Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Axiomatic/Asymptotic Method and Best Theory Diagram for Composite Plates and Shells

  • Erasmo CarreraEmail author
  • Marco Petrolo
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_140-1

Synonyms

Definitions

Plates and shells are 2D structural models; in fact, the unknown, primary variables depend on two coordinates and are assumed along the third one. Plates and shells can model the structural behavior of 3D bodies in which the third dimension – the thickness, h – is much smaller than the other two. To define a plate or shell is useful to use segments of height h and the surface containing the midpoints of the segments, namely, the mid-surface. In plates, the mid-surface is flat; in shells the mid-surface is curved. Figure 1 shows the geometry of a shell, the mid-surface, or reference surface embodies the 3D features of the structure. From a mathematical standpoint, in plates and shells, expansions of the thickness coordinate (z) defines the behavior of the unknown variables along the thickness. Each term of the expansion is a generalized unknown variable. The more the terms of the expansion, the higher the accuracy and the computational cost.
Fig. 1

2D shell modeling of a 3D curved body

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 asymptotic-like 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.

The axiomatic method makes use of hypotheses that cannot be mathematically proved. Hypotheses reduce the mathematical complexity of the 3D differential equations of elasticity to find solutions for a wide variety of boundary conditions, geometries, and materials. Axiomatic theories stem from the intuition of a scientist who can determine the most important variables for a given problem. Kirchhoff (1850), Reissner (1945), and Mindlin (1951) are typical examples of plate and shell models. For instance, the Reissner-Mindlin model, also referred to as first-order shear deformation theory (FSDT), can be defined via its displacement field as follows:
$$\displaystyle \begin{aligned} \begin{aligned} u_{x}(x,y,z) &= u_{x_1}(x,y) + zu_{x_2}(x,y)\\ {}u_{y}(x,y,z) &= u_{y_1}(x,y) + zu_{y_2}(x,y)\\ {}u_{z}(x,y,z) &= u_{z_1}(x,y) \end{aligned} \end{aligned} $$
(1)
The unknown variables are five, \(u_{x_1}\), \(u_{x_2}\), \(u_{y_1}\), \(u_{y_2}\), and \(u_{z_1}\). The in-plane FSDT displacement field is linear along the thickness (z), whereas the transverse displacement is constant, as shown in Fig. 2. The assumptions leading to such a displacement field are:
  1. 1.
    Straight lines perpendicular to the mid-surface, also referred to as transverse normals, remain straight in the deformed configuration.
    Fig. 2

    FSDT displacement field along the thickness

     
  2. 2.

    The transverse normals cannot elongate or compress, i.e., are inextensible.

     
In particular, the first assumption leads to the linear distribution of the in-plane displacements, the third to the constant transverse displacement. The constant terms are necessary to detect the membranal behavior, whereas the linear terms for bending. In-plane stresses are linear along z, whereas transverse shear stresses are constant. The transverse normal stress is neglected.
As explained in next sections, the capabilities of a structural model and its applicability can extend as soon as the displacement field has more variables. For instance, a third-order (N = 3) model has a cubic expansion over z:
$$\displaystyle \begin{aligned}u_{x}(x,y,z) &= u_{x_1}(x,y) + zu_{x_2}(x,y) \\ &\quad + z^2u_{x_3}(x,y) + z^3u_{x_4}(x,y)\\ u_{y}(x,y,z) &= u_{y_1}(x,y) + zu_{y_2}(x,y) \\ &\quad + z^2u_{y_3}(x,y) + z^3u_{y_4}(x,y)\\ u_{z}(x,y,z) &= u_{z_1}(x,y) + zu_{z_2}(x,y) \\ &\quad + z^2u_{z_3}(x,y) + z^3u_{z_4}(x,y)\end{aligned} $$
(2)
In this case, the model has 12 unknown variables. In other words, the overcoming of the assumptions leads to higher computational costs. Figure 3 shows the displacement field along the thickness for various order models. The expansion of the unknown variables defines univocally the structural theory; therefore, hereinafter, expansion and structural theory or model are synonymous.
Fig. 3

Various order displacement fields along the thickness

Many models have been developed over the years and most of them exploit various orders of the expansion for each unknown. For instance, Pandya and Kant (1988) proposed the following:
$$\displaystyle \begin{aligned}u_{x}(x,y,z) &= u_{x_1}(x,y) + zu_{x_2}(x,y)\\ &\quad+ z^2u_{x_3}(x,y) + z^3u_{x_4}(x,y)\\ u_{y}(x,y,z) &= u_{y_1}(x,y) + zu_{y_2}(x,y)\\ &\quad+ z^2u_{y_3}(x,y) + z^3u_{y_4}(x,y)\\ u_{z}(x,y,z) &= u_{z_1}(x,y)\end{aligned} $$
(3)
in which constant and cubic expansions model the transverse and in-plane displacements, respectively.
An important shortcoming of axiomatic methods is the impossibility to evaluate a priori the accuracy of the approximated theory with respect to the exact 3D solution. Let A and B be two generic theories with B a theory adopted to enhance A through additional terms in the expansion. A fundamental issue is the effectiveness of the additional terms in B. In other words, do the additional terms improve the accuracy of A? Usually, structural analysts answer by selecting the necessary terms by their knowledge and experience. A mathematically rigorous way to tackle such an issue is the asymptotic method which provides approximated theories with known accuracy with respect to the 3D exact solution (Gol’denweizer, 1961; Cicala, 1965). Let the following expansion be an exact solution:
$$\displaystyle \begin{aligned}f_{\textit{exact}} &= f_1(x,y) + f_2(x,y)\;z + \frac{1}{2}f_3(x,y)\;z^2 \\ &\quad+ \frac{1}{6}f_4(x,y)\;z^3 + \ldots\end{aligned} $$
(4)
The assumption here is that A contains all the terms that have the same effectiveness as f 1(x, y) and f 2(x, y) in the solution. It is necessary to evaluate whether B has all the terms with the same effectiveness as f 1(x, y), f 2(x, y) and f 3(x, y). Many axiomatic theories from the last decades missed some fundamental terms, therefore, lacking effective terms. The asymptotic method overcomes this drawback by introducing some controls on the order of magnitude of the effectiveness of each term introduced into an expansion. As soon as the order of magnitude is set, the asymptotic method guarantees the presence of all the significant terms of that given order. In an asymptotic model, the reference solution is the limit of a function with respect to a characteristic feature of the problem, e.g., a characteristic length. For instance, in a 2D model, this parameter can be the ratio between the thickness and the width of a plate:
$$\displaystyle \begin{aligned} \delta = \frac{h}{L} = \frac{\mathrm{thickness}}{\mathrm{reference\;\, length}} \end{aligned} $$
(5)
When δ → 0, the 3D solid plate becomes a 2D surface, and the 2D plate theory is exact. A typical procedure to build an asymptotic theory is the following:
  1. 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. 2.

    The expansion is introduced in the problem governing equations, and the thickness parameter is isolated.

     
  3. 3.

    The 3D equations are then written as a series expansion with respect to the thickness parameter δ.

     
  4. 4.

    All the terms in the equations that multiply δ by exponents that are lower or equal to a given order n are retained.

     
The development of asymptotic theories is more difficult than the development of axiomatic ones and requires as many as single analyses as the number of problem parameters to handle. The main advantage of these theories is that they contain all the terms whose effectiveness is of the same order of magnitude. Moreover, these theories are exact as δ → 0.
The number of expansion terms to add depends on the problem characteristics. In the framework of composite plates and shells, the typical factors making classical models inefficient, i.e., refined models necessary, are the following (Carrera, 2001):
  1. 1.

    Moderately thick or thick structures, i.e., \(\frac {L}{h}<50\).

     
  2. 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. 3.

    Transverse anisotropy due, for instance, to the presence of contiguous layers with different properties.

     
Factors 1 and 2 make the influence of shear and normal transverse stresses not negligible and their distribution along z not constant. Factor 3 causes rapid variations of the displacement field at the interface between two layers with different mechanical properties, i.e., the zigzag effect. In other words, at the interface, the displacement field is continuous but its first derivative not. Moreover, according to the Cauchy theorem, the transverse shear and normal stresses must be continuous at the interface, i.e., interlaminar continuity is necessary. Classical models, originally developed for isotropic materials, cannot handle such effects.

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)

The CUF has been developed over the last two decades as a tool to generate any structural model via arbitrary expansions of the unknown variables (Carrera, 2003; Carrera et al, 2014). For the sake of simplicity, the displacement-based version of CUF for 2D models is considered here; in other words, the primary unknown variables are generalized displacements and the equations are valid for plate and shell models. The main steps undertaken by CUF are the following:
  1. 1.

    The displacement field definition makes use of a unified index notation taking into account any-order expansions.

     
  2. 2.

    The unified formulation of the displacement field enables the definition of the geometrical and constitutive equations valid for any expansion order.

     
  3. 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.

     
Such a procedure leads to a formulation in which the order of the expansion, but also the expansion itself, e.g., polynomial or exponential, is a free parameter of the problem. In other words, one of the inputs of the analysis is the structural model, and no limitations on it are in place. Here, for the sake of brevity, only the finite element formulation of CUF based on the PVD is used.
The CUF defines the displacement field for a 2D model as
$$\displaystyle \begin{aligned} \mathbf{u}(x, y, z)=F_{\tau}(z)\mathbf{u}_{\tau}(x, y)\qquad\tau=1, \dots, M \end{aligned} $$
(6)
where the Einstein notation operates on the index τ. u is the displacement vector (u x u y u z ). F τ are the so-called thickness expansion functions and u τ is the vector of the generalized unknown displacements. M is the number of expansion terms In the case of polynomial, Taylor-like expansions, a third-order model, hereinafter referred to as ED3, has the following displacement field:
$$\displaystyle \begin{aligned} \begin{aligned} u_{x}=u_{x_{1}}+z\,u_{x_{2}}+z^{2}\,u_{x_{3}}+z^{3}\,u_{x_{4}}\\ u_{y}=u_{y_{1}}+z\,u_{y_{2}}+z^{2}\,u_{y_{3}}+z^{3}\,u_{y_{4}}\\ u_{z}=u_{z_{1}}+z\,u_{z_{2}}+z^{2}\,u_{z_{3}}+z^{3}\,u_{z_{4}}\\ \end{aligned} \end{aligned} $$
(7)
In this case, τ ranges from 1 to 4, and F 1 = 1, F 2 = z, F 3 = z 2, and F 4 = z 3. The FE formulation makes use of the shape functions N i (x, y) to interpolate the displacement field along the reference surface of the structure:
$$\displaystyle \begin{aligned} \mathbf{u}(x, y, z)=F_{\tau}(z)N_i(x,y)&\mathbf{u}_{\tau i}\quad\tau=1, \dots, M; \\&\quad \ i=1, \dots, N_{\textit{nodes}} \end{aligned} $$
(8)
where N nodes indicates the number of nodes per element. The virtual variation is
$$\displaystyle \begin{aligned}\delta \mathbf{u}(x, y, z)\,{=}\,F_{s}(z)N_j(x,y)\delta &\mathbf{u}_{sj}\quad s{=}1, \dots, M;\\ &\quad j{=}1, \dots, N_{\textit{nodes}} \end{aligned} $$
(9)
Via the introduction of the geometrical and constitutive relations, it can be proved that the assemblage of the element stiffness matrix can make use of a 3 × 3 block, referred to as the fundamental nucleus:
$$\displaystyle \begin{aligned} \begin{array}{ccc} k^{\tau sij} & = & \left[ \begin{array}{ccc} k_{xx}^{\tau sij} & k_{xy}^{\tau sij} & k_{xz}^{\tau sij} \\ k_{yx}^{\tau sij} & k_{yy}^{\tau sij} & k_{yz}^{\tau sij} \\ k_{zx}^{\tau sij} & k_{zy}^{\tau sij} & k_{zz}^{\tau sij} \\ \end{array} \right] \end{array} \end{aligned} $$
(10)
The formal expression of the nine components does not depend on the expansion order or type. Moreover, only two expressions lead to the definition of the matrix; in the case of isotropic material, they are
$$\displaystyle \begin{aligned} k^{\tau sij}_{xx}&= (\lambda\,{+}\,2G) \int_{\varOmega}N_{i,x}N_{j,x}dx\;dy \int_h F_{\tau}F_s dz \\ &\quad+ G \int_{\varOmega} N_i N_j dx \;\;\int_h F_{\tau,z} F_{s,z} dz \\ &\quad + G \int_V N_{i,y} N_{j,y}dx\;dy \;\; \int_h F_{\tau} F_s dz \end{aligned} $$
(11)
$$\displaystyle \begin{aligned} k^{\tau sij}_{xy} &= \lambda \int_{\varOmega} N_{i,y}N_{j,x} dx\;dy \;\; \int_h F_{\tau} F_s dz\\&\quad + G \int_{\varOmega} N_{i,x} N_{j,y} dx\;dy \;\; \int_h F_{\tau} F_s dz \end{aligned} $$
(12)
where λ and G are the Lamé coefficients, and Ω is the reference surface domain. All the diagonal and non-diagonal terms stem from index permutations of \(k^{\tau sij}_{xx}\) and \(k^{\tau sij}_{xx}\), respectively. As an example, we may consider an ED1 model; that is, τ, s range from 1 to 2, F 1 = 1, and F 2 = z. In this case, the nodal stiffness matrix is where the superscripts indicate the expansion functions that are involved in each component of the stiffness matrix, i.e., 1 and z. The explicit expression of a component is
$$\displaystyle \begin{aligned} \begin{aligned} k_{xx}^{1,1} &= C_{11}\int_{h} 1 \cdot 1\;dz\int_{\varOmega} N_{i,x}N_{j,x}d\varOmega \\&\quad + {C}_{66}\int_{h} 1 \cdot 1\;dz\int_{\varOmega} N_{i,y}N_{j,y}d\varOmega \end{aligned} \end{aligned} $$
(14)
Any theory from literature is a particular case of a full expansion. For instance, the model proposed by Pandya and Kant (1988) is a particular case of the ED3:
$$\displaystyle \begin{aligned} \begin{array}{l} u_{x}=u_{x_{1}}+z\,u_{x_{2}}+z^{2}\,u_{x_{3}}+z^{3}\,u_{x_{4}}\\ u_{y}=u_{y_{1}}+z\,u_{y_{2}}+z^{2}\,u_{y_{3}}+z^{3}\,u_{y_{4}}\\ u_{z}=u_{z_{1}}\\ \end{array} \end{aligned} $$
(15)
The extension to multilayered structures is straightforward. In particular, equivalent single-layer (ESL) and layer-wise (LW) approaches are of interest. The former reduces the multilayered structure to an equivalent single one. The latter keeps the properties of each layer and operates the homogenization at the interface level via the imposition of the continuity of the displacements. Figure 4 shows the stiffness matrix assembly procedures for both cases. In LW, differently from ESL, the number of unknowns depends on the number of layers leading to higher computational costs than ESL. However, LW provides more accurate transverse stress fields and an improved interlaminar continuity. A way to obtain LW models is via Legendre polynomial expansions in each layer k:
$$\displaystyle \begin{aligned} \begin{array}{l} \mathbf{u}^{k}=F_{t}\cdot\mathbf{u}_{t}^{k}+F_{b}\cdot\mathbf{u}_{b}^{k}+F_{r}\cdot\mathbf{u}_{r}^{k}=F_{\tau}\mathbf{u}_{\tau}^{k} \\ \tau=t, b, r\,\,\,r=2,3,\dots, N \quad k=1,2,\dots,N_{L} \end{array} \end{aligned} $$
(16)
where N L is the number of the layers. Subscripts t and b correspond to the top and bottom surfaces of the layer. Functions F τ depend on the nondimensional thickness coordinate ζ k , − 1 ≤ ζ k  ≤ 1. F τ are linear combinations of the Legendre polynomials:
$$\displaystyle \begin{aligned} \begin{array}{l} F_{t}=\frac{P_{0}+P_{1}}{2} \qquad F_{b}=\frac{P_{0}-P_{1}}{2} \\ F_{r}=P_{r}-P_{r-2}\qquad r=2,3,\dots,N \end{array} \end{aligned} $$
(17)
LDN indicates an LW model, where N is the expansion order. For instance, LD3 is
$$\displaystyle \begin{aligned} \begin{array}{l} u_{x}^{k}=F_{t}\,u_{xt}^{k}+F_{2}\,u_{x2}^{k}+F_{3}\,u_{x3}^{k}+F_{b}\,u_{xb}^{k} \\ u_{y}^{k}=F_{t}\,u_{yt}^{k}+F_{2}\,u_{y2}^{k}+F_{3}\,u_{y3}^{k}+F_{b}\,u_{yb}^{k} \\ u_{z}^{k}=F_{t}\,u_{zt}^{k}+F_{2}\,u_{z2}^{k}+F_{3}\,u_{z3}^{k}+F_{b}\,u_{zb}^{k} \\ \end{array} \end{aligned} $$
(18)
The unknown variables at the top and bottom surfaces are actual displacements making the continuity imposition at the interface straightforward.
Fig. 4

ESL and LW assemblage for a three-layer structure via ED2 and LD2

Axiomatic/Asymptotic Method (AAM) and Best Theory Diagram (BTD)

A convergence analysis allows one to select the proper expansion order for a given problem. As stated by Washizu (1968), an infinite expansion would guarantee the exact 3D solution. However, in most cases, a fourth-order expansion can provide very accurate results. In an ESL fourth-order model (ED4), the unknown displacement variables are 15. Each variable has a different influence on the solution, and, in some cases, the influence is null or negligible. An asymptotic analysis provides the influence of each variable. Recently, in the CUF framework, an alternative method – the axiomatic/asymptotic method (AAM) – has been developed to provide the same information by exploiting an axiomatic-like theory and enable to handle the variation of many problem parameters easily, e.g., thickness, orthotropic ratio, stacking sequence, and boundary conditions (Carrera and Petrolo, 2010, 2011). The AAM leads to the definition of reduced models with a lower computational cost than full models but with the same accuracy. A typical AAM analysis consists of the following steps:
  1. 1.

    Parameters, such as the geometry, boundary conditions, materials, and layer layouts, are fixed.

     
  2. 2.

    A starting theory is fixed (axiomatic part). That is, the displacement field is defined; usually, a theory which provides 3D-like solutions is chosen, and a reference solution is defined.

     
  3. 3.

    The CUF is used to generate the governing equations for the theories considered.

     
  4. 4.

    The effectiveness of each term of the adopted expansion is evaluated by measuring the error due to its deactivation.

     
  5. 5.

    The most suitable structural model for a given structural problem is then obtained discarding the noneffective displacement variables.

     
A graphical notation is introduced to show the results. It consists of a table of three lines and columns equal to the number of the variables used in the expansion. For example, if an ED4 model is considered with \(u_{y_3}\) deactivated, the displacement field is
$$\displaystyle \begin{aligned} \begin{array}{lccccc} u_{x} = &u_{x_1} + &z\;u_{x_2} + &z^2\;u_{x_3} + &z^3\;u_{x_4} + &z^4\;u_{x_5} \\ u_{y} = &u_{y_1} + &z\;u_{y_2} + &\ \ \ \ \ \ \ \ +&z^3\;u_{y_4} + &z^4\;u_{y_5} \\ u_{z} = &u_{z_1} + &z\;u_{z_2} + &z^2\;u_{z_3} + &z^3\;u_{z_4} + &z^4\;u_{z_5} \end{array} \end{aligned} $$
(19)
Such a displacement field is depicted by Table 1.
Table 1

ED4 model with \(u_{y_3}\) inactive

z 0

z 1

z 2

z 3

z 4

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\vartriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

\(\blacktriangle \)

The use of the AAM can be extended to all the possible combinations of active/inactive variables of a starting theory. Each reduced model can be related to the number of the active terms and its error computed on a reference solution as reported in Fig. 5. The error values are reported on the abscissa, and the number of active terms is reported on the ordinate. A model is defined considering its error and the number of the active terms. In such a Cartesian plane, the best theory diagram is the curve composed by all those models providing the minimum error with the least number of variables. As an example, Fig. 6 shows the BTD based on an ED4 model. Each dot represents a structural model with various accuracies. For a given accuracy, there are no models based on ED4 with fewer variables than those on the BTD. Also, for a given number of variables, there are no models based on ED4 with better accuracy than those on the BTD.
Fig. 5

Best theory diagram (BTD)

Fig. 6

BTD for an ED4 model

The BTD represents a Pareto front. This curve varies as materials, geometries, and boundary conditions vary. An ED4 has 215 models obtained combining the 15 terms of the expansion. The evaluation of the accuracy of all these models may lead to excessive computational costs. Genetic algorithms are helpful to obtain the BTD and avoid such a computational overhead (Carrera and Miglioretti, 2012). In fact, each structural theory is an individual, the genes are the terms of the expansion, and each gene can be active or inactive as in Fig. 7. Each individual has a number of active terms and an error computed on a reference solution. Through these two parameters, it is possible to apply the dominance rule to evaluate the individual fitness. Copies and mutations lead to new individuals to find the Pareto front, that is, the subset of individuals not dominated by the others.
Fig. 7

Displacement variables as genes of a genetic algorithm

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 quasi-3D 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

The first case considers a square simply supported plate with two layers – 0/90 – and a bi-sinusoidal load. Figure 8 shows the influence of each of the 15 unknown variables of an ED4 on σ xx for various thickness ratios. A layer-wise fourth-order model is the reference solution (LD4); in fact, for this type of problems, LD4 provides quasi-3D accuracy (Carrera, 2003). The continuous horizontal line is the error of ED4, hence, when all the 15 variables are active. Each dot provides the error caused by the absence of each variable. The main outcomes from this case are the following:
  • 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.
    Fig. 8

    Influence of ED4 primary variables on σ xx for various thickness ratios

  • For thin plates, a/h = 50, the higher-order terms are negligible.

  • For moderately thick and thin plates, the higher-order terms are as effective as the FSDT ones.

In the second case, the plate has one layer, and Fig. 9 shows the influence of u x4 on the transverse shear stress for various orthotropic ratios. As expected, the influence is higher as the orthotropic ratio increases.
Fig. 9

Influence of u x4 on σ xz for various orthotropic ratios

Reduced Models Providing Quasi-3D Accuracy

Reduced models have fewer unknown variables than full ones. The accuracy of such reduced models is an input of the AAM analysis. In other words, the error against a reference solution is an input of the analysis. In this section, the error is set to null, and the reference solution is the LD4. Therefore, the reduced models provide results as accurate as the full LD4. The numerical case deals with the same 0/90 plate of the previous section. Table 2 shows the reduced models as accurate as the full LD4. Each row refers to an output variable, and each column to a thickness ratio. M e /M indicates the number of active terms over the total LD4 variables. For instance, the reduced LW plate model providing u z as accurately as LD4 for a thin plate has 11 unknown variables and the following displacement field:
$$\displaystyle \begin{aligned} \begin{array}{l} u_{x}=F_{t}\,u_{xt}^{1}+F_{b}\,u_{xb}^{1}+F_{b}\,u_{xb}^{2} \\ u_{y}=F_{t}\,u_{yt}^{1}+F_{b}\,u_{yb}^{1}+F_{b}\,u_{yb}^2 \\ u_{z}=F_{t}\,u_{zt}^{1}{+}F_{2}\,u_{z2}^{1}{+}F_{b}\,u_{zb}^{1}\,{+}\,F_{2}\,u_{z2}^{2}\,{+}\,F_{b}\,u_{zb}^{2} \\ \end{array} \end{aligned} $$
(20)
The last row of the table presents the reduced models required to obtain all the considered output variables as accurately as the LD4. The results suggest that:

Best Theory Diagrams

As mentioned in previous sections, the BTD is the Pareto front of an optimization problem involving all the expansion terms. The first case deals with the shell geometry, boundary conditions, and material of Varadan and Bhaskar (1991). The BTD considers the ED4 terms and LD4 is the reference solution. The shell has two layers, 0/90, and σ αα is the output parameter, in which α is a curvilinear coordinate. Figure 10 shows the BTD in the case of thick shell, R β /h = 4, in which R β is the radius of curvature with respect to the curvilinear coordinate β. The same plot shows the FSDT and Pandya and Kant (1988) models together with the BTD model with nine terms. The results suggest that:
  • 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.
    Fig. 10

    BTD for a thick shell, 0/90

  • At least eight unknown variables are necessary to have errors lower than 5%.

The second case deals with plate models with non-polynomial terms in the expansion (Filippi et al, 2016). The complete starting expansion for u x is the following: The other two components have the same starting expansion; therefore, the initial variables are 54. The sixth term is the zigzag term as in Murakami (1986). This case considers a square simply supported plate with a bi-sinusoidal load and three layers, 0/90/0. The plate is thick, a/h = 5. All the three displacement and six stress components are the output variables, and LD4 is the reference. Figure 11 shows the BTD with the 21 and 15 term models shown. For example, the BTD with 15 unknown variables is the following:
$$\displaystyle \begin{aligned}u_{x}&= u_{x_{1}}+z\,u_{x_{2}}+z^{2}\,u_{x_{3}}+z^{3}\,u_{x_{4}}\\&\quad +(-1)^k\zeta_k\,u_{x_{6}}+\mathrm{sin}(\frac{\pi z}{h})\,u_{x7}\\ u_{y}&= u_{y_{1}}+z\,u_{y_{2}}+z^{3}\,u_{y_{4}}+(-1)^k\zeta_k\,u_{y_{6}}\\&\quad +\mathrm{sin}(\frac{2\pi z}{h})\,u_{y8}\\ u_{z}&= u_{z_{1}}+z\,u_{z_{2}}+z^{2}\,u_{z_{3}}+z^{4}\,u_{z_{5}}\\\end{aligned} $$
(22)
The results suggest that:
  • Models with less than 50% of the initial variables provide very high accuracy.
    Fig. 11

    BTD for a thick plate, 0/90/0

  • The non-polynomial 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 in-plane 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 use of AAM and BTD leads to the definition of guidelines for the development of structural models. In fact,
  • 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 layer-wise 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.

Cross-References

References

  1. 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
  2. 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
  3. 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
  4. 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
  5. Carrera E, Petrolo M (2011) On the effectiveness of higher-order terms in refined beam theories. J Appl Mech 78. https://doi.org/10.1115/1.4002207
  6. Carrera E, Cinefra M, Petrolo M, Zappino E (2014) Finite element analysis of structures through unified formulation. Wiley, ChichesterCrossRefzbMATHGoogle Scholar
  7. Cicala P (1965) Systematic approximation approach to linear shell theory. Levrotto e Bella, TorinoGoogle Scholar
  8. 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
  9. Gol’denweizer AL (1961) Theory of thin elastic shells. International series of monograph in aeronautics and astronautics. Pergamon Press, New YorkGoogle Scholar
  10. Kirchhoff G (1850) Uber das gleichgewicht und die bewegung einer elastischen scheibe. Journal fur reins und angewandte Mathematik 40:51–88CrossRefGoogle Scholar
  11. Mindlin RD (1951) Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. J Appl Mech 18:1031–1036zbMATHGoogle Scholar
  12. Murakami H (1986) Laminated composite plate theory with improved in-plane response. J Appl Mech 53:661–666CrossRefzbMATHGoogle Scholar
  13. Pandya B, Kant T (1988) Finite element analysis of laminated compiste plates using high-order displacement model. Compos Sci Technol 32:137–155CrossRefGoogle Scholar
  14. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12:69–76MathSciNetzbMATHGoogle Scholar
  15. 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
  16. Washizu K (1968) Variational methods in elasticity and plasticity. Pergamon, OxfordzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.MUL2 Group, Department of Mechanical and Aerospace EngineeringPolitecnico di TorinoTorinoItaly

Section editors and affiliations

  • Erasmo Carrera
    • 1
  1. 1.Mechanical and Aerospace EngineeringPolitecnico di TorinoTorinoItaly