Fracture Assessment of Notched Severely Plastic Deformed 7075 Al Alloy under Mode I Loading

Equal channel angular extrusion is an extrusion to refine the microstructure of metals and alloys. This paper presents for the first time the experimental results on effects of the equal channel angular pressing (ECAP) on tensile mechanical properties and ductile fracture strength of 7075 Al alloy. The 7075 Al was subjected to equal channel angular pressing after solution treatment, and significant grain refinement and enhancement in tensile strength and hardness are obtained after equal channel angular pressing. Then, the ductile static strength of ECAP 7075 Al alloy weakened by U-notch is investigated under mode I loading. The criterion based on the averaged value of the strain energy density over a control volume at the notch edge in combination with the equivalent material concept is applied to assess the static strength of specimens. A sound agreement between experimental data and the results obtained from strain energy density criterion is found.


INTRODUCTION
Aluminum alloys are widely used in engineering applications. There is, nevertheless, much significance to improve mechanical properties of the alloys using engineering processes. Although mechanical properties of all crystalline materials are determined by several factors, the average grain size plays a very significant, and often a dominant role. In order to decrease the grain size of a material using metal-working procedure, it is necessary to impose an exceptionally high strain in order to introduce a high density of dislocations subsequently rearranged to form an array of grain boundaries. In practice, because of the limitations of conventional metal-working procedures on overall imposed strains, attentions have been devoted to techniques based on application of severe plastic deformation [1].
Equal channel angular pressing (ECAP) as a severe plastic deformation process, produces significant plastic strain into materials without reducing the cross sectional area. The sample, in the form of a rod or bar, is placed so that the sample can be pressed through the die using a plunger. The nature of the imposed deformation is simple shear which occurs as the sample passes through the die. As shown schematically in Fig. 1, the theoretical shear zone is shown between two elements within the sample numbered 1 and 2, and these elements are transposed by shear as depicted in the lower part of the diagram [1]. Despite the introduction of a very intense strain as the sample passes through the shear plane, the sample ultimately emerges from the die without experiencing any change in the cross-sectional dimensions. ECAP process also introduces nonequilibrium condition in the microstructure of the alloys such as high dislocation density and large number of low angle grain boundaries [2]. Although, equal channel angular pressing as a method of severe plastic deformation improves strength with decreasing grain size and increasing dislocation density, but the elongation to failure associated with the movement of dislocations decreased after the first pass.
Al alloys are widely used in various industrial and scientific engineering as a structural and/or functional material. In most of industrial products, some notches of U-or V-shapes are viewed as desirable entities under different loadings. The notches are weak points that may generate cracks and lead to fracture. Several criteria are present in the literature for fracture assessment of engineering notched components.
The failure criterion proposed by Novozhilov [3] and developed by Seweryn [4] called as theory of critical distances suggests that failure occurs when the average normal stress along the characteristic length scale denoted by d 0 equals a material dependent stress at failure without the presence of a notch. The successful application of theory of critical distances on the notched components is widely investigated in [5,6]. Leguillon [7,8] proposed a criterion for the failure initiation at a sharp V-notch based on a combination of the Griffith energy criterion for a crack, and the strength criterion for a straight edge. Neuber [9] first suggested the idea of linking the stress averaging to the fictitious notch rounding approach and other researchers investigated the influence of plane stress and plane strain conditions on the application of the fictitious notch rounding approach and in particular on the calculation of the multi-axiality factor s. Marsavina et al. [10] investigated the dynamic and static fracture toughness of polyurethane rigid foams and in another work [11], four fracture criteria (maximum circumferential tensile stress, minimum strain energy density, maximum energy release rate, equivalent stress intensity factor) were applied to evaluate the mixed mode fracture of polyurethane foams using asymmetric semicircular specimens. In particular, the authors showed that the equivalent stress intensity factor criterion predicted well the mixed mode fracture more precisely.
The other worth mentioning approach is based on the cohesive zone model. The major advantages of cohesive zone model over the conventional methods in fracture mechanics like those including linear elastic fracture mechanics, crack tip open displacement, etc. is that it is able to adequately predict the behavior of uncracked structures, including those with blunt notches. Moreover, the size of the non-linear zone has not to be negligible in comparison with other dimensions of the cracked geometry in cohesive zone model. This approach was first proposed for concrete and later successfully extended to brittle or quasi-brittle failure of a large bulk of materials and in particular poly-methyl-methacrylate (PMMA) specimens tested at room and low temperature [12]. In those works, both sharp and blunt U-and V-notches were considered.
A recent approach, based on the strain energy density (SED), was proposed and successfully applied for the fracture assessment of notched components. The strain energy density approach is based on the evaluation of the averaged strain energy density over a control volume [13,14]. The criterion was applied to assess the fracture behaviour of different materials. Radaj [15,16] made a review on the local strain energy density concept and its relation to the J-integral and peak stress method. Recently, the strain energy density criterion has been developed to fracture assessment of notched functionally graded materials numerically and experimentally [17][18][19][20][21][22][23][24][25][26].
Although the strain energy density criterion can be used to predict the brittle or quasi-brittle fracture of notched components, any applied materials must be typically engineered as ductile structures like those of metallic materials, including steel, aluminum and titanium alloys, as well as polymers or polymer-based composites, most of which may contain notches of various shapes [27]. Therefore, we need elastic-plastic analysis for fracture assessment of ductile materials.
Two of the first attempts to utilize elastic analysis instead of elastic-plastic one has been made by Glinka and Molski [28] and Glinka [29]. They made use of the strain energy density approach to determine the elastic-plastic stress distribution around some notched components. In their works, first, the elastic stress concentration factor has been used to formulate the elastic stresses at the notch tip and then, the strain energy density at the notch tip has been equated for elastic and elastic-plastic components with the aim to determine the stress distribution in the notched component made of ductile material. With the aim of applying the strain energy density to nonlinear elastic conditions, but keeping its simple linear elastic formulation, the approache will be combined with the equivalent material concept. Based on the equivalent material concept, the strain energy density criterion of the area under the stress-strain curve in uniaxial tension is assumed to be the same for ductile and virtual brittle materials with similar moduli of elasticity and fracture toughness [30,31].
In this study, the alloy is subjected to equal channel angular pressing after solution treatment. Experimental results on effects of equal channel angular pressing on microstructure and mechanical properties (yield stress, ultimate stress and hardness) of finegrained 7075 Al alloy are also presented. Also the well-known strain energy density criterion was employed in combination with the equivalent material concept to predict the fracture behavior of U-notched ductile ECAP 7075Al components without the need for performing an elastic-plastic analysis. Moreover, a new set of experimental data was provided by U-notched specimens made of ECAP 7075Al with different values of notch depth and notch radius which can be useful to engineers engaged with the static strength analysis of ECAP 7075Al components. Our experimental data showed good agreement with those obtained via strain energy density approach.

Material
Initial rods of 7075 Al alloy were immediately pressed through an ECAP die after solution treatment at 490°C for 4 hours and water quenching. The ECAP die was consisted of channels with equal cross sections having intersection angle of Φ = 90° and outer angle of Ψ = 20° so as to introduce the effective strain of approximately 1 on a single pass. The ECAP process was carried out through only one pass at room temperature with pressing speed of 1 mm/s. The ECAP samples were then subjected to natural aging. In order to study effect of equal channel angular pressing on strength and ductility, hardness and tensile tests were also carried out. Tensile tests were performed at room temperature according to ASTM E8, using a tensile-compression testing machine operating at a constant rate of crosshead displacement with an initial strain rate of 4  10 −3 1/s. Microstructural examination and fracture surface characterization of the tensile test segmented specimens were performed by optical microscope and scanning electron microscope respectively. In longitudinal direction, shear lines can be detected. Comparing the micrographs of transverse directi-  on, remarkable grain size reduction is obvious as a consequence of ECAP process. Figure 3a shows H V hardness values of the samples at every stage of the process, while Fig. 3b shows H V variation of ECAP samples versus natural aging time. Significant enhancement in hardness up to 165 H V resulted immediately after a single pass of equal channel angular pressing. The increase in hardness for the ECAP specimen can be attributed to strain hardening effects including increase in dislocation density and grain refinement and also dynamic aging (strain induced aging) which are in agreement with [32]. Furthermore, since the alloy was in super saturated solidsolution state before equal channel angular pressing, the hardness increased to about 195 H V after aging at room temperature for three months, due to age hardening effect. Comprehensive description on influences of equal channel angular pressing and aging on mechanical properties of 7075 Al alloy has been presented in previous paper [33].

Hardness and Tensile Strength
Stress-strain behavior of the alloy before and after ECAP and natural aged for three months alloy are shown in Fig. 4. Remarkable increase in strength was obtained after a single ECAP pass (increase by 100% in yield stress from 225 to 450 MPa). Ultimate stresses of the post-ECAP and the as-received specimens are 340 and 530 MPa, respectively. Along with the improvement of strength, the ductility decreased after equal channel angular pressing from 15% to 12%. Decrease of the ductility is attributed to relatively small strain hardening after yielding in the ECAP processed material [34].
Comparing the results with strength of pre-ECAP annealed specimens, it can be indicated that the strength of the pre-ECAP solutionised specimen after only one pass of equal channel angular pressing is significantly greater than the strength of pre-ECAP annealed specimen after 4 passes, where the ultimate tensile stress and the ductility have reported to be 425 MPa and 7.9%, respectively [34]. Figure 5 shows the detail of tensile test specimens. Table 1 represents mechanical properties of 7075 Al alloy under different conditions.

Three Point Bending Tests
In this study, the specimens drawn from the ingots were characterized by 40 mm in length, 10 mm in width and 5 mm in the thickness direction as shown in Fig. 6. The effects of the notch tip radius ρ and notch depth a on the fractures of the specimens under the mode I loading was investigated. Two values of the notch radius ( = 0.2 and 0.5 mm) and two values of the notch depth (a = 1 and 2 mm) were considered for the test specimens.
The tests were carried out by a ZWICK 1494 testing device under displacement control with a constant displacement rate of 0.05 mm/min. A U-notched specimen during the three point bending loading and a broken specimen within the machine are shown in Fig. 7. The values of the critical load of each experimental test (F exp1 and F exp2 ) are presented in Table 2.

The Averaged Strain Energy Density Criterion
The averaged strain energy density criterion was employed to predict the fracture loads of U-notched specimens under mode I. Based on this criterion, brittle or quasi-brittle fracture occurs when the averaged value of strain energy density over a well-defined control volume reaches a critical value W c . For brittle materials, W c can be evaluated as follow [14]: For a U-notch under mode I loading, the control volume assumes the crescent shape which is centered with respect to the notch bisector line. R c is the critical length which is measured along the notch bisector line as shown in Fig. 8.
The critical length R c can be evaluated as follow under plane strain conditions [13,14]: where K Ic is the fracture toughness,  ut is the ultimate tensile stress and  is the Poisson's ratio. The outer radius of the control volume is equal to R c + /2 and  is the notch tip radius. In this paper, linear elastic analyses were performed to evaluate fracture loads. Two procedures were employed for prediction. First, by neglecting the plastic behavior of the material, fracture loads were predicted by simply using the mentioned criterion assuming the material is ideally brittle. Second, the idea of equivalent material concept [30,31] was used to define virtual brittle materials. Considering the equi-    F exp2 ) and theoretical ones. F th1 is obtained assuming the material is ideally brittle, F th2 is obtained by using equivalent material concept valent brittle materials instead of ductile ones, fracture loads were predicted by the averaged strain energy density criterion.

Equivalent Brittle Materials
In this paper, the equivalent material concept as a novel concept was employed to equate the ductile and virtual brittle materials from the strain energy density viewpoint. The equivalent material concept provides an imaginary consideration of a virtual brittle instead of a ductile material to investigate a linear elastic rather than an elastic-plastic behavior in fractures. The simple criteria for brittle fractures could be ultimately utilized in the study of the fracture phenomenon occurring to ductile materials.
Based on the equivalent material concept, the strain energy density values of the existing virtual brittle and ductile materials with similar moduli of elasticity could be assumed to be the same. Strain energy density actually represents the strain energy absorbed by the unit volume of a material. The following equation can be written for a ductile material un- (3) where  and  indicate the plastic stress and strain, respectively. The parameters of K and n demonstrate the strain-hardening coefficient and exponent, which depend on the material properties, respectively. Figure 9 depicts a schematic representation of a tensile stress-strain curve for a typical ductile material, in which E, Y, u, and  f denote the elastic modulus, yield strength, ultimate tensile strength, and strain at rupture, respectively. The total strain energy density (SED) can be expressed in the following general elastic-plastic form: As defined in the equivalent material concept, the equivalent virtual brittle material has the same values of E and K Ic respectively representing the elastic modulus and plane-strain fracture toughness, but an undetermined value of the ultimate tensile strength.  In Fig. 10, a typical uniaxial stress-strain curve is schematically shown for the virtual brittle material, in which the parameters of  f * and  f * stand for the strain (final fracture) occurring with the crack initiation due to the brittleness and the ultimate tensile strength, respectively. Upon the crack initiation, the strain energy density for this material can be calculated as follows: Assuming the equality of the SED values for both the virtual brittle and real ductile materials according to the equivalent material concept, we have: The parameter of  f * presented in Eq. (7) can be used together with the material fracture toughness (K Ic or K I ) as the two necessary inputs of different brittle fracture criteria for predicting crack initiation from notches in ductile structures subjected to tension (pure mode I loading condition) [30,31,35]. Figure 11 shows the stress-strain curves of ductile and the equivalent brittle materials. The tensile strength of as received and ECAP aluminum were determined to be 989 and 1283 MPa, respectively.

Finite Element Analysis
In order to obtain the averaged value of strain energy density over the control volume, some finite element analyses were carried out using ABAQUS software version 6.11 under plane strain conditions and linear elastic hypothesis with eight-node elements. Figure 12 shows a sample maximal principal stress and SED contour lines. The critical fracture load could be evaluated by determining the SED mean value over the control volume using the following expression [28]: where F cr indicate critical fracture loads, W cr is the critical strain energy density, F ap is the applied load that in computer simulation, is voluntary less than the F ut (ultimate force) and in this work was considered 405 N, and ap W is the strain energy density averaged over the control volume relevant to, ap W is equal W ap /V, W ap (strain energy density over the control volume) and V (control volume) were calculated with finite element method in Abaqus software.
In Table 2, the theoretical values of fracture loads are summarized. As can be seen from the table, for such a ductile material, the averaged strain energy density is able to predict the fracture loads with a moderate accuracy by neglecting the plastic behavior of the material. However, such an assumption causes losing well prediction of the trend of fracture loads with respect to the notch tip radius. In contrast, using the equivalent material concept results in a good accuracy of the prediction of fracture loads and the trends. Moreover, Table 2 shows that these is a significant increase in critical fracture load for the ECAP alloy comparing to the as-received alloy.

CONCLUSION
In the present work, the strain energy density averaged over a well-defined control volume ahead at the notch edge in combination with the equivalent material concept was utilized to obtain the critical fracture loads of the U-notched specimens made of 7075 ECAP Al under mode I loading condition.
The main results of this investigation are summarized as follows: Processing a solutionised aluminum alloy 7075 specimen by equal channel angular pressing, even by a single pass, significantly improves the hardness. This finding may have important practical significance because of its advantage in industrial applications with considerable saving in time.
The yield strength of solutionised specimen increased from 240 to 451 MPa after equal channel angular pressing and two months natural aging. The ductility decreased significantly after equal channel angular pressing and two months natural aging.
The strain energy density criterion in combination with the equivalent material concept model provided a suitable approach to the prediction of the ductile fracture behavior of ECAP Al.
The limited average deviation (7.8%) between the theoretical and experimental values based on the critical fracture loads indicated the model accuracy.
The critical fracture load for the ECAP alloy comparing to the as-received alloy increases about 30%.