Abstract
The main aim of the work presented in this paper was treatment of damage and deformation localisation observed in the finite element method (FEM) analysis of strain softening materials combined with local constitutive models where damage is represented using continuum damage mechanics (CDM). The CDM/FEM approach typically suffers from a number of shortcomings, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). The approach proposed here is still based on the local constitutive model including damage, but introduces an alternative representation of damage effects in the system of linear momentum balance equations. The damage effects are included through equivalent damage force (EDF), which contributes to the right-hand side of the momentum balance equations. The main advantages of this approach are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening; numerical stability is preserved without a need for regularisation measures; and significantly reduced mesh dependency. In addition, the EDF approach can be used in combination with existing local CDM damage models and does not violate symmetry of the material stiffness tensor.The EDF approach is applicable to modelling of strain softening typically observed in damaged quasi brittle materials such as fibre reinforced composites and concrete.The EDF model was implemented in the in-house developed coupled FEM-SPH code, where an explicit FEM code is coupled with a stable Total-Lagrange form of SPH. Its performance is demonstrated in the analysis of a dynamic one dimensional (1D) stress wave propagation problem, which was analytically solved by Bazant and Belytschko in 1985. For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be experimentally determined and is an input parameter independent of the discretisation density.
Highlights
Strain softening is deterioration of material strength with increasing strain, which is a phenomenon typically observed at a continuum level in damaged quasi brittle materials, including fibre reinforced composites and concrete
When local Continuum Damage Mechanics (CDM) constitutive models are used with the finite element method (FEM), the strain softening leads to numerical instability, as the tangent stiffness tensor loses positive definiteness and violates the material stability criterion by Hadamard [5]
As already stated the SPH method is inherently non-local, with the smoothing length related to the material damage/failure characteristic length, and not sensitive to interparticle distance
Summary
Strain softening is deterioration of material strength with increasing strain, which is a phenomenon typically observed at a continuum level in damaged quasi brittle materials, including fibre reinforced composites and concrete. The strain-softening instabilities have been of large interest to research in recent decades and have been investigated, among many others in [7,8,9,10,11,12,13,14,15], leading to a development of a number of regularisation methods, including non-local, gradient-enhanced and viscous methods These methods are based on the introduction of a characteristic length scale into constitutive equations through higher-order spatial derivatives or viscous effects, see for instance the models developed by Dillon [16], Bazant [17,18], Aifantis [19,20], Needleman [21,22,23], Pijaudier-Cabot [24,7], Sluys [25,26], and de Borst [27,28,29].
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