Ramberg-Osgood Stress-strain Relation Research Articles

Analyses of delaminations in non-linear elastic multilayered inhomogeneous beam configurations

This paper presents investigation of delamination fracture behavior of multilayered non-linear elastic beam configurations by using the Ramberg-Osgood stress-strain relation. It is assumed that each layer exhibits continuous material inhomogeneity along the width as well as along thickness of the layer. An approach for determination of the strain energy release rate is developed for a delamination crack located arbitrary along the multilayered beam height. The approach can be applied for multilayered beams of arbitrary cross-section under combination of axial force and bending moments. The layers may have different thickness and material properties. The number of layers is arbitrary. The approach is applied for analyzing the delamination fracture behavior of a multilayered beam configuration subjected to four-point bending. The beam has a rectangular cross-section. The delamination crack is located symmetrically with respect to the beam midspan. The strain energy release rate is derived assuming that the modulus of elasticity varies continuously in the cross-section of each layer according to a hyperbolic law. In order to verify the solution to the strain energy release rate, the delamination fracture behavior of the multilayered non-linear elastic four-point bending beam configuration is studied also by applying the method of the J-integral. The solution to the strain energy release rate derived in the present paper is used in order to perform a parametric study of delamination.

Longitudinal fracture of inhomogeneous beams with material non-linearity: An analytical investigation

This paper describes an analytical approach for analysing of longitudinal fracture in beam structures which exhibit smooth material inhomogeneity along the height and length. A solution to the strain energy release rate is derived for a general loading scheme assuming non-linear mechanical behaviour of the material. The Ramberg-Osgood stress-strain relation is used to model the material non-linearity. It is assumed that the three material properties which are involved in the Ramberg-Osgood stress-strain relation vary continuously along the height and length of the beam (exponential laws are applied for describing the distribution of these properties along the height and length of the beam). The solution derived is valid for a longitudinal crack located arbitrary along the beam height. It is shown that the approach developed can be applied for evaluating the effects of material inhomogeneity along the height and length of the beam, non-linear mechanical behaviour of the material, crack location along the beam height and the crack length on the longitudinal fracture.

Inhomogeneous Beams with Different Behaviour in Tension and Compression: A Longitudinal Crack Study

An approach for investigating the strain energy release rate for longitudinal cracks in nonlinear elastic inhomogeneous beams is derived. The Ramberg-Osgood stress-strain relation is applied to describe the unsymmetrical mechanical behaviour of the material with respect to tension and compression. It is assumed that the beams have continuous material inhomogeneity in the height direction. Cracks can be located arbitrarily along the beam height. The approach is applied to investigate a longitudinal crack in a cantilever beam. The J-integral is used for verification of the solution.

Damage controlled by brittle particles crush in AA7075-T6 beneath spherical indenter

The distribution of damage controlled by brittle particles crush in AA7075-T6 beneath spherical indenter was obtained experimentally and predicted analytically. Micro hardness tests were conducted on the vertical sections across the indentation impression and by comparing it with the estimated ideal micro hardness, the damage beneath spherical indenter can be quantified. In the present study, for a spherical indenter with 794 μm radius and a 117 μm penetration, maximum damage is 0.29, locating in the middle of the contact arc. The damage along path 1, which is right below the indentation impression, has generally smaller damage and the path 3, which is below the middle of the contact arc, has more extensive damage. Through a model which can describe the damage controlled by second phase particles crush, a modified expanding cavity model (ECM) coupling with the Ramberg-Osgood stress-strain relation was applied to predict the distribution of damage in spherical indentation deformation. By comparing with the experimental results, it proved that the method could give a prediction with reasonable accuracy.

Longitudinal Fracture Analysis of Nonlinear Elastic Circular Shafts Loaded in Torsion

The longitudinal fracture of circular shafts loaded in torsion is analytically investigated. It is assumed that a longitudinal crack with a circular crack front is arbitrarily located to the radial direction of the shaft cross section. The shafts are made of nonlinear elastic material, which exhibits continuous inhomogeneity in the radial direction. The material nonlinearity is described by the Ramberg–Osgood stress–strain relation. The longitudinal fracture is studied in terms of the strain energy release rate. The developed analysis is applied to evaluate the longitudinal fracture behavior of a clamped nonlinear elastic inhomogeneous shaft loaded in torsion by a torque applied to a free end of internal crack arm. For the method verification, the strain energy release rate is also assessed by analyzing the energy balance in the clamped shaft. A parametric study is performed to assess the effects of material inhomogeneity, its nonlinearity, and crack orientation to the shaft radial direction on the longitudinal fracture behavior of the clamped shaft.

Analysis of the Strain Energy Release Rate for a Delamination Crack in a Multilayered Beam with Material Non-Linearity

The strain energy release rate for a delamination crack in a multilayered cantilever beam which exhibits material non-linearity is derived by considering the energy balance. The material non-linearity is described by the Ramberg-Osgood stress-strain relation. A cantilever beam made of adhesively bonded lengthwise vertical layers is studied. Each layer has individual widths and material properties. Besides, the delamination is located arbitrary between layers. The strain energy release rate is derived also by differentiating the complementary strain energy with respect to the crack area for verification. It is shown that the solution derived is useful for parametric investigations of delamination in multilayered beams.

Longitudinal cracks in non-linear elastic beams exhibiting material inhomogeneity

Longitudinal fracture behavior of non-linear elastic beam configurations is studied in terms of the strain energy release rate. It is assumed that the beams exhibit continuous material inhomogeneity along the width as well as along the height of the cross-section. The Ramberg-Osgood stress-strain relation is used for describing the non-linear mechanical behavior of the inhomogeneous material. A solution to strain energy release rate is derived that holds for inhomogeneous beams of arbitrary cross-section under combination of axial force and bending moments. Besides, the solution may be applied at any law of continuous distribution of the modulus of elasticity in the beam cross-section. The longitudinal crack may be located arbitrary along the beam height. The solution is used to investigate a longitudinal crack in a beam configuration of rectangular cross-section under four-point bending. The crack is located symmetrically with respect to the beam mid-span. It is assumed that the modulus of elasticity varies continuously according a cosine law in the beam cross-section. The longitudinal fracture behavior of the inhomogeneous beam is studied also by applying the J-integral approach for verification of the non-linear solution to the strain energy release rate derived in the present paper. Effects of material inhomogeneity, crack location along the beam height and non-linear mechanical behavior of the material on the longitudinal fracture behavior are evaluated. Thus, the solution derived in the present paper can be used in engineering design of inhomogeneous non-linear elastic structural members to assess the influence of various material and geometrical parameters on longitudinal fracture.

Non-linear fracture in bi-directional graded shafts in torsion

PurposeThe purpose of this paper is to develop an analysis of longitudinal fracture behaviour of a functionally graded non-linear-elastic circular shaft loaded in torsion. It is assumed that the material is functionally graded in both radial and longitudinal directions of the shaft (i.e. the material is bi-directional functionally graded).Design/methodology/approachThe Ramberg–Osgood stress-strain relation is used to describe the non-linear mechanical behaviour of the functionally graded material. The fracture is studied in terms of the strain energy release rate by analysing the balance of the energy. The strain energy release rate is obtained also by differentiating of the complementary strain energy with respect to the crack area for verification.FindingsParametric studies are carried out in order to evaluate the influence of material gradients in radial and longitudinal directions, the crack location in radial direction and the crack length on the fracture behaviour of the shaft. It is found that by using appropriate gradients in radial and longitudinal directions, one can tailor the variations of material properties in order to improve the fracture performance of the non-linear-elastic circular shafts to the externally applied torsion moments.Originality/valueA longitudinal cylindrical crack in a bi-directional functionally graded non-linear-elastic circular shaft loaded in torsion is analysed by using the Ramberg–Osgood stress-strain relation.

Generalization of mixed mode crack behavior on the base of nonlinear fracture resistance parameters

Nonlinear crack-tip fracture resistance parameters for two types of cruciform specimens and a compact tension–shear specimen subjected to mixed mode loading are studied by using an elastic–plastic finite element (FE) analysis. FE analysis is performed for two types of steel and titanium and aluminum alloys with different elastic–plastic properties. A Ramberg–Osgood stress–strain relation is used to characterize the elastic-plastic properties of considered materials. Different degrees of mode mixity from pure Mode I to pure Mode II are realized in all considered specimens by combinations of the nominal stresses σn, remote biaxial stress ratio η, and the initial crack angle α with respect to the loading direction. For the specified geometry of the specimens considered, the governing parameter of the elastic– plastic crack-tip stress field In-factor, the stress triaxiality, J-integral are determined as a function of mode mixity and elastic–plastic material properties, described by strain hardening exponent. As a result influence of the specimen configuration on all considered nonlinear fracture resistance parameters is evaluated.

Delamination analysis of a layered elastic-plastic beam

PurposeThe purpose of this paper is to perform a theoretical analysis of delamination fracture behaviour of the Crack Lap Shear layered beam configuration taking into account the material non-linearity. A delamination crack located arbitrarily along the beam height was considered in this study.Design/methodology/approachThe beam mechanical behaviour was described by using the Ramberg-Osgood stress-strain relation. Fracture was analysed by applying the J-integral approach. Besides by using symmetric Ramberg-Osgood stress-strain curve, fracture was investigated also by Ramberg-Osgood stress-strain curve that is not symmetric with respect to tension and compression. The J-integral solutions were verified by performing elastic-plastic analyses of the strain energy release rate.FindingsThe effects of crack location and material properties on the non-linear fracture behaviour were evaluated. It was found that the material non-linearity leads to increase of the J-integral values. Therefore, the material non-linearity has to be taken into account in fracture mechanics based safety design of structural members composed by layered materials. The analytical solutions derived are very useful for parametric investigations of delamination fracture with considering the material non-linearity. The results obtained can be applied for optimisation of the beam structure with respect to fracture performance.Originality/valueThe present study contributes for the understanding of delamination fracture in layered beams that exhibit non-linear material behaviour.

Generalization of mixed mode crack behaviour by the plastic stress intensity factor

An elastic–plastic finite element analysis is performed for cruciform specimens of two configurations and a compact tension–shear specimen subjected to mixed Mode I/II loading. A Ramberg–Osgood stress–strain relation is used to characterise the properties of two types of steel and titanium and aluminium alloys. For the specified geometry of the specimen considered, the governing parameter of the elastic–plastic crack-tip stress field In factor, the stress triaxiality, and the plastic stress intensity factor are determined as a function of mode mixity and elastic–plastic material properties. Special emphasis is put on the analysis of the effect of specimen geometry. Analytical and numerical results are compared for the complete range of mixed-mode loading. A correlation between the plastic stress intensity factor and the constraint parameter based on the numerical analysis is found. Coupling between mode mixity and material nonlinearity is indicated. The applicability of the plastic stress intensity factor approach to large-scale yielding analysis is also discussed.

Springback analysis in sheet metal forming by using the Ramberg–Osgood stress–strain relation

The Ramberg–Osgood stress–strain relation is used to perform a theoretical springback analysis in the problem of bending of a narrow rectangular strip made of a strain-hardening material. The maximum strip thickness is 5 mm, and its length is significantly greater than the thickness. Based on the elasticity and plasticity deformation theory and also on the Tresca and von Mises yield criteria, an expression for the springback ratio is derived. The springback ratio depends on the ratio of the yield stress to Young’s modulus, Poisson’s ratio, strain hardening coefficient, and sheet thickness.

An Investigation of Winding Curvature Effect on the Mechanical Strength of Transformer Windings

During the winding process of a transformer, a residual strain is induced in its windings conductors. In this paper, the effect of the strain on the mechanical strength of transformer windings under the worst case condition of the short circuit has been investigated. The distribution of the residual strain in the conductors during its winding process has been determined analytically. The strain induced in the conductors under the action of radial short-circuit forces has been computed. The two strains are then added to obtain a resulting strain. By using the Ramberg-Osgood stress-strain relation of the conductor (copper) material, resulting stress is determined. A case study has been given where the presence of residual strain in the conductors of a power transformer has been taken into account for determining its mechanical strength under the worst case condition of short circuit.

Low cycle thermomechanical fatigue of reactor steels: Microstructural and fractographic investigations

The fatigue life of the structural materials 15Ch2MFA (CrMoV-alloyed ferritic steel) and 08Ch18N10T (CrNi-alloyed austenitic steel) of a VVER-440 reactor pressure vessel were investigated under fully reversed total strain controlled low cycle fatigue tests. The measurements were carried out in isothermal conditions at 260°C and with thermal-mechanical conditions in the range 150–270°C using a GLEEBLE-3800 servo-hydraulic thermal-mechanical simulator. The low cycle fatigue results were evaluated with the Coffin–Manson law, and the parameters of the Ramberg–Osgood stress–strain relation were investigated. Fracture mechanics behavior was observed using scanning electron microscopic analysis of the crack shapes and fracture surfaces. Crack propagation was assessed in relation to the actual crack size and the loading level. Interrupted fatigue tests were also carried out to investigate the kinetics of the fatigue evolution of the materials. Microstructural evaluation of the samples was performed using light, scanning and transmission electron microscopy as well as X-ray diffraction, and measurement of dislocations was completed using TEM and XRD. The course of dislocation density in relation to cumulative usage factor was similar for both steels. However, the nature and distribution of dislocations were different in the individual steels and this resulted in different mechanical behaviors. The nature of the fracture surfaces of both steels appeared similar despite differences in dislocation arrangement. The distances between striation lines initially increased with increasing crack length and then became saturated. The low cycle fatigue behavior investigated can provide a reference for the remaining life assessment and lifetime extension analysis of nuclear power plant components.

On the application of the method of fundamental solutions for the study of the stress state of a plate subjected to elastic–plastic deformation

The plane elastoplastic problem for the stress state of a plate with a narrowing subjected to uniaxial extension is considered. The behaviour of the material that hardens with plastic deformation is characterised by the Ramberg–Osgood stress–strain relation. Since this relation provides a smooth continuous curve over the whole elastic and plastic deformation range, the same governing equation can be used for both deformation regions. The paper provides a method for solving the resulting nonlinear boundary-value problem. The algorithm is based on meshless methods, i.e. the method of fundamental solutions and the method of particular solutions, together with a Picard iteration process. The approximate solution, i.e. the stress function, obtained in each iteration step is a linear combination of fundamental and particular solutions. It can thus be further used to compute the values of stresses and some effective material parameters (i.e. the Young modulus and the Poisson ratio) at any point of the domain.

Analysis of Buckling Strength of Inner Windings in Transformers Under Radial Short-Circuit Forces

The buckling of conductors of inner windings in transformers is one of the major causes of their failures. It can occur when a large magnitude of radial short-circuit electromagnetic force acts on them. In this paper, initially, mechanical strains developed during winding processes and due to radial short-circuit forces have been determined. The two mechanical strains viz. the short-circuit induced strain and the winding process-induced strain are algebraically added to obtain their resulting strain. The stress corresponding to the resulting strain has been determined by using the Ramberg-Osgood stress-strain relation. The critical buckling stress has been calculated and compared with the resulting stress. The analytically obtained result of the strain induced in the winding conductor during its winding process has been verified using the finite-element method. A case study has been described in which the factor of safety against the buckling strength is determined.

Determination of Two-Dimensional Plastic Zone Shape and SIF at Crack-Tip Using RKPM

Reproducing kernel particle Method (RKPM) is a meshless technology which has proven very useful for solving problems of elastic-plastic fracture mechanics. The mode I plastic zone shape at the crack-tip in a work-hardening material is obtained using RKPM. Ramberg-Osgood stress-strain relation is assumed and the crack-tip stress intensity factor (SIF) before and after formation of the plastic zone are examined. To impose the essential boundary conditions, penalty method is used. To construct the shape functions in the vicinity of the crack and crack-tip, both the diffraction and visibility criteria are employed. A comparison between two conventional treatments, visibility and diffraction, to crack discontinuity is conducted. The effects of different dilation parameters on SIF under plane-stress and plane-strain conditions are studied. Results including plastic zone shape are compared with finite element method (FEM) to show the accuracy of RKPM. The main objective is to study the effects of different dilation parameters on SIF under plane stress and plane strain conditions and to obtain the mode I plastic zone shape at the crack-tip in a work-hardening material using RKPM.

Plastic-Buckling of Rectangular Plates under Combined Uniaxial and Shear Stresses

This paper is concerned with the plastic-buckling of rectangular plates under uniaxial compressive and shear stresses. In the prediction of the plastic-buckling stresses, we have adopted the incremental theory of plasticity for capturing the inelastic behavior, the Mindlin plate theory for the effect of transverse shear deformation, the Ramberg-Osgood stress–strain relation for the plate material, and the Ritz method for the bifurcation buckling analysis. The interaction curves of the plastic uniaxial buckling stress and the plastic shear buckling stress for thin and thick rectangular plates are presented for various aspect ratios. The effect of transverse shear deformation is examined by comparing the interaction curves obtained based on the Mindlin plate theory and the classical thin plate theory.

Numerical investigation of the plastic behaviour of short welded aluminium double-T beams

The present work is a numerical approach to the mechanical behaviour of short aluminium beams, both extruded and welded at the web–flange junction. The studied beams were taken to be short so as to ensure that their design is dominated by shear. In order to investigate the effects of the weld and the consequent existence of a heat-affected zone (HAZ), both welded and extruded beams of identical geometric characteristics were studied. Three alloys, 6063-T6, 6005A-T6 and 7020-T6, were chosen because of their varying strength characteristics, as well as the different severity of mechanical degradation that each one undergoes in the HAZ. The numerical investigation is performed in the framework of small displacements, and the possibility of lateral buckling is excluded. All the studied cases qualify as Class-I cross sections for normal actions. The Ramberg–Osgood stress–strain relation is used to describe the hardening of the material. The results obtained by means of finite element models are compared to those of classical beam theory and to the resistance checks of Eurocode 9.

Plastic buckling analysis of thick plates using p-Ritz method

This paper is concerned with the application of the p-Ritz method for the plastic buckling analysis of thick plates. In order to allow for the effect of transverse shear deformation in thick plates, the Mindlin plate theory is adopted. The plastic buckling behaviour of the plate is captured by using the incremental and deformation theories of plasticity. The material property of the plate is assumed to obey the Ramberg–Osgood stress–strain relation. The p-Ritz method will be applied to obtain the governing eigenvalue equation for the plastic buckling analysis of uniformly stressed plates with edges defined by polynomial functions. In the p-Ritz method, the displacement functions of the plate are approximately represented by the product of mathematically complete two-dimensional polynomial functions and boundary equations raised to appropriate powers that ensure the satisfaction of the geometric boundary conditions. The validity, convergence and accuracy of the method were demonstrated for various plate shapes such as rectangular, triangular and elliptical shapes. A parametric study was also undertaken to study the plastic buckling behaviour and the effect of transverse shear deformation.