Power-law Hardening Materials Research Articles

The Improved Irwin’s Model for Crack Problems in Power-Law Hardening Materials

In this paper, an improved Irwin’s model for power-law hardening materials is established through the static equivalence between the linear elastic stress solution and the stress redistribution in the plasticity-affected zone (PAZ) under small-scale yielding (SSY) conditions. HRR solution is adopted to describe the stress field within the crack-tip plastic zone (CTPZ). Analytical expressions of the effective stress intensity factor (SIF) and the size of PAZ are derived to illustrate the effect of CTPZ on the fracture behavior for power-law hardening materials. Influences of external load and hardening exponent on fracture behavior are explored. The proposed model can assess the toughening effect of CTPZ accurately using the SIF ratio, which is always smaller than 1. This model can be applied to various materials obeying power-law hardening constitutive equations. It will reduce to the extended Irwin’s model for elastic-perfectly plastic materials in the limiting case.

Stress distribution model for the entire plastic zone of conical indentation with power-law hardening materials

<p indent="0mm">The classical contact mechanics provides an elastic conical indentation solution, and verifies that the strain field of elastic-plastic indentation with an arbitrary blunt indenter mostly satisfies the expanding cavity model. This study reviews the stress solution of the elastic zone for conical indentation using power-law hardening materials derived from predecessors based on the expanding cavity model and elastic-plastic mechanics. Furthermore, the expression of von-Mises equivalent stress in the plastic zone of conical indentation is determined according to the plastic stress solution of an internally pressurized thick spherical shell. The numerical verification revealed that the calculated equivalent stresses via finite element analysis are consistent with the theoretical solution of the elastic zone for conical indentation. However, a clear deviation in the entire plastic zone for the power-law hardening materials is still observed. Furthermore, to accurately describe the equivalent stress in the entire plastic zone of conical indentation for power-law hardening materials, a novel model for the indented stress distribution is established by combining the dimensional analysis method and finite element simulation. Finally, the stress model was verified using a wide range of ductile materials. The comparison results revealed that all of the calculated equivalent stresses are consistent with the model predictions for the entire plastic zone.

Determination of biaxial residual stresses via indentation energy difference method

<p indent="0mm">In this study, a novel method is proposed to estimate biaxial residual stress (RS) by using the indentation energy difference between the stressed and the stress-free samples. A Knoop indenter is applied along the directions of principal stresses, and the indentation energy obtained from the load-depth (<italic>F</italic>-<italic>h</italic>) curves is used as the calculation parameter without observing the indentation morphology. A work-energy transformation model is constructed to explain the change in the <italic>F</italic>-<italic>h</italic> curve with the RS in terms of energy. Then, the finite element analysis of different power-law hardening materials under different RS combinations is implemented to study the energy evolution. Consequently, the formula for calculating the indentation energy difference under a uniaxial RS is established. Meanwhile, the indentation energy difference of the Knoop indenter exhibits superposition and direction characteristics, which can be used to convert the indentation energy difference under a biaxial RS into the sum of that under two equivalent uniaxial stresses. Through this conversion, the two stress components can be calculated from the two indentation energy difference formulas. An indentation testing device is developed to verify the proposed method. The results obtained using the proposed method agree well with those obtained using the indentation strain-gage method.

Unified theoretical solutions for describing the crack-tip stress fields of finite specimens with mode-I crack under fully plastic conditions

The analytical equation to express the equivalent stress of the representative volume element (RVE) is derived from the energy density equivalence. Based on the analytical expression for equivalent stress of the RVE, extensive analyses are conducted for the stress fields in the forward sector near the crack-tip of finite specimens with mode-I crack for pure power-law hardening materials. Further, unified theoretical solutions with various special functions for describing the stress fields are proposed under fully plastic and plane-strain conditions. Comparisons of radial stress distributions in various angles, angular stress distributions, and contour lines of the equivalent stress between the results predicted by the finite element analysis, HRR (Hutchinson and Rice and Rosengren) solutions, and the proposed unified theoretical solutions are performed. The excellent capability of the proposed unified theoretical solutions for predicting the crack-tip stress fields is verified.

Semi-analytical expressions to describe stress fields near the tip of Mode-I crack under plane-strain conditions

In this study, the stress distribution near a mode-I crack tip under plane-strain conditions for power-law hardening materials and its semi-analytical expression are investigated. The analytical equations to express the equivalent stress and strain of the representative volume element are derived based on energy density equivalence and dimensional arguments. After acquiring the correspondence between the equivalent stress and the derived J-integral for fully plastic materials, the present work proposes a mathematical formula to characterize the stress fields near the crack tip that is closely related to the Hutchinson–Rice–Rosengren asymptotic fields. Furthermore, semi-analytical expressions are proposed to describe the stress fields in the close vicinity of the crack tip. The findings of this study will be useful in structural safety assessment and can promote theoretical advancement in the field of fracture mechanics.

Asymptotic Solutions of Plastic Stress and Displacement at V-Notch Tips Under Anti-Plane Shear

An efficient method was developed to determine the first- and high-order terms of asymptotic solutions of plastic stress and displacement near V-notch tips under anti-plane shear in power-law hardening materials. Through introduction of the asymptotic series expansions of stress and displacement fields around the V-notch tip into the fundamental equations of the elastoplastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions were established. Then the interpolating matrix method was employed to solve the resulting nonlinear and linear ODEs. Consequently, the high-order stress exponents and the associated eigen-solutions were obtained. The presented method, being capable of dealing with the V-notches with arbitrary opening angles and strain hardening indexes under anti-plane shear, has the advantages of great versatility and high accuracy. Typical examples were given to demonstrate the accuracy and effectiveness of this method.

Dimensional transitions in creeping materials due to nonlinearity and microstructural disorder

Abstract The transition from extremely brittle to very ductile behaviours of creeping materials is discussed, where analogies with power-law hardening materials are pointed out. Considering Norton's Law as a viscous constitutive law, it is possible to define a generalized stress-intensity factor Kc ―characterizing the intermediate asymptotic behaviour under steady-state creep conditions― with physical dimensions depending upon the Norton stress exponent n. In the two limit cases of creep resistant materials (n≅1) and creep sensitive materials (n ≫ 1), Kc assumes respectively the dimensions of an elastic stress-intensity factor ( F L − 3 / 2 ) and of a stress ( F L − 2 ). Such a dimensional transition, with consequent stress-singularity attenuation, is completely analogous to that occurring through the introduction of a fractal stress-intensity factor (Kc)*, when the influence of microstructural disorder is considered.

Complete elastic–plastic stress asymptotic solutions near general plane V-notch tips

An efficient method is developed to determine the multiple term eigen-solutions of the elastic–plastic stress fields at the plane V-notch tip in power-law hardening materials. By introducing the asymptotic expansions of stress and displacement fields around the V-notch tip into the fundamental equations of elastic–plastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions are established. Then the interpolating matrix method is employed to solve the resulting nonlinear and linear ODEs. Consequently, the first four and even more terms of the stress exponents and the associated eigen-solutions are obtained. The present method has the advantages of greater versatility and high accuracy, which is capable of dealing with the V-notches with arbitrary opening angle under plane strain and plane stress. In the present analysis, both the elastic and the plastic deformations are considered, thus the complete elastic and plastic stress asymptotic solutions are evaluated. Numerical examples are shown to demonstrate the accuracy and effectiveness of the present method.

Estimation of stress field for sharp V-notch in power-law creeping solids: An asymptotic viewpoint

The characterization of the stress field for notch contained structures and components at elevated temperature is a preliminary to assess their safety and integrity reasonably. In this paper, the characteristics of the sharp V-notch tip field in power-law creeping materials are presented from the asymptotic viewpoint. Through asymptotic analysis and solution, the stress exponents and the angular distribution functions for various notch angles and creep exponents are presented. The asymptotic structure of the stress field for the sharp V-notch tip field is presented by proposing the amplitude factor K(t) which is dependent on the creep time, specimen geometry, notch angle as well as loading condition. The amplitude factor K(t) can bridge the fracture parameters between notch and crack in power-law creeping solids. A reasonable and accurate estimation method for the evaluation of the K(t) factor is also proposed. With the determination of K(t), the notch tip stress fields of single edged sharp V-notch specimens in power-law creeping materials with various notch angles and different depths are presented, compared and analyzed. It has been verified that the stress components for sharp V-notch specimens can be estimated accurately and reasonably with the theory and method proposed in this paper. The differences of the asymptotic solutions for sharp V-notch tip in power-law creeping solids and power-law hardening materials are also discussed and clarified.

Semiempirical Formulae for Mechanical Properties Controlled by Strength and Ductility of Power-Law Hardening Metallic Materials

Some subordinate mechanical properties, such as the strain hardening exponent (n) and the fracture true stress ( $$\sigma_{\text{f}}$$ ), are not available in general material property handbooks, although they are important in choosing materials. This paper investigated the relations between the subordinate mechanical properties to the basic ones for power-law hardening materials. Our semiempirical analysis shows that, in uniaxial tensile experiment, the strain hardening exponent is controlled by the ratio of the yield strength ( $$\sigma_{\text{s}}$$ ) and the ultimate tensile strength ( $$\sigma_{\text{b}}$$ ), and the value also equals to the maximum uniform strain. The absolute calculation errors of n on various metal materials are approximately within ± 0.03. Physically, n expresses the ability of the strength rising from the yield strength to the ultimate tensile strength during uniform deformation; it may also denote the maximum ability of uniform deformation before necking. The fracture strength is controlled directly by n, $$\sigma_{\text{b}}$$ and the reduction in area (ψK) in our concise expression. The relative errors of the fracture strength between the calculations and the measurements on metal materials are approximately within 20%. Results also indicate that the fracture true stress represents the increased stress during uniform deformation plus the concentrated stress caused by necking.

A special notched tensile specimen to determine the flow stress-strain curve of hardening materials without applying the Bridgman correction

Structural integrity assessment of weldments requires the input of flow stress-strain curve of each individual material zone. To cope with these challenges, a cylindrical cross weld tensile specimen with a notch located either in the weld metal, base metal or possibly heat affected zone has been previously developed by the authors to determine the true stress-strain curve for the material zone of interest. The disadvantage of this notched tensile testing method as well as the standard tensile testing method using a smooth specimen, is that the well-known Bridgman correction still has to be applied in order to obtain material’s equivalent or flow stress-strain curves. In this study, tensile specimens with various notch geometries have been scrutinized and a ‘magic’ specimen with a special notch geometry has been identified. By using this special notched tensile specimen, material’s flow stress-strain curve can be directly calculated from the recorded load versus diameter reduction curve and no Bridgman correction is needed. The method is very accurate for power-law hardening materials and becomes less accurate for materials with significant Lüders plateau in the initial yield region.

Semi-analytical and numerical analysis of sliding asperity interaction for power-law hardening materials

The study of the sliding process between asperities on rough surfaces can improve the understanding of wear mechanisms. The sliding interaction between asperities is analyzed in this paper using both a semi-analytical model and a finite element model. Power-law hardening materials are considered, and the asperity profiles are assumed to be a parabolic approximation to the cylinder. The effects of strain hardening exponents on some contact parameters are explored with the finite element model. Results show that the faster semi-analytical model agrees well with the finite element model for materials with larger hardening exponents, while for materials with smaller exponents, the errors would preclude its use. As the exponent decreases, the dragging effect in sliding becomes more notable and influences the contact parameters more significantly. Friction shows a significant effect in the sliding process after preliminary consideration, which should be explored in detail further.

Analytical asperity interaction model and numerical model of multi-asperity contact for power hardening materials

In the present work, an analytical interaction model of multi-asperity contact for the power-law hardening materials is proposed. The real surface topography including the asperity locations, heights and radii of summit is considered. Meantime, a numerical model is built based on the finite element method, to study the contact of a rigid flat and a deformable rough surface which is simplified and reconstructed with the measured original surface using the wavelet transform. The analytical results are close to the numerical results. Finally, the effect of the power-law hardening material properties on the asperity interaction is studied.

Loading and unloading of a power-law hardening spherical contact under stick contact condition

The loading and unloading process of a deformable sphere pressed by a rigid flat is a primary problem in contact mechanics. This work studies the contact of an elastic–plastic sphere with a rigid flat under stick condition during loading and unloading process by the finite element method. The sphere material is assumed isotropic with power-law hardening. The contact load and contact area are calculated at various Poisson׳s ratios and strain hardening exponents, with the sphere material properties varying from purely elastic to elastic-perfectly plastic. Analytical dimensionless expressions of the contact load, contact area and the residual interference after fully unloading for the power-law hardening materials in the elastic–plastic case are proposed for a wide range of interferences.

On the relationships between hardness and the elastic and plastic properties of isotropic power-law hardening materials

A comprehensive understanding of the relationship between the hardness and the elastic and plastic properties for a wide range of materials is obtained by analysing the hardness characteristics (that are predicted by experimentally verified indentation analyses) of over 9000 distinct combinations of material properties that represent isotropic, homogeneous, power-law hardening metallic materials. Finite element analysis has been used to develop the indentation algorithms that provide the relationships between the elastic and plastic properties of the indented material and its indentation hardness. Based on computational analysis and virtual testing, the following observations are made. The hardness (H) of a material tends to increase with an increase in the elastic modulus (E), yield strength (σy) and the strain-hardening exponent (n). Several materials with different combinations of elastic and plastic properties can exhibit identical true hardness (for a particular indenter geometry/apex angle). In general, combinations of materials that exhibit relatively low elastic modulus and high yield strength or strain-hardening exponents and those that exhibit relatively high elastic modulus and low yield strength or strain-hardening exponents exhibit similar hardness properties. Depending on the strain-hardening characteristics of the indented material, (i.e. n = 0 or ), the ratio H/σy ranges, respectively, from 2.2 to 2.6 or 2 to 20 (for indentations with a cone angle of 70.3°). The materials that have lower σy/E and higher n exhibit higher H/σy ratios. The commonly invoked relationship between hardness and the yield strength, i.e. H ≈ 3σy, is not generally valid or applicable for all power-law hardening materials. The indentation hardness of a power law hardening material can be taken as following the relationship H ≈ (2.1–2.8)σr where σr is the representative stress based on Tabor’s representative strain for a wide range of materials.

Stress singularity of symmetric free-edge joints with elasto-plastic behaviour

The stress singularity for symmetric free-edge joint is presented for isotropic elastic and elasto-plastic materials. Complex functions were used to describe the elastic stress and displacement fields and an eigenvalue problem was solved. The singularity orders for the elastic materials were plotted versus joint angles for different material combinations investigating the influence of Poisson’s ratio.For Ramberg–Osgood power-law hardening materials the shooting method was implemented to obtain the unknown boundary conditions at free-edge, necessary for solving numerically the eigenvalue problem, using a fourth-order Runge–Kutta method. The obtained numerical results are compared with those of a highly focused finite element (FE) analysis.The authors proposed a new formula to estimate the singularity order for the elasto-plastic stress singularity based on the hardening parameter and the computed elastic singularity for incompressible material.

Asymptotic Analysis for the Singular Stress Behaviour around an Interface Edge of Dissimilar Power-Law Hardening Materials Joint

An asymptotic analysis for singular stress fields around an interface-edge of dissimilar power-law hardening materials joint has been presented under plane strain condition and J2 deformation plasticity theory. Both the balance of force and the continuity of displacement are satisfied on the interface. In the higher order approximation, the nonlinear effective stress term was expanded by Taylor series. An iteration method is proposed for the determination of singular fields around interface edge. Multiple stress singular terms exist for in the higher order approximation. The order of stress singularity has a dependency with the combination of hardening exponents, .

The equivalent axisymmetric model for Berkovich indenters in power-law hardening materials

Nix and Gio [Nix, W.D., Gao, H.J., 1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids 46, 411–425] established an important relation between the micro-indentation hardness and indentation depth for axisymmetric indenters. For the Berkovich indenter, however, this relation requires an equivalent cone angle. Qin et al. [Qin, J., Huang, Y., Xiao, J., Hwang, K.C., 2009. The equivalence of axisymmetric indentation model for three-dimensional indentation hardness. Journal of Materials Research 24, 776–783] showed that the widely used equivalent cone angle from the criterion of equal base area leads to significant errors in micro-indentation, and proposed a new equivalence of equal cone angle for iridium. It is shown in this paper that this new equivalence holds for a wide range of plastic work hardening materials. In addition, the prior equal-base-area criterion does not hold because the Berkovich indenter gives much higher density of geometrically necessary dislocations than axisymmetric indenter. The equivalence of equal cone angle, however, does not hold for Vickers indenter.

Plastic J-integral calculations using the load separation method for the double edge notch tension specimen

The load separation method was used to determine the plastic work factor ( η pl) for the double edge notch tension (DENT) geometry for power-law hardening materials. The resulting expression for η pl compares favorably to the results of Rice, Paris, and Merkle. The results were also compared to η pl calculated from the plastic J results in EPRI Handbook NP-1931 and substantial differences were noted. A dependence on gage length was identified in the load separation results which indicate a gage length to half-width ratio of at least 3 is required to ensure that all of the plastic displacement is fully sensed and that η pl is independent of gage length. Additionally, a coupling term between crack size and material hardening exponent was identified in the geometric load separation function G.

On the uniqueness and sensitivity issues in determining the elastic and plastic properties of power-law hardening materials through sharp and spherical indentation

A systematic study of the uniqueness, reversibility and sensitivity issues associated with seven indentation-based methods of property extraction demonstrates that: (i) The indentation algorithms generally identify the elastic and plastic properties of materials uniquely for most materials. (ii) The indentation forward algorithms (wherein the indention responses are determined from the elastic and plastic properties of the indented materials) and the reverse algorithms (wherein the elastic and the plastic properties of materials are extracted from the indentation responses) are distinct for each indentation method and are internally consistent in that the differences in the elastic and plastic properties determined through the reverse analysis and the ‘true’ material properties are generally small for a large number of materials, for each of the seven methods. (iii) While the differences in the indentation response parameters predicted by each of the seven indentation methods (for a particular material) could be small, there could be considerable dispersion in the elastic and plastic properties predicted by the reverse algorithms of the seven methods (for a particular set of indentation response parameters). (iv) In the forward analysis, small uncertainties in the elasto-plastic properties lead to small uncertainties in the predictions of the indentation response of materials. The sensitivity distribution is generally heterogeneous and symmetric across positive and negative variations in the material elasto-plastic properties. (v) In the reverse analysis, the elastic modulus exhibits low sensitivity, while the yield strength and the strain-hardening exponent generally exhibit high sensitivity to uncertainties in the indentation response parameters. The sensitivity distribution is heterogeneous and asymmetric across positive and negative variations in the indentation response parameters. (vi) The representative stresses are fairly robust to uncertainties in the indentation response parameters. Consequently, dual sharp and spherical indentation methods, which identify multiple representative stresses, exhibit reduced sensitivity in the determination of the plastic properties.