# BRJ – Volume 60, Number 3–4, 2012 (with fulltext)

**Cover
Editor’s Note** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 – 110

**(fulltext)**

Vladimir Sladek, Jan Sladek and Ladislav Sator:

**Mesh-free formulations for solution of bending problems for thin elastic plates with variable bending stiffness.** **Part I: mathematical formulation** . . . . . . . . . . . . . . . . . . . . . . . . . 111 – 148 **(fulltext)**

*
*Ladislav Sator, Vladimir Sladek and Jan Sladek:

**Mesh-free formulations for solution of bending problems for thin elastic plates with variable bending stiffness.**

**Part II: numerical solutions**. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 – 170

**(fulltext)**

*Slavomír Krahulec, Jan Sladek, Vladimir Sladek and Michael Wünsche:*

**Analysis of interface crack between elastic and piezoelectric solids**

by meshless method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 – 184

by meshless method

**(fulltext)**

Peter Stanak, Antonio Tadeu, Jan Sladek and Vladimir Sladek:

**Meshless MLPG approach for analysis of piezoelectric sensors
under compressive load** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 – 198

**(fulltext)**

Alexander Tesar:

**Failure process of carbon fiber composites** . . . . . . . . . . . . . . . . . . . . . . . . . 199 – 210 **(fulltext)
**

Martin Križma, Jaromír Petržala and Marek Kovačovic:

**Resistance between concrete surfaces of composite members** . . . . . . . . . . . 211 – 222 **(fulltext)
**

Martin Palou, Vladimír Živica, Ľubomír Bágeľ and Tomáš Ifka:

**Influence of hydrothermal curing on g-oilwell cement properties** . . . . . . . . . 223 – 230 **(fulltext)
**

Vladimír Živica, Martin Palou and Tomáš Ifka:

**High strength metakaolin based geopolymer** . . . . . . . . . . . . . . . . . . . . . . . 231 – 246 **(fulltext)
**

##
**Abstracts**

**Mesh-free formulations for solution of bending problems for thin elastic plates with variable bending stiffness. Part I: mathematical formulation**

*Vladimir Sladek, Jan Sladek and Ladislav Sator*

The first part of the paper is devoted to a mathematical formulation of the plate bending problems within the classical Kirchhoff-Love theory for bending of thin elastic plates. The functional gradations of material coefficients due to the in-plane and transversal continuous variation of the Young modulus are considered separately. The plate thickness is allowed to be variable with respect to the in-plane coordinates. The effects associated with the transversal gradation are discussed. The decomposed formulation is proposed in order to decrease the order of the derivatives in the governing equations and the boundary densities playing role in boundary conditions. In order to get the exact solutions usable as benchmark solutions in numerical tests, we developed also the formulation for circular plates with central circular hole. Finally, the exact solutions are derived for the angular symmetric plate bending problems with variable bending stiffness.

**Keywords:** Kirchhoff-Love theory, variable thickness, functionally graded materials, governing equations, exact solutions, decomposed formulation, plane stress state, coupling effects

**Mesh-free formulations for solution of bending problems for thin elastic plates with variable bending stiffness. Part II: numerical solutions**

*Ladislav Sator, Vladimir Sladek and Jan Sladek*

The second part of the paper is devoted to the development of mesh-free computational methods and numerical solutions of boundary value problems for thin plate bending problems treated in Part I within the Kirchhoff-Love theory. The in-plane gradation of the Young modulus and/or the variability of the plate thickness lead to governing equations which are the partial differential equations with variable coefficients. On the other hand, the transversal gradation of the Young modulus gives rise to coupling between the deflections and in-plane deformations. To facilitate the numerical solution of the boundary value problems for such rather complex governing equations, we propose the strong formulation combined with meshless approximations for field variables. Two alternative meshless approximation schemes are employed: the Moving Least Square (MLS) approximation and the Point Interpolation Method (PIM). The modified differentiation is proposed to increase the accuracy of higher order derivatives of field variables. Several numerical examples are presented to investigate the accuracy, convergence of accuracy and computational efficiency of studied mesh-free formulations for various boundary value problems for circular plate with central hole and variable bending stiffness.

**Keywords**: Kirchhoff-Love theory, variable thickness, functionally graded materials, decomposed formulation, strong formulation, meshless approximations

**Analysis of interface crack between elastic and piezoelectric solids by meshless method**

*Slavomír Krahulec, Jan Sladek, Vladimir Sladek and Michael Wünsche*

The paper discusses a crack analysis with the central crack on the interface between elastic and piezoelectric solids under static or dynamic mechanical loading. Due to the specific boundary conditions, the original 3-D crack problem is reduced into a 2-D problem with plane deformation. The crack opening displacements, electric potential and intensity factors are investigated. The intensity factors are computed from the generalized crack-opening-displacements. For the solution of general boundary value problem, the meshless local Petrov-Galerkin (MLPG) method is applied. Here, the local integral equations are derived from the weak form of the governing partial differential equations. Nodal points are uniformly spread on the analyzed domain and each of them is surrounded by a small local circular subdomain. The Heaviside function is used as a test function. On the interface, two sets of nodes are introduced. For the approximation of unknown variables, the Moving Least-Squares (MLS) approximation scheme is used. In the time dependent analysis, the Houbolt finite difference scheme is used. Various electromagnetic boundary conditions are applied on the crack-faces. Two extreme cases are analysed; fully permeable and fully impermeable crack surfaces.

**Keywords**: the meshless local Petrov-Galerkin (MLPG) method, the Moving Least-Squares (MLS) approximation, the Houbolt finite difference scheme, intensity factors, crack opening displacement

**Meshless MLPG approach for analysis of piezoelectric sensors under compressive load**

*Peter Stanak, Antonio Tadeu, Jan Sladek and Vladimir Sladek*

The paper presents a numerical analysis of a three-dimensional piezoelectric solid using the meshless local Petrov-Galerkin (MLPG) method. The mechanical and electrical response of a piezoelectric sensor is investigated for a static and transient compressive load. In the proposed meshless formulation nodal points are uniformly distributed in the analyzed domain, but without loss of generality. Each node is the center of a small spherical subdomain. The local integral equation (LIE) formulation is derived from the equations of motion for the mechanical fields and quasi-static Maxwell’s equations on the appropriate subdomains. The unit step function is used as a test function in derivation of the LIEs. The moving least-squares (MLS) method is used to approximate the spatial variations of all physical fields. The Houbolt finite difference scheme is used to solve a time-dependent system of ordinary differential equations for mechanical displacements and electric potential.

Keywords: piezoelectric sensor, meshless local Petrov-Galerkin method, moving least-squares approximation, Houbolt method

**Failure process of carbon fiber composites**

*Alexander Tesar*

Some research results of failure behaviour of carbon fiber composites are presented. The solution of material instability on the basis of fiber kinking theory is adopted for the treatment of the failure process. The micromechanical modeling adopting the FETM-approach is used for a numerical analysis of the problem. Some numerical and experimental results with an actual application are submitted in order to demonstrate the efficiency of the approaches suggested.

**Keywords**: FETM-approach, fiber kinking, carbon fiber composite, failure process, material instability, transformaton strain, ultimate behaviour

**Resistance between concrete surfaces of composite members**

*Martin Križma, Jaromír Petržala and Marek Kovačovic*

The use of prefabricated structural elements and monolithic constructions made in different times has been ordinary for long decades mainly in the bridge construction. A bearing construction formed by a system of prefabricated pre-stressed beams and an additional monolithic slab constitutes a composite heterogeneous element with various features and with discontinuity at a contact area of the joined members. The mentioned issue relates to the new structures. In the case of already used objects, we meet this topic at damaged and consequently strengthened elements of structures. A loading history change of structural engineering objects and also an enhancement of traffic volume on bridge structures are frequent and relevant reasons of bearing structures strengthening. It concerns mainly linear concrete bearing elements. The main objective of such a formed construction is its monolithic behaviour and an enhancement of its resistance, while a shear resistance in coupling should exceed its flexural resistance and its transversal shear resistance too. The aim of the research program was an experimental and theoretic analysis of shear resistance in coupling on experimental damaged beams, which are needed to be repaired. The introductory part of the paper deals with the basis of coupling concrete/concrete and with factors determining its resistance. In the consequent experimental part, there are evaluated two-stage tests of beams. The first stage simulated the process of damage; the second stage simulated the process of strengthening. Within the both stages, the characteristics determining the both limit states were recorded. In the second stage, the accent was put on shear factors in coupling. The experimental results are compared with the results obtained by numerical simulations on the FEM basis. In the whole topic range, an attention was paid to an influence of a contact gap modification on shear resistance in coupling. After the theoretic-experimental analysis, we can state that the normative relations for the resistance calculations are markedly on the safety side.

**Keywords:** linear elements of concrete structures, coupling, strengthening, identification of failure processes on contact gap, numerical modeling

**Influence of hydrothermal curing on G-Oil Well cement properties**

*Martin Palou, Vladimír Živica, Ľubomír Bágeľ and Tomáš Ifka*

G-Oil Well well cement has been cured under standard and hydrothermal conditions with different steam pressures and temperatures. Compressive strength, pore structure parameters, microstructure and hydrated products were evaluated after 7 days` curing by using SEM, MIP and Simulated TG A/ DSC. Obtained results showed that the 7 days aged sample cured under standard conditions has the highest compressive strength with compact pore structure and hydrated products similar to those found after hydration of Ordinary Portland cement. With increasing the temperature and pressure from standard conditions (25 °C, 10 125 Pa) to hydrothermal ones (150 °C and 0.3 MPa, 200 °C and 1.2 MPa), compressive strength has drastically decreased from 77.5 ± 2.0 MPa to 20.5 ± 1.0 MPa due to the transformation of original hydrated products (C-S-H) to crystallized α-C2SH and C6S2H3. The crystallization has led, under hydrothermal curing, to the increase of permeability and pore structure depletion. The final compressive strength, after curing for 7 days at 150 °C (51.8 ± 2.0MPa) and 200 °C (20.5 ± 1.0 MPa) which significantly exceeds the recommended values of 3.45 MPa according to API to hold many casings of oil wells is questionable for application in geothermal ones.

**High strength metakaolin based geopolymer**

*Vladimír Živica, Martin Palou and Tomáš Ifka*

The unique properties of aluminosilicates have made them valuable in the wide range of industrial applications. One branch of application is represented by building materials. A significant factor of the level of the quality of engineering properties of building materials anyhow is the water/binder ratio. It is well known that its decreasing value effectively increases strength and quality of other engineering properties of the material. Due to the accompanying effect impairing workability of the mixture for processing the pressure compaction is needed. The subject of the paper are the results of the study of the properties of metakaolin based geopolymer prepared under the use of the combination of very low water/metakaolin ratio (0.08), pressure compaction of the fresh mixture by applying an uniaxial compressive stress of 300 MPa and alkali activation. The effect of preparation conditions was systematically studied by a thermal analysis (DT A, GTA), mercury intrusion porosimetry, coupling of scanning microscopy and EDX, and compressive strength estimation of the geopolymer. The used metakaolin was a product of calcination of the source material at 650 °C for 4 hours. It was an amorphous material showing an increased thermodynamic instability and herewith an increased reactivity in comparison with the unheated solid. The pressure compacted geopolymer reached after 24 hrs of the hardening the compressive strength of 146.6 MPa whereas the reference paste only 0.03 MPa. It represents 2540 times increase in behalf of the pressure compacted paste. High compressive strength was evidently the consequence of the found high homogeneous and dense pore structure of the pressure compacted paste.

**Keywords**: geopolymer, microstructure, metakaolin, pore structure, compressive strength