The paraboloid failure surface for the general orthotropic material

Summary It has been established that the elliptic paraboloid failure criterion (EPFC) is a potential means of describing the yielding and failure behavior of transversely isotropic (transtropic) materials presenting also strength differential effects along all principal stress axes. The cases studied correspond to symmetric forms of loading which are parallel to the principal strength directions of the material.In this paper the general case of an orthotropic material not presenting any kind of strength symmetry was encountered. It has been shown that an elloptic parabloid surface may continue to represent the failure surface of such materials. However the symmetry of this surface relatively to the principal diagonal plane containing the strong ₃-axis and the bisector of the two remaining axes is destroyed. For these materials the elliptic paraboloid is leaning toward the intermediate strength ₂-axis and the angle of inclination of the plane of symmetry of the paraboloid with the ₁₂-plane depends on the difference in strengths between the ₂- and ₁-axes. The characteristic properties of the inclined paraboloid were studied and important differences between the inclined and the symmetric surfaces were established. Examples with orthotropic polymers and composites were included exemplifying the characteristic properties of orthotropic materials.With 7 Figures     

On a family of quasi-isotropic fiber-reinforced composites

Summary The spectral decomposition of the compliance and the stiffness tensors for a transversely isotropic body (fiber-reinforced composite), and their eigenvalues derived from, define in a simple and efficient way the respective elastic eigenstates of loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components ₁ and ₂, which are shears, with ₂ a simple shear, and ₁ a suprposition of simple and pure shears, and they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃ and ₄ are the orthogonal supplement to the shear subspace of ₁ and ₂, and consist of an equilateral stressing in the plane of isotropy, superimposed with a prescribed tension or compression along the symmetry axis of the material.In this paper a particular class of transversely isotropic materials is studied, whose mechanical properties are conveniently selected to reduce the two last eigenstate components in such a manner, that the one is identified to contribute only a dilatational type of strain energy, whereas the other a distortional one. In this way the four eigenstates are clearly separated in two distinct groups creating either distortional or dilatational types of strain energy. It was shown that this family of materials behave like the isotropic body, in spite of their differences in the elastic constants along their principal axes of anisotropy.Since with fiber-composites it is possible to arrange their mechanical properties by selecting the appropriate ratios between matrix and inclusions, according to their properties, this possibility of selecting in advance the properties of the composite is feasible. Taking into consideration that quasi-isotropic materials develop the smallest stress concentrations in the structures, the development and selection of such composites with quasi-elastic properties becomes very important.     

Failure criteria for weak-axis quasi-orthotropic woven fabric composites

Summary The failure behavior of woven fabric composites in the form of plain weave fiber unidirectional laminae is studied in this paper as defined by their failure stresses in simple tension and compression along the three principal stress directions. Since the transverse weave plane is the strong and isotropic plane of the composite, while the normal to it direction the weak one, the material is approximated as a weak-axis transversely isotropic composite.The elliptic paraboloid failure surface (EPFS) criterion, as introduced by the author [1], was shown to describe satisfactorily this type of interesting modern materials. It was shown that such weak-axis transversely isotropic composites correspond to tension strong composites and their failure surfaces consist of a single-sheet convex surface open to the tension-tension-tension octant of the principal stress space.The main characteristic of such surfaces is that they are oblate along the normal direction to the isotropic plane, in contrast with the typical (EPFS)-criterion for fiber composites, which, all of them, are prolate along the same direction. While the intersection of this (EPFS)-criterion by the (₁,₃) stress plane (₃ is the weak axis) resembles closely the respective intersection for the unidirectional fiber composites the (₁,₂)-isotopic plane intersection, which coincides with the weaving strong plane approaches very closely a circle thus indicating that along this isotropic plane the failure stress is hydrostatic and independent of its orientation inside this plane. This property constitutes a significant and most promising property which makes this type of woven composites very attractive in applications.Experimental evidence of failure of such materials, which is very sparse, as derived from tests in a woven T-300 Carbon-epoxy composite corroborated excellently with the theory based on the (EPFS)-criterion.     

Evaluation of biaxial stress failure surfaces for a glass fabric reinforced polyester resin under static and fatigue loading

In an attempt to evaluate failure theories for a glass fabric reinforced polyester resin over 370 tests have been conducted on thin-walled tubes under combined axial loading and internal pressure, both for static and fatigue loading. For plane stress the results are considered in relation to imaginary failure surfaces in ₁, ₂, ₆ space. A limited measure of agreement between theories and results can be obtained after highly subjective selection of data. Only those theories which involve complex stress properties provide a reasonable fit. The behaviour of tubular specimens is strongly influenced by the presence of joints in the reinforcements.Nomenclature _x , _y nominal hoop and axial (principal) stresses in a thin-walled tube - ₁, ₂, ₆ normal and shear stresses in the direction of the principal material axes - F ₁,F ₂,F ₆ strengths in the principal material directions and the in-plane shear strength - F _1t,F _2t tensile strengths in the principal material directions - F _1c,F _2c compressive strengths in the principal material directions - K ₂ a constant evaluated from a combined stress test - H ₁₂ normal stress interaction component of a strength tensor - off-axis angle - S-N curve conventional stress-log life fatigue curve - R principal stress ratio, _y /gs _x      

On the measurement and physical meaning of the cleavage fracture stress in steel

Elastic-plastic two-dimensional (2D) and three-dimensional (3D) finite element models (FEM) are used to analyze the stress distributions ahead of notches of four-point bending (4PB) and three-point bending (3PB) specimens with various sizes of a C-Mn steel. By accurately measuring the location of the cleavage initiation sites, the local cleavage fracture stress _f and the macroscopic cleavage fracture stress _F is accurately measured. The _f and _F measured by 2D FEM are higher than that by 3D FEM. _f values are lower than the _F, and the _f values could be predicted by _f=(0.8––1.0)_F. With increasing specimen sizes (W,B and a) and specimen widths (B) and changing loading methods (4PB and 3PB), the fracture load P _f changes considerably, but the _F and _f remain nearly constant. The stable lower boundary _F and _f values could be obtained by using notched specimens with sizes larger than the Griffiths–Owen specimen. The local cleavage fracture stress _f could be accurately used in the analysis of fracture micromechanism, and to characterize intrinsic toughness of steel. The macroscopic cleavage fracture stress _F is suggested to be a potential engineering parameter which can be used to assess fracture toughness of steel and to design engineering structure.     

Investigation of the thermodynamic criterion of crack resistance of steels

The article shows the validity of the previously proposed criterion K_Ic = K_Ic ^* + _0.2iK_Ic/_0.2 (K_Ic is reduced crack resistance, _0.2i is internal component of yield strength _0.2) for a wide range of body-centered cubic metals such as iron—carbon alloys (cast iron; low-, medium-, and high-strength carbon and alloy steels). It examines the relationship between this criterion and energy and force concepts of the fracture micromechanism. Existence of a common temperature dependence of the effective yield strength component in ferrite—pearlite carbon and alloy steels in the annealed, normalized, and heat-treated states is established. It is shown that the fraction of effective stress In the total yield strength, i.e., _0.2 ^*/_0.2 controls crack resistance K_Ic over a wide range of temperatures and deformation rates. For impact strength KCV, linear dependences of KCV — KCV/_0.2 and KCV-KCV/HB are observed In the zone of transition temperatures and cold brittleness threshold. A correlation equation connecting KCV and K_Ic over the indicated range is obtained. An applied software package has been worked out for computer-aided prediction of crack resistance and impact strength.Translated from Problemy Prochnosti, No. 8, pp. 14–22, August, 1993.     

Ionic and electronic conductivity in some simple lithium salts

The electrical conductivity () and thermoelectric power (S) of Li₃VO₄, Li₃PO₄ and Li₃BO₃ solidified melts are presented in the temperature range from 415 K to the melting point of each solid. The ionic ( _i,) and electronic ( _e) contributions to have been separated over the entire temperature range with the help of a time-dependence study of the d.c. electrical conductivity. Superionic phases in all three solids have been observed below their melting points in which the conductivity is almost purely ionic. The value of the phase transition temperature below which the solid transforms from superionic to normal phase has been obtained. It has been shown that in the normal phase, these solids are mixed conductors. Data for the temperature variations of both _i, and _e are also presented and discussed.     

Deformation of a carbon-epoxy composite under hydrostatic pressure

This paper describes the behaviour of a carbon-fibre reinforced epoxy composite when deformed in compression under high hydrostatic confining pressures. The composite consisted of 36% by volume of continuous fibres of Modmur Type II embedded in Epikote 828 epoxy resin. When deformed under pressures of less than 100 MPa the composite failed by longitudinal splitting, but splitting was suppressed at higher pressures (up to 500 MPa) and failure was by kinking. The failure strength of the composite increased rapidly with increasing confining pressure, though the elastic modulus remained constant. This suggests that the pressure effects were introduced by fracture processes. Microscopical examination of the kinked structures showed that the carbon fibres in the kink bands were broken into many fairly uniform short lengths. A model for kinking in the composite is suggested which involves the buckling and fracture of the carbon fibres.List of symbols d diameter of fibre - E _f elastic modulus of fibre - E _m elastic modulus of epoxy - G _m shear modulus of epoxy - k radius of gyration of fibre section - l length of buckle in fibre - P confining pressure (= ₂ = ₃) - R radius of bent fibre - V _f volume fraction of fibres in composite - _t, _c bending strains in fibres - angle between the plane of fracture and ₁ - ₁ principal stress - ₃ confining pressure - _c strength of composite - _f strength of fibre in buckling mode - _n normal stress on a fracture plane - _m strength of epoxy matrix - shear stress - tangent slope of Mohr envelope - slope of pressure versus strength curves in Figs. 3 and 4.     

Residual life calculations for frame structures working in bending

Summary A method is proposed for calculating the residual working life of a frame structure from the viewpoint of failure mechanics. The method has been tested on the load-bearing frame of a KS-3575 track crane.The size of a permissible defect has been calculated for the working life of the crane N=2.10⁴ loading cycles and the maximum stress in the lower wall of the frame is =05_y (material steel 20).Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 6, pp. 35–40, November–December, 1992.     

Electrical conductivity,thermoelectric power and dielectric constant of NiWO4

The a.c. electrical conductivity ( _ac), thermoelectric power () and dielectric constant () of antiferromagnetic NiWO₄ are presented. _ac and have been measured in the temperature range 300 to 1000 K and in the temperature range 600 to 1000 K. Conductivity data are interpreted in the light of band theory of solids. The compound obeys the exponential law of conductivity = ₀ exp (–W/kT). Activation energy has been estimated as 0.75eV. The conductivity result is summarized in the following equation =2.86 exp (–0.75 eV/kT)^–1 cm^–1 in the intrinsic region. The material is p-type below 660 K and above 950 K, and is n-type between 660 and 950 K.     

On internal stress and activation volume in polymers

The two-site model is developed for the analysis of stress relaxation data. It is shown that the product of d In (– )/d and (- _i) is constant where is the applied stress, _i is the (deformation-induced) internal stress and = d/dt. The quantity d In ( )/d is often presented in the literature as the (experimental) activation volume, and there are many examples in which the above relationship with (- _i) holds true. This is in apparent contradiction to the arguments that lead to the association of the quantity d In (– )/d with the activation volume, since these normally start with the premise that the activation volume is independent of stress. In the modified theory presented here the source of this anomaly is apparent. Similar anomalies arise in the estimation of activation volume from creep or constant strain rate tests and these are also examined from the standpoint of the site model theory. In the derivation presented here full account is taken of the site population distribution and this is the major difference compared to most other analyses. The predicted behaviour is identical to that obtained with the standard linear solid. Consideration is also given to the orientation-dependence of stress-aided activation.     

Stress relaxation in hot-drawn low density polyethylene

The tensile stress relaxation behaviour of hot-drawn low density polyethylene, (LDPE), has been investigated at room temperature at various draw ratios. The drawing was performed at 85° C. The main result was an increase in relaxation rate in the draw direction, especially at low draw ratios when compared to the relaxation behaviour of the isotropic material. This is attributed to a lowering of the internal stress. The position of the relaxation curves along the log time axis was also changed as a result of the drawing, corresponding to a shift to shorter times. The activation volume, , varied with the initial effective stress ₀ ^* according to ₀ ^* 10kT, where ₀ ^* =₀ — _i, is the difference between the applied initial stress, ₀, and the internal stress _i. This result supports earlier findings relating to similarities in the stress relaxation behaviour of different solids.     

Shock response of stainless steel at high temperature

Measurements of the dynamic tensile strength of HR-2 (Cr-Ni-Mn-N) stainless steel have been carried out over the initial temperature range of 300 K–1000 K at shock stress of 8 GPa, the corresponding spall strength _f and Hugoniot elastic limit _HEL are determined from the wave profiles. In the temperature range of 300 K–806 K, _f and _HEL decrease linearly with increasing temperature T, i.e., _f = 5.63-4.32 × 10^–3T, _HEL = 2.08-1.54 × 10^–3T, but when heated to 980 K, _HEL increases from 0.84 GPa at 806 K to 0.93 GPa at 980 K and _f keeps at an almost fixed value of 2.15 GPa. The TEM analysis on recovery samples identified the existence of intermatallic compound Ni₃Al and the carbide Cr₂₃C₆ in the sample of 806 K, another intermatallic compound Ni₃Ti was found in the sample of 980 K. All these products emerge along crystal boundary. While no such products were found in the samples of 300 K and 650 K.     

Fatigue behaviour of borosilicate glass-ceramic matrix,nicalon (silicon carbide) fibre composites

Uniaxial fatigue damage analyses were performed on borosilicate glass-ceramic matrix, Nicalon (silicon carbide) fibre reinforced unidirectional composites. The fibre volume fraction varied from about 0.25 to 0.60. Load-controlled tension-tension fatigue tests (R ratio = 0.1) were conducted at room temperature and 540°C (1000°F). The fatigue life was found to decrease with increasing cyclic stress level and a power-law relationship of the form _app = _uts(2N _f)^b was established where _app is the applied maximum stress, _uts the monotonic tensile strength, N _f is the number of cycles to failure and b is the fatigue strength exponent. The fatigue damage evolution manifested itself as a decrease in stiffness of the composite with fatigue cycles. This stiffness drop was associated with matrix cracking followed by fibre-matrix debonding and fibre sliding breakage/pull-out, and final failure, respectively at 540°C. The damage evolution at room temperature was associated with degradation of the matrix followed by steady breakage of fibres with no debonding/pull-out, leading to eventual failure of the net section of the composite. In general, quantitative microscopic observations of debonded and pulled-out fibres showed a good correlation with the observed reduction in stiffness. A predictive model to interpret the drop in stiffness is presented and validated using experimental results from the current study.     

Two parameter approach for elastic-plastic fracture of short cracked specimens under mixed mode loading

For mode-I loading, in order to describe the near-tip stress field in a specimen under large scaled yielding, two parameter approaches such as J-T, J-Q and J-A₂ theories have been developed and proved well for their validity and limit. In this work elastic-plastic finite element analysis were performed to investigate the effects of mode mixity and T-stress upon near-tip stress distribution for a small-scale-yield model with the modified boundary layer and CTS (Compact Tension-Shear) configuration under large-scale-yield state. As the results, some peculiar characteristics were found as follows; As the mode mixity increases, normal stresses _rr and near the crack tip in the small-scale-yield model get significantly affected by the positive T-stress as well as the negative T-stress, while the shear stress _r is little affected by T-stress. Also, the near-tip stress distribution of short cracked CTS specimens under the large-scale-yield state agree fairly well with that of the small-scale-yield model with an appropriate positive T-stress. The two parameters approach with J-integral and T-stress seems to be a good tool for describing the near-tip stress field under a mixed mode loading and large-scale-yield state.     

Statistical analysis of the brittle fracture of sintered tungsten

A Weibull analysis was applied to the fracture data of sintered tungsten round bar specimens. Fracture data were obtained by performing flexural and tensile tests on these components. Two quantities were obtained which characterized the material variability and strength for each test method. The correlation between these quantities for the two test methods was found to be close.Nomenclature C least square line intercept - P _f failure probability - P _f ⁱ experimental failure probability - V specimen volume - W applied load - X horizontal coordinate of least square line - Y vertical coordinate of least square line - d specimen diameter - L distance between supporting knife edges - m Weibull modulus - v unit volume - stress - ₀ normalizing stress - _fV failure stress of specimen - _fv unit volume failure stress of specimen - _fv ^B unit volume failure stress in bending - ⁱ experimental stress - gamma function     

Surface Cracking of Steels in the Process of Cyclic Deformation in Aqueous Media

We study the conditions of surface cracking of steels for pipelines of power-generating equipment subjected to cyclic deformation in aqueous media. A new parameter is proposed for the evaluation of the corrosion fatigue of materials, namely, a certain characteristic level of stresses within the limits of the loading cycle ( = _s ) above which the deformed surface undergoes noticeable electrochemical activation. A model scheme of the surface corrosion-fatigue cracking regarded as a result of the synergetic action of cyclic stresses and electrochemical processes is proposed. A criterion of initiation of a surface crack of length a relating the characteristic stress _s to the maximum stress in a cycle _max, the strength of corrosion current I _cor, and the constants of electrochemical dissolution of the metal on the deformed surface is established. On this basis, we propose an engineering estimate for the period of initiation of surface corrosion-fatigue cracks.     

Studies in the cold drawing of Pd77.5Cu6Si16.5 metallic glass and 316 stainless steel crystalline wires

The stress necessary to draw Pd_77.5Cu₆Si_16.5 metallic glass wires has been measured using a tensile machine, and compared with that of 316 stainless steel crystalline wires. To assess the elastic back-pull of a drawn material, variable static back-stresses have been loaded on a drawing wire. In the case of the metallic glass wires, the draw stress, _d, under different applied back-pulls, _ab, increases linearly with the reduction in area, R _a, up to 23% and then increases less rapidly. This tendency of the curve _d versus R _a is also the same in 316 crystalline wires. On the other hand, the elastic back-pull of the glassy wires is 2.05 kg mm^–2 at zero applied back-pull and then decreases monotonically with _ab, while that of the 316 crystalline wires is 6.62 kg mm^–2 at _ab=0 and decreases linearly with the increase in _ab. The empirical maximum reduction in Pd_77.5Cu₆Si_16.5 metallic glass wire for one-pass drawing is measured to be 40% at _ab=0 and it then decreases with increasing _ab. The resultant curves of drawing stress versus reduction in area are discussed in the light of plastic theory. The theoretical curves calculated under the assumption of small strain-hardening fits quite well with the experimental curves of Pd_77.5Cu₆Si_16.5 glassy wires.     

Theory of the Surface Tension Maximum of Liquid 3He

Recently Matsumoto et al. performed very precise measurements of the surface tension of liquid ³He, (T), at low temperatures and found that (T) exhibits a small maximum at about 100 mK. Existing theories are unable to explain this anomaly. On the basis of a local approximation for the entropy in which the Fermi liquid effect is included, we can evaluate the variation of (T), (T)=(T)–(0), as a function of T and of the number density (or the interaction strength). It is found that (T) consists of two terms; a T² term and T⁴ ln T term. We predict that, for the density of real liquid ³He, exhibits a tiny minimum and a small but relatively larger maximum. This prediction explains qualitatively and quantitatively all salient features of the observed (T).     

Effect of magnesium content on the stress exponent and effective stress in the steady state creep of Al-Mg alloys

The stress exponent of steady state creep,n, and the internal ( _i) and effective stresses ( _e) have been determined using the strain transient dip test for a series of polycrystalline Al-Mg alloys creep tested at 300° C and compared with previously published data. The internal or dislocation back stress, _i, varied with applied stress,, but was insensitive to magnesium content of the alloy, being represented by the empirical equation _i=1.084 ^1.802. Such an applied stress dependence of _i can be explained by using an equation for _i of the form _i (dislocation density)^1/2 and published values for the stress dependence of dislocation density. Values of the friction stress, _f, derived using the equation _e/=(1–c) (1– _f/), indicate that _f is not dependent on the magnesium content. A constant value of _f can best be rationalized by postulating that the creep dislocation structure is relatively insensitive to the magnesium content of the alloy.On leave from Engineering Materials Department, University of Windsor, Windsor, Ontario N9B 3P4, Canada.