Finite-size Volume Research Articles

Fatigue Strength Reduction Factors Based on Strain Energy Density Applied to Sharp and Blunt Notches under Multiaxial Loading

Real mechanical assemblies favor the initiation and propagation of fatigue cracks due to stress concentration phenomena arising from the geometrical features such as notches, corners, holes, welding toes, etc. Classical fatigue analysis of notched specimens is done using an empirical formula and a fitted fatigue strength reduction factor, which is experimentally expensive and lacks physical scene. In the present paper, a simple and meaningful methodology is proposed to assess notched components against multiaxial fatigue. In this method, by precisely defining a finite-size volume surrounding the fatigue crack initiation site (notch tip), over which the strain energy is averaged, the morphological effect on the process zone is fully addressed. Such a method takes into account the effect of combination of different modes (I, II, III) and the load ratio. In order to implement it for components with a sharp or blunt notch, it is enough to analyze a linear elastic finite element model and to know only the properties of materials obtained from simple uniaxial tests. New relationships for determining an effective (tensile-type) stress and fatigue strength reduction factors are derived for notched specimens. The accuracy of the proposed model is validated by experimental data available in the literature, related to tubular specimens weakened with sharp/blunt notches under combined bending-torsion loading. Such a situation widely appears in equipment used for various branches of industry such as piping, automotive, power plant, drilling, etc.

Dynamics of Microscale Precursors During Brittle Compressive Failure in Carrara Marble

AbstractMicroscale heterogeneities influence failure mechanisms in the crust. To track the microstructural changes in rock samples when loaded until failure, we employed a novel experimental technique that couples dynamic X‐ray microtomography imaging with a triaxial deformation apparatus. We studied the brittle failure of Carrara marble under triaxial compression. Dynamic tomographic data revealed the spatial organization of microfractures and damage increments characterizing the precursory activity toward catastrophic failure. We quantified the emergence of scaling relationships between microstructural parameters, including total damage volume, incremental damage volume, the largest connected microfracture, and the applied differential stress. The total volume of microfractures accumulated from the beginning of the experiment as well as the incremental damage showed power law increase. The growth of the largest connected microfracture was related to differential stress as a power law with divergence at failure. The microfracture volume increments were distributed according to a power law with an upper cutoff that itself spanned the entire volume toward failure. These characteristic features of brittle failure in Carrara marble under compression are in agreement with theoretical models that consider failure as a critical phase transition. We also observed that, very close to failure, several power law relationships broke down, which we interpret to be related to the coalescence of the largest microfractures in a finite size volume. Scaling laws and associated exponents computed from our data are compared with predictions made from theoretical and numerical models. Our results show that precursors of macroscopic brittle failure in Carrara marble follow predictable trends.

Study of stress–strain state of elastic body with hyperbolic notch

The paper considers elastic stress distributions in infinite space with hyperbolic notch when normal or tangential stresses are given on the boundary of notch. The work considers plane deformation. So, exact (analytical) solution of two-dimensional boundary value problems of elasticity in the domain with hyperbolic boundary in the elliptic coordinate system is constructed using the method of separation of variables. The stress–strain state of a homogeneous isotropic infinite body with a hyperbolic cut is studied when there are non-homogeneous (nonzero) boundary conditions given on the hyperbolic cut. Finally, the numerical simulation is performed to the stress and displacement distributions over a finite size volume surrounding the notch and relevant graphs for the numerical results of some test problems are presented.

Constructal Design of Y-Shaped Conductive Pathways for Cooling a Heat-Generating Body

This paper applies constructal design to obtain numerically the configuration that facilitates the access of the heat that flows through Y-shaped pathways of a high-conductivity material embedded within a square-shaped heat-generating medium of low-conductivity to cooling this finite-size volume. The objective is to minimize the maximal excess of temperature of the whole system, i.e., the hot spots, independent of where they are located. The total volume and the volume of the material of high thermal conductivity are fixed. Results show that there is no universal optimal geometry for the Y-shaped pathways for every value of high conductivity investigated here. For small values of high thermal conductivity material the best shape presented a well defined format of Y. However, for larger values of high thermal conductivity the best geometry tends to a V-shaped (i.e., the length of stem is suppressed and the bifurcated branches penetrates deeply the heat-generating body towards the superior corners). A comparison between the Y-shaped pathway configuration with a simpler I-shaped blade and with X-shaped configuration was also performed. For constant values of area fraction occupied with a high-conductivity material and the ratio between the high thermal conductivity material and low conductivity of the heat-generating body (φ = 0.1 and = 100) the Y-shaped pathways performed 46% and 13% better when compared to I-shaped and X-shaped pathway configuration, respectively. The best thermal performance is obtained when the highest temperatures (hot spots) are better distributed in the temperature field, i.e., according to the constructal principle of optimal distribution of imperfections.

Constructal design of non-uniform X-shaped conductive pathways for cooling

This paper applies constructal design to discover the configuration that facilitates the access of the heat that flows through non-uniform X-shaped pathways of high-conductivity material embedded within a square-shaped heat-generating medium of low-conductivity to cooling this finite-size volume. The objective is to minimize the maximal excess of temperature of the whole system, i.e. the hot spots, independent of where they are located. The total volume and the volume of the material of high thermal conductivity are fixed, but the lengths of the pathways and the angles between the pathways can vary. The configuration was optimized for four degrees of freedom: the two angles between the pathways, and the two ratios between the lengths. It was found numerically that the performance of the non-uniform X-shaped pathways is approximately 10% better than the performance of the uniform X-shaped pathways, i.e. X-shaped with equal lengths and thicknesses. When compared to the configuration calculated for a simpler I-shaped blade (i.e. a single pathway of high thermal conductivity material beginning in the isothermal wall and ending of such a way that there is spacing between the tip of the pathway and the insulated wall) the non-uniform X-shaped pathways configuration performs 56% better.

Constructal design of X-shaped conductive pathways for cooling a heat-generating body

This paper applies constructal design to discover the configuration that facilitates the access of the heat that flows through X-shaped pathways of high-conductivity material embedded within a square-shaped heat-generating medium of low-conductivity to cooling this finite-size volume. The objective is to minimize the maximal excess of temperature of the whole system, i.e. the hot spots, independent of where they are located. The total volume and the volume of the material of high thermal conductivity are fixed, but the lengths can vary. It was found numerically that the performance of X-shaped pathways is approximately the same as that the one calculated for a simpler I-shaped blade (i.e. a single pathway of high thermal conductivity material beginning in the heat sink and ending of such a way that there is spacing between the tip of the pathway and the insulated wall) for small values of the high thermal conductivity material and area fraction. However, the X-shaped conductive pathway configuration performs approximately 51% better when compared to I-shaped pathway configuration for larger values of the high-conductivity material and area fraction.

Topology Optimization Using the SIMP Method for Multiobjective Conductive Problems

The efficient cooling of a finite-size volume generating heat, including adiabatic boundary conditions with the exception of a small heat sink, poses the problem of optimal allocation of high-conductivity material. Among the structural optimization methods, this article couples solid isotropic material with penalization parametrization (SIMP) with an aggregated objective function approach (AOF) to tackle this topology optimization problem through a multiobjective strategy. Both average and variance temperature-reduction problems is solved by the identification of Pareto fronts, which are highly dependent on the quantity of the high-conductivity material. This study also underlines the link between the sensitivity analysis of both objective functions, which is required by the method of moving asymptotes (MMA). Furthermore, additional calculations have been done to include variations in heat-generation rate between two conductive materials by means of an additional penalization strategy. The main conclusion deals with the possibility of finding an acceptable trade-off between average and variance objective functions thanks to the convex shape of Pareto frontiers.

Various estimates of Representative Volume Element sizes based on a statistical analysis of the apparent behavior of random linear composites

This article aims at proposing various estimates of the size of the Representative Volume Element (RVE) of random linear elastic matrix–inclusion composites. These estimates are derived from the computation of the apparent behavior of finite size volume elements (VE) by a new procedure presented in [18] by Salmi et al. (2012) and briefly recalled. Two different points of view to define an RVE are considered: the RVE is defined as being the smallest VE required either to evaluate numerically the considered effective property of the composite by appropriate statistical averaging of apparent ones, or to be allowed to replace any instance of the heterogeneous material by a unique homogeneous equivalent one in structural mechanics problems. In order to introduce the fluctuations of the apparent properties within such definitions of the RVE size, we first study the statistics of the apparent properties. Then, relying on the results of this statistical study, several proposals of RVE criteria are presented and applied to random linear elastic fiber–matrix composites for several contrasts and inclusion (or pore) volume fractions.

A new perspective of constructal networks cooling a finite-size volume generating heat

Constructal theory has been used in several scientific fields since it has been established a decade ago. It mainly treats of the “area to point” flow problem, which has been first-written in the frame of the cooling of a finite-size volume generating heat. This problem, widely discussed in literature, has been optimized thanks to constructal theory formulated by Bejan, with a deterministic approach. However, some of the physical and mathematical assumptions made to simplify the problem are questionable, especially the ones regarding the high-conductivity material constituting the tree-shaped structure. In this paper, the performances of constructal designs are evaluated using a finite volume method, which allows making a comparison between different constructal geometries. An algorithm has been developed which has the ability for restraining constructal networks inside a finite-size area, without overconstraining the structure. All things considered, this paper examines constructal theory from a new point of view thanks to analytical and numerical investigations, in the frame of a finite-size volume generating heat.

A review of the volume-based strain energy density approach applied to V-notches and welded structures

A large bulk of experimental data from static tests of sharp and blunt V-notches and from fatigue tests of welded joints are presented in an unified way by using the mean value of the Strain Energy Density (SED) over a given finite-size volume surrounding the highly stressed regions. When the notch is blunt, the control area assumes a crescent shape and R 0 is its width as measured along the notch bisector line. In plane problems, when cracks or pointed V-notches are considered, the volume becomes a circle or a circular sector, respectively. The radius R 0 depends on material fracture toughness, ultimate tensile strength and Poisson’s ratio in the case of static loads; it depends on the fatigue strength Δ σ A of the butt ground welded joints and the Notch Stress Intensity Factor (NSIF) range ΔK 1 in the case of welded joints under high cycle fatigue loading (with Δ σ A and ΔK 1 valid for 5 × 10 6 cycles). Dealing with welded joints characterised by a plate thickness greater than 6 mm, the final synthesis based on SED summarises nine hundred data taken from the literature while a new synthesis from spot-welded joints under tension and shear loading, characterised by a limited thickness of the main plate, is presented here for the first time (more than two hundred data). Dealing with static tests, about one thousand experimental data as taken from the recent literature are involved in the synthesis. The strong variability of the non-dimensional radius R/ R 0, ranging from about zero to about 1000, makes the check of the approach based on the mean value of the SED severe.

The volume-based Strain Energy Density approach applied to static and fatigue strength assessments of notched and welded structures

A large bulk of experimental data from static tests of sharp and blunt V-notches and from fatigue tests of welded joints are presented in an unified way by using the mean value of the Strain Energy Density (SED) over a given finite-size volume surrounding the highly stressed regions. When the notch is blunt, the control area assumes a crescent shape and R0 is its width as measured along the notch bisector line. In plane problems, when cracks or pointed V-notches are considered, the volume becomes a circle or a circular sector, respectively, with R0 being their radius. R0 depends on material fracture toughness, ultimate tensile strength and Poisson’s ratio in the case of static loads; it depends on the fatigue strength ΔσA of the butt ground welded joints and the Notch Stress Intensity Factor (NSIF) range ΔK1 in the case of welded joints under high cycle fatigue loading (with ΔσA and ΔK1 valid for 5×106 cycles). Dealing with static tests, about nine hundred experimental data as taken from the recent literature are involved in the synthesis. The strong variability of the non-dimensional radius R/R0, ranging from about zero to about 1000, makes the check of the approach based on the mean value of the SED severe. In parallel, dealing with welded joints, nine hundred experimental data are here summarised in terms of the local SED.

Elastic stress distributions for hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings

Closed-form solutions are developed for the stress fields induced by circumferential hyperbolic and parabolic notches in axisymmetric shafts under torsion and uniform antiplane shear loading. The boundary value problem is formulated by using complex potential functions and two different coordinate systems, providing two classes of solutions. It is also demonstrated that some solutions of linear elastic fracture and notch mechanics reported in the literature can be derived as special cases of the general solutions proposed herein. Finally the analytical frame is used to link the Mode III notch stress intensity factor to the maximum shear stress at the notch tip, as well as to give closed-form expressions for the strain energy averaged over a finite size volume surrounding the notch root.

Some Expressions for the Strain Energy in a Finite Volume Surrounding the Root of Blunt V-notches

The paper gives some closed form expressions for the strain energy averaged in a finite size volume surrounding the root of blunt V-shaped notches under Mode I loading. The control volume, reminiscent of Neuber’s concept of elementary structural volumes, is thought of as dependent on the ultimate tensile strength and the fracture toughness KIC in the case of brittle or quasi-brittle materials subjected to static loads. Expressions for strain energy density under plane strain conditions and Mode I loading have been derived from an analytical frame recently reported in the literature, which matches Williams and Creager-Paris’ solutions in the particular cases of plates weakened by sharp V-notches or blunt cracks (U-notches), respectively. In order to validate a local-strain-energy based approach, a well-documented set of experimental data recently reported in this journal by Gomez and Elices has been used. Data refer to blunt and sharp V-specimens of PMMA subjected to static tension loads and characterised by a large variability of notch root radius (from 0 to 4.0 mm) and notch angle (from 0° to 150°). Critical loads obtained experimentally have been compared with the theoretical ones, estimated here by keeping constant the mean value of the strain energy in a well-defined small size volume.

Constructal Trees of Convective Fins

This paper extends to the field of convective heat transfer the constructal theory of optimizing the access of a current that flows between one point and a finite-size volume, when the volume size is constrained. The volume is bathed by a uniform stream. A small amount of high-conductivity fin material is distributed optimally through the volume, and makes the connection between the volume and one point (fin root) on its boundary. The optimization proceeds in a series of volume subsystems of increasing sizes (elemental volume, first construct, second construct). The shape of the volume and the relative thicknesses of the fins are optimized at each level of assembly. The optimized structure emerges as a tree of fins in which every geometric detail is a result of minimizing the thermal resistance between the finite-size volume and the root point (source, sink). Convection occurs in the interstitial spaces of the tree. The paper shows that several of the geometric details of the optimized structure are robust, i.e., relatively insensitive to changes in other design parameters. The paper concludes with a discussion of constructal theory and the relevance of the optimized tree structures to predicting natural self-organization and self-optimization.

Constructal Three-Dimensional Trees for Conduction Between a Volume and One Point

This paper extends to three-dimensional heat conduction the geometric “constructal” method of minimizing the overall thermal resistance between a finite-size volume and a small heat sink. The volume contains (i) low-conductivity material that generates heat at every point, and (ii) a small amount of high-conductivity material that must be distributed optimally in space. The given volume is covered in a sequence of building blocks (volume sizes) that starts with the smallest volume element, and continues toward larger assemblies. It is shown that the overall shape of each building block can be optimized for minimal volume-to-point resistance. The relative thicknesses of the high-conductivity paths can also be optimized. These optima are developed analytically and numerically for the smallest elemental volume and the first assembly. The high-conductivity paths form a tree network that is completely deterministic.

Constructal tree networks for the time-dependent discharge of a finite-size volume to one point

This paper shows that the time needed to discharge a volume to a concentrated sink can be minimized by making appropriate changes in the geometry of the flow path. The time-dependent flow of heat between a volume and one point is chosen for illustration, however, the same geometric optimization method (the constructal principle) holds for other transport processes (fluid flow, mass transfer, conduction of electricity). There are two classes of geometric degrees of freedom in designing the flow path: the external shape of the volume, and the distribution (amount, location, orientation) of high-conductivity inserts that facilitate the volumetric collection of the discharge. The optimization of flow path geometry is executed in a sequence of steps that starts with the smallest volume elements and proceeds toward larger and more complex volume sizes (first constructs, second constructs, etc.). Every geometric feature is the result of minimizing the time of discharge, or the resistance in volume-to-point flow. The innermost details of the structure have only a minor effect on the minimized time of discharge. The high-conductivity inserts come together into a tree-network pattern which is the result of a completely deterministic principle. The interstices are equally important in this optimal design, as they are occupied by the low-conductivity material in which the energy charge was stored initially. The paper concludes with a discussion of the relevance on this deterministic principle—the constructal law—to predicting structure in natural flow, and to understanding why the geometry of nature is not fractal.

Streets tree networks and urban growth: Optimal geometry for quickest access between a finite-size volume and one point

The geometric form of the tree network is deduced from a single mechanism. The discovery that the shape of a heat-generating volume can be optimized to minimize the thermal resistance between the volume and a point heat sink, is used to solve the kinematics problem of minimizing the time of travel between a volume (or area) and one point. The optimal path is constructed by covering the volume with a sequence of volume sizes (building blocks), which starts with the smallest size and continues with stepwise larger sizes (assemblies). Optimized in each building block is the overall shape and the angle between constituents. The speed of travel may vary from one assembly size to the next, however, the lowest speed is used to reach the infinity of points located in the smallest volume elements. The volume-to-point path that results is a tree network. A single design principle – the geometric optimization of volume-to-point access – determines all the features of the tree network.

Deterministic Tree Networks for Fluid Flow: Geometry for Minimal Flow Resistance Between a Volume and One Point

The function of many natural flow systems is to connect by a fluid flow a finite-size volume and one point. This paper outlines a strategy for constructing the architecture of the volume-to-point path such that the flow resistance is minimal (constructal theory1). The given volume is viewed as an assembly of volume elements of various sizes. The main discovery is that the shape of each element can be optimized such that the elemental volume-to-point flow resistance is minimal. This optimization principle applies at every volume scale. The smallest volume element contains a fluid saturated porous medium with Darcy flow, which is collected by and channeled through a high permeability path (e.g., fissure) to one point on the element boundary. The geometric optimization is repeated for larger volume elements, which are constructs (assemblies) of optimized smaller volumes. The flow integrated over each new assembly is channeled through a high-permeability path to a point on the side of the assembly. One remarkable feature of the emerging minimal-resistance flow path is that the high-permeability channels of the various volume elements form a tree network which is completely deterministic. The interstices of the network are filled with low permeability porous medium. The method is extended to applications where the high-permeability paths are empty spaces (e.g., parallel-plate channels). It is shown that when the total void volume is constrained it can be distributed optimally among the volume elements to further decrease the overall flow resistance.

Constructal tree network for fluid flow between a finite-size volume and one source or sink

The ‘constructal theory’ of formation of structure in nature is extended to fluid-flow systems. The fluid flow path between one point and a finite-size volume (an infinite number of points) is optimized by minimizing the overall flow resistance when the flow rate and the duct volume are fixed. The solution is constructed as a sequence of optimization and organization steps. The sequence has a definite time direction: it begins with the smallest building block (elemental system, with flow by volumetric diffusion), and proceeds toward larger building blocks (assemblies, with flow collected in ducts). Optimized at each level are the shape of the assembly, the number of constituents (ie, smaller assemblies), and the distribution of the duct volume. It is shown that the ducts of the optimized assemblies form a tree-like structure, in which every architectural detail is deterministic. It is also shown that the structure cannot be determined when the time direction is reversed, from large elements toward smaller elements. The general importance of the constructal law (access-optimization principle) in physics, biology and economics is discussed.

Constructal tree networks for heat transfer

This paper addresses the fundamental problem of how to connect a heat generating volume to a point heat sink by using a finite amount of high-conductivity material that can be distributed through the volume. The problem is one of optimizing the access (or minimizing the thermal resistance) between a finite-size volume and one point. The solution is constructed by covering the volume with a sequence of building blocks, which proceeds toward larger sizes (assemblies), hence, the “constructal” name for this approach. Optimized numerically at each stage are geometric features such as the overall shape of the building block, its number of constituents, and the internal distribution of high-conductivity inserts. It is shown that in the optimal design, the high-conductivity material has a distribution with the shape of a tree. Every aspect of the tree architecture is deterministic: the shapes of the largest assembly and all its constituents, the number of branches at each level of assembly, the relative position of building blocks in each assembly, and the relative thicknesses of successive branches. The finer, innermost details of the tree architecture (e.g., the branching angle) have a negligible effect on the overall thermal resistance. The main conclusion is that the structure, working mechanism, and minimal resistance of the tree network can be obtained deterministically, and that the constrained optimization of access routes accounts for the macroscopic structure in nature.