Effects of constitutive parameters on adiabatic shear localization for

ductile metal based on JOHNSON-COOK and gradient plasticity models

WANG Xue-bin(王学滨)

Department of Mechanics and Engineering Sciences, Liaoning Technical University, Fuxin 123000, China

Received 6 March 2006; accepted 8 May 2006

Abstract: By using the widely used JOHNSON-COOK model and the gradient-dependent plasticity to consider microstructural effect beyond the occurrence of shear strain localization, the distributions of local plastic shear strain and deformation in adiabatic shear band(ASB) were analyzed. The peak local plastic shear strain is proportional to the average plastic shear strain, while it is inversely proportional to the critical plastic shear strain corresponding to the peak flow shear stress. The relative plastic shear deformation between the top and base of ASB depends on the thickness of ASB and the average plastic shear strain. A parametric study was carried out to study the influence of constitutive parameters on shear strain localization. Higher values of static shear strength and work to heat conversion factor lead to lower critical plastic shear strain so that the shear localization is more apparent at the same average plastic shear strain. Higher values of strain-hardening exponent, strain rate sensitive coefficient, melting point, thermal capacity and mass density result in higher critical plastic shear strain, leading to less apparent shear localization at the same average plastic shear strain. The strain rate sensitive coefficient has a minor influence on the critical plastic shear strain, the distributions of local plastic shear strain and deformation in ASB. The effect of strain-hardening modulus on the critical plastic shear strain is not monotonous. When the maximum critical plastic shear strain is reached, the least apparent shear localization occurs.

Key words: adiabatic shear band; ductile metal; shear localization; JOHNSON-COOK model; constitutive parameters

Adiabatic shear band(ASB) is a very narrow zone with a high concentration of shear strain. It is believed that ASB is formed by a process of thermo-mechanical instability. ASB can be observed in the process of dynamic deformation of various ferrous and nonferrous metals (such as titanium, aluminum and steels), alloys (such as titanium alloys, aluminum alloys and metallic glass), single crystals, polycrystals, polymers and geomaterials (such as rock and soil).

The various problems related to ASB were studied by experimental tests[1-11], numerical simulations [9,12-14] and theoretical analyses[1,9,15], such as the critical conditions for ASB formation, the sites for initiation of ASB, the propagation of ASB, the microstructural characteristics in ASB and outside, the nucleation, growth and coalescence of voids and microcracks in ASB, the length, number, width, spacing, pattern, orientation of ASB, the development or evolution of stress, strain, strain rate, deformation and temperature in ASB.

Different from the previous investigations, on the aspect of theoretical analysis, the work by WANG et al [16-22] was focused on the distributions of plastic strain, deformation, temperature and damage variable in ASB based on gradient-dependent plasticity where an internal length parameter was included in the yield function to describe the interactions and interplay among microstructures in ductile metals, such as titanium and its alloys as well as low-carbon steel. Considering the effects of strain-hardening, strain-rate sensitivity, thermal-softening and microstructures, JOHNSON- COOK model and gradient-dependent plasticity were used to calculate the temperature distribution in ASB[22].

In this study, by using the widely used JOHNSON- COOK model and the gradient-dependent plasticity to consider microstructural effect beyond the occurrence of shear strain localization, the characteristics of local plastic shear strain and deformation in ASB are analyzed. A detailed parametric study is carried out to study the influence of constitutive parameters on the flow shearstress-average plastic shear strain curve, the critical plastic shear strain, the distributions of local plastic shear strain and deformation in ASB.

In JOHNSON-COOK model[13], the flow shear stress τ and the temperature T are

    (1)

                         (2)

where  is the average plastic shear strain;  is the reference shear strain rate;  is the imposed shear strain rate; T₀ is the initial temperature; T_m is the melting temperature; β is the work to heat conversion factor; c_p is the heat capacity; ρ is the density; A, B, C, m, and n are static shear strength, strain-hardening modulus, strain rate sensitive coefficient, thermal-softening exponent and strain-hardening exponent, respectively.

The occurrence of ASB is usually attributed to the thermal-plastic shear instability: the thermal softening due to the dissipation of part of the mechanical work just overcomes the strain-hardening effect. Therefore, the condition for the onset of ASB is

dτ=0                                       (3)

where  d is a differential sign.

Average plastic shear strain corresponding to the peak flow shear stress τ_max is called the critical plastic shear strain γ_c.

In the strain-hardening stage prior to the peak stress τ_max, there is

dτ＜0                                      (4)

In the strain-softening stage beyond the occurrence of ASB, there is

dτ＞0                                      (5)

Using gradient-dependent plasticity, WANG[22] derived the thickness w of ASB and the distribution γ_p(y) of local plastic shear strain in ASB:

w=2πl                                      (6)

              (7)

where  l is the internal length parameter reflecting the heterogeneous extent of ductile metal material; y is the coordinate whose original point is set at the center of ASB.

We can determine the maximum value of γ_p(y) using Eqn.(7):

                (8)

The minimum value of γ_p(y) is reached at the two boundaries of ASB:

                        (9)

The first-order gradient of γ_p(y) with respect to the coordinate y is

                  (10)

When

,

When

,

Using Eqns.(8) and (9), we can establish the relation for the nonlocal variable :

                  (11)

According to the geometrical equation (the relation between strain and deformation), the local plastic shear strain γ_p(y) is related to the local plastic shear deformation s_p(y):

                             (12)

Thus, s_p(y) is expressed as

 (13)

It is found from Eqn.(13) that s_p(y) is an odd function with respect to the coordinate y:

                            (14)

At the center of ASB, i.e., y=0, the local plastic shear deformation is

s_p(0)=0                                    (15)

At the top of ASB, i.e., y=w/2, the local plastic shear deformation is

            (16)

At the base of ASB, i.e., y=-w/2, the local plastic shear deformation is

      (17)

Thus, the relative plastic shear deformation is

                    (18)

Eqns.(7) and (13) show that the distribution of local plastic shear strain in ASB is highly nonuniform, and the distribution of local plastic shear deformation in ASB is highly nonlinear.

4.1 Effect of static shear strength

Fig.1 shows the influence of static shear strength Aon the flow shear stress—average plastic shear strain curve (τ—curve), the critical plastic shear strain γ_c, the local plastic shear strain γ_p(y) and deformation s_p(y) in ASB with l=3.18 μm, B=500 MPa, C=0.014, m=1.03, n=0.26, T_m=1 790 K, ρ=7 850 kg/m³, T₀=300 K, =  3 300 s^-1, c_p=473 J/(kg?K), =11 000 s^-1 and β=0.9.

It is found from Fig.1 that higher value of A leads to higher peak flow shear stress τ_max and lower γ_c. The γ_c—A curve is concave upward. Fig.1(c) shows that higher A results in steeper profile of γ_p(y) and higher peak local plastic shear strain γ_{p, max}, as also seen from Eqns.(7) and (8). This means that the distribution of γ_p(y) becomes more nonuniform and the concentration of plastic shear strain in ASB becomes more apparent as A increases.

According to Eqn.(9), the local plastic shear strain at the two boundaries of ASB is equal to γ_c. Higher γ_c is expected at lower value of , as seen from Figs.1((a)- (c)).

As can be seen from Fig.1(d), the distribution of s_p(y) in ASB is nonlinear. The extent of the nonlinear distribution is more and more apparent at higher A. As seen from Fig.1(d) or Eqns.(15) and (18), the local plastic

Fig.1 Effects of static shear strength A on flow shear stress—average plastic shear strain curve (a), critical plastic shear strain (b), local plastic shear strain (c) and deformation (d) in ASB

shear deformation at the center of ASB is zero, and the relative plastic shear deformation between the two ends of ASB is only dependent on the average plastic shear strain  and the thickness w of ASB.

4.2 Effect of strain-hardening modulus

The effect of parameter B on τ—curve, γ_c, γ_p(y) and s_p(y) is shown in Fig.2 with l=3.18 ?m, A=800 MPa, C=0.014, m=1.03, n=0.26, T_m=1 790 K, ρ=7 850 kg/m³, T₀=300 K, =3 300 s^-1, c_p=473 J/(kg?K), =11 000 s^-1 and β=0.9.

It can be found from Fig.2 that higher value of B leads to higher τ_max. As can be seen from Fig.2, higher γ_c is expected at higher B. Moreover, the γ_c—B curve is concave downward.

Fig.2(c) shows that lower B causes the profile of γ_p(y) to be steeper; lower B results in higher γ_{p, max} and lower γ_c, as also seen from Eqns.(8) and (9). From Fig.2(d), the profile of s_p(y) is more curved at lower B.

It is noted that the γ_c—B curve is omitted in Fig.2(b) when B＞1 GPa. It is found from numerical calculation that when B exceeds a critical value, γ_c decreases as B increases. Thus, for much higher B, γ_{p, min}(equal to γ_c) will be lower and γ_{p, max} will be higher. Moreover, the profile of γ_p(y) will be steeper and the distribution of s_p(y) will be more curved.

4.3 Effect of sensitive coefficient of strain rate

The effect of parameter C on τ—curve, γ_c, γ_p(y) and s_p(y) is shown in Fig.3 with l=3.18 ?m, B=500 MPa, A=800 MPa, m=1.03, n=0.26, T_m=1 790 K, ρ=7 850 kg/m³, T₀=300 K, =3 300 s^-1, c_p=473 J/(kg?K), =11 000 s^-1 and β=0.9.

Figs.3(a) and (b) show that higher τ_max is reached at higher value of C. γ_c slightly decreases as C increases so that the distributions of γ_p(y) and s_p(y) overlap for different values of C, as shown in Figs.3(c) and (d).

4.4 Effect of strain-hardening exponent

The effect of parameter n on τ—curve, γ_c, γ_p(y) and s_p(y) is shown in Fig.4 with l=3.18 ?m, B=500 MPa, A=800 MPa, m=1.03, C=0.014, T_m=1 790 K, ρ=7 850 kg/m³, T₀=300 K, =3 300 s^-1, c_p=473 J/(kg?K), =11 000 s^-1 and β=0.9.

Figs.4(a) and 4(b) reveal that the higher the parameter n is, the higher τ_max and γ_c are. The γ_c—n curve is approximately linear.

Fig.2 Effects of strain-hardening modulus B on flow shear stress—average plastic shear strain curve (a), critical plastic shear strain (b), local plastic shear strain (c) and deformation (d) in ASB

Fig.3 Effects of strain rate sensitive coefficient C on flow shear stress—average plastic shear strain curve (a), critical plastic shear strain (b), local plastic shear strain (c) and deformation (d) in ASB

Fig.4 Effects of strain-hardening exponent n on flow shear stress—average plastic shear strain curve (a), critical plastic shear strain (b), local plastic shear strain (c) and deformation (d) in ASB

From Figs.4(c) and 4(d), it can be seen that the effect of n on γ_p(y) and s_p(y) is similar to the influence of B (less than a critical value), as shown in Figs.2(c) and 2(d).

4.5 Effect of thermal-softening coefficient

The effect of parameter m on τ—curve, γ_c, γ_p(y) and s_p(y) is shown in Fig.5 with l=3.18 ?m, B=500 MPa, A=800 MPa, n=0.26, C=0.014, T_m=1 790 K, ρ=7 850 kg/m³, T₀=300 K, =3 300 s^-1, c_p=473 J/(kg?K), =11 000 s^-1 and β=0.9.

Fig.5(a) shows that higher m causes τ_max to be higher. Fig.5(b) reflects that γ_c is linearly increased with the increase of m.

It can be seen from Figs.5(c) and 5(d) that the effect of m on γ_p(y) and s_p(y) is very similar to the effects of n and B (less than a critical value), as shown in Figs.4(c) and (d), Figs.2(c) and (d), respectively.

4.6 Effect of melting point

The effect of parameter T_m on τ—curve, γ_c, γ_p(y) and s_p(y) is shown in Fig.6 with l=3.18 ?m, B=500 MPa, A=800 MPa, n=0.26, C=0.014, m=1.03, ρ=7 850 kg/m³, T₀=300 K, =3 300 s^-1, c_p=473 J/(kg?K), =11 000 s^-1 and β=0.9.

It can be seen from Figs.6((a)-(b)) that higher T_m leads to higher τ_max and γ_c.

It can be seen from Figs.6((c)-(d)) that the influence of T_m on γ_p(y) and s_p(y) is similar to the effects of m, n and B (less than a critical value), as can be seen from Figs.5((c)- (d)), Figs.4((c)-(d)) and Figs.2((c)-(d)), respectively.

4.7 Effects of heat capacity, density and work to heat conversion factor

The effect of parameter c_p on τ—curve, γ_c, γ_p(y) and s_p(y) is shown in Fig.7 with l=3.18 ?m, B=500 MPa, A=800 MPa, n=0.26, C=0.014, m=1.03, ρ=7 850 kg/m³, T₀=300 K, =3 300 s^-1, T_m=1 790 K, =11 000 s^-1 and β=0.9.

Eqn.(2) shows that ρ and c_p are denominator while β is numerator. Consequently, the influence of ρ and c_p is similar and the effects of c_p and β are opposite.

It is found from Figs.7 ((a)-(b)) that higher c_p leads to higher τ_max and γ_c.

Figs.7((a)-(b)) show that the influence of c_p on γ_p(y) and s_p(y) is similar to the effects of parameters T_m, m, n and B (less than a critical value), as it can be seen from Figs.6((c)-(d)), Figs.5((c)-(d)), Figs.4((c)-(d)) and Figs.2(c)-(d)), respectively.

Fig.5 Effects of thermal-softening exponent m on flow shear stress—average plastic shear strain curve (a), critical plastic shear strain (b), local plastic shear strain (c) and deformation (d) in ASB

Fig.6 Effects of melting point T_m on flow shear stress—average plastic shear strain curve (a), critical plastic shear strain (b), local plastic shear strain (c) and deformation (d) in ASB

Fig.7 Effects of thermal capacity c_p on flow shear stress—average plastic shear strain curve (a), critical plastic shear strain (b), local plastic shear strain (c) and deformation (d) in ASB

5 Conclusions

1) The peak local plastic shear strain is proportional to the average plastic shear strain, while it is inversely proportional to the critical plastic shear strain corresponding to the peak flow shear stress. The relative plastic shear deformation between the top and base of ASB is only dependent on the thickness of ASB and the average plastic shear strain.

2) Effects of constitutive parameters, such as static shear strength, strain-hardening modulus, strain- hardening exponent, strain rate sensitive coefficient, thermal-softening exponent, melting point, thermal capacity, mass density and work to heat conversion factor, on the flow shear stress—average plastic shear strain curve, the critical plastic shear strain, the distributions of local plastic shear strain and deformation in ASB are investigated by a detailed parametric study.

3) Higher static shear strength and work to heat conversion factor lead to lower critical plastic shear strain so that shear localization is more apparent at the same average plastic shear strain. Higher strain- hardening exponent, strain rate sensitive coefficient, melting point, thermal capacity and mass density result in higher critical plastic shear strain, leading to less apparent shear localization at the same average plastic shear strain. The strain rate sensitive coefficient has a minor influence on the critical plastic shear strain, the distributions of local plastic shear strain and deformation in ASB. At a critical value of strain-hardening modulus, the critical plastic shear strain is maximum, leading to the least apparent shear localization.

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(Edited by LI Xiang-qun)

Foundation item: Project(2004F052) supported by the Education Department of Liaoning Province, China

Corresponding author: WANG Xue-bin; Tel: +86-418-3351351; E-mail: wxbbb@263.net