Link¨
oping Studies in Science and Technology
Dissertations No. 973
On failure modelling in finite element
analysis – material imperfections and
element erosion
Mattias Unosson
Division of Solid Mechanics
Department of Mechanical Engineering
On failure modelling in finite element analysis – material imperfections and element erosion
ISBN 91–85457–34–5 ISSN 0345–7524
Link¨oping Studies in Science and Technology. Dissertations No. 973 Printed in Sweden by UniTryck, Link¨oping 2005
Preface
This work was carried out at the Swedish Defence Research Agency, FOI under the scientific supervision by Prof. Larsgunnar Nilsson and Dr. Kjell Simonsson at Link¨oping Institute of Technology and Dr. Lars Olovsson at FOI.
I want to thank FOI, the Swedish Armed Forces and the Fortifications Corps Research Fund for their financial support during these six years.
The realization of this dissertation would not have been possible with-out the support from my supervisors, especially the close co-operation and friendship with Lars. It has been an honour and a privilege.
Emma and Gustav, thank you for constituting the private domain of my life in which power can be restored. Gustav, nu ¨ar boken f¨ardig.
Huddinge, September 2005 Mattias Unosson
Abstract
This dissertation concerns failure modelling with material imperfections and element erosion in finite element analyses. The aim has been to improve the element erosion technique, which is simple to use and implement and also computationally inexpensive. The first part of the dissertation serves as an introduction to the topic and as a summary of the methodologies presented in the following part. The second part consists of seven appended papers. In paper A the standard element erosion technique is used for projectile pene-tration. In papers B and C a methodology that accounts for size effects is de-veloped and applied to crack initiation in armour steel and tungsten carbide. A methodology to better predict the stress state at crack tips with coarse meshes is presented and applied to armour steel in paper D. Papers E and F concern the development of selective mass scaling which allows for larger time steps in explicit methods. Finally, in paper G the previously presented methodologies are used in combination and validated against experimental results on tungsten carbide. The computations show good agreement with the experimental results on failure initiation for both materials, while the computational results on the propagation of cracks show better agreement for the armour steel than for the tungsten carbide.
Dissertation
This dissertation for the degree of Teknologie doktor (Doctor of Philosophy) at Link¨opings universitet consists of an introductory part and the following appended papers:
Paper A
M. Unosson and L. Nilsson, Projectile penetration and perforation of high performance concrete: Experimental results and macroscopic modelling, Ac-cepted for publication in International Journal of Impact Engineering. Paper B
M. Unosson, L. Olovsson and K. Simonsson, Failure modelling in finite el-ement analyses: Random material imperfections, Mechanics of Materials, 37 (12) (2005) 1175–1179.
Paper C
M. Unosson, L. Olovsson and K. Simonsson, Weakest link model with im-perfection density function: Application to three point bend of a tungsten carbide, Submitted for publication.
Paper D
M. Unosson, L. Olovsson and K. Simonsson, Failure modelling in finite el-ement analyses: Elel-ement erosion with crack-tip enhancel-ement, Accepted for publication in Finite Elements in Analysis and Design.
Paper E
L. Olovsson, M. Unosson and K. Simonsson, Selective mass scaling for thin walled structures modeled with tri-linear solid elements, Computational Mechanics, 34 (2) (2004) 134–136.
Paper F
L. Olovsson, K. Simonsson and M. Unosson, Selective mass scaling for ex-plicit finite element analyses, International Journal for Numerical Methods in Engineering, 63 (10) (2005) 1496–1505.
Paper G
M. Unosson, L. Olovsson and K. Simonsson, Imperfection density function and crack-tip enhancement: Validation against symmetrical bending of cir-cular tungsten carbide plates, Submitted for publication.
Introduction
Nonlinear explicit finite element analyses of transient dynamic processes are common in different domains of the engineering community. In defence ap-plications the processes frequently involve material failure that has to be encompassed by the analyses.
The initiation and propagation of cracks can be studied at different spa-tial scales, as illustrated in Figure 1. However, even though the governing mechanisms are well known on the molecular level, the corresponding models quickly become too large to handle numerically, cf. [1] and [2]. Thus, some degree of homogenization is necessary to analyse structures of engineering applications and the prevailing choice is the macroscopic approach in com-bination with field theory, cf. [3]. In this context a crack can be described either as part of the boundary or as a first order singular surface with respect to the motion over which jump conditions replace the field equations, cf. [4]. Recently there have been efforts made to incorporate such discontinuities into the finite element approximations, cf. [5] [6] [7], but these methods are still under development and are not commonly used for engineering applications. For strain-softening materials so called cohesive zone models are often used, cf. [8], [9], [10] and [11], in which energy is dissipated during relaxation of the elements that subsequently only are allowed to withstand compressive stresses. With this method the inelastic deformation localizes to one or a few elements and the fracture energy release can be controlled. However, the large gradients near the crack tip are not resolved.
In engineering applications the material is commonly modelled as uni-form and initiation of failure is governed by a local strain or stress-based criterion. Failed elements are eroded, i.e. removed from the model, to rep-resent cracks and their propagation. However, there are problems related to this technique. Firstly, it does not take into account the well known phe-nomenon that the strength of a material volume is influenced by its size, cf. [12]. The assumption of material homogeneity can render inaccurate failure behaviour for some applications. Secondly, the element meshes used in engi-neering applications are commonly too coarse to capture gradients near the
Figure 1: Material failure at different spatial scales
crack-tip, which leads to an over-estimation of the energy required to prop-agate the crack. For element sizes smaller than the physical crack-tip radius the solution may under-estimate the fracture energy.
In this dissertation a twofolded methodology is proposed to remedy the above discussed deficiencies; random material imperfections and crack-tip enhancement. The objective is to be able to predict the global behaviour of a structure rather than to correctly describe the local state at the crack-tip. Two papers on selective mass scaling are also included in this dissertation. Even though not a central topic here, the papers represent an important contribution to the wider context of explicit finite element methods.
The effect of size on the macroscopic strength
The expectation and variance of the macroscopic strength of a material in-creases with decreasing size of the affected volume, see Figures 2–3. One
3 0 500 1000 1500 2000 2500 3000 3500 0 0.2 0.4 0.6 0.8 1
tensile fracture strength (MPa)
diameter (mm)
Figure 2: Tensile fracture strength of glass fibres. From [12]
philosophy that has been successful in explaining this size effect is the prob-abilistic approach in combination with the assumption of random material imperfections, cf. [13] and [14]. Crystalline metals, which are in focus in this dissertation, contain imperfections in the form of point imperfections (vacan-cies, interstitials, chemical impurities), line imperfections (dislocations) or plane imperfections (grain boundaries and micro-cracks), cf. [15]. Depend-ing on the manufacturDepend-ing process the material also contains micro-stresses. In this work a macroscopic weakest link model is developed that character-izes the effect of the initial material imperfections. An inherent feature to this model is that only one finite element analysis is needed to predict the local and global probability of failure initiation for a component exposed to a certain load or deformation history.
Crack-tips and finite element discretization
A failure criterion that gives accurate predictions of failure initiation, does not necessarily work well in describing the propagation phase. Unless the mesh is extremely dense, it is not possible to capture the solution at the crack-tip and the deformation at the integration points will be significantly
0 0.2 0.4 0.6 0.8 1 Distribution fracture strength 0 0.2 0.4 0.6 0.8 1 Distribution fracture strength P P P P P P P P P
Figure 3: Schematic illustration of the size effect
used to represent the crack the fracture energy release will be mesh dependent and converge to zero as the element size is decreased, cf. [16]. The zone of inelastic deformation around a crack-tip in steel is of the order 10−6–10−4m, cf. [17], but with a reasonably fine mesh, the inelastic deformation around the crack-tip is smeared out over a much larger volume and the globally dissipated energy will be larger than the true one. Commonly, the crack-tip itself and the near crack deformation field is of little or no interest for the analyst. It is often sufficient if the model reproduces experimental results on those global loads and energies that are necessary to propagate the crack. In this work a methodology is developed in which the solution for the elements adjacent to the crack-tip, see Figure 4, is enhanced with a scaling function.
Outlook
The demand for finite element analyses of material failure in engineering applications will most certainly increase. Existing methodologies will have to evolve and new methodologies will have to be developed to meet this demand. Nevertheless, the main challenge will stay the same; to merge dis-crete and field theoretical descriptions. The methodologies developed in this work increase the applicability of todays common practice in engineering to
5
Real crack with an inelastic zone at the crack-tip HH HH HH HHHj Crack-tip neighbourhood
Figure 4: FE-representation of a crack with crack-tip neighbours
handle failure. But, there are still limitations in their current form, such as the applicability to handle cyclic loading and mixed modes of crack open-ing. Furthermore, the imperfection methodology developed may also have a potential for use in micromechanical analyses where spatial scales are small
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