Mechanics Of Deformable Solids Research Articles

Influence of Initial Stresses on the Fracture of Composite Material Weakened by a Subsurface Mode III Crack

By using the relations of linearized mechanics of deformable solids, we study the space problem of fracture of prestressed semibounded composites weakened by subsurface disk-shaped torsion (mode-III) cracks. With the help of representations of the general solutions of linearized equilibrium equations in terms of harmonic potential functions and Hankel integral transforms, we reduce the problem to a system of dual integral equations and then to the resolving Fredholm integral equation of the second kind. From the analysis of the stress distribution in the vicinity of the crack, we obtain the values of the stress intensity factors and study their dependences on the initial stresses, mechanical characteristics of the components of the composite, and geometric parameters of the problem.

Improved Finite Element Approach for Modeling Three-Dimensional Linear-Elastic Bodies

Objectives: This paper proposes a general formula for computation of the stiffness matrix of three-dimensional linearelastic bodies by the method of Moment Schema of Finite Elements (MSFE). Methods: Numerical methods based on Lagrange variational principle, such as Finite Element Method (FEM) and the method of Moment Schema of Finite Elements (MSFE). The paper focuses on the problem of obtaining the stiffness matrices of Finite Elements (FE), which ensure effectiveness of the developed version of FEM. Variational principle for calculation of energy functional of rectangular three-dimensional finite element is used. Results: Analyses of various problems of mechanics of deformable solids shows slow convergence of traditional FEM, especially for rigid bodies and surfaces of complex curved shapes. This paper proposes a scheme for inference for FEM ratios, which takes into account the basic properties of rigid displacements for isoparametric and curvilinear FE. The presented version of FEM allows us to obtain the stiffness matrix of FE, taking into account the effect of a false shift and low compressibility of elastomers. Conclusion/Application: The general method to find the potential energy of three-dimensional linear-elastic bodies is proposed, which allows us to solve the problems of mechanics of complex structures with higher accuracy.

ТЕХНОЛОГИИ И ЗАДАЧИ МЕХАНИКИ КОМПОЗИЦИОННЫХ МАТЕРИАЛОВ ДЛЯ СОЗДАНИЯ ЛОПАТКИ СПРЯМЛЯЮЩЕГО АППАРАТА АВИАЦИОННОГО ДВИГАТЕЛЯ

The goal of this work is an analysis of modern technologies and problem definitions of mechanics of composite materials for production of outlet guide vane for new domestic aviation propulsion PD-14, planned for installation on the short-range and mid-range jet aircraft MS-21. Outlet guide vanes (OGV), also called flow straightening vanes, are radially disposed behind the fan to straighten out the airflow to reduce losses in the outer contour of the engine. Considering a large number of vanes in the engine, the weight reduction can be very significant due to using polymer composite materials (PCM) instead of the metal in OGV. It is predicted that the weight reduction of each vane, at certain approach to design, can reach 40 %. The results of literature search of composite materials application in the details of fan for aircraft jet engines of world top manufacturers are presented in this paper. The analysis of advanced techniques for production of composite outlet guide vane, such as prepreg technology with autoclave molding, resin transfer molding (RTM), the use of thermoplastic binders and press-materials was carried out. It was noted that the formulation and solution of complex problems of engineering mechanics of composite materials are necessary for high-rate realization of mechanical properties in constructions and reducing the occurrence of defects. The possible problem definitions of deformable solid body mechanics, describing such processes as filtration, physical and chemical conversion, visco-elastic-plastic deformation in heterogeneous medium, typical for composite materials production cycle were considered. The influence of autoclave molding parameters on the mechanical properties of several types of carbon fiber reinforced plastics (CFRP) was analyzed. Using the equations of mechanics of laminated composite plates and shells we calculated effective elastic and strength properties of quasi-isotropic CFRP. A comparative estimate of the effectiveness of composite materials application in the design of OGV was obtained.

Fracture model for structured quasibrittle materials

We analyze the applicability of a modified Leonov-Panasyuk-Dugdale model to the description of the propagation of a mode I crack in structured materials under plane stress conditions. For quasi-brittle materials, refined formulas of the critical length of the prefracture zone and the critical load containing a structural parameter are proposed. The Kornev model is extended to the case of quasi-ductile materials. Numerical simulation of plastic zones in square plates of a bimetal and a homogeneous material under quasi-static loading is performed. In the numerical model, the equations of deformable solid mechanics are expressed in the Lagrangian formulation, which is the most preferred for large-strain deformations of elastoplastic materials. The results of the numerical experiments are consistent with the results of calculations using the analytical model for the fracture of structured materials.

Influence of periodic change in the yield strength in a plate on the development of plastic zones near a crack tip

An experimental study of plastic deformation zones in the vicinity of stress concentrators found a deviation of the shapes of plastic zones at the crack tip from the shapes obtained using conventional models. Moreover, the systems of bands of slip lines observed in the experiment and predicted by linear fracture mechanics are significantly different. Mathematical modeling of the propagation of plastic zones based on a finite element numerical solution of the equations of deformable solid mechanics was performed with the framework of the theory of large elastoplastic deformations. The basis of the mathematical model is the hypothesis that the structural heterogeneity of the material has a substantial effect throughout the volume of the test sample on the formation of plastic bands. Nonuniformity of the yield strength of the material is specified as a checkerboard distribution and as a system of horizontal and vertical bands. The results of the numerical calculations are compared with the experimental data, and it is shown that they are in qualitative agreement.

On projectile fragmentation at high-velocity perforation of a thin bumper

By means of 3D numerical simulations, we study the statistical properties of the fragments cloud formed during high-velocity impact of a spherical projectile on a mesh bumper. We present a quantitative description of the projectile fragmentation, and study the nature of the transition from the damage to the fragmentation of the projectile when the impact velocity varies. A distinctive feature of the present work is that the calculations are carried out by smoothed particle hydrodynamics (SPH) method applied to the equations of mechanics of deformable solids (MDS). We describe the materials behavior by the Mie–Grüneisen equation of state and the Johnson–Cook model for the yield strength. The maximum principal stress spall model is used as the fracture model. It is shown that the simulation results of fragmentation based on the MDS equations by the SPH method are qualitatively consistent with the results obtained earlier on the basis of the molecular dynamics and discrete element models. It is found that the power-law distribution exponent does not depend on energy imparted to the projectile during the high-velocity impact. At the same time, our calculations show that the critical impact velocity, the power-law exponent and other critical exponents depend on the fracture criterion.

Method of molecular dynamics in mechanics of deformable solids

The basic principles of the method of molecular dynamics are analyzed. Symplectic difference schemes for the numerical solution of molecular dynamics equations are considered. Stability is studied, and the errors in the energy conservation law, which are induced by using these schemes, are estimated. Equations of mechanics of continuous media are derived by means of averaging over the volume of an atomic system. Expressions for the stress tensor are obtained by using the virial principle and the method of averaging over the volume. The principles of construction of EAM and MEAM potentials of atomic interaction in crystals are analyzed. Two problems of fracture of copper-molybdenum composites are solved by the method of molecular dynamics.

ボクセル有限要素法の代替手法としてのオイラー型有限差分法による大変形固体力学解析

ボクセル有限要素法の代替手法としてのオイラー型有限差分法による大変形固体力学解析

813 生体力学への応用を目指した有限差分法による大変形固体力学解析手法に関する研究(OS01-3 材料力学と計算力学の新展開,オーガナイズドセッション1:「材料力学と計算力学の新展開」)

813 生体力学への応用を目指した有限差分法による大変形固体力学解析手法に関する研究(OS01-3 材料力学と計算力学の新展開,オーガナイズドセッション1:「材料力学と計算力学の新展開」)

Effective constitutive relations for inelastic composites

The first special boundary value problem in the mechanics of deformable solids is considered to derive the effective constitutive relations for a heterogeneous inelastic body. The problem is reduced to a number of auxiliary boundary value problems for functions dependent on the shape of the body and on the form of constitutive relations. In the case of a layer of nonuniform thickness, the problem of finding the effective constitutive relations is reduced to an operator equation whose solution is sought by an iterative method of successive approximations. An approximate analytical formula is proposed to find the effective constitutive relations for a laminated composite on the basis of known inelastic constitutive relations for its components. This approximate formula takes into account the character of structural anisotropy in a laminated composite and, in the elastic case, yields the exact values of the effective elastic modulus.

Thermo-mechanical processes

The theory of thermo-mechanical processes, as an apparatus in the development of equations of state and thermodynamics of continuous media, includes all the main sections of the fundamental mechanics of deformable solids. The paper considers two aspects of the overall problem: 1) non-monotonic processes in visco-plastic flows; 2) panel flutter of viscoelastic plates. In the first case there is released a fundamentally new problem - the definition of the functional of contact friction and the construction of the theory of plasticity under complex loading of bodies with non-uniform initial stress-strain state. The solution is supposed to look by using the SN-EVM (CH-ЭВМ) method together with forming a data bank. In the second case there is risen the problem of definitive explaination of the well-known paradox of the definition of the critical parameters in the problem of flutter of viscoelastic plate. The article also addresses the issues of scientific ethics and morality; driving force behind this were the publications appeared in which subjective and biased historical facts relating to the development of the science of thermo-mechanical processes are interpreted subjective and interestedly.

The rift origin of the Vilyui basin (East Siberia), from reconstructions of sedimentation and mechanical mathematical modeling

Results of modeling of the formation of the Vilyui sedimentary basin are presented. We combine backstripping reconstructions of sedimentation and thermal regime during the subsidence with a numerical simulation based on the deformable solid mechanics. Lithological data and stratigraphic sections were used to “strip” the sedimentary beds successively and calculate the depth of the stratigraphic units during the sedimentation. It is the first time that the evolution of sedimentation which is nonuniform over the basin area has been analyzed for the Vilyui basin. The rift origin of the basin is proven. We estimate the spatial distribution of the parameters of crustal and mantle-lithosphere extension as well as expansion due to dike intrusion. According to the reconstructions, the type of subsidence curves for the sedimentary rocks of the basin depends on the tectonic regime of sedimentation in individual basins. The backstripping analysis revealed two stages of extension (sediments 4–5 km thick) and a foreland stage (sediments >2 km thick). With the two-layered lithosphere model, we conclude that the subcrustal layer underwent predominant extension (by a factor of 1.2–2.0 vs. 1.1–1.4 in the crust). The goal of numerical experiments is to demonstrate that deep troughs can form in the continental crust under its finite extension. Unlike the oceanic rifting models, this modeling shows no complete destruction or rupture of the continental crust during the extension. The 2D numerical simulation shows the possibility of considerable basement subsidence near the central axis and explains why mafic dikes are concentrated on the basin periphery.

New invariants in the mechanics of deformable solids

A system of invariants of symmetric three-dimensional tensors of second order is proposed. The system contains three classical invariants of a tensor and three new invariants which depend on the components of two or three tensors. The system proposed allows one to express the strain energy density of a linear elastic anisotropic body and virtual work done by internal stresses in terms of the invariants for any constitutive law of the material. Application of the invariants to the derivation of the tetrahedron finite element of anisotropic solids is discussed.

On the Treatment of Statical Indeterminacy in the Elementary Statics of Particles and Rigid Bodies

The inability to determine reactions beyond a maximum of six for a three-dimensional rigid body (or three for a planar rigid body) is termed statical indeterminacy and represents a flaw in the elementary statics of particles and rigid bodies because the indeterminate reactions cannot be obtained by methods developed from within the subject. As noted in virtually every modern textbook on statics, statical indeterminacy may be resolved by relaxing the assumption of structural rigidity. This theory forms the content of the mechanics of deformable solids, which considers the stretching, twisting and bending of elastic bars. The indeterminacy may also be resolved by maintaining structural rigidity but relaxing the assumption of support rigidity. Because deformable supports may be modeled by lumped linear and torsional springs (which are traditionally part of the statics of particles and rigid bodies), analysis of statically indeterminate reactions in such systems utilizes no new concepts except that of rigid body translation and rotation. The purpose of this paper is to demonstrate, by means of example, that this class of statically indeterminate problem is amenable to simple solution and therefore should be treated in standard presentations of the subject.

Homogenization of nonlinear problems in the mechanics of composites

A further development of the homogenization method is proposed to solve the physically nonlinear equilibrium problems for the laminated plates or the plates made of functionally graded materials. In the linear case, according to this method, the corresponding solution is a superposition of the solution to the global problem in the entire domain and the solution to the local problem in a representative domain, e.g., in a periodicity cell. In the nonlinear case, such a superposition is not valid, which complicates the application of the homogenization method. In order to eliminate this difficulty, it is possible to combine the homogenization method and the linearization method when solving a boundary value problem or a variational problem. In the mechanics of deformable solids, the constitutive relations can be considered as equations with respect to velocities or the stress and strain differentials in time or in the loading parameter. When these equations are linear with respect to velocities, it is possible to use the homogenization method. In this paper such an approach is illustrated by the example of a symmetric laminated plate bent under a uniformly distributed time-dependent load.

Problems concerning the surface quality of continuous-cast slabs

This article examines the main types of surface defects on slabs and their causes. A general explanation is given for the nucleation and growth of surface cracks. An analysis is made of the possibility of applying criteria from the fracture mechanics of deformable solids to the conditions that exist during the continuous casting of steel slabs.

Influence of initial stresses on the stressed state of a composite with a periodic system of parallel coaxial normal tensile cracks

Within the framework of the linearized mechanics of deformable solids, we have investigated the axisymmetric problem of fracture of a prestressed composite material with a periodic system of parallel coaxial normal tensile cracks. Using the representations of the general solutions of linearized equilibrium equations via the harmonic potential functions and the technique of Hankel integral transformations, we have reduced the problem to a system of paired integral equations and then to a resolving second-kind Fredholm integral equation. We have obtained expressions for the stress intensity factors near the crack contours and for the case of a layered composite with isotropic layers. We have also analyzed the dependence of stress intensity factors on the initial stresses, the mechanical characteristics of components of the material, and the geometric parameters of the problem.

Surface topography formation in a region of plate collision: Mathematical modeling

The collision of earth’s crustal plates is modeled mathematically based on a numerical solution of the equations of deformable solid mechanics using a finite element method with the MSC software. The interaction of the plates with each other and with the mantle is described by the solution of the contact problem with an unknown contact boundary between the solids considered. The mantle material is assumed to be ideal elastic-plastic with the Huber-Mises yield surface, and the properties of the plate material are described using an elastic-plastic model with the Drucker-Prager parabolic yield function which takes into account fracture in the tensile stress region. The results of the mathematical modeling show that the surface profiles of the plates in the region of their collision are consistent, both qualitatively and quantitatively, to the surface topography observed in nature under similar conditions.

Fundamental system of solutions of the axially symmetric problem of the theory of elasticity for a body with a plane sheet of volume moment dipoles and forces

UDC 539.3 We have constructed the fundamental system of solutions of the axially symmetric problem of the theory of elasticity for an unbounded body with a sheet of volume forces, normal to a chosen plane, and moment dipoles, which is a mathematical model of the internal boundary layer of a certain type. With the help of such layers, one succeeds in formulating some inverse problems of elasticity and, hence, the related problems of the control of stress-strain state on the corresponding surfaces. We have also formulated and solved the generalized Kelvin problem and, according to it, for the plane of distributed normal load, and established the law of distribution of the moment dipoles (sheet parameters), which provides the vertical displacements assigned for points of the plane, in particular, zero, by the corresponding tension. Based on the definition of zero- and first-order singular material surfaces of discontinuity of the field characteristics [9], we investigate a variant of such surface as a mathematical model of the plane sheet of (distributed according to certain laws, determined by two generating functions) volume moment dipoles and volume, normal to the plane of distribution, forces, which are represented via the Hankel integrals. These distributions correspond to the fundamental system of solutions of the equations of statics of an elastic body in a cylindrical coordinate system, according to which the radial displacement, volume strain, and normal stresses have jumps in passage across the sheet plane [6, 10], and the tangential stress contains a term concentrated only in this plane. According to the definition in [9], we consider such a plane (zero- and first-order material singular surface) as an internal boundary layer, which is a material carrier of certain properties, different from those predicted by the equations of the linear theory of elasticity, and enables one to model certain types of planar defects of the structure of a perfect elastic material. The constructed solutions for the displacement field and all the other field characteristics of stress-strain state in a cylindrical coordinate system are written in terns of the Hankel integrals and contain the generating functions of the distribution densities of volume moment dipoles and volume forces unknown beforehand. Their appropriate choice enables one to obtain the solutions of a certain class of axially symmetric problems of the mechanics of deformable solids. Note that the fundamental system of solutions of the equations of statics with a sheet of volume forces, tangent to the plane, and momentless dipoles in a cylindrical coordinate system was constructed in [3]. 1. Fundamental System of Solutions of the Second-Type Equations of Statics

Variationally consistent force-based finite element method for the geometrically non-linear analysis of Euler–Bernoulli framed structures

The principles of complementary energy in the field of small deformation solid mechanics problems are well known. They only involve stress-like variables and, furthermore, under the subsidiary conditions of equilibrium, they render maximum principles. However, in the geometrically non-linear framework, the existence of pure complementary energy principles has been subject of discussion since the early 1900s. In this work, a pure complementary energy principle, only involving internal forces and bending moments as fundamental variables, is formulated for geometrically non-linear Euler–Bernoulli beams. This principle is derived from the well-known principle of stationary total potential energy by resorting to the Legendre transformation and also to the so-called Lagrangian multiplier method. In addition, a force-based finite element method is derived based on this principle. This method is capable of producing numerical solutions that satisfy in strong form the equilibrium differential equations, as well as the interelement equilibrium and Neumann boundary conditions. To validate and assess the accuracy of the proposed method, some numerical examples are analyzed and their solutions compared with those obtained using a standard displacement-based finite element method. A numerical study of convergence of the proposed method is also carried out.