[Paper Review] Strain localization and shear banding in ductile materials

N. Bordignon, A. Piccolroaz+2 more|arXiv (Cornell University)|Jan 24, 2015

Force Microscopy Techniques and Applications6 references3 citations

TL;DR

This paper proposes a zero-thickness nonlinear interface model for shear bands in ductile materials, using finite element simulations to show that shear bands propagate rectilinearly under shear loading and exhibit strong stress concentration at their tips. The model, based on an imperfection approach with a pre-existing compliant shear band, confirms that shear bands behave fundamentally differently from cracks, with rectilinear growth and tip singularities due to localized plasticity.

ABSTRACT

A model of a shear band as a zero-thickness nonlinear interface is proposed and tested using finite element simulations. An imperfection approach is used in this model where a shear band, that is assumed to lie in a ductile matrix material (obeying von Mises plasticity with linear hardening), is present from the beginning of loading and is considered to be a zone in which yielding occurs before the rest of the matrix. This approach is contrasted with a perturbative approach, developed for a J$_2$-deformation theory material, in which the shear band is modelled to emerge at a certain stage of a uniform deformation. Both approaches concur in showing that the shear bands (differently from cracks) propagate rectilinearly under shear loading and that a strong stress concentration should be expected to be present at the tip of the shear band, two key features in the understanding of failure mechanisms of ductile materials.

Motivation & Objective

• To develop a mechanical model of shear bands as zero-thickness interfaces to study their propagation and stress fields.
• To compare the imperfection approach (pre-existing shear band) with the perturbative approach (shear band emerging during deformation).
• To investigate why shear bands grow rectilinearly under mode II loading, unlike cracks.
• To quantify stress concentration at shear band tips, a key factor in ductile failure mechanisms.
• To provide a theoretical and computational framework for understanding shear band evolution in ductile materials under extreme loading.

Proposed method

• A zero-thickness nonlinear interface is modeled as a compliant shear band with reduced yield stress embedded in a ductile matrix obeying von Mises plasticity with linear hardening.
• The imperfection approach assumes the shear band exists from the start of loading, acting as a pre-existing defect that yields before the matrix.
• Finite element simulations are used to analyze deformation and stress fields, comparing results with analytical solutions from the perturbative approach.
• The perturbative approach models the shear band as a sliding surface emerging during uniform deformation, using incremental boundary conditions to describe discontinuities in stress and displacement.
• The J2-deformation theory of plasticity is applied, with the incremental shear stiffness moduli related by μ* = Nμ, where N is the strain hardening exponent.
• A stream function formulation is used to satisfy incompressibility and solve the incremental boundary value problem, revealing square-root singularities at shear band tips.

Experimental results

Research questions

• RQ1 Why do shear bands propagate rectilinearly under mode II shear loading, unlike cracks?
• RQ2 Does a stress concentration develop at the tip of a shear band, and if so, how strong is it?
• RQ3 How do the imperfection and perturbative approaches to shear band modeling compare in predicting shear band behavior?
• RQ4 What is the asymptotic behavior of a thin, highly compliant layer in a ductile matrix, and how does it justify the zero-thickness interface model?
• RQ5 How does the strain hardening exponent N influence the evolution and stability of shear bands?

Key findings

• Shear bands propagate rectilinearly under mode II loading, a behavior not observed in brittle fracture, due to the localized plasticity and compliance of the band.
• A strong square-root singularity in incremental deformation and stress is found at the tips of pre-existing shear bands, indicating significant stress concentration.
• The perturbative approach confirms the imperfection model’s prediction of rectilinear growth and tip singularities, validating the results across different modeling frameworks.
• For perfect plasticity (N → 0), the shear band becomes extremely compliant (μ* ≪ μ), leading to pronounced localization and stress concentration.
• Finite element simulations show that deformation is highly focused along the shear band direction, supporting the rectilinear propagation mechanism.
• The analytical solution from the perturbative approach matches numerical simulations, confirming the presence of a stress concentration field similar to that in fracture mechanics.