Composites redistribute forces from overloaded zones to neighboring ones by microstructural fractures. This is a reserve effect of a complex microsystem (Machutov, Koksharov). Stress-strain-damage state (SSDS) of fibre composites is characterized by damage parameters such as: the fraction of broken fibres ( the ratio of the number of broken structural elements (SE) to the sum of SEs), and the relative length of delaminations ( the ratio of the average length of delaminations to the total length of fibres).

The strength of the composite does not depend on SSDS at one critical point but in a certain zone V that involves n SEs. Suppose that two cases - damaged composite (Fig.1a) and volume with eliminated parts (Fig.1b) are equivalent. The length of the eliminated fibre C0 is the material constant and has an order of the average length of the delamination. With 'volume' reduction its specific energy uv increases. It is assumed that the failure occurs when uv reaches its critical value uc. Using the terms of nominal stress and equivalent stress acting in the volume gives the direct equation for the equivalent stress. The fibre property is known and can be described by the Weibull distribution with parameters such as the average strength of the fibre SE. The theory provides the nominal stress level at which the breakdown of the multitude of the breakages takes place (Fig.2).

If the number of SEs in the volume V is finite then the binomial distribution provides an estimate of the confidence interval for damage parameter (Fig.3) and critical stress at the prearranged level of the result trustworthiness [P]. The numerical image of the probability distribution function for the nominal critical stress is obtained by analyses at different values of [P]. Compared with the law for fibre SE this function has a lower average value and narrower restricted scatter band. In the low value area, the probability of failure in the volume V (curve 2, Fig.4) has a qualitative distinction from the fibre property (line 1). For a composite structure that involves 1,000,000 V the curve shifts to position (3). By specifying [P] it is possible to estimate a value for the decrease in the carrying ability of the composite fibres (Fig.4). These approaches allow us to not only estimate reliability indices, but also to analyze the effect of the component properties, structure sizes, and initial damages on the composite structure reliability.

Fig.5 Reliability function for cyclic loading

Fig.1 Composite and model of eliminated fibres

Fig.2 SSDS parameter (equivalent stress) vs nominal stress

Fig.3 Probability of composite failure

Fig.4 Critical nominal stress for different level of probability


  1. Koksharov I.I. Prediction of mechanical properties of unidirectional composite with brittle fibres under tension.- Zavodskaja Laboratorija (Factory Laboratory) , 1990, N 1, p. 46-49. (in russian)
  2. Machutov N.A., Koksharov I.I. Model notions of fracture of a unidirectional composite material with brittle fibres in tension.- Mechanika Kompozitnux Materialov (Mechanics of composite materials), 1991, N 5, p. 804 - 811. (in russian)


a) numbers

b) analytic density function

c) interval analysis

d) histograms

e) numerical images of density functions

An example of the transformation of two numbers (addition) is shown in Fig. 1a. Fig. 1b shows the graphical performance of the transformation of two analytic functions. The result of the transformation is also an analytic function. The addition of two variables with normal density functions is an example. The sum has a normal density function with known parameters. The use of analytic functions is possible for a limited set of probability functions and simple transformations. Interval estimations of the transformation results have an important place in engineering calculations. For known initial data of the lower and upper bounds of the interval (Fig. 1c), the mathematical apparatus of interval analysis allows us to estimate the bounds of the transformation results. For complex transformations with numerous parameters there are cases when a large interval of the results has no practical significance. The use of histograms in the engineering calculations can be convenient and visual. Fig.1d shows an example of the operation of addition. In this case, the histograms of the initial data were experimentally received. The result of the transformation was also a histogram at the taken intervals of the numerical axis. There are no problems storing probability functions in the form of a set of its values at the chosen interval of the numerical axis (Fig.1e). The value of the probability density functions at arbitrary points can be calculated by an approximation of data in the neighboring points. In contrast to using histograms with fixed steps, the numerical image of the probability function can have a different number of points and various positions on the numerical axis. For a description of the different initial data a storage method with a suitable support of maintenance for the resulting accuracy is more universal, since the numerical image can describe and approximate the data of all other methods of information storage. All transformations of the numerical images are carried out by these numerical methods. In contrast to known packages of processing all the statistical information in the program, basic attention is devoted to the possibility of fulfilling different types of transformations of the probability functions: arithmetic, algebra, integration, and others. For the program a special language for describing the initial data and transformations was created. This language allows an engineer to input initial data and fulfill the calculations of the probability parameters.
Density function transformation for fatigue crack length
The figure shows the characteristics of the transformations. It provides graphs of the distribution functions of crack length depending on time (number of loading cycles). The peaks of the density functions are the results of the fact that the SE is destroyed completely. The dependence of the probability distribution functions of the fracture on the number of cycles is determined by these results. For normal and uniform (Figure) distributions of the size of the initial crack the peculiarities of the form changes are different. The programs for the evaluation of reliability indices of damaged SEs are powerful instruments for designers. These programs are necessary to check the calculations of structures, which rupture is connected with great technical and human losses.

  1. Koksharov I. Codes for estimation of reliability indices of structural elements // Scientific Siberian, AMSE Press, France, vol. 11, ser. A (Numerical and Data Analysis), 1994, pp. 83-93.
  2. Makhutov N.A., Koksharov I.I., Lepikhin A.M. Applications of numerical methods of calculation of reliability indices for construction elements with damages} - Problems of Strength, 1991, N 5, p.3-8. (in Russian)


Fig.1 Local stress responsible for failure initiation versus nominal stress. All values divided by its critical values.
Fig.2 Ratio of the local stresses for homo-geneous and composite materials versus nominal stress.
Fig.3 Qualitive picture for probability distribution density function of homogeneous and composite materials.
Fig.4 Changes in remaining strength of homogeneous and composite materials.
According to fracture criteria for homogeneous material the stresses responsible for failure initiation are proportional to nomi-nal stresses. The deformation criteria are exceptions. The theory of eliminated fibres for unidirectional composites /1/ also suggests other dependence between nominal and local stresses. Fig. 1 shows the dependence of the local stress responsible for failure initiation from nominal stress. Both parameters are divided by its critical values. The theory provides an explicit formula for nominal stress as a function of local stress, where b is the shape parameter of the Weibull distribution for stronger components of the composite. The coefficient of variation depends on b (1/b). The ratio of lo-cal stress responsible for the fracture to its critical value is equal to carrying ability exhaustion factor CAEF k /2/. The CAEF shows how close the mechanical system comes to its critical point. The coefficient and its probability function can be considered one of the quantitative parameters of risk.

The comparison of the CAEFs for homogeneous and composite materials (Fig. 2) shows that under other equal conditions (the same critical stresses) in the process of loading the local stress the for composite is less than the same value for homogeneous materials. This means that only at the moment of fracture are the stresses equal and that during the life of the structure the local stresses in the composite are less. The relative CAEF is minimal at the initial stage of loading.

The distribution density function for redundant systems (composite materials) cannot be used in the form of a well-known analytical function (for example, normal or Weibull distributions). The function can be described by a two-mode distribution on a finite interval (Fig.3, curve 2). The same function for homogeneous material also has a finite interval. In the low probability zone the difference between the functions is result of the strength reservation (redundancy) effect /1/ and depends on the technology of material production. The mean value for critical stress is defined by the components properties and mechanisms of fracture. The difference between these materials is sufficient for long-time exploitation .

The degradation of the remaining strength depends on the fracture mechanisms. The slope and form of the degradation curves for homo-geneous (fracture mechanism: single crack growth) and composite (fracture mecha-nism : multiple damages) differ (Fig.4). The rate of delamination growth (b) is proportional to the range of nominal stresses.
The number of fibre breakages (a) depends on local stress s and the probability function for strong fibres (parameters: coefficient of variation, mean value). The theory of eliminated fibres /1/ shows the thresholds of insensitivity of the remaining strength to the damages of different types (delamination, fibre breakages). The rate of single crack growth (c) is described by the Paris law. In the last case the remaining strength defined by stress intensity factor depends on crack length.

Bimodal and more complicated experimental data for probability function distribution can be used to predict the reliability of composite structures by using special computing methods /3/, /4/.

  1. Kokcharov I. An estimation of reliability of unidirectional composites by catastrophe theory // Mechanics of Composite Materials, 1996 , vol.32, N 4, p.539-548.
  2. Kokcharov I. A difference between reliabilities of homogeneous and composite materials. Proceedings of the International Conference on Composites Engineering (ICCE/3), July 21-26, 1996, New Orleans, USA, p. 453-454.
  3. Koksharov I.I., Burov A.E. Comparative analysis of carrying ability for metalwork units by automated system for strength and crack resistance evaluations // Problemi Prochnosti (Problems of Strengths), 1994, N 4, pp.84-88 (in russian).
  4. Koksharov I. Codes for estimation of reliability indices of structural elements // Scientific Siberian, AMSE Press, France, vol. 11, ser. A (Numerical and Data Analysis), 1994, pp. 83-93.
  5. Koksharov I., Burov A.E. Crack resistance of structural elements of AL-B composite under tension // Proceedings of the ICCM-9, Madrid, 12-16 July, 1993, p.171-178.