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Open Circular Hole in a Finite Plate Under Tension Treated by Airy Stress Function Method
Abstract
Open circular holes are an essential design feature and are often placed in narrowplates for reasons of limited space raising high stress concentrations. To create lightweight optimal structures precise analysis tools are vital, which can be based on analytical means providing efficient computation. That is why focus of the present paper is the determination of the stress field for the isotropic finite-width open-hole problem under uniform tension using the Airy stress function. This is performed by taking the stress field of corresponding infinite domain problem and supplementing it with auxiliary functions enabling to continuously model traction-free edges. The results are eventually validated against Finite Element analyses.
... Hence, applicability to other problems cannot be concluded. For instance, it has been revealed that this heuristic approach leads to erroneous results for both isotropic bolted joints [45,47] and open holes with finite dimensions [44] when assessing the stresses along the entire net section plane. ...
... More recent studies that employ an approach to continuously fulfil stress-free boundary conditions of finite domains without the necessity of optimisation routines are the following. The open hole is treated by Nguyen-Hoang and Becker [44] for isotropic plates, which is extended for bolted joints with isotropic [47] and orthotropic plate Open holes in composite laminates with finite dimensions material [48]. The corresponding stress results show good agreement to FE data. ...
... Nguyen-Hoang and Becker [45] further use FFM to analyse the size effect for w/d = {2; 3} and to investigate to which extend the TCD may be applied in bolted-joints assessment. This is based on a preliminary stress calculus based on analytical means, which is enhanced yielding good agreement for isotropic bolted joints [47] and excellent correlation for open holes [44]. This accurate stress calculation methodology shall be extended to the orthotropic case and then be used to conduct a failure analysis by means of TCD and FFM subsequently. ...
Open circular holes are an important design feature, for instance in bolted joint connections. However, stress concentrations arise whose magnitude depends on the material anisotropy and on the defect size relative to the outer finite plate dimensions. To design both safe and light-weight optimal structures, precise means for the assessment are crucial. These can be based on analytical methods providing efficient computation. For this purpose, the focus of the present paper is to provide a comprehensive stress and failure analysis framework based on analytical methods, which is also suitable for use in industry contexts. The stress field for the orthotropic finite-width open-hole problem under uniform tension is derived using the complex potential method. The results are eventually validated against Finite-Element analyses revealing excellent agreement. Then, a failure analysis to predict brittle crack initiation is conducted by means of the Theory of Critical Distances and Finite Fracture Mechanics. These failure concepts of different modelling complexity are compared to each other and validated against experimental data. The size effect is captured, and in this context, the influence of finite width on the effective failure load reduction is investigated.
... Laminate plates can be found in various maritime and aerospace structures, wherein plates are frequently drilled with holes or apertures for practical purposes, such as reducing the joining parts' weight and admittance to structural access [1]. When a plate with a hole is exposed to tension or shear pressure, considerable stresses rise around the holes, reducing component strength and causing premature failure. ...
Laminated plates with holes are often used in industrial applications such as aeronautics, automobiles, and marine. It is necessary to present a study of the combined effect of deformation theories and thickness variation for the laminated plate with a hole. This manuscript considers analysis theories named: Kirchhoff, layer-wise, and Reissner-Mindlin, to study their validity for different thickness aspect ratios under transverse compression loading. Studies are conducted on CFRP laminate (symmetric cross-ply and quasi-isotropic), and the performed numerical (FEM) analysis processes are validated through existing literature. Transverse shear, circumferential, radial, and radial-hoop stress variations with stress concentration factors are presented along the thickness and around the hole configurations (18 cases are covered) of the plate. Also, the resultant effects are discussed on in-plane, and out-of-plane stresses for laminates to specify the selection of design conditions (theory, thickness, model, and laminates). This research may provide engineers and researchers with various assessments and design insight for laminate structures with a hole.
... This leads to the widespread use of 3D braided composites in different fields, such as aeronautics, marine, transportation, and other industries. Different holes or openings are typically drilled into these composites' plates to reduce the system's weight and allow access to system equipment [1]. However, due to specified cut-outs, significant stresses are formed around the holes or openings when a plate is subjected to tension or shear pressure. ...
Composite plates with holes are common in engineering applications, such as the automotive and aerospace industries. Three-dimensional braided carbon/epoxy polymers are an advanced textile composite and are used in various structures due to their high damage resistance and relatively low manufacturing cost. When a braided polymer plate with a hole is used in engineering applications, it is necessary to know its mechanical behavior under loading conditions using analysis theory to design it better. However, the effects of stress distribution with shear deformation theories on the variable thickness of the braided polymer plate (carbon/epoxy) with a hole under tensile loading have not been reported yet. In this paper, a study is conducted to evaluate shear deformation theories for a braided polymer plate with variable thickness and a hole in the center, analyzing the stresses and their concentration variations. First, multiscale modeling and analysis are performed to determine the mechanical properties of the plate. Then, finite element analyses are performed on a homogenized macro plate with a hole. The analysis process is verified by comparison with the available literature. Results show that the first-order shear deformation theory calculates 37, 56, and 70 percent less maximum transverse shear stress than the high-order shear deformation theory (Reissner–Mindlin) and the elasticity theory for thin, moderately thick, and thick braided polymer plates, respectively. Additionally, changing the theory has no significant effect on circumferential stress, radial stress, Von Mises stress, and stress concentration factor. As a result, this research can provide researchers and designers with structural intuition for a braided polymer plate with a center hole.
Bolted joints are frequently used to connect rather thin parts in lightweight structures, e.g. air- and spacecraft. This is also motivated by inexpensive manufacturing and the ability to disassemble. However, holes need to be drilled and stress concentrations arise. Usually, multiple bolts/rows of bolts are placed. Then, the load is partly introduced into one bolt while the remaining rest stays in the plate. This problem setting is also referred to as bolted joint under combined bearing-bypass load. When there are negligible secondary bending effects a 2D model can be employed. Further, a linear approach without contact elasticity is commonly chosen, which is provided by superimposing the open- and pin-loaded hole/filled-hole problem. The latter is then idealised by sinusoidal radial tractions along half of the hole edge. The structural assessment shall be done using precise analysis means since safety-critical parts are connected by bolted joints. These means can be based on analytical methods, which are beneficial in terms of computational effort and shall be scope of the present paper. For stress field representation, use is made of the Airy stress function method. First, the solutions of the special cases open and filled hole are determined. These are then superimposed. Finite plate dimensions are taken into account using auxiliary and correction stress functions, which are based on a novel periodic arrangement technique. Effects of raised stress concentrations due to finite dimensions as well as the ratio between bearing and bypass load are extensively discussed. Validation by means of Finite Element analyses reveals excellent agreement.
Bolted joints are commonly used to connect safety–critical parts in aeronautics, which can be composite laminates. However, drilling holes also means introducing stress raisers. For narrow connections, fatal tension failure mode is likely to be triggered. Accurate stress assessment methods are crucial to create safe and lightweight optimal designs. These tools can be based on analytical means beneficial in terms of efficient computation and shall be scope of this paper. In this context, a plate model idealising orthotropic composite laminates without bending extension coupling is chosen. This type of laminates is commonly used in industry contexts. Special consideration is given to finite plate dimensions. Their impact on the critical stress concentrations is studied for different lay-ups of the composite laminate. This further allows to investigate the effects of material orthotropy. Validation to Finite Element analyses shows good agreement for commonly used lay-ups and plate dimensions.
Bolted joints are often used for the connection of safety-critical parts in the aeronautical industry. However, drilling holes also means introducing stress concentrations and precise assessment tools are essential to design both safe and lightweight optimal structures. These tools can be based on analytical means providing efficient computation. That is why the present work is dedicated to determine the stress field of finite dimensions bolted joints with isotropic plate material by means of the Airy stress function. The boundary value problem is solved by taking the stress field of the corresponding infinite domain problem and supplementing it with auxiliary functions enabling the continuous modelling of traction-free edges. The results are eventually validated against Finite Element analyses revealing good agreement.
In the present paper, solution of moment distribution around elliptical hole in symmetric infinite laminated plate is obtained. The solution is based on Lekhnitskii and Savin formulation considering in-plane bending moment at infinity and hygrothermal effects. The present solution is in close agreement with the existing literature. The parametric study to learn the effect of parameters like in-plane bending loading, volume fraction of fiber, temperature, moisture content and hole parameter (a/b ratio) on moment distribution around elliptical hole is presented. The unidirectional lamina properties are obtained using a micromechanical approach. The study revealed, under hygrothermal environment these factors affects the moment concentration.
Crack initiation in brittle materials is not covered by classical fracture mechanics that deals only with the growth of pre-existing cracks. In order to overcome this deficiency, the Finite Fracture Mechanics concept assumes the instantaneous formation of cracks of finite size at initiation. Within this framework, a coupled criterion was proposed at the beginning of the 2000’s requiring two necessary conditions to be fulfilled simultaneously. The first one compares the tensile stress to the tensile strength, while the other uses an energy balance and the material toughness. The present analysis is restricted to the 2D case, and, through a wide list of references, it is shown that this criterion gives predictions in agreement with experiments in various cases of stress concentration, which can be classified in two categories: the singularities, i.e. indefinitely growing stresses at a point, and the non-singular stress raisers. It is applied to different materials and structures: notched specimens, laminates, adhesive joints or embedded inclusions. Of course, a lot of work remains to do in these domains but also in domains that are almost not explored such as fatigue loadings and dynamic loadings as well as a sound 3D extension. Some ideas in these directions are issued before concluding that FFM and the coupled criterion have filled a gap in fracture mechanics.
Bolted joints are widely used to connect safety-critical parts in the aeronautical industry and require precise structural assessment tools. Focus of this paper is the development of an overall analytical calculation method to determine the stresses of a bolted joint with quasi-isotropic composite material, which is then used in a tension failure analysis. The stress solution is obtained by means of the Airy stress function. To predict crack initiation in the tension failure analysis use is made of the Theory of Critical Distances. Crack initiation occurs in the net section plane and its assessment requires precisely calculated net section stresses. For this purpose, the stress boundary conditions in load direction at the straight free edges must be fulfilled, which is reached by supplementing the field modelling the load introduction with auxiliary functions. This technique is a physically motivated means to deal with symmetric finite geometry problems involving straight free edges. However stress boundary conditions perpendicular to the load direction are not covered and some inaccuracies in the stress solution may arise. A failure analysis is conducted to investigate their impact on the predicted failure load using literature values based on Finite Fracture Mechanics as reference.
Elasticity: Theory, Applications, and Numerics, Third Edition, continues its market-leading tradition of concisely presenting and developing the linear theory of elasticity, moving from solution methodologies, formulations, and strategies into applications of contemporary interest, such as fracture mechanics, anisotropic and composite materials, micromechanics, nonhomogeneous graded materials, and computational methods. Developed for a one- or two-semester graduate elasticity course, this new edition has been revised with new worked examples and exercises, and new or expanded coverage of areas such as spherical anisotropy, stress contours, isochromatics, isoclinics, and stress trajectories. Using MATLAB software, numerical activities in the text are integrated with analytical problem solutions. These numerics aid in particular calculations, graphically present stress and displacement solutions to problems of interest, and conduct simple finite element calculations, enabling comparisons with previously studied analytical solutions. Online ancillary support materials for instructors include a solutions manual, image bank, and a set of PowerPoint lecture slides. ? Thorough yet concise introduction to linear elasticity theory and applications ? Only text providing detailed solutions to problems of nonhomogeneous/graded materials ? New material on stress contours/lines, contact stresses, curvilinear anisotropy applications ? Further and new integration of MATLAB software ? Addition of many new exercises ? Comparison of elasticity solutions with elementary theory, experimental data, and numerical simulations ? Online solutions manual and downloadable MATLAB code.
Abstract Plates notched with open-holes that are subject to combined tensile and in-plane bending loading are studied in this analysis of crack onset. A finite fracture mechanics approach is employed to study the onset of asymmetric crack patterns in the notched structure. The stress field and the energy release rates required to solve the coupled stress and energy criterion are obtained from linear elastic finite element analyses. Closed-form expressions for the dependence of the solution on governing parameters as e.g. the structural size, the Young’s modulus or the ratio of bending to normal loading are given significantly reducing the computational effort required for the failure model. A comparison to experimental results and to numerical analyses using a cohesive zone model highlights the model’s ability to render brittle failure of the notched structure and to provide reliable failure load estimates. The results reveal that a certain threshold of the hole diameter exists, below which the hole does not affect the failure load. The threshold and the corresponding bearable loads are discussed using explicit solutions.
Peterson's Stress Concentration Factors establishes and maintains a system of data classification for all of the applications of stress and strain analysis and expedites their synthesis into CAD applications. Substantially revised and completely updated, this book presents stress concentration factors both graphically and with formulas. It also employs computer-generated art in its portrayal of the various relationships between the stress factors affecting machines or structures. These charts provide a visual representation of the machine or structure under consideration as well as graphs of the various stress concentration factors at work. They can be easily accessed via an illustrated table of contents that permits identification based on the geometry and loading of the location of a factor. For the new third edition, new material will be added covering finite element analyses of stress concentrations, as well as effective computational design. The book explains how to optimize shape to circumvent stress concentration problems and how to achieve a well-balanced design of structures and machines that will result in reduced costs, lighter products, and improved performance.
The anisotropy of composite plates often poses difficulties for stress field analysis in the presence of notches. The most common methods for these analyses are: (i) analytical means (AM), (ii) finite element analysis (FEA), and (iii) semi-analytical means (SAM). In industry, FEA has been especially popular for the determination of stresses in small to medium size parts but can require a considerable amount of computing power and time. For faster analyses, one can use AM. The available solutions for a given problem, however, can be quite limited. Additionally, AM implemented in commercial computer software are scarce and difficult to find. Due to this, these methods are not widespread and SAM were proposed. SAM combine the (easy) implementation of complex problems from FEA and the computational efficiency from AM to reduce the difficulty on mathematical operation and increase computational speed with respect to FEA. AM, however, are still the fastest and most accurate way to determine the stress field in a given problem. Complex problems, however, e.g., finite width plates with multiple loaded/unloaded notches, require a significant amount of mathematical involvement which quickly discourages, even seasoned, scientists, and engineers. To encourage the use of AM, this paper gives a brief review of the mathematical basis of AM followed by a historic perspective on the expansions originating from this mathematical basis. Specifically the case of a two-dimensional anisotropic plate with unloaded cut-outs subjected to in-plane static load is presented.
Results of applying the boundary collocation method to finite geometry plates with stress concentrations is reported. A boundary point least-squares fit approach in conjunction with a complex variable technique is used to satisfy boundary conditions in a doubly-connected anisotropic region. The effect of finite geometry on stress distributions around unloaded and loaded fastener holes in isotropic and anisotropic plates are investigated and compared with known solutions.
A new model based on finite fracture mechanics is proposed to predict the open-hole tensile strength of composite laminates. Failure is predicted when both stress-based and energy-based criteria are satisfied. The material properties required by the model are the ply elastic properties, and the laminate unnotched strength and fracture toughness. No empirical adjusting parameters are required. Using experimental data obtained in quasi-isotropic carbon–epoxy laminates it is concluded that the model predictions are very accurate, resulting in improvements over the traditional strength prediction methods. It also is shown that the proposed finite fracture mechanics model can be used to predict the brittleness of different combinations of materials and geometries.
Stress concentrations in the vicinity of cutouts can often be regarded to be the limiting factor for a whole structure. As a further development of prior research at the Institute of Lightweight Engineering and Polymer Technology, an analytical method for the determination of the whole stress-strain fields in the vicinity of holes in multilayered textile-reinforced composites has been developed, which takes into consideration the influences of a finite outer boundary of the specimen. The analytical method is based on the classical laminate theory and the use of complex-valued potential functions. To account for the shape of the specimen, the method of conformal mappings is applied for the inner boundary, while a combination of boundary collocation and least squares method is used for the outer boundary. The method allows a layer-by-layer analysis of stress concentrations. For the verification of the developed calculation model, extensive experimental and numerical finite-element (FE) studies have been carried out on multilayered GF/PP plates with different laminate layups, notches, and specimen dimensions. The comparison of the experimentally or numerically determined results with the analytically calculated ones shows a very good correlation, of which the numerical studies are presented here for the first time. In a second step, the applicable boundary conditions on the outer boundary have been extended in such a way that varying stress and moment resultants can be applied, so that the calculation method can be used as an analytical sub-model in combination with FE techniques.
A Finite Fracture Mechanics model is proposed to predict the net-tension strength of composite mechanically fastened joints. A modification of coupled stress–energy criteria is suggested to take properly into account the effect of the R-curve in determining the crack length corresponding to the unstable crack propagation. The proposed model works for quasi-isotropic laminates and it requires as input the longitudinal strength and the fracture toughness of the laminate. The predictions obtained show good agreement when compared with the experimental results.
The complex variable method of Lekhnitskii and Savin can be applied for solving stress concentration problems in symmetric laminates. Although significant amount of literature is available on heterogeneous aelotropic plates, the stress concentration problems have not been addressed in it until Becker gave his complex potential method for plate problems with bending extension coupling. The present work is an extension of Becker's solution for elliptical hole problem for unsymmetric laminates. In this paper, it is aimed to bring out a general solution to determine the stress resultants and moments around holes of any shape with a simple mapping function under arbitrary biaxial loading condition.
Adapting the formulation given by Savin for stresses around holes in anisotropic plates under inplane loading, a general solution is obtained to consider an arbitrary shape of hole and arbitrarily oriented uniaxial, biaxial, and shear stresses at infinity as well as uniform tangential force, and uniform pressure around the hole. This is achieved by introducing a general form of mapping function and an arbitrary biaxial loading condition into the boundary conditions. In addition, the basic formulation is extended for multilayered plates. By introducing into the computer program, the constants of the mapping function, the arbitrary biaxial loading factor, the orientation angle and the complex parameters for the laminate, the stresses around the hole can be easily obtained. This solution is useful to study the effect of the above parameters on stress distribution. To illustrate the usefulness of the solution, results are obtained for various shapes of holes, loading conditions and laminate geometry. Failure strength is determined using different failure criteria. It is noted that the stress distribution around a given a shape of hole depends on the combined effect of hole geometry, type of loading and laminate geometry.
An analytical procedure for the determination of stress concentrations and strength pre diction of finite composite laminates with elliptical holes is proposed. The Airy stress function expressed in terms of full Laurent's series and a complex variable technique in conjunction with the least-squares "boundary collocation method" are employed in this study. Laminate strength is predicted by using the concept of characteristic curve and the Yamada-Sun failure criterion. Numerical solutions for various hole sizes, layups, material properties, and loading conditions are shown in graphical form and comparisons are made with some other available results.
The finite-width correction factors for anisotropic and orthotropic plates containing an elliptical opening are presented in a tractable and closed-form solution. Examining with experimental data and finite element solutions shows that the present theory is very ac curate for a broad range of opening-to-width ratio, 2a/W, and opening aspect ratio, a/b. The application of isotropic finite-width correction factors to estimate the anisotropic or orthotropic finite-width correction factors can cause significant error in many cases. The sensitivity of the finite-width correction factors as a function of the opening aspect ratio is also discussed.
Two related criteria based on stress distribution are presented for predicting the uniaxial tensile strength of laminated composites containing through the thickness discontinuities of a general shape. The criteria result in two parameter (unnotched tensile strength and a characteristic dimension) models which are capable of predicting observed discontinuity size effects without resorting to classical concepts of linear elastic fracture mechanics. As a direct consequence of the stress criteria, however, a relationship between Mode I fracture toughness and unnotched laminate tensile strength is determined. Limited comparison of theory to experimental data for circular holes and straight cracks yields good results. The simplicity of the analytical approach coupled with its generality make it of practical value to the designer.
Within the framework of linear-elastic classical laminated plate theory, the problem of an elliptical hole in an infinitely extended unsymmetric laminate is treated. For the underlying non-symmetric layup arbitrary bending extension coupling is admitted and is taken into account by means of a new complex potential approach. The corresponding analytical solution is given for the case of homogeneous in-plane and bending loading of the laminate. The derived solution describes all essential plate quantities in any vicinity of the elliptical hole and it reveals interesting features of the considered bending extension coupling.Im Rahmen der linear-elastischen klassischen Laminattheorie wird das Problem eines ellipsenfrmigen Loches im unendlich ausgedehnten unsymmetrischen Laminat behandelt. Dabei wird ein nicht-symmetrischer Lagenaufbau mit beliebiger Biege-Dehn-Kopplung zugelassen und mittels einer neuen komplexen Methode fr den Fall homogener Membranund Biegebelastung die entsprechende analytische Lsung hergeleitet. Die erstellte Lsung beschreibt alle wesentlichen Plattengren in einer beliebigen Umgebung des elliptischen Loches und zeigt interessante Eigenheiten der jeweiligen Biege-Dehn-Kopplung.
The strain concentrations of orthotropic composite laminates containing a circular hole and subject to tensile loading were
measured experimentally using strain gages. Then the stress concentrations were calculated using the strain distributions
in the initial region of the stress-strain curve before any microdamages were developed. The graphite/epoxy AS4/3502 [O2/±45]2s
and [45]4s
were chosen to represent fiber-dominated and matrix-dominated laminates, respectively. Several combinations of hole-diameter/plate-width
ratio were designed to show the width effect. The conditions of the laminates, after the holes were drilled, were examined
using X-ray techniques. Good correlation was obtained between theory and experimental result using specimens in good condition
(without machining damages). A procedure for accurately determining the strain and stress concentrations is given.
A solution method for stress concentration problems of fibre- and textile-reinforced multilayered composites with account
of the influence of a circular or elliptical cut-out and of the finite outer boundary of a composite plate is presented. The
method is based on complex-valued displacement functions and conformal mappings in combination with the boundary collocation
and least squares methods. This allows a layer-by-layer calculation of full stress, strain, and displacement fields in a generally
multilayered anisotropic plate. To verify the calculation model, extensive experimental studies have been carried out. For
all the combinations of multilayered GF/PP plates, laminate lay-ups, and notch and specimen dimensions investigated so far,
a very good agreement between the analytical calculations and experimental results is found to exist.
Keywordsreinforced multilayered composites-stress concentrations-notch-finite boundary
Notch-induced stress concentrations in anisotropic composite materials depend on their directional material properties, especially
for uniaxially reinforced composites with high-modulus fibres. The design of notched high-performance composites requires
therefore a special proof of their notched strength, which includes the structural parameters of the fibre/matrix combination,
fibre orientation and layer arrangement. The assessment of the effects of the finite outer boundary is of practical importance
when dimensioning critical notched regions. An anisotropic plate with finite dimensions and a hole in its center will be used
here to model stress concentrations. The calculation is based on conformal mappings combined with complex-valued stress functions.
The outer boundary is described using point-matching and the least-squares method. The solutions are conducive to the assessment
of the essential influencing factors of material properties, geometry and loads. Notched finite plates made of fibre/matrix
composites, mainly carbon-fibre reinforced polymers, will be presented as illustrations.
Both energy and stress criteria are necessary conditions for fracture but neither one nor the other are sufficient. Experiments by Parvizi et al. on transverse cracking in cross-ply laminates corroborate this assumption. Thanks to the singularity at the tip of the notch, the incremental form of the energy criterion gives a lower bound of admissible crack lengths. On the contrary, the stress criterion leads to an upper bound. The consistency between these two conditions provides a general form of a criterion for crack nucleation. It enjoys the desirable property of coinciding with the usual Griffith criterion to study the crack growth and with the stress criterion for the uniform traction along a straight edge. Comparisons with experiments carried out on homogeneous notched materials and on bimaterial structures show a good agreement.
This paper presents a new methodology to predict the onset of damage, final failure and failure mode of mechanically fastened joints in composite laminates. The stress distribution at each ply is obtained using semi-analytical or numerical methods. The elastic limit of the joint is predicted using the ply strengths and stress distribution in failure criteria. Final failure and failure mode are predicted using point or average stress models. Standardized procedures to measure the characteristic distances used in the point or average stress models are proposed. The methodology proposed is applicable in double-shear joints using quasi-isotropic laminates. The predictions are compared with experimental data obtained in pin- and bolt-loaded joints, and the results indicate that the methodology proposed can accurately and effectively predict ultimate failure loads as well as failure modes in composite bolted joints.
The aim of the present paper is to introduce a new failure criterion in the framework of Finite Fracture Mechanics. Criteria assuming that failure of quasi-brittle materials is affected by stress or energy flux acting on a finite distance in front of the crack tip are widely used inside the scientific community. Generally, this distance is assumed to be small compared to a characteristic size of the structure, i.e. to any length describing the macroscale. A key point of the present paper is to analyse what happens if the smallness assumption does not hold true. The proposed approach relies on the assumption that the finite distance is not a material constant but a structural parameter. Its value is determined by a condition of consistency of both energetic and stress approaches. The model is general. In order to check its soundness, an application to the strength prediction for three point bending tests of various relative crack depths and of different sizes is performed. It is seen that, for the un-notched specimens, the present model predicts the same trend as the Multi-Fractal Scaling Law (MFSL). Finally, a comparison with experimental data available in the literature on high strength concrete three point bending specimens is performed, showing an excellent agreement. It is remarkable to observe that the method presented herein is able to provide the fracture toughness using test data from un-notched specimens, as long as the range of specimen sizes is broad enough.
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