The generalised local model applied to Fibreglass

In the present research work, the Generalised Local Model (GLM) is employed to characterize the failure of a commercial Fibreglass by determining the Primary Cumulative Distribution Function of Failure (PCDFF) through an experimental campaign and numerical computations. Four criteria used to define the failure condition are based on the following Generalised Parameters (GPs): (i) Maximum Principal Stress; (ii) von Mises Stress; (iii) Tsai-Hill GP; (iv) Hoffman GP. For each GP adopted, the PCDFF is derived in two different ways: (a) by examining data obtained from tension tests only (GLM single application); (b) by collecting data, determined through tension tests and three-point bending tests, into a unique set (GLM joint application). Finally, a comparison between the obtained PCDFFs and the experimental failure results is presented.


INTRODUCTION
The application fields of composites are various, due to their high strength-to-weight and stiffness-to-weight ratios, good thermal properties, and potential for tailor-made design in mechanical, civil, marine and aerospace engineering. Their manufacture process Uncertainties in composite performance may be of three types [2,3]: aleatory uncertainty, epistemic uncertainty, and errors.
The first type of uncertainty is related to the randomness of the system, i.e. due to fibres and matrix characteristics, their reciprocal interaction, manufacturing variations and so on. The epistemic uncertainty is the consequence of the limited knowledge of the composite behaviour, and is related to the experimental procedures and modelling methods adopted (for instance, the assumptions regarding the material load-response). Finally, the third cause of uncertainty refers to errors, such as those due to instrumentation, regression method applied and so on, that some In the present work, the GLM is employed to analyse the GPF of a commercial Fibreglass. The novelty of this paper is that the most suitable failure Generalised Parameter (GP) to determine the GPF of a commercial Fibreglass is identified among four alternatives: Maximum Principal Stress, von Mises Stress, Tsai-Hill GP, and Hoffmann GP.

Formulation
The flowchart describing the Generalised Local Model [15,32-33] is shown in Figure 1.
The GLM allows us the determine the PCDFF, which is a function of a Generalised Parameter characterising the failure of the structural component being examined.
and  are three Weibull parameters (shape, scale, and location), and GP is the above-mentioned Generalised Parameter.
More precisely, by starting from the data obtained from an experimental campaign, the is an arbitrary reference size (for instance, an area or a volume), ij S  represents the size (area or volume) of the j-th finite element, and initial values are assumed for .
The global cumulative failure probability, is the maximum value of the GP evaluated for the i-th model. In such a way, the PCDFF obtained at the end of the GLM procedure is independent of the size of the finite elements used in the discretisation.
The global probability just obtained is equal to that computed by means of Eqs (3) and (4).
Up to this step of the procedure, Finally, the PCDFF can be evaluated by using Eq. (1) and last values of the three Weibull parameters.

Single and joint application of GLM
The GLM described in Section 2.1 can be applied in two alternative ways: (i) The first one, named GLM single application, consists in applying such a model to a homogeneous set of experimental data obtained from the same type of experimental test, such as tension test or three-point bending (3PB) test. The procedure described by the flowchart in Figure 1 is essentially the GLM single application. ( and the global cumulative failure probability can be expressed as follows: In both cases above (i and ii), the PCDFF obtained can be applied to estimate the probability of material failure for any other test type.

GENERALISED PARAMETERS EXAMINED
The

GP related to the Maximum Stress Failure Criterion
According to the maximum stress theory, failure occurs when the maximum principal stress exceeds the ultimate material strength.
By assuming the material to be macroscopically homogeneous, the condition of failure is given by [38]: where 1  is the maximum principal stress, where , , , ,

GP related to the Tsai-Hill Failure Criterion
Such a failure criterion was proposed for orthotropic material [39,41]. By defining a Failure Index FI , the condition of failure can be written in tensor notation:

GP related to the Hoffman Failure Criterion
Such a failure criterion was proposed by Hoffman [40] in order to take into account the different material strengths exhibited under tension and compression. Hoffman considered also the linear terms, i F , in Eq. (13) adopting the parameters reported in Table 1.
By assuming the material to be isotropic and macroscopically homogeneous, the parameters in Table 1 can be re-written as follows: The experimental campaign has consisted of: (1) tension tests on 33 specimens; (2) three-point bending (3PB) tests on 38 specimens; (3) four-point bending (4PB) tests on 34 specimens.

Single GLM application
The GLM is applied to the test results presented in the above Sub- In order to verify which one of the four adopted Generalised Parameters most suitably describes the failure behaviour of the tested material, the Weibull parameters related to the PCDFF for tension (listed in Table 3) are used to evaluate the PCDFF for both three-and four-point bending. Table 3.  Table 3).  Table 3).  Table 3). for four-point bending condition (see Table 3).

Joint GLM application
The GLM is here applied by considering the results obtained from the specimens subjected to both tension and three-point bending as a unique set of data, as is described in Section 2.2. The Weibull parameters, computed for each Generalised Parameter, are listed in Table 3.  Table 3).    Table 3). In conclusion, the present paper has proved that the GLM is a promising tool to assess the cumulative failure probability of structural components made of short-fibre reinforced materials, although more complex load configurations need to be processed in order to devise a robust procedure suitable for practical component design.             four-point bending.