scholarly journals Full elastic strain and stress tensor measurements from individual dislocation cells in copper through-Si vias

IUCrJ ◽  
2015 ◽  
Vol 2 (6) ◽  
pp. 635-642 ◽  
Author(s):  
Lyle E. Levine ◽  
Chukwudi Okoro ◽  
Ruqing Xu
Keyword(s):  
Stress Tensor ◽  
Elastic Strain ◽  
Accurate Determination ◽  
Strain Tensor ◽  
Dislocation Cell ◽  
Sample Volume ◽  
Dislocation Cells ◽  
Individual Dislocation ◽  
Individual Grains ◽  
Nondestructive Measurements

Nondestructive measurements of the full elastic strain and stress tensors from individual dislocation cells distributed along the full extent of a 50 µm-long polycrystalline copper via in Si is reported. Determining all of the components of these tensors from sub-micrometre regions within deformed metals presents considerable challenges. The primary issues are ensuring that different diffraction peaks originate from the same sample volume and that accurate determination is made of the peak positions from plastically deformed samples. For these measurements, three widely separated reflections were examined from selected, individual grains along the via. The lattice spacings and peak positions were measured for multiple dislocation cell interiors within each grain and the cell-interior peaks were sorted out using the measured included angles. A comprehensive uncertainty analysis using a Monte Carlo uncertainty algorithm provided uncertainties for the elastic strain tensor and stress tensor components.

scholarly journals On continuum damage mechanics

2019 ◽  
Vol 52 (3) ◽  
pp. 125-147
Author(s):  
Kari Juhani Santaoja
Keyword(s):  
Stress Tensor ◽  
Effective Stress ◽  
Elastic Strain ◽  
Damage Mechanics ◽  
Volume Fraction ◽  
Strain Tensor ◽  
Continuum Damage Mechanics ◽  
Continuum Damage ◽  
The Matrix ◽  
Elastic Strain Tensor

A material containing spherical microvoids with a Hookean matrix response was shown to take the appearance usually applied in continuum damage mechanics. However, the commonly used variable damage D was replaced with the void volume fraction f , which has a clear physical meaning, and the elastic strain tensor \Bold {ε}^e with the damage-elastic strain tensor \Bold {ε}^{de}. The postulate of strain equivalence with the effective stress concept was reformulated and applied to a case where the response of the matrix obeys Hooke’s law. In contrast to many other studies, in the derived relation between the effective stress tensor \Bold {\Tilde{σ}} and the stress tensor \Bold {σ}, the tensor \Bold {\Tilde{σ}} is symmetric. A uniaxial bar model was introduce for clarifying the derived results. Other candidates for damage were demonstrated by studying the effect of carbide coarsening on creep rate.

Wolff’s Law of Trabecular Architecture at Remodeling Equilibrium

1986 ◽  
Vol 108 (1) ◽  
pp. 83-88 ◽  
Author(s):  
S. C. Cowin
Keyword(s):  
Stress Tensor ◽  
Bone Tissue ◽  
Constitutive Relation ◽  
Cancellous Bone ◽  
Strain Tensor ◽  
Constitutive Relationship ◽  
Trabecular Architecture ◽  
Fabric Tensor ◽  
Principal Axes ◽  
Wolff’S Law

An elastic constitutive relation for cancellous bone tissue is developed. This relationship involves the stress tensor T, the strain tensor E and the fabric tensor H for cancellous bone. The fabric tensor is a symmetric second rank tensor that is a quantitative stereological measure of the microstructural arrangement of trabeculae and pores in the cancellous bone tissue. The constitutive relation obtained is part of an algebraic formulation of Wolff’s law of trabecular architecture in remodeling equilibrium. In particular, with the general constitutive relationship between T, H and E, the statement of Wolff’s law at remodeling equilibrium is simply the requirement of the commutativity of the matrix multiplication of the stress tensor and the fabric tensor at remodeling equilibrium, T* H* = H* T*. The asterisk on the stress and fabric tensor indicates their values in remodeling equilibrium. It is shown that the constitutive relation also requires that E* H* = H* E*. Thus, the principal axes of the stress, strain and fabric tensors all coincide at remodeling equilibrium.

Submicrometre-resolution polychromatic three-dimensional X-ray microscopy

2012 ◽  
Vol 46 (1) ◽  
pp. 153-164 ◽  
Author(s):  
B. C. Larson ◽  
L. E. Levine
Keyword(s):  
Spatial Resolution ◽  
Elastic Strain ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Direct Determination ◽  
Phase Identification ◽  
Sn Whisker ◽  
X Ray ◽  
Elastic Strain Tensor

The ability to study the structure, microstructure and evolution of materials with increasing spatial resolution is fundamental to achieving a full understanding of the underlying science of materials. Polychromatic three-dimensional X-ray microscopy (3DXM) is a recently developed nondestructive diffraction technique that enables crystallographic phase identification, determination of local crystal orientations, grain morphologies, grain interface types and orientations, and in favorable cases direct determination of the deviatoric elastic strain tensor with submicrometre spatial resolution in all three dimensions. With the added capability of an energy-scanning incident beam monochromator, the determination of absolute lattice parameters is enabled, allowing specification of the complete elastic strain tensor with three-dimensional spatial resolution. The methods associated with 3DXM are described and key applications of 3DXM are discussed, including studies of deformation in single-crystal and polycrystalline metals and semiconductors, indentation deformation, thermal grain growth in polycrystalline aluminium, the metal–insulator transition in nanoplatelet VO2, interface strengths in metal–matrix composites, high-pressure science, Sn whisker growth, and electromigration processes. Finally, the outlook for future developments associated with this technique is described.

Fractional Derivative Constitutive Models for Finite Deformation of Viscoelastic Materials

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Stress Tensor ◽  
Finite Deformation ◽  
Dynamical Behavior ◽  
Strain Tensor ◽  
Constitutive Models ◽  
Cauchy Stress ◽  
Viscoelastic Materials ◽  
Kirchhoff Stress Tensor ◽  
Biot Stress

A methodology to derive fractional derivative constitutive models for finite deformation of viscoelastic materials is proposed in a continuum mechanics treatment. Fractional derivative models are generalizations of the models given by the objective rates. The method of generalization is applied to the case in which the objective rate of the Cauchy stress is given by the Truesdell rate. Then, a fractional derivative model is obtained in terms of the second Piola–Kirchhoff stress tensor and the right Cauchy-Green strain tensor. Under the assumption that the dynamical behavior of the viscoelastic materials comes from a complex combination of elastic and viscous elements, it is shown that the strain energy of the elastic elements plays a fundamental role in determining the fractional derivative constitutive equation. As another example of the methodology, a fractional constitutive model is derived in terms of the Biot stress tensor. The constitutive models derived in this paper are compared and discussed with already existing models. From the above studies, it has been proved that the methodology proposed in this paper is fully applicable and effective.

The grain refinement mechanism of electrodeposited copper

2009 ◽  
Vol 24 (10) ◽  
pp. 3226-3236 ◽  
Author(s):  
Guoyong Wang ◽  
Zhonghao Jiang ◽  
Jianshe Lian ◽  
Qing Jiang
Keyword(s):  
Grain Refinement ◽  
Large Angle ◽  
Xrd Analysis ◽  
Dislocation Cell ◽  
X Ray Diffraction ◽  
X Ray ◽  
Grain Sizes ◽  
Transmission Electron ◽  
Refinement Mechanism ◽  
Dislocation Cells

Microstructure features of five electrodeposited coppers with different grain sizes were systematically characterized by using transmission electron microscopy (TEM) observations and x-ray diffraction (XRD) analysis. Based on the experimental observations, two mechanisms for the grain refinement in electrodeposited copper were identified: (i) twin–twin intersection can directly create grains with large-angle boundaries as small as 10 nm and (ii) grains can also be refined via formation of dislocation cells, transformation of dislocation cell walls into sub-boundaries with small misorientations, and evolution of sub-boundaries into highly misoriented grain boundaries. Besides, dislocations are also effective to cut twin lamellas into pieces and make twin boundaries curved and round.

Towards Crack Trajectory Identification via the Direct Strain Imaging Method

2015 ◽  
Author(s):  
Athanasios Iliopoulos ◽  
John G. Michopoulos
Keyword(s):  
Mechanical Load ◽  
Deformable Body ◽  
Accurate Determination ◽  
Strain Tensor ◽  
Strain Imaging ◽  
Imaging Method ◽  
Extended Finite Element ◽  
Strain Compatibility ◽  
Crack Trajectory

In this paper we present an investigation on the feasibility of exploiting the Direct Strain Imaging (DSI) method for the purpose of tracking propagating discontinuities on the surface of a deformable body under mechanical load. The proposed approach is based on a strain compatibility functional that does not require any assumptions about the continuity conditions of the underlying medium. The proposed approach is based on the recently introduced Direct Strain Imaging method that is used to identify with high accuracy the full fields of strain tensor components that are required to define the strain compatibility functional. We performed synthetic numerical experiments based on the exercising the eXtended Finite Element Method solution for simulating a propagating crack of a particular problem in order to assess the feasibility and potential of the proposed approach. We demonstrated that indeed our DSI-based approach can achieve a very accurate determination of the crack trajectory even under noisy conditions.

Measurement of Elastic Lattice Distortion in PbTe/PbSnTe - Strained Layer Superlatices by Asymmetric High Angle X-ray interfernces

1985 ◽  
Vol 29 ◽  
pp. 367-374
Author(s):  
E. J. Fantner
Keyword(s):  
Elastic Strain ◽  
Lattice Mismatch ◽  
Strain Tensor ◽  
Misfit Strain ◽  
High Angle ◽  
Double Crystal ◽  
X Ray ◽  
Strained Layer ◽  
Thermal Expansion Coefficients ◽  
Quaternary Compounds

AbstractElastic strain significantly affects the electric and optical properties of PbTe/Pb1-xSnxTe - strained-layer superlattices. In the range of 10 - 350K the temperature dependence of the elastic strain present in these superlattices was measured by double-crystal x-ray diffraction. For superlattice periods smaller than 100nm High-angle x-ray interferences were observed. Using a novel method, which makes use of the High-angle interferences both for symmetrical as well as for asymmetrical reflections in a theta-twotheta scan with a narrow detector slit, the relative inclination of equivalent lattice planes due the elastic strain was measured. The components of the complete strain tensor of the constituent layers can be determined seperately even if their unstrained lattice constants are not known with sufficient accuracy as is the case in ternary and quaternary compounds. The lattice mismatch of up to 0.4% for Sn-contents smaller than 20% was found to be accommodated almost completely by elastic misfit strain. The amount of strain is shared between the constituent layers inversely to their relative thicknesses as long as the superlattice as a whole is much thicker than the buffer layer. Below room temperature an additional temperature dependent tensile strain due to differnt thermal expansion coefficients of the film and the BaF2-substrate is measured quantitatively.

Stressed-Strain State of Multi Layer Foil under One-Axis Tension

2015 ◽  
Vol 756 ◽  
pp. 540-545
Author(s):  
V.N. Demidov ◽  
Anna G. Knyazeva
Keyword(s):  
Analytical Solution ◽  
Stressed State ◽  
Stress Tensor ◽  
Strain State ◽  
Strain Tensor ◽  
Stress And Strain ◽  
Plane Stressed State ◽  
Diagonal Component ◽  
Space Coordinate ◽  
Properties Of Materials

Multilayer protective materials and coatings attract attention last years because the possibility appears to obtain unique properties of materials due to combination of properties and sizes of layers. In his paper analytical solution is constructed for the problem on multilayer foil tension. It is assumed that stress and strain tensor components depend on one space coordinate in the axis direction perpendicular to layers and generalized plane stressed state is realized. Interesting effect is detected for two-layer material: thickness of the coating exists when one of diagonal stress tensor components is minimal, that ensure minimal break in the corresponding stresses in interface. This effect is observed previously experimentally by many authors. However, other diagonal component of stress tensor has maximal value for other value of coating thickness. The positions of the maximal and minimal values of stresses and breaks depend on combination of properties and thicknesses of layers.

scholarly journals On the Structure of Stress-Strain Relations for Time-Dependent Plastic Deformation in Metals

1970 ◽  
Vol 37 (3) ◽  
pp. 728-737 ◽  
Author(s):  
J. R. Rice
Keyword(s):  
Potential Function ◽  
Strain Tensor ◽  
Inelastic Strain ◽  
Time Dependent ◽  
Strain Rate Tensor ◽  
Stress Strain ◽  
Potential Surfaces ◽  
Discrete Dislocation ◽  
Flow Potential ◽  
Individual Grains

The paper is concerned with the structure of multiaxial stress-strain relations in time-dependent metal plasticity, as for transient creep and rate sensitive yielding. First, a general kinematical relation is developed between the macroscopic inelastic strain tensor and microstructural slip displacements, as modeled either by continuum shearing on crystallographic planes of individual grains or by the motion of discrete dislocation lines. It is assumed that at any given slipped state, the rate of slipping on a particular system is governed by the resolved shear stress on that system (or by the local “forces” on dislocation lines). This leads to the primary result of the paper: Components of the macroscopic inelastic strain rate tensor are derivable, at each instant in the course of deformation, from a potential function of stress. General features of the flow potential surfaces in stress space are discussed, and some specific functional forms are examined. Linear viscoelasticity and time-independent plasticity are developed as limiting cases of the flow potential formulation, and the appropriateness of a potential function for stationary creep is discussed.

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