Delayed retardation phenomena of fatigue crack growth in various steels and alloys

The delayed retardation phenomena of fatigue crack growth resulting from a single application of overload were investigated for five steels, two aluminium alloys and a titanium alloy. As long as the small scale yielding condition was satisfied at the overloaded crack tips, the retardation behaviour of these materials was expressed consistently by four parameters; the peak/baseline stress ratio, r, the exponent in the Paris equation, m, the overload-affected zone size, _D, and the crack distance at the minimum rate of crack growth, _B. Then the parameters, _B and _D, characterizing the retardation phenomena for these materials were determined. The retardation of aluminium alloys was stronger than that of the other materials. This is attributed to the lower value of _B/ _D in aluminium alloys than in the other materials. In the case of r=2, the overload-affected zone sizes, _D, were nearly equal to 1.5 ₀ in HT80 steel and aluminium alloys, slightly lower than 1.5 ₀ in SNCM8 steel and much larger than 1.5 ₀ in A553 steel and the titanium alloy, where ₀ is the monotonic plastic zone size calculated according to the Dugdale model. The dependence of retardation on baseline stress intensity, K ₁, appeared somewhat complicated. In the cases of A553 steel and A2017 aluminium alloy the amount of retardation increased with increasing K ₁ value, while in the cases of HT80 steel and Ti-6A1-4V titanium alloy the tendency appeared in the reverse direction. The former behaviour was related to the change in the stress state from plane strain to plane stress at the overloaded crack tips and the latter was related to the threshold of stress intensity.     

Radiation of internal waves during vertical motion of a body through a nonuniform liquid

Energy losses to radiation of internal waves during the vertical motion of a point dipole in two-dimensional and three-dimensional cases are computed.Notation _o(z), p_o(z) density and pressure of the ground state - z vertical coordinate - v, p, perturbed velocity, pressure, and density - H(d 1n _o/dz)^–1 characteristic length scale for stratification - N=(gH^–1–g²c _o ^–2 )^1/2 Weisel-Brent frequency - g acceleration of gravity - c_o speed of sound - vertical component of the perturbed velocity - V vector operator - k wave vector - frequency - d vector surface element - W magnitude of the energy losses - (t), (r) (x)(y)(z) Dirac functions - v_o velocity of motion of the source of perturbations - d dipole moment of the doublet - _o,l length dimension parameters - _o intensity of the source Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 619–623, October, 1980.     

Heating of a newtonian liquid when extruded through a matrix

Calculations are carried out of the dynamic forces and viscous heating when a layer of Newtonian liquid at the bottom of a rigid container is extruded through a matrix in the form of a circular opening or a narrow slit.Notation r, z axes of a Cartesian (k=0) or a cylindrical (k=1) system of coordinates - u, v velocity components of the liquid in the directions r and z, respectively - V velocity of the liquid in the opening - p pressure - ₁ and ₂ temperature of the liquid under the die and in the opening - m, n rheological constants of the material - tangential stress - shear velocity - _o, c_p, _o density, heat capacity, and thermal conductivity of the liquid - S, M, w area of the working surface, mass and velocity of motion of the die - r_o,l,a, R linear dimensions of the matrix and the container - F operating force on the instrument - Re, Pe similitude criteria Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 710–715, October, 1980.     

Effect of crack shape,inclination angle,and friction coefficient in crack surface contact problems

In rolling/sliding contact fatigue, it is known that the crack propagates at a characteristic angle =15–30 deg to the surface. To analyze the mechanism, however, the body force method has been widely used assuming 3D crack models for =45–90. In this study, therefore, the unknown body force densities are newly approximated by using fundamental density functions and polynomials. Then, a semi-elliptical crack model is analyzed for =15–90 under compressive residual stresses and Hertzian contact loads. The stress intensity factors K _II, K _III are calculated with varying the crack shape b/a, inclination crack angle , and crack face friction coefficient . The calculations show that the present method is useful for the analysis for =15–30 deg with high accuracy. It is seen that the K _II-values when b/a0 are larger than the ones when b/a=1 by 0–24% for both under compressive residual stress and Hertzian contact load. Regarding the maximum K _II values under Hertzian contact load, the results of =15 deg are smaller than the ones of =45 deg by 23–34%. Regarding the amplitude of (K _{II max}–K _{II min}), the results of =15 deg are smaller than the ones of =45 deg by 4–24%. With increasing the value of friction coefficient for crack faces the value of K _II decreases significantly. When the crack is short and the inclination angle is small, the value of friction coefficient f for Hertzian contact load largely affect the K _II value.     

Equations of generalized thermoelasticity of a Cosserat medium in stresses

Thermoelasticity equations in stresses are derived in this paper for a Cosserat medium taking into account the finiteness of the heat propagation velocity. A theorem is proved on the uniqueness of the solution for one of the obtained systems of such equations.Notation u displacement vector - small rotation vector - absolute temperature - ₀ initial temperature of the medium - relative deviation of the temperature from the initial value - , , , , , ,, m constants characterizing the mechanical or thermophysical properties of the medium - density - I dynamic characteristic of the medium reaction during rotation - k heat conduction coefficient - ₀ a constant characterizing the velocity of heat propagation - X external volume force vector - Y external volume moment vector - w density of the heat liberation sources distributed in the medium - E unit tensor - T force stress tensor - M moment stress tensor - nonsymmetric strain tensor - bending-torsion tensor - s entropy referred to unit volume - V volume occupied by the body - surface bounding the body - (T)_ki, (M)_ki components of the tensorsT andM - q thermal flux vector Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 482–488, March, 1981.     

Analysis of a wavy crack in sandwich specimens

A crack in a brittle adhesive layer joining two substrates can grow in a variety of ways. The crack can grow along one of the interfaces, within the adhesive or alternate between the two interfaces. In this paper, we consider a crack that grows along an alternating path between the two interfaces. A quantitative analysis of this elastic problem is carried out using the finite element method to predict the wavelength of the alternating crack. The joint is loaded remotely by the singular stress field for a cracked homogeneous solid, parameterised by K _I and K _II , and by an in-plane tensile residual stress ₀ in the layer, parallel to the interface. The induced interfacial stress intensity factor and its phase angle are evaluated and used to predict the onset of kinking out of the interface. The wavelength of the alternating crack is found to depend on the elastic mismatch parameters, and , and on the level of residual stress in the layer, parameterised by {ie29-1}, where h is the adhesive layer thickness, {ie29-2} is a modulus quantity and {ie29-3} is the toughness of the interface.     

An investigation of the micromechanisms of fatigue crack growth in structural gas turbine engine alloys

This paper presents an overview of fatigue fracture modes in selected structural alloys employed in gas turbine engines. These include the mechanisms of fatigue crack growth in the near-threshold, Paris and high-K regimes obtained from Ti-6Al-4V, Inconel 718 and PWA 1472 (a single crystal nickel-based superalloy of similar chemical composition to Inconel 718). Fatigue fracture modes in these materials are shown to be strong functions of the stress intensity factor range, K, and the maximum stress intensity factor, K _max. Fatigue mechanism maps are also presented to show the parametric ranges of K and K _max corresponding to the different fatigue fracture modes.     

Breakup of an anomolously viscous liquid film in a centrifugal force field

An equation is obtained for the breakup radius with consideration of tipping moments and Laplacian pressure forces acting on the liquid ridge at the critical point.Notation K, n rhenological constants - density - surface tension - r current cup radius - R maximum cup radius - r_c critical radius for film breakup - ¯r=¯r=r/R dimensionless current radius - ¯r_c=r_c/R dimensionless critical radius - ₀, _c actual and critical film thicknesses - current thickness - R_r ridge radius - h₀ ridge height - h current ridge height - ₀ limiting wetting angle - current angle of tangent to ridge surface - angle between axis of rotation and tangent to cup surface - angular velocity of rotation - q volume liquid flow rate - v₁ and v meridional and tangential velocities - =4vv _lm/r,=4v_m/r dimensionless velocities - M moments of surface and centrifugal forces - M_v moment from velocity head - p_r pressure within ridge - P_vm pressure from velocity head - p_m, p_pm pressures from centrifugal force components tangent and normal to cup surface - deviation range of breakup radius from calculated value - ¯r_max, ¯r_min limiting deviations of breakup radius - _c angle of tangent to curve _c/°₀=f(¯r) at critical point - t random oscillation of ratio _c/_c Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 51–56, July, 1980.     

Creep crack growth in brittle materials

During steady state crack growth by diffusive cavitation at grain boundaries, crack tip fields are relaxed due to the presence of a cavitation zone. In the present analysis, analytic solutions for the actual crack tip stress fields and the crack velocity in the presence of cavitation zone consisting of continuously distributed cavities ahead of the crack tip are derived using the smeared volume concept. Results indicate that the r ^–1/2 singularity is now attenuated to r ^{–1/2 +} (0<<1/2) singularity. The singularity attenuation parameter is a function of the crack velocity and material parameters. The crack growth rate is related to the mode I stress intensity factor K by K ² at relatively high load, K ⁿ at intermediate load, and approaches zero at small load near K _th. Meanwhile, the cavitation zone extends further into the material due to the stress relaxation at the crack tip and the subsequent stress redistribution. Such relaxation effects become very distinct at low crack velocity and low applied load. Key words: Creep crack growth, brittle material, diffusive cavity growth, sintering stress, crack tip stress field.     

A method of determining the thermal conductivity of powdered and fibrous insulation as a function of filler gas pressure

An examination is made of the theoretical basis and implementation of a nonstationary method of rapid measurement of the thermal conductivity of powdered and fibrous insulation under conditions of monotonic change of filler gas pressure.Notation t temperature - ,a thermal conductivity and diffusivity of test material - k, k_a relative temperature coefficients of anda - thickness of test layer - x variable layer coordinate reckoned from shell - =(x), _c excess temperature of material at section x and of core over shell - b_c, b_v rate of cooling of core and of variation of volume-mean temperature of layer - c_c, c total heat capacity of core and material - f_s, F_c area of working surfaces of shell and core - d diameter of particles of bulk material - p material porosity - volume density of material     

Boundary element study on the optical method of caustic for measuring fast crack propagation toughness K ID

The dynamic stress field near the tip of a crack tip which is accelerated and decelerated in an elastic plate with finite width under impact loading is analyzed by the boundary element method, and a simulation of measuring fast crack propagation toughness K _ID by the caustic method is performed. The results of the simulation agree qualitatively with the experimental results by Arakawa and Takahashi, and indicate the dependence of the measured K _ID not only on crack acceleration but also plate width. The dependence of measured K _ID on crack acceleration may result from the fact that under the condition of high loading rate or abrupt change in crack velocity, the transient stress field near the initial curve of caustic can not be represented fully by the dynamic stress intensity factor K _I(t, v), as suggested by Rosakis et al. The dependence of measured K _ID on plate width may be attributable to the fact that the transient stress field near the initial curve is affected directly by the reflected stess wave and also indirectly through crack acceleration which depends on the reflected stress wave. The possible dependence of the measured K _ID on loading rate, loading history, crack propagation history is also discussed.     

Bifurcation of crack pattern in arrays of two-dimensional cracks

Theoretical calculations based on simple arrays of two-dimensional cracks demonstrate that bifurcation of crack growth patterns may exist. The approximation used involves the dipole asymptotic or pseudo-traction method to estimate the local stress intensity factor. This leads to a crack interaction parametrized by the crack length/spacing ratio =a/h. For parallel and edge crack arrays under far field tension, uniform crack growth patterns (all cracks having same size) yield to nonuniform crack growth patterns (bifurcation) if is larger than a critical value _cr. However, no such bifurcation is found for a collinear crack array under tension. For parallel and edge crack arrays, respectively, the value of _cr decreases monotonically from (2/9)^1/2 and (2/15.096)^1/2 for arrays of 2 cracks, to (2/3)^1/2/ and (2/5.032)^1/2/ for infinite arrays of cracks. The critical parameter _cr is calculated numerically for arrays of up to 100 cracks, whilst discrete Fourier transform is used to obtain _cr for infinite crack arrays. For infinite parallel crack arrays under uniaxial compression, a simple shear-induced tensile crack model is formulated and compared to the modified Griffith theory. Based upon the model, _cr can be evaluated numerically depending on (the frictional coefficient) and c ₀/a (c ₀ and a are the sizes of the shear crack and tensile crack, respectively). As an iterative method is used, no closed form solution is presented. However, the numerical calculations do indicate that _cr decreases with the increase of both and c ₀/a.     

On the determination of the cohesive zone parameters for the modeling of micro-ductile crack growth in thick specimens

The paper deals with the determination of the cohesive zone parameters (separation energy, , and cohesive strength, T _max) for the 3D finite element modeling of the micro-ductile crack growth in thick, smooth-sided compact tension specimens made of a low-strength steel. Since the cohesive zone parameters depend, in general, on the local constraint conditions around the crack tip, their values will vary along the crack front and with crack extension. The experimental determination of the separation energy via automated fracture surface analysis is not accurate enough. The basic idea is, therefore, to estimate the cohesive zone parameters, and T _max, by fitting the simulated distribution of the local crack extension values along the crack front to the experimental data of a multi-specimen J _IC-test. Furthermore, the influence of the cohesive zone parameters on the crack growth behavior is investigated. The point of crack growth initiation is determined only by the magnitude of . Both and T _max affect the crack growth rate (or the crack growth resistance), but the influence of the cohesive strength is much stronger than that of the separation energy. It turns out that T _max as well as vary along the crack front. In the center of the specimen, where plane strain conditions prevail, the separation energy is lower and the cohesive strength is higher than at the side-surface.     

Fatigue crack growth behaviour of tungsten carbide-cobalt hardmetals

Fatigue crack propagation studies have been carried out on a range of WC-Co hardmetals of varying cobalt content and grain size using a constant-stress intensity factor double torsion test specimen geometry. Results have confirmed the marked influence of mean stress (throughK _max), which is interpreted in terms of static modes of fracture occurring in conjunction with a true fatigue process, the existence of which can be rationalized through the absence of any frequency effect. Dramatic increases in fatigue crack growth rate are found asK _max approaches that value of stress intensity factor ( 0.9K_IC) for which static crack growth under monotonic load (or static fatigue) occurs in these materials. Lower crack growth rates, however, produce fractographic features indistinguishable from those resulting from fast fracture. These observations, and the important effect of increasing mean free path of the cobalt binder in reducing fatigue crack growth rate, can reasonably be explained through a consideration of the mechanism of fatigue crack advance through ligament rupture of the cobalt binder at the tip of a propagating crack.     

An energy approach to crack closure

Crack closure is analyzed using an energy approach whereby it is shown that crack closure does not completely shield the input mechanical energy to the crack tip at a load below the crack opening load P _op if the compliance below P _op is non-zero. An equivalent shielding stress intensity range is defined by the energy release rate against crack closure. From this energy standpoint, the true effective stress intensity range should be defined as K _eff=K _max–K _op, where is the shielding factor. The conventional definition (K _eff=K _max–K _op) is equivalent to the new definition only when the compliance below P _op is zero such that =1, i.e., for a fully closed crack. The corrected K _eff is found to be effective in correlating fatigue crack growth rates (FCGRs) generated in 8090-T8771 aluminum-lithium alloy with and without crack closure. In contrast, the conventional K _eff fails to reconcile the FCGR data within an acceptable scatter band.The Canadian Government's right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

The edge interface crack

An analysis of a crack lying along an interface between two elastically dissimilar quarter-planes, and breaking the free surface is given. The method of distributed dislocations is employed and the nominal stress field is taken to be uniform along the length of the line of the crack. Models for crack tip behaviour are discussed and it is shown that a simplified quadrature can be used to extract the crack extension force for cases where small-scale contact obtains. Values of the crack extension force are given, displayed on the , diagram.     

A criterion for plane strain,fully plastic,quasi-steady crack growth

Plane strain fracture by hole growth in ordinary-sized parts of low-to-medium strength steels is essentially rigid-plastic, and may be approximated as non-hardening. Quasi-steady crack growth for such materials is predicted for crack-tip fields approximated by a pair of slip lines, such as unequally grooved specimens in tension and deep singly-face-cracked specimens under combined bending and tension. The crack growth increment a is given in terms of material parameters, far-field geometry, and loadings and their increments.For the rigid-plastic, non-hardening approximation, stress and strain increment fields for growing cracks are identical to those for stationary cracks. For fields with a pair of symmetric slip-lines, the flanks of the decohering zone turn out to be rigid, and the decohering zone does not affect the crack-tip opening angle (CTOA), which then depends only on the micromechanisms of hole nucleation, growth and linkage by flow localization or fine cracking. These mechanisms are in turn approximately controlled by the near-field plasticity parameters: the angle of the slip plane _s, and the normal stress and displacement increment across the slip plane _s and u_s. Note the three-parameter characterization of the near-tip fields, in contrast to the one- or two-parameter characterization in elastic or nonlinear elastic fracture mechanics.A sliding off and shear-cracking model for a growing crack, based on a hole growth equation, gives an approximate CTOA in terms of _s, _s, and material parameters. When hole nucleation strain is negligible, the estimated CTOA exhibits an inverse exponential dependence on _s and a higher order parabolic dependence on _s. For a given material, a series of fully plastic crack growth experiments is suggested to determine the approximate material parameters needed to characterize the dependence of CTOA on _s and _s, or from kinematics, of the shear strain behind the slip plane, _f, on _s.     

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.     

The stress dependence of the crack growth rate during creep

An equation is derived for the crack growth rate under creep conditions. In the model, the propagation of a grain boundary crack is controlled by the plastic growth of cavities located in the grain boundaries ahead of the crack. It is assumed that the cavities grow by power law creep in the elastic crack tip stress field. Hence, the stress dependence of the crack velocity is provided through the elastic stress intensity factor, i.e., dC/dt=BK _I ^p .The cavity spacing, , appears as an important factor in the coefficient,B ^–(p–2)/2. At large values of , corresponding to less severe creep damage in the grain boundaries, the above equation would predict very low values for the crack velocity. Under such conditions, we suggest that another mechanism, whose stress dependence is provided through the net section stress, becomes active, i.e., dC/dt=B _net ^p . Since increases with decreasing applied stress, one should observe the _net correlation at low stresses. The results of recent creep crack growth experiments which tend to support this hypothesis are presented.

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Résumé On dérive une équation décrivant la vitesse de propagation d'une fissure dans les conditions de fluage. Dans ce modèle, la propagation d'une fissure aux frontières de grains est contrôlée par la croissance dans le domaine plastique de cavités situées aux frontières de grains en avant de la fissure.On suppose que les cavités s'étendent dans le champ de contraintes élastiques situées à l'extrémité de la fissure en suivant une loi de fluage parabolique. Dès lors, la dépendance de la vitesse de la fissuration en fonction de la contrainte est fournie par un facteur d'intensité de contrainte élastique, c'est-à-dire dC/dt=BK _I ^p .L'espace entre les cavités, , apparaît être un facteur important dans les coefficientsB. Pour de grandes valeurs de , qui correspondent à un dommage moins sévère par fluage aux frontières des grains, l'équation ci-dessus permettrait de prédire des valeurs très faibles de la vitesse de fissuration.Sous ces conditions, il est suggèré qu'un autre mécanisme, dont la dépendance de la contrainte est fournie par la contrainte agissant sur la section droite, devient plus actif; on a alors dC/dt=B _nette ^p .Comme augmente lorsque la contrainte appliquée diminue, on devrait observer une corrélation de nette à basses contraintes.Les résultats d'essais de croissance de fissure sous des conditions de fluage effectués récemment tendent à supporter cette hypothèse et sont présentés.

     

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