The First Shell Theory: The Love-Kirchhoff-Love Model

Improved Essays
From the historical point of view, the first shell models (like many other structural models) have been developed on the basis of engineering evaluation of experiments and heuristic assumptions. The first consistent theory of thin shells goes back to August E.H. Love in 1888 [Lov1888]. As Love’s publication is based on Kirchhoff’s work, this shell model became known as the Kirchhoff-Love model, Kirchhoff is credited to be the founder of modern plate theory with his publication in 1850 [Kir1950].
Several theories have been proposed and each has its application areas and limitations. In general, there are two groups of basic assumptions on which the theories of the shells are based. The first group includes the assumptions for the theories of
…show more content…
[Tas1965], Martin and Drew [Mar1989], whose theory was based on Donnell’s equations, and the work done by Chao [Cha1960] whose analysis was based on Timoshenko’s buckling equations. All the theories discussed above are based on the classical shell theory where the Love-Kirchhoff assumptions are used. The Love-Kirchhoff assumptions amount to treating shells as infinitely rigid in the transverse direction by neglecting transverse strains. The theory underestimates deflections and stresses and overestimates natural frequencies and buckling loads. Since the transverse shear moduli of advanced composite materials are usually very low compared to the in-plane moduli, the transverse shearing strains must be taken into account for an accurate representation of the response of laminated plates and shells. Numerous plate and shell theories which account for transverse shear deformations are documented in the …show more content…
Introducing a polynomial description of the displacements in thickness direction, he opened the door for shell models that match the three-dimensional theory with arbitrary exactness. Nevertheless, these shear-deformable shell models are commonly called shells with Reissner-Mindlin kinematics due to the origin of the assumption of shear-deformable cross-sections. Following the naming of Bischoff [Bis1999], the pure displacement-based shell formulations with shear deformation are called five-parameter formulation. This name relates to the use of five kinematic degrees of freedom (3 displacements and 2 rotational degrees of freedom).
The fundamental idea of shell models is based on the dimensional reduction of the 3D continuum: Taking advantage of the disparity in length scale, the semi-discretization of the continuum is performed to end up in a two-dimensional problem description. This discretization in thickness direction is completely independent from the (later) discretization within the shell mid-surface. Starting from the mechanical model, two general approaches can be

Related Documents

  • Great Essays

    Cellulose Case Study

    • 2188 Words
    • 9 Pages

    After resin impregnation in the TBC, the gel was stretched in the same way as the BC and the mechanical properties of the final composites were investigated to understand the efficacy of the stretching process. It is well-known that porosity should be reduced as much as possible in a composite material since the stress transfer between fiber and resin is negatively influenced by the pores. [24] The porosity in the composites was calculated according to the equation: Porosity=(1-ρ_c/(w_f ρ_f+w_r ρ_r ))×100% , where ρc, ρf, ρr, wf, and wr are the bulk density of composite, density of TBC, density of SPI, weight fraction of TBC, and weight fraction of SPI, respectively. [25] The porosities in both composites were calculated to be in an acceptable range of 2.0-3.5%. The fiber content in both composites was maintained at about 42 ± 2 wt% based on the previous studies by controlling the viscosity of the resin.…

    • 2188 Words
    • 9 Pages
    Great Essays
  • Improved Essays

    If the contact surfaces are free from lubricant, then it is called dry friction. Otherwise it is called fluid friction. The friction is commonly controlled by kinematics of the surfaces in contact, external load, environmental conditions, texture of surface and properties of the materials. Figure 1.7 Friction force required to cause relative motion by (a) sliding (b) rolling 1.6.2 Mechanism of Sliding Friction Always the engineering surfaces are not perfectly smooth and flat. It has asperities (ie.,short range perturbations from the mean) and waviness (i.e., long range perturbations from the mean).…

    • 2169 Words
    • 9 Pages
    Improved Essays
  • Improved Essays

    This means, in this scenario the electron motion is diffusive and green functions are nearly isotropic. Usadel’s approach not only deals to ‘strongly dirty’ samples but also ‘moderately dirty’ samples. The theory of upper critical field for MgB2 was developed separately by Alex Gurevich [133], and Golubov and Koshelov [134]. Their theoretical models are based on multiband generalization of the quasiclassical Usadel equations. The main difference between these two approaches is that Gurevich model considers the small band anisotropy where as Golubov and Koshelov model addresses the case of very high anisotropy in quasi-2D band [134].…

    • 889 Words
    • 4 Pages
    Improved Essays
  • Great Essays

    Nano Tribology

    • 1685 Words
    • 7 Pages

    If we would do the scan in high speed then we have choose low stiffness material but in case of low speed we should always choose high elastic moduli probe. Adhesion force measurement: Adhesion force measurement can be done during recording of the force vs displacement curve. This measurement has to be done in force calibration way. In the below plot point A indicates that the distance between the tip and the sample is too high, hence there is no interaction between sample and probe. After that, probe approaching the sample surface and the distance between the probe and sample is decreases.…

    • 1685 Words
    • 7 Pages
    Great Essays
  • Improved Essays

    Channel Transition Essay

    • 970 Words
    • 4 Pages

    Author plotted the curves in between the Froude number and loss coefficient. The plotted curve showed that the geometry and Froude number both influence the loss of energy in the expansion channel. The author compared the head losses between the different types of transition and concluded that the smooth S-shape curve transition had low head loss coefficient but S-shape curve with vanes obtained the highest head loss coefficient which was unexpected by the author. With the exception of the S-shape curve all other transition displayed some separation. The result indicated that the optimum flow pattern obtained in S-shape walls rather than straight segment channel transition.…

    • 970 Words
    • 4 Pages
    Improved Essays
  • Improved Essays

    The strain energy release rate can also be computed from J for pure hardening plastic materials that undergo small-scale yielding at the crack tip. The quantity J is not path-independent for monotonic mode I and mode II loading of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. Also, Rice showed that J is path-independent in plastic materials when there is no non-proportional loading. Unloading is a special case of this, but non-proportional plastic loading also invalidates the path-independence. Such non-proportional loading is the reason for the path-dependence for the in-plane loading modes on elastic-plastic materials.…

    • 1300 Words
    • 6 Pages
    Improved Essays
  • Great Essays

    In this expansion, Φ_1 is the first order theory (linear theory), Φ_2 is the second order theory and so on. The Stokes expansion method is formally valid under the conditions that H⁄d≪ (kd)^2 for kd<c, the wave would become unstable and break. Stokes found that a wave having a crest angle less than 〖120〗^° would break. Michell (1893) found that in deep water the theoretical limit for wave steepness is (H_0/λ_0 )_max=0.142 Miche (1944) gives the limiting steepness for waves travelling in depths less than λ_0⁄2 without a change in form as; (H/λ)_max= 0.142…

    • 1092 Words
    • 5 Pages
    Great Essays
  • Improved Essays

    2.9 Material Properties 2.9.1 Hardness Test Hardness is defined as a measure of a material’s resistance to contained plastic deformation (e.g., a small dent or scratch). Hardness testing comprises a small indenter being enforced into the surface of the material being confirmed under organized circumstances of load and rate of application. The depth or size of the resulting indentation is measured, which in turn is linked to a hardness number; weaker the material, indentation will be the superior and deeper, and the lower the hardness index number. Measured hardness’s are simply relative (rather than absolute), therefore maintenance must be taken when comparing values determined by dissimilar techniques. However, Hardness is not a ultimate…

    • 1108 Words
    • 5 Pages
    Improved Essays
  • Great Essays

    Spatial features of the individual disturbance are merged into Fourier components however, rotating disturbance can be detected. Results improve only after the application of filters. A free from filtering T-F analysis for the varying spectral characteristics of pressure signals is recommended. In STFT, window size limits to either fine spectral or fine temporal resolutions. Although, compromizing on time resolution in our case, STSF identified rotating stall frequency on waterfall plot and spectrogram but left a question mark on temporal resolution and hence in detecting the rotating deisturbance.…

    • 1521 Words
    • 7 Pages
    Great Essays
  • Superior Essays

    SWAN Model Analysis

    • 1459 Words
    • 6 Pages

    The first three source terms are significant in deep water and the last three terms are significant in shallow water. The SWAN default values for tuneable parameters are Br=1.75 ×〖10〗^(-3) and C_ds=5.0×〖10〗^(-5). Transfer of wind energy to waves, denoted S_in in SWAN, is described by different authors [7,8,12]. Bottom friction (S_br) also has several different formulations: the empirical JONSWAP model of Hasselmann et al. [19], the drag law model of Collins [20] and the eddy viscosity model of Madsen et al.…

    • 1459 Words
    • 6 Pages
    Superior Essays