Spherical Part Research Articles

Deviatoric couple stress theory and its application to simple shear and pure bending problems

Classical couple stress theory is indeterminate since the number of independent basic equations is inconsistent with that of field variables and the corresponding differential equation is not closed. The purpose of this paper is to remedy this gap and it is proven that the spherical part of the couple stress tensor vanishes. With the vanishing trace of the couple stress tensor as a premise or the couple stress tensor being deviatoric (or traceless), a deviatoric couple stress theory is named and proposed. Besides basic equations, the governing equation along with appropriate boundary conditions are given for a three-dimensional problem. Two special cases of plane problems and anti-plane problems are also provided. A simple shear problem is considered to show the advantage of the DCST. An elastic layer with a clamped surface under uniform shear loading on the other surface is solved. Exact solution of the in-plane and anti-plane shear problems of an elastic strip is determined and however, it has no solution if classical elasticity is used. The results indicate that there exists a boundary layer near the clamped surface of the strip or a great stress gradient occurs near the clamped surface when the characteristic length is sufficiently small. An annulus subjected to anti-plane shear loading on the inner edge and fixed on the outer edge and the pure bending of a 3D bar with rectangular cross-section are analyzed to illustrate the size-dependent effect. Modified couple stress theory and consistent couple stress theory can be reduced as two extreme cases of the present theory.

Research on dimple suppression in multi-point flexible stretch forming of parabolic spherical part

Multi-point flexible stretch-forming process applies the idea of discretization to the die and clamp of stretch-forming, so that the die and clamp have high flexibility at the same time. Because the surface of the multi-point die is discontinuous, the sheet metal will produce the dimple during the forming process, which will affect the forming effect. To suppress the dimple, this paper takes parabolic spherical parts as the research object, and discusses the influence of swinging unit and discrete elastic cushion on the dimples of forming parts by numerical simulation. The results show that the strain and thickness distribution are more uniform and the dimples on the surface of the forming parts are obviously reduced when the swinging unit is used. The use of discrete elastic cushions on the fixed unit and the swinging unit can further suppress the dimples of the forming part and improve the forming accuracy. To verify the correctness of the numerical simulation, the multi-point die for swinging unit with discrete elastic cushion was used to carry out the flexible stretch-forming test of the parabolic spherical part. The test results are in agreement with the numerical simulation results, which shows that the methods are reliable and feasible.

Refined linearly anisotropic couple-stress elasticity

A recently developed, refined version of the conventional linear couple-stress theory of isotropic elasticity is extended to include the influence of anisotropic material effects. With this development, the implied refined theory (1) retains ability to determine the spherical part of the couple-stress and (2) is further furnished with constitutive ability to embrace modelling of linearly elastic solids that exhibit inherent polar material anisotropy of advanced levels that reach the class of locally monoclinic materials. This type of anisotropy embraces most of the structural material subclasses met in practice, such as those of general and special orthotropy, as well as the subclass of transverse isotropy. The thus obtained, enhanced version of the refined theory is furnished with ability to also handle structural analysis problems of polar fibrous composites reinforced by families of perfectly flexible fibres or, more generally, polar anisotropic solids possessing one or more material preference directions that do not possess bending resistance. A relevant example application considers and studies in detail the subclass of polar transverse isotropy caused by the presence of a single family of perfectly flexible fibres. By developing the relevant constitutive equation, and explicitly presenting it in a suitable matrix rather than indicial notation form, that application also exemplifies the way that the spherical part of the couple-stress is determined when the fibres are straight. It further enables this communication to initiate a discussion of further important issues stemming from (1) the positive definiteness of the full, polar form of the relevant strain energy function and (2) the lack of ellipticity of the final form attained by the governing differential equations.

Optimized expression of Peptidyl-prolyl cis/transisomerase cyclophilinB with prokaryotic toxicity from Sporothrix globosa.

The recombinant cyclophilin B protein of Sporothrix globosa was expressed successfully by retaining extracellular domain with peptidyl-prolyl cis-trans isomerase activity to avoid toxicity to the host bacterium.

New trends in couple-stress hyperelasticity

This communication reconsiders and reorganises parts of existing knowledge that refers to Cosserat-type hyperelasticity formulations in a manner that respects and pays tribute to the early development pattern of the couple-stress theory and, also, associates with it more recent findings and relevant discoveries related to a certain type of polar hyperelastic behaviour of fibrous composites. It further shows that these different branches of couple-stress theory can both emanate, as special cases, from a common, more general, advanced theoretical hyperelasticity framework. It thus reveals that proper completion of any relevant theoretical formulation requires in advance specification of a pair of kinematic parameters, each of which (1) is related, in a virtual or actual manner, to the observed global, macroscopic elastic deformation of the material, but (2) is principally relevant to the polar part of the observed material response. The first of these parameters represents a pseudovector field whose gradient is energetically reciprocal to the emerging couple-stress field, while the second represents either a vector or a pseudovector field that serves specific constitutional needs characterising the source of the anticipated polar material response. Different versions of couple-stress theory formulations, thus, are obtained by appropriately choosing or suitably tuning that pair of kinematic parameters. It is also seen that, regardless of the employed couple-stress theoretical model, full solution of a relevant well-posed boundary value problem is generally achievable with use of a two-step solution process. The first step includes determination of the deviatoric couple-stress and the actual spin vector of the global material deformation. In the second step, these initial findings enable formation of an additional differential equation whose solution leads to determination of the spherical part of the couple-stress.

Stress intensity factor of a cracked molar restored with different materials and designs: A 3D-FEA

ObjectiveThis work used 3D finite element analysis (FEA) to analyze and directly compare the stress intensity factor (SIF) and stress distribution at the crack tip of identical cracked tooth models restored with different materials and crown parameters. MethodsA 3D model of the cracked tooth was generated. Then, we applied 25 restorative models, including three parameters (shoulder height, width, and degree of polymerization), five restorative materials (GC, IPS, LU, ZC, VE), and two combinations of types of cement (RMGIC and GIC). An occlusal load of 800N was applied to the spherical part along the longitudinal axis. The stress distribution of the preparation and the SIF of the crack tip was analyzed. ResultsThe crack tip SIF was minimal for a shoulder height offset of 0.8 mm (P = 0.032), a shoulder width of 0.6 mm (P = 0.045), a crown material of ZC (P < 2e-16), and a cement material of RMGIC (P < 0.05), respectively. In contrast, the effect of different polymerization degrees on SIF was insignificant (P = 0.95). ConclusionOur results suggest that the selection of a larger modulus of elasticity (MOE) material for the crown, the preparation of a smaller shoulder width within a safe range, a reasonable increase in the crown length, and the selection of adhesive materials with high fracture toughness are favorable methods to prevent further crack extension.

Droplet evaporation on pillar structured surface: A 3D lattice Boltzmann numerical study

In this paper, a three-dimensional symmetric lattice Boltzmann (LB) model is built up to investigate evaporation of a sessile droplet on pillar structured surfaces. Parametric effects on droplet evaporation, including pillar structure size, roughness, temperature and wettability, are studied comprehensively, with emphasis on the variations of contact angle, contact radius and velocity of triple-phase contact line (TPCL) as well as heat and mass fluxes during evaporation. Pillar structures are found to induce the droplet to split into spherical cap part and liquid film part, resulting in an evaporation mode different from that on a plain surface. The extent of pinning-depinning is strongly associated with structures when TPCL retracts between pillars. Increasing surface temperature can accelerate evaporation by breaking the constant contact angle (CCA) mode, and increasing surface roughness accelerates evaporation by decreasing the duration of CCA mode. The majority of heat and mass transfer takes place around TPCL. High wettability can facilitate the evaporation of spherical cap due to the meniscus at liquid–vapor interface in the vicinity of pillar top surface. The substrate heat flux directly affected by structure size is higher in the region where pillars are distributed, and the solid–liquid interfacial heat flux varies with the specific interface location within droplet. The total mass flux primarily depends on the TPCL length determined by the droplet evaporation mode. When the droplet evaporates by partial penetration in the Cassie state, the amount of evaporation at liquid–vapor interface between pillars is greater than that above pillars. This paper provides more details and insights for a deep understanding of droplet evaporation on pillar structured surfaces.

Determination of the Spherical Couple-Stress in Polar Linear Isotropic Elasticity

It is well known that the conventional couple-stress theory leaves the spherical part of the couple-stress indeterminate. This indeterminacy problem is recently resolved for fibrous composites subjected to either small or large deformations and containing a single family of fibres resistant in bending (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595). However, the problem remains still unsolved in simpler cases where the implied preference material direction is not related to fibre bending resistance, and even in the simplest possible case where the polar material of interest is linearly elastic and isotropic. This communication aims (i) to show that a relevant virtual spin concept employed (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595) is further applicable in the latter case of polar linear isotropic elasticity, (ii) to demonstrate the process in which that concept thus leads to determination of the spherical part of the couple-stress, (iii) to exemplify this process by providing a couple of simple illustrative examples, (iv) to specify and discuss the reason that the outlined method meets a hurdle in cases of linear anisotropic elasticity that is due to one or more preferential material directions, and, hence, (v) to further discuss the manner in which that newly identified difficulty is currently confronted, and may thus be handled successfully.

Validation of Neutron Flux and Activation in the Nagra AMAC model of the Mühleberg Power Plant

The decommissioning of the Mühleberg Nuclear Power Plant (KKM) is supported by Nagra (National Cooperative for the Disposal of Radioactive Waste) through computational analysis of neutron activation in the reactor components and structures using the Nagra A dvanced M ethodology for A ctivation C haracterization (AMAC). To validate these calculations, measurements of neutron flux and neutron-induced activity were carried out in the form of two foil activation campaigns (72 samples) and three concrete sampling campaigns (118 samples). Generally, the ratio of calculated to measured activities is in the order of 2 in the proximity of the reactor pressure vessel and 3 in the upper sections of the drywell. As expected, the flux in the lower (spherical) part of the drywell is significantly overestimated due to the simplified modelling of this space. The overestimation, which ratios of up to 10, can be corrected by scaling the calculated radionuclide activity distributions in the vicinity of the validation locations.

A novel triple-layer hot stamping process of titanium alloy TC4 sheet for enhancing formability and its application in a plug socket part

In this paper, a novel method was proposed to improve the production efficiency and formability of titanium alloy, which was named triple-layer sheet hot stamping with a cold die cooling (THF). The titanium alloy sheet was clamped by two steel sheets with no adhesive added, then heated, transferred and formed together. In order to verify the ability in enhancing formability of this process, spherical part forming tests were carried out to study the formability under the single-layer sheet hot stamping (SHF) and THF. Finally, a plug socket part was formed to verify the practicability of the novel forming process. The finite element (FE) model was established to analyze the forming process finding that the formability of the parts were significantly improved after using THF process. Comparing with SHF process, the stamping depth increases by 135.7% at 900 °C in the THF process. The forming limit strain increases from 0.18 of the single-layer sheet to 0.51 of the triple-layer sheet. The forming limit strain obtained by the THF process is close to that of isothermal forming. The friction state is one of the main reason for improving formability. Moreover, for plug socket parts, there is almost no thinning of the part in the THF process at 900 °C. In conclusion, it has great advantages in improving the formability of titanium alloy using the THF process.

Application of the variational EIV approach to linear viscoelastic phases governed by several internal variables - Examples with the generalized Maxwell law

New developments regarding the variational approach originally proposed by Lahellec, N., and Suquet, P., Int. J. Solids Struct. 44, 507–529, 2007, also called EIV approach, are presented in the context of linear viscoelasticity. In previous work (Tressou et al., Eur. J. Mech./A Solids. 68, 104–116, 2018), we have demonstrated the efficiency of this approach as associated to different linear schemes and, as a result, its possible application to various microstructures. In the present paper, our aim is to enlarge the application domain of the EIV model in terms of constituents' viscoelastic laws that may be considered. Initially applied to linear viscoelastic phases described by a single, traceless, internal variable, the original formulation of the EIV model is here generalized to phases described by several internal variables with both spherical and deviatoric parts. The resulting generalized EIV approach is evaluated by comparison of the estimated phase average and effective responses to reference solutions obtained by full-field finite element (FE) simulations for a fiber-reinforced microstructure. Different linear viscoelastic laws with increasing complexity are considered to this aim. At first, a classical Maxwell model is considered for the matrix as in Lahellec and Suquet's original work, but now the internal variable (viscous strain) has a spherical part. Bulk and shear moduli have identical or different characteristic times, the matrix only or both constituents are viscoelastic. Then, the matrix is described by a generalized Maxwell model with two and four internal variables. Different distributions of relaxation times are studied for bulk and shear moduli in order to test the robustness of the model. In every studied situation, the agreement with reference solutions is satisfactory.

A thermodynamical model for paleomagnetism in Earth’s crust

A thermodynamically consistent model for soft deformable viscoelastic magnets is formulated in actual space (Eulerian) coordinates. The possibility of a ferro-paramagnetic-type (or ferri-antiferromagnetic) transition exploiting the Landau phase-transition theory as well as mechanical melting or solidification is considered, being motivated and applicable to paleomagnetism (involving both thermo- and isothermal and viscous remanent magnetization) in rocks in Earth’s crust and to rock-magma transition. The temperature-dependent Jeffreys rheology in the deviatoric part combined with the Kelvin-Voigt rheology in the spherical (volumetric) part is used. The energy balance and the entropy imbalance behind the model are demonstrated, and its analysis is performed by time discretization, proving existence of weak solutions.

Effectiveness Study of Using HAp Milkfish Bone to Synthesize Ag&lt;sub&gt;3&lt;/sub&gt;PO&lt;sub&gt;4 &lt;/sub&gt;with Ion-Exchange Method for Methylene Blue Degradation

The development of Ag3PO4 photocatalyst for dye waste treatment is necessary because of its superiority compared to other materials. In this research, Ag3PO4 was synthesized with AgNO3 and HAp from milkfish bones. Syntesis was done by simple ion-exchange method. In this research, composition of AgNO3:HAp are 1:0.1; 1:0.3; and 1:0.5 to determine the optimum composition. Based on XRD results, it shows that ion-exchange process between phosphate ions from HAp milkfish bones and Ag+ from AgNO3 was imperfect. It was shown from the presence of HAp peaks in synthesis results. Furthermore, the diffraction pattern showed that AgNO3:HAp 1:0.1 composition has the best cristalinity. Based on SEM characterization, it shows that all compositions have majority shape as spherical and small part at AgNO3/HAp 1:0.5 composition has nanorod morphology. It was also known that material with composition of AgNO3:HAp 1:0.1; 1:0.3 and 1:0.5 have average size are 100.76 nm, 92.31 nm and 107.51. From UV-Vis spectroscopy data processing, it represent the value of direct band gap of material is 3.83 eV and the value of indirect band gap is 3.6 eV. It also shows the degradation value of methylene blue with the best value was shown in AgNO3:HAp 1:0.3 composition with 47.4%.

A quantum rotator on a three-dimensional sphere

In this work, the quantum-mechanical problem of the motion of two material points of different masses on a three-dimensional sphere with a non-fixed position of the center of mass of the system is formulated on the basis of the previously solved classical problem. It is shown that the established Schrödinger equation includes two different reduced masses, depending on the distance between the points. For the case of the interaction potential of points, depending only on the distance between them, this equation allows the separation of variables into a radial, depending on the relative distance and both the reduced masses and the spherical part. The equation for the spherical part depends only on one of the above reduced mass and allows one to formulate and solve the problem of a rigid rotator - the distance between the points is fixed. The solution and spectrum of the problem of a rigid rotator are found. It is shown that the spectrum of the system has an upper limit that does not depend on the distance between points, in contrast to the spectrum in a flat space.

Turbulence kinetic energy transfers in direct numerical simulation of shock-wave–turbulence interaction in a compression/expansion ramp

A direct numerical simulation is performed for a supersonic turbulent boundary layer interacting with a compression/expansion ramp at an angle $\alpha =24^{\circ }$ , matching the same operating conditions of the direct numerical simulation by Priebe &amp; Martín (J. Fluid Mech., vol. 699, 2012, pp. 1–49). The adopted numerical method relies on the high-order spectral difference scheme coupled with a bulk-based, low-dissipative, artificial viscosity for shock-capturing purposes (Tonicello et al., Comput. Fluids, vol. 197, 2020, 104357). Filtered and averaged fields are evaluated to study total kinetic energy transfers in the presence of non-negligible compressibility effects. The compression motions are shown to promote forward transfer of kinetic energy down the energy cascade, whereas expansion regions are more likely to experience backscatter of kinetic energy. A standard decomposition of the subgrid scale tensor in deviatoric and spherical parts is proposed to study the compressible and incompressible contributions in the total kinetic energy transfers across scales. On average, the correlation between subgrid scale dissipation and large-scale dilatation is shown to be caused entirely by the spherical part of the Reynolds stresses (i.e. the turbulent kinetic energy). On the other hand, subtracting the spherical contribution, a mild correlation is still noticeable in the filtered fields. For compressible flows, it seems reasonable to assume that the eddy-viscosity approximation can be a suitable model for the deviatoric part of the subgrid scale tensor, which is exclusively causing forward kinetic energy cascade on average. Instead, more complex models are likely to be needed for the spherical part, which, even in statistical average, provides an important mechanism for backscatter.

A mathematical model for thinning rate prediction of sheet double hydro-forming

Double sheet blank hydro-forming (DSBH) is a technology for forming hollow parts with complex shapes. The pair of workpieces is deformed and shaped by the high-pressure liquid inside. The material is thinned after hydraulic forming, which significantly affects the quality of the product, especially the fields with high requirements, such as the automotive and aerospace industry. The goal of optimizing input process parameters to ensure that the level of thinning into a product is within the allowable limit is posed by this study. This study considered blank holder force, forming fluid pressure, and relative thickness as candidate factors for optimization using Response Surface Method (RSM). The spherical parts were formed by the DSBH method of welding blank pairs of DC04 carbon steel material based on theoretical analysis, experiment solution, and experiment to verify the results. Experiments were performed with different combinations of parameters using the Box-Behnken design. This paper presented a mathematical model that helps determine material thinning rate according to these three process parameters in the hydro-forming of spherical parts from welded sheet metal pairs. The research results can be applied to control the input parameters in the DSBH to achieve the wall thickness of the spherical part as desired by the manufacturer.

Imaging Electron Optics of Electrostatic Focusing Concentric Spherical System Part C: Electron Optics of Concentric Spherical System Composed of Multiple Electrodes

Imaging Electron Optics of Electrostatic Focusing Concentric Spherical System Part C: Electron Optics of Concentric Spherical System Composed of Multiple Electrodes

Imaging Electron Optics of Electrostatic Focusing Concentric Spherical System Part B: Paraxial Lateral Chromatic Aberration and Geometrical Lateral Spherical Aberration

Imaging Electron Optics of Electrostatic Focusing Concentric Spherical System Part B: Paraxial Lateral Chromatic Aberration and Geometrical Lateral Spherical Aberration

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness – Part II: determination of the spherical part of the couple-stress

The indeterminacy of the spherical part of couple-stress is a well-known drawback of any theoretical formulation stemming from the Cosserat couple-stress theory of elasticity. The relevant theory of finite elastic deformations of solids reinforced by a family of fibres that resist bending is not an exception. The present communication extends and completes that theory in a manner that enables it to measure the spherical part of the couple-stress tensor outside the conventional equilibrium considerations. To achieve this, the present study reconsiders an extra piece of information that has surprisingly emerged already but, so far, has been left unexplained and unexploited; namely, the fact that the energy stored in a fibrous composite elastic solid with fibre-bending stiffness involves a couple-stress generated term that does not influence the relevant couple-stress constitutive equation. The thus resulting new theoretical development complements the theory previously presented without dismissing any of the theoretical results detailed or the conclusions drawn there. Its validity embraces boundary value problems concerning both finite and infinitesimal elastic deformations of polar fibrous composites. In the latter case, its applicability is also tested and verified through the successful determination of the spherical couple-stress of a polar transversely isotropic elastic plate subjected to pure bending.

Modeling and simulation of ductile damage in forged steel used to encapsulate high-level radioactive waste

Andra, the French national radioactive waste management agency, is in charge of studying the disposal of high-level and long-lived intermediate-level waste (HLW and ILW-LL) in a deep geological repository. According to the reference concept, it is planned to encapsulate high-level waste in non-alloy P285NH steel overpacks before inserting them into horizontal steel cased micro-tunnels. This work is a part of the study about the long-term behavior of a welded steel overpack subjected to external hydrostatic pressure and several localized loading paths. Indeed, the main objective of this work is to develop the most suitable model for non-alloy steel P285NH to be used in the prediction of the long-term overpack behavior. Dealing with a ductile steel, elastoplastic constitutive equations accounting for mixed nonlinear isotropic and kinematic hardening strongly coupled with ductile isotropic damage are adopted. They are formulated based on the classical thermodynamics of irreversible processes framework with state variables at the macroscopic scale, (Germain, 1973) (Lemaitre 1985, Saanouni 2012). In this paper, a new coupling formulation between the scalar isotropic ductile damage and the deviatoric and spherical part of the Cauchy stress and elastic strain tensors is proposed. In order to calibrate the developed model on P285NH steel, multiple tensile tests are performed using classical cylindrical specimens, notched specimens and double notched specimens. In the last part, some experimental fields are measured using digital image correlation. Application is made to a simplified overpack represented by thick walled cylinder subject to compressive loading path. A FEM (Finite Element method) crushing operation of an overpack’s cylindrical part has simulated and analysed.