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 K_IC 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 Gómez 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.     

Effect of material plasticity on fatigue crack propagation under complex stress state

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. However, this fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, a modification has been made to G _max criterion to implement the consideration of the plastic strain energy. This criterion is extended to study the fatigue crack growth characteristics of mixed mode cracks in steel pipes. To predict crack propagation due to fatigue loads, a new elasto-plastic energy model is presented. This new model includes the effects of material properties like strain hardening exponent n, yield strength σ_y and fracture toughness and stress intensity factor ranges. The results obtained are compared with those obtained using the commonly employed crack growth law and the experimental data.     

Mixed-mode fracture load prediction in lead-free solder joints

Double cantilever beam (DCB) fracture specimens were made by joining copper bars with both continuous and discrete SAC305 solder layers of different lengths under standard surface mount (SMT) processing conditions. The specimens were then fractured under mode-I and various mixed-mode loading conditions. The loads corresponding to crack initiation in the continuous joints were used to calculate the critical strain energy release rate, J_ci, at the various mode ratios using elastic–plastic finite element analysis (FEA). It was found that the J_ci from the continuous joint DCBs provided a lower bound strength prediction for discrete 2 mm and 5 mm long joints at the various mode ratios. Additionally, these J_ci values calculated from FEA using the measured fracture loads agreed reasonably with J_ci estimated from measured crack opening displacements at crack initiation in both the continuous and discrete joints. Therefore, the critical strain energy release rate as a function of the mode ratio of loading is a promising fracture criterion that can be used to predict the strength of solder joints of arbitrary geometry subject to combined tensile and shear loads.     

Failure mechanics in ternary composites of polypropylene with inorganic fillers and elastomer inclusions

The effects of phase morphology, interfacial adhesion and filler particle shape and volume fraction on the fracture toughness of polypropylene (PP) filled with CaCO₃ or Mg(OH)₂ and ethylene-propylene elastomer (EPR) were investigated. Separation of the inorganic filler and elastomer particles was achieved using maleic-anhydride-grafted PP (MPP) to enhance the inorganic filler-matrix adhesion. Encapsulation of the rigid filler by the elastomer was achieved by using maleic-anhydride-grafted EPR (MEPR) to increase the inorganic filler-elastomer adhesion. The two limiting morphologies differed significantly in fracture toughness under impact loading at the same material composition. A model for a mixed mode of failure, accounting for the plane strain and plane stress contributions to the strain energy release rate,G _c, was used to predict the upper and lower limits forG _c for the two limiting morphologies over an interval of elastomer volume fractions,v _e, from 0–0.2 at a constant filler volume fraction,V _f = 0.3, and over the filler volume fraction from 0–0.4 at constant EPR content. The role of material yield strength in controlling fracture toughness has been described successfully using Irwin's analysis of plastic zone size. The presence of elastomer enhances both the critical strain energy release rate for crack initiation,G _c, and the resistance to crack propagation as expressed by Charpy notched impact strength for the two limiting morphologies. Satisfactory agreement was found between the experimental data and predictions of upper and lowerG _c limits.     

Experimental and theoretical study on large ductile transverse deformations of rectangular plates subjected to shock load due to gas mixture detonation

The main purpose of this study is to show that metal plate forming by direct application of gas mixture detonation loads can be considered as an alternative high‐velocity forming method for structures instead of a conventional one. Therefore, in this investigation, a series of experimental tests have been conducted on aluminium alloy and mild steel plates with different thicknesses to examine large ductile transverse deformations of rectangular plates with clamped edge conditions subjected to gas mixture detonation loading. The main aim of the experimental section is to investigate the effects of predetonation pressures of acetylene (C₂H₂) and oxygen (O₂) gasses and different mixture ratios on the dynamic response of specimens. The permanent deflections of plates have widely varied from 21.66 up to 56.31 mm. In theoretical analysis, according to an upper bound solution and energy method, theoretical models have been presented by assuming a zero‐order Bessel function of the first kind in the x and y directions for a transverse displacement profile to predict permanent deflections. To account for material strain rate sensitivity, a Cowper–Symonds model has been used, whereas the material coefficients of this equation are constant values or functions of plate thickness. A comparison of the present models with Jones' theoretical model shows that a good agreement with experimental results can be obtained when constant values are used for material coefficients in the Cowper–Symonds equation.     

Multiaxial fatigue of V‐notched steel specimens: a non‐conventional application of the local energy method

The paper deals with the multi‐axial fatigue strength of notched specimens made of 39NiCrMo3 hardened and tempered steel. Circumferentially V‐notched specimens were subjected to combined tension and torsion loading, both in‐phase and out‐of‐phase, under two nominal load ratios, R=?1 and R= 0, also taking into account the influence of the biaxiality ratio, λ=τ_a/σ_a. The notch geometry of all axi‐symmetric specimens was a notch tip radius of 0.1 mm, a notch depth of 4 mm, an included V‐notch angle of 90° and a net section diameter of 12 mm. The results from multi‐axial tests are discussed together with those obtained under pure tension and pure torsion loading on plain and notched specimens. Furthermore the fracture surfaces are examined and the size of non‐propagating cracks measured from some run‐out specimens at 5 million cycles. Finally, all results are presented in terms of the local strain energy density averaged in a given control volume close to the V‐notch tip. The control volume is found to be dependent on the loading mode.     

Stacking sequence effects on fracture of notched carbon fibre/epoxy composites

The purpose of this study was to investigate the ability of the so-called damage zone model (DZM) to predict the influence of stacking sequence on the strength of notched carbon fibre/epoxy composites. The DZM is in essence based on the unnotched tensile strength, σ₀, and the apparent fracture energy, G_c^*, and the damage zone is modelled as a crack with cohesive forces acting on the crack surfaces. The DZM predicts fracture loads for three-point bend (TPB) specimens and specimens with circular holes quite accurately. As an attempt to explain the difference in strengths, the damage zone extension in the TPB specimens with different stacking sequence was examined.     

Fracture investigation of U‐notch made of tungsten–copper functionally graded materials by means of strain energy density

The averaged strain energy density over a well‐defined control volume was employed to assess the fracture of U‐notched specimens made of tungsten–copper functionally graded materials under prevalent mode II loading. The boundary of control volume was evaluated by using a numerical method. Power law function was employed to describe the mechanical properties (elasticity modulus, Poisson's ratio, fracture toughness and ultimate tensile stress) through the specimen width. The effect of notch tip radius and notch depth on notch stress intensity factors and mode mixity parameter χ were assessed. In addition, a comparison based on fracture load between functionally graded and homogeneous W–Cu was made. Furthermore, in this research, it was shown that the mean value of the strain energy density over the control volume can be accurately determined using coarse meshes for functionally graded materials.     

Comparative analysis of fatigue life predictions in central notched specimens of Al–Mg–Si alloys

The fatigue behaviour of an Al–Mg–Si alloy was studied using notched specimens. Fatigue tests were conducted at two stress ratios R= 0 and R= 0.4 on thin plates with a central hole. Constant and block variable loading amplitudes were applied to the specimens using a servo‐hydraulic machine. The applicability of the local strain approach method to the prediction of the fatigue life was investigated for this type of discontinuity. Two methods, the equivalent strain energy density approach and a modified stress–strain intensity field approach, were used to predict the fatigue strength. For the second one an elastic–plastic finite element analysis was carried out in order to obtain the local strain and stress distributions near the notch root. Based on Miner's rule an equivalent stress was used to correlate the fatigue lives for the variable amplitude histories. The experimental results were compared with the predicted results obtained by the two methods investigated and better agreement was found with the stress–strain field intensity approach, while the strain energy approach gave more conservative results. Miner's rule gives a good correlation between the variable amplitude and constant amplitude results.     

Energy density zone model and fatigue life prediction considering microscopic effects

An energy density zone (EDZ) model is developed for the prediction of fatigue life. The microscopic effects can be involved in the EDZ model. Three scale transitional functions in the model are utilized to describe the trans‐scale behaviours of fatigue failure from micro‐scale to macro‐scale. Fatigue failure behaviours of a low‐alloy and ultra‐high‐strength steel material (i.e. 40CrNi2Si2MoVA steel) is investigated. Two fatigue parameters in the model are determined from the experimental S–N curves for the smooth cylinder specimens (the stress concentration factor, SCF, K_t = 1). Then, fatigue lives of notched specimens with SCFs K_t = 2 and K_t = 3 are predicted respectively by the proposed model. The predicted S–N curves are satisfactory in comparison with the experimental results. Scatter of the fatigue test data can be depicted when the microscopic effects are considered. Influences of microscopic effects on the fatigue behaviours are explored by means of numerical simulations.     

An FEM study on crack tip blunting in ductile fracture initiation

Ductile fracture is initiated by void nucleation at a characteristic distance (I_c) from the crack tip and propagated by void growth followed by coalescence with the tip. The earlier concepts expressed I_c in terms of grain size or inter-particle distance because grain and particle boundaries form potential sites for void nucleation. However, Srinivas et al. (1994) observed nucleation of such voids even inside the crack tip grains in a nominally particle free Armco iron. In an attempt to achieve a unified understanding of these observations, typical crack-tip blunting prior to ductile fracture in a standard C(T) specimen (Mode I) was studied using a finite element method (FEM) supporting large elasto-plastic deformation and material rotation. Using a set of experimental data on Armco iron specimens of different grain sizes, it is shown that none of the locations of the maxima of the parameters stress, strain and strain energy density correspond to I_c. Nevertheless, the size of the zone of intense plastic deformation, as calculated from the strain energy density distribution ahead of the crack tip in the crack plane, compares well with the experimentally measured I_c. The integral of the strain energy density variation from the crack tip to the location of void nucleation is found to be linearly proportional to J_IC. Using this result, an expression is arrived at relating I_c to J_IC and further extended to CTOD_c.     

Microstructural aspects and modeling of failure in naturally occurring porous composites

Results from an experimental investigation on the compression behavior of balsa wood are presented. Specimens with varying densities, ranging from 55 to 380 kg/m³, are loaded in the grain (fiber, cell) direction using a screw-driven material testing system at a strain rate of 10⁻³ s⁻¹. The results indicate that compressive strength of balsa wood increases with increasing density. Post-test scanning electron microscopy is used to identify the failure modes. The failure of low-density specimens is governed by elastic and/or plastic buckling, while kink band formation and end-cap collapse dominate in higher density balsa specimens. Based on the experimental results and observations, several analytical models are proposed to predict the compressive failure strength of balsa wood under uniaxial loading conditions.     

A synthesis of the push‐pull fatigue behaviour of plain and notched stainless steel specimens by using the specific heat loss

The energy dissipated to the surroundings as heat in a unit volume of material per cycle, Q, was recently proposed as fatigue damage index, and it was successfully applied to rationalise fatigue data obtained by carrying out stress‐controlled and strain‐controlled fatigue tests on AISI 304 L stainless steel plain and hole specimens. In this paper, it is shown that the Q parameter is independent on thermal and mechanical boundary conditions occurring during experiments. After that, additional stress‐controlled fatigue tests on plain and notched specimens characterised by smaller notch tip radii than those tested previously have been performed. Present data have been compared with previous ones, and it was found that all available results can be synthesised in terms of the energy parameter Q into a unique scatter band, independently on the testing conditions (stress‐controlled or strain‐controlled) and on the specimens' geometry (plain or notched). About 100 data were included in the statistical analysis to characterise the energy‐based scatter band of the material. Finally, some limitations of applicability of the experimental technique adopted in the present paper are discussed.     

A method to develop a unified fatigue life prediction model for filled natural rubbers under uniaxial loads

Rubber components are widely used in many fields because of their superior elastic properties. Fatigue failures, commonly encountered in rubber components, however, remain a critical issue. In this study, the effect of strain ratio R on the fatigue life of filled natural rubbers used in automotive mounts is investigated experimentally and numerically. A uniaxial tension/compression fatigue experiment was conducted on dumb‐bell cylindrical rubber specimens subject to loads representing different R ratios. The experimental fatigue data are used to formulate two preliminary fatigue models based on peak strain and strain amplitude as the damage parameters. The deficiencies of these two models in predicting fatigue life over a wide range of R ratios are discussed, and an alternative life prediction model is proposed. The proposed model incorporates the effect of R ratio using an equivalent strain amplitude. It is shown that the proposed model could effectively predict fatigue life over a wide range of R ratios with an improved accuracy.     

Dynamic impact factors of strain hardening UHP-FRC under direct tensile loading at low strain rates

Enhanced matrix packing density and tailored fiber-to-matrix interface bond properties have led to the recent development of ultra-high performance fiber reinforced concrete (UHP-FRC) with improved material tensile performance in terms of strength, ductility and energy absorption capacity. The objective of this research is to experimentally investigate and analyze the uniaxial tensile behavior of UHP-FRC under various strain rates, ranging from 0.0001 to 0.1 1/s. A direct tensile test set up is used. The experimental parameters encompass three types of steel fibers, each in three different volume fractions at four different strain rates resulting in 36 test series. Elastic and strain hardening tensile parameters, such as, cracking stress, elastic and strain hardening modulus, composite tensile strength and strain, energy absorption capacity, and crack spacing of the UHP-FRC specimens, are recorded and analyzed. Explanation of the material’s strain rate sensitivity is mainly based on the inertia effect of matrix micro cracking. Potential contributions of other mechanisms include viscosity of water within nanopores and confinement effects. Dynamic impact factor (DIF) formulas are provided based on the experimental data to illustrate the relationship between DIF and strain rate for UHP-FRC.     

A notch stress intensity approach to assess the multiaxial fatigue strength of welded tube-to-flange joints subjected to combined loadings

Full penetration T butt weld joints between a tube and its flange are considered, subjected to pure bending, pure torsion and a combination of these loading modes. The model treats the weld toe like a sharp V‐notch, in which mode I and mode III stress distributions are combined to give an equivalent notch stress intensity factor (N‐SIF) and assess the high cycle fatigue strength of the welded joints. The N‐SIF‐based approach is then extended to low/medium cycle fatigue, considering fatigue curves for pure bending and pure torsion having the same slope or, alternatively, different slopes. The expression for the equivalent N‐SIF is justified on the basis of the variation of the deviatoric strain energy in a small volume of material surrounding the weld toe. The energy is averaged in a critical volume of radius R_C and given in closed form as a function of the mode I and mode III N‐SIFs. The value of R_C is explicitly referred to high cycle fatigue conditions, the material being modelled as isotropic and linear elastic. R_C is thought of as a material property, independent in principle of the nominal load ratio. To validate the proposal, several experimental data taken from the literature are re‐analysed. Such data were obtained by testing under pure bending, pure torsion and combined bending and torsion, welded joints made of fine‐grained Fe E 460 steel and of age‐hardened AlSi1MgMn aluminium alloy. Under high cycle fatigue conditions the critical radius R_C was found to be close to 0.40 mm for welded joints made of Fe E 460 steel and close to 0.10 mm for those made of AlSi1MgMn alloy. Under low/medium cycle fatigue, the expression for energy has been modified by using directly the experimental slopes of the pure bending and pure torsion fatigue curves.     

Effects of thermal aging on fracture performance of polychloroprene

Measurements of tensile and tearing resistance have been conducted as a function of aging time and aging temperature for a polychloroprene (CR). The steady increase in stiffness indicates that crosslinking is dominant during the aging of CR. In the early stage of aging, tensile strength and tearing energy increase thanks to an optimized balance between the strength enhancement from the crosslink network and the network capability in dissipating energy. Prolonged aging after a characteristic time results in a gradual decrease in strength and tearing energy and this change is more pronounced with higher aging temperatures. The superposition principle between aging time and aging temperature was used to determine the activation energy controlling the change of tearing energy in thermal aging. This value agrees with the activation energy E_a = 91 kJ/mol reported in current literature by monitoring the oxygen consumption rates during the aging of CR, suggesting that the change in tearing energy is controlled by chemical reactions much the same as the change in oxygen consumption dissolved in the material. It was also found that the variation in the strain energy density to break is also controlled by the same activation energy.     

Multiple film cracking in film/substrate systems with residual stresses and unidirectional loading

Multiple film cracking in film/substrate systems is analyzed in the present study. Specifically, the experimental measurements of multiple cracking of SiO _x films of various thicknesses on polyethylene terephthalate substrates are analyzed. The system is subjected to both residual stresses and unidirectional tensile loading. Considering a three- dimensional geometry, an analytical model is developed to derive the stress distribution in the system, and the film-cracking problem is analyzed using both the strength and the energy criteria. Compared to the strength criterion, the energy criterion shows better agreement with the measurements of the crack density versus applied strain relation.     

Constant strain- and stress-rate compressive strength of columnar-grained ice

Unconfined compressive strength of transversely isotropic columnar-grained ice has been investigated for loads applied normal to the longitudinal axis of the columns at the high homologous temperature of 0.96 T _m (T _m is the melting temperature) under truly constant strain and stress rates. A closed-loop, servo-hydraulic test system inside a cold room was used. Both the strain- and stress-rate dependences of upper yield stress can be expressed in terms of power laws. The observed strain-rate dependence of strength was found to be numerically the same as the dependence of viscous-flow rate on stress in constant stress creep tests at the same temperature. It is shown that the strain-rate sensitivity of yield strength compares well with previous results (obtained under constant cross-head rates using a conventional machine) only if the average strain rate to yield is used as the independent variable instead of the conventional nominal strain rate. The paper also discusses the strain and time aspects of the tests. It shows interdependence among values for compressive yield strength, strain rate, failure strain and time very similar to the interdependence among the corresponding values in tensile creep failures in metals, alloys and other polycrystalline materials at high temperatures. It is emphasized that the splitting type of brittle-like premature failure depends on the stiffness of the test system and should not be considered to be a fundamental material property. The concept of failure modulus is proposed for examining the ductile to brittle transition.