Porosity dependence of material elastic moduli

A new equationE =E ₀(1+aP+bP ²)/(1+cP), whereE andE ₀ are Young's moduli at porosityP and zero, respectively, anda, b, c are constants, has been derived. Our theoretical derivation is based on the dependence of sound velocity on the Young's modulus of the material.     

Elastic properties of porous thermosetting polymers

A new equationE =E ₀ (1 –bp) ⁿ has been derived semi-empirically to describe the porosity dependence of elastic properties of thermosetting polymers. The material constantb is defined as a pore distribution geometry factor and the other material constantn is dependent on pore geometry. The equation shows good agreement with the data on porous polyester and epoxy resins.     

Morphologies and mechanical properties of polyblends of polyurethane with poly(4,4′-diphenylsulphone terephthalamide)

Two polyurethane (PU) elastomers were blended physically with various ratios of poly(4,4-diphenylsulphone terephthalamide) (PSA) to form eight PU/PSA polyblends in order to modify their mechanical properties. The stress-strain and stress-relaxation behaviour of PU/PSA polyblends were studied. The measured values of Young's modulus,E, Mooney-Rivlin elastic parameters, C₁ and C₂, relaxation moduli,E(10 s) andE(100 s), as well as relaxation speed,V _r , were used to estimate the effect of semi-rigid PSA molecules on the mechanical behaviour of polyblends. It was found that PU/PSA polyblends still displayed good elastic properties, although PU/PSA polyblends had a dispersed phase structure. At lower of PSA contents (below 10 wt % PSA), PU/PSA polyblends showed improved stress-relaxation properties.     

Open pore description of mechanical properties of ceramics

A dependence of Young's modulus of elasticity on open porosity in ceramics is derived from an open-porosity model, which in the literature, is applied to salinity conductivity and fluid permeability in rocks. A random distribution of grain and pore size is assumed. The relation developed,E(p)=E _o(1–"p)^m, whereE is the modulus of elasticity of the porous ceramic,E _o is the theoretical elastic modulus,p is the porosity andm is an exponent dependent on the tortuosity of the structure of the ceramic, adequately describes the dependence of the modulus of elasticity on porosity. The model is applied to the experimental data from several ceramics such as alumina, silicon nitride, silicon carbide, uranium oxide, rare-earth oxides, and YBa₂Cu₃O_7– superconductor, and the value ofm is obtained for each case. We have shown thatm has a value of nearly 2 for sintered ceramics, unless sintering aids or hot pressing have been used during fabrication of the ceramic. Such additional procedures approximately double the magnitude ofm. On joint appointment with: Materials Laboratory, Physics Department, University of the West Indies, Mona, Kingston 7, Jamaica,     

Ultrasonic evaluation of elastic parameters of sintered powder compacts

The variation of elastic moduli, M, of sintered powder compacts with porosity, p, has been analysed in terms of an equation M = M ₀ (1–p) ⁿ , where M ₀ is the elastic modulus of non-porous material and n is a constant. The variation of ultrasonic velocities has also been described in terms of a similar equation derived from the relations given by physical acoustics theory. It has been shown that the parameter n is related to a stress concentration factor around pores in the material and is dependent on pore geometry and its orientation in the material. The observed variation in moduli and velocities with porosity has been compared with the theoretically predicted values based on self-consistent oblate spheroidal theory.     

Piezoelectricity and polarization studies in unstretched san copolymer films

The piezoelectricity and charge storage in unstretched corona poled poly (styrene-co-acrylonitrile) films have been investigated under different poling conditions. The piezoelectric coefficient d₃₁ has been measured as a function of poling field (E_p) and poling temperature (T_p) and the maximum value of d₃₁ 1.0 pC/N has been obtained at E_p 46 MV/m and T_p = 85 °C. The Frolich equation for dipolar polarization has been applied for quantitative analysis of the thermally stimulated current (TSC). The g-factor = 1.7 has been obtained, suggesting cooperative dipolar relaxation. The Young's modulus has been measured to be 0.45 GPa. The dipolar polarization from the TSC and the elastic modulus of the films have been correlated to the piezoelectricity through the modified equation for d₃₁ for amorphous SAN films. It has been concluded that the mechanisms of the dimensional and the local field effects are involved in the piezoelectric phenomenon.     

Brittle matrices reinforced with polyalkene films of varying elastic moduli

The inclusion of polyalkene films of different moduli in a cement-based matrix has shown the benefits to be gained, in terms of increased stress at a given strain, from the use of films of high elastic modulus. Further, the concept of load-bearing cracks is used to explain the transition region between the limit of proportionality and the bend-over point on the tensile stress-strain curve, which is found to exist with high film modulus composites. This transition region could be an important factor affecting the choice of film to be used in a commercial composite.Nomenclature E _c uncracked composite modulus - E _m matrix modulus - E _f film modulus - V _m matrix volume-fraction - V _t film volume-fraction - V _f(crit) (E _{c mu})/ _fu - A _c cross-sectional area of composite - (E _m V _m/E _fV_f) - _m matrix strain - _mu matrix cracking strain - _mu average matrix cracking strain, (#x03C3;_co)/E _c - _mc strain at end of multiple cracking - _fu ultimate fibre stress - _cu ultimate composite stress - _co average composite cracking stress (assumed at a strain of _mc/2) - S4 81 draw ratio polypropylene film - S8 181 draw ratio polypropylene film - E3H polyethylene film - LOP limit of proportionality (stress at first crack, assumed to be a departure from linearity of the tensile stress-strain curve of a perfectly straight and uniform test specimen. However, this point cannot be reliably determined from the stress-strain curve because of the clamping strains induced in warped specimens) - BOP bend-over point (stress at which the approximately horizontal portion of multiple cracking region commences. The BOP is generally higher than the LOP and is a much more reliable point to determine experimentally than the LOP)     

Comparison of the uniaxial tensile modulus and dynamic shear storage modulus of a filled hydroxy-terminated polybutadiene and GAP propellant

The uniaxial tensile moduli for two filled composite solid propellants, one based on a hydroxy-terminated polybutadiene with an ammonium perchlorate oxidizer and the other on a glycidyl azide polymer with a phase-stabilized ammonium nitrate oxidizer, were measured at various temperatures and over a three decade range of strain rates. Dynamic shear moduli were measured at various strains and temperatures over a three decade range of frequencies. Values of the tensile modulus, E _T/3 (assuming Poisson's ratio equals 0.5), were compared with the dynamic storage modulus, G, for each of the dynamic strain levels investigated. The results demonstrate that it is possible to compare 3G measured at different dynamic strains and frequencies with the incremental tangent moduli obtained at corresponding uniaxial strains from constant strain rate tests on specimens with a JANNAF geometry. The comparisons are most favourable when the concave-up region of the stress-strain curve extends only up to approximately 2.0% strain. It was observed that G obtained at a dynamic shear strain of 2.0% provided the best overall correlation with E _T/3 measured over a range of temperatures and strain rates.     

Fracture mechanism in short fibre reinforced thermoplastic resin composites

The properties of two types of short carbon fibre (CF) reinforced thermoplastic resin composites (CF-PPS and CF-PES-C), such as strength (_y). Young's modulus (E) and fracture toughness (K _1c), have been determined for various volume fractions (V _f) of CF. The results show that the Young's modulus increases linearly with increasingV _f with a Krenchel efficiency factor of 0.05, whereas _y andK _1c increase at first and then peak at a volume fraction of about 0.25. The experimental results are explained using the characteristics of fibre-matrix adhesion deduced from the load-displacement curves and fractography. By using a crack pinning model, the effective crack tensions (T) have been calculated for both composites and they are 57 kJ m^–1 for CF-PPS and 4.2 kJ m^–1 for CF-PES-C. The results indicate that the main contribution to the crack extension originates from localized plastic deformation of the matrix adjacent to the fibre-matrix interface.     

Laser-target technique for determination of elastic constants of glass over a wide temperature range

Optical methods have been developed for high temperature application in the determination of the basic quantities, E, Young's modulus and , Poisson's ratio. The methods are simple and the techniques are inexpensive. The methods presented have been tried on commercially available soda-lime-silica glass and consistent results were obtained.     

Elastic constants of MoSi2 and WSi2 single crystals

Single crystals of MoSi₂ and WSi₂ with a body-centred-tetragonal C 1 1_b structure were fabricated using a floating-zone method. The elastic wave velocity was measured for samples with various orientations using a simple pulse echo method at room temperature, and six elastic stiffness constantsc _ij were calculated. The stiffness constants were a little higher for WSi₂ than for MoSi₂.c ₁₁ andc ₃₃ of these compounds were approximately equal toc ₁₁ of tungsten and molybdenum, respectively, althoughc _ij (i j) was a little higher for these compounds than for molybdenum and tungsten. Young's modulus 1/s ₁₁ was the highest in the <0 0 1> direction, and the lowest in the <1 0 0> direction. The shear modulus 1/s ₆₆ was high on the {0 0 1} plane and independent of shear direction. It was generally low on the close-packed {1 1 0} plane and largely dependent on shear direction. The elastic constants for the polycrystalline materials were estimated fromc _ij ands _ij . Poisson's ratiov was 0.15 for MoSi₂ and for WSi₂, and these values were much lower than for ordinary metals and alloys. The Debye temperature _D was estimated using the elastic-wave velocity of the polycrystalline materials via the elastic constants such as Young's modulus and shear modulus: it was 759 K for MoSi₂ and 625 K for WSi₂.     

Mechanical properties of some plant materials

A study of mechanical properties of some plant materials, particularly vegetable flesh and cultural plant stalks is reported. It is shown that the tensile (compressive) strength, _m, of these and other plant materials is controlled by a relatively close exponential regression relation _m=0.2E ^0.75, where E is Young's modulus (r=0.975). More significant deviations from this relation are explained by the participation of buckling strength in the deformation by compression of the materials consisting of large thin-walled cells filled with air. A marked dependence of Young's modulus and strength of plant tissues on the crude fibre content is also demonstrated.     

Application of homogenization FEM analysis to regular and re-entrant honeycomb structures

Honeycomb structures are widely used in structural applications because of their high strength per density. Re-entrant honeycomb structures with negative Poisson's ratios may be envisaged to have many potential applications. In this study, an homogenization finite element method (FEM) technique developed for the analysis of spatially periodic materials is applied for the analysis of linear elastic responses of the regular and re-entrant honeycomb structures. Young's modulus of the regular honeycomb increased with volume fraction. Poisson's ratio of the regular honeycomb structure decreased from unity as volume fraction increased. The re-entrant honeycomb structure had a negative Poisson's ratio, its value dependent upon the inverted angle of cell ribs. Young's modulus of the re-entrant honeycomb structure decreased as the inverted angle of cell ribs increased. The results are in good agreement with previous analytical results. This homogenization theory is also applicable to three-dimensional foam materials — conventional and re-entrant.Nomenclature b _i Body force - E, E _ijkl Young's modulus, elasticity tensor - E _e Effective Young's modulus - E _ijkl ^H Homogenized elasticity tensor - t _i Traction - u _i , u Displacement - v _i , v Virtual displacement - x _i , x Macroscale coordinate - y _i , y microscale coordinate - Microscopic/macroscopic ratio - Volume fraction - v Poisson's ratio - v_e Effective Poisson's ratio - _ij Stress - _P ^KL Microscale parameter of separation of variables     

Effect of fibre misalignment on fracture behaviour of fibre-reinforced composites

When a matrix crack encounters a fibre that is inclined relative to the direction of crack opening, geometry requires that the fibre flex is bridging between the crack faces. Conversely, the degree of flexing is a function of the crack face separation, as well as of (1) the compliance of the supporting matrix, (2) the crossing angle, (3) the bundle size, and (4) the shear coupling of the fibre to the matrix. At some crack face separation the stress level in the fibre bundle will cause it to fail. Other bundles, differing in size and orientation, will fail at other values of the crack separation. Such bridging contributes significantly to the resistance of the composite to crack propagation and to ultimate failure. The stress on the composite needed to produce a given crack face separation is inferred by analysing the forces and displacements involved. The resulting model computes stress versus crack-opening behaviour, ultimate strengths, and works of failure. Although the crack is assumed to be planar and to extend indefinitely, the model should also be applicable to finite cracks.Glossary of Symbols a radius of fibre bundle - C 2 _f /aE _f - ^* critical failure strain of fibre bundle - _b bending strain in outer fibre of a bundle - _c background strain in composite - _f axial strain in fibre - _s strain in fibre bundle due to fibre stretching = _f - () strain in composite far from crack - E Young's modulus of fibre bundle - E _c Young's modulus of composite - E _f Young's modulus of fibre - E _m Young's modulus of matrix - f() number density per unit area of fibres crossing crack plane in interval to + d - F total force exerted by fibre bundle normal to crack plane - F _s component of fibre stretching force normal to crack plane - F _b component of bending force normal to crack plane - G _m shear modulus of matrix - h crack face opening relative to crack mid-point - h _m matrix contraction contribution to h - h _f fibre deformation contribution to h - h _max crack opening at which bridging stress is a maximum - I moment of inertia of fibre bundle - k fibre stress decay constant in non-slip region - k ₀ force constant characterizing an elastic foundation (see Equation 7) - L exposed length of bridging fibre bundle (see Equation 1a) - L _f half-length of a discontinuous fibre - m, n parameters characterizing degree of misalignment - N number of bundles intersecting a unit area of crack plane - P _b bending force normal to bundle axis at crack midpoint - P _s stretching force parallel to bundle axis in crack opening - Q() distribution function describing the degree of misalignment - s _f fibre axial tensile stress - s _f ^* fibre tensile failure stress - S stress supported by totality of bridging fibre bundles - S _max maximum value of bridging stress - v fibre displacement relative to matrix - v elongation of fibre in crack bridging region - u _coh non-slip contribution to fibre elongation - U fibre elongation due to crack bridging - v overall volume fraction of fibres - v _f volume fraction of bundles - v _m volume fraction matrix between bundles - w transverse deflection of bundle at the crack mid-point - x distance along fibre axis, origin defined by context - X distance between the end of discontinuous fibre and the crack face - X ^* threshold (minimum) value of X that results in fibre failure instead of complete fibre pullout - y displacement of fibre normal to its undeflected axis - Z() area fraction angular weighting function - tensile strain in fibre relative to applied background strain - ^* critical value of to cause fibre/matrix debonding - angle at which a fibre bundle crosses the crack plane - (k ₀/4EI)^1/4, a parameter in cantilever beam analysis - v_m Poisson's ratio of matrix - L (see Equation 9) - shear stress - _* interlaminar shear strength of bundle - _d fibre/matrix interfacial shear strength - _f frictional shear slippage stress at bundle/matrix interface - angular deviation of fibre bundle from mean orientation of all bundles - angle between symmetry axis and crack plane     

The role of thermal contraction stresses associated to inclusions in the formation of the final microstructure of weld metal deposits

The effect of deoxidation products on the formation of the final microstructure of weld deposits has been studied on a ferritic-austenitic stainless steel. Results show that the product phase austenite nucleates on dislocations set up by thermal contraction stresses. The presence of these defects has been ascertained by transmission electron microscopy and calculation confirms the existence of a plastic zone which surrounds the inclusions. This mechanism seems to be more effective than the direct nucleation of the second phase on the inclusion interface and has important implications for the problem of acicular ferrite formation in carbon steels.Nomenclature mean linear coefficient of thermal expansion over the given temperature range - E Young's modulus - Poisson's ratio - Y ₂ yield stress of matrix - R distance from inclusion centre - R ₁ radius of inclusion - R ₂ outer radius of the elastic zone of matrix - R outer radius of the elasto-plastic zone of matrix - T temperature change (increase is positive)     

Mechanical properties in the initial stage of sintering

Silicon nitride-based ceramics with different compositions were sintered in the 60%–90% range of theoretical density. Linear correlations between the apparent density and the modulus of elasticity, the three- and four-point bend strengths or the Vickers hardness, were observed. The slopes of the straight lines were nearly the same for all compositions. Furthermore, the modulus of elasticity, hardness, fracture toughness and strength were calculated as functions of density by modelling the structure as a random arrangement of spheres as suggested by Fischmeister and Arzt. The relationships obtained have been compared with the measured ones.Nomenclature a average contact area - a _c increase of the area of a crack - A area of the reference plane - b size of the critical defect - c constant in Equation 4 - D density - D ₀ density before shrinkage - D _T theoretical density - e direction of macroscopic strain - E modulus of elasticity - E ₀ modulus of elasticity of the dense material - f force loading a contact - f() projection of force f to e - F force loading the reference plane - g geometry parameter in the Griffiths relationship - H hardness - K _IC fracture toughness - N number of particles in unit volume - N() the fraction of N in a given spherical angle - n() number of particles in the volume around the reference plane - P porosity - R initial particle radius - R particle radius after fictitious growth - R particle radius after redistribution of material - R _SQ shared correlation coefficient - S surface energy of the defect - vector connecting the centres of neighbouring particles - W work necessary for increase the area of a crack - Z average coordination number - Z ₀ initial coordination number - strain - _T strain at theoretical strength - strength - _T theoretical strength (limit of elasticity) - angle between v and e     

The physics and mechanics of fibre-reinforced brittle matrix composites

This review compiles knowledge about the mechanical and structural performance of brittle matrix composites. The overall philosophy recognizes the need for models that allow efficient interpolation between experimental results, as the constituents and the fibre architecture are varied. This approach is necessary because empirical methods are prohibitively expensive. Moreover, the field is not yet mature, though evolving rapidly. Consequently, an attempt is made to provide a framework into which models could be inserted, and then validated by means of an efficient experimental matrix. The most comprehensive available models and the status of experimental assessments are reviewed. The phenomena given emphasis include: the stress/strain behaviour in tension and shear, the ultimate tensile strength and notch sensitivity, fatigue, stress corrosion and creep.Nomenclature a _i Parameters found in the paper by Hutchinson and Jensen [33], Table IV - a _o Length of unbridged matrix crack - a _m Fracture mirror radius - a _N Notch size - a _t Transition flaw size - b Plate dimension - b _i Parameters found in the paper by Hutchinson and Jensen [33], Table IV - c _i Parameters found in the paper by Hutchinson and Jensen [33], Table IV - d Matrix crack spacing - d _s Saturation crack spacing - f Fibre volume fraction - f _l Fibre volume fraction in the loading direction - g Function related to cracking of 90 ° plies - h Fibre pull-out length - l Sliding length - l _i Debond length - l _s Shear band length - m Shape parameter for fibre strength distribution - m _m Shape parameter for matrix flaw-size distribution - n Creep exponent - n _m Creep exponent for matrix - n _f Creep exponent for fibre - q Residual stress in matrix in axial orientation - s _ij Deviatoric stress - t Time - t _p Ply thickness - t _b Beam thickness - u Crack opening displacement (COD) - u _a COD due to applied stress - u _b COD due to bridging - v Sliding displacement - w Beam width - B Creep rheology parameter _o/ _o ⁿ - C _v Specific heat at constant strain - E Young's modulus for composite - E _o Plane strain Young's modulus for composites - Unloading modulus - E _* Young's modulus of material with matrix cracks - E _f Young's modulus of fibre - E _m Young's modulus of matrix - E _L Ply modulus in longitudinal orientation - E _T Ply modulus in transverse orientation - E _t Tangent modulus - E _s Secant modulus - G Shear modulus - G Energy release rate (ERR) - G _tip Tip ERR - G _tip ^o Tip ERR at lower bound - K Stress intensity factor (SIF) - K _b SIF caused by bridging - K _m Critical SIF for matrix - K _R Crack growth resistance - K _tip SIF at crack tip - I _o Moment of inertia - L Crack spacing in 90 ° plies - L _f Fragment length - L _g Gauge length - L _o Reference length for fibres - N Number of fatigue cycles - N _s Number of cycles at which sliding stress reaches steady-state - R Fibre radius - R R-ratio for fatigue (_max/_min) - R _c Radius of curvature - S Tensile strength of fibre - S _b Dry bundle strength of fibres - S _c Characteristic fibre strength - S _g UTS subject to global load sharing - S _o Scale factor for fibre strength - S _p Pull-out strength - S _th Threshold stress for fatigue - S _u Ultimate tensile strength (UTS) - S _* UTS in the presence of a flaw - T Temperature - T Change in temperature - t Traction function for thermomechanical fatigue (TMF) - t _b Bridging function for TMF - Linear thermal coefficient of expansion (TCE) - _f TCE of fibre - _m TCE of matrix - Shear strain - _c Shear ductility - _c Characteristic length - Hysteresis loop width - Strain - _* Strain caused by relief of residual stress upon matrix cracking - _e Elastic strain - _o Permanent strain - _o Reference strain rate for creep - Transient creep strain - _s Sliding strain - Pull-out parameter - Friction coefficient - Fatigue exponent (of order 0.1) - Beam curvature - Poisson's ratio - Orientation of interlaminar cracks - Density - Stress - _b Bridging stress - ¯_b Peak, reference stress - _e Effective stress = [(3/2)s _ijs_ij]^1/2 - _f Stress in fibre - _i Debond stress - _m Stress in matrix - _mc Matrix cracking stress - ^o Stress on 0 ° plies - _o Creep reference stress - _rr Radial stress - ^R Residual stress - _s Saturation stress - _s ^* Peak stress for traction law - Lower bound stress for tunnel cracking - ^T Misfit stress - Interface sliding stress - _f Value of sliding stress after fatigue - _o Constant component of interface sliding stress - _s In-plane shear strength - ¯_c Critical stress for interlaminar crack growth - _ss Steady-state value of after fatigue - ^R Displacement caused by matrix removal - _p Unloading strain differential - _o Reloading strain differential - Fracture energy - _i Interface debond energy - _f Fibre fracture energy - _m Matrix fracture energy - _R Fracture resistance - _s Steady-state fracture resistance - _T Transverse fracture energy - Misfit strain - _o Misfit strain at ambient temperature     

Deformation transitions

All solids with given mechanical properties will fracture brittly when of large enough size; vice versa it is difficult to comminute solids below certain sizes. Both effects are caused by the fracture stress changing with size (according to cube/square scaling principles) whereas the flow stress is essentially independent of size. Again, a fixed size of body, made of different materials, can respond in quite different ways: simple elasticity, elastic fracture, elastoplastic flow, elastoplastic fracture, plastic flow, plastic fracture or plastic collapse are all possible, depending upon the different mechanical properties of the different materials from which it may be made. This review shows that such deformation transitions are controlled by the relative values of size and a material parameter given byER/ _Y ² whereE is Young's modulus,R the specific work of fracture and _Y the flow stress. At fixed size of body, made of given material, transitions occur when one or more of the mechanical property terms are altered by rate, temperature, environment, superimposed hydrostatic stress and so on. A wide range of examples is used to illustrate these effects, and their role in load-bounding methods in elastoplastic design of structures is considered.     

Microstructure,chemistry, elastic properties and internal friction of Silceram glass-ceramics

The temperature dependence of both Young's modulus (E) and internal friction (Q ^–1) from room temperature to 700° C has been determined by Förster's forced-resonance method for three Silceram glass-ceramics, produced by the direct controlled cooling of glass melts in the quaternary system CaO-MgO-Al₂O₃-SiO₂. These results are correlated with microstructural and phase chemistry data as well as calculated viscosity against temperature data. In particular, the viscosity of the residual glass is shown to predominate over its volume fraction in deter mining the temperature dependence ofE andQ ^–1 for a given Silceram. A simple model which enables the residual glass content of Silceram glass-ceramics to be estimated from a know ledge of the proportions of silicon, iron and magnesium in the corresponding glass melts is also proposed. Furthermore, the room-temperature bulk modulus (K) and Poisson's ratio of two Silceram glass-ceramics are calculated using experimentalE and shear modulus (G) values obtained using both Förster's method and another forced-vibration technique.     

‘When can we apply LEFM principles to elastic softening materials?’

The paper focusses on the determination of R, the size of the fully developed softening zone associated with a semi-infinite crack in a remotely loaded infinite elastic softening solid. R is a characteristic length for a material, and is important in that if R is less than an appropriate characteristic dimension of a structure, then LEFM principles can be used to describe the structure's failure. With p _c and _c being respectively the maximum stress and displacement within the softening zone, then provided the softening is not particularly pronounced, i.e. the area under the stress (p)-displacement (v) curve is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaacaaeaacq% GH+aGpaiaawoWaaiaabccacaqGWaGaaeOlaiaabkdacaqG1aGaamiC% aSGaam4yaOGaeqiTdq2ccaWGJbaaaa!3FB5!\[\widetilde > {\text{ 0}}{\text{.25}}pc\delta c\], it is shown that R 0.4E _0c/P _c and R is relatively insensitive to the precise p-v softening behaviour (E ₀ = E/(1 – v ²) where E is Young's modulus and is Poisson's ratio. However, when the area under the curve is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaacaaeaacq% GH8aapaiaawoWaaiaabccacaqGWaGaaeOlaiaabkdacaqG1aGaamiC% aSGaam4yaOGaeqiTdq2ccaWGJbaaaa!3FB1!\[\widetilde < {\text{ 0}}{\text{.25}}pc\delta c\], then R increases above this 0.4E _0c/P _c value. For this case, and provided most of the area under the p-v curve is not associated with the tail in the softening law, a more appropriate expression for R is R 0.1E ₀ ² / ₀ ² /K ² , with K ² /E ₀ being the area under the p-v curve and K being the stress intensity associated with the full development of a softening zone.