Local Strain Energy Density Research Articles

Rapid parametric analysis of SEN(T) specimens using algorithmic modelling: Evaluation of strain energy density and notch stress intensity factors

Abstract The aim of this paper is to present a methodology to rapidly evaluate the strain energy density (SED) and the notch stress intensity factor (NSIF) of a wide range of sharp V-notched SEN(T) specimens. A dislocation based method is used to find the solution of the boundary value problem. As a post-processing stage, computation of the local strain energy density over a circular sector surrounding the point of singularity is performed. NSIFs are then assessed by means of the strain energy approach (SEA) for several opening angles and specimen widths. Comparisons of local strain energy results with full-term solutions obtained by finite element analysis (FEA) are provided in order to validate the method and to establish the limitations of the NSIF approach for the case under consideration. The procedure presented in this work is shown to provide accurate results for the range of notch depth to specimen width ratios a/L

Modified theory of local strain energy density for sharp V-notches

A new approach is proposed to estimate the fracture of bodies with V-shaped notches under loading of two types. Two examples of quasibrittle fracture are considered: with an initial straight crack and with a sharp V-notch. Experimental data are obtained on the fracture of V-notched specimens under bending.

Some relationships between the peak stresses and the local strain energy density for cracks subjected to mixed-mode (I+II) loading

In this work, a link between the averaged strain energy density (SED) approach and the peak stress method in the case of cracks subjected to mixed mode (I+II) loading has been investigated. Some closed-form expressions of the strain energy density, averaged in a volume of radius R0, as function of the Stress Intensity Factors are provided for plane strain conditions under mixed mode I+II loadings, the material being thought of as isotropic and linear elastic. On the basis of the peak stress method (PSM) some expressions useful to estimate the mode I and mode II stress intensity factors (SIFs) have been recently derived. These relationships take advantage of the elastic peak stresses from FE analyses carried out by using a given mesh pattern where the element size and type are kept constants. The evaluation of the SIFs from a numerical analysis of the local stress field usually requires very refined meshes and then large computational effort. The usefulness of the PSM-based expressions is that (i) only the elastic peak stresses numerically evaluated at the crack tip are needed and not a set of stress–distance data; (ii) the employed meshes are rather coarse if compared to those necessary for the evaluation of the whole local stress field. By substituting the PSM-based relationships in the closed-form expressions of the averaged SED it appears that the latter can be directly estimated by means of the elastic peak stresses evaluated at the crack tip. Several FE analyses have been carried out on cracked plates subjected to tension loading considering different geometrical combinations, varying the length 2a and the inclination ? of the crack (i.e. the mode mixity) as well as the size d of the adopted finite elements, with the aim to evaluate the local SED and the elastic peak stress components ?peak and ?peak. In all cases the numerical values of the SED derived from the FE analyses have been compared with those analytically obtained by using the expressions for the SED based on the elastic peak stresses, in order to verify the range of applicability of the proposed relationships.

Multiaxial fatigue strength of severely notched titanium grade 5 alloy

The multiaxial fatigue strength of severely notched titanium grade 5 alloy (Ti-6Al-4V) is investigated. Experimental tests under combined tension and torsion loading, both in-phase and out-of-phase, have been carried out on axisymmetric V-notched specimens considering different nominal load ratios (R = -1, 0). All specimens are characterized by a notch tip radius less than 0.1 mm, a notch depth of 6 mm and a notch opening angle equal to 90 degrees. The experimental data from multiaxial tests are compared with those from pure tension and pure torsion tests on un-notched and notched specimens, carried out at load ratio ranging from R = -3 to R = 0.5. In total, more than 160 new fatigue data are examined, first in terms of nominal stress amplitudes referred to the net area and then in terms of the local strain energy density averaged over a control volume surrounding the V-notch tip. The dependence of the control radius on the loading mode is analysed showing a very different notch sensitivity for tension and torsion. For the titanium alloy Ti-6Al-4V, the control volume is found to be strongly dependent on the loading mode.

Experimental and theoretical investigation of brittle fracture in key-hole notches under mixed mode I/II loading

The Brazilian disk specimen weakened by a central dumbbell-shaped slit with two key-shaped ends and made of PMMA is employed to perform mixed mode I/II brittle fracture experiments on key-hole notches for various notch lengths and radii and, also, different mode mixity ratios. The load-carrying capacity of the specimen resulting from the test machine at the onset of sudden fracture is theoretically predicted by means of a well-known brittle fracture criterion, namely the local strain energy density criterion. A good agreement is found to exist between the theoretical and the experimental results.

A criterion based on the local strain energy density for the fracture assessment of cracked and V-notched components made of incompressible hyperelastic materials

The present contribution is devoted to the theoretical and numerical study of the elasto-static fields at a vertex notch under mode I loading. The analysis is based on the plane deformation hyperelasticity theory for an incompressible Mooney–Rivlin material. While for cracked components some contributions can be found in the recent and past literature, studies on V-notched components are instead very limited. The aim of this paper is to partially fill this lack, providing a fracture criterion for the assessment of components weakened by sharp V-notches. In the first part of the paper, a brief description of the analytical frame available for V-notches in hyperelastic material is reported. A Williams’ type diagram reporting the degree of singularity for a material obeying a Mooney–Rivlin behavior is present. The asymptotic stress field and the local strain energy density are investigated by means of non-linear finite element analyses. In the second part of the paper, a criterion based on the local energy is proposed and successfully applied to a set of experimental data taken from the literature. Future works are surely necessary to validate the criterion considering more sets of data from sharp and blunt V-notches.

РЕАЛИЗАЦИЯ МОДИФИЦИРОВАННОЙ ТЕОРИИ ЛОКАЛЬНОЙ ПЛОТНОСТИ ЭНЕРГИИ ДЕФОРМАЦИИ ДЛЯ ОСТРЫХ V-ОБРАЗНЫХ ВЫРЕЗОВ

РЕАЛИЗАЦИЯ МОДИФИЦИРОВАННОЙ ТЕОРИИ ЛОКАЛЬНОЙ ПЛОТНОСТИ ЭНЕРГИИ ДЕФОРМАЦИИ ДЛЯ ОСТРЫХ V-ОБРАЗНЫХ ВЫРЕЗОВ

Local strain energy density to predict mode II brittle fracture in Brazilian disk specimens weakened by V-notches with end holes

Abstract The present paper reports some experimental, theoretical and numerical results on brittle fracture of notched components made of Polymethylmethacrylate (PMMA). First, fracture experiments were carried out at room temperature on V-notches with end holes (VO-notches) under mode II loading by means of a recently proposed disk-type specimen, called the Brazilian disk containing central VO-notches (VO-BD specimen) made of PMMA for different notch angles and various notch radii. Then, the well-established averaged strain energy density (SED) criterion over the specified control volume which embraces the notch border was formulated and employed to predict theoretically the experimental fracture loads. The center of the control volume is located on the notch edge, where the principal stress reaches its maximum value. The correct orientation is obtained by a rigid rotation of the crescent-shaped volume while the size depends on the fracture toughness and the ultimate strength of the material. This methodology has been already used in the literature to analyze U- and V-shaped notches subject to different loading modes with very good results and advantages with respect to classic approaches. In this contribution it was demonstrated that SED works well also on VO-notches under pure mode II loading.

Local Strain Energy Density Applied to Bainitic Functionally Graded Steels Plates Under Mixed-Mode (I + II) Loading

In this paper, the averaged value of the strain energy density (SED) over a control volume is used to predict the critical load of V-notched specimens made of functionally graded steels (FGSs) under mixed-mode loading. The studied FGSs contain ferritic and austenite phases in addition to bainitic layer produced by electroslag remelting. The mechanism-based strain gradient plasticity theory is used to determine the flow stress (yield stress or ultimate stress) of each layer. The Young’s modulus and the Poisson’s ratio have been assumed to be constant, while other mechanical properties vary exponentially along the specimen width. The control volume is centered in relation to the maximum principal stress present on the notch edge and assumes a crescent shape. The points belonging to the volume perimeter are obtained numerically. In the present contribution, the effects of notch radius and notch depth on the SED and the critical load are studied. The notch radius varies from 0.2 to 2.0 mm, and the notch depth varies from 5 to 7 mm. By using the SED approach and finite element simulations, the critical load is determined, and the obtained results show a sound agreement with the experimental results.

The effects of different boundary conditions on three-dimensional cracked discs under anti-plane loading

Abstract Accordingly to the recent multi-scale model proposed by Sih and Tang, different orders of stress singularities are related to different material dependent boundary conditions associated with the interaction between the V-notch tip and the material under the remotely applied loading conditions. This induces complex three-dimensional stress and displacement fields in the proximity of the notch tip, which are worthy of investigation. Starting from Sih and Tang's model, in the present contribution the authors propose some analytical expressions for the calculation of the strain energy density (SED) averaged over a control volume embracing the V-notch tip. The expressions vary as a function of the different boundary conditions. Dealing with the specific crack case, the results from the analytical frame are compared with those determined numerically under linear-elastic hypotheses, by applying different constraints to the through-the-thickness crack edges in three dimensional discs subjected to Mode III loading. Free–free and free–clamped cases are considered. Due to three-dimensional effects, the application of a nominal Mode III loading condition automatically provokes coupled Modes (I and II). Not only the intensity of the induced modes but also their degree of singularity depend on the applied conditions on the crack flanks. The variability of local SED through the thickness of the disc is investigated by numerical analyses and compared with the theoretical trend. The capability of the SED to capture the combined three-dimensional effects is discussed in detail showing that this parameter is particularly useful when the definition of the Stress Intensity Factors (SIFs) is ambiguous or the direct comparison between SIFs with odd dimensionalities is not possible.

State‐of‐the‐art review on the local strain energy density concept and its relation to the J‐integral and peak stress method

ABSTRACTThe local average strain energy density (SED) approach has been proposed and elaborated by Lazzarin for strength assessments in respect of brittle fracture and high‐cycle fatigue. Pointed and rounded (blunt) V‐notches subjected to tensile loading (mode 1) are primarily considered. The method is systematically extended to multiaxial conditions (mode 3, mixed modes 1 and 2). The application to brittle fracture is documented for PMMA flat bar specimens with pointed or rounded V‐notches inclusive of U‐notches. Results for other brittle materials (ceramics, PVC, duraluminum and graphite) are also recorded. The application to high‐cycle fatigue comprises fillet‐welded joints, weld‐like shaped and V‐notched base material specimens as well as round bar specimens with a V‐notch. The relation of the local SED concept to comparable other concepts is investigated, among them the Kitagawa, Taylor and Atzori–Lazzarin diagrams, the Neuber concept of fictitious notch rounding applied to welded joints and also the J‐integral approach. Alternative details of the local SED concept such as a semicircular control volume, microrounded notches and slit‐parallel loading are also mentioned. Coarse FE meshes at pointed or rounded notch tips are proven to be acceptable for accurate local SED evaluations. The peak stress method proposed by Meneghetti, which is based on a notch stress intensity factor consideration combined with a globally even coarse FE mesh and is used for the assessment of the fatigue strength of welded joints, is also presented.

Averaged strain energy density and J-integral for U- and blunt V-shaped notches under torsion

Analytical solutions for the J-integral and the strain energy density (SED) averaged over a control volume are presented for U- and blunt V-shaped notches under torsion. The influence of the notch tip radius and the notch opening angle are fully included in the new proposed relationships. These expressions take advantage of some recent solutions for the stress fields ahead of blunt notches under Mode III loading and can be seen as a synthesis of the efforts carried out during the last years by the present authors on this topic. Afterwards, the expressions are applied to a recent set of experimental data from cracked, notched and plain specimens tested under torsion at low temperature $$(-60\,^{\circ }\hbox {C})$$ . The large scatter shown by notched specimens in terms of maximum elastic stresses strongly reduces when the J-integral or the local SED are used in combination with the specific control volumes.

The local strain energy density approach applied to pre‐stressed components subjected to cyclic load

ABSTRACTAhead of sharp V‐notches, residual stresses, arising from the solidification of a fusion zone, have the same asymptotic nature of the stress field induced by mechanical loads. This stress field significantly affects the engineering properties of structural components, notably fatigue life and corrosion resistance of welded joints. Tensile residual stresses can reduce the fatigue strength of welded joints particularly in the high‐cycle regime, where no stress redistribution due to local plasticity phenomena is expected to be present. The aim of this work is to analyse, by means of the numerical simulation, the residual stress redistribution near a V‐notch tip induced by cyclic loads and to propose a method, based on the local strain energy approach, for the fatigue resistance estimation of pre‐stressed components. The numerical solutions of the problem were carried out under the hypothesis of generalized plane strain conditions by means of SYSWELD and SYSTUS codes.

Multiaxial fatigue strength of severely notched cast iron specimens

The present work deals with multiaxial fatigue strength of severely notched cast iron (EN-GJS400). Circumferentially V-notched specimens are tested under combined tension and torsion loading, both in-phase and out-of-phase, with two nominal load ratios, R=−1 and R=0. All axis-symmetric specimens are characterized by a constant notch tip radius (less than 0.1mm), a notch depth of 4mm and a V-notch angle of 90°. The results from multiaxial tests are discussed together with those obtained under pure tension and pure torsion loading from notched specimens with the same geometry. Altogether more than eighty new fatigue data (10 Wöhler curves) are summarised in terms of normal and shear stresses. Afterwards, all fatigue strength data are given also in terms of the local strain energy density averaged over a material-dependent control volume surrounding the V-notch tip. The dependency of the control volume size as a function of the loading mode is investigated showing the necessity to introduce for the cast iron EN-GJS400 a different value of the control volume radius under tension and under torsion.

The Clinical Biomechanics Award 2012 — Presented by the European Society of Biomechanics: Large scale simulations of trabecular bone adaptation to loading and treatment

BackgroundMicrostructural simulations of bone remodeling are particularly relevant in the clinical management of osteoporosis. Before a model can be applied in the clinics, a validation against controlled in vivo data is crucial. Here we present a strain-adaptive feedback algorithm for the simulation of trabecular bone remodeling in response to loading and pharmaceutical treatment and report on the results of the large-scale validation against in vivo data. MethodsThe algorithm follows the mechanostat principle and incorporates mechanical feedback, based on the local strain-energy density. For the validation, simulations of bone remodeling and adaptation in 180 osteopenic mice were performed. Permutations of the conditions for early (20th week) and late (26th week) loading of 8N or 0N, and treatments with bisphosphonates, or parathyroid hormone were simulated. Static and dynamic morphometry and local remodeling sites from in vivo and in silico studies were compared. FindingsFor each study an individual set of model parameters was selected. Trabecular bone volume fraction was chosen as an indicator of the accuracy of the simulations. Overall errors for this parameter were 0.1–4.5%. Other morphometric indices were simulated with errors of less than 19%. Dynamic morphometry was more difficult to predict, which resulted in significant differences from the experimental data. InterpretationWe validated a new algorithm for the simulation of bone remodeling in trabecular bone. The results indicate that the simulations accurately reflect the effects of treatment and loading seen in respective experimental data, and, following adaptation to human data, could be transferred into clinics.

Research on structure emergence based on cellular automata

In order to reveal the complexity regularity of structure emergence, this paper adopts cellular automata to describe its local interactions, and a continuous structure emergence model based on cellular automata is proposed. At first, the model uses the unit with regular shape to make mesh generation of continuous structure, and adopts the finite element method to calculate the local strain energy density. Then it directly maps the mesh to the evolutionary environment of the cellular automata, and uses PID control theory and fully stressed design criterion to reconstruct local evolutionary rule of the cellular based on the strain energy. At last, it obtains the variation coefficients of the unit density through cellular evolution, thus getting a new design domain. The control parameters of the interaction rule and neighbourhood structure are also analysed. Furthermore, an optimal topology emergence example of the square plate dissipating heat is given. Finally, the characteristics of the proposed structure emergence model based on cellular automata are summarises.

Fatigue strength of Al7075 notched plates based on the local SED averaged over a control volume

When pointed V-notches weaken structural components, local stresses are singular and their intensities are expressed in terms of the notch stress intensity factors (NSIFs). These parameters have been widely used for fatigue assessments of welded structures under high cycle fatigue and sharp notches in plates made of brittle materials subjected to static loading. Fine meshes are required to capture the asymptotic stress distributions ahead of the notch tip and evaluate the relevant NSIFs. On the other hand, when the aim is to determine the local Strain Energy Density (SED) averaged in a control volume embracing the point of stress singularity, refined meshes are, not at all, necessary. The SED can be evaluated from nodal displacements and regular coarse meshes provide accurate values for the averaged local SED. In the present contribution, the link between the SED and the NSIFs is discussed by considering some typical welded joints and sharp V-notches. The procedure based on the SED has been also proofed to be useful for determining theoretical stress concentration factors of blunt notches and holes. In the second part of this work an application of the strain energy density to the fatigue assessment of Al7075 notched plates is presented. The experimental data are taken from the recent literature and refer to notched specimens subjected to different shot peening treatments aimed to increase the notch fatigue strength with respect to the parent material.

Local strain energy density near sharp V-notches in plates

Conditions of fracture of the local strain energy density, which were first formulated by Sih for a sharp V-notch with an arbitrary tip angle, are proposed. The edges of the considered V-notch are free from loading. If loading schemes of type I and type II are used, it is shown that the known brittle fracture conditions proposed by Sih contradict one of the basic postulates in fracture mechanics: the greater the intensity of stresses or elastic energy near the V-notch tip, the greater the probability of crack propagation. The proposed new conditions of fracture (in a polar coordinate system) are obtained as a result of independent determination of the energy densities of changes in volume and shape. In this case, the above-mentioned contradiction is eliminated.

Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density

The main aim of the present work is to check the suitability of the brittle fracture model, namely the local strain energy density (SED), in predicting the experimental results on mode I fracture of blunt V-notched graphite components. For this purpose, a wide range of test results reported in the recent literature on brittle fracture of V-notched test specimens characterized by different geometries is considered. The specimens are made of the same type of coarse-grained polycrystalline graphite. The fracture assessment is carried out predicting theoretically the fracture loads by means of the SED criterion. The SED parameter is evaluated by averaging the local energy over a well-defined control volume which embraces the notch edge. It is found that the SED criterion allows assessing the fracture behavior of graphite specimens characterized by different notch angles and tip radii.

A synthesis of data from steel spot welded joints of reduced thickness by means of local SED

In some previous works two fatigue scatterbands were given in terms of the local strain energy density (SED) for welded joints made of structural steel and aluminium alloy. The material-dependent radius R0 of the control volume was identified with reference to conventional arc welding processes. It resulted equal to 0.28mm for welded joints made of structural steel tested under prevailing mode I, with fatigue failures both at the weld root and the weld toe. The scatterband did not include data from thin welded joints, the main plate thickness being greater than 6mm. The SED approach is used here to summarise more than 400 fatigue data from steel spot welds in lap–shear joints, coach–peel specimens and cross–tension specimens of small thickness. Due to the typical crack initiation and propagation phases, an appropriate choice to summarise the data in a narrow scatterband seems to be the use of a toroidal control volume. The volume circumferentially encapsulates the slit tip along the weld spot nugget. The SED value at 2 million cycles is found to match that of the previous scatterband for welded joints of higher thicknesses.