Probabilistic modeling of stiffness degradation for fiber reinforced polymer under fatigue loading

Highlights

• The inherent correlation between stiffness degradation and strength degradation of FRP is explored.

• A probability model of stiffness degradation is proposed based on the same damage state principle.

• The proposed model is flexible enough to fit the stiffness degradation data of FRP laminates with good agreement.

Abstract

The stiffness degradation of fiber reinforced polymer (FRP) has significant effects on its fatigue life and reliability. In this study, the inherent correlation between stiffness degradation and strength degradation is explored, and a probability model that describing stiffness degradation of FRP is presented. Firstly, the evolution of fatigue damage in FRP is analyzed, and then it is characterized by stiffness degradation and strength degradation, respectively. Secondly, the randomness of initial stiffness and critical stiffness of FRP is considered, and a probability model of stiffness degradation is proposed based on the same damage state principle. Finally, the test data of FRP laminates are used to verify the proposed model, and the fitting accuracy is compared with several existing models. The results show that the stiffness degradation of FRP presents a fast-slow-fast trend during fatigue failure process, and the proposed model is capable of fitting stiffness degradation data of FRP with good agreement.

Introduction

Fiber reinforced polymer (FRP) is a kind of composite material, which is usually made of fiber material and matrix material through compounding technology. FRP has been widely used in engineering due to it has many superior properties, such as light weight, high specific strength and stiffness, excellent resistance to corrosion, impact and fatigue [1]. As a result of the complexity and diversity of load conditions, FRP is often subjected to the cyclic loading in practical applications. Under such circumstances, fatigue fracture becomes one of the most common failure modes of FRP [2]. Generally speaking, fatigue fracture of FRP components will cause serious economic losses and even lead to catastrophic accidents. Therefore, the fatigue resistance of FRP is a serious issue from the perspectives of economic and safety.

To avoid economic losses and catastrophic accidents, fatigue fracture of FRP under cyclic loading has been extensively studied over the past few decades, and significant progress has been made. In general, the existing theoretical models can be classified into three categories: (1) fatigue life models [3], [4], which do not take into account the development of fatigue damage, but use $\sigma \text{-}N$ curve and macroscopic fatigue failure criterion to predict fatigue life; (2) phenomenological models [5], theories in this category usually regard residual stiffness or residual strength as a damage metric to quantify fatigue damage accumulation in FRP; and (3) mechanism models [6], [7], theories in this category aim to reveal the physical mechanisms of fatigue failure based on actual damage modes, such as matrix cracking, interfacial debonding and fiber breakage.

The mechanical properties (such as stiffness and strength) of FRP will gradually decrease with the accumulation of fatigue damage. The stiffness degradation and strength degradation have significant effects on fatigue life and reliability of FRP. In recent years, many studies that focused on residual stiffness and residual strength models have been carried out [8]. Farahani and Shirazi [9] considered the influence of orientation angle of fibers on fatigue damage, and a stiffness degradation model of FRP was developed. Pakdel and Mohammadi [10] investigated the mechanisms of matrix cracking and induced delamination, and the stiffness degradation law of FRP under tension–tension loading was analyzed. To account for the variations in failure modes and damage accumulation rates, Suzuki et al. [11] developed a residual stiffness model of FRP based on Weibull distribution. Mousavi and Behrooz [12] presented a creep-fatigue stiffness degradation model for FRP, which was composed of a linear elastic component and a nonlinear time-dependent component. Roundi et al. [13] considered the influences of loading types and stress levels on residual stiffness, and the stiffness degradation behaviors of FRP with different stacking sequences were explored. Shiri et al. [14] made a brief comment on the limitations of existing residual stiffness models, and a new stiffness degradation model was established based on fatigue tests.

Residual strength is a commonly used damage metric to quantify fatigue damage accumulation in FRP, which refers to the resistance of material to external load after a given loading times. Llobet et al. [15] assumed that the ultimate strain remains unchanged after fatigue damage, and a residual strength model was developed based on continuum damage mechanics. To reduce the required experimental efforts, Stojković et al. [16] proposed a two-parameter residual strength model based on the concept of strength reserve. Talreja [17] used strength degradation to quantify the accumulated fatigue damage in FRP, and a reliability model was developed based on three-parameter Weibull distribution. To explore the fatigue damage mechanisms of FRP, Yao and Himmel [18] presented a residual strength model to describe the development of fatigue damage. Cheng and Hwu [19] proposed a strength degradation model to calculate fatigue life of FRP, and a reliability assessment approach was presented based on probability and statistics theory. D'Amore and Grassia [20] reviewed the existing residual strength models, and a comparative study was conducted based on the same data set of FRP.

The degradation laws of stiffness and strength of FRP have been extensively studied, and many theoretical models have been developed over the past decades [21], [22]. But it should be noted that the existing residual strength and residual stiffness models have their own inherent limitations. Residual strength model is usually established based on strength degradation data obtained from destructive tests. Besides, only one strength degradation data can be obtained from one FRP specimen, which means that a large number of destructive tests are required to establish a residual model. Therefore, the application of residual strength model is severely limited due to the constraints of test time and cost in practical engineering. By comparison, residual stiffness model can be established based on sufficient stiffness degradation data obtained from nondestructive tests, which makes it possible to monitor the fatigue damage development of FRP continuously during fatigue test. However, most of the existing residual stiffness models can not accurately describe the evolution process of fatigue damage in FRP. In addition, the inherent correlation between stiffness degradation and strength degradation has not been completely explored in the existing studies. In this study, the evolution of fatigue damage in FRP is analyzed, and then it is quantified by the degradation of stiffness and strength, respectively. The inherent correlation between stiffness degradation and strength degradation is explored based on the same damage state principle. A probability model is presented to characterize the stiffness degradation, and its validity is verified by using the test data of four kinds of FRP laminates.

The rest of this paper is organized as follows. In Section 2, the evolution process of fatigue damage in FRP is analyzed, and the same damage state principle is elaborated. In Section 3, the randomness of initial stiffness and critical stiffness of FRP is considered, and a probability model of stiffness degradation is proposed from the perspectives of probability and statistics. In Section 4, the test data of FRP laminates with different stacking sequences are applied to verify the proposed model, and the fitting accuracy is compared with several existing models. Some corresponding conclusions are presented in Section 5.