Mathematical Problems in Engineering

Volume 2015, Article ID 145629, 11 pages

http://dx.doi.org/10.1155/2015/145629

## 3D Gradual Material Degradation Model for Progressive Damage Analyses of Unidirectional Composite Materials

^{1}School of Astronautics, Beihang University, Beijing 100191, China^{2}Institute of Solid Mechanics, Beihang University, Beijing 100191, China

Received 15 September 2014; Accepted 1 December 2014

Academic Editor: Chenfeng Li

Copyright © 2015 Libin Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new 3D constitutive model for progressive damage analyses of unidirectional composite materials is presented, in which several important damage phenomena for the composite materials, such as the interfiber crack orientation, coupling of fiber failure and interfiber failure under longitudinal loads, closure effect for interfiber cracks, and longitudinal compressive behaviors under transversal constraints, have been considered comprehensively. A modified maximum stress failure criterion has been used for the damage onset prediction and a linear damage model has been adopted to establish the evolution rules of different damage. Numerical analyses with the model proposed have been implemented by using the subroutine UMAT in commercial software ABAQUS. Progressive damage analyses and static tensile experiments of a group of double-lap composite bolted joints have been carried out to validate the model proposed. Good agreements between the numerical and experimental results have been obtained.

#### 1. Introduction

With the increasing application of composite materials in aircrafts, automobiles, ships, and so forth, there is an urgent requirement for composite structure design methodologies. During the past decades, the structural design has mainly relied on experimental data by using the building block approach [1], which results in extremely expensive cost due to a large number of tests at each structural level. Thus, initiatives bridging the tests and analyses to reduce the design cost have become a common view in the world [2].

Great efforts have been taken to develop reliable and accurate analysis methods for composite structures. Empirical methods [3–6] obtained from plenty of tests and experiences are usually only suitable for certain structure types or load cases, though they are convenient and beneficial to the structure design. Besides, series of tests are required to determine some parameters for the empirical methods when the materials, ply sequences, or structural configurations change, which also lead to expensive cost. Theoretical analysis methods can significantly reduce the composite structural design and analysis costs and always draw lots of attention in the engineering. Among these methods, some predict the structural strength only with failure criteria. These methods will result in too conservative predictions since they ignore the structural nonlinear damage propagation process and fail to reveal the structural failure mechanism conveniently [2, 7]. In the later of the 20th century, a progressive damage method, which builds nonlinear mechanics models for composite materials and has capacity of accurately simulating the structural failure process from initial damage to ultimate failure, attracts widespread attention in composite structure analyses [8–11]. A progressive damage model contains three parts: a stress analysis model to obtain structural stress distribution, a failure criterion to estimate the structural damage and failure, and a material degradation model to control the property changes of damaged materials. Furthermore, the last one has been a hotspot in the composite research during decades since it directly represents the nonlinear damage propagation.

The material degradation models can fall into two categories, sudden degradation models and gradual degradation models. In the sudden degradation models, the material properties degrade to zero or a certain proportion of original values, which is called a degradation ratio. A total discounting model in which all the material properties degrade to zero when the damage is predicted is the simplest sudden degradation model. Based on the total discounting model, conservative predictions of structural strength are always obtained. Additionally, a usually used ply discounting model is one of extreme editions [12–15]. Another average practice is that the properties are degraded with targeted selection according to the failure modes. For example, Engelstad et al. [16] only reduced one material modulus at a time corresponding to a failure mode dominated by a stress component. More commonly, several material properties are reduced simultaneously for a failure mode, which is called interaction models [17–19]. Camanho and Matthews [10] distinguished the material degradation model by the tensile and compressive stress for the first time. Obviously, the sudden degradation models neglect the damage accumulation during the composite material failure process since they simply treat the material as undamaged and totally damaged.

To establish more accurate models, gradual degradation rules are good choices to describe the postfailure behavior of composite materials. The micromechanics theories are usually used for establishing gradual degradation models. Chang et al. [20–27] adopted the fiber bundle theory, while Lee et al. [28–30] used the shear lag theory to build the degradation model for the fiber failure. In contrast to the micromechanics methods, more researchers set the material property degradation rules with a predefined function as a matter of experience. In Lin and coworker’s [31, 32] gradual degradation models, the material degradation models were chosen to make the stress reduce linearly after the damage initiation. Hwang et al. [33] adopted an exponential function to establish the composite material degradation model.

The continuum damage mechanics is another way to propose the gradual damage model for composite materials, in which the damage variable is used to represent the damage scale in the materials. With the increasing damage variable from zero to unit, the material properties degrade from the original value to zero. In a strict sense, the evolution laws for damage variables should obey the thermodynamics principles of irreversible processes. Since the continuum damage mechanics has a rigorous theory basis, establishing suitable 3D continuum damage model for composite materials has been the major goal of the Third World-Wide Failure Exercise [34]. In addition, the continuum damage mechanics based gradual degradation models are more appealing. However, due to the complexity and intricacy of composite material failure mechanisms, the developments of continuum damage model for composite materials are slow. Existing continuum damage mechanics based gradual degradation models commonly focused on partial characters of the failure. Maimí et al. [2] and Raimondo et al. [35] proposed 2D gradual degradation models for composite laminates separately, in which the crack direction and crack closure were considered. Pinho et al. [36] developed a 3D gradual degradation model for composite materials. The fiber failure was focused on, and the crack direction was taken into account. However, the crack closure was not mentioned.

For cases where the composite structures surfer from 3D complex stresses such as bolted composite joints, the more accurate 3D gradual degradation model that can consider more damage phenomena of composite materials is still required. In this paper, a 3D gradual degradation model that takes account of the matrix crack direction, matrix crack closure, interaction of the fiber damage, and matrix damage as well as mechanical behaviors of compressive fiber failure is proposed based on the physical damage phenomena of composite materials. Furthermore, the progressive damage analysis is implemented for composite structures by using subroutine UMAT in ABAQUS and validated by the experimental results of double-lap composite bolted joints.

#### 2. 3D Progressive Damage Model Using Gradual Degradation Model

The degradation model of unidirectional composite materials is developed in this paper based on the characteristics of damage mechanisms of different failure modes. The modified maximum stress failure criteria [37] are used to predict the damage modes, initiation, and crack angles of composite materials. The linear damage evolution law is chosen to define the softening relationship for all failure modes. Finally, the model is embedded into the software ABAQUS with UMAT.

##### 2.1. Failure Mechanisms of Composite Materials

The composite materials have complex damage mechanisms in the mesoscale such as fiber breakage, fiber buckling, matrix cracking, interface debonding, and fiber bridging [2, 38]. The propagation and connection of damage in the mesoscale lead to the damage in the macroscale. According to the loads inducing the damage, the composite damage in the macroscale include five modes: fiber tensile damage, fiber compressive damage, matrix tensile damage, matrix compressive damage, and fiber-matrix shearing damage. In virtue of the regular fiber distribution in the mesoscale, some characteristic of the damage in the macroscale can be tracked, which is helpful to establish the mechanics model of composite materials.

Under the longitudinal loading, fiber damage initiates by isolated fiber fractures in weak zones [38]. The localized fractures will induce the matrix cracking, interface debonding, and matrix shearing failure in the adjoining fibers on account of local stress concentrations caused by the localized fractures, as shown in Figure 1. With the increment of the longitudinal loading, more fibers fail and further lead to the ultimate failure of the material. For the longitudinal tensile case, a macroscopic crack perpendicular to the fiber will be formed [2], while for the longitudinal compressive case a damaged band as a result of fiber buckling, fiber kinking, matrix crack, and interface debonding is formed [39, 40]. The macroscopic damaged band is assumed to be perpendicular to the fiber for simplification.