Solid Mechanics Problems Research Articles

Phase-field modeling of brittle fracture using an efficient virtual element scheme

This work addresses an efficient low order virtual element method (VEM) for the phase-field modeling of isotropic brittle fracture. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with constitutive terms rooted in fracture mechanics, see Miehe et al. [23] and Miehe et al. [29]. In the presented contribution, we propose a rigorous variational-based framework for the phase-field modeling of brittle fracture in elastic solids undergoing small strains. The key goal here, is the extension towards the recently developed virtual element formulation due to the flexible choice of nodes number in an element which can be changed easily during the simulation process, as outlined in Wriggers et al. [11] and Wriggers et al. [18]. To this end, the potential density is formulated in terms of suitable polynomial functions, instead of computing the unknown shape functions for complicated element geometries, e.g. arbitrary convex or concave polygonal elements. An important aspect of this work is the introduction of an incremental minimization principle, with a novel construction of the stabilization density for the coupled multi-field problem. On the computational side, a robust and efficient monolithic scheme is employed using the software tool AceFEM program in the numerical implementation to compute the unknowns (displacement and crack phase-field), see Korelc and Wriggers [52]. The performance of the formulation is underlined by means of representative examples.

A low order 3D virtual element formulation for finite elasto–plastic deformations

This work addresses an efficient low order 3D virtual element method for elastic–plastic solids undergoing large deformations. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The successful application of the method to non-linear problems such as finite strain elasticity and plasticity in 2D leads naturally to the question of its effectiveness and robustness in the third dimension. This work is concerned with the extensions of the virtual element method to problems of 3D finite strain plasticity. Low-order formulations for problems in three dimensions, with elements being arbitrary shaped polyhedra, are considered. The formulation is based on minimization of a pseudo energy expression, with a generalization of a stabilization techniques, introduced for two dimensional polygons, to the three-dimensional domain. The resulting discretization scheme is investigated using different numerical examples that demonstrate efficiency, accuracy and convergence properties. For comparison purposes, results of the standard finite element method are also demonstrated.

Group-Theoretic Exploitations of Symmetry in Novel Prestressed Structures

In recent years, group theory has been gradually adopted for computational problems of solid and structural mechanics. This paper reviews the advances made in the application of group theory in areas such as stability, form-finding, natural vibration and bifurcation of novel prestressed structures. As initial prestress plays an important role in prestressed structures, its contribution to structural stiffness has been considered. General group-theoretic approaches for several problems are presented, where certain stiffness matrices and equilibrium matrices are expressed in symmetry-adapted coordinate system and block-diagonalized neatly. Illustrative examples on structural stability analysis, force-finding analysis, and generalized eigenvalue analysis on cable domes and cable-strut structures are drawn from recent studies by the authors. It shows how group theory, through symmetry spaces for irreducible representations and matrix decompositions, enables remarkable simplifications and reductions in the computational effort to be achieved. More importantly, before any numerical computations are performed, group theory allows valuable and effective insights on the behavior or intrinsic properties of a prestressed structure to be gained.

A Three-Node Triangular Element with Continuous Nodal Stress (Trig3-CNS) for Geometry Nonlinear Solid Mechanics Problems

This paper investigates the performance of the three-node triangular element with continuous nodal stress (Trig3-CNS) for geometry nonlinear solid mechanic problems. This Trig3-CNS element was recently proposed to improve accuracy of the finite element method (FEM). By synergizing the individual strengths of meshfree method and FEM, the Trig3-CNS element achieves higher accuracy and convergence rate. Furthermore, Trig3-CNS presents high tolerance to mesh distortion. Therefore, it is potentially useful for geometry nonlinear solid mechanics problems in which mesh distortion takes place. Compared with the traditional hybrid “FE-Meshfree” elements, Trig3-CNS naturally processes CNS without requiring any extra operation in post-processing. Numerical tests in the present work show that for geometry nonlinear analysis, the results of the Trig3-CNS element are better than the 3-node triangular element (Trig3) and 4-node isoparametric quadrilateral element (Quad4). In addition, the performance of Trig3-CNS is comparable to that of traditional hybrid “FE-Meshfree” elements.

Development of the method for calculation of cantilever construction's oscillations taking into account own weight

A new method for calculating bending oscillations of vertical cantilever structures with allowance for their own weight is proposed. The method is based on the exact solution of the corresponding partial differential oscillation equation with variable coefficients. In the analytical form with the help of dimensionless fundamental functions, formulas for dynamic parameters – motion, angle of rotation, bending moment and shear force, which completely characterize the state of the rod, are written out. In general, the frequency equation is written out and the method for finding its roots is determined. It is shown that the problem of determining natural frequencies can be reduced to finding the corresponding dimensionless coefficients from the frequency equations. The formulas for determining mode shapes are found. The algorithm that allows determining natural frequencies and mode shapes of cantilever structures with any given accuracy is described. The algorithm is implemented on the example of a through lattice tower. It is found that the numerical values obtained by the author’s method coincide with the results obtained with the help of the software system that implements the finite element method. In comparison with approximate methods, this method allows obtaining a more reliable picture of oscillations of cantilever structures, since it is the exact solution that carries information of a qualitative nature and forms the most complete picture of the physical phenomenon under consideration. Using explicit analytical formulas, the accuracy of calculation of bending oscillation is increased. The proposed method does not require the discretization of the structure and is a real alternative to the use of approximate methods when solving this class of problems of solid mechanics.

An Implicit Coupling Finite Element and Peridynamic Method for Dynamic Problems of Solid Mechanics with Crack Propagation

An implicit coupling finite element and peridynamic (PD) method is developed in this paper for the dynamic problems of solid mechanics with crack propagation. In this method, an implicit PD formulation is derived from the bond-based pairwise force that is described as a linear function of the displacements by using the first-order Taylor’s expansion technique. The equivalent incremental equations of the PD method and the finite element method are obtained on the basis of the Newmark and the Newton–Raphson schemes. To combine these two methods, the system is partitioned into two subregions and a convenient and efficient coupling strategy is proposed to form the coupling equivalent equation. The coupling domain is achieved by considering that the nodes and material points share the common information. Furthermore, displacement and load control-based incremental-iterative methods are adopted to solve the nonlinear equations. Several representative numerical examples are carried out and the results demonstrate the effectiveness and accuracy of the proposed coupling method.

Algorithms for Solving Boundary Value Problems of Deformable Solid Mechanics in View of Creep Strain

Further development of power and, primarily, engine engineering is associated with significantly increasing specific indicators. For example, the main trend in development of gas turbine engines is to increase gas parameters before the turbine. At the same time, there is an intensive growth of thermal and mechanical tension, and first of all this applies to the parts and components of the flow range. The destruction of these structural elements may have grave consequences. Increasing reliability and durability of responsible components of engines under operating conditions of complex cyclic thermo-mechanical loading is one of the priority tasks of modern engine engineering.One of the factors to determine a design performance is high-temperature creep. When solving the problems of deformable solid mechanics (DSM) in terms of creep, various options of the theory of hereditary creep and three main technical theories of aging, flow and hardening are widely used. There are also theories known that use an apparatus of the structural models and mechanical analogues to describe the creep. Most theories satisfactorily describe the creep strain under constant or slowly changing loads. Analysis of stress-strain states under variable loads is better described by the theories of flow and hardening, and the theory of hardening has some advantages over the theories of flow, as it gives more exact approximation for experiment results. From the point of view of the computing cycle arrangement, the technical theories have well-known advantages over the hereditary ones.When using the finite element method (FEM) to solve the boundary value problems of DSM considering the creep strain, an explicit or implicit Euler scheme is very often used. Depending on the features of the problem under consideration, a solution algorithm is constructed either in accordance with the method of initial stress, or by the method of initial strains. The method of initial strains when solving the problems in terms of creep is used more often, because the application of an initial stress method for this class of problems is technically much more complicated. The paper examines the explicit and implicit Euler schemes in combination with FEM. Both schemes are formulated in accordance with the method of initial strains. A constitutive relation was chosen in the form of the theory of flows.

Smoothed FE-Meshfree method for solid mechanics problems

This paper presents a smoothed FE-Meshfree (SFE-Meshfree) method for solving solid mechanics problems. The system stiffness matrix is calculated via a strain-smoothing technique with the composite shape function, which is based on the partition of unity-based method, combing the classical isoparametric quadrilateral function and radial-polynomial basis function. The corresponding Gauss integration in the element is replaced by line integration along the edges of the smoothing cells, so no derivatives of the composite shape functions are needed during the field gradient estimation process. Several numerical examples including an automobile mechanical component are employed to examine the presented method. Calculation results indicate that SFE-Meshfree can obtain a high convergence rate and accuracy without introducing additional degrees of freedom to the system. In addition, it is also more tolerant with respect to mesh distortion. The volumetric locking problem is also explored in this paper under a selective smoothing integration scheme.

A Four-Noded Triangular (Tr4) Element for Solid Mechanics Problems with Curved Boundaries

Linear triangular elements with three nodes (Tr3) were the earliest, simplest and most widely used in finite element (FE) developed for solving mechanics and other physics problems. The most important advantages of the Tr3 elements are the simplicity, ease in generation, and excellent adaptation to any complicated geometry with straight boundaries. However, it cannot model well the geometries with curved boundaries, which is known as one of the major drawbacks. In this paper, a four-noded triangular (Tr4) element with one curved edge is first used to model the curved boundaries. Two types of shape functions of Tr4 elements have been presented, which can be applied to finite element method (FEM) models based on the isoparametric formulation. FE meshes can be created with mixed linear Tr3 and the proposed Tr4 (Tr3-4) elements, with Tr3 elements for interior and Tr4 elements for the curved boundaries. Compared to the pure FEM-Tr3, the FEM-Tr3-4 can significantly improve the accuracy of the solutions on the curved boundaries because of accurate approximation of the curved boundaries. Several solid mechanics problems are conducted, which validate the effectiveness of FEM models using mixed Tr3-4 meshes.

A variational method to avoid locking—independent of the discretization scheme

SummaryWe present a variational method for problems in solid and structural mechanics that is designed to be intrinsically free from locking when using equal‐order interpolation for all involved fields. The specific feature of the formulation is that it avoids all geometrical locking effects (as opposed to material locking effects, for instance Poisson locking) for any type of structural or solid model, independent of the underlying discretization scheme. The possibility of employing equal‐order interpolation for all involved fields circumvents the task of finding particular function spaces to remove locking and avoid artificial stress oscillations. This is particularly attractive, for instance, for isogeometric analysis using unstructured meshes or T‐splines. Comprehensive numerical tests underline the promising behavior of the proposed method for geometrically linear and nonlinear problems in terms of displacements and stress resultants using standard finite elements, isogeometric finite elements, and a meshless method.

Modification of the quadratic 10‐node tetrahedron for thin structures and stiff materials under large‐strain hyperelastic deformation

SummaryThe concept of energy‐sampling stabilization is used to develop a mean‐strain quadratic 10‐node tetrahedral element for the solution of geometrically nonlinear solid mechanics problems. The development parallels recent developments of a “composite” uniform‐strain 10‐node tetrahedron for applications to linear elasticity and nonlinear deformation. The technique relies on stabilization by energy sampling with a mean‐strain quadrature and proposes to choose the stabilization parameters as a quasi‐optimal solution to a set of linear elastic benchmark problems. The accuracy and convergence characteristics of the present formulation are tested on linear and nonlinear benchmarks and compare favorably with the capabilities of other mean‐strain and high‐performance tetrahedral and hexahedral elements for solids, thin‐walled structures (shells), and nearly incompressible structures.

On the rate dependent behaviour of epoxy adhesive joints: Experimental characterisation and modelling of mode I failure

The increasing use of adhesive joints in dynamic applications require reliable measurements of the rate-dependent stress-displacement behaviour. The direct measurement of the stress-displacement curve is necessary when using cohesive models in discretised solutions of boundary value problems in solid mechanics. This paper aims to investigate the rate-dependent tensile failure of adhesive joints by using a new experimental methodology – it relies upon the combination of the stress wave propagation theory and digital image correlation methods on high speed footage to quantify the tensile stress and the dissipated energy respectively. For this purpose, the Split Hopkinson Bar methodology was employed – the experimental configuration was optimised using numerical modelling. To prove the sensitivity of our framework, two different adhesives are characterised at different loading rates: the adhesive failure strength was found to increase considerably with the strain rate, while the plastic deformation of these adhesives was reduced. The film adhesive showed superior performance over the particle toughened one. In the final part, a rate-dependent cohesive zone model is proposed, one which captures the measured behaviour and which has the potential to be used in industrial applications.

BEYOND BLACK-BOXES IN BAYESIAN INVERSE PROBLEMS AND MODEL VALIDATION: APPLICATIONS IN SOLID MECHANICS OF ELASTOGRAPHY

The present paper is motivated by one of the most fundamental challenges in inverse problems, that of quantifying model discrepancies and errors. While significant strides have been made in calibrating model parameters, the overwhelming majority of pertinent methods is based on the assumption of a perfect model. Motivated by problems in solid mechanics which, as all problems in continuum thermodynamics, are described by conservation laws and phenomenological constitutive closures, we argue that in order to quantify model uncertainty in a physically meaningful manner, one should break open the black-box forward model. In particular we propose formulating an undirected probabilistic model that explicitly accounts for the governing equations and their validity. This recasts the solution of both forward and inverse problems as probabilistic inference tasks where the problem's state variables should not only be compatible with the data but also with the governing equations as well. Even though the probability densities involved do not contain any black-box terms, they live in much higher-dimensional spaces. In combination with the intractability of the normalization constant of the undirected model employed, this poses significant challenges which we propose to address with a linearly-scaling, double-layer of Stochastic Variational Inference. We demonstrate the capabilities and efficacy of the proposed model in synthetic forward and inverse problems (with and without model error) in elastography.

Analytic formulas for the natural frequencies of hinged structures with taking into account the dead weight

Free bending vibrations of hinged vertical uniform rod with taking into account the dead weight are investigated. The research is based on the exact solution of the partial differential vibration equation with variable coefficients. This approach allows to get more reliable picture of rod’s vibration because only the exact solution carries information of a qualitative nature and forms the most complete picture of the physical phenomenon under consideration. The frequencies equation of problem was written in dimensionless form and the way of its root finding is shown. It has been shown that the problem of determination the nature frequencies of structures is reduced to finding corresponding dimensionless vibration coefficients from equation. The formulas for the first three vibration frequencies of structures were obtained in analytical form. An analytic relationship between the frequencies with and without taking into account the dead weight of the structures was established. The nature of the dependence of frequencies on the value of the longitudinal load was revealed. The presence of conclusive analytical formulas for determining the vibration frequencies of hinged vertical structures with taking into account the dead weight is a real alternative for using the approximate methods for solving this class of problems of solid mechanics.

Decomposition of Design Analysis Problems of Precision Structures of Large Reflectors

The concept and content of design analysis problems of precision structures of large reflectors are formulated within a systemic approach to the ensurement of their geometric stability and strength, as well as to the search for rational constructive options. Design analysis of the deformation of large reflectors is based on multimodel and multivariate computational modeling by numerical solution of the conjugate problems of solid mechanics, thermal conductivity, thermoelasticity, and aerodynamics. Applied problems are formulated with an accounting for the uncertainty of the initial data and the availability of information to computational models.

A novel node-based smoothed radial point interpolation method for 2D and 3D solid mechanics problems

This paper presents a novel node-based radial point interpolation method (NS-RPIM), which has two different versions termed as NS-RPIM−Tr4−Cd (for 2D problems) and NS-RPIM−Tr5−Cd (for 3D problems). These NS-RPIMs are created using edge-based Tr4-scheme and face-based Tr5-scheme, respectively. In the formulation, we use the generalized smoothed Galerkin (GS-Galerkin) weak-form which requires only value of shape functions. Because W2 formulation allows the use of discontinuous functions, RPIM can now be used to create proven stable and accurate models. The computational efficiency of the NS-RPIM−Tr4−Cd is rigorously examined against other NS-RPIMs and FEM. It is found that our NS-RPIM produce highly accurate solutions at low computational cost, due to the use of the condensed RPIM shape functions. Numerical results for 2D and 3D problems demonstrate that the NS-RPIMs possess the following important properties: (1) upper bound solution in the strain energy; (2) volumetric locking free; (3) superconvergence in strain energy solution; (4) insensitive to node distribution.

A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities

The SCRBE (Static-Condensation Reduced-Basis-Element) method is a component-to-system model order reduction approach for efficient many-query and real-time treatment of linear partial differential equations characterized by many spatially distributed constitutive, geometry, and topology parameters. In this paper we incorporate the SCRBE approach into a framework for analysis of problems in solid mechanics which are largely linear with the exception of local nonlinearities. In particular, we exploit a linear–nonlinear domain decomposition to develop a hybrid formulation: we consider a SCRBE approximation over the (assumed predominant) part of the domain associated to a linear elasticity model; we revert to a full finite element (FE) approximation over the part of the domain associated to the locally nonlinear model. We adapt the SCRBE port training procedures to anticipate the behavior of the field over the linear–nonlinear interface and hence ensure an accurate solution over the entire domain. We choose a globally conforming approximation which exploits the intrinsic port structure of the SCRBE method and yields a decoupled “non-invasive” formulation of the respective linear and nonlinear blocks of the residual vector and (Newton) Jacobian matrix. We present numerical results for several local nonlinearities – elastic contact with and without friction, plastic (contact) – which demonstrate substantial savings relative to standard FE approximation over the entire domain, in particular in the regime in which the linear part of the domain represents the majority of the (FE) degrees of freedom.

Four-Node Quadrilateral Element with Continuous Nodal Stress for Geometrical Nonlinear Analysis

In this paper, the performance of a hybrid ‘FE-Meshfree’ quadrilateral element with continuous nodal stress (Quad4-CNS) is investigated for geometrical nonlinear solid mechanic problems. By combining finite element method (FEM) and meshfree method, this Quad4-CNS synergizes the individual strengths of these two methods, which leads to higher accuracy, better convergence rate, as well as high tolerance to mesh distortion. Therefore, Quad4-CNS is attractive for geometrical nonlinear solid mechanic problems where excessive distorted meshes occur. For geometrical nonlinear analysis, numerical results show that the results of Quad4-CNS element are much better than those of four-node isoparametric quadrilateral element (Quad4), and are comparable to quadratic quadrilateral element (Quad8) and other hybrid ‘FE- Meshfree’ elements.

SOLUTION OF NON-LINEAR PROBLEMS OF STRUCTURAL MECHANICS BY METHOD OF STEEPEST DESCENT

The algorithm of application of the method of steepest descent to the solution of problems of structural mechanics and solid mechanics, described by nonlinear differential equations. For application of this method to nonlinear operators are described by a sequence of linear operators in incremental form, unlimited and complex linear operator, in line with the idea of L. V. Kantorovich is limited to a simple linear unbounded operator.

СИНТЕЗ МЕТОДУ ІДЕНТИФІКАЦІЇ КРАЙОВИХ УМОВ І МЕТОДУ СКІНЧЕННИХ ЕЛЕМЕНТІВ. ПРЯМОКУТНІ ПЛАСТИНИ

The key problem in the use of previously proposed boundary conditions identification method is the solution of intermediate boundary-value problems. These are quite complex boundary-value problems of structural elements conjugation of branched thin-walled spatial systems, such as bearing surfaces and hulls of aircraft structure. In solving the boundary conditions identifying problem for selected block-element the industrial finite element method can substantially help. On the other hand, boundary conditions identification method as such can help the finite element method to choose the effective type of finite element typical for the considered class of solid mechanics problems. This idea “boundary conditions identification method ↔ finite element method” is the basis of the study. The fruitfulness of combining two essentially different methods is shown by carrying out numerous calculations on a particular example of a rather simple but not trivial problem.