Homogeneous Isotropic Material Research Articles

An anisotropic plastic‐damage model for 3D nonlinear simulation of masonry structures

AbstractPredicting the structural response of masonry structures with acceptable accuracy is paramount to safeguard the historical heritage and build new constructions with safety margins adequate to modern standards. However, due to the heterogeneous nature and anisotropic response of masonry, such prediction is still difficult to achieve, where most current masonry representations are based upon homogeneous isotropic material models or even more simplified masonry macro‐elements. In this article, a novel anisotropic constitutive model to be used in detailed 3D continuum FE representations is described. This is based upon the application of the transformed‐tensor method to an isotropic uncoupled plastic‐damage model, which is further enhanced by additional novel features enabling the proper definition of the shear behavior both in terms of yielding surface and damage evolution while increasing local computational robustness. Illustrative examples at different scales are presented, highlighting the characteristics and potential of the developed masonry material model. Focus is placed on the mechanical behavior under uniaxial and biaxial stress states considering pure compression on wallets with varying inclination of the material principal axes and the out‐of‐plane response of wall components. The numerical results confirm the ability of the proposed constitutive model to predict typical masonry anisotropic response characteristics, which cannot be accurately represented by standard isotropic representations commonly used in professional practice and research.

Concept of Equivalent Stress Criterion For Predicting Failure of Brittle Materials

Purpose: A paucity of proven failure criteria for brittle engineering materials exists, and this paper intends to present and validate a novel concept of equivalent stress criterion for predicting the failure of brittle isotropic homogeneous materials based on the concept of effective causative failure stress.
 Design/Methodology/Approach: Mathematical modelling is first performed based on strain-state equivalence, followed by conversion to the equivalent causative stress. The model is then validated with experimental and other data and with comparisons to traditional models. The material studied is BS 1452 Grade 250 continuous-cast grey cast iron with a Young’s Modulus of 39 000 MPa and ultimate tensile strength of 290 MPa. The test samples were prepared square in shape 12 mm x 12 mm to enable stresses in two perpendicular directions. Data is generated from uniaxial and bi-axial tests, performed using a standard universal testing machine, INSTRON 880, improvised to enable bi-axial recordings.
 Findings: Results point consistently to higher fidelity and transparency of the new model in representing the state of stress, especially in the second and fourth quadrants of the principal stress diagram, where Rankine’s criterion completely ignores stress differences and Mohr handles shear stresses in a suboptimal fashion. Both the maximum principal stresses and maximum shear stresses predicted by the proposed model are found to be somewhat greater than those from the traditional models, indicating higher accuracy and greater aggressiveness in prediction. The findings have further revealed that shearing effects play a greater role in the failure of engineering brittle materials than traditional failure theories have considered.
 Research Limitation: The study involved improvisation to enable biaxial stress recordings. This process was not perfect, resulting in smaller-than-ideal values of the lateral stresses.
 Practical implication: The study recommended process and equipment development toward perfecting multiaxial tests.
 Social implication: The survey will enrich the literature with pertinent design methodology to help in product design, including social-interest products. 
 Originality / Value: Since truly homogeneous materials are known to withstand very high hydrostatic pressures, direct stresses alone do not constitute valid failure criteria for all loading conditions.

An Improved Mathematical Representation of Mohr’s Failure Criterion For Brittle Materials

Purpose: This paper addresses issues bearing on accuracy neglected by earlier failure theories such as Rankine’s and Mohr’s. The overall aim is thus to present a thorough analysis of Mohr’s failure criterion and offer an improved model.
 Design/Methodology/Approach: The foundation of the methodology is Mohr’s criterion for predicting the failure of brittle isotropic homogeneous materials, built on the foundation of test results from three simple cases namely, pure tension, pure compression, and pure torsion. Thus the methodology involves first carrying out a critical analysis of Mohr’s model, followed by encapsulation of Mohr’s three simple monolithic cases in one generic equation of a circle whose parameters can be varied to match specific principal loading conditions more correctly. Experimental data are then used to validate the improved model.
 Findings: The work’s output is a material evaluation procedure that consists of a set of simple mathematical tests, any one of which predicting failure first, would then indicate the overall failure of the structural component under investigation. Results show clearly that this approach, i.e. using one parametric generic equation to represent material strength, is not only feasible but also robust. It offers an accurate method for predicting the failure of a brittle material under complex stresses.
 Research Limitation: Improvised conditions for biaxial data collection were less than ideal.
 Practical implication: The study recommended that other brittle materials beyond cast iron be included in any further studies to broaden the scope of applicability of the findings.
 Social implication: The research adds new literature and findings to an old subject. With this new knowledge, bookmakers could shape the way brittle materials are used in engineering design. 
 Originality / Value: The value of the study lies in the fact that to date very few failure theories exist that cater fully satisfactorily to brittle materials. The rigour of the methodology confers potential for its application beyond brittle materials.

A systematic investigation on the minimum tensile strengths and size effects of 3D printing polymers

Unlike isotropic homogenous materials, 3D printing polymers are anisotropic in terms of their strength and fracture toughness properties, according to previous research. Therefore, the lowest strength must be accurately measured before the extensive application of any 3D printing materials. Certain 3D printing materials have been increasingly employed in large structures; therefore, the size effect of their strengths has become very important. In this work, we performed a systematic experimental investigation on the minimum strengths and size effects of four common 3D printing polymers made by fusion deposition modeling and injection molding. Three types of tensile specimens with very different cross-sectional areas were selected. For each material system, two types of specimens with different printing surface angles were employed. The results from 200 specimens demonstrated that the minimum tensile strengths in the build direction of some specimens were only 25% that of the other directions in the same specimens, and 59% of the injection-molded specimens composed of the same material. No general conclusion on the size effects of the tensile strengths could be made for most specimens. Only the strengths along the build direction decreased up to 72% in two types of specific specimens if their cross-sectional areas increased by 60 times. Although their strengths were anisotropic, the stiffness properties of the 3D printing polymers were isotropic. Digital image correlation measurements showed that the Young's moduli of all specimens composed of the same material were similar. A dual-notch mechanics model was employed to explain the anisotropic strengths, and a fractography analysis supported the size effect.

On the low temperature creep controlling mechanism in a high strength spring steel

Despite of the well-known fact that high strength steels are exhibiting low temperature creep deformation, its rate controlling mechanism is yet to be understood. The strain hardening theory and the exhaustion creep model were proposed almost seven decades ago to unravel the low temperature creep mechanism in a single phase homogeneous isotropic material. However, their applicability to low temperature creep deformation in a technical material like a modern high strength steel is still a matter of investigation owing to their nature of multi-phase, in-homogeneous, and an-isotropic behavior. The authors have grabbed this chance to experimentally validate the exhaustion creep model based on low temperature creep tests of the SAE 9254 spring steel. The ultimate tensile strength and yield strength σe of SAE 9254 were determined in a temperature range of 298K≤T≤353K at a constant strain rate of 2.5∙10−4s−1. Low temperature creep deformation behavior was studied at the above-mentioned temperatures T at a constant stress σ=1634MPa, and at stresses 1071MPa≤σ≤1634MPa at a constant temperature T=353K for a duration of 1hr, respectively, at each condition. The retained austenite quantity was determined prior and post low temperature creep testing by means of X-ray diffraction. A reworked exhaustion creep model well describes the stress σ and temperature T dependency of low temperature creep deformation behavior in SAE 9254. Consequently, our hypothesis of a low temperature creep rate controlling mechanism reads: Low temperature creep strain is predominantly contributed by dislocation glide in the retained austenite phase. The reduction in creep rate with time can then be attributed to the exhaustion of this specific deformation mechanism by strain induced martensitic transformation of the retained austenite.

Hyperbolic two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to the non-Gaussian laser beam

The present work is devoted to a study of the induced temperature and stress fields in an elastic infinite medium with a cylindrical cavity under the purview of hyperbolic two-temperature thermoelasticity. The medium is an isotropic homogeneous thermoelastic material. The bounding plane surface of the cavity is loaded thermally by a non-Gaussian laser beam with a pulse duration of two ps. An exact solution to the problem is obtained in the Laplace transform space and the inversions of the Laplace transform have been carried out numerically. The derived expressions are computed numerically for copper and the results are presented in graphical form. The two-temperature parameter has significant effects on all the studied functions and plays a vital role in the speed propagation of the thermomechanical waves.

Finite Element Analysis for the Static Response of Functionally Graded Porous Sandwich Beams

In this paper, the finite element method is used to analyze the static response of the functionally graded porous (FGP) sandwich beams subjected to uniformly distributed loads along the beam span. The core of the beam is made up of functionally graded porous material while the top and bottom layers are made up of isotropic homogenous materials. Uniform distribution and symmetric distribution of pores are used as two different types of porous material. Shear deformation is considered in the analysis by utilizing BEAM189 in ANSYS which is a finite element package program. This element is based on the first-order shear deformation theory. The influence of porosity coefficient, boundary conditions, and type of the porous material on the static response of the considered structures is presented in detail. The results demonstrate that the porosity coefficient has an important impact on the static response of the FGP sandwich beams.

Cohesive Model of the Process Zone Near the Tip of a Crack Originating from a Corner Point of the Broken Interface

Within the framework of a cohesive model, we determine the parameters of the process zone in the binding material at the tip of an interface crack originating from the corner point of the broken interface of two different homogeneous isotropic materials. The indicated zone is modeled by the line of discontinuity of displacements where the stresses satisfy the Mises–Hill fracture criterion. By the Mellin integral transformation, we reduce the problem to the Wiener–Hopf vector equation whose analytic solution is found by the method of successive approximations. We deduce the equations for the length of the process zone and the phase length of stresses in this zone, as well as the expression for the crack-tip opening displacement. Finally, we perform the numerical analysis of parameters of the process zone.

Energy-based universal failure criterion and strength-Poisson's ratio relationship for isotropic materials

The prediction of material failures has been one of the most important topics in engineering design since the onset of human civilization. To date, a plethora of failure criteria have been developed, particularly for homogeneous isotropic materials. However, many of them have limited usage for specific groups of materials, are obscure in their origin, or different from the actual phenomena at the microscopic and macroscopic scales. Therefore, based on the most fundamental concept of energy, this study aims to establish a universal failure criterion for homogeneous full-density isotropic materials. The developed failure criterion requires three independent load-bearing capabilities of the material, that is, tensile (T), compressive (C), and shear (S) strengths, as criterion calibrators for applicability to a wide range of materials, including those with extremely ductile to exceedingly brittle behaviors. Energy and polynomial invariants expansion approaches were used to formulate the universal failure criterion, and both yielded a perfect dual relationship. The criterion in the energy form revealed a new physical insight into the zero-energy bound or fracture criterion, which corresponds to an intrinsic cut-off plane in a failure envelope. Comparisons between the experimental data across a broad spectrum of material types and the proposed failure criterion were in excellent agreement in both the stress and energy spaces. The strength-Poisson's ratio relationship was also derived for the first time, demonstrating that only two strength properties are independent and that three important bounding conditions for isotropic materials exist: (1) S/C = 0 when T/C = 0; (2) when T/C = 1 and S/C = 1/3, Poisson's ratio v = 0.5; and (3) v = −1 when T/S and C/S approach zero.

Finite element analysis of the seepage problem in the dam body and foundation based on the Galerkin’s approach

In this study, the 2D steady-state seepage analysis of the dam body and its base is investigated using the finite element method based on the Galerkin’s approach. The body and foundation soil are considered as homogeneous isotropic and anisotropic materials and the effects of horizontal drainage length and cutoff wall on seepage are investigated. The differential equation governing the response of water in the soil is obtained with the help of Darcy's law and continuity equations. A program has been prepared in Fortran programming language to find the hydraulic load, pressure values and phreatic line pressure values at the points inside the dam body and the quantity of seepage on the bottom of dam. It has been shown that the obtained values are in good agreement by comparing them with the results of existing studies in the literature.

Analysis of vibration characteristics of auxetic sandwich cylindrical shells resting on elastic foundation

The main objective of this research work is focused on the vibration analysis of auxetic sandwich cylindrical shell structures resting on an elastic foundation. In the analysis, the sandwich shell structure is composed of three layers in which the middle layer consists of auxetic material with a negative Poisson’s ratio, and the two skin layers are isotropic homogeneous materials. The motion equation is extracted according to the first-order shear deformation theory (FSDT) and the Hamilton principle. The governing equations of coupled partial differential equations are solved by the generalized differential quadrature (GDQ) method, and the natural frequencies are determined. By comparing the experimental results with the numerical results calculated by commercial finite element software, the validity of the proposed theoretical model is verified. Finally, the influences of geometrical parameters and elastic foundation on the vibration behavior of sandwich shell structures have been investigated.

Surface Wave Propagation in a Rotating Doubly Coated Nonhomogeneous Half Space with Application

The current study examines the propagation of surface waves in an asymmetric rotating doubly coated nonhomogeneous half space. The coating layers are assumed to be made of different homogeneous isotropic materials, while the overlaying nonhomogeneous half space layer is considered to be of exponentially varying material properties. The consequential exact vibrational displacements and dispersion relation are determined analytically, in addition to the approximate validation of the dispersion relation via the application of an asymptotic procedure within the long wave limit. Two cases of unloaded and loaded end surface scenarios are analyzed by examining the posed fundamental modes. More precisely, an elastic Winkler foundation was considered in the case of a mechanically loaded end surface condition and was found to proliferate the transition between having a fundamental mode over the frequency axis to the wave number axis as the angular velocity increased. Moreover, the rotational effect was found to have a direct impact on the surface wave propagation with a long wave and low frequency. Aside from that, an increase in the nonhomogeneity parameter resulted in propagation with a relatively long frequency.

Convective Cloak in Hele-Shaw Cells with Bilayer Structures: Hiding Objects from Heat and Fluid Motion Simultaneously

Thermal convection is one of the three basic mechanisms of heat transfer, which profoundly influences the natural environment, social production, and our daily life. However, the high complexity of the governing equation, which describes the coupling of heat and mass transfer, makes it difficult to manipulate thermal convection at will in both theory and experiment. Here, we consider the heat transfer in Hele-Shaw cells, a widely used model of Stokes flow between two parallel plates with a small gap, and apply the scattering-cancellation technology to construct convective thermal materials with bilayer structures and homogeneous isotropic materials. By tailoring thermal conductivities and viscosities, we demonstrate cloaking devices that can hide obstacles from heat and fluid motion simultaneously, and verify their robustness under various thermal-convection environments by numerical simulations. Our results show that about 80% of the temperature and pressure disturbances in the background caused by obstacles can be eliminated by the cloak. The developed approach can be extended to control other convection-diffusion systems or even other multiphysics processes, and the results pave a promising path for designing various metadevices such as concentrators or sensors.

Thermoelectric and stress distributions around a smooth cavity in thermoelectric material

• In this paper the thermoelectric coupling problem of an infinite homogeneous isotropic thermoelectric material with a smooth cavity and exerting a uniform current density and/or uniform energy flux is studied. • New analytical solutions are completely exact, and operable for complex shapes. • New solutions have a finite form for cavity characterized by Laurent polynomial. • The curvature of the cavity contour affects the distribution of the fields. • The maximum stress concentration appears near the points with extremal curvature. Thermoelectric materials have attracted more and more attention since they are friendly to the environment and have potentials for sustainable and renewable energy applications. As typically brittle semiconductors with low mechanical strength and always subjected to defects and damages, to clarify the stress concentration is very important in the design and implement of thermoelectric devices. The two-dimensional thermoelectric coupling problem due to a cavity embedded in an infinite isotropic homogeneous thermoelectric material, subjected to uniform electric current density or uniform energy flux, is studied, where the shape of the cavity is characterized by the Laurent polynomial, and the electric insulated and adiabatic boundary around the cavity are considered. The explicit analytic solutions of Kolosov-Muskhelishvili (K-M) potentials and rigid-body translation are carried out through a novel tactic. Comparing with the reported results, the new obtained are completely exact and possess a finite form. Some results of three typical cavities are presented to analyze the electric current densities (energy fluxes) and stresses around the tips. The main conclusions include: the distribution of thermoelectric field and stress at the tip obviously depends on the curvature of the contour and load directions; for triangle and square with symmetrical tips, the thermoelectric and stress concentration reach the maximum or minimum when the load direction is parallel to or perpendicular to the symmetry axis of the tip, which is distinct to the extremum characteristics of pentagram with bimodal of curvature around the tip; the maximum thermoelectric and stress concentration appear near the maximum curvature point for most load directions, but not at the maximum curvature point.

Nondestructive measurements of elastic constants of thin rods based on guided waves

Thin rods have been widely used in mechanical systems. The elastic constants, such as Young’s modulus and Poisson’s ratio, are of great interest for evaluating the quality of thin rods. This paper proposes a nondestructive method, utilizing longitudinal guided waves at low frequency and flexural guided waves at high frequency, to approximate bar waves and transverse bulk waves, respectively. According to the approximate velocities of bar waves and transverse bulk waves, the elastic constants can be determined inversely. Both simulation and experiment are conducted on thin aluminum rods with known elastic constants. The measurement errors are less than 2.03%, demonstrating the correctness of proposed method. Then, the proposed method is applied on 20 thin basalt rods and 30 basalt fibers. The incident frequency ranges from 0.3 to 1.8 MHz, and the diameter of thin rods ranges from 0.58 to 1.51 mm. The determined Young’s modulus is 4.30% higher than the value from the destructive tensile tests, and the determined Young’s moduli increase linearly with the chemical content (Al2O3, MgO, TFe2O3, and TiO2). The proposed method and associated experimental and simulation techniques can be used in the quality control and nondestructive testing of rod-like structures made by homogenous isotropic linear elastic materials.

ТОЧНОЕ РЕШЕНИЕ ЗАДАЧИ О РЕЛАКСАЦИИ НАПРЯЖЕНИЙ В ТОЛСТОСТЕННОЙ ТРУБЕ ИЗ НЕЛИНЕЙНО ВЯЗКОУПРУГОГО МАТЕРИАЛА, ПОДЧИНЯЮЩЕГОСЯ СООТНОШЕНИЮ РАБОТНОВА

We constructed and studied analytically the exact solution of the quasi-static boundary value problem for a hollow cylinder (a tube) made of physically nonlinear homogeneous isotropic viscoelastic material obeying the Rabotnov constitutive equation with two arbitrary material functions (a creep compliance and a function which governs physical nonlinearity). A time-dependent displacement and zero shear stresses are given on the inside cylinder surface, a pressure (normal stress) is given on outside surface and zero axial displacements and zero shear stresses are preset on the end cross sections of the tube (thus, cylinder stress and plain strain are realized in a tube). We supposed that a material is incompressible and that given boundary displacement and pressure vary slowly enough with time to neglect inertia terms in the equilibrium equations. We obtained explicit expressions for strains and stresses at any point via the ratio of a tube radii, given external pressure and integral operators involving composition of two material functions of the constitutive relation and displacement preset on the internal cylindrical surface. In particular, assuming given boundary displacement and pressure are constant we considered the stress relaxation problem. For arbitrary material functions, we calculated all the hereditary integrals involved in the general representation for the stress field and reduced it to simple algebraic formulas convenient for analysis and use. We studied analytically general properties of the stress relaxation curves at any point of a tube and features of stress distributions along radius. We proved that the first material function (relaxation modulus) of the constitutive equation governs completely stresses dependence on time and the second one (non-linearity function) governs only stresses dependence on radial coordinate and found out that hoop stress and axial stress can change sign and can increase, decrease and be non-monotone with respect to radial coordinate. In case of zero external pressure, we proved that all stress relaxation curves at any point of a tube are proportional to each other (ratio of stresses at different points doesn't depend on time), decrease with time and have no flexure points. We also calculated integral mean values of axial and hoop stresses and discovered that their ratio depends on thickness/radius ratio only and doesn't depend on time and material functions although mean stresses do.

Development of Validated Numerical Model for Study of Mechanical Properties and Permeability of Carbon Nanotube-Bioactive Glass Scaffolds

Abstract This study aims to predict mechanical properties of scaffolds made of bioactive glass-carbon nanotube (CNT) composite through finite element analysis (FEA) and their permeability using computational fluid dynamics (CFD) simulations. We start with constructing a three-dimensional model for the complete scaffold using cleaned/denoised images obtained from microcomputed tomography. To save computational effort, a representative volume element (RVE) is carved out from this model such that geometric properties like porosity and tortuosity are preserved. FEA requires material properties for which we have assumed that the CNTs are uniformly dispersed and hence, the composite behaves as a homogeneous isotropic material whose mechanical properties are experimentally obtained from a standard specimen. FEA has been performed on converged mesh for the RVE to obtain the compressive strength of the scaffolds. These computationally obtained compressive strengths compared well with those obtained experimentally, justifying our use of a homogeneous isotropic material model. We repeat the comparison for another geometry fabricated using additive manufacturing and find similarities in computational and experimental results. Hence, the compressive strength of bioactive glass-CNT composite scaffolds can be nondestructively predicted from our bulk identified mechanical properties irrespective of the geometry. For the CFD analysis, fluid flow is simulated in the porous region of the RVE and the estimated permeability of the scaffold is found to be satisfactory for nutrient and oxygen supply. Our study suggests that computational tools can help gain insights into the efficient design of scaffolds by obtaining the geometry having the right balance between strength and permeability for optimum performance.

Numerical wave speed sensitivity study for assessment of myocardial elasticity in a simplified linear elastic and isotropic left ventricle model

Since tissue elasticity can change with pathology, noninvasive assessment of elasticity has received increasing attention. Emerging methods for assessing cardiac elasticity utilize either an external source to induce propagating shear waves or intrinsic longitudinal waves created by natural cardiac events such as left ventricle stretching that occurs due to atrial kick during late diastole. However, the effect of morphological variations that occur in diseased hearts on this longitudinal stretch wave and the corresponding estimate of elasticity is not well understood and is an active area of research. This study investigated the sensitivity of longitudinal wave speed to material properties and chamber geometry parameters through numerical simulations using a finite element model of a bullet-shaped chamber with homogeneous isotropic linear elastic material properties. A longitudinal impulse displacement was applied to the base edge of the model to investigate wave propagation from this boundary. Parametric studies were performed for variables of interest related to geometry and material properties. The wave speeds estimated from simulation results were used to determine wave speed sensitivity to each variable. Wave speed was found to be a strong function of material elasticity and a weak function of chamber geometry and viscous damping. Simulated wave speed as a function of elasticity was in good agreement with wave speeds determined from an analytical expression for longitudinal wave speed in elastic thin plates. These promising preliminary results increase our understanding of how these parameters affect intrinsic longitudinal wave speed and warrant future studies addressing the impact of patient-specific model geometry, material anisotropy and hyperelasticity, and boundary conditions on wave speed.

Solution of some three-dimensional boundary value problems for thermoelastic bodies with voids

The paper considers a three-dimensional system of differential equations describing the thermoelastic static equilibrium of homogeneous isotropic materials with voids. In the Cartesian coordinate system the general solution of the mentioned system of equations is constructed using harmonic and metaharmonic functions. On the basis of the constructed general solution using the method of separation of variables boundary value problems for a semi-infinite prism and a rectangular parallelepiped are analytically solved. The corresponding boundary-contact problems for a multilayer rectangular parallelepiped are also considered.

Computational tools for the analysis of circular failure of rock slopes

Among other types of failure in slopes created by excavation or filling, circular (also referred to as rotational) type of failure plays an important role in the design and stability analysis of high rock slopes, such as those encountered in deep open pit mines. When a rock slope of significant height involves predominantly one type of jointed rock mass, the scale of the problem is frequently such that the rock mass (intact rock and rock discontinuities) can be regarded as a homogeneous isotropic material. In such case, if the rock slope is unstable and collapse occurs, circular (or rotational) failure as in failed slopes in soils is also observed for the rock slope. One main goal in practical design of deep open pit mines is to determine safe inclination angles for the pit faces to be excavated to avoid failure. Computational tools for quick assessment of the stability conditions, in particular determination of factor of safety and location of the critical circular failure surface, are useful during the preliminary stages of open pit design. Among others, such tools allow the engineer to gain insight in the role that the different variables controlling the problem have in defining the stability conditions for the rock slope. This paper provides first an overview of typical numerical techniques used to assess the stability of high rock slopes. For the case of slopes in rock masses that are assumed to obey the Mohr-Coulomb or the Hoek-Brown failure criteria, the paper discusses how application of transformation rules leads the criteria to be re-casted in dimensionless form, so when these are applied to the solution of slope problems, the results (factor of safety and coordinates that define the location of the critical circular failure surface) can be summarized in dimensionless compact forms. The paper presents then computational tools in the form of dimensionless charts and computer spreadsheets for stability analysis. These are developed from numerical limit equilibrium models that use the mentioned transformation rules. Among others, the computational tools allow the concept of mechanical similarity of rock slopes to be naturally introduced. Finally, the paper presents various rock slope examples that illustrate the use of the computational tools. In these examples, results from limit equilibrium models are also compared with results obtained with full numerical analysis, in particular with finite difference models implementing the shear strength reduction technique.