# Relationship between elastic constants controversy

### Charting the complete elastic properties of inorganic crystalline compounds

23 by permission of Ohmsha]. schools quarreled over these elastic constants. that such a polycrystal has Poisson's ratio of l/3 on an average instead of l/4. Thus, through the elastic constant controversy, the mathematical theory of elasticity. Another such relation is the subject of Problem To prove (1) we proceed in two steps. Consider first a cube of side length a that is exposed to shear stress. The chapter discusses the bulk modulus, Poisson's ratio, Young's modulus, . of the motion connected with the nutatioii is still controversial (see Section ).

Despite the importance of the elastic tensor, experimental data for this quantity is available for only a very small subset of all known inorganic compounds. This presents a fundamental bottleneck for the discovery and design of materials with targeted thermal and mechanical properties, or for performing continuum simulations of mechanical response that require elastic moduli as input. Considering only materials for which the full tensor of elastic coefficients is available, the classical works have references that sum up to a total number of around independent systems for which experimental measurements have been compiled 18— Considering papers that have investigated elastic constants of particular systems, this number might be twice as large, which is a very small fraction of the approximately 30, to 50, entries for ordered compounds in the inorganic crystal structure database 27— Among the systems for which experimental data is available are approximately 70 pure elements, with the remainder consisting of binary systems and—to a much smaller extent—ternary systems and a variety of complex minerals.

## Poisson's ratio over two centuries: challenging hypotheses

Among the binary materials are solid solutions and compounds, the latter often being ordered intermetallic compounds. One challenge associated with using published experimental data for elastic moduli is that the spread in the reported values for a given system can be quite large, depending on the details of the experimental conditions and techniques employed.

Efforts aimed at developing databases of elastic moduli from first-principles computational methods have been undertaken in previous work e. Such a computational approach provides an advantage that all of the data can be derived in a consistent manner, facilitating comparisons across materials chemistries.

In the present work we expand on this approach. Specifically, we present here the to-date largest database of calculated elastic properties of crystalline inorganic compounds, ranging from metals and metallic compounds to semiconductors and insulators.

These calculations are part of a high-throughput HT effort 36undertaken within the framework of the Materials Project MP www. The database of elastic tensors currently consists of over 1, materials and is being updated regularly.

The elastic properties are obtained using first-principles quantum-mechanical calculations based on Density Functional Theory DFT. The remainder of the paper is organized as follows. We then give an overview of the structure of the data, followed by a description of our results. Finally, we describe the verification and validation tests to assess the precision and accuracy of the chosen density functional and the HT algorithms employed in the calculations.

Methods Generation of elasticity data In this launch of the elastic constant database we tabulate results for a subset of 1, compounds chosen from those present in the current MP database.

This subset includes 2 broad categories: These properties have been calculated previously by DFT using the standard HT-procedure and chosen MP parameters suitable for ground-state energy, lattice structure, and band structure 37 The definition of these behaviours in an FEA model relies on the usage of constants, such as the elastic modulus Ein the constitutive equations of the material. These constants have a specific numerical value for each different material.

The most common and simplest way to obtain the constants is by applying destructive tests to the specimens see a review of mechanical tests for the study if the mechanical behaviour of bones in Sharir et al. Destructive tests were performed until failure to record the materials behaviour under various loads and to make it possible to define the constitutive equation of the material.

Nevertheless, in biological sciences, and especially in vertebrate structures, preserving the integrity of the specimens is mandatory when we have limited specimens to analyse. In this case, nondestructive tests characterised by no damage should be used. In line with this, ultrasonic testing is a nondestructive, successful technique that has been used in engineering fields to determine the elastic constants of materials, included the elastic constants of bone, to determine the constitutive equations of materials Rose et al.

In spite of the development of these technologies for determining the materials properties of bone, knowledge is lacking about most of the extant species in the animal kingdom, and it is obviously impossible to test the mechanical properties of all of the fossil record. This fact leads researchers to rely on hypotheses about materials considerations in the FEA models.

It is common to use the materials properties of reliable, related data - for example Neenan et al. In addition, these assumptions must be qualified by lengthy discussion, justification and references citing the assumed materials properties. Particularly, this lack of information means that, for example in Fortuny et al. At this point, we demonstrate that in most of the cases, there is no necessity to hypothesise about materials or create justifications.

In certain cases, the mechanical performance characterised by the stresses, strains and displacements is not directly affected by the type of material employed in the simulations.

Finally, we elucidate when and why materials data are relevant and when the selection of these data can be ignored.

Elasticity is the tendency of solid materials to return to their original shape, after being deformed, when this load is released. Elastic problems are driven by known equations of equilibrium Mase and Mase, with three relevant quantities that meet the equations: The desired situation for biological systems is the elastic behaviour when no permanent deformation appears after loading. For elastic bodies, in the absence of temperature variations, the strain energy - or internal work - represents the energy stored by the system undergoing deformation Equation 1: On the other side, the total external work is performed by the external forces that are loading the body Equation 2: Therefore, the total energy E of the body is defined in Equation 3 balancing the internal work strain energy U and the external work W: Therefore, equilibrium exists when there is a field of displacements u that fulfils the boundary conditions imposed in the problem and minimises the total energy.

The weak form is also known as the principle of virtual work Equation 4: Discretisation in Finite Elements There are few analytical solutions for a limited number of theoretical and unrealistic elastic problems that solve the principle of virtual work.

These solutions are mainly in very simple geometries and, hence, numerical solutions are necessary to solve the complex geometries that biological systems usually present. The finite element method is a good mathematical method to solve the weak form of the elastic problem Zienkiewicz, in any geometry and is adequate for complex geometries.

The method divides the continuum model into a discrete model by meshing the body with the so-called finite elements and enabling a matrix formulation. The displacement field u can be represented by Equation 5: Equation 5 makes it possible to compute the displacement at any point in the body given the values of its coordinates x, y, z. According to the elastic hypothesis, the strains are the derivatives of the displacements and can be obtained from Equation 6, where B is the matrix of the shape function derivatives: It is defined by Equation 7 for a continuum and in the discrete form: Therefore, rewriting the continuum equation Equation 4 in a discrete form using the matrix formulation, a system of equations results Equation 8.

Details can be found in Zienkiewicz et al. A material is orthotropic when there are three orthogonal planes of materials symmetry.

In this case, the material acts similarly in certain directions.

As mentioned earlier, most works performed in palaeobiology consider bone to have isotropic behaviour. This matrix form is relating all the stresses and the strains in Equation 7. Nevertheless, the value of E largely determines the solution of the system Equation 8. Therefore, the study developed in this work analyses the impact of using the two different models for bone properties: Bones as an isotropic, homogeneous material.

When a body is considered to have homogeneous properties, the entire body has the same constitutive equation, and Equation 11 can be defined with a proper elastic modulus E see Fortuny et al.

The solution of the equilibrium equation Equation 8 is: Bone as an isotropic, inhomogeneous material. Inhomogeneous properties mean that the body is composed of distributed masses with different isotropic properties See Aquilina et al. In this case, the expressions Equations must consider that the model is composed of i different materials. Every elastic modulus Ei can be referenced to a material E for which n i is a scalar value that is different for each i-material: This means that the reference value of the elastic modulus in the simulation of an inhomogeneous material is not relevant.

Nevertheless, notice that a strong condition has been used: This necessitates that the relative value between the materials must be maintained.

The value of n i must be known. Thus, when inhomogeneous materials are simulated, certain limitations over the values of the elastic modulus are assumed. The change in the value of the elastic modulus also induces an inverse proportion change for the displacements and the strains as in the previous analysis.

Principal stresses and strains.

### Relationships Between Elastic Constants

If stress values are identical according to Equations 20 and 32, therefore maximum and minimum values eigenvalues will perfectly match, and eigenvectors will be the same. Hence, the maximum and minimum values are scaled and eigenvalues depend on this scalar. Because values are scaled by a real number, eigenvectors will coincide and principle directions will not be affected.

Therefore, strain orientations will maintain and comparison can also be done. Another interesting metric that has been used in comparing the performance of different vertebrate models is the strain energy: Strain energy was defined previously in Equation 1 and for a material with elastic modulus E is: As points of interest, two points P the most mesial point of the first premolar at the alveolus and Q the most distal point of the third molar at the alveolus were placed in the jaws to record the numerical values of the results Figure 1.

Although there are several results produced by FEA different types of stresses, strains, displacements, etc. For this reason, in this work, we will display the results of the equivalent von Mises stress, the equivalent von Mises strain and the displacements when studying the influence of the elastic material properties.

- Charting the complete elastic properties of inorganic crystalline compounds

To study this influence, two FE analyses were performed. The first had a homogeneous, linear, elastic and isotropic material for the entire jaw, and the second shown in Figure 2 divided the bovine jaw into three different areas with three different linear, elastic and isotropic materials to generate an inhomogeneous materials distribution inside the jaw.

Homogeneous Material Plane models of Connochaetes taurinus and Alcelaphus buselaphus were solved with nine different values of elastic modulus E between 10 GPa and 50 GPa, which are normal values for bone Sharir et al. The von Mises stress distribution, von Mises strain distribution and displacements in the entire jaw are shown in Figure 2 for three different E values. The change in the coloured scale of the legend is according to the inverse proportionality between strains and displacements Equations 18 and 19and the same scale is retained for the von Mises stress Equation The results show that the displacement field, the strain distribution and the stress distribution are qualitatively identical and quantitatively proportional.

In the supplementary information Figure S1the von Mises stress and the von Mises strain distributions and the displacement field for the entire jaws are also given when the coloured scale in the legend is the same for the three cases. In this figure, it is clear that the values are dissimilar. The von Mises stress, strain and displacements at points P and Q figure 3 were recorded, and the variation in these values preceding the change in E values are shown in Figures 3.

This value and the same value, except for strains and displacements, are drawn in the figure preceding the variation in the elastic modulus. This shows that the relationship between the reference value of the elastic modulus MPa and the other values maintain the same proportion for all cases, which is constant for stresses and the inverse linear proportion of the elastic modulus for the strains and displacements.

The numerical results obtained for maximum and minimum principal stress and strains and for strain energy are also available in Table S3 and Table S4 of supplementary information. They show the same proportion with the material established previously. Inhomogeneous Material Plane models of Connochaetes taurinus and Alcelaphus buselaphus were divided into three areas where different bone properties were applied to define an inhomogeneous material in the entire jaw see Figure 1 with the three areas differentiated.

The von Mises stress distribution, von Mises strain distribution and displacements in the whole jaw are shown in Figure 4 for three different E values. The change in the coloured scale of the legend is according to the inverse proportionality between the strains and displacements Equations 30 and 31and the same scale is kept for the von Mises stress Equation The results show that the displacement field, the strain distribution and the stress distribution are qualitatively the same and quantitatively proportional.

In the supplementary information Figure S2 the von Mises stress and von Mises strain distributions and the displacement field for the entire jaws are also given when the coloured scale is the same for the three cases. In this figure, it is clear that the values are different.

The von Mises stress, strain and displacements in points P and Q were recorded, and the variation of these values preceding the change in the E values is shown in Figures 5. These values and the same values, except for strains and displacements, are drawn in the figure showing the variation in the elastic modulus.

This shows that the relationship between the reference value of the elastic modulus MPa and the other values maintains the same proportion for all cases, which is constant for the stresses and the inverse linear proportion of the elastic modulus for strains and displacements. The numerical results obtained for maximum and minimum principal stress and strains and for strain energy are also available in Table S7 and Table S8 of supplementary information.

In case study A, for an isotropic homogeneous material, the value of the elastic modulus is determinant for the values of the displacements and strains and is not determinant for the values of the stresses. Notice that the stress values of the examples Von Mises stresses and principal stresses in supplementary information are identical for any elastic modulus.

Moreover, for a comparison between the patterns of the strains and displacements, the elastic modulus might also be considered irrelevant. The displacement and strain values change Von Mises strains and principal strains and strain energy in supplementary informationbut these magnitudes maintain a common distribution pattern as Strait et al.

The proportionality of the values enables the comparison of computational models. This is important when comparing the von Mises stress distribution of two different bovine jaws as Connochaetes taurinus and Alcelaphus buselaphus in the absence of information on the material properties of the jaws.

A fair approach is to use relevant information from Reilly and Burstein, about the properties of bovine haversian bone. Nevertheless, assuming elastic behaviour, it is not necessary to have a close materials value because the results of the stress values and distribution will be identical regardless of the value chosen.

The stress results are not affected by the value of the elastic modulus. In the example cited in the introduction Fortuny et al.

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The decision to use this materials property in the FEA model was not crucial for the results because the values and the distribution of the von Mises stresses would be the same regardless of the materials definition.

And, although the values of the displacements obtained in the jaws will be different as a function of the values of the elastic modulus, in the different jaws, the displacement values maintain proportionality, and the pattern distribution can also offer interesting insights.

In case study B for an isotropic inhomogeneous material, the elastic modulus must be defined for each of the materials. As in the other case, a free choice of the elastic modulus values will induce an incorrect displacement and the exact value of the strain but not of the stress. The value of the stress does not depend on the elastic modulus.