## No 2 (2016)

**Year:**2016**Articles:**11**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/21**DOI:**https://doi.org/10.15593/perm.mech/2016.2

Manufacturing defects in composite flanges: modelling and analysis of influence on static strength

#### Abstract

This work presents an approximate methodology based on the numerical simulation aimed at assessing how defects affect static strength of polymer composite flanges in load bearing elements used in aeronautical engineering. Various defects may occur in the process of flange forming in the areas of bending, e.g. curvature of the layers, resin pockets, voids, delamination and others. Thus, there is an actual problem of assessing how technological defects affect the strength of this part. The main defects in composite laminates are defined and a review of literature related to strength problems for such structures are presented in this work. The numerical stress-strain analysis of composite flange with main types of defects under force loading condition was carried out with ANSYS software. A two - dimensional axisymmetric finite element model was used. For the development of the structural model a parametric modelling approach was applied, including defect size, configuration and location parameters. The problem was solved in a general statement for an anisotropic elastic body. In order to keep the original part profile, we reduced the overall thickness of the layers in the local area near the defect by the size of the defect, so the overstated value of structural strength was obtained in simulation. The safety factor of the flange was estimated by stress components using the maximum stress criterion. The stresses in the material principal directions and interlaminar shear and normal stresses were determined for analysis. A comparison of the obtained data with the results for a defect-free flange allows estimating the impact of each defect or its combination on the static strength of flange.

**PNRPU Mechanics Bulletin**. 2016;(2):5-21

Special features of numerical simulation for elastic-plastic buckling of hemispherical shells under loading with the rigid indenter

#### Abstract

The three-dimensional problem of elastic-plastic deformation and buckling of the hemispherical shells under contact with the rigid bodies is considered. Current Lagrangian formulation is applied to describethe shell deformation. The equation of motion is derived from thebalance of possible work powers. The ratios of the flow theory with isotropic hardening are used as the equations of state. Contact interaction of a rigid body and shell is modeled based on non-penetration conditions. We use a 8-node isoparametric finite element with multilinear form functions for a digitization of the constitutive system of equations. The solution of the problem under given boundary and initial conditions is based on the moment scheme of finite element method and an explicit cross type finite-difference scheme of time integration. Calculations of elastic-plastic deformation and buckling of hemispherical steel shell, located on the stationary plate at quasi-static indentation by nondeformable indenter, are done. Indenter has a form of a cylindrical shell with longitudinal notches in the contact area. Number of notches is varied in the calculations. A numerical study of the influence of the indenter form on the level of plastic deformation and the value of the critical load is carried out. As shown by the results of calculations, significant local deformations of hemispherical shell in the contact area with the indenter in the process of loading are formed. It is characterized by large displacements and rotation angles of finite element as a rigid body. The reliability of the results of numerical solution of the problem is confirmed by a good agreement with the experimental data of other authors. It is shown that the application of moment scheme of finite element method to determine the rate of strain and stress in the local basis is justified at low shear deformations and large angles of rotation torque by using explicit cross type finite-difference scheme of time integration with small time steps.

**PNRPU Mechanics Bulletin**. 2016;(2):22-33

Separation process modeling of composite with adhesive layer

#### Abstract

The deformation model of a composite with a thin adhesive layer is examined. The consideration of a layer’s stress state is based on the relationship between the average stresses by thickness and the stresses on the layer’s border. The layer’s medium strains are expressed in terms of its boundary displacements. The average stresses and strains are used to avoid the stress-strain state dependence on the shape of end faces. The variational condition for the equilibrium state of two bodies linked through an adhesive layer is obtained within small strains. The problem is considered in the framework of linear theory of elasticity. The Hooke’s law relates the strain and stress fields in the matched bodies. As a result, the system of variational equations is reduced to the equations with respect to the displacements fields in the matched bodies including the layer’s bounds. The system of variational equations with respect to displacements contains the adhesive layer thickness as a parameter. It is significant that the current equations system is not a discrete one since the displacement fields are supposed to be continuous. Various approximations for displacements may be used to obtain approximate solutions. In particular, the finite element method with a quadratic approximation for displacement fields is used for the case of plane strain. The influence of the characteristic size of a finite element on the convergence of the solution is studied. It is found that the numerical convergence is present when the ratio between the finite element faces and the layer’s thickness is four or more. The proposed approach allows to use the well-known local failure criteria under the absence of stress singularity at the points of conjugation of the adhesive layer with the bodies. The analysis of the possible forms of composite destruction due to the destruction of a material layer as well as due to bonds breaking between the layer and adjacent materials is carried out.

**PNRPU Mechanics Bulletin**. 2016;(2):34-44

NON-AXISYMMETRIC EDGE BUCKLING OF CIRCULAR PLATES WHEN HEATED

#### Abstract

The paper presents the results of simple experiments on uniform heating of thin polymeric circular plates which edges are fixed on rigid rings; and the formation of non-axisymmetric buckling with different numbers of waves around the circumference localized near the plates’ edges. At the same time, buckling modes with a large number of waves around the circumference are observed for plates with a smaller thickness. The classical model of stability for a circular plate under radial compressive forces distributed on a plate contour is used for non-axisymmetric buckling of this kind. The model also describes the temperature swelling of the plate. It became possible to get the interrelation between the minimum critical load and number of waves around the circumference in the plate buckling mode. We compared a multiwave form of buckling for the plate with the modes of eigenoscillations similar to it, which are localized at the edge of the plate. The comparison is carried out based on the transitional line position which mathematically separates the area with an active oscillation from the plateau with an almost nondeformed central region of the plate (which is displaced as a rigid body); and also based on the location of the buckling mode extrema and oscillations modes with respect to the plate center. Using the same parameters, we compared non-axisymmetric plate buckling modes (observed in the experiment) with calculation results related to the theoretical model for rigid, hinged and elastic clamping of the plate edge. In the latter case, we show that it is possible to determine the support rigidity based on a good fit of theoretical and experimental values related to the transitional line radius and a circle radius where the plate buckling extrema are located. The given data illustrates a tendency of a shift of buckling extrema to the plate contour when the boundary conditions are weakened.

**PNRPU Mechanics Bulletin**. 2016;(2):45-53

Rationale of the use of the constructive anisotropy method in the calculation of shallow shells of double curvature, weakened holes

#### Abstract

The paper presents the geometrically nonlinear mathematical model of deformation related to isotropic shallow shells with double curvature weakened by holes. The model is based on the hypotheses of Kirchhoff-Love theory of shells which is presented in the form of geometric and physical relationships and the functional of the total potential energy of deformation. Expressions for the forces and moments are also provided. We consider two methods of introducing holes: discrete and constructive anisotropy methods, which allow the most accurate "smudge" zero stiffness of holes through the shell field. Ritz method is used to minimize the functional; it reduces the problem to solving a system of nonlinear algebraic equations, which is solved by Newton's method. The algorithm is implemented in the medium 2015 Maple analytical calculations. The stability analysis of shallow shell structure with double curvature, made of steel, is performed when exposed to an external uniformly distributed lateral load using hinged-fixed method to place the shell contour. Calculations are carried out in for different numbers of holes, with a fixed ratio of the total area of holes to the shell area. Thus, on increasing the number of holes, their sizes decreased. Hole distribution along the shell is made in two different ways. The critical loads of buckling are given for all designs. We compare the values obtained with the discrete introduction of holes and those of the method of constructive anisotropy. For several variants of designs, the "load - deflection" graphs are shown. For one embodiment of the shell weakened by a large number of holes, we provide a deflection field before and after buckling; the intensity of the stress is also highlighted with colors. Based on the data, it is shown that the increase in the number of holes results in the lost of discreteness of their input; hence, it becomes possible to use a specially developed method of constructive anisotropy which justifies the use of this method in calculating the stability of shallow shells weakened by a large number of holes.

**PNRPU Mechanics Bulletin**. 2016;(2):54-68

The method for analyzing mechanics of thin elastic rod systems using the differential approach

#### Abstract

This paper proposes a method for analyzing geometrically nonlinear dynamic deformation of rod system (several interconnected rods) using differential model of a thin elastic rod. The deformation of each rod is defined by 12 differential equations with boundary conditions. Boundary conditions for each rod are formed by kinematics, loads and interconnections between rods. Appropriate nonlinear boundary value problem is solved numerically. It needs many times to solve a large system of linear equations with matrix formed by a Jacobi matrix of differential equations, boundary conditions and conditions of interconnections between the rods. On the main diagonal of this matrix the blocks are formed by differential equations and boundary conditions for each rod. These blocks are linked by elements formed by the equations of connection between the rods. The paper proposes the method for solving sparse linear systems with blocks on the main diagonal, while these blocks are related with each other by a small number of equations. The solution of original sparse system is splitted to few solutions of smaller systems (for each block on the main diagonal) and to the solution of SLAE size of equal to number of nonzero lines linking the matrix blocks. Therefore, the proposed method is effective with a comparatively small (compared with size of the original matrix) count of nonzero lines linked with isolated matrix blocks on the main diagonal. The paper presents the solution of the test problems of large displacements loaded frames. The numerical results of the proposed method are compared with the results of the calculation in the finite element program ANSYS. The results of calculations are practically the same; the accuracy of matches depends on models discretization.

**PNRPU Mechanics Bulletin**. 2016;(2):69-88

Transformation and interaction of microcracks in metal under high-energy pulsed electromagnetic field

#### Abstract

Simulation of processes occurring in metals when they are treated with short pulses of high density electric current is of interest primarily due to studying the phenomenon of electroplasticity; the physical mechanism of which is still unknown to researchers. The effect of healing micro-defects in metals is one of existing explanations for this phenomenon. The present paper considers the processes of transformation and interaction related to flat microcracks with linear sizes of about 10 microns when processing metal samples with short pulses of high-density currents. The investigation is carried out numerically on basis of coupled quasi-stationary model of impact using high-energy electromagnetic field on the pre-damaged thermal elastoplastic material with defects. The model accounts for melting and evaporation of the metal and the dependence of its physical and mechanical properties on the temperature [1]. The system of equations is solved numerically by finite elements method with adaptive mesh using alternative Euler-Lagrange’s method. The calculations show that the crack welding and micro-defects healing occur under the short pulse of current. The healing occurs due to a simultaneous reduction in length, the ejection of the molten metal into the cracks and closing of micro-crack shores which leads to the fact that the shores of the crack come into contact with the jet stream; and in the end of these processes the jet’s material is completely jammed by the cracks shores [1]. This paper studies the influence of distance between the cracks and their relative position with respect to each other on deformation and healing of micro-defects; also the choice of the integration regions and conditions at its boundaries. Numerical modeling shows that it is enough to study microcracks healing by considering one representative element (or one-quarter of the axisymmetric representative element) as a region of integration by setting the electrical potential which is certain for the element without defects (when it is "unperturbed" by the presence of microcracks) on its borders that are the axes of symmetry. When the distances between the cracks exceed their lengths by 5-6 times, the healing processes will occur in the same manner regardless of the fact that we model them in the region of integration consisting of one or several representative elements. When the distance between the cracks increases, the influence of mutual arrangement of micro-cracks on the healing process is decreased. Thus, if the distances between the microcracks exceed their lengths by six times, in fact, the healing of microcracks is the same for any position of cracks compared to each other. Interaction between microcracks begins to significantly affect their healing processes when the distance between them is reduced to 5-6 lengths of microcracks. If the distance between the cracks exceeds their six lengths, the processes of microcracks healing become practically independent of the distance between the defects or the position of defects with regard to each other. Decreasing the distance between the cracks up to 1-2 of their linear sizes (taking into account their relative position changes) does not qualitatively change the described healing process of microcracks, however it results in a considerable slowing down: the ejection of a molten material in a crack is retained, but the crack reduction is significantly reduced especially in the transverse direction.

**PNRPU Mechanics Bulletin**. 2016;(2):89-110

INTEGRATED APPROACH TO STUDYING CHARACTERISTICS OF DYNAMIC DEFORMATION ON FLEXIBLE PAVEMENT SURFACE USING NONDESTRUCTIVE TESTING

#### Abstract

A comprehensive approach to studying dynamic deformation characteristics on the surface of non-rigid pavements using nondestructive inspection techniques was proposed. It includes the analysis of the calculated and experimental bowls of maximum dynamic deflection and analysis of the dynamic response characteristics of a non-rigid road design in the frequency domain with a short-term shock loading. The results of the numerical simulation experiment were given related to the effect of adhesion loss between individual elements of non-rigid pavements on the form of amplitude-frequency characteristics (AFC) of acceleration on the pavement surface. It is established that there is only one frequency extreme point present during a complete interlaminar adhesion between all elements of the non-rigid road structure in the amplitude-frequency characteristic of acceleration frequency. With the loss of interlayer adhesion between individual elements of the non-rigid road construction, some local frequency extremes occur in the frequency response. The adequacy of numerical simulation results is based on a good match between the forms of acceleration amplitude-frequency characteristics which were calculated using an analytical model of dynamic stress-strain state and amplitude-frequency characteristics registered on the surface of non-rigid pavement coatings in natural conditions. The paper considers the issues of constructing bowls with a maximum dynamic deflection on the surface of a non-rigid pavement under the influence of a short-term shock loading. There are examples of applying the integrated approach to address the evaluation elasticity modules of structural elements in non-rigid pavements that are arranged on highways sections. This approach might be further developed to improve methods and tools for analyzing road structure behavior in the frequency domain under dynamic loading.

**PNRPU Mechanics Bulletin**. 2016;(2):111-130

On delamination of a strip along the boundary between two elastic layers. Part 2, Сase of shear crack

#### Abstract

The problem of a strip, composed by two isotropic elastic layers of different elastic properties and thicknesses, separated by a semi-infinite crack located along the line between the layers is considered. The mechanical load is supposed to be applied at infinity. In the first part of the study [1] the mathematical formulation of the problem and its reduction to a homogeneous Riemann-Hilbert problem by application Laplace transform was presented. Under the assumption of possibility to neglect the cross-terms related to the influence of the normal stresses to the shear displacements and the shear stresses to the normal displacements, the problem is reduced to two scalar Riemann-Hilbert problems. Such a formulation may be considered as an approximation for the general case (which is not worse than the traditional beam or rode approximation) and as the exact one for the case, where the two layers may slide but may not separate due to cohesion. By means of factorization procedure the exact analytical solution has been obtained for one of the formulated in [1] scalar problems, namely, the problem of a shear crack. The asymptotical expression has been derived for the relative displacements of the crack faces far from the crack tip. It is shown, that the leading asymptotic terms of these relative displacements correspond to a rode under the boundary condition of the type of elastic clamping. i.e. the proportionality of the displacement of the clamping point to the applied force. The analytical expression for this coefficient has been obtained under the accepted assumptions. The asymptotical expression for the stress field near the crack tip (stress intensity factor and energy release rate) is also derived.

**PNRPU Mechanics Bulletin**. 2016;(2):131-142

Theoretical-experimental study of mechanical behavior in 3D composites under quasi-steady damage

#### Abstract

Composite materials are used to manufacture all modern aircrafts and aircraft engines. The development of these materials is labour-intensive and expensive, as it is necessary to carry out extensive experimental studies of anisotropic material properties. There are also certain constraints which make it impossible to obtain some material properties via experiments. There are various numerical techniques to reduce development time and costs for such materials, e.g. construction of a virtual prototype, a series of numerical experiments and use of different varieties of continuous damage mechanics method (CDM). Continuous damage mechanics method (CDM) is a popular tool for modelling damage development in reinforced composites. It is based on the assumption that any given volume of a damaged material can be equivalently changed by the undamaged volume of material with a decreased elastic modulus. Thus, material damage is only reflected by lowered elastic properties; and damage growth is reflected by a degree of material’s elastic modulus reduction. APDL macros was developed for the application of a variant of this method in finite elements analysis problem. The method of building an idealized virtual prototype of a 3D-reinforced composite material using tomography is considered in the paper. The method of determining effective elastic properties of a 3D-reinforced composite material by a series of numerical experiments is presented here. Continuous damage mechanics is used for obtaining materials’ strength parameters. Acoustic emission diagrams and stress-strain relations for material specimens in uniaxial tension and shear stress states are shown. Numerical modelling results are compared with the experimental data obtained.

**PNRPU Mechanics Bulletin**. 2016;(2):143-158

O geometricheski nelineynykh opredelyayushchikh sootnosheniyakh uprugogo materiala

#### Abstract

**PNRPU Mechanics Bulletin**. 2016;(2):159-160