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Figure 1 Multiplicative decomposition of the deformation gradient

The volumetric inelastic deformation gradient __F__^{p}_{v}, is assumed to have the form __F__^{p}_{v} = Φ__I,__ where Φ is a function to be determined from kinematics (or conservation of mass).

The Jacobian of Eq. (6.1) is related to the change in volume or change in density for constant mass as

$$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$$

where Φ is the damage variable.

The velocity gradient associated with the deformation gradient

$$L=\mathrm{det}F$$

where

$$D=\mathrm{det}F$$

and

$D=\frac{1}{2}\mathrm{det}F$

The deformation rate D is additively decomposed into an elastic part and plastic part (volumetric and deviatoric).

$D=\mathrm{det}F$

The plastic volumetric rate of deformation is defined as

$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$

Also note here that__F__^{p}_{v} vanishes. The trace of the volumetric part (Eq. (6.9)) is given as

$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$

The formulation of damage Φ is described in terms of void nucleation and void growth in the unstressed intermediate configuration:

$$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$$

Where v_{v} is the void average volume and η the number of voids per unit volume or void nucleation density in the intermediate configuration

### Void Nucleation, Growth, and Coalescence Aspects

The process of ductile fracture of most metals and alloys occurs mainly due to the void nucleation, growth and finally coalescence into a micro-crack (Figure 2). If the extent of void growth up to fracture is small, it is possible to ignore its effects on the constitutive equations. However, a realistic model on ductile fracture prediction must include void nucleation, void growth and a condition for void coalescence.

The Jacobian of Eq. (6.1) is related to the change in volume or change in density for constant mass as

$$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$$

where Φ is the damage variable.

The velocity gradient associated with the deformation gradient

$$L=\mathrm{det}F$$

where

$$D=\mathrm{det}F$$

and

$D=\frac{1}{2}\mathrm{det}F$

The deformation rate D is additively decomposed into an elastic part and plastic part (volumetric and deviatoric).

$D=\mathrm{det}F$

The plastic volumetric rate of deformation is defined as

$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$

Also note here that

$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$

The formulation of damage Φ is described in terms of void nucleation and void growth in the unstressed intermediate configuration:

$$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$$

Where v

Figure 2 – Sequences in a ductile fracture mechanism.

The damage formulation is shown conceptually in Figure 3. The number density of voids can change and growth of voids can occur independently or simultaneously.

Figure 3 – Schematic of fictitious material with increasing void nucleation density and void growth to be captured conceptually in a fracture prediction model.

$$J=\mathrm{det}FeFee=\frac{1}{1=\mathrm{\Phi}}$$

(A.30) where η(t) is the void nucleation density, ε is the strain at time t, Ccoeff is a material constant. T is temperature in the absolute scale, and CTh is the temperature dependent material constant determined from experiments. The volume fraction of the second phase material is f, the average silicon particle size is d, and the bulk fracture toughness is KIC. The material parameters a, b, and c relate to the volume fraction of nucleation events arising from local microstresses in the material. These constants are determined experimentally from tension, compression, and torsion tests in which the number density of void sites is measured at different strain levels. The stress state dependence (compression, tension, and torsion) on damage evolution is captured in Eq. (6.17) by using the three stress invariants denoted by I1, J2, and J3, respectively [Horstemeyer and Gokhale, 1999].

$${\u1f48}^{\dot{}}=\frac{\sqrt{3}\phantom{\rule{0.1em}{0ex}}{R}_{0}}{2(1-n)}[{\mathrm{sinh}}^{}(\frac{\sqrt{3}(1-n)}{\sqrt{2}I{1}_{}}\frac{3}{\sqrt{J{2}_{}}})]{\u1f55}^{\dot{}}$$

In Eq. (6.18) the void volume grows as the strain and/or stress triaxiality increases. The material constant n is related to the strain hardening exponent and is determined from the tension tests. R0 is taken to be the initial radius of the voids.

Another void growth model in terms of void radius is given by Rice and Tracey [1969] is given byAnother void growth model by Budiansky et al. [1982] in terms of void volume rate is given by

$${r}^{\dot{}}=0.283{R}_{0}{\mathrm{exp}}^{\frac{3}{2}}(\frac{\sqrt{2}}{3}){\u1f55}^{\dot{}}$$

The last one worth mentioning in this context is that by Cocks and Ashby [1980] given in terms of the void volume fraction rate,

$${\varphi}^{\dot{}}=[\frac{1}{{(1-\varphi )}^{\mathrm{\u215f}m}}-(1-\varphi )sinh[\frac{2(2-m)}{2(2-m)+1}\frac{\sqrt{2}}{3}]{\u1f55}^{\dot{}}$$

Figure 4 – The damage model encompasses the limiting cases shown by (a) just void nucleation in and (b) a single void growth.

$$C=[C{\mathrm{D1}}_{}+C{\mathrm{D2}}_{}\eta v](\frac{{\mathrm{DCS}}_{0}}{\mathrm{DCS}}){z}^{}(\mathrm{TC}\mathrm{TC})$$

which gives $$\varphi =(\nu v+{\varphi}_{\mathrm{pore}})C$$

Figure 5 – Two different coalescence mechanisms observed in various materials.

In the limiting case when the function in Eq. (A.35) equals zero, simple coalescence occurs. In this case, the model reflects the growing of two voids into one. When , microvoid linking is reflected and the rate of damage is increased [Garrison and Moody, 1987].
The dependence on dendrite cell size (DCS) is based on the work of Major et al. [1994], who noted a dependence of the elongation to failure with DCS.
The constant CTC is characterizes the temperature dependence of the coalescence term based on a multitude of mesomechanical simulations.

### Visco-Elastoplasticity Aspects

The internal state variable (ISV) plasticity model [Bammann et al., 1993] is modified to account for stress state dependent damage evolution.
**Elastic Relation**

The elastic relation is given by,

$${|}_{\sigma}={(1-\phi )}^{\xi}{\u1d60}^{e}:{\epsilon}_{e}=\lambda (1-\phi ){\xi}^{}\mathrm{tr}({\epsilon}_{e}){\mathit{A}}_{2}+2\mu (1-\phi ){\xi}^{}{\epsilon}_{e}$$

Where σ is the stress tensor, ε^{e} is the elastic strain, C^{e} is the elastic stiffness, and the elastic Lame constants, and ξ a material parameter.
**Flow Rule**

The deviatoric inelastic flow rule, ${\bm{j}}_{\mathrm{in}}$, encompasses the regimes of creep and plasticity and is a function of the temperature $T$, the kinematic hardening internal state variable $T$, the isotropic hardening internal state variable $\kappa $, the volume fraction of damaged material $\varphi $, and the functions $\mathrm{F(T)}$, $\mathrm{V(T)}$, and $\mathrm{Y(T)}$, which are related to yielding with Arrhenius-type temperature dependence (Table A.7).

The flow rule is given by:

$${\bm{j}}_{\mathrm{in}}=f(T)\mathrm{sinh}(\frac{\Vert {\Sigma}_{\prime}\Vert -{\mathit{\alpha}}_{\kappa}+Y(T))}{(1-\varphi )V(T)}\u1d45$$

where ${\Sigma}_{\prime}$ denotes the deviatoric stress tensor, and $n$ the plastic normal tensor defined as

$$n=\frac{\Vert {\Sigma}_{\prime}\Vert -{\alpha}_{\epsilon}}{\Vert}$$

The plastic flow rule can be inverted to

$$\frac{\Vert}{{\Sigma}_{\prime}}(1-\varphi )\u2013Y(T)\u2013{\kappa}_{\epsilon}\u2013V\mathrm{sinh}\u207b1(D\u0304{\mathrm{in}}^{\dot{}})\u2013f(T)=0$$

This constitutive relation clearly shows the decomposition of the stress $\Vert {\Sigma}_{\prime}\Vert $ as the sum of the yield stress $Y$, the isotropic hardening $\kappa $, the kinematic hardening $\Vert {\alpha}_{\epsilon}\Vert $, and a viscosity term, often called the viscous stress (Figure 6).

$${|}_{\sigma}={(1-\phi )}^{\xi}{\u1d60}^{e}:{\epsilon}_{e}=\lambda (1-\phi ){\xi}^{}\mathrm{tr}({\epsilon}_{e}){\mathit{A}}_{2}+2\mu (1-\phi ){\xi}^{}{\epsilon}_{e}$$

Where σ is the stress tensor, ε

The flow rule is given by:

$${\bm{j}}_{\mathrm{in}}=f(T)\mathrm{sinh}(\frac{\Vert {\Sigma}_{\prime}\Vert -{\mathit{\alpha}}_{\kappa}+Y(T))}{(1-\varphi )V(T)}\u1d45$$

where ${\Sigma}_{\prime}$ denotes the deviatoric stress tensor, and $n$ the plastic normal tensor defined as

$$n=\frac{\Vert {\Sigma}_{\prime}\Vert -{\alpha}_{\epsilon}}{\Vert}$$

The plastic flow rule can be inverted to

$$\frac{\Vert}{{\Sigma}_{\prime}}(1-\varphi )\u2013Y(T)\u2013{\kappa}_{\epsilon}\u2013V\mathrm{sinh}\u207b1(D\u0304{\mathrm{in}}^{\dot{}})\u2013f(T)=0$$

This constitutive relation clearly shows the decomposition of the stress $\Vert {\Sigma}_{\prime}\Vert $ as the sum of the yield stress $Y$, the isotropic hardening $\kappa $, the kinematic hardening $\Vert {\alpha}_{\epsilon}\Vert $, and a viscosity term, often called the viscous stress (Figure 6).

Figure 6 – Decomposition of the stress tensor into a yield, isotropic hardening, kinematic hardening and viscous stresses.

The first form of yield surface evolution is called isotropic hardening κ, which reflects the effect of the global dislocation density. For isotropic hardening, the yield surface grows in size while the center remains at a fixed point in stress space (Figure 7).

The second form of yield surface evolution is called kinematic hardening α, also called Bauschinger effect (Figure 7), which reflects the effect of anisotropic dislocation density. For kinematic hardening, the center of the yield surface translates in stress space, while the size remains fixed. For both isotropic and kinematic hardening, the orientation of the yield surface remains fixed.

The third type of yield surface evolution is called mixed hardening where both isotropic and kinematic hardening characteristics are evident. For mixed hardening, the orientation (not considered here) of the yield surface may also change as well.

Although isotropic hardening is the most common form of yield surface evolution assumed in finite element models for metal forming simulation, it is not necessarily the most accurate. The mixed hardening model is most likely the most accurate of the three models.

The evolution of the kinematic hardening is defined by:

$${\alpha}^{\u0307}=\{h(T)\phantom{\rule{thinmathspace}{0ex}}D{\mathrm{in}}^{\u0302}-[\sqrt{\frac{2}{3}}\phantom{\rule{thinmathspace}{0ex}}r{d}_{\u201d}(T)\Vert \Vert D{\mathrm{in}}^{\u0302}\Vert \Vert +r{s}_{\u201d}(T)\Vert \alpha \Vert (\alpha )+\}[(\mathrm{DCS}{0}_{\u201d})/\mathrm{DCS}]{(}^{z}$$

The evolution of the isotropic hardening is defined by:

$${\kappa}^{\u0307}=\{H(T)\phantom{\rule{thinmathspace}{0ex}}D{\mathrm{in}}^{\u0302}-[\sqrt{\frac{2}{3}}\phantom{\rule{thinmathspace}{0ex}}R{d}_{\u201d}(T)\Vert \Vert D{\mathrm{in}}^{\u0302}\Vert \Vert +R{s}_{\u201d}(T)\Vert \kappa {2}^{\u0302}\}[(\mathrm{DCS}{0}_{\u201d})/\mathrm{DCS}]{(}^{z}$$

Figure 7 – Evolution of the yield surface in case of (a) isotropic hardening and (b) kinematic hardening.

Mechanism | Description | Term Definition |
---|---|---|

Yield Stress | Rate-Independent Yield Stress | $Y(T)={C}_{3}{e}^{/T}={C}_{3}{e}^{/T}$ |

Magnitude of Rate-Dependence on yielding | $Y(T)={C}_{3}{e}^{/T}={C}_{3}{e}^{/T}$ | |

Rate-Dependence on Initial Yielding | $V(T)={C}_{1}{e}^{/T}={C}_{1}{e}^{/T}$ | |

Kinematic Hardening | Modulus | $f(T)={C}_{5}{e}^{/T}={C}_{5}{e}^{/T}$ |

Dynamic Recovery | $f(T)={C}_{5}{e}^{/T}={C}_{5}{e}^{/T}$ | |

Static Recovery | ${r}_{d}(T)={C}_{7}(1+{C}_{19}[\frac{4}{27}-\frac{{J}_{3}^{2}}{{J}_{2}^{3}}]){e}^{/T}={C}_{7}(1+{C}_{19}[\frac{4}{27}-\frac{{J}_{3}^{2}}{{J}_{2}^{3}}]){e}^{/T}$ | |

Isotropic Hardening | Modulus | ${r}_{s}(T)={C}_{11}({e}^{/T}{C}_{12})={C}_{11}({e}^{/T}{C}_{12})$ |

Dynamic Recovery | $H=\{({C}_{15}(1+{C}_{22}[\frac{4}{27}-\frac{{J}^{3}}{{\U0001f4a4}^{2}}])\}-{C}_{16}T$ | |

Static Recovery | ${R}_{s}(T)={C}_{17}{e}^{/T}={C}_{17}{e}^{/T}$ |

Component Definition | SDV |
---|---|

Kinematic Hardening b11 | Statev(1) |

Kinematic Hardening b22 | Statev(2) |

Kinematic Hardening b33 | Statev(3) |

Kinematic Hardening b12 | Statev(4) |

Kinematic Hardening b13 | Statev(5) |

Kinematic Hardening b23 | Statev(6) |

Isotropic Hardening κ | Statev(7) |

Temperature | Statev(8) |

Effective Plastic Strain | Statev(9) |

McClintock Void Growth (second phase pores) | Statev(10) |

Rate of Change of M Porosity | Statev(11) |

Stress Triaxiality | Statev(12) |

Nucleation | Statev(13) |

Damage | Statev(14) |

Nucleation Rate | Statev(15) |

Damage Rate | Statev(16) |

Nucleation from previous time step | Statev(17) |

Cocks-Ashby (CA) Void Growth (large pores) | Statev(18) |

Rate of Change of CA Porosity | Statev(19) |

Kinematic Hardening α11 | Statev(20) |

Kinematic Hardening α22 | Statev(21) |

Kinematic Hardening α33 | Statev(22) |

Kinematic Hardening α12 | Statev(23) |

Kinematic Hardening α13 | Statev(24) |

Kinematic Hardening α23 | Statev(25) |

blank | Statev(26) |

Constant Definition | PROPS | Constant Definition | PROPS |
---|---|---|---|

Shear Modulus G0 | Props(1) | Initial Temperature T0 | Props(28) |

a | Props(2) | Heat Generation Coefficient | Props(29) |

Bulk Modulus K0 | Props(3) | McClintock Damage Constant n | Props(30) |

b | Props(4) | Initial Void Radius | Props(31) |

Melting Temperature Tmelt | Props(5) | Torsional constant a (nucleation) | Props(32) |

BCJ Plasticity Parameter c01 | Props(6) | Tens/Comp constant b (nucleation) | Props(33) |

BCJ Plasticity Parameter c02 | Props(7) | Triaxiality constant c (nucleation) | Props(34) |

BCJ Plasticity Parameter c03 | Props(8) | Coeff. constant c (nucleation) | Props(35) |

BCJ Plasticity Parameter c04 | Props(9) | Fracture Toughness KIC (nucleation) | Props(36) |

BCJ Plasticity Parameter c05 | Props(10) | Average Particle Size (nucleation) | Props(37) |

BCJ Plasticity Parameter c06 | Props(11) | Particle Volume Fraction (nucleation) | Props(38) |

BCJ Plasticity Parameter c07 | Props(12) | Nearest Neighbor distance (coal.) | Props(39) |

BCJ Plasticity Parameter c08 | Props(13) | Chi Exponent (coalescence) | Props(40) |

BCJ Plasticity Parameter c09 | Props(14) | Reference Dendrite Cell size DCS0 | Props(41) |

BCJ Plasticity Parameter c10 | Props(15) | Dendrite Cell size DCS | Props(42) |

BCJ Plasticity Parameter c11 | Props(16) | Dendrite Cell size Exponent zz | Props(43) |

BCJ Plasticity Parameter c12 | Props(17) | Initial Void Volume Fraction for CA | Props(44) |

BCJ Plasticity Parameter c13 | Props(18) | BCJ Plasticity Parameter c21 | Props(45) |

BCJ Plasticity Parameter c14 | Props(19) | BCJ Plasticity Parameter c22 | Props(46) |

BCJ Plasticity Parameter c15 | Props(20) | BCJ Plasticity Parameter c23 | Props(47) |

BCJ Plasticity Parameter c16 | Props(21) | BCJ Plasticity Parameter c24 | Props(48) |

BCJ Plasticity Parameter c17 | Props(22) | BCJ Plasticity Parameter c25 | Props(49) |

BCJ Plasticity Parameter c18 | Props(23) | BCJ Plasticity Parameter c26 | Props(50) |

BCJ Plasticity Parameter c19 | Props(24) | Nucleation Temperature Dependence | Props(51) |

BCJ Plasticity Parameter c20 | Props(25) | Elastic Modulus Porosity Exponent β | Props(52) |

BCJ Plasticity Parameter ca | Props(26) | Flag to use vvfr.dat file (0=no,1=yes) | Props(53) |

BCJ Plasticity Parameter cb | Props(27) | Cacon | Props(54) |

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