Finite element analysis of mode III steady crack growth in anisotropic hardening materials

In this paper, the steady crack growth of mode III under small scale yielding conditions is investigated for anisotropic hardening materials by the finite element method. The elastic-plastic stiffness matrix for anisotropic materials is given. The results show the significant influences of anisotropic hardening behaviour on the shape and size of plastic zone and deformation field near the crack tip. With a COD fracture criterion, the ratio of stress intensity factorsk _ss/k_c varies appreciably with the anisotropic hardening parameterM and the hardening exponentN.     

An approach to the elastic-plastic analysis of the plane strain mode-I crack

This paper presents an approach to the solution of the approximate elastic-plastic analysis for the plane strain mode-I crack.     

Elastic-plastic crack growth on plane strain bimaterial interface

Finite element computation are carried out to simulate plane strain crack growth on a bimaterial interface under the assumption of small scale yielding. The modified Gurson constitutive equation and the element vanish technique introduced by Tvergaard et al. are used to model the final formation of an open crack. It is found from the calculation that the critical fracture toughness for crack growth is much lower in bimaterials than that in homogeneous material. The critical fracture toughness is strongly dependent on material properties of the bimaterial pair and the mixed mode of remote loads. The interface crack grows in the more compliant (lower hardening) material or in the weaker (lower yield strength) material. In Mode-I loading, the crack grows zigzag along the interface. Project supported by Fok Ying-Tung Education Foundation and National Natural Science Foundation of China.     

A line plastic-zone model for steady mode III crack growth in an elastic-plastic material

Steady-state quasi-static growth of a crack in the anti-plane shear mode through an elastic-plastic material is analyzed. The material is non-hardening and small-scale yielding conditions are assumed. The essential feature of the model is that the active plastic-zone is assumed to be a pair of discrete lines emanating from the crack tip out of the crack plane on which a suitable yield condition is satisfied. An exact solution is obtained for the plastic strain left in the wake of this active line plastic-zone. The extent of the plastic zone from the tip is determined to be 0.071 $\text{(}\text{k}\text{τ}{}_0\text{)}{}^2$ where k and τ₀ are the remote elastic stress intensity factor and the shear flow stress, respectively, and it is found that 36% of the elastic energy flowing into the crack-tip region during growth is dissipated through plastic work and 64% is trapped as residual elastic energy in the plastic-zone wake.     

Subcritical crack growth in strain-hardening materials with allowance for strain anisotropy

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 2, pp. 90–94, February, 1988.     

Pseudo plane stress mode II crack growth in plastic materials

The near tip field of mode II crack that grows in thin bodies with power hardening or perfectly plastic behavior is analyzed. It is shown that for power hardening behavior, the pseudo plane stress field possesses the logarithm singularity, i.e. σ (ln r)²/(n−1), (ln r)^{2n/(n − 1)}, where r is the distance from the crack tip, n the hardening exponent is σⁿ. When n → ∞ the solution reduced to that for the perfectly plastic case.     

Near tip quasi-static crack growth behavior in strain hardening and softening material

Analyzed in this work is a semi-infinite crack that grows slowly in a steady-state. The assumed constitutive relation for the material permits strain hardening and softening as it is damaged in time. Four distinct regions divided angularly are identified for the asymptotic expressions of the quasi-static crack-tip stress field. They refer to material degraded in front of the crack; undergone elastic unloading; reloading of degraded material; and material completely by exhausted in its load carrying capacity.     

A directional crack path criterion for crack growth in ductile materials subjected to shear and compressive loading under plane strain conditions

A directional crack growth criterion in a compressed elastic perfectly plastic material is considered. The conditions at the crack-tip are evaluated for a straight stationary crack with a small incipient kink. Remote load is a combined hydrostatic pressure and pure shear applied via a boundary layer. Crack surfaces in contact are assumed to develop homogenous Coulomb friction.The crack opening displacement of an extended kink is examined in a finite element analysis to judge the risk of opening mode failure. It has been found that the direction that maximizes the crack opening displacement of an extended kink tip coincides very well with a prediction of the crack growth direction obtained by using a criterion for continued crack growth direction discussed by the authors elsewhere [Int. J. Fract. 108 (2001) 351].Moreover, the by the model predicted incipient crack growth directions are qualitatively comparable with reported crack paths obtained in ductile materials in a limited number of experiments performed under a combined load of in-plane shear and compression.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

Three dimensional microstructural effects on plane strain ductile crack growth

Ductile crack growth under mode I, plane strain, small scale yielding conditions is analyzed. Overall plane strain loading is prescribed, but a full 3D analysis is carried out to model three dimensional microstructural effects. An elastic-viscoplastic constitutive relation for a porous plastic solid is used to model the material. Two populations of second-phase particles are represented, large inclusions with low strength, which result in large voids near the crack tip at an early stage, and small second-phase particles, which require large strains before cavities nucleate. The larger inclusions are represented discretely and the effects of different three dimensional distributions on the crack path and on the overall crack growth rate are analyzed. For comparison purposes, a two dimensional distribution of cylindrical inclusions is analyzed. Crack growth occurs off the initial crack plane in all 3D computations, whereas straight ahead crack growth occurs with the two dimensional cylindrical inclusions. As a consequence, the three dimensional distributions of spherical inclusions exhibit an increased crack growth resistance as compared to the two dimensional distribution of cylindrical inclusions.     

Nonproportional loading effects on elastic-plastic behavior based on stress resultants for thin plates of strain hardening materials

A stress resultant constitutive law in rate form is constructed for power-law hardening materials. The change of plate thickness is considered in the constitutive law. The elastic-plastic behavior of a plate element based on the stress resultant constitutive law under uniaxial combined tension and bending is determined under a limited number of nonproportional and unloading paths. The results based on the stress resultant constitutive law and the through-the-thickness integration method are compared within the context of both the small-strain and finite deformation approaches. The results indicate that the selection of the normalized equivalent stress resultant and the corresponding work-conjugate normalized equivalent generalized strain is appropriate for describing the hardening behavior in the stress resultant space. However, the hardening rule in a power law form must be modified for low hardening materials at large plastic deformation when finite deformation effects are considered.     

Non-self-similar crack growth in elastic-plastic finite thickness plate

This work is concerned with non-self-similar crack growth in medium strength metal plates while the loading step, plate thickness and material properties are altered. The three-dimensional elastic-plastic finite element stress analysis is combined with the strain energy density criterion for modeling the material damage process from crack initiation to final global instability including the intervening stage of slow crack growth. Both inelastic deformation and crack growth are accounted for each increment of loading such that the redistribution of stresses and strains are made for each new crack profile. Numerical results are obtained for the center cracked plate configuration under uniform extension with twenty-seven (27) different combinations of specimen thickness, loading step and material type. The fracture toughness S_c being related to K_1c for three different materials are predicted analytically from the corresponding uniaxial tensile test data. Effective strain energy density factor and half crack length are defined so that the results can be compared with their two-dimensional counterparts. Crack growth resistance curves (R-curves) are constructed by plotting as a function of . The condition is found to prevail during slow crack growth. Translation and/or rotation of the lines can yield results other than those calculated and serve a useful purpose for scaling component size and test time. The minimum thickness requirement for the ASTM valid K_1c test is also discussed in connection with predictions based on the strain energy density criterion. The corresponding K_1c for smaller specimens that exhibit moderate ductility and nonlinearity can also be obtained analytically. In such cases, the influence of loading step can be significant and should not be neglected. Notwithstanding the shortcomings of the theory of plasticity, the qualitative features of non-self-similar crack growth are predicted by the strain energy density criterion. Any refinements on the analytical modeling of the material damage process would only affect the results qualitatively, a subject that is left for future investigation.     

THE EXACT SOLUTIONS OF ELASTIC－PLASTIC CRACK LINE FIELD FOR MODE ⅡPLANE STRESS CRACK

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Analytical solutions of mixed mode plane strain crack problems in elastic perfectly plastic materials

In the present paper analytical solutions concerning the stress state at the tip of a crack in an elastic-perfectly plastic body, subjected to mixed mode loadings under plane strain conditions, are presented. Analytical solutions of the nonlinear ordinary differential equations are obtained and the dominant singularity is completely determined with the aid of suitable boundary conditions. The obtained results are in perfect agreement with those given by other investigators, both analytical and numerical. The novel aspect here is the methodology used for the solution, as well as the direct determination of the plastic zones. As a consequence, the resulting analytical solutions cover many more problems in the mathematical theory of plasticity compared to similar existing methods and they may be proved of importance in various applications.     

Asymptotic fields in steady crack growth with linear strain-hardening

The asymptotic stress and velocity fields of a crack propagating steadily and quasi-statically into an elastic-plastic material are presented. The material is characterized by J₂-flow theory with linear strain- hardening. The possibility of reloading on the crack flanks is taken into account. The cases of anti-plane strain (mode III), plane strain (modes I and II), and plane stress (modes I and II) are considered. Numerical results are given for the strength of the singularity and for the distribution of the stress and velocity fields in the plastic loading, elastic unloading and plastic reloading regions, as functions of the strain-hardening parameter. An attempt is made to make a connection with the perfectly-plastic solutions in the limit of vanishing strain-hardening.     

Variational formulations for the plane strain elastic-plastic problem for materials governed by the von Mises criterion

Path-dependent materials, complying with Drucker's postulate requirements and governed by an internal variable rate plasticity model, are considered. A variational principle for the small strain, rate plasticity problem is established in this context and extended to cover finite loading steps. Results are subsequently specialized to plane strain solids made of elastically isotropic materials with a plastic behavior governed by the von Mises criterion, accounting for combined isotropic and kinematic hardening. By exploiting previous results, the formulation is fully reduced to the plane. Further generalizations of the statements are also provided, which can be regarded as extensions to the elastic-plastic, plane strain problem of the Hu-Washizu principle in elasticity.