A modification to Brown and Miller's critical plane approach is proposed to predict multiaxial fatigue life under both in-phase and out-of-phase loading conditions. The components of this modified parameter consist of the maximum shear strain amplitude and the maximum normal stress on the maximum shear strain amplitude plane. Additional cyclic hardening developed during out-of-phase loading is included in the normal stress term. Also, the mathematical formulation of this new parameter is such that variable amplitude loading can be accommodated. Experimental results from tubular specimens made of 1045 HR steel under in-phase and 90" out-of-phase axial-torsional straining using both sinusoidal and trapezoidal wave forms were correlated within a factor of about two employing this approach. Available Inconel 718 axial-torsional data including mean strain histories were also satisfactorily correlated using the aforementioned parameter. NOMENCLATURE b = fatigue strength exponent c = fatigue ductility exponent E = modulus of elasticity G = modulus of rigidity K:, = cyclic shear strength coefficient n: = cyclic shear strain exponent n = constant Nf = cycles to failure R = strain ratio, minimum strain/maximum strain S = constant a, = crack angles c, , c2, c3 = principal strains t, = Ac /2 = axial strain amplitude tT = total strain ti = fatigue ductility coefficient AZ/2 = effective strain amplitude ya = Ay/2 = shear strain amplitude y,,, = maximum shear strain 1 = biaxial strain ratio, Ay/At v = Poisson's ratio v, = elastic Poisson's ratio vD = plastic Poisson's ratio u, = A u / 2 = axial stress amplitude (ry = yield strength ( u , ) , , ,~~ = maximum value of the largest principal stress u; = alternating normal stress on yma, plane u; = mean (static) normal stress on ymax plane u; = fatigue strength coefficient 4 = phase angle u y = maximum normal stress on yma, plane T~ = AT/^ = shear stress amplitude 149 F F E M S lI/%A

Two multiaxial fatigue damage models are proposed: a shear strain model for failures that are primarily mode II crack growth and a tensile strain model for failures that are primarily mode I crack growth. The failure mode is shown to be dependent on material, strain range and hydrostatic stress state. Tests to support these models were conducted with Inconel 718, SAE 1045, and AISI Type 304 stainless steel tubular specimens in strain control. Both proportional and non-proportional loading histories were considered. It is shown that the additional cyclic hardening that accompanies out of phase loading cannot be neglected in the fatigue damage model.

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This paper describes a multiaxial low cycle fatigue parameter for correlating Hues under nonproportional loadings. Constant amplitude low cycle fatigue tests were carried out under 14 proportional and complex nonproportional cyclic strain paths using type 304 stainless steel hollow cylinder specimens at room temperature. In nonproportional loading tests, fatigue lives are decreased by as much as a factor of 10 in comparison with those in proportional loading tests with the same strain range. Reduction in fatigue life due to nonproportional loading is closely related to additional nonproportional cyclic hardening. The product of the maximum principal stress and strain ranges correlated the nonproportional fatigue data. A nonproportional cyclic hardening parameter computed from the strain path is also proposed that allows life estimates to be obtained directly from the strain history without the need for a cyclic plasticity model.

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This paper reviews the evolution of the critical plane damage models and traces their origins from the early work such as that of Guest. Physical justification in the form of detailed observations of crack nucleation and early growth are provided for the models. A common feature of all successful models is that they consider both cyclic stresses and strains. Material-dependent failure models are needed to account for the differences in crack nucleation and early growth. Shear strain-based models are appropriate for materials that have substantial Mode II growth. Tensile strain-based models are needed for materials that have predominantly Mode I growth. Problems and inconsistencies in interpreting the damage models for variable amplitude nonproportional loading are discussed. Critical experiments for evaluating and discriminating between proposed damage models are suggested.

Biaxial fatigue tests were conducted on Inconel 718 specimens at room temperature. Thin-walled tubular specimens were subjected to tension, torsion, and combined tension and torsion loading in strain control. Two strain ratio's Rε = 0 and Rε = −1.0, were investigated. Effective strain amplitudes of 1.0 and 0.5% were employed and resulted in fatigue lives ranging from 103 to 104 cycles. Fatigue lives were determined in terms of life to 0.1 and 1.0-mm cracks as well as specimen failure. Fatigue lives were correlated in terms of the Lohr and Ellison parameter for plastic strain which is based on the plane of maximum shear strain that causes the crack to grow into the thickness of the specimen and the normal strain to that plane. Good correlation was also obtained for the Kandel, Brown, and Miller parameter which is based on maximum shear strain and the normal strain to the maximum shear plane. Mean stress effects for the Rε = 0 tests were observed. These data could be correlated with the Rε = −1 data by introducing a mean stress term in the parameters which were then combined with the Coffin-Manson equation for estimating fatigue lives. Data on 24 tests correlated within a factor of 1.5 on life when fatigue lives were determined on the basis of 1-mm cracks.

The dislocation substructures created in 1100 aluminum, OFHC copper, and type 304 and 310 stainless steels by in-phase (proportional) and 90 deg out-of-phase (nonproportional) tension-torsion cyclic loading were examined with a transmission electron microscope. Multislip structures (cells and subgrains) are observed in aluminum under both in-phase and 90 deg out-of-phase tension-torsion loading. For copper and stainless steel, single-slip structures (planar dislocations, matrix veins, and ladders) are observed after proportional loading, whereas multislip structures (cells and labyrinths) are observed after nonproportional loading. The increased cyclic hardening of copper and stainless steels under nonproportional loading is attributed to the change of dislocation substructures. Based on these observations, formulation of a nonproportionality parameter for constitutive modeling is discussed.