The Constitutive Equation for Silicon and Its Use in Crystal Growth Modeling

1990 ◽  
Vol 112 (2) ◽  
pp. 183-187 ◽  
Author(s):  
C. T. Tsai ◽  
O. W. Dillon ◽  
R. J. De Angelis
Keyword(s):  
Crystal Growth ◽  
Dislocation Density ◽  
Constitutive Equation ◽  
Constitutive Model ◽  
Material Model ◽  
Internal Variable ◽  
Dislocation Motion ◽  
Silicon Crystals ◽  
Initial Dislocation Density ◽  
Temperature Strain

A stress analysis that describes the crystal growing process requires a material model that is valid over a wide temperature range and includes dislocation motion and multiplication. The stresses developed in the growing process could induce residual stresses, changes in dislocation density and buckling into the growing crystals. The dislocation density is introduced as an internal variable in the constitutive model. The stress-strain and dislocation density-strain characteristics of silicon crystals are discussed as a function of temperature, strain rate, and initial dislocation density.

Mechanics of shaped crystal growth from the melt

1996 ◽  
Vol 11 (9) ◽  
pp. 2163-2176 ◽  
Author(s):  
John C. Lambropoulos ◽  
Chien-Hsing Wu
Keyword(s):  
Crystal Growth ◽  
Dislocation Density ◽  
Growth Parameters ◽  
Growth Direction ◽  
Liquid Interface ◽  
Dislocation Generation ◽  
Crystalline Anisotropy ◽  
Initial Dislocation Density ◽  
Solid Liquid Interface ◽  
Solid Liquid

We present the numerical formulation of the thermal stress driven steady-state dislocation generation during the growth of shaped crystals from the melt, with Czochralski (CZ) growth of solid cylinder III–V compound semiconductors as an example. We use and compare the Haasen–Alexander model, coupling dislocation multiplication and creep strain rates, and the Jordan model, based on thermoelastic stresses. Growth parameters may be chosen so as to produce an overall approximately flat interface, leading to reduced dislocation density in the majority of the crystal's cross section. Calculation of final dislocation density requires the initial dislocation density and all stress components along the solid-liquid interface, microstructural features which depend on the physical processes leading to solidification. The final dislocation density is not sensitive to the initial dislocation density along the solid-liquid interface, but strongly depends on the interface stress. Significant stress relaxation at the interface is required to produce experimentally observed “W” shaped dislocation patterns. Crystal growth direction and crystalline anisotropy couple elastic (lattice) and plastic (slip systems) crystalline anisotropy.

A Constitutive Model for Jointed Rock Mass With Orthogonal Sets of Joints

1989 ◽  
Vol 56 (1) ◽  
pp. 25-32 ◽  
Author(s):  
E. P. Chen
Keyword(s):  
Constitutive Model ◽  
Material Model ◽  
Jointed Rock ◽  
Rate Equations ◽  
Strain Partitioning ◽  
Additional Material ◽  
The Subject ◽  
Nonlinear Normal ◽  
Elastic Rock ◽  
The Continuum

The development and numerical implementation of a constitutive model for jointed rock media is the subject of investigation in this paper. The constitutive model is based on the continuum assumption of strain-partitioning among the elastic rock matrix and joint sets with nonlinear normal and shear responses. Rate equations for the stress-strain response of the jointed media have been formulated. A numerical incremental solution scheme to these equations has been developed. It has been implemented into the finite element code JAC as an additional material model. Several sample problems have been solved for demonstration purposes. Interpretation and discussion of these results are presented.

Establishment of an ANSYS-Based Constitutive Modeling for Age Forming of Aluminum Alloy

2012 ◽  
Vol 217-219 ◽  
pp. 1497-1500 ◽  
Author(s):  
Xiao Jun Zuo ◽  
Jun Chu Li ◽  
Da Hai Liu ◽  
Long Fei Zeng
Keyword(s):  
Aluminum Alloy ◽  
Constitutive Equation ◽  
Material Model ◽  
Tensile Tests ◽  
Material Constants ◽  
Tensile Process ◽  
Simulation Based ◽  
Optimal Material ◽  
Two Stages ◽  
Good Agreement

Constructing accurate constitutive equation from the optimal material constants is the basis for finite element numerical simulation. To accurately describe the creep ageing behavior of 2A12 aluminum alloy, the present work is tentatively to construct an elastic-plastic constitutive model for simulation based on the ANSYS environment. A time hardening model including two stages of primary and steady-state is physically derived firstly, and then determined by electronic creep tensile tests. The material constants within the creep constitutive equations are obtained. Furthermore, to verify the feasibility of the material model, the ANSYS based numerical scheme is established to simulate the creep tensile process by using the proposed material model. Results show that the creep constitutive equation can better describe the deformation characteristics of materials, and the numerical simulations and experimental test points are in good agreement.

A dislocation density-based single crystal constitutive equation

2010 ◽  
Vol 26 (7) ◽  
pp. 925-938 ◽  
Author(s):  
M.G. Lee ◽  
H. Lim ◽  
B.L. Adams ◽  
J.P. Hirth ◽  
R.H. Wagoner
Keyword(s):  
Dislocation Density ◽  
Constitutive Equation ◽  
Single Crystal

A Dislocation Density Based Constitutive Model for Cyclic Deformation

1996 ◽  
Vol 118 (4) ◽  
pp. 441-447 ◽  
Author(s):  
Y. Estrin ◽  
H. Braasch ◽  
Y. Brechet
Keyword(s):  
Cyclic Loading ◽  
Dislocation Density ◽  
Constitutive Model ◽  
Constitutive Equations ◽  
Cyclic Deformation ◽  
Internal Variables ◽  
Density Evolution ◽  
Alloy 800H ◽  
Material Response ◽  
Dislocation Densities

A new constitutive model describing material response to cyclic loading is presented. The model includes dislocation densities as internal variables characterizing the microstructural state of the material. In the formulation of the constitutive equations, the dislocation density evolution resulting from interactions between dislocations in channel-like dislocation patterns is considered. The capabilities of the model are demonstrated for INCONEL 738 LC and Alloy 800H.

The Concept of Physical Metric in the Thermomechanical Modeling of Phase Transformations With Emphasis on Shape Memory Alloy Materials

2013 ◽  
Vol 135 (2) ◽  
Author(s):  
Vassilis P. Panoskaltsis ◽  
Lazaros C. Polymenakos ◽  
Dimitris Soldatos
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory ◽  
Phase Transformations ◽  
Shape Memory Effect ◽  
Memory Effect ◽  
Material Model ◽  
Internal Variable ◽  
Theoretical Interest ◽  
Numerical Examples ◽  
Thermomechanical Modeling

In this work we derive a new version of generalized plasticity, suitable to describe phase transformations. In particular, we present a general multi surface formulation of the theory which is capable of describing the multiple and interacting loading mechanisms, which occur during phase transformations. The formulation relies crucially on the consideration of the intrinsic material (“physical”) metric as a primary internal variable and does not invoke any decomposition of the kinematical quantities into elastic and inelastic (transformation induced) parts. The new theory, besides its theoretical interest, is also important for application purposes such as the description and the prediction of the response of shape memory alloy materials. This is shown in the simplest possible setting by the introduction of a material model. The ability of the model in simulating several patterns of the experimentally observed behavior of these materials such as the pseudoelastic phenomenon and the shape memory effect is assessed by representative numerical examples.

A Molecular Structure-Informed Viscoelastic Constitutive Model for Natural Rubber Materials

2021 ◽  
Author(s):  
Jiwon Jung ◽  
Chanwook Park ◽  
myungshin RYU ◽  
Gunjin Yun
Keyword(s):  
Molecular Structure ◽  
Constitutive Equation ◽  
Molecular System ◽  
Loss Modulus ◽  
Low Frequency ◽  
Viscoelastic Behavior ◽  
Material Model ◽  
Coarse Grained ◽  
Diffusivity Coefficient ◽  
Element Analysis

Abstract This paper presents a molecular structure-informed viscoelastic constitutive equation that adopts the Doi-Edward’s tube model with coarse-grained molecular dynamics (MD) simulation and primitive path analysis. Since this model contains polymer physics-related parameters directly obtained from molecular simulations, it can reflect molecular information in predictions of the viscoelastic behavior of elastomers, unlike other empirical models. The proposed incremental formulations and constitutive stiffness matrix were implemented into implicit finite element analysis (FEA) codes as a user-supplied material model and viscoelastic properties (storage, loss modulus, and tan⁡δ) were calculated from the constitutive equation. While obtaining polymer dynamics parameter of the molecular system, a relationship between self-diffusivity coefficient (D_c) and the polymerization degree of the polymer was confirmed. Furthermore, a series of parametric studies showed that increase of the primitive path length (L) and decrease of D_c have led to the strengthening of moduli and decrease of tan⁡δ peak. Moreover, under the same condition, the shift of tan⁡δ peak to low-frequency domain was observed, which implies a decline in free volume in the molecular system and an increase in the glass transition temperature.

Sign in / Sign up

Export Citation Format

Share Document