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On the behavior of crack surface ligaments
Nuclear Engineering and Design 184 (1998) 145 – 153
On the behavior of crack surface ligaments P. Nilsson, P. Sta˚hle *, K.G. Sundin Di6ision of Solid Mechanics, Lulea˚ Uni6ersity of Technology, SE-971 87 Lulea˚, Sweden Received 13 March 1996; received in revised form 31 July 1997; accepted 20 January 1998
Abstract Small ligaments connecting the fracture surfaces just behind a moving crack front are assumed to exist under certain conditions. The ligaments are rapidly torn as the crack advances. Inelastic straining of such ligaments influences the energy balance in the fracture process. The rapid tearing of a single ligament is studied both numerically and experimentally. An elastic visco-plastic material model is adopted for finite-element calculations. The results show that relatively large amounts of energy are dissipated during the tearing process. Further, the energy needed to tear a ligament increases rapidly with increasing tearing rate. The computed behavior is partly verified in a few preliminary experiments. The implications for slow stable crack tip speeds during dynamic fracture are discussed. © 1998 Elsevier Science S.A. All rights reserved.
1. Introduction Often during fast crack growth in structural steels, the influence of rate-dependent material behavior during plastic straining cannot be ignored (Brickstad, 1983; Lo, 1983). Increasing stress with increasing strain rate has been observed. When this effect is sufficiently strong, the plastic-strains have a minor effect on the stress distribution. Such a case has been treated by Freund and Hutchinson (1985), who considered the energy rate balance necessary for continuous steady-state crack growth. The elastic energy release rate G for dynamic crack propagation was assumed to be balanced by the sum of the work rate due to inelastic deformation in the plastic
* Corresponding author. Tel.: + 46 920 72188; fax: + 46 920 91047; e-mail: [email protected]
zone, Gvp, and the rate of the work dissipated at the crack tip. It is assumed here that tearing of ligaments accounts for the essential part of the work rate in the process zone. Therefore, it is suggested that G is divided into three parts, namely, Glig associated with ligament tearing, Gvp associated with deformation in the plastic zone, and Gcl associated with cleavage. The relation is indicated schematically in Fig. 1. Freund and Hutchinson (1985) found that stable crack growth is possible only when the crack speed is greater than 0.55cr, where cr is the Rayleigh wave speed. Below that, in their analysis the speed will be unstable and the crack tip will immediately accelerate to a speed higher than 0.55cr or come to an arrest. In structural steels, both extended crack propagation and crack arrest often occur at speeds as low as from 0.05cr to 0.1cr. Observations are,
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P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
however, qualitatively in accordance with the results predicted by Freund and Hutchinson (1985). The higher arrest speed predicted by Freund and Hutchinson (1985) is obtained on the assumption that the crack tip energy release rate is independent of the strain rate. It is believed that the rate sensitivity of the ligament tearing process may partly explain the quantitative discrepancy between observations and predictions. The crack tip is assumed to propagate through a coalescence of transgranular micro cracks. Due to mismatches of geometry and mechanical properties at grain boundaries, unbroken parts will remain, connecting the upper and lower crack surfaces. During increasing separation of the crack surfaces, the unbroken parts will become ligaments, bridging the gap between the separating crack surfaces. Studies of cleavage fracture surfaces show that parts of the surfaces consist of plastically formed ridges, presumably traces of plastically torn ligaments. These ligaments are assumed to be formed as previously described. A large part of the energy consumed in the fracture process region is suggested to be due to the disruption of such ligaments. In this paper a ligament model is studied as it deforms plastically to a state where very little remains of its initial cross-sectional area. The energy consumption is calculated as a function of the ligament extension rate. At crack tip speeds of practical interest, the energy consumption rate in
the ligaments is found to be comparable to total energy release rates in structural steels. In the discussion part of this paper, implications for stable crack growth are considered. An attempt is made to experimentally verify the assumed ligament model. Two high loading rate tensile tests were performed on small test specimens. The preliminary results lend some confidence to the numerical results.
2. Model of a single ligament A large body described in the coordinate system x= x1, y= x2 and z= x3 is considered. The body consists of a lower y0 0 and an upper half-plane y\ 0 separated along the entire plane y= 0 except in the region x 0 a, y= 0. The surfaces at x \a, y= 0 are assumed to be traction free. The extent of the body in the z-direction is assumed to be large, thus creating plane strain conditions. A large strain theory is adopted. Thus, the rate of Green’s strain tensor, o; ij, is used. The employed Kirchhoff stress tensor, sij, is the work rate conjugate to o; ij. An elastic visco-plastic material model is used to describe the mechanical behavior of the body (Malvern, 1951). The strain rate is decomposed into an elastic part, o; eij, and a visco-plastic part, o; pij : o; ij = o; eij + o; pij
(1)
The elastic strain rates are given by Hooke’s law using Young’s modulus E and Poisson’s ratio n= 0.3. Let se =
3 sij sij 2
1/2
(2)
define the von Mises effective stress, where sij is the stress deviator. The visco-plastic strain rates are given by the following constitutive relationship
o; pij = g; 0
Fig. 1. Energy balance during fast crack growth.
se −1 s0
n
sij se
(3)
where g; 0 is the strain rate sensitivity, s0 is the yield stress and n is the strain rate exponent (Perzyna, 1963). The ratio between Young’s modulus and the yield stress E/s0 = 502 is used, which
P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
147
represents a common structural steel. The strain rate exponent, n =5 is chosen as proposed by Malvern (1951). The strain rate sensitivity, g; 0, does not have to be quantified for the analysis. The dissipation of energy during plastic deformation will lead to an increased temperature. However, here the mechanical behavior is assumed not to be affected by the increased temperature. The temperature rate due to plastic deformation is calculated as follows T: =
sijo; pij cvr0
(4)
where r0 is the mass density and cv is the heat capacity. It is assumed that the loading of the specimen is fast and, thus, produces predominantly adiabatic conditions. Hence, the effects of heat conduction are neglected. The structure is subjected to a remotely applied tension parallel with the y-axis. Symmetry is assumed across x =0 and across y =0. The study is therefore limited to the part x E 0 and yE 0. A polar coordinate system, r and u, is attached to x =y=0, as Fig. 2(a) shows. An elastic solution is employed for a boundary layer analysis of a part bounded by r0R. A prescribed displacement rate is used to control the load. The following boundary conditions are thus applied: u =0
and
Fig. 2. (a) Geometry and (b) finite-element mesh.
6 =6; 0t ln(jR/a),
at r =R, 00 u 0p/2
(5)
where j = 2 exp{ − 1/[2(1 −n)]} is obtained by assuming that cos2(u)/log(R/a) :0 (cf. Appendix A). Here, t is the time. Hence, the displacement becomes independent of u, which simplifies the numerical analysis. The effect of this is assumed to be small and decreasing as the width of the ligament decreases during the loading process. Symmetry across x =0 and across y =0 requires txy =0
and
u=0
at x=0, 00 y B R
(6)
and
6 =0
at 0B x 0a, y = 0
(7)
and txy =0
Finally, traction-free surfaces require txy =sy = 0
at a Bx B R, y = 0
(8)
The ratio R/a is chosen to be 40, which is assumed to be sufficiently large to meet the requirements for a boundary layer solution.
3. Numerical model The commercial finite-element code ABAQUS was used for the numerical calculations. An updated Lagrangian method for large deformation elastic visco-plastic materials was used (ABAQUS, 1989). Full integration of element stiffnesses was used. To obtain a reasonably rapid convergence of the calculations, two modifications of the proposed model were made. Firstly, the suggested sharp notch tips formed at x = a, y= 0 were
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P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
each modeled as a notch with a small finite rootradius, r= 0.2a. The center of the half-circle forming the notch bottom is at x = 1.2a and y = 0 (see Fig. 2(b)). This improves the convergence rate considerably. The modification was shown to have little influence on the solution as long as the radius during the loading increases several times beyond the radius of the original notch (cf. McMeeking, 1977). Secondly, a small amount of inertia was introduced. This was verified not to have any significant effect on the result but has a beneficial effect on the convergence rate. A disadvantage of this modification is that the prescribed constant displacement rate boundary condition could not be applied instantly at the beginning of the calculation. Instead, the displacement rate was increased linearly to the prescribed level during the few first increments. The effect on the results of this is believed to be negligible. The body selected for the calculations is covered by a mesh containing 481 nodes and 432 plane strain, isoparametric, bilinear displacementbased constant-pressure elements. During the analyses, the elements become distorted. To keep the side-to-side aspect ratios within reasonable limits, the original near-tip element shape is chosen so that an element experiences an extending strain along the side originally short, and a compressive strain along the side originally long. The dependence of element size was investigated by changing the number of elements in a small region containing the ligament. The employed mesh was therefore found to produce results with reasonable accuracy. The objective is to study the energy dissipation in the ligaments as a function of the ligament extension rate. Calculations were performed for different constant extension rates, 6; 0. The load increases to a maximum and thereafter drops due to the reduction of the cross-sectional area. At the phase when the loads decrease during increasing displacement, the energy dissipation is neglected at loads lower than a critical level. That level is taken to be 30% of the maximum load. The calculations are performed for a large contraction of the ligament that gives a correspondingly large distortion of the elements in the ligament region. The difference between the work
done by the external forces and the total strain energy is not at any time larger than 5% of the total energy. 4. Experiments Notched specimens of mild steel (E= 206 GPa, s0 = 216 MPa, n= 0.3) with a geometry according to Fig. 3(a) are chosen for the experiments. The aim is to simulate the behavior of a ligament by rapidly straining the notched part of the specimen in tension. An elastic wave in an impacted rod is used to generate the transient loading.
Fig. 3. (a) Specimen and (b) experimental set-up.
P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
The experimental set-up is shown in Fig. 3(b). A loading rod (steel with diameter 27 mm, length 1500 mm) is impacted at its far end (not shown in the figure) by a projectile (steel rod with diameter 27 mm and length 150 mm) fired from an air gun. The impact generates a compressive elastic wave in the loading rod, travelling towards the instrumented end. As the incident compressive wave reflects, it closes a small gap and the end of the loading rod impacts a light and stiff yoke which is rapidly accelerated. The head of the T-shaped specimen (Fig. 3(b)) is fitted to the yoke and a transient tensile force is thus applied to the specimen. The yoke is carefully aligned with the rod and, as the gap closes, a good surface contact is established, thereby facilitating a rapid loading of the specimen. A 500 mm long measuring rod of steel is attached to the specimen by means of an adhesive joint. This rod has the same cross-sectional dimensions as the end of the specimen (3× 5 mm2) and the joint is very small. Undistorted uniaxial wave propagation along the measuring rod is therefore assumed and the strain history in the wave represents the force history at the tensile fracture in the notch. At a position 100 mm from the joint a strain gauge with an active length of 3 mm is attached to each side of the rod and the pair is coupled in a Wheatstone bridge so that contributions from bending strains are suppressed. The strain signal is fed to an amplifier (Measurement Group model 2210). At a position just below the head of the specimen a target for a non-contacting displacement transducer (Zimmer 100D) is attached. This is a so-called opto-follower that measures the transient axial displacement of the specimen head. Both equipments have wide band characteristics with 3 dB limits of 100 and 400 kHz, respectively. Samples of the strain and displacement signals were recorded by a digital transient recorder (Lucas Datalab DL6034) at a rate of 2.5 × 106 s − 1 and the digital data were then transferred (using the software Labview) to a computer. These data thus represent the time histories of the ligament force, Pexp, and the ligament extension, 6exp. The force, Pexp, is calculated from the strain gauge measurements. The extension over the specimen,
149
Fig. 4. Plastic zone for different extensions 60 at the extension rate 6; 0 =7.79g; 0a
26exp, is obtained from the response of the optofollower on one side of the specimen and displacements computed using the strain gauge measurements. A simple uniaxial wave propagation theory is used to correlate the strain gauge response with the displacement at a short distance from the notch center. Thus, the extension 26exp is assumed to be the difference in displacement between two points, A and B, one 2.5 mm above and one 2.5 mm below the plane of the minimum sectional area in the specimen (see Fig. 3(a)).
5. Results The energy dissipation in the ligaments and its dependence on the ligament extension rate, 6; 0 is studied. Calculations are performed for constant extension rates from 6; 0 = 0.08g; 0a to 80g; 0a. The results for the lowest extension rates were computed primarily for comparison with the experimental results.
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P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
Fig. 4 shows the development of the calculated plastic zone at an extension rate 6; 0 =7.79g; 0a. At low loads, plastic deformation is confined to the notch bottom. As the extension, 60, increases, the plastic zone assumes a circular shape. The maximum height is around 7a and is obtained when 6; 0 = 0.03a. After that, the height of the active plastic zone decreases and the plastic zone gradually assumes a flatter shape. The active plastic zone signifies the part of the zone where plastic strains increase. When the calculation is interrupted at the cut-off load, the extent of the active plastic zone is still considerable (see Fig. 4, 6; 0 = 0.115a). Seemingly, the active plastic zone does not vanish for vanishing loads, which indicates substantial residual strains. The maximum heights for other extension rates vary from around 6a at the lowest rate to around 15a at the highest rate. The force history in the ligament can be studied in Fig. 5(a) for five ligament extension rates. The ligaments remain essentially elastic during an initial phase. During this phase the response is nearly rate independent for the considered extension rates. One observes that the load at onset of substantial plastic deformation, i.e., when a substantial deviation from the linear elastic response occurs, depends on the extension rate. Further, it is noted that the peak loads occur at a load fairly close to the load at onset of substantial plastic deformation. The peak load is reached approximately when the plastic zone reaches its maximum height. Fig. 5(b) shows the force in the ligament versus extension for two experiments. A rapid decrease in load is observed after the peak load, indicating an early fracture in the experiments. The surfaces of the ruptured ligaments revealed little transverse contraction before final rupture. Growth and coalescence of voids are assumed to have occurred, thus changing the geometry and the conditions during the experiments. A dependence on the ligament extension rate is seen in the force (see Fig. 5(b)). The distribution of the plastic deformation work density, q = t0 sijo; pij dt%, in the vicinity of the ligament at the cut-off load, is displayed in Fig. 6. Here, t is the time when the cut-off load is met. By assuming that the process is adiabatic and that
all energy is transformed into heat the temperature change, DT, can be calculated from DT =q/ (cvr). Hence, Fig. 6 identically shows the non-dimensional temperature change in the ligament at the cut-off load for the ligament extension rate 6; 0 = 7.79g; 0a. The temperature is high, e.g., for a common structural steel with a heat capacity of 460 J (kg · °C) − 1, a density of 7850 kg
Fig. 5. (a) Calculated ligament extension force vs. extension for different values of extension rates, and (b) experimental force vs. extension for two values of extension rates.
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The accumulated dissipated energy in a ligament at the cut-off load, Glig, is displayed in Fig. 7, for different extension rates, 6; 0. This energy is computed as Glig = V q dV%, where V is the volume of the entire body under consideration. The plastically dissipated energy is observed to increase with increasing ligament extension rates 6; 0.
6. Discussion and conclusions
Fig. 6. The distribution of the plastic deformation work density and temperature distribution in the vicinity of the ligament at the cut-off load.
m − 3 and a yield stress of 400 MPa, 30% of the ligament on the symmetry line y = 0 reaches temperatures above 500°C. Maximum temperatures resulting from the numerical calculations are unreliable since the strain energy density at sharp notches is unbounded. Maximum temperatures therefore depend on the selected finite notch radius and the size of the finite elements.
The choice of material data for the calculations is based on data given in the literature (Campbell and Ferguson, 1970; Huang and Clifton, 1985) (see Fig. 8). The tests made by Campbell and Ferguson are dynamic shear tests done with a drop weight testing technique. The temperature increase during those tests is estimated to be approximately 1°C in the medium strain rate region (1–104 s − 1). The test results presented by Huang and Clifton (1985) are pressure–shear impact tests on high-purity iron. Numerical simulations of those experiments indicate temperature increases of several hundred degrees centigrade due to large strains. Huang and Clifton (1985) observe that tests done on high-strength steel (4340 VAR) do not indicate the same strong strain rate sensitivity as the tests done on high-pu-
Fig. 7. Accumulated dissipated energy in a ligament at the cut-off load.
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P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
Fig. 8. Tests made by different investigators compared with the chosen constitutive relation.
rity steel. It is believed that the chosen constitutive relation is useful as long as the temperature increase in the ligament is low compared to the melting point of the material. To illustrate the significance of the rate sensitivity of ligament extension during fast fracture the following was assumed: ligaments cover 10% of the crack surface, n = 5, g; 0 =7000 s − 1, E= 206 GPa, s0 =400 MPa and a =5 mm. An average crack surface displacement rate can be calculated for a strip yield model (Dugdale, 1960; Sta˚hle and Freund, 1990). The results are well known. The length of the yielding zone is d =(p/8)GE/s 20 and the crack tip opening displacement is h =G/s0. Thus, as the crack grows to d, the separation of the crack surfaces increases from zero to h. This suggests that the crack tip speed as an average is d/h =(p/8)E/s0 :200 times larger than the rate of separation of the crack surfaces, e.g., if the crack tip speed is 300 m s − 1 then 26; 0 is approximately 1.5 m s − 1. Fig. 7 then gives Glig = 7.5s0a 2 = 0.075 Nm per ligament. Based on observation, ligaments covering 10% of the crack surface could be a reasonable assumption (Sta˚hle and Freund, 1990). Linear scaling for such a case gives Glig =530 N m − 1. If it is assumed that the ligaments alone are responsible for practically the entire fracture toughness, the following estimation
is made KIc = (GligE)1/2 = 10.4 MPa m1/2. Since 90% of the fracture surface is created through cleavage, in the case above the fracture mode would definitely be characterized as cleavage. Considering this, the fracture toughness KIc = 10.4 MPa m1/2 is not at all a small value for steel during strain-rate-induced cleavage fracture. The increasing energy dissipation with increasing speeds will stabilize the crack tip speed provided that the decreasing energy dissipated in the plastic zone is compensated for. Without making any definite assumptions about the energy dissipated in the plastic zone, it is believed that it is of the same order of magnitude as Glig, and that the ligaments in many cases are responsible for low arrest speeds and difficulties in experiments to obtain crack tip speeds of the order of the Rayleigh wave speed.
Appendix A The elastic solution is given by the potential function C given as follows: C=
P p a 2 − z 2
(A1)
P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
where z= x +iy (Muskhelishvili, 1953). The load P is defined by P=
&
a
sy y = 0 dx
(A2)
−a
The stresses in the entire body are given by sx + sy = 2Re(C)
sy +itxy =Re(C) − iyC% (A3)
A polar coordinate system is attached to x= y=0 as Fig. 2(a) shows. Eq. (A3) can be expanded for r/a , where r = x 2 +y 2 and u = tan − 1(y/x). The dominating term for the displacement 6 in the y-direction is obtained as follows: 6=
2r P(1+n) 2(1 −n) ln +cos2(u) − 1 a pE
n
(A4)
References ABAQUS, 1989. User’s manual v. 5.3-1, Hibbit, Karlsson and Sorensen, Providence, RI, USA. Brickstad, B., 1983. A viscoplastic analysis of rapid crack
propagation experiments in steel. J. Mech. Phys. Solids 31, 307 – 327. Campbell, J.D., Ferguson, W.G., 1970. The temperature and strain rate dependence of shear strength of mild steel. Philos. Mag. 21, 63 – 82. Dugdale, D.S., 1960. Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100 – 108. Freund, L.B., Hutchinson, J., 1985. High-strain rate crack growth in rate-dependent plastic solids. J. Mech. Phys. Solids 33, 169 – 191. Huang, S., Clifton, R.J., 1985. Dynamic response of OFHC copper at high shear strain rates. In: IUTAM Symp. Macro- and Micro-Mechanics of High Velocity Deformation and Fracture, Tokyo, Japan. Lo, K.K., 1983. Dynamic crack tip fields in rate sensitive solids. J. Mech. Phys. Solids 31, 287 – 305. Malvern, L.E., 1951. Plastic wave propagation in a bar of material exhibiting a strain rate effect. Q. J. Appl. Math. 8, 405 – 411. McMeeking, R.M., 1977. Finite deformation analysis of cracktip opening in elastic – plastic materials and implications for fracture. J. Mech. Phys. Solids 25, 357 – 381. Muskhelishvili, N.I., 1953. Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen, Holland. Perzyna, P., 1963. The constitutive equations for rate sensitive plastic materials. Q. J. Appl. Math. 20, 321 – 332. Sta˚hle, P., Freund, L.B., 1990. Process region characteristics during fast crack growth — an examination of wide plate crack arrest experiment. Division for Solid Mechanics Report, Brown University, Providence, RI, USA.
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On the behavior of crack surface ligaments P. Nilsson, P. Sta˚hle *, K.G. Sundin Di6ision of Solid Mechanics, Lulea˚ Uni6ersity of Technology, SE-971 87 Lulea˚, Sweden Received 13 March 1996; received in revised form 31 July 1997; accepted 20 January 1998
Abstract Small ligaments connecting the fracture surfaces just behind a moving crack front are assumed to exist under certain conditions. The ligaments are rapidly torn as the crack advances. Inelastic straining of such ligaments influences the energy balance in the fracture process. The rapid tearing of a single ligament is studied both numerically and experimentally. An elastic visco-plastic material model is adopted for finite-element calculations. The results show that relatively large amounts of energy are dissipated during the tearing process. Further, the energy needed to tear a ligament increases rapidly with increasing tearing rate. The computed behavior is partly verified in a few preliminary experiments. The implications for slow stable crack tip speeds during dynamic fracture are discussed. © 1998 Elsevier Science S.A. All rights reserved.
1. Introduction Often during fast crack growth in structural steels, the influence of rate-dependent material behavior during plastic straining cannot be ignored (Brickstad, 1983; Lo, 1983). Increasing stress with increasing strain rate has been observed. When this effect is sufficiently strong, the plastic-strains have a minor effect on the stress distribution. Such a case has been treated by Freund and Hutchinson (1985), who considered the energy rate balance necessary for continuous steady-state crack growth. The elastic energy release rate G for dynamic crack propagation was assumed to be balanced by the sum of the work rate due to inelastic deformation in the plastic
* Corresponding author. Tel.: + 46 920 72188; fax: + 46 920 91047; e-mail: [email protected]
zone, Gvp, and the rate of the work dissipated at the crack tip. It is assumed here that tearing of ligaments accounts for the essential part of the work rate in the process zone. Therefore, it is suggested that G is divided into three parts, namely, Glig associated with ligament tearing, Gvp associated with deformation in the plastic zone, and Gcl associated with cleavage. The relation is indicated schematically in Fig. 1. Freund and Hutchinson (1985) found that stable crack growth is possible only when the crack speed is greater than 0.55cr, where cr is the Rayleigh wave speed. Below that, in their analysis the speed will be unstable and the crack tip will immediately accelerate to a speed higher than 0.55cr or come to an arrest. In structural steels, both extended crack propagation and crack arrest often occur at speeds as low as from 0.05cr to 0.1cr. Observations are,
0029-5493/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0029-5493(98)00160-5
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P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
however, qualitatively in accordance with the results predicted by Freund and Hutchinson (1985). The higher arrest speed predicted by Freund and Hutchinson (1985) is obtained on the assumption that the crack tip energy release rate is independent of the strain rate. It is believed that the rate sensitivity of the ligament tearing process may partly explain the quantitative discrepancy between observations and predictions. The crack tip is assumed to propagate through a coalescence of transgranular micro cracks. Due to mismatches of geometry and mechanical properties at grain boundaries, unbroken parts will remain, connecting the upper and lower crack surfaces. During increasing separation of the crack surfaces, the unbroken parts will become ligaments, bridging the gap between the separating crack surfaces. Studies of cleavage fracture surfaces show that parts of the surfaces consist of plastically formed ridges, presumably traces of plastically torn ligaments. These ligaments are assumed to be formed as previously described. A large part of the energy consumed in the fracture process region is suggested to be due to the disruption of such ligaments. In this paper a ligament model is studied as it deforms plastically to a state where very little remains of its initial cross-sectional area. The energy consumption is calculated as a function of the ligament extension rate. At crack tip speeds of practical interest, the energy consumption rate in
the ligaments is found to be comparable to total energy release rates in structural steels. In the discussion part of this paper, implications for stable crack growth are considered. An attempt is made to experimentally verify the assumed ligament model. Two high loading rate tensile tests were performed on small test specimens. The preliminary results lend some confidence to the numerical results.
2. Model of a single ligament A large body described in the coordinate system x= x1, y= x2 and z= x3 is considered. The body consists of a lower y0 0 and an upper half-plane y\ 0 separated along the entire plane y= 0 except in the region x 0 a, y= 0. The surfaces at x \a, y= 0 are assumed to be traction free. The extent of the body in the z-direction is assumed to be large, thus creating plane strain conditions. A large strain theory is adopted. Thus, the rate of Green’s strain tensor, o; ij, is used. The employed Kirchhoff stress tensor, sij, is the work rate conjugate to o; ij. An elastic visco-plastic material model is used to describe the mechanical behavior of the body (Malvern, 1951). The strain rate is decomposed into an elastic part, o; eij, and a visco-plastic part, o; pij : o; ij = o; eij + o; pij
(1)
The elastic strain rates are given by Hooke’s law using Young’s modulus E and Poisson’s ratio n= 0.3. Let se =
3 sij sij 2
1/2
(2)
define the von Mises effective stress, where sij is the stress deviator. The visco-plastic strain rates are given by the following constitutive relationship
o; pij = g; 0
Fig. 1. Energy balance during fast crack growth.
se −1 s0
n
sij se
(3)
where g; 0 is the strain rate sensitivity, s0 is the yield stress and n is the strain rate exponent (Perzyna, 1963). The ratio between Young’s modulus and the yield stress E/s0 = 502 is used, which
P. Nilsson et al. / Nuclear Engineering and Design 184 (1998) 145–153
147
represents a common structural steel. The strain rate exponent, n =5 is chosen as proposed by Malvern (1951). The strain rate sensitivity, g; 0, does not have to be quantified for the analysis. The dissipation of energy during plastic deformation will lead to an increased temperature. However, here the mechanical behavior is assumed not to be affected by the increased temperature. The temperature rate due to plastic deformation is calculated as follows T: =
sijo; pij cvr0
(4)
where r0 is the mass density and cv is the heat capacity. It is assumed that the loading of the specimen is fast and, thus, produces predominantly adiabatic conditions. Hence, the effects of heat conduction are neglected. The structure is subjected to a remotely applied tension parallel with the y-axis. Symmetry is assumed across x =0 and across y =0. The study is therefore limited to the part x E 0 and yE 0. A polar coordinate system, r and u, is attached to x =y=0, as Fig. 2(a) shows. An elastic solution is employed for a boundary layer analysis of a part bounded by r0R. A prescribed displacement rate is used to control the load. The following boundary conditions are thus applied: u =0
and
Fig. 2. (a) Geometry and (b) finite-element mesh.
6 =6; 0t ln(jR/a),
at r =R, 00 u 0p/2
(5)
where j = 2 exp{ − 1/[2(1 −n)]} is obtained by assuming that cos2(u)/log(R/a) :0 (cf. Appendix A). Here, t is the time. Hence, the displacement becomes independent of u, which simplifies the numerical analysis. The effect of this is assumed to be small and decreasing as the width of the ligament decreases during the loading process. Symmetry across x =0 and across y =0 requires txy =0
and
u=0
at x=0, 00 y B R
(6)
and
6 =0
at 0B x 0a, y = 0
(7)
and txy =0
Finally, traction-free surfaces require txy =sy = 0
at a Bx B R, y = 0
(8)
The ratio R/a is chosen to be 40, which is assumed to be sufficiently large to meet the requirements for a boundary layer solution.
3. Numerical model The commercial finite-element code ABAQUS was used for the numerical calculations. An updated Lagrangian method for large deformation elastic visco-plastic materials was used (ABAQUS, 1989). Full integration of element stiffnesses was used. To obtain a reasonably rapid convergence of the calculations, two modifications of the proposed model were made. Firstly, the suggested sharp notch tips formed at x = a, y= 0 were
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each modeled as a notch with a small finite rootradius, r= 0.2a. The center of the half-circle forming the notch bottom is at x = 1.2a and y = 0 (see Fig. 2(b)). This improves the convergence rate considerably. The modification was shown to have little influence on the solution as long as the radius during the loading increases several times beyond the radius of the original notch (cf. McMeeking, 1977). Secondly, a small amount of inertia was introduced. This was verified not to have any significant effect on the result but has a beneficial effect on the convergence rate. A disadvantage of this modification is that the prescribed constant displacement rate boundary condition could not be applied instantly at the beginning of the calculation. Instead, the displacement rate was increased linearly to the prescribed level during the few first increments. The effect on the results of this is believed to be negligible. The body selected for the calculations is covered by a mesh containing 481 nodes and 432 plane strain, isoparametric, bilinear displacementbased constant-pressure elements. During the analyses, the elements become distorted. To keep the side-to-side aspect ratios within reasonable limits, the original near-tip element shape is chosen so that an element experiences an extending strain along the side originally short, and a compressive strain along the side originally long. The dependence of element size was investigated by changing the number of elements in a small region containing the ligament. The employed mesh was therefore found to produce results with reasonable accuracy. The objective is to study the energy dissipation in the ligaments as a function of the ligament extension rate. Calculations were performed for different constant extension rates, 6; 0. The load increases to a maximum and thereafter drops due to the reduction of the cross-sectional area. At the phase when the loads decrease during increasing displacement, the energy dissipation is neglected at loads lower than a critical level. That level is taken to be 30% of the maximum load. The calculations are performed for a large contraction of the ligament that gives a correspondingly large distortion of the elements in the ligament region. The difference between the work
done by the external forces and the total strain energy is not at any time larger than 5% of the total energy. 4. Experiments Notched specimens of mild steel (E= 206 GPa, s0 = 216 MPa, n= 0.3) with a geometry according to Fig. 3(a) are chosen for the experiments. The aim is to simulate the behavior of a ligament by rapidly straining the notched part of the specimen in tension. An elastic wave in an impacted rod is used to generate the transient loading.
Fig. 3. (a) Specimen and (b) experimental set-up.
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The experimental set-up is shown in Fig. 3(b). A loading rod (steel with diameter 27 mm, length 1500 mm) is impacted at its far end (not shown in the figure) by a projectile (steel rod with diameter 27 mm and length 150 mm) fired from an air gun. The impact generates a compressive elastic wave in the loading rod, travelling towards the instrumented end. As the incident compressive wave reflects, it closes a small gap and the end of the loading rod impacts a light and stiff yoke which is rapidly accelerated. The head of the T-shaped specimen (Fig. 3(b)) is fitted to the yoke and a transient tensile force is thus applied to the specimen. The yoke is carefully aligned with the rod and, as the gap closes, a good surface contact is established, thereby facilitating a rapid loading of the specimen. A 500 mm long measuring rod of steel is attached to the specimen by means of an adhesive joint. This rod has the same cross-sectional dimensions as the end of the specimen (3× 5 mm2) and the joint is very small. Undistorted uniaxial wave propagation along the measuring rod is therefore assumed and the strain history in the wave represents the force history at the tensile fracture in the notch. At a position 100 mm from the joint a strain gauge with an active length of 3 mm is attached to each side of the rod and the pair is coupled in a Wheatstone bridge so that contributions from bending strains are suppressed. The strain signal is fed to an amplifier (Measurement Group model 2210). At a position just below the head of the specimen a target for a non-contacting displacement transducer (Zimmer 100D) is attached. This is a so-called opto-follower that measures the transient axial displacement of the specimen head. Both equipments have wide band characteristics with 3 dB limits of 100 and 400 kHz, respectively. Samples of the strain and displacement signals were recorded by a digital transient recorder (Lucas Datalab DL6034) at a rate of 2.5 × 106 s − 1 and the digital data were then transferred (using the software Labview) to a computer. These data thus represent the time histories of the ligament force, Pexp, and the ligament extension, 6exp. The force, Pexp, is calculated from the strain gauge measurements. The extension over the specimen,
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Fig. 4. Plastic zone for different extensions 60 at the extension rate 6; 0 =7.79g; 0a
26exp, is obtained from the response of the optofollower on one side of the specimen and displacements computed using the strain gauge measurements. A simple uniaxial wave propagation theory is used to correlate the strain gauge response with the displacement at a short distance from the notch center. Thus, the extension 26exp is assumed to be the difference in displacement between two points, A and B, one 2.5 mm above and one 2.5 mm below the plane of the minimum sectional area in the specimen (see Fig. 3(a)).
5. Results The energy dissipation in the ligaments and its dependence on the ligament extension rate, 6; 0 is studied. Calculations are performed for constant extension rates from 6; 0 = 0.08g; 0a to 80g; 0a. The results for the lowest extension rates were computed primarily for comparison with the experimental results.
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Fig. 4 shows the development of the calculated plastic zone at an extension rate 6; 0 =7.79g; 0a. At low loads, plastic deformation is confined to the notch bottom. As the extension, 60, increases, the plastic zone assumes a circular shape. The maximum height is around 7a and is obtained when 6; 0 = 0.03a. After that, the height of the active plastic zone decreases and the plastic zone gradually assumes a flatter shape. The active plastic zone signifies the part of the zone where plastic strains increase. When the calculation is interrupted at the cut-off load, the extent of the active plastic zone is still considerable (see Fig. 4, 6; 0 = 0.115a). Seemingly, the active plastic zone does not vanish for vanishing loads, which indicates substantial residual strains. The maximum heights for other extension rates vary from around 6a at the lowest rate to around 15a at the highest rate. The force history in the ligament can be studied in Fig. 5(a) for five ligament extension rates. The ligaments remain essentially elastic during an initial phase. During this phase the response is nearly rate independent for the considered extension rates. One observes that the load at onset of substantial plastic deformation, i.e., when a substantial deviation from the linear elastic response occurs, depends on the extension rate. Further, it is noted that the peak loads occur at a load fairly close to the load at onset of substantial plastic deformation. The peak load is reached approximately when the plastic zone reaches its maximum height. Fig. 5(b) shows the force in the ligament versus extension for two experiments. A rapid decrease in load is observed after the peak load, indicating an early fracture in the experiments. The surfaces of the ruptured ligaments revealed little transverse contraction before final rupture. Growth and coalescence of voids are assumed to have occurred, thus changing the geometry and the conditions during the experiments. A dependence on the ligament extension rate is seen in the force (see Fig. 5(b)). The distribution of the plastic deformation work density, q = t0 sijo; pij dt%, in the vicinity of the ligament at the cut-off load, is displayed in Fig. 6. Here, t is the time when the cut-off load is met. By assuming that the process is adiabatic and that
all energy is transformed into heat the temperature change, DT, can be calculated from DT =q/ (cvr). Hence, Fig. 6 identically shows the non-dimensional temperature change in the ligament at the cut-off load for the ligament extension rate 6; 0 = 7.79g; 0a. The temperature is high, e.g., for a common structural steel with a heat capacity of 460 J (kg · °C) − 1, a density of 7850 kg
Fig. 5. (a) Calculated ligament extension force vs. extension for different values of extension rates, and (b) experimental force vs. extension for two values of extension rates.
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The accumulated dissipated energy in a ligament at the cut-off load, Glig, is displayed in Fig. 7, for different extension rates, 6; 0. This energy is computed as Glig = V q dV%, where V is the volume of the entire body under consideration. The plastically dissipated energy is observed to increase with increasing ligament extension rates 6; 0.
6. Discussion and conclusions
Fig. 6. The distribution of the plastic deformation work density and temperature distribution in the vicinity of the ligament at the cut-off load.
m − 3 and a yield stress of 400 MPa, 30% of the ligament on the symmetry line y = 0 reaches temperatures above 500°C. Maximum temperatures resulting from the numerical calculations are unreliable since the strain energy density at sharp notches is unbounded. Maximum temperatures therefore depend on the selected finite notch radius and the size of the finite elements.
The choice of material data for the calculations is based on data given in the literature (Campbell and Ferguson, 1970; Huang and Clifton, 1985) (see Fig. 8). The tests made by Campbell and Ferguson are dynamic shear tests done with a drop weight testing technique. The temperature increase during those tests is estimated to be approximately 1°C in the medium strain rate region (1–104 s − 1). The test results presented by Huang and Clifton (1985) are pressure–shear impact tests on high-purity iron. Numerical simulations of those experiments indicate temperature increases of several hundred degrees centigrade due to large strains. Huang and Clifton (1985) observe that tests done on high-strength steel (4340 VAR) do not indicate the same strong strain rate sensitivity as the tests done on high-pu-
Fig. 7. Accumulated dissipated energy in a ligament at the cut-off load.
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Fig. 8. Tests made by different investigators compared with the chosen constitutive relation.
rity steel. It is believed that the chosen constitutive relation is useful as long as the temperature increase in the ligament is low compared to the melting point of the material. To illustrate the significance of the rate sensitivity of ligament extension during fast fracture the following was assumed: ligaments cover 10% of the crack surface, n = 5, g; 0 =7000 s − 1, E= 206 GPa, s0 =400 MPa and a =5 mm. An average crack surface displacement rate can be calculated for a strip yield model (Dugdale, 1960; Sta˚hle and Freund, 1990). The results are well known. The length of the yielding zone is d =(p/8)GE/s 20 and the crack tip opening displacement is h =G/s0. Thus, as the crack grows to d, the separation of the crack surfaces increases from zero to h. This suggests that the crack tip speed as an average is d/h =(p/8)E/s0 :200 times larger than the rate of separation of the crack surfaces, e.g., if the crack tip speed is 300 m s − 1 then 26; 0 is approximately 1.5 m s − 1. Fig. 7 then gives Glig = 7.5s0a 2 = 0.075 Nm per ligament. Based on observation, ligaments covering 10% of the crack surface could be a reasonable assumption (Sta˚hle and Freund, 1990). Linear scaling for such a case gives Glig =530 N m − 1. If it is assumed that the ligaments alone are responsible for practically the entire fracture toughness, the following estimation
is made KIc = (GligE)1/2 = 10.4 MPa m1/2. Since 90% of the fracture surface is created through cleavage, in the case above the fracture mode would definitely be characterized as cleavage. Considering this, the fracture toughness KIc = 10.4 MPa m1/2 is not at all a small value for steel during strain-rate-induced cleavage fracture. The increasing energy dissipation with increasing speeds will stabilize the crack tip speed provided that the decreasing energy dissipated in the plastic zone is compensated for. Without making any definite assumptions about the energy dissipated in the plastic zone, it is believed that it is of the same order of magnitude as Glig, and that the ligaments in many cases are responsible for low arrest speeds and difficulties in experiments to obtain crack tip speeds of the order of the Rayleigh wave speed.
Appendix A The elastic solution is given by the potential function C given as follows: C=
P p a 2 − z 2
(A1)
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where z= x +iy (Muskhelishvili, 1953). The load P is defined by P=
&
a
sy y = 0 dx
(A2)
−a
The stresses in the entire body are given by sx + sy = 2Re(C)
sy +itxy =Re(C) − iyC% (A3)
A polar coordinate system is attached to x= y=0 as Fig. 2(a) shows. Eq. (A3) can be expanded for r/a , where r = x 2 +y 2 and u = tan − 1(y/x). The dominating term for the displacement 6 in the y-direction is obtained as follows: 6=
2r P(1+n) 2(1 −n) ln +cos2(u) − 1 a pE
n
(A4)
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