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Research Papers

# Overview of Enhanced Continuum Theories for Thermal and Mechanical Responses of the Microsystems in the Fast-Transient Process

[+] Author and Article Information

Boyd Professor
Computational Solid Mechanics Laboratory,
Department of Civil and
Environmental Engineering,
Louisiana State University,
Baton Rouge, LA 70803

Danial Faghihi

Computational Solid Mechanics Laboratory,
Department of Civil and
Environmental Engineering,
Louisiana State University,
Baton Rouge, LA 70803

Referred to quasi-conservative by Ziegler and Wehrli [53].

While in the context of strain gradient theory, the stored energy can be due to both homogeneous and nonhomogenous plastic deformation—in spirit of the conventional plasticity—the term “gradient independent stored energy of cold work” refers to the stored energy connected with the homogenous plastic deformation in this paper.

According to Gurtin [31] “the plastic spin vanishes identically when the free energy is independent of curl of plastic strain, but not generally otherwise”. He noted that sufficiently far from boundaries solutions should not be affected by plastic spin, but close to microscopically hard boundaries, boundary layers characterized by a large Burgers vector and large plastic spin should form.

Using direct notation, Eq. (6) reads as $α=(curl ɛp)T$. It should be noted that the transpose of $α$ is referred to as Burgers tensor, $G$, in the works of Gurtin (e.g., [31]), therefore: $G=(curl ɛp)$.

The examples of such micro-free and micro-clamped boundary conditions can be found in thin films with unpassivated and passivated surfaces (e.g., [167]).

In this formulation the term “elastic energy” is used to imply the recoverable energy associated with the stretching of the atomic bonds in a crystal lattice. The term “defect energy” is used to denote the energy stored and dissipated in the lattice due to the presence of defects such as dislocations (both SSDs and GNDs associated with plastic strain and gradient of plastic train) and entanglements.

It should be noted that TY1 is assumed to be the melting temperature of the specific material by some other researchers in order not to introduce an additional material parameter. However, a general constant is included here as a normalizing parameter that needs to be calibrated against careful experimental data.

Gurtin and Reddy [70] showed that “the classical isotropic hardening rule, which is dissipative in nature, may equally well be characterized via a defect energy and, what is striking, this energetically based hardening rule mimics dissipative behavior by describing loading processes that are irreversible.” As it will be shown later in this paper, such manipulation is taken into consideration here in order to derive the plastic strain gradient independent stored energy of cold work with no additional material parameters that required to be calibrated against experimental data [156].

Specifically, densities of edge and screw GNDs according to Gurtin (e.g., [30, 31]). It should be noted that the kinematic hardening in the current theory does not account for certain interval of the plastic strain during the loading reversal where dislocation density remains constant (Hasegawa et al., 1975). This is because at the onset of reverse loading, the dislocation structure is first annihilated and then rebuilt in the slip systems that correspond to the reverse direction [154].

Such correlation between GNDs will quickly decrease to zero within a few dislocation distances [173].

Starting from Eq. (17) and considering virtual variations of the temperature and entropy states—which obeys the energy equation—the relation between second-order variation of $Ψ$ and $e$ can be derived such as $∂2Ψ/∂T2(δT)2=-∂2e/∂s2(δs)2$. From this relation, one can conclude that at the state where $e$ is a convex function of entropy, $Ψ$ is a concave function of temperature $∂2e/∂s2>0=>∂2Ψ/∂T2<0$ [175]. Moreover, according to [176] one can assume that entropy is a monotonically increasing function of temperature, thus: $∂s/∂T=-∂2Ψ/∂T2$.

Conventionally, dislocations impinging transversely on a slip plane are termed forest dislocations and are thought to be responsible for the second stage of hardening [177,178].

e.g., [179] and [180] for an inclusive treatment of the configurational (material) forces.

Assuming the interaction between only two GNDs (or pile-ups), this stress acts on them in the opposite direction of slip.

As it is discussed by Bardella [160], ensuring $m1>0$ and $m2>0$ not only preserve the convexity of $Daccp$ and $Daccg$, respectively, but also abolishes the requirement to implement any yield criterion in the moving elastic–plastic boundaries (i.e., imposing any higher-order boundary condition at the internal surfaces between elastic and plastic domains).

For further details regarding the effect of the interaction coefficient value, see Dahlberg and Faleskog [187].

According to the second paragraph after Eq. (39).

Both Kuroda and Tvergaard [195] and Gurtin and Ohno [85] discussed the crystal plasticity case which can also be applied to the continuum plasticity.

Gurtin [204]—Eq. (3.14), p. 647.

Where $ΨI$ is convex with respect to $ɛijpI$.

Corresponding to the condition that dislocations are nucleated within the bulk and did not reach the interface yet.

Since it is not possible to program a priori the unloading point exactly at the onset of the strain burst, hence there are no experimental observations to prove the recoverable nature of this energy. However, this can be interpreted as the vanishing of the repulsive stress of pile-up by removing the applied load.

Obviously, the calibrated parameters will be more accurate by means of a 3D finite element modeling of the indentation using the gradient theory proposed by Faghihi and Voyiadjis [156]. However, such a calibration along with implementation of the theory to 3D finite element model is far too complex for practical purposes.

Detailed set of indentation tests at various temperatures and strain rates are required to calibrate the full model parameters which does not currently exist in the literature. The experiments at elevated temperatures are more challenging since specialized equipment and sample preparation is necessary and some indenter parts are not vacuum-compatible. One may use the hardening-softening observed in the experiments (e.g., Voyiadjis et al. [247] and Faghihi and Voyiadjis [206] and the references herein) without accounting for the exact distance between the indenter and the grain boundary to determine the average rate and temperature dependent grain boundary parameters of the material.

The unloading (i.e., reverse straining) is only applied for the stress–strain curves and the other results are extracted at the end of loading.

This agrees with the experimental observations of Xiang and Vlassak [20] for the passivated thin film.

The thermal softening will be more pronounced in the absence of the energetic length scale. However, a large value of the backstress is considered here in order to investigate the capability of the model to address the effect of temperature rise on overcoming the kinematic hardening mechanisms.

$r=1$ is considered for the results presented in this section.

In order to emphasize the variation, for the results of the stress-strain curves extracted by assuming $Y=YI=20 (MPa)$ and 0.03% applied stress.

It should be noted that since the experiments are conducted under the low strain rate, the isothermal condition is preserved for the obtained results. For simplicity, the effect of the gradient independent stored energy of cold work is not incorporated and the parameters $TY1$ and $TY2$ are assumed to be the Al melting temperature.

The average grain sizes of the films with $1.00$, $1.90$, and $4.20 μm$ are $1.5 ± 0.05$, $1.51 ± 0.04$, and $1.5 ± 0.05 μm$, respectively.

A well-adhered passivating layer prevents dislocation from exiting the film and results in significant plastic strain gradient. The plastic flow constrain due to the presence of the passivation layer, cannot be described neither by the classical plasticity nor the first-order gradient theories.

1Corresponding author.

2Present address: Institute for Computational Engineering and Science, The University of Texas at Austin, Austin, TX 78712.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received July 17, 2014; final manuscript received July 28, 2014; published online August 22, 2014. Assoc. Editor: Mohammed Zikry.

J. Eng. Mater. Technol 136(4), 041003 (Aug 22, 2014) (36 pages) Paper No: MATS-14-1145; doi: 10.1115/1.4028121 History: Received July 17, 2014; Revised July 28, 2014

## Abstract

The recently growing demand for production and applications of microscale devices and systems has motivated research on the behavior of small volume materials. The computational models have become one of great interests in order to advance the manufacturing of microdevices and to reduce the time to insert new product in applications. Among the various numerical and computational techniques, still the approaches in the context of continuum theories are more preferable due to their minimum computational cost to simulation on realistic time and material structures. This paper reviews the methods to address the thermal and mechanical responses of microsystems. The focus is on the recent developments on the enhanced continuum theories to address the phenomena such as size and boundary effects as well as microscale heat transfer. The thermodynamic consistency of the theories is discussed and microstructural mechanisms are taken into account as physical justification of the framework. The presented constitutive model is calibrated using an extensive set of microscale experimental measurements of thin metal films over a wide range of size and temperature of the samples. An energy based approach is presented to extract the first estimate of the interface model parameters from results of nanoindentation test.

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## Figures

Fig. 1

Illustration of phonon interactions and scattering mechanisms of free electrons within metal lattice

Fig. 2

Schematic illustration of grain boundary separating grains A and B along with single slip system at each grain and the misalignment angles [122,127]

Fig. 4

Expansion of the plastic zone under the indenter and the effect of grain boundary. According to the figure: (a) nucleation of dislocations (point E); (b) expansion of plastic zone with no grain boundary influence (line EA); (c) dislocation pile-up at the interface (line AB); and (d) dislocation transmission to the adjacent grain (line BC) [127].

Fig. 3

A schematic nano-indentation response near the grain boundary. The solid line represents the test far from the grain boundary (grain interior response) and the dashed line denotes the response when indentation is conducted near the grain boundary (L < 1 μm) [122,127].

Fig. 5

(a) Qualitative stress–strain response from 1D solution of the Faghihi [127] model and (b) one-dimensional model of grain and grain boundary

Fig. 6

One-dimensional model for a single crystal bounded by two grain boundaries

Fig. 7

Size effect due to the dissipative length scale when the energetic backstress vanishes (i.e., ℓen→0): (a) plastic strain distribution across the grain; (b) stress–strain; (c) temperature distribution across the grain; and (d) evolution of temperature with plastic strain at the midpoint of the grain [122]

Fig. 8

Effect of isotropic hardening constants on the stress–strain responses when the energetic backstress vanishes (i.e., ℓen→0): (a) hardening parameter, h and (b) hardening power, r

Fig. 9

Size effect due to the energetic length scale when the dissipative strengthening vanishes (i.e., ℓdis→0): (a) plastic strain distribution across the grain; (b) stress–strain; (c) temperature distribution across the grain; and (d) effect of backstress power, a, on the stress–strain responses (r = 1) [122,127]

Fig. 10

Effect of temperature and strain rate power (n and m) on ((a) and (c)) stress–strain responses; ((b) and (d)) evolution of temperature with plastic strain at the midpoint of the grain (ℓen/L = 0.1 and ℓdis/L = 1)

Fig. 11

Effect of thermal parameter (TY) and initial temperature (T0) on ((a) and (c)) stress–strain responses; ((b) and (d)) evolution of temperature with plastic strain at the midpoint of the grain (ℓen/L = 0.1, ℓdis/L = 1 for all curves, m = 0.05 and strain rate of 2000/s are considered for (c) and (d)).

Fig. 14

Grain size effect in case of nonvanishing energetic and dissipative interfacial length scales: (a) plastic strain distribution across the grain and (b) stress–strain behavior [122]

Fig. 15

Effect of interfacial thermal parameters (nI and TYI) on stress–strain responses and plastic strain distribution across the grain (ℓen/L = ℓenI/L = 0.1 and ℓdis/L = ℓdisI/L = 1)

Fig. 16

Effect of initial temperature T0 on: (a) plastic strain distribution across the grain and (b) stress–strain responses (ℓen/L = ℓenI/L = 0.1, ℓdis/L = ℓdisI/L = 1, nI = 0.1)

Fig. 12

Size effect due to the energetic interfacial length scale when energy dissipation vanishes (i.e., ℓdisI,ℓdis→0). The interface is considered as compliant (ℓenI/ℓen = 0.1) in the left column and stiff (ℓenI/ℓen = 10) in the right column: ((a) and (b)) stress–strain; ((c) and (d)) plastic strain distribution across the grain; ((e) and (f)) temperature distribution across the grain (127).

Fig. 17

Model predictions and experimental measurements of (a) the film thickness effect on the stress–strain curves of sputter-deposited Al films (experimental data taken from Ref. [255]) and (b) the temperature effect on the stress–strain curves of sputter-deposited Ni films (experimental data taken from Refs. [256,122])

Fig. 13

Size effect due to the energetic interfacial length scale when energy storage vanishes (i.e., ℓenI,ℓen→0). The interface is considered as compliant (ℓdisI/ℓdis = 0.01), intermediate (ℓdisI/ℓdis = 0.2), and stiff (ℓdisI/ℓdis = 1.5): ((a) and (b)) stress–strain; ((c) and (d)) plastic strain distribution across the grain; ((e) and (f)) temperature distribution across the grain.

Fig. 18

Model predictions and experimental measurements of the film thickness effect on the stress–strain curves of electroplated Cu films with both surface passivated by 20 nm of Ti (experimental data taken from Refs. [21,156])

Fig. 19

Model predictions and experimental measurements of the actual yield stress of the electroplated Cu films as a function of inverse film thickness (experimental data taken from Refs. [21,156])

Fig. 20

Distribution of (a) plastic strain and (b) plastic strain gradient, across the film thickness [156]

Fig. 21

Model predictions and experimental measurements of the surface conditions (passivated-freestanding) on the stress–strain curves of sputter-deposited Al films (experimental data taken from Refs. [20,156])

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