Direct modeling and simulation of engineering problems with various irregularities are computationally very inefficient and in some cases impossible, even in these days of massively parallel computational systems. As a result, in recent times, a number of schemes have been put forward to tract such problems in a computationally efficient manner. Needless to say, such schemes are still going through evolutionary stages. This paper addresses direct solution based on the selective use of different dimensional models at different regions of the problem domain. For the multidimensional approach, a higher-order transition element is developed to connect the different element types where two- and three-dimensional laminated elements based on higher-order subparametric concept are considered. Modeling simplicity and calculation efficiency of the multidimensional approach are shown for the analysis of cantilever plates with stepped section and patch-repaired plates.

A finite element method is one of versatile numerical tools to solve some differential equations which model physical phenomena of various engineering problems. In last few decades, lower-order finite elements have been conventionally used to solve a wide range of practical problems. However, these elements are unable to provide accurate distributions of stress resultants in structures with free edges and stress singularities. Even though one uses a highly refined mesh to model problems having significant stress gradients, the accuracy of stresses obtained by using lower-order finite elements is rather poor [

Meanwhile, some examples of irregularities in a structure, say, in plates, can appear as step change in thickness, presence of reentrant corner due to skewness, cutouts of various shape, cracks, and notches, free edges of laminates, local damage involving plasticity, strain-hardening, patch repair at damage locations, and so forth. The primary objective of multidimensional approach is to achieve computational efficiency by accurately capturing the local three-dimensional behavior using, as needed, three-dimensional models and to use lower dimensional model away from such locations. This requires special modeling as zone of transition between the two different dimensional models. In the case of composite materials, the concept of weight-averaged apparent constituent properties is used. As needed, orthotropic and transversely isotropic properties are considered. In the case of patch-repaired plates, the repair can be bonded or riveted and the patch material can be same as the parent material, or more conveniently, composite material bonded to the area with damage in the form of, say, holes, cracks, and so forth. Physically the patch repair work is yet another source of irregularity over and existing irregularities caused by the damage. Furthermore, the situation gets more complex due to the potential for interfacial failure in patch repair leading to debonding. Further consideration of the first model will, therefore, be focused with respect to patch-repaired problems. In addition, the scheme will consider layer by layer modeling of laminated composites in the context of both two and three dimensions.

For more efficient analysis in terms of solution accuracy and computational efficiency, techniques using sequential methods or methods based on a combination of different mathematical models have been proposed. For sequential methods [

Over the years, different

In this study, discussion for this multidimensional analysis will be confined to the higher-order approximation based on Lobatto shape functions with hierarchical properties. In many industrial applications the structures with laminated composite materials are composed of three-dimensional solid continuum with the thin shell-like portions connected to them. When the structures are modeled by function refined mesh technique to reduce negative effect from various irregularities, this multidimensional analysis proposed in this work can be considered. Also, this proposed method can be attempted to distribute limited computational resources in an optimal manner to achieve maximum solution accuracy with minimal solution cost, subject to certain problem-specific constraints.

One-dimensional shape functions are classified into two groups as shown in Figure

One-dimensional element on standard domain.

In (

Two-dimensional shape functions can be built from the one-dimensional shape functions defined above. They are divided by three groups as shown in Figure

Two-dimensional

For three-dimensional elements, displacement fields are given by

For two-dimensional modeling, an element is formulated by the dimensional reduction from three-dimensional solid to two-dimensional surface which satisfy the first-order shear deformation plate theory and plane stress theory. Each of in-plane displacement fields,

To carry out connection of two different types of elements, the transition elements are developed. From the aforementioned (

Connection between two- and three-dimensional elements.

For demonstrating the scheme, the simple problem of a 12 m long × 1 m wide stepped cantilever plate shown in Figure

Cantilever plate problem (

For convenience in selecting different trial finite element models, the problem domain is divided into four segments, as shown in Figure

Finite element models.

Model | Segment 1 | Segment 2 | Segment 3 | Segment 4 |
---|---|---|---|---|

A | 3D (3 layers) | 3D (1 layer) | 3D (1 layer) | 3D (1 layer) |

B | 2D | 2D | 2D | 2D |

C | 3D (1 layer) | 3D (1 layer) | Transition | 2D |

Partitioning the problem domain.

The center line variation of axial displacement is found to be identical with Models A and C. However, the results by purely 2D modeling (Model B) show discrepancy, as is evident from the plot of Figure

Variation of axial displacement near step.

Axial displacement variation along

Variation of normal stress,

In this case, the transverse displacement profile for Cases A and C shows perfect agreement, whereas, Case B tends to underestimate it, especially in the thinner part. It is evident from Figure

Axial displacement variation along

Stress fringes for

A uniaxial stretched single-edge-cracked aluminum plate is shown in Figure

Single-edge-cracked plate with one-sided composite patch repair.

Multidimensional mesh for patched plate with edge crack.

Multidimensional analyses with example problems were presented. In the case of multidimensional modeling three-dimensional and two-dimensional

Future works would consider that the application of the proposed method will be investigated on problems composed of other composite materials like functionally graded materials which have a continuous variation of material properties from one surface to another. While present work was implemented to show mechanical response on geometrical irregularities and material irregularities of laminated systems, future works on functionally graded materials will consider mechanical response on high temperature environments including thermal shock.

The author declares that there is no conflict of interests regarding the publication of the paper.

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01060909).