The influence of loading rate on the fracture mode for the viscous-elastic composites

The study of failure processes of composites should be accompanied with the analysis of the nature of failure, three main categories of which may be identified as: the defect accumulation; macrocracks propagation; instantaneous rupture, being a consequence of the cleavage of a weak link. Usually in these type of investigations the consideration is given to one or another type of failure while the conditions of transition from one type of failure to another or a possible change in its character are not discussed. However, the determination of these conditions is crucial for an accurate analysis of the performance of a composite structural component. Various types of failure for the viscous-elastic composites can be classified into three major categories: 1) the subsequent accumulation of defects, with certain concentration of which resulting in catastrophic structural break; 2) the propagation of the macro cracks or systems of cracks previously occurring in the material; 3) complete and practically instantaneous failure – the result of the cleavage of the weak link. There are multiple publications with consideration given to one or another type of failure while the conditions of transition, i.e., the problem of possible change of the fracture character are not significantly discussed and covered. However, the determination of such conditions for transition is necessary for an accurate analysis of the performance of a structural composite component. Besides the structural characteristics such as the volume fraction of the fiber [1] that undoubtfully might affect the conditions of the fracture type transition the variable mechanical loading rate as well as temperature are very important factors to consider. The analysis of the effect of loading rate on the nature of the failure is performed on two types of carbon fibers reinforced composites. The matrix material was the same: epoxyanilinophenol formaldehyde type binder. As the reinforcement a carbon tape was used with normal (material I) and nitric acid-activated (material II) surfaces. Nitric acid etching promotes an increase in the specific surface (roughness) of the carbon fibers, resulting in the increased shear strength of the materials with treated components by the factor of 1.5-2 [2]. Some physical and mechanical characteristics for both materials are given in table 1. The experimental program on tensile tests was carried out at a constant loading rate up to failure in the following range of rates (kg/mm2·sec): 6·10−1, 2.5·100, 1.25·101, 1.25·102, (6− 8)·102, (1− 1.6)·104. The values of the maximum loading were fixed in the experiments and provided the data for the graphs of the dependence of the tensile strength on the loading rate, given in figure 1.The results on durability testing at the constant loading level are given in figure 2 together with the constructed model durability curves. Analysis of the graphs in figure 1 suggests strongly nonmonotonic character of the dependence of σ∗ on the loading rate. It can be assumed that the occurrence of extrema peaks is due to the change of material failure type. The first region AB is due to a progressive accumulation of defects with strength increasing in synch with the loading rate increase. The maximum point for this region corresponds to the moment when the cracks or defects start to grow. To the right from the maximum point the failure takes place by the propagation of the macro cracks and hence is determined by the value of KIc for the given material, which falls both with the fiber volume content increase and with the rise of the loading rate, illustrated by decreasing region II in figure 1.The extrema minimum for this decreasing region corresponds to the appearance of cleavage of the weakened link in the composite material resulting in its practically instantaneous failure; from this moment the strength values tend to increase again with the growth of dynamical component of the loading rate. The developed mathematical model describes the process of damage accumulation in material and allows to establish the dependence of the ultimate breaking stress on time (or the rate) of loading. The failure criterion is formulated as following

Various types of failure for the viscous-elastic composites can be classified into three major categories: 1) the subsequent accumulation of defects, with certain concentration of which resulting in catastrophic structural break; 2) the propagation of the macro cracks or systems of cracks previously occurring in the material; 3) complete and practically instantaneous failure -the result of the cleavage of the weak link.There are multiple publications with consideration given to one or another type of failure while the conditions of transition, i.e., the problem of possible change of the fracture character are not significantly discussed and covered.However, the determination of such conditions for transition is necessary for an accurate analysis of the performance of a structural composite component.
Besides the structural characteristics such as the volume fraction of the fiber [1] that undoubtfully might affect the conditions of the fracture type transition the variable mechanical loading rate as well as temperature are very important factors to consider.
The analysis of the effect of loading rate on the nature of the failure is performed on two types of carbon fibers reinforced composites.The matrix material was the same: epoxyanilinophenol formaldehyde type binder.
As the reinforcement a carbon tape was used with normal (material I) and nitric acid-activated (material II) surfaces.Nitric acid etching promotes an increase in the specific surface (roughness) of the carbon fibers, resulting in the increased shear strength of the materials with treated components by the factor of 1.5-2 [2].Some physical and mechanical characteristics for both materials are given in table 1.
The experimental program on tensile tests was carried out at a constant loading rate up to failure in the following range of rates (kg/mm 2 •sec): 6•10 −1 , 2.5•10 0 , 1.25•10 1 , 1.25•10 2 , (6 − 8)•10 2 , (1 − 1.6)•10 4 .The values of the maximum loading were fixed in the experiments and provided the data for the graphs of the dependence of the tensile strength on the loading rate, given in figure 1.The results on durability testing at the constant loading level are given in figure 2 together with the constructed model durability curves.
Analysis of the graphs in figure 1 suggests strongly nonmonotonic character of the dependence of σ * on the loading rate.It can be assumed that the occurrence of extrema peaks is due to the change of material failure type.
The first region AB is due to a progressive accumulation of defects with strength increasing in synch with the loading rate increase.The maximum point for this region corresponds to the moment when the cracks or defects start to grow.To the right from the maximum point the failure takes place by the propagation of the macro cracks and hence is determined by the value of K Ic for the given material, which falls both with the fiber volume content increase and with the rise of the loading rate, illustrated by decreasing region II in figure 1.The extrema minimum for this decreasing region corresponds to the appearance of cleavage of the weakened link in the composite material resulting in its practically instantaneous failure; from this moment the strength values tend to increase again with the growth of dynamical component of the loading rate.
The developed mathematical model describes the process of damage accumulation in material and allows to establish the dependence of the ultimate breaking stress on time (or the rate) of loading.The failure criterion is formulated as following where M* is hereditary type operator, and σ 0 * is the material constant, corresponding to the limiting value of the cross-sectional stress, which for composites might be taken proportional to the product of the mean strength by the volume fraction for the fibers.With appropriate choice of kernel in (1) the defining equation for the value of the ultimate stress σ * and the time to failure for any loading condition is obtained.This model combined with equation ( 1) is used to describe the rate dependence for σ * , characteristic for the parts A 1 B 1 and A 2 B 2 of the diagram in figure 1 as well as to construct the durability curves.
For the line segments A 1 B 1 and A 2 B 2 the parameters for equation (1) were determined based on the kernel taken in the form: M(t − τ) = m(1 − α)/(t − τ) α .The following  In figure 1, the solid line segments A 1 B 1 and A 2 B 2 represent the estimated results modeled by equation ( 1) for the loading regime σ = const.Subsequently the same parameters were used to construct the durability curves (the solid lines in figure 2).It is clear that the estimated set of parameters appear to establish rather good approximation to the available experimental data shown by dots and circles in figure 2.
A certain disparity in the results obtained for two different materials should be noted.The weak bonding between the fibers and the matrix characteristic for material I tends to facilitate more rapid accumulation of defects and therefore line A 1 B 1 (figure 1) proceeds more steeply than A 2 B 2 : the same applies also to the analysis of the durability curves (figure 2).On the other hand, the occurrence of weaker adhesion, with corresponding delamination, retards crack propagation and therefore point B 1 , representing the point of transition from failure due to accumulation of defects to failure from crack propagation, is situated further to the right than is B 2 .
To analyze the second segment of the rate dependence strength diagram we turn again to equation ( 1), the righthand side of which has the value σ 0 * characterizing the actual ultimate value proportional to the true stress in the cross section [3].
It is known that if a crack starts to grow, the stress σ is related to the intensity coefficient in the following way: σ = K Ic / √ πl, where l is the size of the crack (or defect).If the material defects do not appear substantially large then, for not too high rates of loading the inequality holds: σ 0 * <K Ic / √ πl (figure 3).This condition also determines the initial section A 1 B 1 (A 2 -B 2 ) (figure 1).During loading, the value of the true stress in cross section attains its limiting value earlier than appears possible for crack movement.In the case when the rate of loading is quite large K Ic decreases so much that K Ic / √ πl can become equal to or less than σ 0 * , hence, this case failure is now determined by the value of K Ic .Unfortunately, there was no data available to establish the change of K Ic with the loading rate, therefore the dependences given in figure 3 were calculated for points B 1 and C 1 and B 2 and C 2 (figure 1).The values of the loading rates corresponding to the points of transition from defect accumulation to crack propagation, i.e., points B 1 and B 2 (figure 1) must correspond to the values B 1 and B 2 (figure 3).Thus, for the conditions when σ 0 * ≤ K Ic / √ πl, it is necessary to use equation ( 1), defining the damage accumulation process in order to determine the loading rate dependences of the strength.
Subsequently, for the following stage the equation should be written in the form The diagrams in figure 3 are illustrating the dependences of K Ic / √ πl on the loading rate for two studied materials.Values of K Ic , shown in table 1, were determined by the method discussed in [4] at rates of loading of 2.5 kgf/mm 2 •sec; in figure 3 they are shown by arrows.By making rather crude assumption of a linear dependence of K Ic / √ πl on σ it is possible to extend it to the value

Fig. 1 .
Fig. 1.Tensile strength versus loading rate for two tested materials (I and II).Solid lines -modelled results, dots -experimental data.
* The values were determined under loading with a constant machine grip rate at v = 2 mm/min ( σ ∼ 2.5 kgf/mm 2 •sec).