Plane Stress Loading Conditions Research Articles

Dual series solution for the standardized ISRM Brazilian disc test modelled as an advancing contact problem

A new analytical solution is found for the stress and displacement fields in a standardized ISRM Brazilian disc test with curved jaws, based on the solution of a set of dual trigonometric series. The disc is assumed to be linear elastic and isotropic and in frictionless advancing contact with the two jaws assumed as rigid, under plane stress loading conditions. Use is made of the Michell solution for an elastic disc in polar coordinates, whose coefficients are found by imposing the mixed boundary conditions along the disc rim, both along the free surface and the contact region. The problem is first reduced to a set of dual trigonometric series and then to a linear system of infinite equations, which is solved by truncation. The non-linear relations providing the contact angular extent and pressure distribution in terms of the applied load or jaw displacement, consequent to the progressive advance of the contact, are derived by using an inverse method. The obtained results are validated by comparison with previous theoretical and experimental results available in the literature. The study indicates that the method of dual series is simpler and more straightforward as compared to the analytical methods proposed in the literature for treating the Brazil disc test as an advancing contact problem.

Experimental studies on the extent of 3D and 2D stress–strain states at sharp V-notches

A sharp V-notched plate under plane stress loading conditions is always accompanied by a strong three-dimensional (3D) stress–strain region close to the notch tip, followed by a 3D-2D transition state and a region dominated by the 2D stress–strain field. Experimental studies on planar V-notched bodies constitute an important part of notch fracture mechanics and therefore it is important to know the extents of the above fields for accurate measurement of field variables. In the present work, experimental verification of the extent of the 3D stress–strain field, 3D-2D transition zone and plane stress zone ahead of the tip of a sharp V-notch has been carried out in order to suggest a minimum radius required for identification of the plane stress dominant zone. Knowing this value apriori is vital for accurate sampling of the 2D field variables in any experimental study on the sharp V-notched configurations. In order to achieve this, strain gage experiments on the single-edge notched tensile (SENT) specimens have been conducted. The above zones have been identified by studying the error in the measured mode I NSIF of the SENT specimen. The experimental results of the current study agree well with the results of published numerical works. The results clearly show that the minimum radius of the plane stress dominant zone has been found to be 1.25 times the thickness of the plate. Results also show that 1.5 times the thickness is the most conservative and dependable value for this radius. The results of the present investigation also indicate that not only the identification of the plane stress region, but also the locations of the sampling points within the plane stress region are important for achieving the required accuracy.

User-friendly anisotropic hardening function with non-associated flow rule under the proportional loadings for BCC and FCC metals

This paper presents a user-friendly function under non-associate flow rule to capture anisotropic hardening (or evolution of yield surface) and the curvature of yield loci for body-centred cubic (BCC) and face-centred cubic (FCC) under the proportional loading conditions. The calibrated model for BCC and FCC metals is applied to illustrate the hardening behavior of uniaxial tension along different loading directions and of the equibiaxial tension for several steels and aluminum alloys (718AT, QP980, AA5182-O and AA 6xxx-T81) to evaluate its accuracy. The prediction is compared with the experimental results and modeling results of several popular plasticity models. The comparison demonstrates that the proposed model is pressure-insensitive, relatively simple for numerical implementation, capable of illustrating the different yielding behaviors between BCC and FCC metals and, capable of accurate modeling of flow curves under directional loading directions and equibiaxial tension both under the plane stress and triaxiality loading conditions. The proposed model is also flexible on adjusting the curvature of the yield surface between uniaxial tension and equibiaxial tension. Accordingly, the proposed anisotropic hardening model is useful to illustrate anisotropic hardening behaviors under uniaxial and equibiaxial tension.

Evolution of multiple Martensite variants in a SMA thick-walled cylinder loaded by internal pressure

The stress and deformation fields in a SMA ring or a thick-walled cylinder loaded by internal pressure at constant temperature (over the start temperature of the martensitic transformation) are determined in closed form under plane stress loading conditions. The phenomenological SMA constitutive model incorporates the volume fractions of multi-variants Martensite, which are assumed to evolve linearly with the Tresca effective stress, according to the associative flow rule and the corner flow rule. Initially, the cylinder is everywhere in a state of Austenite. The application of an internal pressure then triggers the martensitic transformation starting from the inner radius of the cylinder wall and extending towards the outer radius. If the wall thickness is large enough, the tangential stress may vanish at the inner radius and correspondingly the stress state may reach a corner of the Tresca transformation condition, thus originating two different Martensite variants according to the corner transformation rule. The admissible phase partitions within the wall thickness originating during the loading process have been systematically investigated according to the ratio between the outer and inner radii. The results obtained here suggest that the loading process should be interrupted soon after the complete martensitic transformation is achieved at the inner radius of the cylinder to avoid permanent plastic deformations.

A theoretical resistance-curve based on nonproportional plastic strain

Using an exact solution for an elliptical hole in a perfectly plastic material, an expression is derived for the resistance of a ductile material undergoing subcritical crack propagation in the plane. This resistance curve is based on an analogy to the J-integral where an energy dissipation rate is determined rather than an energy release rate. The Tresca yield condition under plane stress loading conditions is employed in this derivation as well as finite deformation theory. This resistance curve is applicable to the initial stage of subcritical crack growth for a ductile material subject to crack tip blunting.

A new test for determining the mechanical and fracture behavior of materials in sheet-bulk metal forming

This paper presents a new experimental test for determining the stress–strain curve and the fracture toughness of sheets to be used in sheet-bulk metal forming (SBMF) applications. The test is based on the utilization of double-notched specimens loaded in shear and combines the plane stress loading conditions of sheet metal forming with the three-dimensional plastic flow conditions of bulk metal forming, which are commonly found in SBMF processes. The methodology to obtain the stress–strain curve involves calculation of the shear stresses and strains along the two symmetric plastic shear zones of the test specimens up to point where cracks start to propagate along the ligaments that connect each pair of opposite notches. The determination of fracture toughness involves characterization of the evolution of load with displacement for a number of test cases performed with specimens having different ligaments between the two symmetric opposite notches. The work is performed on aluminium alloy EN AW 5754 H111 sheets with 5 mm thickness and the results obtained by means of the new proposed test are compared against those from conventional mechanical and fracture characterization tests.

The Interaction of the Plane Stress Mode I Crack with an Inhomogeneity Undergoing a Stress-Free Transformation Strain

Thin composite films containing inclusions are commonly being developed as corrosion resistant, wear resistant coatings and so forth. For systems where the composite film is under residual stresses properly designed, a controlled debonding process of the inclusions can be used to reduce the stress levels in the film lowering the risk of through cracks in the film as well as the risk for the film's delamination from the substrate. In this paper, on the basis of the Eshelby equivalent inclusion theory, a general solution is derived by treating an inhomogeneous inclusion as a homogeneous one with transformation strain for the configuration force (CF) and stress intensity factor (SIF) between mode I crack and an inhomogeneous inclusion of arbitrary shape which undergoes some degree of stress-free transformation strain under plane stress loading conditions. Then the CF associated with the transformation is calculated from the work done during the transformation, from which some simplified approximate formulae are also presented for common inclusion shapes in order to provide a quick estimate for the effects of inclusion shape, location and size on the CF of plane stress model I crack. In comparison to conventional numerical approaches, the present solution provides a novel approach to explain the behavior of crack deflection/penetration and can be used for the optimization design of the composite films.

The plasticity-corrected stress intensity factor for plane stress mode I and mode II cracks

An approximate analytical solution for plasticity-corrected stress intensity factor (PC–SIF) is developed on the basis of Eshelby equivalent inclusion theory and transformation toughening theory both for plane stress mode I and mode II cracks. As validated by detailed finite element analysis the approximate solution has good accuracy. The proposed PC–SIF provides a new mechanical parameter to describe the fatigue and fracture behaviors associated with crack tip plastic zone for mode I and mode II cracks under plane stress loading conditions.

Contact Transform for the Biharmonic Equation Applicable to Plane Stress Elastoplastic Elliptical Hole Problems

An analytical technique is developed that reduces the unknown elastic-plastic boundary of a linear elastic-perfectly plastic material containing an elliptical hole under tensile plane stress loading conditions into an equivalent mathematical problem with known boundaries. This mathematical transformation may facilitate this problem’s solution by either analytical or numerical means. Yield is assumed to occur in this analysis under the Tresca yield criterion. An example elastic-plastic problem illustrating this method is drawn from existing literature in the form of a perturbation solution for an elliptical hole derived by a series expansion about a circular boundary.

The yield condition strongly influences the formation of Dugdale plastic strips ahead of crack tips under tensile plane stress loading conditions

AbstractA finite element analysis indicates a good correlation between the Dugdale plastic strip model and a linear elastic/perfectly plastic material under plane stress loading conditions for a flow theory of plasticity based on the Tresca yield condition. A similar analysis under the von Mises yield condition reveals no plastic strip formation.

Investigation on the strain-path dependency of stress-based forming limit curves

Path-dependent forming limits have been computed for sheet metals undergoing various combinations of plane stress loading conditions. This paper presents a theoretical model for prediction of stress-based forming limit curves (SFLC) based on the Marciniak and Kuczynski (MK) model. Acceptable agreement was observed between calculated forming limit curves (FLC) and experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 alloys (Graf and Hosford Metallurgical Trans 24A:2503–2512, 1993). In this paper, the path dependency of SFLCs predicted for different non-proportional loading histories has been investigated. For a range of prestrain values in different bilinear loading paths, the SFLC remains practically unchanged. However, some strain path dependency is observed for large values of prestrain (\( \bar{\varepsilon } \geqslant 0.35 \) for AISI-1012 steel) and for abrupt changes in strain path. Nevertheless, the SFLC remains a good failure criterion for virtual forming simulations because the path dependency of SFLCs is much less significant than that of strain-based FLCs.

Path Independent Integral for an Elliptical Hole in a Plate under Tension for Plane Stress Deformation Theory

An exact expression is derived for a path independent integral surrounding an elliptical hole in an infinite plate with tensile tractions at infinity for plane stress loading conditions. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under proportional loading and small strain assumptions. This problem may serve as a simple classroom example for the derivation of a relationship between crack opening displacement and path independent integral for a nonlinear elastic material satisfying the Tresca yield condition.

Numerical plane stress elastic–perfectly plastic crack analysis under Tresca yield condition with comparison to Dugdale plastic strip model

A number of plane stress numerical analyses of the mode I elastoplastic fracture mechanics problem have been performed in the past using the Huber–Mises yield criterion. This study employs instead the Tresca yield condition using an incremental theory of plasticity for a stationary crack. A commercial finite element program is used to solve the opening mode of fracture problem (mode I) for a square plate containing a central crack under generalized plane stress loading conditions. A biaxial uniform tensile traction is applied to the edges of a thin plate composed of a linear elastic non-work hardening material under small strain assumptions. The finite element results are compared with the analytical predictions of the Dugdale plastic strip model for a crack in an infinite plate subject to a biaxial uniform load at infinity.

A Complete Perfectly Plastic Solution for the Mode I Crack Problem Under Plane Stress Loading Conditions

A continuous stress field for the mode I crack problem for a perfectly plastic material under plane stress loading conditions has been obtained recently. Here, a kinematically admissible velocity field is introduced, which is compatible with the continuous stress field obtained earlier. By associating these two fields together, it is shown that they constitute a complete solution for the uncontained plastic flow problem around a finite length internal crack, having a positive rate of plastic work. The yield condition employed is an alternative criterion first proposed by Richard von Mises in order to approximate the plane stress Huber-Mises yield condition, which is elliptical in shape, to one that is composed of two intersecting parabolas in the principal stress plane.

Perfectly plastic caustics for the opening mode of fracture

Exact expressions for the caustics generated by the reflection of light surrounding crack tips in perfectly plastic materials under plane stress loading conditions and tensile tractions at infinity (mode I) are derived. Two individual cases are examined involving two different yield criteria. The first case uses an approximation of the Mises yield condition, where in the principal stress plane two intersecting parabolas replace the standard ellipse. The second case uses the Tresca yield condition where the mode I caustic is obtained as a limit of an elliptical hole in a perfectly plastic material. In both cases, kinematically admissible velocity fields are employed to obtain strain fields from which the theoretical caustics are predicted.

Crack growth in a new nickel-based superalloy at elevated temperature

Crack growth at elevated temperature has been simulated using the finite element method for sustained and cyclic loading conditions, representative of time-dependent and time-independent crack growth. Elastic-creep (EC) and elastic-plastic-creep (EPC) models have been used to simulate the crack growth under sustained loads at 650 and 725°C. Crack mouth opening displacements as well as the evolution of the inelastic zones due to creep and plasticity have been obtained. Elastic-plastic finite element analysis has been carried out to simulate the crack growth under cyclic load using a constitutive model. Fatigue crack growth was simulated for plane stress, plane strain and generalized plane strain loading conditions. The influence of plasticity on the effective crack driving force was also examined. Creep damage was found to be very limited at both temperatures for this alloy. Plasticity-induced crack closure was found to be absent in plane strain or generalized plane strain conditions, overestimated in plane stress loading conditions by the conventional compliance method.

Scale dependent crystal plasticity framework with dislocation density and grain boundary effects

The geometrically non-linear scale dependent response of polycrystal FCC metals is modelled by an enhanced crystal plasticity framework based on the evolution of several dislocation density types and their distinct physical influence on the mechanical behaviour. The isotropic hardening contribution follows from the evolution of statistically stored dislocation (SSD) densities during plastic deformation, where the determination of the slip resistance is based on the mutual short range interactions between all dislocation types, i.e. including the geometrically necessary dislocation (GND) densities. Moreover, the GND's introduce long range interactions by means of a back-stress measure, opposite to the slip system resolved shear stress. The grain size dependent mechanical behaviour of a limited collection of grains under plane stress loading conditions is determined using the finite element method. Each grain is subdivided into finite elements and an additional expression, coupling the GND densities to spatial crystallographic slip gradients, renders the GND densities to be taken as supplemental nodal degrees of freedom. Consequently, these densities can be uncoupled at the grain boundary nodes, allowing for the introduction of grain boundary dislocations (GBD's) based on the lattice mismatch between neighbouring grains and enabling the obstruction of crystallographic slip perpendicular to the grain boundary.

Stress across an elastoplastic boundary of a mode I crack: parabolic to hyperbolic plasticity transition

In previous work, the stresses of a mode I elastic–plastic fracture mechanics problem were analytically continued across a prescribed elastoplastic boundary for plane stress loading conditions involving a linear elastic/perfectly plastic material obeying the Tresca yield condition. Immediately across the elastic-plastic boundary, a nonlinear parabolic partial differential equation governs the plastic stress field. The present solution deals with stresses extending beyond the parabolic region into the hyperbolic region of the plastic zone. This analytical solution is obtained through a tranformation of the original system of nonlinear partial differential equations into a linear system with constant coefficients. The solution, so obtained, is expressible in terms of elementary transcendental functions. It also exhibits a limiting line which passes through the crack tip. This feature of the solution suggests the formation of a plastic hinge in the material.

In-plane electronic speckle pattern shearing interferometry

An analysis of the out-of-plane shearing interferometer has been performed which shows that production of in-plane strain partial derivatives is possible, which are not affected by out-of-plane displacement function components. The in-plane data are represented as subtraction correlation fringes. This interferometer employs a single diverging illumination beam and is applicable to object plane stress and plane strain loading conditions. The interferometer was tested and compared using a compact tension crack specimen and the results are correlated with finite element software predictions of strain distributions across modelled specimens. This experimental validation was chosen because we had an existing test rig and finite element models which had been independently verified.

Influence of the Microstructure on the Crack Resistance of Cast Irons / Einfluß des Gefüges auf den Rißwiderstand von Gußeisen

The morphology of the surface formed after crack propagation for five nodular and four lamellar graphite cast irons both with ferritic and ferritic-pearlitic microstructure has been studied. Single edge notched three point bending specimens were fatigue precracked and then tested. The critical stress intensity factor K C in plane stress loading conditions has been determined. The crack paths and the morphology of the fracture surfaces were investigated by light microscopy and scanning electron microscopy. The influence of the microstructural components on the type of fracture and the K c values has been studied. It is established that the presence of a plastic deformation zone in the crack tip vicinity is typical for the nodular cast irons where relatively ductile type of fracture takes place. The graphite and pearlite constrain the plastic deformation in the crack tip vicinity thus reducing the K C values. The graphite plays a principal role in this phenomenon. The behaviour of the graphite and the pearlite is entirely different in the brittle type of fracture typical for the lamellar graphite cast irons - both these phases favour the crack propagation.