On the elastic modulus degradation in continuum damage mechanics

doi:10.1016/S0045-7949(99)00187-X

Computers & Structures

Volume 76, Issue 6, 30 July 2000, Pages 703-712

https://doi.org/10.1016/S0045-7949(99)00187-X Get rights and content

Abstract

To measure accurately the elastic modulus of a metal, E, can be a difficult task when a specimen undergoes plastic strains. Moreover, some failure criteria, such as those associated with Continuum Damage Mechanics, require the change of elastic modulus with strain to define a measure of damage, D, in a material or structure. Thus, it is important to assess the possible geometrical influence of a specimen on the measurement of the elastic modulus at different deformation levels. It is shown in this article, with the aid of a numerical simulation, that any plastic strains induce important geometrical effects in the evaluation of E, which have a significant influence on the evaluation of the scalar damage parameter, D.

Introduction

Continuum Damage Mechanics (CDM) is sometimes used to predict the phenomenon of failure in structures loaded statically [6], [7], [12] and dynamically [11], [14]. The seminal idea of this method is due to Kachanov [16], who introduced a damage variable, D, to model the phenomenon of creep. Since then, many publications have been produced on this subject and formal theories embracing damage and plasticity [5], [13] have been developed.

In the simplest case of isotropic and homogeneous damage, the damage variable, D, is related to the surface density of micro-defects in the material. Clearly, the successful use of CDM to predict failure is related closely to accurate measurements of the damage.

The postulate of strain equivalence, due to Lemaitre [18], [19], states that a constitutive equation for a damaged material can be obtained by replacing the stress σ in a virgin material by the effective stress $σ ̃ =σ/(1−D)$ , where $σ ̃$ is the force divided by the area that effectively sustains the load. Thus, the damage may be represented by an elastic modulus change $D=1− E ̃ E,$ where $E ̃$ is the elastic modulus of the damaged material.

Another postulate, known as energy equivalence [8], also relates damage to the change of the elastic modulu but now through the equation $D=1− E ̃ E .$

Other expressions relating damage to the change in elastic modulus can be found in Luo et al. [22]. Though other techniques can be used to estimate D [1], [2], [20], it appears that the change in elastic modulus is the most convenient one, both for metals [17], [20] and for composites [9], [21], [26].

Lemaitre and Dufailly [20] and Dufailly [10] were the first researchers to measure D through the degradation of the elastic modulus. A specimen similar to the one depicted in Fig. 1, which is called here a damage specimen, was loaded in tension up to some plastic strain recorded by tiny local strain gauges fixed at the minimum cross-section. The specimen is then unloaded and the elastic modulus is obtained from the slope of the unloading stress-strain curve. After some level of deformation, typically producing strains less than 0.10, the strain gauges fail and are replaced by a new set. The specimen is loaded again to produce further plastic strain and then unloaded to obtain a new value for the elastic modulus. This process is repeated until a visual crack is detected. For ductile metals with failure strains of around one, it is necessary to use at least ten pairs of expensive strain gauges, which require a time-consuming installation.

The test specimens used by Lemaitre and Dufailly [20] and Dufailly [10] had a radius (R₀) of 80 mm which restricts the plastic deformation and, consequently, the damage to a small zone where any changes in the elastic modulus can be monitored. It is acknowledged by Dufailly [10] that this geometry leads to a non-uniform stress field, but it appears that no further attention has been paid in the literature to the influence of this non-uniformity on the measurement of the elastic modulus. Because most of the measurements of D through the elastic modulus degradation are made using similar specimen geometries, it is important to determine whether the measured change in the elastic modulus is due only to material damage or is due partly to specimen geometry effects.

To investigate this aspect, one could test several specimens from the same material with different radii (R₀) and compare the measured elastic moduli at various plastic strains. Alternatively, one could perform a numerical simulation of such a test, investigating possible geometrical effects on the accuracy of the elastic modulus measurement. This last procedure is followed here and it is shown that the initial geometry and the distribution of strains cause an error in the evaluation of the elastic modulus.

Section snippets

Method of analysis

The specimen geometry in Fig. 1 was represented by 20 node isoparametric finite-elements with reduced integration type C3D20R, available in the ABAQUS programme. A very fine mesh was used near the smallest cross-section with the initial size of the smallest element being 0.3125×0.05×0.1875 mm³ (x,y,z). Thus, a quarter of the cross section includes 64 elements and one-half of a strain gauge with a gauge length of 0.381 mm spans 8 nodes. Advantage was taken of symmetry to analyse only one-eighth

Results

An initial numerical simulation was carried out for the specimen in Fig. 1 but with an infinite radius, i.e. R₀=∞. This specimen was loaded and unloaded elastically and the output load, area and strain on the external surface were used to calculate the elastic modulus in the usual way. As in a parallel strip with no initial plastic deformation, the stress field is uniform and the calculated elastic modulus must coincide with the input value. This is exactly what was obtained, leading to the

Discussion

It is evident that the specimen geometry in Fig. 1, which is traditionally used to measure the degradation of the elastic modulus, induces non-uniformities in the stress field. Thus, the ratio F/(Ae_e) has a magnitude different from the actual elastic modulus for the material. This non-uniformity is illustrated in Fig. 7; the elastic strains at the middle of a specimen (x=0) are lower than at the boundary (x=5 mm) for the first four steps. This makes the apparent elastic modulus, at those

Conclusions

Apart from experimental errors, the technique currently used to measure the damage variable, D, through the degradation of the elastic modulus, suffers from some limitations. The specimen geometry is such that it introduces an error even during the first elastic loading due to a nonhomogeneous strain field. This error is not large but it grows as the plastic strains increase.

Different strain measures were defined and used for the evaluation of D. The data gives strong support to the use of an

Acknowledgements

M. Alves was funded by the Brazilian Research Agency, CAPES and the Impact Research Centre and J.L. Yu by the Royal Society, the Chinese Academy of Sciences and the Impact Research Centre. This study forms part of EPSRC grant GR/J 69998 to the Impact Research Centre at the University of Liverpool.

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On the elastic modulus degradation in continuum damage mechanics

Abstract

Introduction

Section snippets

Method of analysis

Results

Discussion

Conclusions

Acknowledgements

Acta Metallurgica

International Journal of Solids and Structures

Computers & Structures

Engineering Fracture Mechanics

International Journal of Solids and Structures

Engineering Fracture Mechanics

Engineering Fracture Mechanics

Engineering Fracture Mechanics

Engineering Fracture Mechanics

Damage measurement in aluminium using the Moiré method

Damage initiation and development in chopped strand mat composites

Studies in large plastic flow and fracture

Damage induced elastic anisotropy