Shell Modes Research Articles

In Situ Liquid Cell Transmission Electron Microscopy Observations of Growth and Anticorrosion Behaviors of AuCl3 Shell on Au Nanobipyramids

Comprehensive SummaryDesigning new materials and architectures to maintain activity and stability requires a better understanding on the anticorrosion dynamics of nanoparticles. Under‐coordinated atoms on the surface of nanoparticles can be protected by deposited shells. Real‐time observation on how protective shells grow and play a role is challenging but worthwhile. Here, protective effects of AuCl3 shells on Au nanobipyramids (NBPs) are studied in HAuCl4 aqueous solutions by in‐situ liquid cell transmission electron microscopy (LCTEM). This study is the first to observe the formation of Au‐AuCl3 core‐shell nanostructure and the corresponding anticorrosion behaviors of AuCl3 deposited shell. The presence of CTAB can substantially influence the growth mode and structure of AuCl3 shell, by a direct or indirect way, intervene the dissolution of Au NBP. These growth or dissolution kinetics here revealed at the nanoscale provide insights towards engineering of the surface anticorrosion to pursue Au nanoparticles with improved stability in acidic environment.

Theoretical and simulation study on the vibration modes of shear walls in prefabricated modular construction

Abstract As prefabricated construction advances, the enhanced informatization and the industrialized modular construction method of MIC (Modular integrated construction) shear walls have been widely applied in real projects. MIC shear walls are generally made up of prefabricated wall modules with varying strength grades, using concrete of various ages, and are combined with cast-in-place concrete. Due to the unique construction requirements of these walls, defects such as interfacial delamination and internal voids might occur, arising from factors like concrete shrinkage and creep, complex interfacial structures, inadequate compaction during vibration, and construction process issues. Since these interfacial defects cannot be visually inspected from the exterior, developing non-destructive testing (NDT) methods specifically for detecting such defects in shear walls becomes an urgent necessity. This paper utilizes a combination of theoretical analysis and numerical simulation to explore the vibration modes of MIC shear walls with hidden defects. First, a theoretical model is proposed for the local vibration modes of MIC shear wall shells with consideration of interfacial delamination defects. Then, a numerical simulation model is established based on MIC shear wall construction and defect features to simulate the vibration behavior accurately. Finally, after examining the theoretical and simulated vibration characteristics of MIC shear walls, a significant theoretical foundation is established for identifying interfacial defects in MIC shear walls.

Motion state factor driven for doubly-curved shallow shell deformation reconstruction

The conventional shape sensing method, based on C0 continuity, is recognized for its computational efficiency. However, it is prone to shear locking due to low-order polynomial interpolation in shell analysis, introducing two primary challenges. Firstly, it lacks consideration for C1 continuity. Secondly, existing high-order polynomial interpolation methods significantly increase the number of degrees of freedom (DOF), reducing computational efficiency. Therefore, a brand-new real-time structural health monitoring (SHM) method driven by motion state factor instead of nodal factors is proposed. This method accommodates all deformation modes of shallow shells with C1 continuity and improves the shape function's order without additional DOF. It begins with deriving motion states from compatibility equations, leading to a polynomial shape function expression. Subsequently, conversion equations for real constants and DOF are developed, establishing a least-squares model for theoretical strains and measured values. Following that, the higher-order element formula, optimized for minimal DOF, is then inferred from varied displacement data. Finally, simulations and experiments demonstrate this method's effectiveness, accuracy, and reliability in analyzing shallow shell elements.

Towards high-performance polarimeters with large-area uniform chiral shells: a comparative study on the polarization detection precision enabled by the Mueller matrix and deep learning algorithm

Polarization detection and imaging technologies have attracted significant attention for their extensive applications in remote sensing, biological diagnosis, and beyond. However, previously reported polarimeters heavily relied on polarization-sensitive materials and pre- established mapping relationships between the Stokes parameters and detected light intensities. This dependence, along with fabrication and detection errors, severely constrain the working waveband and detection precision. In this work, we demonstrated a highly precise, stable, and broadband full-Stokes polarimeter based on large-area uniform chiral shells and a post-established mapping relationship. By precisely controlling the geometry through the deposition of Ag on a large-area microsphere monolayer with a uniform lattice, the optical chirality and anisotropy of chiral shells can reach about 0.15 (circular dichroism, CD) and 1.7, respectively. The post-established mapping relationship between the Stokes parameters and detected light intensities is established through training a deep learning algorithm (DLA) or fitting the derived mapping-relationship formula based on the Mueller matrix theory with a large dataset collected from our home-built polarization system. For the detection precision with DLA, the mean squared errors (MSEs) at 710 nm can reach 0.10% (S1), 0.41% (S2), and 0.24% (S3), while for the Mueller matrix theory, the corresponding values are 0.14% (S1), 0.46% (S2), and 0.48% (S3). The in-depth comparative studies indicate that the DLA outperforms the Mueller matrix theory in terms of detection precision and robustness, especially for weak illumination, small optical anisotropy and chirality. The averaged MSEs over a broad waveband ranging from 500 nm to 750 nm are 0.16% (S1), 0.46% (S2), and 0.61% (S3), which are significantly smaller than those derived from the Mueller matrix theory (0.45% (S1), 1% (S2), and 39.8% (S3)). The optical properties of chiral shells, the theory and DLA enabled mapping-relationships, the combination modes of chiral shells, and the MSE spectra have been systematically investigated.

Vibration of cylindrical shells: Design criteria for transition from shell modes to beam modes

The dynamic behavior of the cylindrical shell can be predicted by more simplified beam models for a wide range of applications. The present paper deals with finding design conditions in which the cylindrical shell performs like a beam. Employing the Hamilton’s principle, the governing equations are obtained using both Donnell-Mushtari and Flugge shell theories and the analytical solution is obtained for long cylinders with simply-supported boundary conditions at both ends. Then, by equalizing the shell and beam vibration frequencies, the shell-to-beam transition conditions are obtained for both theories. To account for the effects of shear distortion and rotatory inertia of the shell, the finite element method is applied to find the best transition conditions with less approximating assumptions. Finally, the effects of boundary conditions on the transition parameters as well as the frequency response are studied. The obtained conditions simply define that if the shell can be assumed as a beam in any specific geometrical and material conditions.

Lorentz force detuning in heterodyne gravitational wave experiments

Heterodyne cavity experiments for gravitational wave (GW) detection experience a rising interest since recent studies showed that they allow to probe the ultra high frequency regime above 10kHz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10\\,\ ext {kHz}$$\\end{document}. In this paper, we present a concise theoretical study of the experiment based on ideas from the former MAGO collaboration which already started experiments in turn of the millenium. It extends the former results via deriving an additional term originating from a back-action of the electromagnetic field on the cavity walls, also known as Lorentz Force Detuning. We argue that this term leads to a complex dependence of the signal power Psig\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{\ ext {sig}}$$\\end{document} on the coupling coefficient between the mechanical shell modes and the electromagnetic eigenmodes of the cavity. It turns out that one has to adapt the coupling over the whole parameter space since the optimal value depends on the mechanical mode ωl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _l$$\\end{document} and the GW frequency ωg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _g$$\\end{document}. This result is particularly relevant for the design of future experiments.

Influence of polyurea on dynamic response behaviors of cylindrical composite shells under internal explosion load

The dynamic responses of open-ended cylindrical shells with different polyurea layers under explosion load are studied in the current paper. The influences of polyurea thickness and position on the response characteristics of the composite shell are analyzed in detailed. The results indicate that the polyurea layer has significant influence the failure mode and energy absorption of the composite shell. Both polyurea in the interlayer and on the outer layer will decrease the bending moments of the composite shells of equal mass in the initial deformation process, which is easier to make the increase of the deformation heights of metal liner of composite shells with polyurea layers. The polyurea in the interlayer will weaken the constraint of metal liner, delay the fracture of fiber and restrain the bulking phenomenon of metal liner. With increase of polyurea thickness, the bulking phenomenon becomes less obvious and the deformation height of liner decreases non-monotonously. On the contrary, the outer layer polyurea will cause the bulking phenomenon of metal liner more obviously with increase of the polyurea thickness.But the outerlayer polyurea will restrict the deformation of the inner materials and limit the scattering of fiber fragment, improving the safety distance of the explosion vessel. In the whole process of anti-explosion, both polyurea in interlayer and outer layer absorb less than 10% of total energy, but they can significantly change the energy absorption of metal liner and fiber composite.

Practical tutorial on cylindrical structure vibro-acoustics Part 2 - Acoustics

In part 1 of this tutorial (proceedings of Inter-noise 2022) I explained vibrations in cylindrical shell structures. In that paper I limited the mathematics and focused on the key behavior of shells based on critical parameters like the ring frequency, helical wavenumbers, and mean mobilities over different frequency ranges. I compared measured data to the simple theories. In part 2, I focus on the acoustics of cylindrical shells, including sound within shells and the sound radiated outside them. Exterior sound radiation depends strongly on the circumferential order of the shell modes. Breathing modes near the ring frequency radiate sound extremely well, but have very high impedances, so can be difficult to excite. Beam-like modes, where the entire shell cross-section vibrates transversely, radiate less efficiently, but can be easily driven. Higher order, or 'lobar' modes radiate even less efficiently, but nevertheless are commonly observed in radiated sound spectra due to their low impedances. I also review statistical estimates of radiation efficiency of groups of shell modes, which show clear peaks at both the ring frequency as well as at the critical frequency of bending waves. The mathematics of sound inside cylindrical shells is some of the most challenging in vibro-acoustics. At low frequencies, however, the interior sound is dominated by simple one-dimensional planar acoustic waves. At higher frequencies, the sound depends on how well a shell vibration field matches the interior acoustic field based on proximity of resonance frequencies and the similarity of mode shape orders, as well as the 'cut on' frequencies of higher order internal acoustic modes. Finally, I review the well-known phenomenon of how a low shell wall impedance can reduce the effective acoustic sound speed of one-dimensional waves inside cylindrical shells.

Choice of the shape imperfections model in dynamics problems of a long flexible cylindrical shell subjected to force couples

The issue of modeling geometrical imperfections in the dynamics problems of thin-walled shells was little researched. In cases when the natural modes of shell coincided with its buckling modes, the issue of choosing a dangerous imperfection model did not arise. When these shell modes did not coincide, it was important to investigate and compare the effect of different imperfections models on the static and dynamic characteristics of such shells. The choosing the shape imperfections model of a long flexible cylindrical shell subjected to force couples, the natural and buckling modes of which did not coincide, was studied using procedures of the finite element analysis software NASTRAN. The shell wall as a set of plat rectangular elements with six degrees of freedom at the node in the cylindrical coordinate system was modeled. The action of force couples as the concentrated forces were distributed at the nodes of the shell edges in accordance to the presentation of A.S. Volmir. The linear buckling problem and the geometrical nonlinear static analysis of the perfect shell by the Lanzosh method and the Newton-Raphson one were performed, respectively. The long half-waves buckling mode was taken as the first shell imperfections model. The modeling of the second shape imperfections as the first natural mode of the perfect shell using the natural vibration analysis by the Lanzosh method was performed. The different amplitudes of geometrical imperfections in proportion to the shell thickness using a program adapted to this software were set. The results of the geometrical nonlinear static analysis of the imperfect shell by the Newton-Raphson method showed that the shape imperfection model in the form of long half-waves more reduced the values of critical buckling loads. Investigations of natural shell vibrations by the Lanzosh method revealed the same influence of different imperfections models on the natural frequencies and natural forms. We think that the shape imperfections model in the form of long half-waves in studies of forced vibrations and dynamic stability of a long flexible cylindrical shell subjected to force couples will be more effective.

New buckling solutions of truncated conical shells incorporating pre-buckling nonlinearity

The pre-buckling nonlinearity is found to have a remarkable effect on both the buckling loads and modes of shells according to the previous studies. In this study, we present a novel accurate buckling analysis of the truncated conical shells under broad boundary constraints incorporating the pre-buckling nonlinearity by a quasilinearization-precise integral method (Q-PIM). Specifically, the nonlinear buckling equations of the shells are transformed into several linear ones by the perturbation and quasilinearization, and they are then solved by the PIM. The produced state transition equations by the PIM are assembled into a global matrix equation, involving the boundary conditions (BCs), to yield the buckling solutions of the shells with or without incorporating the pre-buckling nonlinearity. The convergence study and benchmark buckling solutions verified by the refined finite element method are presented. The quantitative effects of the size parameters and BCs on the nonlinear critical buckling loads as well as the pre-buckling nonlinearity are investigated.

Smart controllable wave dispersion in acoustic metamaterials using magnetorheological elastomers

The success in the flexible design of smart acoustic metamaterials is crucially contingent upon the degree of control over the parameters that uniquely define the spectrum of band structure and morphology of dispersion surfaces. In this work, we have studied the driving physical mechanisms, the control of which makes possible operating, in real-time, on the set of band gaps formed in 3D metamaterials based on magneto elastomers. In such acoustic structures, the stiffness of the medium in which unit cells are immersed, as well as the stiffness of the shells surrounding multi-particle cores depend on the induction, B(t), of an external magnetic field. The results obtained are systematized through the qualitative analysis of diversе scenarios for the evolution of the frequency characteristics of the metamaterial and summarized in the following complete physical picture of their dynamics. Variation of the stiffness of the medium/shells changes the wavelength of the shell's surface waves at characteristic frequencies of the core vibrations and, as a result, the level of coupling between the vibration modes of the flexible shells and cores. Consequently, with an increase/decrease in the stiffnesses of the shells/medium, the dispersion surfaces of the entire acoustic system shift up/down along the frequency axis with noticeably different ‘mobilities’ that reversibly lead either to the formation of the band gaps in initially dense frequency spectrum or to the transformation of the band gaps formed into pass bands. The tuning of the set of dispersion surfaces depending on the range of changes in the magnetic field induction can be carried out in dynamic, quasi-stationary, and over-critical regimes when some of the dispersion surfaces degenerate into planes. In anisotropic metamaterials, simultaneously with creating full band gaps, it is possible to create tunable directional band gaps with adjustable frequency and angular widths. The results obtained, within the framework of the idealized discrete mass-spring model, are in good agreement with the data presented in the available experimental works and can be useful in the design of acoustic systems with desirable properties.

Typical fracture modes of metal cylindrical shells under internal explosive loading

The competition and coupling of various fracture modes, including tensile fracture, shear fracture and spallation, exist in explosively-driven metal shells. However, the fracture mechanisms and influencing factors of different fracture modes are still unclear. In this study, three typical fracture modes of explosively-driven metal shells–tensile fracture, tensile-shear fracture and shear fracture are numerically revealed by using both the maximum principal strain and the Johnson-Cook cumulative-damage failure criterion. The selection of failure criteria is explained in detail. The fracture strain and fragment morphology are in good agreement with the experimental results. Combined with failure criterion, the numerical results are analyzed from the perspective of pressure, stress and strain. It is found that the formation of different fracture modes depends on the competition between the shear failure formed in the inner surface of the shell and the tensile micro-cracks formed in the middle of the shell. To further investigate the influencing factors of different fracture modes, three dimensionless parameters affecting the fracture modes are proposed based on dimensional analysis. In addition, by analyzing 29 sets of experimental data, we propose the dimensionless equations that can control different fracture modes. And the distribution diagrams of three typical fracture modes (tensile, tensile-shear and shear fracture) of explosively-driven metal shells are determined. Furthermore, we propose a parameter that can effectively distinguish adiabatic shear fracture from non-adiabatic shear fracture.

Colored textiles based on noniridescent structural color of ZnS@SiO2 colloidal crystals for daytime passive radiative cooling

Passive radiative cooling (PRC) technology, which utilizes the optical properties of materials to achieve electricity-free and spontaneous cooling, holds great promise for energy conservation and combating global warming. Integrating PRC with textiles enables precise regulation of the microclimate of the human body, facilitating sustainable and effective individual thermal comfort. However, such PRC materials typically exhibit a white or mirror-like appearance to minimize solar energy absorption, severely limiting their aesthetic appeal and practical application in real-life scenarios. Herein, we present a colored textile for PRC by employing the spray coating method with noniridescent structural color pigments consisting of carbon-modified ZnS@SiO2 core–shell microspheres (colloidal crystals). This approach simultaneously achieves noniridescent colors and efficient cooling. By using the photonic properties of ZnS and the core–shell structure, the ZnS@SiO2 colored textile selectively reflects visible light, resulting in coloration while reducing solar absorption. Furthermore, the phonon resonance (Fröhlich mode) of the SiO2 shell ensures high emissivity of the ZnS@SiO2 colored textile within the atmospheric window, allowing for substantial radiative heat dissipation. Outdoor experiments demonstrate that both the green and amaranth ZnS@SiO2 textiles could lower the simulated skin temperature by 5 ∼ 6 ℃ in direct sunlight, compared to ZnS and dye textiles of the same hue. Additionally, the ZnS@SiO2 colored textiles also exhibit excellent softness, washability, hydrophilicity, breathability, and tensile properties, making them suitable for practical applications. This work not only provides a practical method for synthesizing noniridescent structural color pigments with high color saturation but also offers new insights into the design of colored passive radiative cooling textiles.

Study on the deformation of inter-rib shell plate of reinforced conical shell structure under deep-water-explosion loading

The reinforced cylindrical and conical column structures are widely used structural forms in underwater vehicle structures. In this paper, the reinforced conical shell structure in the transition compartment of underwater vehicles was used as the research object. Based on the shock wave propagation theory and ABAQUS acoustic structure coupling numerical simulation method, the numerical simulation model of conical shell structure under deep-water explosion load was established.The deep-water explosion simulation of conical shell structure under the conditions of different water depths, charges and explosion distances was carried out, and the deformation and failure mode of small depressions in the shell plate between ribs of stiffened conical shell structure was obtained, that was, independent small pits were formed along the axis direction. According to the theory of elastic-plastic deformation of plate and shell structure, the small pits of shell and plate between ribs were approximated as the deformation of trapezoidal plate, and the deformation function of trapezoidal plate was obtained by coordinate transformation. The deformation energy of plate and shell was divided into bending deformation energy and midplane strain energy. By establishing the correlation between explosion energy and deformation energy, the theoretical calculation model of the small pit deformation mode of shell and plate between ribs was obtained.

Acoustic scattering from spherical shells and its active control

The scattering of sound by a thin spherical shell is considered using an analytic formulation involving spherical harmonics. The integral of the far field scattered intensity, termed the scattered power, can then be expressed in a simple form. At low frequencies the scattered power can be minimised by an appropriate choice of material properties and shell thickness, which is illustrated for both a steel shell and for one in which the mass and stiffness of the shell are equal to those of the displaced fluid. Simulations of feedforward active control are then used to investigate the best possible performance in attenuating the scattered power, although this approach requires knowledge of the incident and scattered sound fields. Feedback control of the shell vibration using structural actuators and sensors is a more practical control strategy since it does not involve the need to separate the incident and scattered contributions. Direct velocity feedback control is considered using collocated and distributed actuators and sensors that spread the applied force and sensed velocity over spherical caps on the surface of the shell. This approach shows effective suppression of the structural shell modes that give rise to significant scattering at their resonant frequencies.

Buckling performance of stiffened polymer composite cylindrical shell

The buckling of carbon fiber composite cylindrical shells (CF-CCSs) with annular ribs under external pressure was investigated numerically and experimentally. The effects of rib cross-sections, rib geometry and distribution on buckling load and buckling mode were analyzed numerically. In addition, the ribbed polymer composite cylindrical shell was fabricated and tested. The geometry, thickness, buckling load and final collapse mode of the ribbed polymer composite cylindrical shell were measured. Finally, a nonlinear buckling analysis with geometrical imperfections was carried out based on the measured data; the numerical results were in agreement with the experimental results. The results show that different rib cross-sections have limited influence on the load carrying capacity of CF-CCSs, but the distribution of the ribs and the geometry of the ribs have a greater influence on CF-CCSs.

Dirac: A command-line γ-matrix calculator

A software for simplification of Dirac matrix polynomials that arise in particle physics problems is implemented. Program summaryProgram Title: DiracCPC Library link to program files:https://doi.org/10.17632/s5w3zyhy58.1Developer's repository link:https://github.com/skutnii/diracLicensing provisions: MITProgramming language: C++20Nature of problem: Particle physics computations require simplification of Dirac matrix polynomials. Non-commutativity and tensorial nature of Dirac matrices makes the task nontrivial. Complexity of the expressions grows quickly with the number of factors making manual computations beyond a few simplest cases unfeasible and highly error-prone. Existing software solutions are either proprietary and highly expensive or have poor if any support for Dirac matrix algebra.Solution method: The problem is solved by considering a representation of Dirac matrix algebra with 5×5pseudo-matrices - matrices made of the algebra's structure constants. In this representation, multiplication of the original matrices corresponds to multiplication of pseudo-matrices and Lorentz index contraction. The problem then gets reduced to simplification of polynomials in Minkowski metric, Kronecker, and Levi-Civita symbols.Additional comments including restrictions and unusual features: The input and output have ▪-like syntax with single-expression command line input and interactive shell modes. Scripting can be used for batched input processing and advanced ▪ formatting of the results. Example scripts demonstrate this feature and provide automated correctness tests. The software supports both rational and floating-point arithmetic.

Numerical and Theoretical Study on Failure Characteristics and Damage Assessment Model of Curved Steel-Concrete-Steel Sandwich Shells Subjected to Impact Load

In this study, damage assessment model of curved steel–concrete–steel (CSCS) sandwich shells subjected to impact loads was investigated. A numerical simulation method was applied to obtain failure modes of CSCS sandwich shells under impact load by changing the impact velocity as well as the mass and diameter of the hammer. From the 108 finite element (FE) models, three failure modes were found: local deformation, top steel plate fracture, and penetration. The failure mechanism for each failure mode was analyzed based on numerical simulation results, such as impact force and displacement-time histories, varying internal energy, and deformation modes of the structures. The law of failure mode distribution of the CSCS sandwich shell was also summarized. Moreover, the impact force–displacement relationship was obtained using an analytical method, and the FE results were in acceptable agreement with experiments. Finally, damage assessment model of the CSCS sandwich shells was obtained by comparing the hammer initial kinetic energy with the maximum absorbed energy of the CSCS sandwich shell. This model can be used as a quick tool to judge the failure status of the CSCS sandwich shell. The work presented in this paper is of significance and can fill the gap of the damage assessment model of CSCS sandwich shells subjected to impact loads.

On the peak acceleration of a beam in response to a shock and its validity for cylindrical pipes

This paper presents a closed-form solution for the peak acceleration in an infinite beam as a response to a half-cycle sine pulse excitation. The limits of its validity for a finite length beam are discussed. Numerical and experimental results are presented for a drilling collar where the excitation of shell modes may lead to underestimating the maximum acceleration if beam theory is used. A condition is suggested to decide if beam theory is appropriate for thick cylinders and the contribution from shell modes can be neglected.

Isogeometric stiffness matrix formulations for buckling analysis of multi-layered shells based on lamination parameters for solid-shell elements

Multi-layered composite shells are extensively used in many industrial fields since they have superior performance in applications. As load-carrying components, buckling failure is one of the most common failure modes of multi-layered shells. Hence, the layup design of multi-layered shells based on optimal buckling factors is widely concerned, where lamination parameters are commonly used in the representation of stiffness. However, the common formulations of laminate stiffness are based on the Classical Laminate Plate Theory (CLT) or the First-order Shear Deformation Plate Theory (FSDT). This paper aims at developing stiffness formulations based on 3D continuum solid theory. The accuracy and efficiency of the stiffness formulations derived in this study are analysed and compared to those obtained by layer-by-layer integration and summation of stiffness calculations.