Bulk Modulus Of Materials Research Articles

A stabilized mixed three‐field formulation for stress accurate analysis including the incompressible limit in finite strain solid dynamics

AbstractIn this work a new methodology for finite strain solid dynamics problems for stress accurate analysis including the incompressible limit is presented. In previous works, the authors have presented the stabilized mixed displacement/pressure formulation to deal with the incompressibility constraint in finite strain solid dynamics. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelastic material model. The incompressible limit is attained automatically depending on the material bulk modulus. This work exploits the concept of mixed methods to formulate stable displacement/pressure/deviatoric stress finite elements. The final goal is to design a finite element technology able to tackle simultaneously problems which may involve incompressible behavior together with a high degree of accuracy of the stress field. The variational multi‐scale stabilization technique and, in particular, the orthogonal subgrid scale method allows the use of equal‐order interpolations. These stabilization procedures lead to discrete problems which are fully stable, free of volumetric locking, stress oscillations and pressure fluctuations. Numerical benchmarks show that the results obtained compare very favorably with those obtained with the corresponding stabilized mixed displacement/pressure formulation.

Bézier extraction based isogeometric approach to multi‐objective topology optimization of periodic microstructures

AbstractThis article presents the Bézier extraction based isogeometric approach to multi‐objective topology optimization of periodic microstructures. The approach utilizes the B‐splines based Bézier elements as the finite element representation at both macro and micro levels. The equivalent elastic properties of the representative volume element (RVE) are computed by the numerical homogenization method with the periodic boundary conditions on the control points (CPs) of the B‐spline mesh. The multi‐objective optimization problems including the material bulk modulus maximization, negative Poisson's ratio, and concurrent topology optimization of the composite structures are formulated by using the Bézier elements based isogeometric analysis. The distance‐based density distribution on the CPs of the B‐spline mesh is proposed as the initialization for designing the RVE, which is more robust than the uniform density distribution towards the optimal results. Several numerical examples are presented to illustrate the effectiveness of the proposed approach, and a variety of isotropic and anisotropic RVEs and composite structures are obtained. Meanwhile, various influences on the optimal design of the RVE and macrostructure are also discussed.

Equation of State and Entropy Theory Approach to Thermodynamic Scaling in Polymeric Glass-Forming Liquids

We show that thermodynamic scaling can be derived by combining the Murnaghan equation of state (EOS) with the generalized entropy theory (GET) of glass formation. In our theory, thermodynamic scaling arises in the non-Arrhenius relaxation regime as a scaling property of the fluid configurational entropy density $s_c$, normalized by its value $s_c^*$ at the onset temperature $T_A$ of glass formation, $s_c / s_c^*$, so that a constant value of $TV^{\gamma}$ corresponds to a \textit{reduced isoentropic} fluid condition. Molecular dynamics simulations on a coarse-grained polymer melt are utilized to confirm that the predicted thermodynamic scaling of $\tau_{\alpha}$ by the GET holds both above and below $T_A$ and to test whether the extent $L$ of stringlike collective motion, normalized its value $L_A$ at $T_A$, also obeys thermodynamic scaling, as required for consistency with thermodynamic scaling. While the predicted thermodynamic scaling of both $\tau_{\alpha}$ and $L/ L_A$ is confirmed by simulation, we find that the isothermal compressibility $\kappa_T$ and the long wavelength limit $S(0)$ of the static structure factor do not exhibit thermodynamic scaling, an observation that would appear to eliminate some proposed models of glass formation emphasizing fluid `structure' over configurational entropy. It is found, however, that by defining a low temperature hyperuniform reference state, we may define a compressibility relative to this condition, $\delta \kappa_T$, a transformed dimensionless variable that exhibits thermodynamic scaling and which can be directly related to $s_c / s_c^*$. Further, the Murnaghan EOS allows us to interpret $\gamma$ as a measure of intrinsic anharmonicity of intermolecular interactions that may be directly determined from the pressure derivative of the material bulk modulus.

Nonlinear and linearized gray box models of direct-write printing dynamics

PurposeThe purpose of this study is to develop nonlinear and linearized models of DW printing dynamics that capture the complexity of DW while remaining integrable into control schemes. Control of material metering in extrusion-based additive manufacturing modalities, such as positive displacement direct-write (DW), is critical for manufacturing accuracy. However, in DW, transient flows are poorly controlled due to capacitive pressure dynamics – pressure is stored and slowly released over time from the build material and other compliant system elements, adversely impacting flow rate start-ups and stops. Thus far, modeling of these dynamics has ranged from simplistic, potentially omitting key contributors to the observed phenomena, to highly complex, making usage in control schemes difficult.Design/methodology/approachThe authors present nonlinear and linearized models that seek to both capture the capacitive and nonlinear resistive fluid elements of DW systems and to pose them as ordinary differential equations for integration into control schemes. The authors validate the theoretical study with experimental flow rate and material measurements across a range of extrusion nozzle sizes and materials. The authors explore the contribution of the system and build material bulk modulus to these dynamics.FindingsThe authors show that all tested models accurately describe the measured dynamics, facilitating ease of integration into future control systems. Additionally, the authors show that system bulk modulus may be substantially reduced through appropriate system design. However, the remaining build material bulk modulus is sufficient to require feedback control for accurate material delivery.Originality/valueThis study presents new nonlinear and linear models for DW printing dynamics. The authors show that linear models are sufficient to describe the dynamics, with small errors between nonlinear and linear models. The authors demonstrate control is necessary for accurate material delivery in DW.

A stabilized mixed finite element approximation for incompressible finite strain solid dynamics using a total Lagrangian formulation

In this work a new methodology for both the nearly and fully incompressible transient finite strain solid mechanics problem is presented. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelastic material model. The incompressible limit is attained automatically depending on the material bulk modulus. The system is stabilized by means of the Variational Multiscale-Orthogonal Subgrid Scale method based on the decomposition of the unknowns into resolvable and subgrid scales in order to prevent pressure fluctuations. Several numerical examples are presented to assess the robustness and applicability of the proposed formulation.

Relaxation of Extracellular Matrix Forces Directs Crypt Formation and Architecture in Intestinal Organoids.

Intestinal organoid protocols rely on the use of extracellular scaffolds, typically Matrigel, and upon switching from growth to differentiation promoting media, a symmetry breaking event takes place. During this stage, the first bud like structures analogous to crypts protrude from the central body and differentiation ensues. While organoids provide unparalleled architectural and functional complexity, this sophistication is also responsible for the high variability and lack of reproducibility of uniform crypt-villus structures. If function follows form in organoids, such structural variability carries potential limitations for translational applications (e.g., drug screening). Consequently, there is interest in developing synthetic biomaterials to direct organoid growth and differentiation. It has been hypothesized that synthetic scaffold softening is necessary for crypt development, and these mechanical requirements raise the question, what compressive forces and subsequent relaxation are necessary for organoid maturation? To that end, allyl sulfide hydrogels are employed as a synthetic extracellular matrix mimic, but with photocleavable bonds that temporally regulate the material's bulk modulus. By varying the extent of matrix softening, it is demonstrated that crypt formation, size, and number per colony are functions of matrix softening. An understanding of the mechanical dependence of crypt architecture is necessary to instruct homogenous, reproducible organoids for clinical applications.

A new method for an old topic: Efficient and reliable estimation of material bulk modulus

Bulk modulus is a key material property which provides fundamentals on the chemical bonding nature of a material as well as for deriving other related material properties, such as Young’s modulus and Grüneinsen constant. Traditional numerical methods of estimating bulk modulus often involve layers of approximations and cost considerably. In this work, we propose a novel and efficient numerical framework to estimate bulk modulus using the ab initio calculated data. Based on the generalized polynomial chaos expansion (gPC) method, our approach does not impose any physical priori and is mathematically rigorous and versatile. We have demonstrated the reliability and efficiency of the proposed method by estimating the bulk modulus of Ti3SiC2 and Sb2Te3 with comparison to conventional methods.

Higher order derivatives of bulk modulus of materials at infinite pressure

Pressure derivatives of bulk modulus of materials at infinite pressure or extreme compression have been studied using some basic principles of calculus. Expressions for higher order pressure derivatives at infinite pressure are obtained that are found to have the status of identities. A generalized formula is derived for the nth-order pressure derivative of bulk modulus in terms of the third-order Grüneisen parameter at infinite pressure.

Interphase zone effect on the spherically symmetric elastic response of a composite material reinforced by spherical inclusions

This work deals with the problem of modeling the effect due to an interphase zone between inclusion and matrix in particulate composites to better estimate the bulk modulus of materials with inclusions. To this end, in this paper the problem of a body containing a hollow or solid spherical inclusion subjected to a spherically symmetric loading is investigated in the framework of the elasticity theory. The interphase zone around the inclusion is modeled by considering the elastic properties varying with the radius moving away from the interface with inclusion and, asymptotically approaching the value of the homogeneous matrix. The explicit solutions are obtained in closed form by using hypergeometric functions and numerical investigations are performed to highlight the localized effects of the graded interphase in the stress transfer between inclusion and matrix. Finally, the exact solutions are used to estimate the effective bulk modulus of a material containing a dispersion of hollow or solid spherical inclusions with graded interphase zone.

Effective bulk moduli of materials containing stochastically distributed nano‐inhomogeneities with surface stresses

AbstractIn the present contribution, a mathematical model for the investigation of the effective properties of a material with randomly distributed nano‐particles is proposed. The surface effect is introduced via Gurtin‐Murdoch equations describing properties of the matrix/nano‐particle interface. They are added to the system of stochastic differential equations formulated within the framework of linear elasticity. The homogenization problem is reduced to finding a statistically averaged solution of the system of stochastic differential equations. These equations are based on the fundamental equations of linear elasticity, which are coupled with surface/interface elasticity accounting for the presence of surface tension. Using Green's function this system is transformed to a system of statistically non‐linear integral equations. It is solved by the method of conditional moments. Closed‐form expressions are derived for the effective moduli of a composite consisting of a matrix with randomly distributed spherical inhomogeneities. The radius of the nano‐particles is included in the expression for the bulk moduli. As numerical examples, nano‐porous aluminum and nano‐porous gold are investigated assuming that only the influence of the interface effects on the effective bulk modulus is of interest. The dependence of the normalized bulk moduli of nano‐porous aluminum on the pore volume fraction (for certain radii of nano‐pores) are compared to and discussed in the context of other theoretical predictions. The effective Young's modulus of nano‐porous gold as a function of pore radius (for fixed void volume fraction) and the normalized Young's modulus vs. the pore volume fraction for different pore radii are analyzed. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

ANALYSIS OF SELECTED CONTACT ALGORITHMS TYPES IN TERMS OF THEIR PARAMETERS SELECTION

The analysis of many engineering problems involves not only deformation of the considered system, but occurrence of the interaction between the individual separate elements of the system as well. The occurrence of friction is the most common phenomenon occurring during this interaction. In the case of intense friction, the heat released in this process is also important. In computer methods of mechanics, the process of interaction between bodies is carried out using special algorithms. The most frequently applied are: the penalty method, the barrier method, direct elimination of constraints, the Lagrange multiplier method, the perturbed Lagrangian method, the augmented Lagrangian method, Nitsche method. Owing to its easy implementation process, an approach based on a penalty function is often applied. In this approach, the contact between the bodies can be identified with the presence of the spring between the elements of the bodies in the contact. The stiffness of the spring depends on: material bulk modulus, face area, volume or shell diagonal and a numerically selected scale factor. The article will present the results of analyses that will allow fast and easy selection of its value. In the analyses there were presented the results considering the basic types of contacts: node to surface, surface to surface and surface to surface mortar.

Structural and Mechanical Properties of RT<sub>2</sub>Zn<sub>20</sub> and RFe<sub>2-x</sub>Co<sub>x</sub>Zn<sub>20</sub>(R=Y, Sm; T=Fe, Ru, Os, Co, Rh and Ir)

The phase stability, crystal structure and mechanical properties of YT2Zn20 and SmT2Zn20 (T=Fe, Ru, Os, Co, Rh and Ir) compounds have been investigated by using interatomic potentials based on the lattice inversion technique. The calculated lattice constants are in good agreement with the experimental data. The lattice constants increase and Bulk modulus decrease as the transition metal T varies from 3d to 5d. The Y-based compounds with lower energies are more stable than the Sm analogs. Also, the Bulk modulus of YT2Zn20 series are larger than SmT2Zn20 series. Moreover, the mechanical properties of the quaternary compounds YFe2-xCoxZn20 and SmFe2-xCoxZn20­, such as the elastic constants and bulk modulus, have been calculated in this work. The substitution of Co atoms would decrease the cohesive energies and increase the bulk modulus of materials.

Structural properties, electric response and electronic feature of perovskite

It has been observed that the semiconducting compound SnO2 presents very good results as gas sensor. One important development has been performed to study perovskite oxides for this relevant application. One oxide material which constitutes an excellent candidate for this technological application is BaSnO3. Polycrystalline samples with single phase of BaSnO3 were synthesized by using the solid state reaction method. Samples were characterized structurally by means of X-ray diffraction (XRD) technique. Rietveld refinement, by using the GSAS code, reveals that this material synthesizes in a cubic perovskite, space group Pm3m (#221), with lattice parameter 4.1190(3)A˚. Electric response was examined through the impedance spectroscopy technique. Results of Bode diagram, from an equivalent circuit, evidence the semiconductor character of material. We carried out a theoretical study by means of the calculation of the bands diagram and the density of states of the BaSnO3. Calculation was performed by employing the density functional theory (DFT), with the generalized gradient approach (GGA). DFT theory permitted to establish that BaSnO3 material has an indirect semiconducting behavior. The calculated gap for this perovskite-like stagnate is at least 0.4 eV. Bulk modulus for material was also determined to be 132 GPa.

Calculations showing a correlation between electronic density and bulk modulus in fcc and bcc metals

The linearized augmented-plane-wave method is used to study the electronic density nw, at the boundary of the Wigner-Seitz cell for bce and fee elemental metals. The calculated n(ws) is generally in good agreement with the results derived from an empirical formula that was introduced by Miedema and co-workers. A formula is introduced to calculate the bulk modulus of materials using n(ws), and from it another formula for calculating the relationship between the external pressure and the volume of materials is derived. As an example, the pressure-volume compression curve for Cu is calculated. The limitations and features of the conventional method and our method for calculating the bulk modulus of materials are discussed.

Consistent finite element procedures for nonlinear rubber elasticity with a higher order strain energy function

The use of higher order terms in the Rivlin's polynomial strain energy density function is necessary to describe the elastic behavior of rubber undergoing very large and complex deformation. In this paper, the material response tensor for general Rivlin's strain energy density function is derived in a consistent manner such that both major and minor symmetries are retained. Lack of minor symmetry in the material response tensor will lead to numerical convergence difficulties, especially in shear dominant problems. The projection method is used to avoid volumetric locking due to the nearly incompressible nature of rubber. The relation between the numerical penalty number and the material bulk modulus is characterized. The importance of this relation is demonstrated in the study of the apparent Young's modulus of bonded rubber units. The need to include higher order terms in the strain energy density function is presented in the numerical examples. Several classical elasticity problems are also analyzed.

On the bulk modulus of particulate composites

The present study contains numerical evaluations of the macroscopic bulk modulus of materials containing spherical particles. The effects of load, volume fraction of particles, and material properties of the constituents with particular emphasis on the effect of plasticity on the bulk modulus are explored. It is shown that the bulk modulus of such composites is often load dependent.

Variation of whole and fractured porous rock permeability with confining pressure

Phenomenological models have been devised to determine the variation with pressure of the permeability of whole and fractures porous rock. For whole porous rock, the permeability variation with pressure is based upon the Hertzian theory of deformation of spheres. This model gives a permeability variation with pressure given by k(P) = k 0 {1 − C 0[(P + P 1)/P 0] sol2 3 } 4 where k 0 is the initial permeability of the loose-grain packing, C 0 is a constant depending upon the packing (and is of the order of 2), P 1 is the ‘equivalent pressure’ due to the cementation and permanent deformation of the grains and P 0 is the effective elastic modulus of the grains (and is of the order of the grain material bulk modulus). The permeability variation of a fracture (or fractured rock) with confining pressure is determined by using a ‘bed of nails’ model for the asperities of the fracture. Its functional dependence is k( P) = k 0 [1 − ( P/ P 1) m ] 3 where k o is the zero pressure permeability, P 1 is the effective modulus of the asperrities (and is of the order of one-tenth to one-hundredth of the asperity material bulk modulus) and m is a constant (0 < m < 1) which characterizes the distributions function of the asperity lengths. The above expression assumes a simple power-law variation for the asperity-length distribution. More complicated asperity-length distributions can be used, but the data quality and the fracture-to-fracture variability does not warrant the use of such distributions. A comparison of experimental data with the theoretical curves shows good correlation between the two and gives reasonable values for the constants k 0, P 1 and m.