Plane Stress Approximation Research Articles

Consequences of plane-strain and plane-stress assumptions in fully coupled chemo-mechanical Li-ion battery models

In Li-ion battery research, it is common to simulate chemo-mechanical phenomena in reduced dimensions (e.g., 2-D) as opposed to fully resolve these complex physics in 3-D. It is common to assume either (1) the out-of-plane strain is negligible (commonly referred to as plane-strain), or (2) the out-of-plane stress is negligible (commonly referred to as plane-stress). However, there is typically little consideration as to the quantitative consequences of these approximations. Furthermore, the influence of these out-of-plane assumptions can be compounded and convoluted when chemo-mechanics models implement so-called “fully coupled” formulations, where the local species concentrations influence the stress-state and the stress-state influences the local species fluxes. The present manuscript explores the implications of using plane-stress and plane-strain assumptions in 2-D as compared to simulating a full 3-D electrode particle. This comparative study includes simulating both isotropic and anisotropic particle intercalation where the particles can be surrounded by either a liquid or solid electrolyte. Additionally, common Li-ion battery-model metrics such as the state-of-stress, intercalation fraction distribution, and specific capacity are compared, while also considering the effects of particle size and C-rate. As alternatives to the pure plane-strain and plane stress approximations, two modified plane-strain assumptions are found to better approximate the fully coupled chemo-mechanical 3-D behavior.

Corrections to the Theory of Elastic Bending of Thin Plates for 2D Models in the Reissner Approximation

Elastic bending stresses and strains in the lithosphere are typically calculated based on the Kirchhoff–Love (thin) plate theory. The criterion for its applicability is the smallness of plate thickness to length ratio. In oceanic plates, due to the buoyant force of the mantle, the main deformations are not uniformly distributed along the plate but are concentrated in the vicinity of the subduction zone. Therefore, the effective length of the bending part of the plate is a few fractions of the actual length, and the criterion of a thin plate is partially violated. In this paper, we analyze the possibility of applying thick plate bending equations. The existing variational theories of 3D bending of thick plates are much more complicated than the Kirchhoff–Love theory, as they involve solving three differential equations instead of one, and have limited application due to their complexity. Since geophysical applications frequently use 2D models, in this paper we analyze in detail the potential and accuracy of the thick plate bending theory for 2D models. After the conversion to the 2D plane strain and plane stress approximation, the original 3D Reissner thick plate bending equations are written out in the form similar to the Kirchhoff equations with additive corrections and are supplemented with the explicit formulas for longitudinal displacement. The comparison of the analytical solutions of the 2D Reissner equations with the exact solutions shows that the 2D approximation only provides a correction for the plate bending function. However, this correction refines the Kirchhoff–Love theory by almost an order of magnitude. At the same time, the solution of the equations in this case turns out to be almost as simple as the solution of the thin plate equations.

Peridynamic modelling of cracking in TRISO particles for high temperature reactors

A linear-elastic computer simulation (model) for a single particle of TRISO fuel has been built using a bond-based peridynamic technique implemented in the finite element code ‘Abaqus’. The model is able to consider the elastic and thermal strains in each layer of the particle and to simulate potential fracture both within and between layers. The 2D cylindrical model makes use of a plane stress approximation perpendicular to the plane modelled. The choice of plane stress was made by comparison of 2D and 3D finite element models. During an idealised ramp to normal operating power for a kernel of 0.267 W and a bulk fuel temperature of 1305 K, cracks initiate in the buffer near to the kernel-buffer interface and propagate towards the buffer-iPyC coating interface, but do not penetrate the iPyC and containment of the fission products is maintained. In extreme accident conditions, at around 600% (1.60 W) power during a power ramp at 100% power (0.267 W) per second, cracks were predicted to form on the kernel side of the kernel-buffer interface, opposite existing cracks in the buffer. These were predicted to then only grow further with further increases in power. The SiC coating was predicted to subsequently fail at a power of 940% (2.51 W), with cracks formed rapidly at the iPyC-SiC interface and propagating in both directions. These would overcome the containment to fission gas release offered by the SiC ‘pressure vessel’. The extremely high power at which failure was predicted indicates the potential safety benefits of the proposed high temperature reactor design based on TRISO fuel.

Shape-programmable cantilever made of a magnetoactive elastomer of mixed content

This work presents an approach to the macroscopic field-controlled mechanics of magnetoactive elastomers of mixed content, which are a special type of smart materials made of an elastic composite and a combination of two essentially different ferromagnetic fillers. High-coercive particles of NdFeB-alloy powder for the magnetically hard (MH) filler and carbonyl iron powder particles with nearly zero coercivity for the magnetically soft (MS) filler are usually used. The MH particles are tens-of-micron in size and impart to the elastomer a remanent magnetisation, whereas due to the MS particles of several microns in size, the elastomer acquires a high magnetic susceptibility. Since large MH particles once magnetised in a strong field possess their own fields to which the MS particles are susceptible, the overall elastomer magnetisation as well as its mechanical response greatly depends on the relative concentration of both fillers. This work particularly studies the bending deformation of horizontally fixed magnetoactive cantilevers with the permanent magnetisation along the length axis under the action of gravity and a vertically applied uniform magnetic field. The cantilevers of the same geometry and fixed NdFeB content but different carbonyl iron concentration are considered. The magnetomechanical model is developed based on the finite-strain theory assuming the plane-stress approximation of the two-dimensional cantilever of infinite width. The magnetic energy comprises two magnetic terms, one of which is qualitatively linear and the other one is quadratic in the applied field strength. The numerically calculated field-programmed equilibrium bending shapes of the cantilevers are compared with the experimentally observed shapes. The model provides good agreement with the experiment up to moderate concentrations of the MS filler, when the coefficients of customary interpolation formulas for the concentration dependencies of elastic modulus and magnetic susceptibility are properly adjusted.

Time scale disparity yielding acoustic nonreciprocity in a two-dimensional granular-elastic solid interface with asymmetry.

We study nonreciprocal wave transmission across the interface of two dissimilar granular media separated by an elastic solid medium. Specifically, a left, larger-scale and a right smaller-scale granular media composed of two-dimensional, initially uncompressed hexagonally packed granules are interfacing with an intermediate linearly elastic solid, modeled either as a thin elastic plate or a linear Euler-Bernoulli beam. The granular media are modeled by discrete elements and the elastic solid by finite elements assuming a plane stress approximation for the thin plate. Accounting for the combined effects of Hertzian, frictional and rotational interactions in the granular media, as well as the highly discontinuous interfacial effects between the (discrete) granular media and the (continuous) intermediate elastic solid, the nonlinear acoustics of the integrated system is computationally studied subject to a half-sine shock excitation applied to a boundary granule of either the left or right granular medium. The highly discontinuous and nonlinear interaction forces coupling the granular media to the elastic solid are accurately computed through an algorithm with interrelated iteration and interpolation at successive adaptive time steps. Numerical convergence is ensured by monitoring the (linearized) eigenvalues of a nonlinear map of interface forces at each (variable) time step. Due to the strong nonlinearity and hierarchical asymmetry of the left and right granular media, time scale disparity occurs in the response of the interface which breaks acoustic reciprocity. Specifically, depending on the location and intensity of the applied shock, propagating wavefronts are excited in the granular media, which, in turn, excite either (slow) low-frequency vibrations or (fast) high-frequency acoustics in the intermediate elastic medium. This scale disparity is due to the size disparity of the left and right granular media, which yields drastically different wave speeds in the resulting propagating wavefronts. As a result, the continuum part of the interface responds with either low-frequency vibrations-when the shock is applied to the larger-scale granular medium, or high-frequency waves-when the shock is applied to the smaller-scale granular medium. This provides the fundamental mechanism for breaking reciprocity in the interface. The nonreciprocal interfacial acoustics studied here apply to a broad class of asymmetric hybrid (discrete-continuum) nonlinear systems and can inform predictive designs of highly effective granular shock protectors or granular acoustic diodes.

Boundary influence on the fields and arrangement of the needle domains in barium titanate single crystals

The superposition theory is adopted to study the boundary value problem of needle domains in barium titanate single crystals, which decomposes into an infinite-medium solution and a complementary solution. The model of an equivalent edge dislocation and line charge is adopted to calculate the fields due to the discrete needle domains in an infinite medium. The complementary solutions are obtained from a linear elastic boundary value problem using the finite element method. The influence of boundaries on the fields and arrangement of the needles in thin barium titanate single crystals is analysed, by considering traction-free boundaries and using the plane stress approximation. According to the principle of minimum potential energy, it is found that pairs of parallel needle domains can be stabilized if the effective charge associated with the needle tips is reduced to 48% of the material’s full polarization charge. The evolution of comb-like arrays of needle domains under compressive mechanical load is simulated, and the results are found to be qualitatively similar to experimental observations which have not previously been explained theoretically.

Positional finite element formulation for two-dimensional analysis of elasto-plastic solids with contact applied to cold forming processes simulation

In this work, we develop a position-based finite element formulation for elasto-plastic solids under contact situation. The proposed positional formulation employs a total Lagrangian description and naturally considers geometric nonlinearities. The employed elasto-plastic model is derived from the dissipation inequality, using the thermodynamic conjugacy between the plastic strain rate and the so-called Mandel stress. The formulation is based on the Kroner–Lee decomposition, in which the deformation gradient is multiplicatively split into its elastic and plastic parts. We apply the backward Euler method to solve the plastic evolutions and von Mises yield criterion to define the elastic limit. The adopted kinematic hardening model is a finite strain generalization of the Armstrong–Frederick law, which uses the objective Jaumann derivative for the evolution equation and the concept of back stress tensor as an internal variable. For the elastic parcel of strains, we adopt a neo-Hookean constitutive law. With respect to the 2D application, plane strain and plane stress approximations are considered, where the latter is solved numerically by a local Newton–Raphson numerical procedure. Regarding the contact problem, a classical node-to-segment algorithm is applied, considering both frictionless and frictional cases, with the introduction of Lagrange multipliers in order to enforce contact constraints. Representative numerical examples are used to validate and show the possibilities of the proposed formulation in macroscale simulation of metal cold forming manufacturing processes.

On the elastic anatomy of heterogeneous fractures in rock

This study examines the feasibility of the full-field ultrasonic characterization of fractures in rock. To this end, a slab-like specimen of granite is subjected to in-plane, O(104 Hz) excitation while monitoring the induced 2D wavefield by a Scanning Laser Doppler Vibrometer (SLDV) with sub-centimeter spatial resolution. Upon suitable filtering and interpolation, the observed wavefield is verified to conform with the plane-stress approximation and used to: (i) compute the maps of elastic modulus in the specimen (before and after fracturing) via a rudimentary application of the principle of elastography; (ii) reconstruct the fracture geometry; (iii) expose the fracture's primal (traction-displacement jump) contact behavior, and (iv) identify its profiles of shear and normal specific stiffness. Through the use of full-field ultrasonic data, the approach provides an unobscured, high-resolution insight into the fracture's contact behavior, foreshadowing in-depth laboratory exploration of interdependencies between the fracture geometry, aperture, interphase properties, and its seismic characteristics.

Thermal stress and end-bulging in monoclinic crystals: the case study of double tungstates.

An analytical description of the thermal stress and end-bulging component of thermal lensing is presented for monoclinic laser crystals, taking into account the anisotropy of their optical, thermal, and elastic properties for the first time, to the best of our knowledge. The geometry of longitudinal diode-pumping (plane stress approximation) is considered. The developed approach is applied to the monoclinic double tungstates (MDTs), Yb:KGW, Yb:KYW, and Yb:KLuW, yielding values of normal and shear stresses, tensile stress, and the end-bulging term. We show that for low-symmetry crystals, end-bulging is responsible for the astigmatism of the thermal lens. The geometry of this astigmatism is explained in terms of the principal meridional planes. We show that the crystal cut of the MDTs along the optical indicatrix axis Ng (propagation direction) provides low thermal stress and a purely positive and weakly astigmatic thermal lens, which is a key condition for power scaling and diffraction-limited laser output.

Photo-elastic effect, thermal lensing and depolarization in a-cut tetragonal laser crystals

We report on analytical description of thermal lensing effect in tetragonal crystals cut along the [1 0 0] crystallographic axis, for the two principal light polarizations, E ┴ c and E || c, under diode-pumping (plane stress approximation). Within this approach, we take into account anisotropy of elastic, photo-elastic, thermal and optical properties of the material. Expressions for the ‘generalized’ thermo-optic coefficient χ are presented. It is shown that astigmatism of thermal lens is determined both by the photo-elastic and end-bulging effects. The sign of the photo-elastic term χ″ can be either positive or negative affecting significantly the sign of the thermal lens. Depolarization loss in a-cut tetragonal crystals is few orders of magnitude lower than that in cubic crystals. Calculations are performed for a-cut tetragonal molybdates, Nd:CaMoO4, Nd:PbMoO4 and Nd:NaBi(MoO4)2.

On the use of photoelastic effect and plane strain or plane stress approximations for the description of thermal lensing

Correct use of the photoelastic effect for the description of thermally induced refractive index change is discussed and the analytical relations between thermo-optic coefficients at zero stresses and zero strains are found for all classes of cubic crystals. These relations may be useful for the investigation of thermal effects in very promising sesquioxide class m3 laser crystals. An accepted set of elasto-optical coefficients of the YAG crystal and an alternative one found in the literature were used in numerical simulations. Significant differences in the calculated thermo-optic coefficients and induced birefringence are found using different sets of these coefficients. Misunderstandings related with the so-called photoelastic coefficients are resolved and new expressions for these coefficients are found. It is shown that the incorrect use of these coefficients for different pump beam distributions can lead to significant discrepancies for thermally induced birefringence. It is also shown that common use of the generalized thermo-optic coefficients significantly overestimates the values of optical power of thermal lenses when they are applied to the laser rods with lengths several times longer than their diameter.

First-order estimate of the Canary Islands plate-scale stress field: Implications for volcanic hazard assessment

In volcanic areas, the existing stress field is a key parameter controlling magma generation, location and geometry of the magmatic plumbing systems and the distribution of the resulting volcanism at surface. Therefore, knowing the stress configuration in the lithosphere at any scale (i.e. local, regional and plate-scale) is fundamental to understand the distribution of volcanism and, subsequently, to interpret volcanic unrest and potential tectonic controls of future eruptions. The objective of the present work is to provide a first-order estimate of the plate-scale tectonic stresses acting on the Canary Islands, one of the largest active intraplate volcanic regions of the World. In order to obtain the orientation of the minimum and maximum horizontal compressive stresses, we perform a series of 2D finite element models of plate scale kinematics assuming plane stress approximation. Results obtained are used to develop a regional model, which takes into account recognized archipelago-scale structural discontinuities. Maximum horizontal compressive stress directions obtained are compared with available stress, geological and geodynamic data. The methodology used may be easily applied to other active volcanic regions, where a first order approach of their plate/regional stresses can be essential information to be used as input data for volcanic hazard assessment models.

Relationships between different expressions of thermo-optic and photoelastic coefficients of YAG crystal

Using correct expressions for piezo-optic and elasto-optic effects, the analytical relations between thermal coefficients of the refractive index at zero stress and zero strain have been established for YAG-type cubic crystals. Using the plane strain and plane stress approximations, analytical expressions for thermal radial and tangential changes of the refractive index have been found. The relations between different expressions for photoelastic coefficients are established. Examples of the inconsistent usage of generalised thermo-optic coefficients are discussed. It is shown that their values are often significantly overestimated when the results of the plane stress approximation are applied to a laser rod with a length larger than its diameter.

Thermal lensing in diode-pumped [001]-cut tetragonal crystals

Abstract Thermal lensing is theoretically studied for laser discs from [001]-cut tetragonal crystals under the plane stress approximation, considering anisotropy of their elastic and photo-elastic properties. Analytical description of the stress-induced birefringence is presented. This allows us to derive analytical expressions for the photo-elastic term χ″, “generalized” thermo-optic coefficient (TOC), sensitivity factors of the thermal lens and its astigmatism degree, as well as discuss orientation of the principal meridional planes of the lens. These parameters are determined for the discs from tetragonal scheelite-type molydbate crystals, Nd:PbMoO4, Nd:CaMoO4 and Nd:NaBi(MoO4)2 at ~1 μm. It is shown that the photo-elastic effect plays an important role in the formation of the thermal lens for tetragonal crystals, and χ″ value can be either positive or negative. We also predict theoretically that for anisotropic crystals the orientation of the principal meridional planes of the thermal lens is not linked to the polarization direction.

Thermal stresses and end-bulging in the laser disc from a tetragonal [1 0 0]-cut crystal

The problem of thermal stresses and strains is considered in relation to the laser disc from a tetragonal crystal cut along [1 0 0] axis under plane stress approximation (for diode-pumped lasers). The formula describing the contribution of end-bulging of the disc to the optical power of the thermal lens is obtained. It is shown that end-bulging for anisotropic crystal is partially responsible for the astigmatism of the thermal lens. The relationship between the anisotropy of thermal expansion and the position of the principal meridional planes of the thermal lens is shown. Thermal stresses, strains and displacements, as well as the end-bulging term of the ‘generalized’ thermo-optic coefficient are calculated for a disc from an a-cut yttrium vanadate, Nd:YVO4. The stress fracture issues for this crystal are also analyzed.

Strain in YBCO Double-Pancake Coil With Stainless Steel Overband Under External Magnetic Field.

This paper deals with the mechanical strain issue in a high-temperature superconducting (HTS) insert for a GHz-class (> 23.5 T) LTS/HTS NMR magnet. We present results, experimental and analytical, of hoop strains in a double-pancake (DP) test coil, wound with 6-mm wide YBCO coated conductor (CC) and equipped with strain gauges at their innermost and outermost turns. To keep the YBCO CC to within a 95% Ic retention, the conductor tensile strain must be limited to 0.6%. To satisfy this strain limit in our test DP coil, we wrapped 0.08-mm thick, 6-mm wide stainless steel strip over its outermost turn of an 4.8-mm overband radial build deemed sufficient by our stress analysis based on force equilibrium and generalized Hooke's law with plane stress approximation. A control test DP coil, actually the same test DP coil, without overbanding, was run under the same experimental condition. In each case the test DP coil was energized up to 350 A at 4.2 K in a background field of 4 T. We report the experiment and analysis, with discussion on the merit of overbanding as a means to limit hoop strain in high-field HTS inserts.

Thermal stresses in the laser disc from a tetragonal c-cut crystal

Analytical expressions for thermal stresses and strains, as well as displacements, are obtained for the laser disc from a tetragonal crystal cut along the [001] axis under plane stress approximation, for the first time, to our knowledge. This study illustrates that, in polar coordinates, the normal stresses, σr and σθ, are angular independent, while the shear one τrθ is zero. The thermal strains, εr and εθ, and displacements, u and υ, depend on both radial and tangential coordinates; this dependence has the shape of a four-leaf rose. For considered crystal cutting with isotropic in-plane thermal expansion, the displacements are not pure radial (υ≠0). The values of stresses, strains and displacements are calculated for the disc from a c-cut yttrium vanadate laser crystal, Nd:YVO4. The thermal fracture issues are analyzed for this crystal.

Thermal stress in through-silicon-vias: Theory-of-elasticity approach

Simple physically meaningful analytical (mathematical) predictive stress models are developed for typical through-silicon-via (TSV) designs using theory-of-elasticity approach. Two extreme cases of the TSV geometry are addressed: disc-vias (with height-to-diameter ratios below 0.25), when plane stress approximation can be used, and rod-vias (with height-to-diameter ratios above 2.5), when plane strain approximation is applicable. The objectives of the analysis is to evaluate the effect of the size of the opening in the silicon material on the induced pressure at the boundary of the opening, when the TSV structure is heated up, and to assess the possible role of a “surrogate” layer, if any, on the level of the longitudinal interfacial shearing stresses. While the pressure at the Si opening and the maximum interfacial stresses determine the reliability of the silicon material, the longitudinal shearing stress at the Si-copper (Cu) interface is critical from the standpoint of the adhesive and cohesive strength of the TSV design. The numerical example is carried out for different opening radii and for the case when Indium is considered as a suitable buffering material. The calculated data indicate that larger openings in the Si result in lower pressures on it and in lower maximum interfacial stresses, especially if disc-like vias are employed. On the other hand, as has been shown earlier (see Suhir and Savastiuk, 2008) elastic stability of disc-vias should be evaluated and ensured. The developed models can be used also to determine if there is sufficient incentive for employing additional, “surrogate”, materials (i.e., materials not needed from the standpoint of the functional performance of the design) for lower thermally induced stresses.

Study of operation dynamics for crystal temperature measurement in a diode end-pumped monolithic Yb:YAG laser

A temperature measurement scheme was proposed in a diode end-pumped thin monolithic Yb:YAG laser by analyzing the red-shifting behaviors of each lasing peak. The amount of peak shift was measured on the basis of the threshold lasing spectrum by using a chopped pump beam. In order to determine the effective scale factor, the ratio between the peak shift and the temperature rise, the dynamics of the spectral shift, the output beam profile, and the output power were investigated. The effective scale factor was determined to be about 0.0114 nm/°C in the case of the crystal sandwiched by copper bocks with a hole, wherein the plane stress approximation is valid. On the other hand, the effective scale factor significantly decreased in the case of the crystal sandwiched by sapphire plates.

Thermal Stresses and End-Bulging in Cylindrical Laser Rods Under Longitudinal Diode Laser Pumping

Three-dimensional thermo-mechanical finite element analysis of conventional and composite cylindrical laser rods under the conditions of longitudinal pumping of the continuous wave diode laser is presented. First, the heat conduction problem with prescribed cooling boundary conditions is solved. Then, the 3D stress and bulging of entrance and exit faces are analyzed. It is shown that the frequently used plane strain and plane stress approximations significantly overestimate the values of face bulging of conventional and particularly composite laser rods under the conditions of homogeneous and inhomogeneous heating.