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.
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 [
Great efforts have been taken to develop reliable and accurate analysis methods for composite structures. Empirical methods [
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
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. [
The continuum damage mechanics is another way to propose the gradual damage model for composite materials, in which the damage variable
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.
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 [
The composite materials have complex damage mechanisms in the mesoscale such as fiber breakage, fiber buckling, matrix cracking, interface debonding, and fiber bridging [
Under the longitudinal loading, fiber damage initiates by isolated fiber fractures in weak zones [
Damage of composite materials under longitudinal loading.
The matrix tensile failure, matrix compressive failure, and fiber-matrix shearing failure will form a crack parallel to the fiber in the macroscale, called an interfiber crack [
Interfiber failure of composite materials: (a) matrix crack in tension; (b) matrix crack in compression; (c) shear-out crack.
The progressive damage model proposed in the paper groups the unidirectional composite material damage into two classes: fiber cracks and interfiber cracks. The fiber cracks include the fiber tensile crack and fiber compressive crack, in which the latter is not an actual crack but a kink band. The interfiber cracks include the matrix tensile crack, matrix compressive crack, and fiber-matrix shearing crack.
The progressive damage model treats the fiber crack as a plane of symmetry because the fiber crack
Definition of damage variables and local coordinate system.
Based on the continuum damage mechanics, three damage variables
Longitudinal stresses will not produce a smooth crack but a damage band. There are not only fiber fracture (tension) and fiber buckling or kinking (compression) in the damage band, but also the matrix crack and fiber-matrix debonding. As a result, the fiber failure will decrease all the material properties. In contrast, interfiber damage seldom influences the load carrying capability in the fiber direction. To simplify the expression of the constitutive model, the effect of fiber failure on the transverse properties is considered by using the damage variables
Longitudinal loadings can induce not only the fiber crack but also the interfiber cracks, while transverse and shearing stresses can only result in the interfiber cracks. Thus, to simplify the constitutive model, this paper assumes that when
The normal stress reversal from the tension to compression will make the interfiber crack close as shown in Figure
Interfiber crack status under reversal load conditions: (a) tension and (b) compression.
On the other hand, this paper assumes the closed cracks cannot recover the shear load-bearing capacity, even though the frictions between the crack surfaces exist. Thus, the effective damage variables of the interfiber cracks
It is assumed that the effect of the fiber crack on the material properties under the longitudinal tensile stress is the same as that when it turns to compressive one. The effective damage variables
Under a longitudinal compressive loading, the composite material failure starts with the fiber kinking or buckling. With the load increases, the fiber kinking or buckling increases and induces the matrix cracks and fiber-matrix debonding. Consequently, the compressive load makes the fibers break into pieces [
The fully damaged composite materials have liquidity like sands [
Compressive behaviors of composite materials.
Due to different mechanical behaviors of damaged composite materials under longitudinal tensile and compressive loads, it is hard to use the same model to depict longitudinal compressive behaviors as that to characterize longitudinal tensile behaviors. Inspired by Chang’s model for the composite material longitudinal compressive behaviors [
Based on the assumptions above, the constitutive model of composite materials is established. The material parameters of the flexibility matrix
Material parameters in 3D progressive damage model.
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The modified maximum criterion proposed by Zhao et al. [
If either of the fiber tensile failure criterion or the fiber compressive failure criterion is satisfied, the fiber crack onset is detected. For fiber tensile (
If any criterion among the matrix tensile failure criterion, matrix compressive failure criterion, and fiber-matrix shearing failure criterion is satisfied, the interfiber crack onset is detected. The criterion is expressed in (
Matrix tensile (
Fiber-matrix shearing failure is
After damage onset, the damage variable
Stress-strain curve of linear damage model.
The damage variable
For interfiber cracks, the normal stress
For a complete continuum damage model, the damage onset strain and final failure strain are key parameters. The former is often obtained from the damage onset stress which is generally referred to the interface strength. Since the interface strength is difficult to be obtained from common tests, appropriate values are often used [
To validate the capability of predicting damage propagation in the composite materials under complex loading conditions, the 3D progressive damage model was embedded into the software ABAQUS and applied to a double-lap composite bolted joint under a tensile load. Meanwhile, the tensile experiments of composite bolted joints were conducted to validate the numerical results. On considering the main object and the limited length of this paper, the implements of the 3D progressive damage model in ABAQUS will be described in a following independent paper.
A double-lap composite bolted joint was selected to verify the model proposed here. Its configuration and dimensions were designed as recommended in engineering and are shown in Figure
Configuration and dimensions of a composite bolted joint: (a) front view and (b) top view.
The experiments were carried out by using a 250 kN Instron8803 testing machine. Five specimens were tested. The displacement load with rate of 1 mm/min was applied up to the catastrophic failure of the joint. The applied load and grip holder displacements were measured by sensors of the testing machine automatically, while the hole deformations were examined by an extensometer mounted on the specimens. More information about the experimental process can be found in [
The FE model of the double-lap composite joint is shown in Figure
FEM of double-lap composite bolted joint.
The elastic properties and strengths of the composite material are
The gradual degradation model and failure criteria were applied to the FE model by developing a user defined subroutine UMAT for ABAQUS. At the beginning of each load increment, the UMAT will be called. The damage state will be checked using failure criteria. Once material damage is predicted, the gradual degradation model will be used to determine each damage variable according to the stress conditions and give the stiffness matrix of damaged material. This operation is repeated time and time again until the structure is fully destroyed.
Figure
Experimental and numerical ultimate loads of composite bolted joints.
Experimental ultimate load/kN | Avg. val./kN | Numerical ultimate load/kN |
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21.18 | 20.82 | 20.50 |
20.80 | SD/kN | Error |
20.78 | 0.35 | −1.5% |
21.06 |
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/ |
20.26 | 1.69 | / |
Load-hole deformation curves of specimens: (a) numerical predictions from different mesh schemes and (b) comparison between experimental and numerical results.
Figure
Failure morphologies of the central plate: (a) from experiments and (b) from numerical predictions.
Good agreements on the joint strength, stiffness, and failure morphology between the experimental outcomes and numerical predictions give evidence that the proposed PDM has great potential on simulating the mechanical behaviors of composite materials under complex loading and constraints.
A new 3D constitutive model for progressive damage analysis of unidirectional composite materials is proposed in this paper. The model has taken account of the possible fiber fracture planes caused by longitudinal tensile and compressive loads as well as interfiber fracture planes at any angles with respect to the ply thickness direction caused by transverse stresses and fiber-matrix shear stresses. Mathematical models are also presented to describe complex phenomena of damaged materials, such as crack closure due to load reversion, interaction of fiber damage and interfiber damage, effective elastic moduli, and Poisson’s ratios of damaged materials. Besides, an approximate model has been created for the longitudinal compressive behaviors. The damage onset has been predicted by using the modified maximum stress failure criterion [
The proposed model has been embedded into ABAQUS using the subroutine UMAT and applied to predict the progressive damage accumulation and collapse failure of a double-lap composite bolted joint under tensile loads. The load-hole deformation curve, strength and stiffness information, progressive damage process, and final failure morphology can be obtained from the progressive damage simulation with the model proposed. The numerical prediction is in good agreement with the static tensile experimental outcomes of the composite bolted joint, which gives evidence of the effectiveness of the model proposed.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research work is supported by the National High Technology Research and Development Program of China (2012AA040209) and the National Science Foundation of China (11372020, 11202012, 11472027, and 10902004).