18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS PLASTIC DAMAGE MODEL FOR PROGRESSIVE FAILURE ANALYSIS OF COMPOSITE STRUCTURES J. F. Chen, E. V. Morozov * , K. Shankar School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, Australia * Corresponding author ( e.morozov@adfa.edu.au ) Keywords : plastic damage model, progressive failure analysis, composite structures, return mapping algorithm various failure mechanisms employing Hashin’s 1 Introduction failure criteria [12]. Laminated composite materials are widely used in The proposed plastic damage model is implemented aerospace, civil, shipbuilding and other industries in Abaqus/Standard using a user-defined subroutine due to their high strength and stiffness to weight (UMAT). The strain-driven implicit integration ratios, good fatigue resistance and high energy procedure for the proposed model is developed using absorption capacity. In many structural applications, equations of continuum damage mechanics, the progressive failure analysis is required to predict plasticity theory and applying the return mapping their mechanical response under various loading algorithm. To ensure the algorithmic efficiency of conditions. the Newton-Raphson method in the finite element The use of appropriate material constitutive models analysis, a tangent operator that is consistent with plays a crucial role in progressive failure analysis of the developed integration algorithm is formulated. composite structures. Most of the damage mechanics The efficiency of the proposed model is verified by based composite material models fall into the performing progressive failure analysis of composite elastic-damage category [1-6]. In these models, laminates containing central hole and subjected to irreversible deformations are not normally in-plane tensile loads. The predicted results agree considered in the unloading stage. Although this well with the test data and provide accurate might be suitable for modelling the mechanical estimates of the failure loads. behaviour of elastic-brittle composites, experimental 2 Plastic damage constitutive model studies [7,8] show that some thermoset or thermoplastic composites exhibit apparent plastic 2.1 Stress-strain relationship response, especially under transverse and/or shear The proposed plastic damage model is formulated stresses. Numerical investigation also reveals that for an elementary orthotropic ply and describes both the model that does not take into account the plastic the plastic response and the damage development nature of composites might be insufficient, in some which is based on the stiffness reduction approach. instances, in the evaluation of energy absorption The damage effects are taken into account by capacity of composite structures [9]. In addition, introducing damage variables in the stiffness matrix damage accumulated within the plies could lead to using the continuum damage mechanics concept. the material properties degradation before the The stress-strain relationships for the damaged and collapse of the composite structures. The undamaged composite materials are written as consideration of material properties degradation follows: improves the predictions of failure loads [10]. � � � � � � � � � � � �� – � � �; � (1) This paper attempts to develop a combined plastic � � ���� � � � � ���� � �� – � � � damage model for composites, which accounts for both the plasticity effects and material properties where bold-face symbols are used for variables of degradation of composite materials under loading. tensorial character and symbol ( : ) denotes inner The plasticity effects are modelled using the product of two tensors with double contraction, e.g. approach proposed by Sun and Chen [11]. The ( ����� � � � � �� ) = S��� ���� � ��� , where the summation prediction of the damage initiation and propagation convention is applied to the subscripts; � , � � are the in the laminated composites takes into account Cauchy stress tensor and the effective stress tensor (both are the second order tensors); � � is the fourth-
order constitutive tensor for linear-elastic plane stress condition proposed by Sun and Chen undamaged unidirectional laminated composites; [11] is adopted in this study: ���� is the one for the corresponding damaged � � 2� � �, �̃ � � � � � materials; �, � � , � � are the total strain, elastic strain, � � – � ���̃ � � � 0 (6) ��� � �� � � � � and plastic strain tensors, respectively; d is the damage variable. The form of the S ( d ) adopted in where � is a material parameter which describes the this model is similar to that presented by level of plastic deformation developed under shear Matzenmiller et al. [2] loading compared to the transverse loading; � � � is the effective stress in the transverse direction, � � � is ���� � the effective in-plane shear stress . Note that the use � � � � � �1 � � � �� � �� �� 0 of this form of yield function improves efficiency � � � � � �1 � � � �� � � � � �� �� 0 � (2) and accuracy of the computational algorithm. � 0 0 ��1 � � � �� �� For the sake of simplicity, an isotropic hardening � � � � �� law expressed in terms of equivalent plastic strain ε � ; parameter where � � 1 � �1 � � � ��1 � � � �� �� is adopted in this work. The following formulation � � �1 � � � ��1 � � � � ; parameters � �, � � , � � of this law proposed by Sun and Chen [11] is used to denote damage developed in the fibre and transverse represent the equivalent stress versus equivalent direction, and under shear (theses damage variables plastic strain hardening curve: � , � � � , are constant throughout the ply thickness); � � � , � �� � and � �� � are elastic moduli and Poisson’s ���̃ � � � � ���̃ � � � ���̃ � � � (7) � �� ratios of undamaged unidirectional composite where � � is the equivalent stress defined as follows: laminae. � In order to differentiate between the effects of � � 2� � � � � � (8) � � � � � � � � � � compression and tension on the failure modes, the damage variables are presented as follows: In Eq.(7), � and � are coefficients that fit the � � � �� �� if � � � 0 � �� if � � � 0 � � � �� �� if � � � 0 experimental hardening curve. These parameters � �� if � � � 0 （ 3 ） together with the material parameter � are determined using an approach based on the linear where � �� , � �� characterise the damage development regression analysis of the results obtained from the caused by tension and compression in the fibre off-axis tensile tests performed on the unidirectional and � �� , � �� direction, reflect the damage composite specimens [11, 13]. development caused by tension and compression in The associated plastic flow rule is assumed for the the transverse direction. It is assumed that the shear plastic evolution in composites. According to this stiffness reduction results from the fibre and matrix law, the plastic strain rate is expressed as: cracking. To take this into account, the �� � � �� � � � corresponding damage variable d � is expressed as: � � (9) where �� � � 0 is a nonnegative plastic consistency � � � 1 � �1 � � � ��1 � � �� � （ 4 ） parameter; hereafter � � � � ��/�x . where � � represents the damage effects on shear Similarly, the associated equivalent plastic flow rule stiffness caused by matrix cracking. is also adopted in the following form: 2.2 Plastic model �̃� � � �� � � � � � � (10) Plasticity is assumed to occur in the undamaged area The equivalent plastic strain rate can be obtained of the composites. The plastic yield function is from the equivalence of the rates of the plastic work expressed in terms of effective stresses as follows: per unit volume � � : �, �̃ � � � � � �� �� � ���̃ � � � 0 ��� (5) � � � � � � �� � � � � �̃� � (11) W where � � is the plastic potential; � is the hardening Making use of Eq. (6), and taking into account Eqs. parameter which depends on the plastic (8) - (11), the following relation is derived: deformations and is expressed in terms of equivalent plastic strain �̃ � . �̃� � � �� � (12) Due to its simplicity and accuracy, an equivalent form of the one-parameter plastic yield function for
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