4 Microstructure and Mechanical Reliability Issues of TSV
97
isotropic material models based on empirical equations are used, even though it has
been known since 1934 that crystalline materials deform plastically by the slip of
dislocations on discrete slip systems [67–69]. However, a physically based CPFE
method is relevant in order to address the polycrystalline nature of Cu TSVs and
their complex deformation and stress relaxation mechanisms.
Early CPFE models used phenomenological constitutive equations and considered
dislocation slip as the only deformation mechanism [67]. In a phenomenological
constitutive model, a critical resolved shear stress, T
α
c , is used as a state variable for
each slip system α. The shear rate, ˙
γ α , is formulated as a function of the resolved
shear stress and the critical resolved shear stress:
˙
γ
α
= f (T
α
, T
α
c )
(4.3)
The evolution of the material state is then formulated as a function of the total
shear, γ , and the shear rate, ˙
γ
α :
T c = g(γ, ˙
γ )
(4.4)
The first CPFE simulations were performed in 1982 using a simplified setup of
two symmetric slip systems to study the tensile behavior of a single crystal [70].
The technique was later extended to a polycrystalline scenario using a 2D setup
with two or three slip systems [71, 72]. Simulations on a face-centered cubic (FCC)
crystal with 12 slip systems was reported in 1991 [73]. Applying the aforementioned
phenomenological constitutive laws to small-scale deformation, interface mechanics,
and twinning and/or deformation-induced phase transformations are often found
inadequate [65]. To address size effects, strain gradient theories [74] can be introduced into the CPFE framework. As strain gradients can be associated with geometrically necessary dislocations (GNDs), new internal-variable constitutive formulations were developed to incorporate dislocation densities as physically-based state
variables replacing the strain variables. This latter class of constitutive models also
allows the flexibility to incorporate additional metallurgical mechanisms such as
grain boundary mechanics or damage initiation into the model [67]. To deal with
additional deformation mechanisms such as occurring in TWIP (twinning-induced
plasticity) or TRIP (transformation-induced plasticity) steels, the CPFE framework
was further extended [75].
The result from a unidirectional compression on a copper single crystal using
the CPFE method is shown in Fig. 4.28 [76]. The simulation was implemented in
the Du¨sseldorf Advanced Material Simulation Kit (DAMASK) framework [77].
Although the mechanical boundary conditions and the material are both different
from the real Cu-TSV structures, the result highlights the usefulness of the CPFE
method in capturing atomistically-informed deformation and deformation induced
crystallographic orientation evolution. The body centered cubic (BCC) tungsten
single crystal subjected to uniaxial loading has also been recently studied using
the DAMASK framework [78].
97
isotropic material models based on empirical equations are used, even though it has
been known since 1934 that crystalline materials deform plastically by the slip of
dislocations on discrete slip systems [67–69]. However, a physically based CPFE
method is relevant in order to address the polycrystalline nature of Cu TSVs and
their complex deformation and stress relaxation mechanisms.
Early CPFE models used phenomenological constitutive equations and considered
dislocation slip as the only deformation mechanism [67]. In a phenomenological
constitutive model, a critical resolved shear stress, T
α
c , is used as a state variable for
each slip system α. The shear rate, ˙
γ α , is formulated as a function of the resolved
shear stress and the critical resolved shear stress:
˙
γ
α
= f (T
α
, T
α
c )
(4.3)
The evolution of the material state is then formulated as a function of the total
shear, γ , and the shear rate, ˙
γ
α :
T c = g(γ, ˙
γ )
(4.4)
The first CPFE simulations were performed in 1982 using a simplified setup of
two symmetric slip systems to study the tensile behavior of a single crystal [70].
The technique was later extended to a polycrystalline scenario using a 2D setup
with two or three slip systems [71, 72]. Simulations on a face-centered cubic (FCC)
crystal with 12 slip systems was reported in 1991 [73]. Applying the aforementioned
phenomenological constitutive laws to small-scale deformation, interface mechanics,
and twinning and/or deformation-induced phase transformations are often found
inadequate [65]. To address size effects, strain gradient theories [74] can be introduced into the CPFE framework. As strain gradients can be associated with geometrically necessary dislocations (GNDs), new internal-variable constitutive formulations were developed to incorporate dislocation densities as physically-based state
variables replacing the strain variables. This latter class of constitutive models also
allows the flexibility to incorporate additional metallurgical mechanisms such as
grain boundary mechanics or damage initiation into the model [67]. To deal with
additional deformation mechanisms such as occurring in TWIP (twinning-induced
plasticity) or TRIP (transformation-induced plasticity) steels, the CPFE framework
was further extended [75].
The result from a unidirectional compression on a copper single crystal using
the CPFE method is shown in Fig. 4.28 [76]. The simulation was implemented in
the Du¨sseldorf Advanced Material Simulation Kit (DAMASK) framework [77].
Although the mechanical boundary conditions and the material are both different
from the real Cu-TSV structures, the result highlights the usefulness of the CPFE
method in capturing atomistically-informed deformation and deformation induced
crystallographic orientation evolution. The body centered cubic (BCC) tungsten
single crystal subjected to uniaxial loading has also been recently studied using
the DAMASK framework [78].