Stroh Formalism Research Articles

New Approach of Dendrite Suppression Using Solid Electrolyte to Enable Li Metal Anodes

The study of dendritic growth and roughening during electrodeposition has gained renewed interest in light of the safety concerns associated with dendritic shorting in the current Li-ion batteries. A precise understanding of the growth of dendrites during electrodeposition could enable the use of metal anodes, especially Li, which could lead to significantly higher energy density batteries. Solid electrolytes present a new avenue for attacking this problem of stable electrodeposition, besides improving the safety of energy storage technology. Replacing the organic liquid electrolyte by a solid electrolyte creates a solid-solid interface whose properties alter the kinetics of electrodeposition. Monroe and Newman analyzed the electrodeposition stability of Li/solid polymer electrolyte system within linear elasticity theory and showed using a kinetic model that solid polymer electrolytes with a sufficiently high modulus are capable of suppressing dendrite growth [1]. However, the propagation of the interface is often accompanied by a change in density of the metal which is thus, an important order parameter that should affect the stability of electrodeposition at the interface. In the theory of stress-driven phase transformation at solid-solid interfaces studied in geological systems, it has been shown that interfacial stability or roughening condition depends on the density change at the interface [2]. In this work, we analyze the stability of electrodeposition at the interface between a solid electrolyte and a metal anode, both of which are assumed to be linearly elastic. Based on the derived stability criteria, we show that the condition for stability is sensitive to the mechanical response (shear modulus) and relative density of metal in the two phases [3]. We illustrate this using the constructed stability diagram which shows stable regimes of shear modulus and molar volume of metal ion. This stability diagram qualitatively resembles the stability diagram for stress-driven phase transition at solid-solid interfaces (encountered in, for example, stylolites) studied by Angheluta et al. [2]. Our analysis shows that it is possible to use a soft solid electrolyte provided the density of metal (e.g. Li) is greater in the solid electrolyte than the metal anode. This finding provides new ways to suppress dendrite growth at metal electrode/solid electrolyte interfaces. Two out of the four regions in the stability diagram represent stable electrodeposition regimes. To determine electrodeposition stability for candidate materials for solid electrolytes, we use first-principles calculations of the elastic tensor [4] and estimate of molar volume from ionic radii tabulated by Shannon [5] and Marcus et al. [6]. We find that typical inorganic solid electrolytes have higher shear modulus, but lower molar volume than that required for stable electrodeposition. On the other hand, solid polymer electrolytes have higher molar volume but lower shear modulus than the requirement, leading once again to unstable electrodeposition. Our results suggest that a composite material with a combination of high (low) Li molar volume and high (low) shear modulus is required to suppress dendrites. Recently, the high anisotropy of Li was proposed to play an important role in determining stability of electrodeposition at different Li surfaces [7]. To investigate anisotropy effects, we performed linear stability analysis of electrodeposition at different metal surfaces in contact with solid electrolyte. The deformation equations were solved using the Stroh formalism with the same boundary conditions. Results show that both high density and low density stable regimes exist, with the critical shear modulus curves shifted depending on the Li surface in contact with the solid electrolyte. This presents an opportunity to realize safer high energy density solid-state batteries with metal anodes. [1] C. Monroe and J. Newman, J. Electrochem. Soc. 152, A396 (2005). [2] L. Angheluta, E. Jettestuen, and J. Mathiesen, Phys. Rev. E 79, 031601 (2009). [3] Z. Ahmad and V. Viswanathan, arXiv:1702.08406. [4] Z. Ahmad and V. Viswanathan, Phys. Rev. B 94, 064105 (2016). [5] R. D. Shannon, Acta Crystallogr., Sect. A: Found. Adv. 32, 751 (1976). [6] Y. Marcus, H. Donald Brooke Jenkins, and L. Glasser, J. Chem. Soc., Dalton Trans., 3795 (2002). [7] C. Xu, Z. Ahmad, A. Aryanfar, V. Viswanathan, and J. R. Greer, Proc. Natl. Acad. Sci. USA 114, 57 (2017). Figure 1: Stability diagram showing stable and unstable electrodeposition regimes depending on shear modulus and molar volume ratio of metal [3]. Change in the critical shear modulus curves due to Poisson's ratio is also shown. Figure 1

On surface waves in a finitely deformed coated half-space

In this paper, the wave propagation in a finitely pre-deformed elastic half-space overlain by a thin coating layer (or surface film) is considered. The coated half-space is subjected to a particular uniform pre-deformation such that it is kept to be tractions-free on its surface. The first-order effective boundary conditions are introduced to approximate the effect of the overlying surface film. Then the Stroh formalism and Barnett-Lothe theory are adopted to study the surface wave characteristics. In particular, general criteria are established to identify the existence of surface waves of different modes by taking advantage of the surface impedance matrix. As an illustration, surface waves in a coated soft half-space under biasing field are investigated. Both the surface film and the half-space are modeled by the Hadamard strain energy function for soft isotropic materials. Explicit conditions for the existence of different surface wave modes (including the first-order Rayleigh waves, second-order Rayleigh waves and Love waves) are obtained, and the corresponding wavenumber ranges are also determined. Our theoretical analysis and numerical simulations show that both the surface film and the pre-deformation could remarkably affect the propagation of surface waves as well as the stability of the coated elastic half-space. Particularly, it is proved that, distinguishing from Rayleigh waves, the velocity of Love waves varies linearly with pre-stretch for a given frequency, a striking feature which is highly desirable in the sensor designs.

Fundamental elastic field in an infinite plane of two-dimensional piezoelectric quasicrystal subjected to multi-physics loads

• Green's functions of 2D decagonal piezoelectric QCs in an infinite plane are obtained. • The effects of the phason field on the generalized displacement and stress are analyzed. • The piezoelectric effect of 2D decagonal piezoelectric QCs is shown. • Mechanical change regulations under different material constants are discussed. By utilizing the extended Stroh formalism, the Green's function of infinite plane is obtained for the problem of two-dimensional decagonal quasicrystals with the piezoelectric effect subjected to multi-physics loads. By numerical computations, the piezoelectric effect of the two-dimensional decagonal quasicrystals is revealed; the changes of the stress and displacement fields with multi-physics loads are discussed. The variation laws of material constants in stress and displacement fields are investigated. The results show that the effect of the phason field on the generalized displacement is larger than that on the generalized stress; and the effects of material parameters are different in diverse field.

Interfacial cracks in piezoelectric bimaterials: an approach based on weight functions and boundary integral equations

The focus of this paper is on the analysis of a semi-infinite crack lying along a perfect interface in a piezoelectric bimaterial with arbitrary loading on the crack faces. Making use of the extended Stroh formalism for piezoelectric materials combined with Riemann–Hilbert formulation, general expressions are obtained for both symmetric and skew-symmetric weight functions associate with plane crack problems at the interface between dissimilar anisotropic piezoelectric media. The effect of the coupled electrical fields is incorporated in the derived original expressions for the weight function matrices. These matrices are used together with Betti’s reciprocity identity in order to obtain singular integral equations relating the extended displacement and traction fields to the loading acting on the crack faces. In order to study the variation of the piezoelectric effect, two different poling directions are considered. Examples are shown for both poling directions with a number of mechanical and electrical loadings applied to the crack faces.

Stroh formalism in evaluation of 3D Green’s function in thermomagnetoelectroelastic anisotropic medium

The paper presents studies on the Green’s function for thermomagnetoelectroelastic medium and its reduction to the contour integral. Based on the previous studies the thermomagnetoelectroelastic Green’s function is presented as a surface integral over a half-sphere. The latter is then reduced to the double integral, which inner integral is evaluated explicitly using the complex variable calculus and the Stroh formalism. Thus, the Green’s function is reduced to the contour integral. Since the latter is evaluated over the period of the integrand, the paper proposes to use trapezoid rule for its numerical evaluation with exponential convergence. Several numerical examples are presented, which shows efficiency of the proposed approach for evaluation of Green’s function in thermomagnetoelectroelastic anisotropic solids.

Thermoelasticity of Anisotropic Bimaterial Solids with Contact Thermal Resistance of the Interface Between Their Components and Thin Inclusions

We use the extended Stroh formalism and the theory of functions of complex variable to construct Somigliana-type integral relations and the corresponding equations for thermoelastic anisotropic bimaterial solids with imperfect thermal contact on the rectilinear interface between their components. Applying the mathematical model of thin deformable thermally insulated inclusion and the obtained integral relations, we solve the thermoelasticity problem for a finite anisotropic bimaterial solid containing a thin inhomogeneity with regard for the thermal resistance on the interface between the main components of the materials.

Numerical estimation of strain intensity factors at the tip of a rigid line inclusion embedded in a finite matrix

In this paper, the strain intensity factor at the tip of a rigid line inclusion, embedded in an isotropic matrix, is studied using analytical and numerical techniques. We first revisit the elasticity solution of a rigid inclusion embedded in an infinite elastic matrix using Stroh formulation as a basic framework. This study reveals that the strain intensity factor is appropriate for quantifying the magnitude of singularities at the inclusion tip. Next, we propose a numerical methodology, based on the reciprocal theorem, to calculate the strain intensity factor of the inclusion problem embedded in a matrix of finite geometry. The input to this method is the actual elasticity solution, which is obtained using finite element analysis (FEA). The FEA model is qualitatively validated by comparing FEA results with that of photoelasticity experiments. Finally, strain intensity factor in the case of finite geometry specimen is also estimated.

Modeling core-spreading of interface dislocation and its elastic response in anisotropic bimaterial

Interfacial dislocation may have a spreading core corresponding to a weak shear resistance of interfaces. In this paper, a conic model is proposed to mimic the spreading core of interfacial dislocation in anisotropic bimaterials. By the Stroh formalism and Green’s function, the analytical expressions of the elastic fields are deduced for such a dislocation. Taking Cu/Nb bimaterial as an example, it is demonstrated that the accuracy and efficiency of the method are well validated by the interface conditions, a spreading core can greatly reduce the stress intensity near the interfacial dislocation compared with the compact core, and the elastic fields near the spreading core region are significantly different from the condensed core, while they are less sensitive to a field point that is 1.5 times the core width away from the center of the spreading core.

Stress intensity factor analysis of a three-dimensional interfacial corner between anisotropic piezoelectric multi-materials under several boundary conditions on the corner surfaces

Asymptotic solutions around an interfacial corner can be obtained by a combination of the Stroh formalism and the Williams eigenfunction method. The H-integral method, which is derived from Betti’s reciprocal theorem, is useful for analyzing the stress intensity factors (SIFs) of cracks and corners. By expanding these theories for a three-dimensional interfacial corner between anisotropic piezoelectric multi-materials, we develop a modified H-integral method. This method has high generality that can deal with a jointed corner with a varied number of materials and boundary conditions on corner surfaces. We proposed a new definition for the SIFs of an interfacial corner between anisotropic piezoelectric multi-materials, which is compatible with the SIF definitions of a crack in a homogeneous material and an interfacial crack, as well as applicable in various coordinate systems. The accuracy of obtained SIFs was confirmed by comparing the asymptotic solutions obtained from the SIFs with the stress/electric-displacement field directly obtained by the finite element method (FEM). We also propose a numerical method for degenerate materials, which cause numerical problems in the Stroh formalism.

Modeling and analysis of loaded multilayered magnetoelectroelastic structures composite materials: Applications

This paper presents the detailed analysis of fiber- reinforced magnetoelectroelastic composite plates. The work is divided into two major sections. The first one deals with the homogenization of the properties of each layer based on the Mori-Tanaka mean field approach where all the needed effective coefficients of each layer are determined. Then, in order to perform analysis of the considered, the Stroh formalism is used to provide solutions for multifunctional multilayered magnetoelectroelastic composites, to predict exactly the mechanical and electrical behaviors near or across the interface of material layers.

Electroelastic Green’s function of one-dimensional piezoelectric quasicrystals subjected to multi-physics loads

Green’s functions of infinite and semi-infinite plane problems of one-dimensional quasicrystals with piezoelectric effect are obtained in a closed form by Stroh formalism. Some numerical examples under different loading conditions, such as line forces, line dislocations, and a line charge, are given to explain the mechanical and electric behaviors of the quasicrystals. Various elastic–electric constants of quasicrystals are analyzed and the coupling effects between the phonon and phason fields are studied. The presented solutions will be useful for many boundary value problems of one-dimensional quasicrystals with piezoelectric effect. Moreover, the numerical results can be used to verify the accuracy of the solutions by some numerical methods, such as the finite element and boundary element methods. Furthermore, the Stroh formalism can be generalized to the researches on more complex problems of quasicrystals.

HOLE PROBLEMS IN A CIRCULAR PIEZOELECTRIC PLATE

Due to the unique electric-mechanical coupling effects, piezoelectric materials are widely used in sensors and actuators. Not only in smart structures but also in structural health monitoring, the developments of piezoelectric materials combined with the micro electro mechanical system (MEMS) techniques make our lives totally different. This paper provides a series of numerical simulations about the circular piezoelectric plate with a hole in the plate by using the special boundary element method (BEM). The Green’s functions of BEM are obtained from the derivation of extended Stroh formalism. By this formalism, the analytical closedform solutions of hole in an infinite piezoelectric medium under various loading conditions are obtained. The results demonstrate the coupling effects with different boundary conditions in the circular plate, which will satisfy the requirements of the designers who need to use the products with the hole in the piezoelectric circular plate.

Causal Stroh formalism for uniformly-moving dislocations in anisotropic media: Somigliana dislocations and Mach cones

In this work, Stroh’s formalism is endowed with causal properties on the basis of an analysis of the radiation condition in the Green tensor of the elastodynamic wave equation. The modified formalism is applied to dislocations moving uniformly in an anisotropic medium. In practice, accounting for causality amounts to a simple analytic continuation procedure whereby to the dislocation velocity is added an infinitesimal positive imaginary part. This device allows for a straightforward computation of velocity-dependent field expressions that are valid whatever the dislocation velocity–including supersonic regimes–without needing to consider subsonic and supersonic cases separately. As an illustration, the distortion field of a Somigliana dislocation of the Peierls–Nabarro–Eshelby-type with finite-width core is computed analytically, starting from the Green’s tensor of elastodynamics. To obtain the result in the form of a single compact expression, use of the modified Stroh formalism requires splitting the Green’s function into its reactive and radiative parts. In supersonic regimes, the solution obtained displays Mach cones, which are supported by Dirac measures in the Volterra limit. From these results, an explanation of Payton’s ‘backward’ Mach cones (Payton, 1995) is given in terms of slowness surfaces, and a simple criterion for their existence is derived. The findings are illustrated by full-field calculations from analytical formulas for a dislocation of finite width in iron, and by Huygens-type geometric constructions of Mach cones from ray surfaces.

A Dual BEM Formulation for Thermo-Magneto-Piezo-Electric 2D Fracture Problems

A hypersingular approach of the boundary element formulation -BEM- is presented for the analysis of the temperature in fracture problems of multifield materials. A fundamental solution based on the extended Stroh formalism is considered. Hypersingular integrals arising in the traction boundary and heat flux integral equations are computed through a regularization technique causing that only regular integrals are computed numerically. The generality of the method allows the study of 2D solids with no shape restrictions. Generalized intensity factors -IFs- are accurately computed from nodal values next to crack tip. Finally, the possibilities of the presented approach are discussed in order to apply it for the assessment of the influence of temperature in fracture problems for materialspresenting mechanical, magnetic and/or electrical coupling

Effects of crack surface electrostatic tractions on the fracture behaviour of magnetoelectric composite materials

The fracture mechanics problem of a Griffith crack embedded in a two-dimensional magnetoelectric composite material subjected to coupling mechanical, electric and magnetic loads at infinity is investigated. The crack surfaces electrostatic tractions are taken into account and the crack is assumed to be electrically and magnetically semi-permeable. Explicit and closed form solutions of electric displacement and magnetic induction inside the crack, stress, electric displacement and magnetic induction intensity factors are derived based on the extended Stroh formalism and continuous distribution of generalized dislocation approach. Numerical computations are also carried out to illustrate the influence of electrostatic tractions on the crack tip field when the crack interior is filled with different dielectric medium. It is found that the electrostatic tractions on the crack surfaces have the tendency to close the crack thus retard the crack propagation, and the traditional traction-free crack model always overestimates the effect of applied magnetoelectric loads on the crack tip filed intensity factors. Crack surfaces electrostatic tractions cannot be neglected for large applied electric or magnetic load to mechanical load ratio, and small dielectric constant and magnetic permeability of the crack interior.

Green's functions of one-dimensional quasicrystal bi-material with piezoelectric effect

Based on the Stroh formalism of one-dimensional quasicrystals with piezoelectric effect, the problems of an infinite plane composed of two different quasicrystal half-planes are taken into account. The solutions of the internal and interfacial Green's functions of quasicrystal bi-material are obtained. Moreover, numerical examples are analyzed for a quasicrystal bi-material subjected to line forces or line dislocations, showing the contour maps of the coupled fields. The impacts of changing material constants on the coupled field components are investigated.

Integral Equations of Plane Magnetoelectroelasticity for a Cracked Bimaterial With Thin Inclusions

Based on the combined application of the Stroh formalism and the theory of functions of complex variable, we deduce dual integral equations for a magnetoelectroelastic bimaterial. For the first time, we construct the integral representations of the Stroh complex potentials and the explicit expressions for all kernels in terms of the parameters and matrices of the applied formalism. This noticeably reduces the amount of computations required to get the governing equations of the boundary-element methods. The explicit formulas are obtained for the principal parts of the complex potentials. These formulas enable us to consider homogeneous magnetoelectromechanical loads applied at infinity. The obtained equations, together with previously developed models of thin deformable inclusions, are introduced in the computational algorithm of the boundary-element method with jump functions. The solution of test problems reveals high accuracy and efficiency of the proposed approach. Some solutions are presented for new problems posed for a magnetoelectroelastic bimaterial with thin inclusion.

An extended boundary element method formulation for the direct calculation of the stress intensity factors in fully anisotropic materials

SummaryWe propose a formulation for linear elastic fracture mechanics in which the stress intensity factors are found directly from the solution vector of an extended boundary element method formulation. The enrichment is embedded in the boundary element method formulation, rather than adding new degrees of freedom for each enriched node. Therefore, a very limited number of new degrees of freedom is added to the problem, which contributes to preserving the conditioning of the linear system of equations. The Stroh formalism is used to provide boundary element method fundamental solutions for any degree of anisotropy, and these are used for both conventional and enriched degrees of freedom. Several numerical examples are shown with benchmark solutions to validate the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.

On the propagation of Rayleigh waves in a strongly elliptic thermoelastic material with microtemperatures

ABSTRACTThis article is concerned with the linear theory of thermoelasticity with microtemperatures, which permits the transmission of heat as thermal waves of finite speed, following the theory of Green and Naghdi. We study the propagation of Rayleigh waves in an anisotropic and strongly elliptic thermoelastic half-space by the Stroh formulation.

Complex variable-based analysis for two semi-permeable collinear cracks in a piezoelectromagnetic media

ABSTRACTThe problem of two equal collinear cracks weakening a poled transversely isotropic piezo-electro-magnetic plate is addressed under semi-permeable electro-magnetic boundary conditions on the crack faces. The problem is formulated employing Stroh formalism and solved using a complex variable technique. Closed form analytical expressions are derived for various fracture parameters. An illustrative numerical case study is presented for poled BaTiO3 − CoFe2O4 ceramic cracked plate to study the effect of prescribed electric load, magnetic load, and inter-crack distance on fracture parameters. Moreover, a comparative study is done of volume fraction, impermeable, and semi-permeable boundary conditions on fracture parameters.