Axiomatic/Asymptotic Method and Best Theory Diagram for Composite Plates and Shells
 268 Downloads
Synonyms
Definitions
The axiomatic/asymptotic method (AAM) is a technique to select the expansion terms of a structural theory for a given problem. The AAM makes use of a set of initial terms axiomatically chosen and, then, provides asymptoticlike results and reduced models with lower computational overheads. One of the outcomes of the AMM is the best theory diagram (BTD), i.e., a curve providing the expansions required to minimize the computational cost and maximize the accuracy for a given problem.
Background
The 3D fundamental equations of continuum mechanics have exact analytical solutions only for a few sets of geometries, material properties, and boundary conditions. This makes necessary approximated solutions which often reduce the 3D problem to 2D or 1D. In a 3D model, the unknown variables f(x, y, z) – displacements u(x, y, z), stresses σ(x, y, z), and strains 𝜖(x, y, z) – are defined at each point P(x, y, z) of the volume V . In a 2D model, each variable lies over a reference surface, whereas the behavior along the third dimension – hereinafter referred to as the thickness – is assumed or derived asymptotically; that is, f(x, y, z) = F(z)f ^{∗}(x, y), in which f ^{∗} is the primary unknown variable, and F is the function defining the behavior of the variable through the thickness; see Fig. 1. Similarly, in a 1D model, each variable lies along a reference axis and is assumed above a surface – from now on referred to as the cross section; in other words, f(x, y, z) = F(x, y)f ^{∗}(z). The choice of F defines the structural theory and the capabilities of the model. The main techniques to select F are:

The axiomatic method.

The asymptotic method.
 1.Straight lines perpendicular to the midsurface, also referred to as transverse normals, remain straight in the deformed configuration.
 2.
The transverse normals cannot elongate or compress, i.e., are inextensible.
 1.
An infinite expansion of an unknown function is introduced, for instance, \(f(x,y,z) = \sum _{i=1}^\infty f_ i(x,y)z^i\).
 2.
The expansion is introduced in the problem governing equations, and the thickness parameter is isolated.
 3.
The 3D equations are then written as a series expansion with respect to the thickness parameter δ.
 4.
All the terms in the equations that multiply δ by exponents that are lower or equal to a given order n are retained.
 1.
Moderately thick or thick structures, i.e., \(\frac {L}{h}<50\).
 2.
Materials with high transverse deformability as in the case of common orthotropic materials in which \(\frac {E_L}{E_T}\), \(\frac {E_L}{E_z}\) > 5, and \(\frac {G}{E_L}\) < \(\frac {1}{10}\). E and G are the Young and shear moduli and L is the fiber direction and T, z are perpendicular to L.
 3.
Transverse anisotropy due, for instance, to the presence of contiguous layers with different properties.
Theory
As mentioned above, the free choice of the number of expansion terms within a structural model is a desirable tool to adapt the model capabilities to the structural problem characteristics. The addition of expansion terms, both axiomatically or asymptotically, is not a trivial task. Specifically optimized expansions are built for a given structural problem, and the problem dependency of such structural models limits their application range.
Carrera Unified Formulation (CUF)
 1.
The displacement field definition makes use of a unified index notation taking into account anyorder expansions.
 2.
The unified formulation of the displacement field enables the definition of the geometrical and constitutive equations valid for any expansion order.
 3.
Depending on the solution method – strong or weak form – and the variational tool, e.g., the principle of virtual displacements (PVD) or mixed formulations, the unified version of problem equations and matrices stems directly from the previous steps.
Axiomatic/Asymptotic Method (AAM) and Best Theory Diagram (BTD)
 1.
Parameters, such as the geometry, boundary conditions, materials, and layer layouts, are fixed.
 2.
A starting theory is fixed (axiomatic part). That is, the displacement field is defined; usually, a theory which provides 3Dlike solutions is chosen, and a reference solution is defined.
 3.
The CUF is used to generate the governing equations for the theories considered.
 4.
The effectiveness of each term of the adopted expansion is evaluated by measuring the error due to its deactivation.
 5.
The most suitable structural model for a given structural problem is then obtained discarding the noneffective displacement variables.
ED4 model with \(u_{y_3}\) inactive
z ^{0}  z ^{1}  z ^{2}  z ^{3}  z ^{4} 

\(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
\(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
\(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
Applications
This section presents some relevant applications of the AAM. In particular, first cases show the influence of unknown variables on the solution per varying characteristic parameters and reduced models as accurate as quasi3D solutions. The aim of these cases is to present the capabilities of AAM in obtaining typical results from asymptotic analyses. The conclusive part of this section presents the BTD for various composite plates and shells to provide guidelines and recommendations for the development of advanced models.
Influence of Primary Variables vs Characteristic Parameters
 Independently of the thickness of the plate, the FSDT variables – u _{ x1}, u _{ y1}, u _{ z1}, u _{ x2}, and u _{ y2} – have a predominant influence on the solution.

For thin plates, a/h = 50, the higherorder terms are negligible.

For moderately thick and thin plates, the higherorder terms are as effective as the FSDT ones.
Reduced Models Providing Quasi3D Accuracy
 As wellknown, the leading factors dominating the choice of the expansion terms are the thickness ratio and the transverse shear stress. In other words, the thicker the plate, the higher the number of variables required to meet the reference accuracy. Also, the transverse shear stress components demand more terms than the transverse displacement and the inplane axial stress.Table 2
Reduced LD4 models for the 0/90 plate
a/h = 100
a/h = 4
M _{ e }/M = 11/27
M _{ e }/M = 16/27
u _{ z }
M _{ e }/M = 11/27
M _{ e }/M = 19/27
σ _{ xx }
M _{ e }/M = 13/27
M _{ e }/M = 27/27
σ _{ xz }
M _{ e }/M = 13/27
M _{ e }/M = 27/27
σ _{ yz }
M _{ e }/M = 15/27
M _{ e }/M = 27/27
COMBINED

For thin plates, the adoption of reduced models leads to significant reductions of the computational costs. In fact, almost 50% of the LD4 primary variables do not contribute to the solution.
Best Theory Diagrams
 The FSDT belongs to the BTD in this case. In other words, the FSDT is the model with five unknown variables providing the best accuracy regarding σ _{ αα }. Pandya’s model is slightly off the BTD.

At least eight unknown variables are necessary to have errors lower than 5%.
 Models with less than 50% of the initial variables provide very high accuracy.

The nonpolynomial terms improve the accuracy, but their contribution is secondary.
Conclusions
One of the classical problems of the theory of structures is the development of 2D models to solve the 3D governing equations of continuum mechanics. In a 2D model, the unknown variables lie on a reference surface, and expansions over the third direction – the thickness – define the mechanical behavior of the structure. Axiomatic or asymptotic methodologies can define such expansions for a given problem. For instance, the classical 2D models of Kirchhoff, Reissner, and Mindlin make use of linear expansions for the inplane displacements and constant for the transverse one. The capability of a model and its computational cost increase as the expansion exploits more terms, i.e., more unknown variables. Various problem characteristics may require the use of more expansion terms, for instance, in the case of composite structures, high shear deformability, and transverse anisotropy.
Axiomatic and asymptotic methods have advantages and shortcomings which may limit their applicability. Recently, a novel technique – the axiomatic/asymptotic method (AAM) – proved to be a valid methodology to overcome the shortcomings of both methods. Via the AAM, in fact, the choice of the expansion terms, i.e., the development of the structural theory, starts from an axiomatically chosen model of any order via the Carrera unified formulation (CUF). Then, the AAM evaluates the effectiveness of each term against the most important problem parameters, such as thickness or orthotropic ratios as in the case of an asymptotic approach. By retaining only the effective terms, the AAM provides reduced models with fewer expansion terms of the starting set but as accurate.
The systematic use of the AAM led to the definition of best theory diagram. The BTD is a curve composed of all those models with the least number of expansion terms and the minimum error. In other words, the BTD serves as a benchmark to verify the computational cost and accuracy of any given structural model against the best solution.

The set of active terms, i.e., the set of the necessary unknown variables, is strongly problem dependent.

The proper choice of the unknown variables should consider all the main output parameters, e.g., displacement and stress components. In fact, unknown variables required by one parameter may be completely ineffective for another.

In the case of layerwise models, the systematic use of the AAM may lead to significant reductions in the computational costs.

The BTD is a powerful tool to evaluate any structural model and provide guidelines for their refinement.
CrossReferences
References
 Carrera E (2001) Developments, ideas and evaluations based upon the Reissner’s mixed variational theorem in the modeling of multilayered plates and shells. Appl Mech Rev 54:301–329Google Scholar
 Carrera E (2003) Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Meth Eng 10(3):216–296Google Scholar
 Carrera E, Miglioretti F (2012) Selection of appropriate multilayered plate theories by using a genetic like algorithm. Compos Struct 94(3):1175–1186. https://doi.org/10.1016/j.compstruct.2011.10.013
 Carrera E, Petrolo M (2010) Guidelines and recommendation to contruct theories for metallic and composite plates. AIAA J 48(12):2852–2866. https://doi.org/10.2514/1.J050316
 Carrera E, Petrolo M (2011) On the effectiveness of higherorder terms in refined beam theories. J Appl Mech 78. https://doi.org/10.1115/1.4002207
 Carrera E, Cinefra M, Petrolo M, Zappino E (2014) Finite element analysis of structures through unified formulation. Wiley, ChichesterCrossRefzbMATHGoogle Scholar
 Cicala P (1965) Systematic approximation approach to linear shell theory. Levrotto e Bella, TorinoGoogle Scholar
 Filippi M, Petrolo M, Valvano S, Carrera E (2016) Analysis of laminated composites and sandwich structures by trigonometric, exponential and miscellaneous polynomials and a mitc9 plate element. Compos Struct 150:103–114CrossRefGoogle Scholar
 Gol’denweizer AL (1961) Theory of thin elastic shells. International series of monograph in aeronautics and astronautics. Pergamon Press, New YorkGoogle Scholar
 Kirchhoff G (1850) Uber das gleichgewicht und die bewegung einer elastischen scheibe. Journal fur reins und angewandte Mathematik 40:51–88CrossRefGoogle Scholar
 Mindlin RD (1951) Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. J Appl Mech 18:1031–1036zbMATHGoogle Scholar
 Murakami H (1986) Laminated composite plate theory with improved inplane response. J Appl Mech 53:661–666CrossRefzbMATHGoogle Scholar
 Pandya B, Kant T (1988) Finite element analysis of laminated compiste plates using highorder displacement model. Compos Sci Technol 32:137–155CrossRefGoogle Scholar
 Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12:69–76MathSciNetzbMATHGoogle Scholar
 Varadan T, Bhaskar K (1991) Bending of laminated orthotropic cylindrical shells – an elasticity approach. Compos Struct 17:141–156. https://doi.org/10.1016/j.compstruct.2011.10.013 CrossRefGoogle Scholar
 Washizu K (1968) Variational methods in elasticity and plasticity. Pergamon, OxfordzbMATHGoogle Scholar