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S. Zacek et al.
degradation of the laminate, such as induced delamination between plies and fiber
breakage [2]. Characterizing and modeling the transverse failure of composites
are complicated by the variability present not only in the material microstructure
(i.e., the fiber size distribution and placement) but also in the local constitutive and
failure properties of the constituents. The interaction between failure mechanisms
such as fiber/matrix interface debonding and matrix cracking further complicates
the prediction of the transverse strength of the composite laminate [1].
Multiple analytical and numerical models have been developed over the past
decades to predict transverse cracking in composite laminates. In analytical models,
it is often assumed that sequential cracks occur midway between existing cracks
[3, 4], while numerical models, which tend to rely on periodic boundary conditions,
simulate only a small portion of the experimental microstructure [5–7] and/or
assume a uniform, structured packing [8, 9]. However, there is an increasing need
to model larger, more realistic composite microstructures, as complex interactions between phases result in effective properties that are highly dependent on
microstructural details [10].
In unidirectional composites with a high fiber volume fraction under transverse
tensile loading, failure typically occurs at the interfaces between the fibers and
the matrix. One of the most successful numerical methods used to capture this
type of failure relies on a cohesive failure law relating the cohesive traction to
the displacement jump along the fiber/matrix interfaces [11, 12]. This approach
is also the basis of the present study, which relies on a nonlinear, discontinuous
extension of a recently introduced interface-enriched generalized finite element
method (IGFEM) [13, 14] that allows for the modeling of transverse failure in
realistic virtual composite microstructures with hundreds of fibers discretized with
nonconforming finite element meshes. Beyond the development of this special form
of the IGFEM, a key goal of this work is to compute the sensitivity of the transverse
failure response of the transverse ply to the cohesive properties of the fiber/matrix
interfaces. To that effect, we present an analytic material sensitivity formulation
based on the direct differentiation method and implement it in the nonlinear,
cohesive IGFEM solver. Related work on IGFEM-based sensitivity analysis in the
context of multi-scale material design can be found in [15, 16].
The manuscript is organized as follows: in Sect. 2, the material system of
interest and experimental observations are presented. Next, Sect. 3 summarizes
the computational method used to simulate the initiation and propagation of the
transverse cracks. Section 4 describes the sensitivity analysis adopted in this work
to capture the dependence of the transverse failure response of the transverse ply on
the cohesive failure properties of the fiber/matrix interfaces. Additional derivations
of the sensitivity to the critical displacement jumps are provided in the Appendix.
The sensitivity formulations are verified against finite difference approximations in
Sect. 5, while Sect. 6 summarizes the results of a sensitivity analysis performed on
a virtual composite laminate composed of hundreds of fibers.
S. Zacek et al.
degradation of the laminate, such as induced delamination between plies and fiber
breakage [2]. Characterizing and modeling the transverse failure of composites
are complicated by the variability present not only in the material microstructure
(i.e., the fiber size distribution and placement) but also in the local constitutive and
failure properties of the constituents. The interaction between failure mechanisms
such as fiber/matrix interface debonding and matrix cracking further complicates
the prediction of the transverse strength of the composite laminate [1].
Multiple analytical and numerical models have been developed over the past
decades to predict transverse cracking in composite laminates. In analytical models,
it is often assumed that sequential cracks occur midway between existing cracks
[3, 4], while numerical models, which tend to rely on periodic boundary conditions,
simulate only a small portion of the experimental microstructure [5–7] and/or
assume a uniform, structured packing [8, 9]. However, there is an increasing need
to model larger, more realistic composite microstructures, as complex interactions between phases result in effective properties that are highly dependent on
microstructural details [10].
In unidirectional composites with a high fiber volume fraction under transverse
tensile loading, failure typically occurs at the interfaces between the fibers and
the matrix. One of the most successful numerical methods used to capture this
type of failure relies on a cohesive failure law relating the cohesive traction to
the displacement jump along the fiber/matrix interfaces [11, 12]. This approach
is also the basis of the present study, which relies on a nonlinear, discontinuous
extension of a recently introduced interface-enriched generalized finite element
method (IGFEM) [13, 14] that allows for the modeling of transverse failure in
realistic virtual composite microstructures with hundreds of fibers discretized with
nonconforming finite element meshes. Beyond the development of this special form
of the IGFEM, a key goal of this work is to compute the sensitivity of the transverse
failure response of the transverse ply to the cohesive properties of the fiber/matrix
interfaces. To that effect, we present an analytic material sensitivity formulation
based on the direct differentiation method and implement it in the nonlinear,
cohesive IGFEM solver. Related work on IGFEM-based sensitivity analysis in the
context of multi-scale material design can be found in [15, 16].
The manuscript is organized as follows: in Sect. 2, the material system of
interest and experimental observations are presented. Next, Sect. 3 summarizes
the computational method used to simulate the initiation and propagation of the
transverse cracks. Section 4 describes the sensitivity analysis adopted in this work
to capture the dependence of the transverse failure response of the transverse ply on
the cohesive failure properties of the fiber/matrix interfaces. Additional derivations
of the sensitivity to the critical displacement jumps are provided in the Appendix.
The sensitivity formulations are verified against finite difference approximations in
Sect. 5, while Sect. 6 summarizes the results of a sensitivity analysis performed on
a virtual composite laminate composed of hundreds of fibers.