# Transient near tip fields in crack dynamics

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## Abstract

Transient effects of stress-strain fields in the vicinity of a stationary crack tip under high rate loads are discussed. Exact analytical solutions to near tip stresses are compared to fields prescribed by leading terms (one or several) of Williams asymptotic expansion. Influence of load application mode, time (or, which is the same, distance from a crack tip) and Poisson’s ratio on this discrepancy is extensively examined. Some effects connected with crack tip propagation speed are also discussed. Significant inconsistencies between real (or received in numerical solutions of state equations - e.g. finite element computations) crack tip fields and stress intensity factor (SIF) singular field observed by numerous researchers are explained. The scope of problems where SIF field can be used for correct prediction of dynamic stress-strain fields in the crack tip region is established. Possibility to correctly approximate fields that are not SIF dominated, accounting additional terms of Williams expansion, is studied.

## Keywords

transient crack tip fields dynamic fracture high-rate loads asymptotic expansions## List of main symbols

*σ*_{ij}components of the Cauchy stress tensor

*t*time

*r,θ*polar coordinates with origin at the crack tip

*K*_{I}the first term of the asymptotic expansion of stresses surrounding the tip of the mode I loaded crack (the mode I stress intensity factor)

*R*_{n}the second and the following terms of the asymptotic expansion of stresses surrounding the crack tip

*ϕ*_{ij}angular functions

*x*(*x*_{1},*x*_{2})Cartesian coordinate

*W=W*(*t*,*x*)displacement field

*u,v*components of displacement

*µ*shear modulus

*ν*Poisson’s ratio

*E*Young’s modulus

- \(c_1 = \sqrt {\frac{{1 - v}} {{\left( {1 + v} \right)\left( {1 - 2v} \right)}}\frac{E} {\rho }} = \sqrt {\frac{{2\left( {1 - v} \right)}} {{1 - 2v}}\frac{\mu } {\rho }}\)
longitudinal wave speed

- \(c_2 = \sqrt {\frac{\mu } {\rho }}\)
transversal wave speed

*c*_{R}Rayleigh wave speed

*f*(*t*)time dependent load

*P*load amplitude

*H*Heaviside step function equal to 0 if the argument is negative and equal to 1 otherwise

*φ*,*ψ*longitudinal and transversal wave potentials

*ρ*mass density

- \(\gamma = {{c_2 } \mathord{\left/ {\vphantom {{c_2 } {c_1 }}} \right. \kern-\nulldelimiterspace} {c_1 }} = \sqrt {\frac{{1 - 2v}} {{2\left( {1 - v} \right)}}}\)
ratio of longitudinal and transverse wave speeds

*γ*_{R}=*c*_{R}/*c*_{1}ratio of Rayleigh and longitudinal wave speeds

*S*_{i}the sum of the first

*i*terms of the asymptotic expansion*σ*_{c}the critical value for tensile stress (ultimate strength)

*K*_{IC}the critical value for stress intensity factor

*τ*the microstructural time of a brittle fracture process (or fracture incubation time) — a parameter characterizing the response of the studied material to applied dynamic loads.

*τ*is constant for a given material and does not depend on problem geometry, the way a load is applied, the shape of a load pulse and its amplitude*d*characteristic size of fracture process zone

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