Enrico Radi

Crack propagation in an orthotropic medium under general loading

Abstract The elastodynamic response of a finite crack steadily propagating in an orthotropic medium, acted upon at infinity by uniform biaxial and shear loads, is studied. In particular, the effects of varying crack velocity as well as the ratio of shear load to tensile load are described. The action of material orthotropy on various quantities describing the crack propagation characteristics is pointed out.

Mode I crack propagation in elastic-plastic pressure-sensitive materials

Abstract Mode I steady-state, quasi-static crack propagation is analysed for elastoplastic pressure-sensitive solids. In particular, reference is made to the incremental small strain elastoplasticity obeying the Drucker-Prager yield condition with associative flow law. The asymptotic crack-tip fields are numerically obtained for the case of linear-isotropic hardening, under plane stress and plane strain conditions.

Effects of pre-stress on crack-tip fields in elastic, incompressible solids

Abstract A closed-form asymptotic solution is provided for velocity fields and the nominal stress rates near the tip of a stationary crack in a homogeneously pre-stressed configuration of a nonlinear elastic, incompressible material. In particular, a biaxial pre-stress is assumed with stress axes parallel and orthogonal to the crack faces. Two boundary conditions are considered on the crack faces, namely a constant pressure or a constant dead loading, both preserving an homogeneous ground state. Starting from this configuration, small superimposed Mode I or Mode II deformations are solved, in the framework of Biots incremental theory of elasticity. In this way a definition of an incremental stress intensity factor is introduced, slightly different for pressure or dead loading conditions on crack faces. Specific examples are finally developed for various hyperelastic materials, including the J2-deformation theory of plasticity. The presence of pre-stress is shown to strongly influence the angular variation of the asymptotic crack-tip fields, even if the nominal stress rate displays a square root singularity as in the infinitesimal theory. Relationships between the solution with shear band formation at the crack tip and instability of the crack surfaces are given in evidence.

Steady-state propagation of dislocations in quasi-crystals

We analyse the steady propagation at constant speed, lower than the shear wave-speed, of a straight dislocation in an unbounded elastic quasi-crystal with five-fold symmetry. We discuss only the ideal elastic behaviour, neglecting the dissipation associated with the atomic rearrangements. Under these conditions, we provide the expressions of phonon and phason fields in closed form. Both phonon and phason stresses appear to be singular near to the dislocation core. We also find the explicit expression of the energy per unit length around a moving dislocation.

Steady crack growth in elastic–plastic fluid-saturated porous media

Abstract An asymptotic solution is obtained for stress and pore pressure fields near the tip of a crack steadily propagating in an elastic–plastic fluid-saturated porous material displaying linear isotropic hardening. Quasi-static crack growth is considered under plane strain and Mode I loading conditions. In particular, the effective stress is assumed to obey the Drucker–Prager yield condition with associative or non-associative flow-rule and linear isotropic hardening is adopted. Both permeable and impermeable crack faces are considered. As for the problem of crack propagation in poroelastic media, the behavior is asymptotically drained at the crack-tip. Plastic dilatancy is observed to have a strong effect on the distribution and intensity of pore water pressure and to increase its flux towards the crack-tip.

Localization of deformation in plane elastic-plastic solids with anisotropic elasticity

Abstract Localization of deformation is analyzed in elastic–plastic solids endowed with elastic anisotropy and non-associative flow rules. A particular form of elastic anisotropy is considered, for which the localization analysis can be performed with reference to an elastic–plastic solid endowed with isotropic elasticity but whose normals to the yield function and plastic potential are not coaxial. On the other hand, so far, available analytical solutions for the onset of strain localization in elastic–plastic solids assume isotropic elasticity and coaxial plastic properties. Here, a new analytical solution is presented when the plastic normals are not coaxial but the analysis is restricted to plane strain and plane stress loadings. As an illustration, for a material with transverse elastic isotropy and with pressure-dependent yield surface and plastic potential, this solution provides explicit expressions at the onset of strain localization for the plastic modulus, for the orientation of the shear-band and for the slip mode. The numerical results highlight the importance of the coupled influence of elastic anisotropy and non-associativity on the onset of strain localization.

Stroh formalism in analysis of skew-symmetric and symmetric weight functions for interfacial cracks

The focus of the paper is on the analysis of skew-symmetric weight functions for interfacial cracks in two-dimensional anisotropic solids. It is shown that the Stroh formalism proves to be an efficient tool for this challenging task. Conventionally, the weight functions, both symmetric and skew-symmetric, can be identified as non-trivial singular solutions of a homogeneous boundary-value problem for a solid with a crack. For a semi-infinite crack, the problem can be reduced to solving a matrix Wiener–Hopf functional equation. Instead, the Stroh matrix representation of displacements and tractions, combined with a Riemann–Hilbert formulation, is used to obtain an algebraic eigenvalue problem, which is solved in a closed form. The proposed general method is applied to the case of a quasi-static semi-infinite crack propagating between two dissimilar orthotropic media: explicit expressions for the weight functions are evaluated and then used in the computation of the complex stress intensity factor corresponding to a general distribution of forces acting on the crack faces.

The effects of inertia on crack growth in poroelastic fluid-saturated media

A closed-form asymptotic solution is provided for the stress, pore pressure and displacement fields near the tip of a Mode I crack, dynamically running in an elastic fluid-saturated porous solid. The Biot theory of poroelasticity with inertia forces is assumed to govern the motion of the medium. At variance with the quasi-static case where the crack-tip is effectively drained, for rapid transient crack propagation, the pore fluid has no time to diffuse away from the crack-tip. Both a qualitative analysis and the obtained asymptotic solution reveal that the pore pressure near the crack-tip displays the same square root singularity as stress in the solid skeleton. Previous analyses have neglected the inertia of the fluid and obtained a regular pore pressure.

Asymptotic fields of mode I steady-state crack propagation in non-associative elastoplastic solids

The quasi-static, steady-state propagation of a crack running in an elastoplastic solid with volumetric-non-associative flow law is analyzed. The adopted constitutive model corresponds to the small strain version of that proposed by Rudnicki and Rice. The asymptotic crack-tip fields are numerically obtained for the case of the incremental theory with linear isotropic hardening, under mode I plane-stress conditions. A relevant conclusion of the study is that the singularity of the near-tip fields appears to be mainly governed by the flow-rule, rather than by the yield surface gradient.

Strain-gradient effects on steady-state crack growth in linear hardening materials

Abstract Steady-state rectilinear crack propagation is analysed in couple stress elastic–plastic solids, displaying linear isotropic hardening. The flow-theory version of couple stress plasticity is adopted for the constitutive description of the material. A higher order asymptotic analysis of crack-tip fields is performed under mode I and mode II loading conditions, both for plane strain and plane stress problems. In particular, the stress and couple stress fields are assumed to display distinct strengths of their singularity, so that, although the most singular term of the velocity field turns out to be irrotational, the leading order terms of couple stress and rotation gradient fields do not vanish, but couple with higher order terms of the strain and velocity fields. It follows that, under mode I crack propagation, the rotation gradients produce a substantial increase of the stress singularity and, thus, of the traction level ahead of the crack-tip, with no need to invoke stretch gradients, whereas an increase in the shear traction ahead of the crack-tip is observed under mode II loading conditions. These results may contribute to explaining the occurrence of cleavage fracture in ductile metals from the point of view of atomistic fracture mechanics.