Mode I and mixed mode elastic-plastic three-dimensional asymptotic crack tip fields

Elastic perfectly-plastic asymptotic mixed mode crack tip fields in plane stress

This paper investigates the stress singularity at the crack tip in a two-dimensional Neo-Hookean hyperelastic material, with a focus on how far-field strain influences the crack tip field distribution. The study demonstrates that, under small far-field strains, the crack tip field can be generally divided into three regions: Region I—the asymptotic neo-Hookean crack tip field as $$r\rightarrow 0$$ r → 0 ; Region II—a finite-radius nonlinear neo-Hookean zone; and Region III—an outer linear elastic region. Within Region III, a subregion may still obey the asymptotic linear elastic solution when the radial distance is sufficiently small. As the far-field strain increases, both the asymptotic linear subregion and the broader linear region shrink and eventually vanish, leaving only the nonlinear zones. This multiscale structure reflects the principle of small-scale nonlinearity, wherein nonlinear effects are confined to an inner core. The inner core consists of Region I, where asymptotic neo-Hookean fields dominate, and Region II, where general nonlinear effects prevail. Initially, this inner core is nested inside Region III. At sufficiently small far-field strains, Region III itself contains an inner core that follows asymptotic linear elastic crack tip fields. As loading intensifies, Regions I and II expand, and Region III—first its asymptotic core, then the broader linear zone – .- progressively diminishes, culminating in a large-scale nonlinearity regime. We also identify and quantify the characteristic length scales over which each region exists and dominates—nonlinear fields in Regions I and II and linear elastic behavior in Region III. An important point is that at the crack tip, Region I always governs the local field, although its extent may be small under small far-field strains, making it difficult to capture in computational simulations. To address this, we introduce a rescaling method to better resolve this near-tip behavior.

A predicting model for three-dimensional crack growth in power-law creeping solids

An extended finite element method (X-FEM) for three-dimensional crack modelling is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three-dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.

Three-dimensional asymptotic stress field at the front of a bimaterial wedge of symmetric geometry under antiplane shear loading

Three-dimensional asymptotic antiplane shear stress fields at the front of interfacial crack/anticrack type discontinuities in trimaterial bonded plates

Effects of anisotropic hardening on crack propagation in porous-ductile materials

Dynamic asymptotic fields near a crack tip growing in a power-law hardening compressible material

This paper discusses the asymptotic fields ahead of mixed mode frictional cohesive cracks in quasi‐brittle materials. These fields have been derived after reformatting the cohesive‐law in a special but universal polynomial containing fractional or integer powers. This special form ensures that the radial and angular variations of the asymptotic fields are separable as in the Williams expansions for a traction‐free crack. The coefficients of the expansions however depend nonlinearly on the softening law and the boundary conditions. As expected, the asymptotic field of a frictional cohesive crack reduces to that of a frictionless cohesive crack with normal cohesive separation when the friction coefficient becomes zero. It is also shown that the asymptotic field of a frictionless cohesive crack can be used to a non‐cohesive crack opened by crack face loading, after including the singular terms corresponding to the eigenvalue 1/2. Furthermore, the coefficients are given explicitly for two special softening laws that are commonly used in practice. The leading terms of the true displacement asymptotic field are also derived explicitly. These are especially useful as the enrichment functions at the tip of a mixed mode cohesive crack for accurate simulation of crack growth in quasi‐brittle materials using the extended finite element (XFEM).

When crack tip constraint is high, the crack tip contour integral J characterises the asymptotic stress, strain and displacement fields of a stationary crack in an elastic-plastic material. In other cases, the crack tip fields can be related to J and a second parameter Q which governs the crack tip constraint. These observations are the basis of J-Q fracture mechanics assessments. In the most usual procedure J is compared to an effective, constraint-corrected fracture toughness Jc which is derived from Q and the fracture toughness of the material. The difference Jc – J is a measure of the margin of safety. The assessment procedure assumes there are no initial inelastic strains in the component or the fracture toughness specimen prior to introducing the crack and subsequent loading. However, plant components may contain inelastic strains prior to cracking arising from welding and other manufacturing or fit-up processes. This initial strain field can be established by a finite element analysis that simulates the welding and/or manufacture sequence. Weld residual stresses develop due to the accumulation of incompatible, inelastic strains, including thermal, plastic and transformation strains in the material. If a crack is inserted into an initial strain field, a procedure is required to calculate J by analysis of the resulting crack tip fields. Moreover, for the fracture assessment method to remain valid, it must be demonstrated that the values of J and Q continue to govern the onset of crack extension or fracture so that a meaningful comparison of J with Jc can be made. This paper describes a domain integral for calculating J when inelastic strains exist prior to cracking, and its implementation in the JEDI computer code. The code is used to determine J for a crack inserted into a three-point bend specimen containing an initial inelastic strain field representative of one that might develop during welding. The extent to which the crack tip stress field is characterised by J and Q is examined by comparing it to the field for high constraint, small-scale yielding conditions.

Mixed mode near-tip fields for cracks in materials with strain-gradient effects

A novel eigenfunction expansion technique, based in part on separation of the thickness-variable and partly on the Eshelby–Stroh type affine transformation, is developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness anticrack reinforcing an infinite orthorhombic single crystal plate, of finite thickness and subjected to far-field mode I/II loadings. Anticrack-face boundary conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the plate are exactly satisfied. The present investigation considers six through-anticrack systems reinforcing orthorhombic single crystal plates. Explicit expressions for the singular stresses in the vicinity of the front of a through-thickness anticrack reinforcing an orthorhombic plate, subjected to far-field mode I/II loadings, are presented. Finally, hitherto largely unavailable results, pertaining to the through-thickness variations of stress singularity coefficients corresponding to symmetric and skew-symmetric sinusoidal loads that also satisfy the boundary conditions on the top and bottom surfaces of an orthorhombic mono-crystalline plate under investigation, bridge a longstanding gap in the stress singularity/fracture mechanics literature.

Plane strain elastic-plastic analyses were performed on a variety of 2-D planar cracked geometries and loading conditions, based on a small strain formulation and de-formation theory plasticity (termed finite boundary (FB) analysis hereafter). For the majority of the cases analyzed, the crack-tip fields in the mechanistically significant region around the crack tip agreed well with those of the corresponding modified boundary layer (MBL) solutions for load levels up to large scale yielding. The crack-tip fields in the MBL solutions were normalized using the elastic-plastic J values and the elastically-scaled T-stress values of the corresponding FB specimens. However, good agreement could be obtained with (J, T) only up to moderate scale yielding for a center-cracked tension (CCT) geometry. At large scale yielding, the MBL solution based on the elastically-scaled T-stress could not capture the effects of crack tip stress and strain redistribution in the CCT. An alternative method was proposed to calculate the T-stress using weight functions and the elastic-plastic remote stress field of the CCT (termed effective T-stress). Incorporating the effective T-stress into the MBL solutions, good agreement in both stress and strain fields was obtained between the MBL solution and the FB CCT solution at load levels up to general yielding. To demonstrate the limits of two-parameter characterizations in accurately describing elastic-plastic cracktip fields, a three-point-bend specimen was analyzed for load levels up to large plasticity. It is shown that although Q-values could be operationally defined at large plasticity, in general, no complete, consistent, two-parameter description was forthcoming to accurately describe the crack-tip stress and strain fields beyond certain limits. The inherent limits of two-parameter characterization of elastic-plastic crack tip fields should be recognized, analogous to those of one-parameter (J) characterization. A locus of two-parameter (J, T) characterization limits is proposed based on the crack-tip stress fields of the geometry and loading configurations analyzed. The J-T based two-parameter approach is shown to greatly extend the validity domain of fracture mechanics analysis for specimens with negative T-stress.

Relationships between crack initiation and crack growth toughness are reviewed by examining the crack tip fields and microscopic (local) and macroscopic (continuum) fracture criteria for the onset and continued quasi-static extension of cracks in ductile materials. By comparison of the micromechanisms of crack initiationvia transgranular cleavage and crack initiation and subsequent growthvia microvoid coalescence, expressions are shown for the fracture toughness of materials in terms of microstructural parameters, including those deduced from fractographic measurements. In particular the distinction between the deformation fields directly ahead of stationary and nonstationary cracks are explored and used to explain why microstructure may have a more significant role in influencing the toughness of slowly growing, as opposed to initiating, cracks. Utilizing the exact asymptotic crack tip deformation fields recently presented by Rice and his co-workers for the nonstationary plane strain Mode I crack and evoking various microscopic failure criteria for such stable crack growth, a relationship between the tearing modulusT R and the nondimensionalized crack initiation fracture toughnessJ Ic is described and shown to yield a good fit to experimental toughness data for a wide range of steels.

The reflection-mode digital gradient sensing (r-DGS) method is extended for visualizing and quantifying crack-tip deformations in solids under quasi-static and dynamic loading conditions. The r-DGS technique employs digital image correlation principles to quantify two orthogonal surface slopes simultaneously in specularly reflective solids by measuring small deflections of light rays. Here, for the first time, r-DGS is implemented to study both mode-I and mixed-mode (I/II) problems and evaluate fracture parameters. Under dynamic loading conditions, r-DGS is employed in conjunction with high-speed digital photography to map surface slopes in edge cracked plates subjected to one-point impact. The measured surface slopes are used to successfully evaluate instantaneous stress intensity factor histories by combining measurements with the corresponding asymptotic crack-tip fields and performing over-deterministic least-squares analyses. Finite element computations are also used to complement experimental measurements as needed.