52 Z.Xia and W.A.Curtin Start Generate fiber strength,(or);i=1...n 出 Apply strain Calculate fiber element stress,o S-8+As Any elements break? N Calculate composite stress,Gcom N Ocom reaches its peak? Y Stop Fig.2.6.Flow chart of simulation procedure for fiber damage evolution in fiber composites process is repeated until no further fiber breaks are found;the damaged composite is then in a stable equilibrium state.The applied displacement or load is then increased by a small increment,and the above process is repeated.In the SLM,which is typically displacement controlled,the ten- sile strength is identified as the maximum stress.In the GFM,which is load controlled,the system undergoes catastrophic failure(all fibers break in some narrow range of the sample cross-section)at the tensile strength. Figure 2.7 shows an example of the simulated stress-strain curve for an Al2O3/Al composite.The fiber damage evolution in the ultimate failure plane is shown via examination of the fiber SCFs in Fig.2.7 at two stages: just at failure and just beyond failure.In Fig.2.7,SCF values less than one indicate that the fiber is broken somewhere within a slip length of the
Fig. 2.6. Flow chart of simulation procedure for fiber damage evolution in fiber composites process is repeated until no further fiber breaks are found; the damaged composite is then in a stable equilibrium state. The applied displacement or load is then increased by a small increment, and the above process is repeated. In the SLM, which is typically displacement controlled, the tensile strength is identified as the maximum stress. In the GFM, which is load controlled, the system undergoes catastrophic failure (all fibers break in some narrow range of the sample cross-section) at the tensile strength. Figure 2.7 shows an example of the simulated stress–strain curve for an Al2O3/Al composite. The fiber damage evolution in the ultimate failure plane is shown via examination of the fiber SCFs in Fig. 2.7 at two stages: just at failure and just beyond failure. In Fig. 2.7, SCF values less than one indicate that the fiber is broken somewhere within a slip length of the Start Calculate fiber element stress, σi Calculate composite stress, σcom Stop Apply strain N Y Y N Any elements break? σcom reaches its peak? Generate fiber strength, (σf )i i=1…n ε=ε+∆ε 52 Z. Xia and W.A. Curtin
Chapter 2:Multiscale Modeling of Tensile Failure 53 failure plane and is carrying a reduced stress in the failure plane due to slip (2.4)while SCF values exceeding one indicate enhanced stresses on un- broken fibers in the plane of view.At low stress levels,isolated breaks occur at weak fiber elements throughout the material,and the stress concentra- tions are not sufficient to drive further failure.With increasing load,clusters of fiber breaks form due to both statistics and to enhanced local stresses. The stress concentrations around these clusters grow with the cluster size. driving further damage.When the load just reaches the tensile strength (Fig.2.7a),a"critical"cluster of fiber breaks forms,consisting of a dis- persed group of fiber breaks leading to local stress enhancements on the unbroken fibers in and around these breaks.With no further increase in applied load,fiber damage continues unabated spreading outward from the critical damage cluster.Figure 2.7b shows the damage configuration after some extent of unstable fiber damage.After some sporadic growth,the damage cluster becomes roughly penny shaped with very high-stress concentrations on its perimeter that drives the continued growth,similar to crack growth in a monolithic material. Critical fiber cluster 2500 2000 可 1500 1000 500 0.004 0.008 0.012 0.016 0.02 Strain (b) Fig.2.7.Predicted stress-strain curve for an alumina fiber/aluminum composite with a matrix yield strength of 100 MPa,with schematics of fiber damage and stress concentrations in the plane of final fracture:(a)just at the failure strength. where a critical damage cluster can be identified and(b)after some unstable damage propagation at the failure strength,where the damage has formed a near penny shape crack.Each node corresponds to a single fiber (reprinted with permission from [35])
failure plane and is carrying a reduced stress in the failure plane due to slip (2.4) while SCF values exceeding one indicate enhanced stresses on unbroken fibers in the plane of view. At low stress levels, isolated breaks occur at weak fiber elements throughout the material, and the stress concentrations are not sufficient to drive further failure. With increasing load, clusters of fiber breaks form due to both statistics and to enhanced local stresses. The stress concentrations around these clusters grow with the cluster size, driving further damage. When the load just reaches the tensile strength (Fig. 2.7a), a “critical” cluster of fiber breaks forms, consisting of a dispersed group of fiber breaks leading to local stress enhancements on the unbroken fibers in and around these breaks. With no further increase in applied load, fiber damage continues unabated spreading outward from the critical damage cluster. Figure 2.7b shows the damage configuration after some extent of unstable fiber damage. After some sporadic growth, the damage cluster becomes roughly penny shaped with very high-stress concentrations on its perimeter that drives the continued growth, similar to crack growth in a monolithic material. 0 500 1000 1500 2000 2500 0 0.004 0.008 0.012 0.016 0.02 Strain Stress, MPa Critical fiber cluster (a) (b) Strain Stress (MPa) 0 500 1000 1500 2000 2500 0 0.004 0.008 0.012 0.016 0.02 Strain Stress, MPa Critical fiber cluster (a) (b) Strain Stress (MPa) Fig. 2.7. Predicted stress–strain curve for an alumina fiber/aluminum composite with a matrix yield strength of 100 MPa, with schematics of fiber damage and stress concentrations in the plane of final fracture: (a) just at the failure strength, where a critical damage cluster can be identified and (b) after some unstable damage propagation at the failure strength, where the damage has formed a near penny shape crack. Each node corresponds to a single fiber (reprinted with Chapter 2: Multiscale Modeling of Tensile Failure 53 permission from [35])