American Journal of Alied Sciences (9): 19-195, 5 ISSN 1546-939 5 Science Publications Pressure-sensitivity Effects on Toughness Measurements of Comact Tension Secimens for Strain-hardening Solids Abdulhamid A. Al-Abduljabbar Deartment of Mechanical Technology, Riyadh College of Technology, Riyadh, Saudi Arabia Abstract: The ath-indeendent J-integral is used as a fracture rediction criterion for loading beyond the elastic limit as Linear Elastic Fracture Mechanics (LEFM) cannot be alied. While J was originally defined from an energy ersective, it was demonstrated that it could be inferred from load-dislacement diagrams. Therefore, the Comact Tension (CT) secimen testing is used to comute J for materials. In this work, the η-factor, an imortant arameter in the comutation of the J-integral, is investigated for comact tension secimen for materials with ressure sensitive yielding. This is achieved by using a lower bound aroach to derive the aroriate exression from η from the test geometry and material roerties. The secimen is considered at fully lastic loading where it is in the state of collase. The effect of ressure sensitivity is accounted for by using a Drucker-Prager yield criterion for solid materials. Since CT testing is usually conducted on metallic materials, strain-hardening behavior of the material is incororated in this analysis. This is done by assuming a simle linear hardening curve of the material. Numerical results comuted for different cases show that as the material strain hardening increases. Key words: J-Integral, lastic yielding, toughness, comact tension testing, ressure-sensitive yielding INTRODUCTION The Linear Elastic Fracture Mechanics (LEFM) theory is useful for estimation of fracture behavior of materials in the elastic range. However, as large inelastic strain behavior is encountered, the LEFM is not adequate to describe the toughness and fracture behavior. Attemts to model such behavior is going as early as Rice [1,] who introduced a ath-indeendent line integral around the ti of the fracture notch to describe the elastic-lastic fracture behavior. Later on, Begley and Landes [3,4] demonstrated that it is ossible to estimate the J integral from the load-dislacement curves of various test secimen geometries. Bucci et al. [5] and Rice et al. [6] roosed ossible estimation methods of J for comact tension secimens among other secimen geometries. Later on, Merkle and Corten [7] imroved the estimation of J by considering the effects of the bending moment resulting from the alied load to the remaining ligament of the comact tension secimen. They used a lower-bound aroach to derive the η factor for the J integral estimation. The η factor aroach has been used to estimate the J integral from the area under the load-dislacement (P- ) curve of a test secimen or structure. The Von-Mises yield criterion assumes an ideal material yield that only deends on the equivalent yield stress. While such assumtion is accetable for many classes of materials esecially metals, it is not the case for other classes of materials such as olymers, hasetransformation ceramics, cast irons and even some classes of steels. In such materials, yielding behavior is affected by the state of hydrostatic stress and the yield enveloe on the equivalent stress versus the hydrostatic stress lane (σ e σ m ) is not horizontal, but rather having a negative sloe. Such henomenon, which is also known as ressure-sensitivity, is also demonstrated by the variety between the values of yield stress in tension and comression. The effect of ressure sensitivity on yield can be accounted for using the Drucker-Prager yield criterion, where, a generalized equivalent yield stress is exressed as a function of the effective yield stress and the hydrostatic stress using a ressure sensitivity factor. The ressure sensitivity factor is a material mechanical roerty deendent on the difference between yield stresses in tension and comression. The reorted values of ressure sensitivity show a wide range of different engineering materials. For examle, for olymers it was reorted by Sternstein and Ongchin [8], Sitzig and Richmond [9], and Kinloch and Young [1], that the ressure sensitivity factor, µ, has a range of.1 ~.5. In cast irons it has a value around., according to Dong et al. [11]. It is also resent, to a lesser degree, in some steels as shown by Richmond and Sitzig [1] with µ has quite a low value Corresondence Address: Abdulhamid A. Al-Abduljabbar, P O Box 19 Riyadh 11455, Saudi Arabia, Tel/Fax +966-1- 493994, 19
Am. J. Alied Sci., (9):19-195, 5 of.64. Phase transformation in some ceramic materials is highly deendent on ressure sensitivity as reorted by exerimental investigations of Yu and Shetty [13] and Chen [14] where µ has a large value u to.93 for zirconia-containing hase-transformation ceramics. Al-Abduljabbar and Pan [15] incororated the material ressure-sensitivity, into the calculation of η to roduce a better estimate of the toughness roerty, using the same aroach and assumtion of rigid erfectly-lastic material behavior used in revious works. Since most metals exhibit strain hardening in various degrees, incororating the effect of strain hardening on the estimate of fracture toughness as described by J will indeed result in an imrovement of this measure of material fracture behavior. Al- Abduljabbar [16] considered strain hardening effect on J for normal solids without ressure sensitivity. In this work, a rocedure similar to those adoted in revious works in order to roduce exressions for the η factor of the J integral for a material with ressure sensitivity and strain hardening. The ressure sensitivity is modeled using a Drucker-Prager yield criterion for solid materials, while strain hardening is deicted by assuming a simle linear hardening model for the material. Fig. 1: Stress-strain for different cases: (a) an idealized erfectly-lastic material, (b) an idealized linearly hardening material and (c) a tyical ductile metal 1 σ c = σ (3) 1 µ where, µ = µ / 3. To describe the material stress strain relation, we adot the Ludwik's exression [19] : PLASTIC LIMIT ANALYSIS The Drucker-Prager yield criterion [17,18] is a generalization of the Coulomb rule is solid mechanics where the shear stress required for simle sli is linearly deendent uon the normal ressure on the sli surface. In the D-P criterion, a generalized yield function is roosed as a combination of the effective yield stress (σ e ) and the mean stress (σ m ) by using the ressure sensitivity factor (µ) as follows: f ( σ ɶ ) = σ + 3µσ = σ (1) e m In equation (1), σ e= (S ij S ij ) 1/, where S ij is the deviatoric stress comonents defined by S ij =σ ij - σ m δ ij. Here, σ ij reresent stress comonents and δ ij is the Kronecker delta. The summation convention is adoted for reeated indices. The mean stress (σ m ) is defined by the relation σ m =σ kk /3. The stress quantity σ is the generalized effective tensile stress at the onset of yield. Considering the uniaxial tensile and comressive loading conditions and alying the yield criterion in equation (1) values for the effective tensile stress (σ t ) and the effective comressive stress (σ c ) can be obtained as follows: 1 σ t = σ () 1 + µ n H σ = σ + ε (4) where, σ is the initial yield stress, H is the hardening constant, ε is the strain and n is the hardening exonent. The exression is further simlified by assuming linear hardening case with n =1. This relation can be used to describe cold worked metals such as high carbon and alloy steels where there is a high amount of restraining [19, ]. Shown in Fig. 1 are three curves describing stressstrain behavior. The first one (a) is in an idealized rigid erfectly-lastic material, where the material exhibits no elastic strain and the stress remains at the constant yield value of σ throughout the deformation and lastic straining. Such behavior is the one assumed in revious works for estimation of the J integral [15]. The second curve is for a rigid linear hardening material as described by equation (4) with n =1. The third curve is for a tyical ductile material. For metals, the elastic strain is usually very small when comared with the lastic strain, esecially at high loading levels, as is considered here. This fact ermits the assumtion of neglecting the contribution of the elastic strain in this analysis. Next, we consider the comact tension (CT) secimen of a rigid strain-hardening material as shown in Fig. [1]. Due to the symmetry of the secimen geometry and loading condition along the horizontal 191
Am. J. Alied Sci., (9):19-195, 5 axis, only the uer half of the secimen is analyzed here. The total width of the secimen is W, the crack length is a, the remaining ligament width is b and the thickness of the secimen (through the aer) is unity. The fully lastic load value alied as shown is P. Also shown in the figure, is the stress distribution in the remaining ligament, where the ortion near the ti is under tensile stress resulting from the tension and bending moment of the load and the other ortion is under comressive loading. At the oint of stress reversal, both comressive and tensile ortions start with initial yield stresses with stress levels increasing linearly u to their maximum values at the edges. On the comressive side, the starting stress level is σ c and the stress at the outer edge of the secimen is σ A and on the tensile side, the starting stress level is σ t and the stress at the inner edge is σ B. The dimensionless arameter, α, is introduced as an indicator of the deviation of the neutral axis from the center of the remaining ligament width b, where the tensile side length is (1+ α)c, because it has lower stress levels and the comressive side length is (1-α)c. Within the factor α lies the effect of the ressure sensitivity of the material resulting from the variation of yield stresses. Another dimensionless arameter, γ, is introduced to account for the hardening effect in the material. Assuming that the sloe of the linear hardening curve on the stress diagram is roortional to γ, the values of the two end stresses are determined as: σ = [1 + γ] σ (5) A B c σ = [1 + γ] σ (6) t For the lastic limit load, we consider the axial force and the bending moment equilibrium due to the in-lane force and stresses. The force equilibrium in the vertical direction requires that: σ + σ σ + σ + α + α = (7) A c B t P c(1 ) c(1 ) Hence, equation (7) is used with equations (5) and (6) to get the lastic limit load P as follows: ( + λ)( α µ ) P = cσ 1 µ (8) Further, the moment equilibrium around stress reversal oint requires that the internal resisting moment balances the moment generated by the alied load M. The internal resisting moment is comosed of two ortions: M t due to the tensile stress and M c due to the comressive stress. This roduces: Mt + Mc MP = (9) The values of these moment comonents are determined, with the hel of equations (-3, 5-6), and Fig., as follows: M M 1 γ (1 + α) t = + c σ 3 1+ µ 1 γ (1 α) c = + c σ 3 1 µ (1) (11) MP = P [ α + (1 + α )c] (1) Substituting values for moments from equations (1-1) into equation (9) and also using equation (8) for the limit load; a quadratic exression for the arameter α is obtained in the form Aα + Bα + C = (13) Where: A = ( λ 1), (14) a B = + 1 µ λ + µ c And: Fig. : Geometry of the uer half of the comact tension secimen and the stress diagram for the a C = + 1 λµ 1 remaining ligament. Unit thickness is assumed c 19 (15) (16)
Am. J. Alied Sci., (9):19-195, 5 In equations (14-16), λ is a dimensionless arameter which is defined by: a 1 + µ α( λ ) α ( λ 1) λ + 1 = c α µ () + γ λ = 3 3 + γ (17) η: Finally, from equations (-), we can determine From the above relations, α can be solved for easily using the standard quadratic equation solution. Recalling the work for ressure sensitive rigid erfectly-lastic materials, resented by Al- Abduljabbar and Pan [15], we can see that once we set the strain hardening coefficient to zero, the constants A, B and C of equation (13) above revert to the result given in equation no. 14 in that work, thereby confirming the validity of relations develoed here. + µ ( λ ) µ λ + (1 µ ) λα η = 1 + µ ( λ ) ( λ 1)( µ α α ) (3) If the material hardening is removed by setting the arameter γ to zero, so that λ =, then the equation (3) reduces to (1 + α)(1 µ ) η = 1 µ α + α (4) THE J INTEGRAL ANALYSIS The exression for the J integral for materials with negligible elastic strain can be adoted from the work of Merkle and Corten [7] as follows: η η P = + b b J Pd dp * (18) where, η and η * are arameters deendent on the secimen geometry and loading conditions. The first and second integrals of the equation (18) are the lastic work and the comlementary lastic work done on the secimen, resectively. The η-factor is defined by: 1 dp b P db η = (19) As will be shown later, η is deendent on the geometry, loading conditions and material behavior. The η-factor can be determined by first differentiating equation (8) with resect to b, noting that dc/db = 1/. Then, using equations (8) and (19): c α η = 1+ α µ c () Differentiating equation (13) with resect to c and using the relation a = W-c; we get an exression for the last term in the equation (): This exression for η is exactly the same one resented for erfectly lastic material [15]. This result for the case without strain hardening serves as a check for the solution obtained from η in the analysis above. DISCUSSION The relations that were develoed for η in the receding section will be used to assess the effects of the strain hardening and ressure sensitivity on the fracture behavior of metals based on the comact tension secimen arameters. Since those relations are based on the aforementioned assumtions in a revious section, they serve as indicative measures of the material behavior, not direct evaluators of the real roerty. The relations develoed for η earlier were resented in terms of a/c. It is aroriate to exress them here in terms of the normalized crack length a/w. This is achieved by recalling from the geometry of the secimen in Fig., that: a a = W (5) c 1 a W We first consider the behavior of the η factor as a function of the normalized crack length for a erfectly lastic material and two cases of strain hardening materials (γ =,.5 and 1., resectively) where all materials having no ressure sensitivity (µ = ). The results for this case are shown in Fig. 3. The figure a λ + c α shows that at low values of crack length the change in c = (1) the hardness factor roduces a change in the value of α µ c a α( λ 1) + λ + 1 µ ( λ ) the η factor in the same direction. An increase in the c value of γ gives an increase in η, resulting in a higher value of the J integral. The case of no hardening Equation (13) is again used to obtain an exression corresonds to the results obtained by revious for λ (a/c+1): works [16, 17]. 193
Am. J. Alied Sci., (9):19-195, 5 Fig. 3: The η factor as a function of the normalized crack length (a/w) for different values of the strain hardening coefficient γ and no ressure sensitivity (µ = ) Fig. 5: The η factor as a function of the normalized crack length (a/w) for different values of the strain hardening coefficient γ and a ressure sensitivity factor µ =.3 Fig. 4: The η factor as a function of the normalized crack length (a/w) for different values of the strain hardening coefficient γ and a ressure sensitivity factor µ =.1 Next, consider the change of η with resect to the normalized crack length for different cases of strain hardening coefficient, γ, for a low value of ressure sensitivity µ =.1 as deicted in Fig. 4. The same observation that at low values of crack length, the change in the hardness factor roduces a change in the value of η in the same direction with the increase in γ resulting in an increase in η, although the effect is milder. It can be seen from the figure that the effect of hardening decreases as the crack extension becomes large and converges to the value of because for deely cracked secimens, the factor aroaches regardless of the material constitutive behavior [6]. Figure 5 is a lot of η as a function of the crack length for a higher value of the ressure sensitivity; µ =.3. The figure shows the change in η as the crack length increases for different cases of strain hardening. By comarison with Fig. 3, we can see that the effect of ressure sensitivity is to reduce the value of η while the main features of the deendence of the strain hardening coefficient are the same. It is noted that the change in η due to the change in hardening generally gets milder as the strain hardening gets higher and ressure sensitivity tends to reduce the counter balance effects roduced by hardening. CONCLUSION In this work, an imroved estimate of the factor involved in the evaluation of the integral was obtained. As a fracture toughness criterion for the comact tension secimen, J estimation for strain-hardening ressure-sensitive materials were ossible by using lastic limit load analysis and assigning a linear hardening behavior of the secimen material; which rovided the basis for derivation of analytical exressions for the relevant arameters. The numerical results for different cases of ressure sensitivity and strain hardening rovide quantification of exected increase in fracture toughness of the material due to hardening when comared with erfectly lastic behavior. The same result is obtained for the decrease due to material ressure sensitivity. The assumtions introduced in the comutation of factor limit the validity of the results, requiring a more rigorous model of hardening and also a comarison with exerimental data. 194
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