool. This article presents a brief overview of one of the simplest continuum strain gradient plasticity theories that reduces to the classical J2 theory when the scale of the deformation becomes large compared to the material length scale. This simple theory captures the essence of the experimental trends observed to date regarding size effects in submicron to micron scale plasticity.
Novel nickel-based microlattice materials with structural hierarchy spanning three different length scales (nm, μm, mm) are characterized microstructurally and mechanically. These materials are produced by plating a sacrificial template obtained by self-propagating photopolymer waveguide prototyping. Ni–P films with a thickness of 120 nm to 3 μm are deposited by electroless plating, whereas thicker films (5–26 μm) are obtained by subsequent electrodeposition of a pure Ni layer. This results in cellular materials spanning three orders of magnitude in relative density, from 0.01% to 8.5%. The thin electroless Ni–P films have ultra-fine grain size (7 nm) and a yield strength of ∼2.5 GPa, whereas the thicker electrodeposited Ni films exhibit a much broader distribution with average grain size of 116 nm and strong (1 0 0) texture in the plating direction, resulting in a yield strength of ∼1 GPa. Uniaxial compression experiments reveal two distinct mechanical responses. At ultra-low densities (<0.1%), these lattices exhibit nearly full recovery after strains up to more than 50%, and damping coefficients an order of magnitude larger than for conventional Ni foams. At higher densities (0.1–10%), the compression behavior is fully plastic, similar to traditional cellular metals. A simple mechanical analysis reveals that the transition occurs when the thickness-to-diameter ratio of the truss elements is of the order of the yield strain of the material, in agreement with experimental observations. Optical and electron imaging of deformed lattices show that the deformation largely localizes around the nodes. In the ultra-light regime, the microlattice materials are stiffer and stronger than any existing alternative.