Uniform Traction Research Articles

Effect of boundary conditions on values of homogenized fabrics’s elastic parameters and inelastic response

The effect of displacement, periodic and surface traction boundary conditions (BCs) on predicted elastic parameters and inelastic response of a homogenized material equivalent to either a unidirectional or a randomly distributed short fibers reinforced composite has not been extensively studied. A few such studies have focused on their elastic analyses. Here, we analyze the effect of the three types of BCs on both elastic and inelastic deformations of weaves and braids with continuum damage mechanics used in the inelastic analyses. We employ simple unit cells, Hill’s theorem, and the displacement, periodic and uniform traction BCs to deduce from the finite element stiffness matrix K the matrix C of homogenized moduli. For both elastic and inelastic problems, it is seen that predictions using the displacement and the periodic BCs are close to those reported in the literature while those from the traction BCs are not. For unidirectional loading, it is found that the nonlinearity in the axial stress - axial strain curve, the localized deformations and the resin damage occurring in regions unsupported by fibers are significantly more for the traction BCs than that for the other two BCs. Also, there is more damage for the periodic BCs than that for the affine displacement BCs.

Pressure-driven micro-poro-mechanics: A variational framework for modeling the response of porous materials

Porous materials are highly relevant in engineering and medical applications due to their enhanced properties and lightweight nature. Current micromechanical models of porous materials can accurately predict the response under the assumptions of small deformations and drained conditions, typically driven by imposed deformations. However, the theoretical framework for the micromechanical modeling of porous material driven by pore pressure in the large-deformation range has been understudied. In this work, we develop a finite-deformation variational framework for pressure-driven foams, i.e., materials where the pore pressure in the cavities produces the deformation. We further consider different kinematical constraints in the formulation of boundary conditions: kinematic uniform displacements, periodic displacements and uniform traction. We apply the proposed model in the numerical simulation of lung porous tissue using a spherical alveolar geometry and an image-based geometry obtained from micro-computed-tomography images of rat lung. Our results show that the stress distributions in the spherical alveolar model are highly dependent on the kinematical constraints. In contrast, the stress distribution in the image-based alveolar model is not affected by the choice of boundary conditions. Further, when comparing the response of pressure-driven versus deformation-driven models, we conclude that hydrostatic stresses experience a marked shift in their distribution, whereas the deviatoric stresses remain unaffected. Our findings of how stresses are affected by the choice of boundary conditions and geometry take particular relevance in the simulation of the lungs, where the pressure-driven and deformation-driven cases are related to mechanical ventilation and spontaneous breathing.

On the efficient enforcement of uniform traction and mortar periodic boundary conditions in computational homogenisation

In this contribution, the enforcement of uniform traction (UT) and periodic boundary (PB) conditions on arbitrarily generated meshes is analysed in detail, and the derivation of the associated macroscopic consistent tangent operators is described. Two different approaches to impose these boundary conditions are examined: the saddle point solution (SPS) resulting from the Lagrange multiplier method and the condensation method (CM). The numerical treatment required to implement the UT and the PB conditions with both methods is presented for rectangular and cuboidal representative volume elements (RVEs). It is shown that the UT condition only needs rigid body motion to be prescribed since it intrinsically restricts rigid body rotations in the finite strain setting. The application of the SPS to the UT condition leads to a formulation where the Lagrange multipliers are the homogenised stresses, and the macroscopic consistent tangent operator for FE2 simulations can be directly derived. The enforcement of PB conditions on non-conforming meshes with the mortar approach is also described for the CM and SPS. The need to modify the Lagrange multiplier dual interpolation functions is emphasised to avoid over-constraints. The proposed implementation recovers the conventional PB condition solution when conforming meshes are used. The computational performance of both methods is compared in terms of time and memory requirements. As anticipated, the results obtained for each boundary condition do not depend on the enforcement method employed. With the direct solvers employed, the SPS is more efficient despite increasing the number of unknowns in the linear system of equations.

Influence of Boundary Conditions on Numerical Homogenization of High Performance Concrete.

Concrete is the most widely used construction material nowadays. We are concerned with the computational modelling and laboratory testing of high-performance concrete (HPC). The idea of HPC is to enhance the functionality and sustainability of normal concrete, especially by its greater ductility as well as higher compressive, tensile, and flexural strengths. In this paper, the influence of three types (linear displacement, uniform traction, and periodic) of boundary conditions used in numerical homogenization on the calculated values of HPC properties is determined and compared with experimental data. We take into account the softening behavior of HPC due to the development of damage (micro-cracks), which finally leads to failure. The results of numerical simulations of the HPC samples were obtained by using the Abaqus package that we supplemented with our in-house finite element method (FEM) computer programs written in Python and the homogenization toolbox Homtools. This has allowed us to better account for the nonlinear response of concrete. In studying the microstructure of HPC, we considered a two-dimensional representative volume element using the finite element method. Because of the random character of the arrangement of concrete’s components, we utilized a stochastic method to generate the representative volume element (RVE) structure. Different constitutive models were used for the components of HPC: quartz sand—linear elastic, steel fibers—ideal elastic-plastic, and cement matrix—concrete damage plasticity. The numerical results obtained are compared with our own experimental data and those from the literature, and a good agreement can be observed.

Considering computational speed vs. accuracy: Choosing appropriate mesoscale RVE boundary conditions

Modeling a material’s microstructure using continuum theories allows for inspection of the relationship between coarse scale and fine scale behaviors. Computational limits generally require selection of a sub-volume from a bulk sample in order to directly model the microstructure. Boundary conditions are applied to the sub-volume to mimic the excluded bulk material. Appropriate selection of boundary conditions helps effectively determine the appropriate spatial scale required of the sub-volume. Applicable boundary conditions include direct displacement, periodic, and uniform traction. While direct displacement and periodic boundary conditions are commonly used, uniform traction boundary conditions have seen limited use due to rigid body stability issues in simulations of compression or shear deformation. A new application of uniform traction boundary conditions was developed through linear constraint equations, similar to approaches employed by direct displacement and periodic boundary conditions, to quench rigid body motions with minimal interference of the relative deformation of the model. These boundary conditions were tested by compressing several synthetically generated periodic microstructures using the finite element method. Evaluating the effective stiffness along the compression axis, the direct displacement boundary condition produced the stiffest response, whereas the uniform traction boundary condition produced the most compliant. Periodic boundary conditions produced the same response for all volumes analyzed and both the direct displacement and uniform traction boundary conditions trended toward the periodic response as the domain volume increased. Computational performance was also evaluated for each boundary condition using implicit and explicit solvers. Direct displacement boundary conditions presented the lowest computational cost of all of the boundary conditions followed by periodic then uniform traction. The computational expense of periodic and uniform traction boundary conditions limited the viable spatial scale and mesh resolutions able to be simulated. Selection of appropriate boundary conditions for specific uses need to be a balance between allowable computational expense and accuracy of the method. Techniques for evaluating which boundary conditions to use are discussed.

Axisymmetric BEM analysis of one-layered transversely isotropic halfspace with cavity subject to external loads

This paper presents an axisymmetric boundary element analysis of one-layered transversely isotropic material of halfspace extent with either ellipsoidal or spherical cavity. This analysis uses the fundamental solution of a transversely isotropic bi-material fullspace subject to the body force uniformly concentrated at a circular ring. The finite and infinite boundary elements are, respectively, used to discretize the finite and semi-infinite regions of the external boundary and the finite boundary elements are used to discretize the internal boundary. Different types of the integrals in the discretized boundary integral equations are efficiently calculated. An improved traction recovered method is applied to calculate the stresses on the boundary. Using one set of BEM mesh, a total of 12 case studies are calculated for the elastic fields of the one-layered transversely isotropic solid by uniform traction on the external horizontal boundary. The solid has either an ellipsoidal, a spherical, or none cavity in the upper layer. Four combinations of two transversely isotropic metals are adopted for the one-layered solid. One metal is the stiff zinc and the other is the soft magnesium. The displacements for the cases with magnesium as the upper layer are larger than those with zinc as the upper layer. Besides, the effects of the lower solid materials to the displacements are limited. The BEM results also reveal the variations of the displacements and stresses within the halfspace with or without cavities for different cases.

Hierarchical finite element-based multi-scale modelling of composite laminates

This paper presents a hierarchic finite element-based computational framework for the multi-scale modelling of composite laminates. Hierarchic finite elements allow changing the approximation order locally or globally without changing the underlying finite element mesh. Both micro- and macro-level structures are discretised with these elements. The macro-level structures of composite laminates are divided into several blocks during the pre-processing stage, and approximation orders are assigned to each block independently. Due to a sharp increase in the interlaminar stresses, higher approximation orders are used in the vicinity of free edges as compared to the rest of the problem domain. This freedom of assigning approximation orders independently to each block provides an efficient and accurate way for modelling composite laminates. The computation framework can either accept the user-defined ply-level homogenised elastic material properties or calculates these directly from the underlying representative volume element consisting of matrix and fibre using the computational homogenisation. The model developed for the computational homogenisation has the flexibility of unified imposition of representative volume element boundary conditions, which allows convenient switching between linear displacement, uniform traction and periodic boundary conditions. The computational framework has additional flexibly of high-performance computing and makes use of state-of-the-art computational libraries including Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Mesh-Oriented datABase (MOAB). Symmetric cross-ply, angle-ply and quasi-isotropic laminates subjected to uniaxial loading are used as test cases to demonstrate the correct implementation and computational efficiency of the developed computational framework.

Boundary element formulation of axisymmetric problems in vertically non-homogeneous solids subject to normal traction

This paper presents an efficient and accurate boundary element method (BEM) for the elastic analysis of axisymmetric problems in vertically non-homogeneous solids without or with cavity. This BEM uses the fundamental solution of the elastic field in a multilayered elastic solid induced by the body force uniformly concentrated at a circular ring. This solution is also called Yue's solution. The effective integration methods are used for dealing with the integrals in the discretized boundary integral equations. The discretization of the boundary surface uses one-dimensional boundary elements. It also adopts the infinite boundary element to take into account the influence of a far-field region on the boundary surface. Numerical verifications of displacements and stresses for three benchmark problems are conducted, which gives the excellent agreement with previously published results. Case studies are presented to numerically illustrate the influences of both vertically non-homogeneous elastic material properties and spherical cavity on the elastic fields induced by uniform tractions on the boundary surface. These numerical results show that this new BEM is a fast and simple numerical algorithm for accurately computing the axisymmetric elastic fields in vertically non-homogeneous solids with or without cavity induced by normal tractions.

A spheroidal inclusion within a 1D hexagonal piezoelectric quasicrystal

In this study, we investigate the induction field and the phonon and phason stress fields of a spheroidal inclusion embedded within an infinite matrix of a one-dimensional (1D) hexagonal piezoelectric quasicrystal subjected to the following three prescribed uniform traction boundary conditions: axisymmetric loading, out-of-plane and in-plane shear. The explicit expressions are derived in the surrounding matrix and the inclusions by setting the correct potential function. The reduced results show that the stresses exhibit a singularity on the crack faces. The obtained results can also serve as a reference for exploring other 1D piezoelectric quasicrystals reinforced by spheroidal inclusions.

Proposal for Introducing Uniform Traction Power System in Context of Efficient Locomotive Use

Since 2010, there has been a strong increase in passenger transport compared to the original forecast. This makes us think about possible ways to improve fixed electric traction installations performance that would enable to increase the transport capacity. To secure modern methods of operation, a quality railroad network with a quality and stable power supply system is necessary. If the existing DC traction system (3 kV) does not ensure sufficient power supply, some other acceptable way must be sought. The problem is technically solvable. If the lines capacity shall be used in the most efficient way, a systematic approach is necessary, in particular using multi-system locomotives with higher performance. Solving the issue of traction power systems development and modernization will be related with the conversion of DC traction system (3kV) into a uniform power supply system (25kV/50Hz).

Measurement of mixed mode interfacial strengths with cementitious materials

In the oil and gas industry, a successful zonal isolation, which is the primary goal of cementing, is achieved by a strong bond at the cement-rock and the cement-casing interfaces. We introduce a universal test in conjunction with an analytical solution to measure the mixed mode interfacial strength of cementitious materials at the casing-cement or rock-cement interfaces. To this end, the most practical method is employed, which is the direct measurement of adhesion at the interface. This experimental setup is consisted of a rotational disk composed of two identical halves. Each half includes a cylindrical hole at its center, which can be filled by cement slurry, a rock cylinder or a steel bar. Therefore, different interfaces i.e. cement-cement, cement-steel, and cement-rock can be examined. After curing, the disk is subjected to two diametral point loads using a compressive frame. By adjusting the angle between the load direction and the overlapping interface, we can examine different combinations of modes I and II. On the other hand, an analytical elasticity solution is developed to calculate the critical shear strength that would initiate sliding between the interfaces. Analytical results show a non-uniform distribution of shear tractions along the sliding interface. Having the peak load obtained from experiment, derived formulations are used to evaluate the bonding strength of cementitious materials corresponding to different loading modes of pure shear, the combination of tension/compression and shear. The value of the current analytical derivation can be appreciated by comparison with the typical approach that assumes a uniform traction distribution along the sliding interface. We find that assuming uniform distribution will overestimate the shear strength at the cement-casing interface. The analytical solution in conjunction with the experimental results can also be used to calibrate cement interfacial friction properties.

A unified framework for the multi-scale computational homogenisation of 3D-textile composites

This paper extends the applications of a novel and fully automated multi-scale computational homogenisation framework, originally proposed by the authors (Ullah et al. (2017)) for unidirectional and 2D-textile composites, to 3D-textile composites. 3D-textile composites offer many advantages over 2D-textile composites but their highly complicated and unpredictable post-cured geometries make their design very challenging. Accurate computational models are therefore essential to the development of these materials. The computational framework described in this paper possesses a variety of novel features which have never been tried for this class of composites and can potentially help to fully automatise and improve their design process. A unified approach is used to impose the representative volume element boundary conditions, which allows convenient switching between linear displacement, uniform traction and periodic boundary conditions. The computational framework is implemented using hierarchic basis functions of arbitrary polynomial order, which allows one to increase the order of approximation without changing the finite element mesh. The yarns' principal directions, required for the transversely isotropic material model are calculated using a potential flow analysis along these yarns. This feature is very useful for 3D-textile composites and can accurately determine fibres’ directions even in the case of very deformed yarns. A numerical example from literature consisting of a 3D-orthogonal woven composite is used to demonstrate the correct implementation and performance of the developed computational framework. Also, the developed computational framework is used to perform a comparative study of the homogenised mechanical properties of five 3D-textile composites with different yarn architectures.

Computational homogenisation of thermo-viscoplastic composites: Large strain formulation and weak micro-periodicity

In this article, a first order two-scale finite element framework for fully coupled thermo-mechanical problems at finite deformations is considered. The scale bridging is achieved under application of Hill–Mandel condition based boundary conditions at the lower scale. Emphasis of the present article is on the extension of the concept of weak micro-periodicity to thermo-mechanically coupled problems. With this formulation at hand, uniform traction, respectively heat flux boundary conditions as well as the classic periodic boundary conditions are captured as well as a transition between them. The governing equations are summarised with a focus on the implementation of the weak periodic boundary conditions for thermo-mechanical problems. The performance of the proposed weak periodic boundary conditions in comparison to the classic boundary conditions is shown by means of simulations of a single representative volume element as well as by means of full multi-scale simulations.

Systematic study of homogenization and the utility of circular simplified representative volume element

Although both computational and analytical homogenization are well-established today, a thorough and systematic study to compare them is missing in the literature. This manuscript aims to provide an exhaustive comparison of numerical computations and analytical estimates, such as Voigt, Reuss, Hashin–Shtrikman, and composite cylinder assemblage. The numerical computations are associated with canonical boundary conditions imposed on either tetragonal, hexagonal, or circular representative volume elements using the finite-element method. The circular representative volume element is employed to capture an effective isotropic material response suitable for comparison with associated analytical estimates. The analytical results from composite cylinder assemblage are in excellent agreement with the numerical results obtained from a circular representative volume element. We observe that the circular representative volume element renders identical responses for both linear displacement and periodic boundary conditions. In addition, the behaviors of periodic and random microstructures with different inclusion distributions are examined under various boundary conditions. Strikingly, for some specific microstructures, the effective shear modulus does not lie within the Hashin–Shtrikman bounds. Finally, numerical simulations are carried out at finite deformations to compare different representative volume element types in the nonlinear regime. Unlike other canonical boundary conditions, the uniform traction boundary conditions result in nearly identical effective responses for all types of representative volume element, indicating that they are less sensitive with respect to the underlying microstructure. The numerical examples furnish adequate information to serve as benchmarks.

Determination of crack opening displacement and critical load parameter within a cohesive zone model

A semi-analytical approach for investigation of the stress and strain state of a cracked body is presented. A cohesive zone model is used to take into account the failure zones which are formed in front of the crack tip. It is proposed to satisfy the crack smooth closure condition approximately. It allows to get rid of solving nonlinear equations for the cohesive lengths. An effective algorithm is developed to find solutions for the uniform traction–separation law. The algorithm is based on the iterative procedure of determination of contact stresses between crack faces, so the possible contact interaction between crack faces can easily be taken into account. In the case of mixed mode cracks, multiple cohesive zone models are considered.

Biaxial Loading of a Plate Containing a Hole and Two Co-Axial Through Cracks

Abstract The paper presents the solution linear elasticity problem for an isotropic plate weakened by a hole and two co-axial cracks. The plate is exerted by uniform traction at infinity. The corresponding 2D problem is solved by the method of Kolosova-Muskhelishvili complex potentials. The method implies reduction of the problem to simultaneous singular integral equations (SIE) for the functions defined the region of the cracks and hole. For particular case the solution the SIE is obtained analytically in a closed form. A thorough analysis of the stress intensity factors (SIF) is carried out for various cases of the hole shape: penny-shaped, elliptical and rectangular.

On the link between fracture toughness, tensile strength, and fracture process zone in anisotropic rocks

This paper presents experimental results on the anisotropy of the fracture toughness, Brazilian tensile strength, and the fracture process zone (FPZ) in granodiorite samples. The fracture toughness is measured using semi-circular bending tests, while Brazilian disk tests were conducted to measure the tensile strength indirectly. Digital image correlation (DIC) was employed to obtain full-field surface deformation associated with the fracture propagation and identify the FPZ. An averaging scheme is proposed to determine the length and width of the FPZ from the strain field. The DIC results confirm a semi-elliptical FPZ developing ahead of the crack tip, with an average length-to-width ratio of approximately two. The results also indicate that the theoretical models such as Irwin and strip-yield with uniform traction, which are based on plastic deformation near the crack tip, underestimate the extent of the inelastic zone, while the strip-yield model with a linear cohesion stress distribution overestimate the length of the process zone. The anisotropy ratio of the FPZ length obtained from the models, however, agrees very well with the ratio obtained from the DIC measurements. This evidence supports the basis of the theoretical models that predict the FPZ length to be proportional to the square of fracture toughness over tensile strength.

Application of low-order potential solutions to higher-order vertical traction boundary problems in an elastic half-space.

New solutions of potential functions for the bilinear vertical traction boundary condition are derived and presented. The discretization and interpolation of higher-order tractions and the superposition of the bilinear solutions provide a method of forming approximate and continuous solutions for the equilibrium state of a homogeneous and isotropic elastic half-space subjected to arbitrary normal surface tractions. Past experimental measurements of contact pressure distributions in granular media are reviewed in conjunction with the application of the proposed solution method to analysis of elastic settlement in shallow foundations. A numerical example is presented for an empirical ‘saddle-shaped’ traction distribution at the contact interface between a rigid square footing and a supporting soil medium. Non-dimensional soil resistance is computed as the reciprocal of normalized surface displacements under this empirical traction boundary condition, and the resulting internal stresses are compared to classical solutions to uniform traction boundary conditions.

Free Streamline Hydrodynamic Analogy for a Linear Elastic Antiplane Slot Problem with Perfectly Plastic Ligaments at Its Ends

It is shown that a two-dimensional analogy exists between an inviscid incompressible fluid flow around two finite-length flat plates (Riabouchinsky problem) and that of a straight slot cut in a linear elastic plate having rounded ends that are on the verge of yielding plastically. For the plates, a uniform flow at infinity is assumed in a direction perpendicular to the side-by-side parallel lines that constitute the two-dimensional geometry of the plates. Correspondingly, the slot has a boundary condition at infinity of a uniform shear traction in the antiplane direction. In the fluid flow problem, two free streamlines form at both ends of the flat plates, while in the slotted plate problem, the rounded ends of the traction-free slot surfaces constitute ligaments of perfectly plastic material. The specific shape of the rounded ends of the slot problem can be inferred by analogy with the fluid flow problem.

Statistics informed boundary conditions for statistically equivalent representative volume elements of clustered composite microstructures

ABSTRACTHomogenized effective constitutive response of arbitrarily dispersed multiphase composites systems are computed using property-based statistically equivalent representative volume elements or P-SERVEs. P-SERVEs are subjected to affine transformation-based displacement boundary conditions, uniform traction boundary conditions or periodic boundary conditions. Predicted P-SERVE sizes are larger as the above boundary conditions ignore the influence of the inclusions exterior to the P-SERVE. Exterior statistics-based boundary conditions (ESBCs) accounting for the presence of fibers and their interactions in the domain exterior to the P-SERVE have been derived in Ghosh and Kubair (2016, J. Mech. Phys. Solids, vol. 96, pp. 1–24). In the present study, ESBCs are derived for statistically inhomogeneous microstructures using the two-point correlation functions used in the statistically informed Greens functions. Comparisons are also made with the statistical volume element approach. It is concluded that the simulations with ESBCs prescribed on the P-SERVEs have a definite advantage over other methods in defining optimal sized P-SERVEs for clustered and matrix-rich microstructures.