Elastic Anisotropy Research Articles

Ab initio calculations of second-, third-, and fourth-order partial and inner elastic constants of diamond.

By means of ab initio calculations, a unified framework is presented to investigate the effect of internal displacement on the linear and nonlinear elasticity of single diamond crystals. The calculated linear and nonlinear elastic constants, internal strain tensor and internal displacement in single diamond crystals are compatible with the available experimental data and other theoretical calculations. The complete set of second-, third- and fourth-order elastic constants and internal strain tensor not only offer a better insight into the nonlinear and anisotropic elasticity behaviors, but also shows us the basic internal mechanical response of diamond. This study provides a route to calculate the nonlinear internal and external elasticity response in a nonprimitive lattice.

Accurate quantification of dislocation loops in complex functional alloys enabled by deep learning image analysis

In-depth statistics of individual defects observed during transmission electron microscopy (TEM) experiments are essential for the thorough characterization of materials. In this study, we aim to quantitatively characterize the population of dislocation loops in ion-irradiated CrFeMnNi alloys. To this end, we propose an efficient guideline to prepare TEM micrographs dataset for deep learning analysis, adapted for accurate characterization of microstructures produced by thousands of overlapping defects, a very common situation in TEM images, unfeasible by previous existing methods. To reduce human effort, we annotate only a few images and complement the database through a two-step process: initially, singular value decomposition to normalize image background, followed by a controlled data augmentation. The performed analysis provides precise quantitative information about the number of loops of different types, as well as their spatial distribution, their size, and the inter-object distances. These characteristics provide insights into the nucleation, mobility, and growth of dislocation loops, as well as the elastic anisotropy of the material. Our results emphasize how accurate analysis of complex microstructures can provide insights into the physical properties of materials and open up many perspectives for attaining quantitative information on materials properties based solely on their image analysis.

Understanding the pressure effect on the physical properties of half-Heusler semiconductor LiCaX (X = As, Sb, N) compounds from Ab-initio calculations

The study delves into the structural, elastic, electronic, thermodynamic, and lattice dynamical and optic properties of LiCaX (X = As, Sb, N) ternary half-Heusler compounds under varying pressure and temperature conditions. Employing density functional theory with the generalized gradient approximation (GGA) in VASP codes, the initial steps involve optimizing the structure of the compound (cubic MgAgAs-type, space group F43m, C1b), determining elastic constants (C11, C12, and C44), and calculating elastic moduli (bulk modulus, shear modulus, Young's modulus). Other mechanical parameters such as Pugh's ratio, Poisson's ratio, and Zener anisotropy factor are also derived, providing insights into the brittle/ductile characteristics and isotropic/anisotropic behavior. The study further explores different vibrational modes in the compounds, and the effects of pressure on elastic anisotropy, vibrational, and optical properties are thoroughly investigated. Utilizing the quasi-harmonic Debye model, the research successfully reports V/Vo, bulk modulus, thermal expansion coefficient, and heat capacity at constant volume over a temperature range from 0 to 1000 K. The variation of photon energy is analyzed concerning the real and imaginary parts of the dielectric constant, refractive indices, extinction coefficient, reflectivity, and energy loss function up to 20 eV. The findings encapsulate the fundamental physical properties of all considered compounds, concluding with suggestions for potential future research avenues.

Novel superhard orthorhombic O12 carbon: a first principle study

Abstract In this study, a new orthorhombic carbon structure with Ama2 symmetry, designated as O12 carbon, has been predicted on basis of the first-principles method. The thermal stability and dynamic stability of O12 carbon were evaluated and confirmed by combining ab initio molecular dynamics and phonon spectra analyses. Meanwhile, the mechanical properties of O12 carbon, compared with other superhard carbon allostrope like diamond, M-carbon, Bct-c4 and Cco-C8, have also been systematically investigated. With a bulk modulus of 351 GPa, a shear modulus of 333 GPa, and a Vickers hardness of 53 GPa, O12 carbon is identified as a superhard material of significantly technological and industrial applications. The existence of elastic anisotropy in O12 carbon has also been confirmed. The electronic band structure calculations have revealed that O12 carbon exhibits a semiconductor characteristic with a direct band gap of 2.5 eV, making it promising candidate in electronic applications. Additionally, the XRD spectra of O12 carbon were obtained to offer further insights for potential experimental observations. Our findings enrich the family of carbon structure and are expected to stimulate additional experimental interest.

MAX phase borides, the potential alternative of well-known MAX phase carbides: A case study of V2AB [A = Ge, P, Tl, Zn] via DFT method

This study predicted four new MAX phase borides via the DFT method, with a comprehensive and thorough approach. The stability of the predicted phases has been thoroughly studied using formation energy, phonon dispersion curve (PDC), and elastic constants (Cij). The metallic nature of the studied phases is confirmed through the computation of the electronic band structure and density of states (DOS). Their bonding nature is disclosed using the partial density of states, Mulliken population analysis, and charge density mapping. The mechanical behavior is investigated in depth by calculating elastic constants, elastic moduli, Poisson's & Pugh ratio, machinability index, and Vickers hardness. Different anisotropic indices are also computed to demonstrate the elastic anisotropy. The Debye temperature (ΘD), Grüneisen parameter (γ), phonon thermal conductivity (kph), minimum thermal conductivity (kmin), thermal expansion coefficient (TEC), and melting temperature (Tm) are all calculated, and the suitability of the studied phases as thermal barrier coating (TBC) materials has been discussed. Finally, the optical constants are calculated and analyzed, further certifying the metallic nature of the herein-studied phases. The reflectivity spectra of all the herein selected compounds reveal their potential as coating materials to lessen solar heating. Among the studied phases, V2PB exhibits the best thermo-mechanical properties for potential applications in diverse fields, such as structural components and TBC materials. The potential applications of our findings are vast, and the obtained results reveal that the predicted phases are indeed potential alternatives to their counterpart carbides.

DFT investigation of thermodynamic, electronic, optical, and mechanical properties of XLiH3 (X= Mg, Ca, Sr, and Ba) hydrides for hydrogen storage and energy harvesting

The present study investigates the hydrogen storage capacity and physical properties of the perovskite hydrides XLiH3 (X = Mg, Ca, Sr, and Ba). The studied compounds MgLiH3, CaLiH3, SrLiH3, and BaLiH3 have lattice constants 3.76, 4.29, 4.64, and 5.04 Å, respectively. These compounds are also observed to be stable in cubic phase under atmospheric pressure and temperature conditions. They have hydrogen storage capacities 8.76 wt%, 5.99 wt%, 3.07 wt%, and 2.03 %, respectively. The SrLiH3 compound has the greatest Debye temperature (θD) of 344.41 K. The analysis of the electronic band structure and density of states reveals that perovskite hydrides have semiconducting characteristics with indirect band gap values of 2.66, 2.34, 1.94, and 1.37 eV, respectively. The materials are semiconductors and have suitable band gaps to be utilized in optical devices. Therefore, the optoelectronic properties of dielectric constants, absorption, and energy loss have been determined to predict the potential of materials for optoelectronic applications. Furthermore, the elastic constants, moduli, and anisotropy are also calculated for the observed materials. The Possion and Pugh ratios indicate these compounds exhibit ductile behaviour and significant anisotropy. Therefore, large values of hydrogen capacities, stabilities, and extraordinary physical behaviour make them important for hydrogen storage systems.

Bending of polymer films: a method for obtaining a compressive modulus of thin films.

Due to the advent of various foldable electric devices, it is becoming increasingly important to understand the bending properties of film materials. Bending of isotropic materials may be trivial if elastic deformation is small within a range of linear elasticity, which is often the case for bending. However, bending of polymer films, often used in recent foldable devices, may not be the case. Polymer films are frequently fabricated with stretching, which induces anisotropic orientation of molecular chains. In addition, there are many studies on bimodulus materials, which suggest the importance of the difference in tensile and compressive elastic moduli, i.e., the importance of elastic asymmetry, considering that bending involves compression and extension. In this study, we extended the standard linear elastic theory to include elastic anisotropy and elastic asymmetry and developed a method for obtaining compressive moduli of films, which cannot be obtained by a simple compression test because thin films under compression buckle at small strains. Our method is based on a bending test combined with a uniaxial tension test, which allows the measurement of Poisson's ratio in addition to Young's modulus, which are both anisotropic and asymmetric. To test our theory and method, we further performed experiments on biaxially stretched poly(ethylene terephthalate) (PET) films. As a result, we found non-negligible anisotropy in Poisson's ratio and non-negligible asymmetry (bimodulus) in tensile and compressive moduli. We further justify our framework by demonstrating a clear data collapse to show agreement between experiment and theory, clarifying limitations. Our results suggest the importance of elastic anisotropy and elastic asymmetry in bending of industrial films and give fundamental knowledge on this subject, which would be useful for applications. Such applications include the control of the position of the neutral plane and precise measurements of elastic moduli and Poisson's ratio, which are crucial, e.g., for the development of tough flexible electric devices and for the structural designs using compliant mechanisms.

First-principles calculations of the elastic anisotropy, thermal conductivity, and optical properties of MAX-phase M2AlC (M = V, Nb, Ta) series carbides

Materials from the MAX phase, such as the M2AlC (M = V, Nb, Ta) series carbides, exhibit superior mechanical, electronic, and thermodynamic properties, making them highly promising for research. Density functional theory (DFT) has been employed to study the elastic anisotropy, thermal conductivity, electronic, and optical properties of these carbides. The results indicate that the formation enthalpies of the M2AlC (M = V, Nb, Ta) series are negative and the phonon spectra do not exhibit imaginary frequencies, confirming the thermodynamic and dynamic stability of these carbides. Additionally, the calculated Poisson's ratios and B/G ratios suggest that these materials are brittle and contain covalent and ionic bonds—a conclusion further corroborated by analyses of electronic structures. The elastic anisotropy is illustrated through anisotropy indices, three-dimensional (3D) surface configurations, and two-dimensional (2D) projections, with the sequence of anisotropy being: Nb2AlC > Ta2AlC > V2AlC. Further investigations into the sound velocities, Debye temperature, anisotropy of sound velocity, thermal conductivity, and optical properties of the M2AlC (M = V, Nb, Ta) series provide substantial theoretical support for the research on MAX phase M2AlC (M = V, Nb, Ta) series carbides.

Anisotropic odd elasticity with Hamiltonian curl forces

Abstract A host of elastic systems consisting of active components exhibit path-dependent elastic behaviors not found in classical elasticity, which is known as odd elasticity. Odd elasticity is characterized by antisymmetric (odd) elastic modulus tensor. Here, from the perspective of geometry, we construct the Hamiltonian formalism to show the origin of the antisymmetry of the elastic modulus that is intrinsically anisotropic. Furthermore, both non-conservative stress and the associated nonlinear constitutive relation naturally arise. This work also opens the promising possibility of exploring the physics of odd elasticity in dynamical regime by Hamiltonian formalism.

Single-crystal elastic moduli, anisotropy, and the B1–B2 phase transition of NaCl at high pressures

Pressure dependences of the highest and lowest possible longitudinal sound velocities in single crystals of the B1 and B2 phases of NaCl were extracted from examination of their polycrystalline samples using the technique of time-domain Brillouin scattering. Based on the data collected up to 41 GPa, we largely extended the pressure range where single-crystal elastic moduli, Cij(P), and elastic anisotropy of the two cubic phases are measured, especially the experimental data for the B2 phase formed upon the reconstructive phase transition. The B1 phase of NaCl exhibits strong and growing anisotropy with increasing pressure, while that of the B2 phase is much weaker. Comparing with the previous experimental Cij(P) of other compounds exhibiting or expected to exhibit the B1–B2 phase transition, such as CaO, MgO, and (Mg1−x, Fex)O, we concluded that the transition is initiated by the shear instability due to violation of the Born stability criterion [C44(P)–P] > 0 and predicted the presently-not-verified transition pressures for MgO and (Mg1−x, Fex)O.

Phase-field approach for precise fracture tracking in anisotropic rocks: Integrating orthotropy-based energy decomposition and two-fold symmetric fracture toughness

Rock formations are known to exhibit material anisotropy, both in terms of elastic and fracture properties. This means that the fracture path in such formations is not a priori known but rather a complex unknown that requires robust numerical techniques to predict accurately. In this context, the phase-field model is considered particularly effective, provided that certain physical considerations are carefully adjusted to align with the physics of the problem. While addressing elastic anisotropy is well-established, the tension–compression asymmetry necessary to inhibit crack interpenetration in phase-field fracture models needs to account for the specific material anisotropy. Additionally, to accurately capture crack propagation, it is critical to simultaneously account for orientation-dependent fracture toughness in such materials. To address this, the present study employs an anisotropic phase-field model that integrates the generalized spectral decomposition proposed in the literature for orthotropic materials with a two-fold symmetric fracture toughness, to predict the fracture trajectories in rock-type samples under fixed mixed-mode loading ratios. While each of the two aspects has primarily been applied to model orthotropic plates under simple tensile and shearing loading conditions in the literature, here we study their applicability in complex loading scenarios. To this end, the experimental data from notched semi-circular specimens of Grimsel Granite undergoing complex mixed-mode loading obtained in our previous work is considered. We focus on two given mode-mixity ratios and perform numerical studies. Our results emphasize the importance of considering this generalized decomposition for phase-field modeling of fracturing in rock-type materials, particularly under loading conditions where the crack might otherwise be unrealistically driven into the compressive region. Although certain features are well captured by considering anisotropy in elasticity alone, our findings demonstrate that incorporating a two-fold symmetric fracture toughness proves to be advantageous for more precise tracking of the fracture path.

Comparison of the Young’s modulus of the lead free solder alloy Sn-Ag3.8-Cu0.7 determined by hot tensile tests, ultrasonic measurements and [formula omitted]-Sn single crystal calculations

Sn-based solders are known for their tendency to form coarse grained microstructures. In combination with the high elastic anisotropy of β-Sn, the overall elastic properties and their interpretation require a careful discussion of the structure–property-relationship as elastic constants are often important input parameters for creep or fatigue models. This study therefore investigates the influence of microstructure and testing method on the Young’s modulus E for the widespread lead free solder alloy Sn-Ag3.8-Cu0.7 (SAC387) using bulk specimens. Due to its already high homologous temperature at room temperature (Thom≈ 0.6 = T/Tm for Tm being the melting temperature), hot tensile tests generally bear the risk of superimposed creep deformation. Mechanical testing becomes even more challenging, since yield strengths are usually low for these alloys. Consequently, in addition to the hot tensile tests executed for the engineering strain rate ϵ̇e=1⋅10−3 s−1, two supplemental methods are used to determine the Young’s modulus comprising of the dynamic resonance frequency measurement and the calculation of the Young’s modulus based on single crystal compliance data of the majority phase β-Sn.Young’s moduli yielded from dynamic resonance frequency measurement and calculations based on single crystal compliance data showed comparable results of (E35 °C≈ 55 GPa, E80 °C≈ 51 GPa and E125 °C≈ 48 GPa). Hot tensile tests showed similar data with the largest deviation at 80 °C, where E80 °C≈ 53 GPa was determined. These absolute values and their temperature dependence can be attributed to the microstructure of the cast specimens which show a general preferred orientation of β-Sn grains close to <110> after analysis via electron backscattered diffraction (EBSD). A comparison with literature data revealed significant differences in the Young’s moduli which are likely to be attributed to differences in the preferred orientation of β-Sn grains.

The influence of substantial intragranular orientation gradients on the micromechanical response of heavily-worked material

In this study, we employ high-energy X-ray characterization to examine the role of relatively large amounts of intragranular lattice misorientation – present after many thermomechanical processes – on the micromechanical response of Al-7085 with a modified T7452 temper. We utilize near-field high energy X-ray diffraction microscopy (HEDM) to measure three-dimensional (3D) spatial orientation fields, facilitated by a novel method that utilizes grain orientation envelopes measured using far-field HEDM to enable reconstruction of grains with intragranular orientation spreads greater than 20°. We then assess the consequences of consideration of intragranular orientation fields on the predicted deformation response of the sample through 3D micromechanical simulations of the forged Al-7085. We construct two virtual polycrystalline specimens for use in simulations: the first a faithful representation of the HEDM reconstruction, the second a microstructure with no intragranular misorientation (i.e., grain-averaged orientations). We find significant differences in the predicted deformation mechanism activation, distribution of stress, and distribution of plastic strain between simulations containing intragranular misorientation and those with grain-averaged orientations, indicating the necessity for consideration of intragranular orientation fields for accurate predictions. Further, the influence of elastic anisotropy is discussed, along with the effects of intragranular misorientation on fatigue life through the calculation and analysis of fatigue indicator parameters.

ECOMAN: an open-source package for geodynamic and seismological modelling of mechanical anisotropy

Abstract. Mechanical anisotropy related to rock fabrics is a proxy for constraining the Earth's deformation patterns. However, the forward and inverse modelling of mechanical anisotropy in 3D large-scale domains has been traditionally hampered by the intensive computational cost and the lack of a dedicated, open-source computational framework. Here we introduce ECOMAN (Exploring the COnsequences of Mechanical ANisotropy), a software package for modelling strain- and stress-induced rock fabrics and testing the effects of the resulting elastic and viscous anisotropy on seismic imaging and mantle convection patterns. Differently from existing analogous software, ECOMAN can model strain-induced fabrics across all mantle levels and is optimised to run efficiently on multiple CPUs. It also enables modelling of shape preferred orientation (SPO)-related structures that can be superimposed over lattice/crystallographic preferred orientation (LPO/CPO) fabrics, which allows the consideration of the mechanical effects of fluid-filled cracks, foliated and lineated grain-scale fabrics, and rock-scale layering. One of the most important innovations is the Platform for Seismic Imaging (PSI), a set of programs for performing forward and inverse seismic modelling in isotropic–anisotropic media using real or synthetic seismic datasets. The anisotropic inversion strategy is capable of recovering parameters describing a tilted transversely isotropic (TTI) medium, which is required to reconstruct 3D structures and mantle strain patterns and to validate geodynamic models.

Determination of elastic modulus and anisotropy for rocks using digital drilling method

The main purpose of this study is to provide a digital drilling-based method for determining rock elastic modulus and evaluating the anisotropy. By considering a concept of crushed zone, the drilling response was characterized as the function of specific energy and drilling strength. On this basis, an equivalent elastic modulus was proposed to evaluate the rock elastic modulus. Moreover, an anisotropy index was proposed according to the difference in specific energy under different borehole directions. Digital drilling experiments were conducted for three lithologies at drilling directions namely 0, 60, and 120° (Based on vertical orientation), revealing the anisotropic effect exhibited in the rock. To validate the measurement of rock elastic modulus and its anisotropy, the compressive tests under same conditions were carried out. The results suggested that the proposed indicator provided an accurate evaluation for the rock's elastic modulus, with an error margin of less than 10%. Drilling-based anisotropy is identified in the drilling response, especially the critical state between cutting and friction. The statical indicator CoV (coefficient of variation) was utilized to characterize the anisotropy along the borehole. This research's innovation lies in providing an approach to evaluating rock anisotropy and determining its elastic modulus by considering the drilling response.

A potential mechanism for abnormal grain growth in Ni thin films on c-sapphire

Normal grain growth (NGG) of a (111) textured Ni film on c-sapphire and abnormal grain growth (AGG) of (100) grains at the expense of this (111) texture has been studied as a function of temperature with and without a capping layer. The grain boundaries (GBs) in the Ni film are controlled by the preferred orientation relationships (ORs) adopted by the Ni grains on the sapphire substrate. The 2 variants of a single OR, Ni(111)<11¯0>//Al2O3(0001)<11¯00>, form a (111) mazed bicrystal with Σ3 GBs. The (100) grains have a single OR, Ni(100)<010>//Al2O3(0001)<11¯00> with 3 variants; their GBs within the (111) grains have the (111)<11¯0>//(100)<010> misorientation.(100) AGG within the (111) mazed bicrystal of the 100 nm Ni film takes place above 1023 K. The orientation transition is driven by the biaxial elastic modulus anisotropy which favors the growth of (100) grains over (111) grains, as this reduces the elastic strain energy induced by the thermal mismatch between Ni and sapphire. (100) AGG is suppressed and the NGG of the (111) texture is slowed down when the film is covered by a 10 nm amorphous alumina layer aimed at inhibiting surface diffusion. Thus, it is proposed that as long as the surface can act as a sink for the point defects diffusing along the GBs, the movement of the GBs is correlated to the diffusivity of atoms and vacancies, which is a function of their misorientation and crystallographic GB structure.

Modeling the effects of initial grain size, martensitic transformation induced dynamic grain refinement, phases, and texture on strength of a high entropy alloy using crystal plasticity

Microstructurally flexible high entropy alloys (HEAs) can be tailored for strength via dual-phase strengthening mechanisms, which originate from the strain-induced phase transformation microstructural phenomena. One such alloy is a recently synthesized Fe42Mn28Co10Cr15Si5 (at%) HEA consisting of metastable gamma austenite (γ), stable epsilon martensite (ε), and stable sigma (σ) phases. In this work, a crystal plasticity homogenization incorporating a physically based phase transformation and hardening models is used to interpret and better understand the effects of strain induced phase transformation phenomena on the overall hardening response of the alloy. The transformation model is conceived based on the grain-scale stress sensitive motion of partial dislocations forming shear bands, while the hardening model is an extended Kocks-Mecking-type dislocation density-based formulation sensitive to grain size and shape. The deformation of constituent grains per phase in the alloy is modeled as a combination of anisotropic elasticity, crystallographic glide, and γ→ε phase transformation. Parameters of the hardening and transformation models within the homogenization are calibrated and validated on a suite of data including flow stress curves and phase fractions measured under compression, while the corresponding texture evolution data is used solely for verification. Good predictions of the model elucidate that the transformation induced dynamic Hall-Petch-type barrier effect is the primary origin of strain hardening along with the increase in the ε-phase fraction and dislocation density, while the individual strength of the phases gives rise to the overall strength. The modeling framework described in the present work is expected to be applicable to other HEA systems.

Designing spongy-bone-like cellular materials: Matched topology and anisotropy

Bone is a natural material with properties such as high specific stiffness and strength. These exceptional mechanical properties are attributed to the meso-scale structure and elastic anisotropy of spongy bone. Replicating the topological traits and mechanical properties of spongy bone presents a novel opportunity to develop high-performance cellular materials. To achieve this, we propose an innovative framework for designing biomimetic cellular materials that match the trabecular structure and elastic anisotropy of spongy bone. This framework introduces a forward-flow design process that utilizes gradient-based feature tuning on a low-dimensional feature vector, transforming the complex inverse design problem into an efficient iterative process. A key innovation in our approach is the use of a pre-trained generative model, SliceGAN, to reconstruct 3D unit cells from 2D micro-CT images. This significantly reduces the cost and time associated with traditional layer-by-layer CT scans typically required for 3D training data. Numerical homogenization is then used to determine the effective elastic stiffness matrix, and a Fourier neural operator is trained to predict these matrices efficiently, greatly enhancing the computational efficiency of the design process. Using this framework, we successfully designed unit cells with topological traits and elastic anisotropy that closely approximate those of natural spongy bone. This opens new avenues for developing spongy-bone-mimetic cellular materials with exceptional mechanical properties. Moreover, the framework's versatility allows it to be extended to the design of other bio-inspired cellular materials.

Impact of Hydrogen Cage Occupancy on the Mechanical Properties and Elastic Anisotropies of sII Hydrates

In this study, we investigate the composition-dependent mechanical properties and elastic anisotropies of sII hydrogen hydrates using first-principles method. The evaluation of the elastic moduli and their direction dependency is achieved by computing the second-order elastic constants (SOECs) of the unit lattice. The various trends of elastic constants with the hydrogen composition of the cages introduces variations in the bonding strength, compressibility, stiffness, and shear properties of the structure which are captured by the Poisson's ratio, bulk, Young, and shear moduli, respectively. Elastic properties were found to be significantly influenced by the system's anisotropy, arising from the geometry of the cages and their unique arrangement within the lattice being affected by the increase in the hydrogen occupancy of the cages. The detailed analysis of elastic anisotropies revealed shifts in the strongest and weakest directions of the material with varying the hydrogen content of the cages. The Poisson's ratio captures the anisotropic bonding strengths within the crystal structure with the hydrogen composition of the lattice, explaining the reason behind the existence of strongest and weakest directions in terms of compression, tension, and shear forces. Taken together the established structure-property-composition relations will be useful in the design and optimization of hydrogen sII hydrates for energy storage applications.

Planar structured materials with extreme elastic anisotropy

Designing anisotropic structured materials by reducing symmetry results in unique behaviors, such as shearing under uniaxial compression. While rank-deficient materials such as hierarchical laminates have been shown to exhibit extreme elastic anisotropy, there is limited knowledge about the fully anisotropic elasticity tensors achievable with single-scale fabrication techniques. No established upper and lower bounds on anisotropic moduli exist. In this paper, we estimate the range of anisotropic stiffness tensors achieved by single-scale two-dimensional structured materials. We first develop a database of periodic anisotropic single-scale unit cell geometries using linear combinations of periodic cosine functions. The database covers a wide range of anisotropic elasticity tensors, which are then compared with the elasticity tensors of hierarchical laminates. We identify the regions in the property space where hierarchical design is necessary to achieve extremal properties. We demonstrate a method to construct various 2D functionally graded structures using this cosine function representation. These graded structures seamlessly interpolate between unit cells with distinct patterns, allowing for independent control of several functional gradients, such as porosity, anisotropic moduli, and symmetry. When designed with unit cells positioned at extreme parts of the property space, these graded structures exhibit unique mechanical behaviors such as selective strain energy localization, compressive strains under tension, and localized rotations.