Residual strain energy in composites containing particles

1996 ◽  
Vol 11 (2) ◽  
pp. 483-494 ◽  
Author(s):  
Toshiaki Mizutani
Keyword(s):  
Strain Energy ◽  
Crystal Lattices ◽  
Composite Systems ◽  
Surrounding Matrix ◽  
Dispersed Particles ◽  
First Approximation ◽  
Bulk Stress ◽  
The Matrix ◽  
Unit Cells ◽  
Physical Validity

Selsing's formula for radial tension at the particle-matrix interface is extended into a general formula which includes the effects of the amount of dispersed particles. A relationship is derived between individual volumes of strained unit cells in the crystal lattices of the particles and of the surrounding matrix. These relationships are used to predict the effect of the particles (2H−TiB2, 2H−ZrB2, and t−WB) on their unit cells and on the unit cell of the surrounding 6H–SiC matrix. The precision of these predictions was 7.1% or better. Hence, in principle, it is possible to investigate the distributions of residual bulk stress/strain. Estimates of characterizing values of the three composite systems are attempted on the rough basis of the elastic constants of the SiC matrix, confirming the physical validity of this approach as a first approximation. Further, the residual bulk strain energies of the particles and the matrix are discussed in connection with the elastic term involved in the fracture energy of such composites.

scholarly journals Quantum mechanics of electrons in crystal lattices

1931 ◽  
Vol 130 (814) ◽  
pp. 499-513 ◽  
Keyword(s):  
Three Dimensional ◽  
Matrix Elements ◽  
Periodic Field ◽  
Crystal Lattices ◽  
One Dimensional ◽  
Physical Problems ◽  
Radiative Transitions ◽  
First Approximation ◽  
The Matrix ◽  
The One

Introduction .–Through the work of Bloch our understanding of the behaviour of electrons in crystal lattices has been much advanced. The principal idea of Bloch’s theory is the assumption that the interaction of a given electron with the other particles of the lattice may be replaced in first approximation by a periodic field of potential. With this model an interpretation of the specific heat, the electrical and thermal conductivity, the magnetic susceptibility, the Hall effect, and the optical properties of metals could be obtained. The advantages and limitations inherent in the assumption of Bloch will be much the same as those encountered when replacing the interaction of the electrons in an atom by a suitable central shielding of the unclear field, as in the work of Thomas and Hartree. In the papers quoted a number of general results were given regarding the behaviour of electrons in any periodic field of potential. To obtain a clearer idea of the details of this behaviour with a view to the application in special problems, however, it appeared worth while to investigate the mechanics of electrons in periodic fields of potential somewhat similar to those met with in practice and of such nature that the energy values W and eigenfunctions Ψ of the wave-equation can actually be computed. It is the purpose of this article to discuss a case where the integration is possible. In Section 1 the energy values and in Section 2 the wave-functions in their dependence on the binding introduced by the potential field are discussed for the one dimensional problem. In Section 3 the matrix elements of the linear momentum, which furnish the electric current associated with the various stationary states, are well as the probability of radiative transitions between these states, are evaluated. In Section 4 the results are extended to the three dimensional case and those features considered which one may expect to find in the case of more general periodic fields of potential. Section 5 deals with some applications to physical problems.

Vacancies for controlling the behavior of microstructured three-dimensional mechanical metamaterials

2018 ◽  
Vol 24 (2) ◽  
pp. 511-524 ◽  
Author(s):  
Z Vangelatos ◽  
K Komvopoulos ◽  
CP Grigoropoulos
Keyword(s):  
Strain Energy ◽  
Three Dimensional ◽  
Crystal Lattices ◽  
Mechanical Metamaterials ◽  
Buckling Behavior ◽  
Unit Cells ◽  
Octet Truss ◽  
Topological States ◽  
Energy Dissipated

Mechanical metamaterials are designed to exhibit enhanced properties not found in natural materials or to bolster the properties of existing materials. The theoretical foundations for tuning the mechanical properties have been established, including topological states, controllable buckling behavior, and quasi-two-dimensional mechanical metamaterials with structures containing vacancies. However, the fabrication and experimental procedures to study these structures at the microscale have not been developed yet and the three-dimensional (3D) architectures examined to date are fairly limited. In this study, 3D mechanical metamaterial structures with select unit cells designed to have vacancies were fabricated by multi-photon lithography, having as the principal objective to control (localize) failure and increase the strain energy capacity of the structure. The metamaterial structure from which all the current designs originate is the octet-truss structure, one of the most widely studied 3D metamaterials. The design of the structures was inspired by the role of vacancies in crystal lattices. Vacancies were introduced in the metamaterial structures to allow localized buckling of lattice members to occur in specific unit cells. The significant increase of the strain energy dissipated in these metamaterials is demonstrated by nanoindentation experiments and finite element results. Vacancy effects on the dynamic response of metamaterial structures are also examined in the light of modal analysis simulations. The findings of this study illustrate the importance of strategically placing vacancies in the microlattices of metamaterial structures to control the overall mechanical behavior and greatly increase strain energy dissipation.

scholarly journals Investigation of the microstructure evolution in TP347HFG austenitic steel at 700°C and its characterization method

2021 ◽  
Vol 40 (1) ◽  
pp. 12-22
Author(s):  
Yuetao Zhang ◽  
Tingbi Yuan ◽  
Yawei Shao ◽  
Xiao Wang
Keyword(s):  
Austenitic Steel ◽  
Aging Time ◽  
Microstructure Evolution ◽  
Precipitate Phase ◽  
Dispersion Strengthening ◽  
Micro Hardness ◽  
Dispersed Particles ◽  
Nonlinear Ultrasonic ◽  
The Matrix ◽  
Solution Strengthening

Abstract This article reports the microstructure evolution in TP347HFG austenitic steel during the aging process. The experiments were carried out at 700°C with different aging time from 500 to 3,650 h. The metallographic results show that the coherent twin and incoherent twin are existed in the original TP347HFG grains, while they gradually vanished with the increase of the aging time. After aging for 500 h, a lot of fine, dispersed particles precipitated from the matrix, but they disappeared after aging for 1,500 h. When the aging time extend to 3,650 h, the precipitates appeared apparently coarse in TP347HFG steel, which include the M23C6 and σ phase; besides, the micro-hardness of TP347HFG also changes during the aging, which was closely related to the effect of dispersion strengthening and solution strengthening. The results of the nonlinear ultrasonic measurement reveal that the β′ of TP347HFG steel was also changed with the aging time. It first increased at 0–500 h, then reduced later, and increased finally at 1,500–3,650 h. The variation of β′ in TP347HFG was influenced by a combined effect of the twin microstructure and the precipitate phase, which indicate that the nonlinear ultrasonic technique can be utilized to characterize the microstructure evolution in TP347HFG.

scholarly journals Prediction of Effective Thermal Conductivities of Four-Directional Carbon/Carbon Composites by Unit Cells with Different Sizes

2021 ◽  
Vol 11 (3) ◽  
pp. 1171
Author(s):  
Chang Xu ◽  
Zhihong Sun ◽  
Guowei Shao
Keyword(s):  
Carbon Fiber ◽  
Longitudinal Direction ◽  
Unit Cell ◽  
Carbon Composites ◽  
Temperature Boundary ◽  
The Matrix ◽  
Unit Cells ◽  
Experimental Values ◽  
Different Diameters ◽  
Thermal Conductivities

Two-unit cells developed to predict the effective thermal conductivities of four-directional carbon/carbon composites with the finite element method are proposed in this paper. The smaller-size unit cell is formulated from the larger-size unit cell by two 180° rotational transformations. The temperature boundary conditions corresponding to the two-unit cells are derived, and the validity is verified by the temperature and heat flux distributions at specific positions of the larger-size unit cell and the smaller-size unit cell. The thermal conductivities of the carbon fiber bundles and carbon fiber rods are measured firstly. Then, combined with the properties of the matrix, the effective thermal conductivities of the four-directional carbon/carbon composites are numerically predicted. The results in transverse direction predicted by the larger-size unit cell and the smaller-size unit cell are both higher than experimental values, which are 5.8 to 6.2% and 7.3 to 8.2%, respectively. In longitudinal direction, the calculated thermal conductivities of the larger-size unit cell and the smaller-size unit cell are 6.8% and 6.2% higher than the experimental results, respectively. In addition, carbon fiber rods with different diameters are set to clarify the influence on the effective thermal conductivities of the four-directional carbon/carbon composites.

Effect of temperature on matrix multicracking evolution of C/SiC fiber-reinforced ceramic-matrix composites

2020 ◽  
Vol 39 (1) ◽  
pp. 189-199
Author(s):  
Longbiao Li
Keyword(s):  
Strain Energy ◽  
Ceramic Matrix Composites ◽  
Critical Value ◽  
Matrix Cracking ◽  
Interface Debonding ◽  
Matrix Composites ◽  
Temperature Dependent ◽  
Damage State ◽  
The Matrix ◽  
Sic Composites

AbstractIn this paper, the temperature-dependent matrix multicracking evolution of carbon-fiber-reinforced silicon carbide ceramic-matrix composites (C/SiC CMCs) is investigated. The temperature-dependent composite microstress field is obtained by combining the shear-lag model and temperature-dependent material properties and damage models. The critical matrix strain energy criterion assumes that the strain energy in the matrix has a critical value. With increasing applied stress, when the matrix strain energy is higher than the critical value, more matrix cracks and interface debonding occur to dissipate the additional energy. Based on the composite damage state, the temperature-dependent matrix strain energy and its critical value are obtained. The relationships among applied stress, matrix cracking state, interface damage state, and environmental temperature are established. The effects of interfacial properties, material properties, and environmental temperature on temperature-dependent matrix multiple fracture evolution of C/SiC composites are analyzed. The experimental evolution of matrix multiple fracture and fraction of the interface debonding of C/SiC composites at elevated temperatures are predicted. When the interface shear stress increases, the debonding resistance at the interface increases, leading to the decrease of the debonding fraction at the interface, and the stress transfer capacity between the fiber and the matrix increases, leading to the higher first matrix cracking stress, saturation matrix cracking stress, and saturation matrix cracking density.

scholarly journals The Influence of Filler Size and Crosslinking Degree of Polymers on Mullins Effect in Filled NR/BR Composites

Polymers ◽  
2021 ◽  
Vol 13 (14) ◽  
pp. 2284
Author(s):  
Miaomiao Qian ◽  
Bo Zou ◽  
Zhixiao Chen ◽  
Weimin Huang ◽  
Xiaofeng Wang ◽  
...  
Keyword(s):  
Particle Size ◽  
Strain Energy ◽  
Mathematical Expression ◽  
Rubber Matrix ◽  
Mullins Effect ◽  
Butadiene Rubber ◽  
Crosslinking Degree ◽  
Test Range ◽  
Two Factors ◽  
The Matrix

Two factors, the crosslinking degree of the matrix (ν) and the size of the filler (Sz), have significant impact on the Mullins effect of filled elastomers. Herein, the result. of the two factors on Mullins effect is systematically investigated by adjusting the crosslinking degree of the matrix via adding maleic anhydride into a rubber matrix and controlling the particle size of the filler via ball milling. The dissipation ratios (the ratio of energy dissipation to input strain energy) of different filled natural rubber/butadiene rubber (NR/BR) elastomer composites are evaluated as a function of the maximum strain in cyclic loading (εm). The dissipation ratios show a linear relationship with the increase of εm within the test range, and they depend on the composite composition (ν and Sz). With the increase of ν, the dissipation ratios decrease with similar slope, and this is compared with the dissipation ratios increase which more steeply with the increase in Sz. This is further confirmed through a simulation that composites with larger particle size show a higher strain energy density when the strain level increases from 25% to 35%. The characteristic dependence of the dissipation ratios on ν and Sz is expected to reflect the Mullins effect with mathematical expression to improve engineering performance or prevent failure of rubber products.

Failure Mechanisms of Particulate Two-Phase Composites

1998 ◽  
Vol 529 ◽  
Author(s):  
T. Antretter ◽  
E D. Fischer
Keyword(s):  
Residual Stresses ◽  
Crack Growth ◽  
Opening Angle ◽  
Geometrical Parameters ◽  
Two Phase ◽  
Absolute Size ◽  
The Matrix ◽  
Unit Cells ◽  
Angle Criterion ◽  
Probability Of Fracture

AbstractIn many composites consisting of hard and brittle inclusions embedded in a ductile matrix failure can be attributed to particle cleavage followed by ductile crack growth in the matrix. Both mechanisms are significantly sensitive towards the presence of residual stresses.On the one hand particle failure depends on the stress distribution inside the inclusion, which, in turn, is a function of various geometrical parameters such as the aspect ratio and the position relative to adjacent particles as well as the external load. On the other hand it has been observed that the absolute size of each particle plays a role as well and will, therefore, be taken into account in this work by means of the Weibull theory. Unit cells containing a number of quasi-randomly oriented elliptical inclusions serve as the basis for the finite element calculations. The numerical results are then correlated to the geometrical parameters defining the inclusions. The probability of fracture has been evaluated for a large number of inclusions and plotted versus the particle size. The parameters of the fitting curves to the resulting data points depend on the choice of the Weibull parameters.A crack tip opening angle criterion (CTOA) is used to describe crack growth in the matrix emanating from a broken particle. It turns out that the crack resistance of the matrix largely depends on the distance from an adjacent particle. Residual stresses due to quenching of the material tend to reduce the risk of particle cleavage but promote crack propagation in the matrix.

Determination of Optimal Unit-Cell Size of Homogeneous Conformal Unit-Cell Lattice Structure Using Solid Shape Matching

2016 ◽  
Author(s):  
Mahmoud A. Alzahrani ◽  
Seung-Kyum Choi
Keyword(s):  
Cell Size ◽  
Solid Body ◽  
Strain Energy ◽  
Lattice Structure ◽  
Weight Ratio ◽  
Unit Cell ◽  
Optimal Size ◽  
Lattice Structures ◽  
Unit Cell Size ◽  
Unit Cells

With rapid developments and advances in additive manufacturing technology, lattice structures have gained considerable attention. Lattice structures are capable of providing parts with a high strength to weight ratio. Most work done to reduce computational complexity is concerned with determining the optimal size of each strut within the lattice unit-cells but not with the size of the unit-cell itself. The objective of this paper is to develop a method to determine the optimal unit-cell size for homogenous periodic and conformal lattice structures based on the strain energy of a given structure. The method utilizes solid body finite element analysis (FEA) of a solid counter-part with a similar shape as the desired lattice structure. The displacement vector of the lattice structure is then matched to the solid body FEA displacement results to predict the structure’s strain energy. This process significantly reduces the computational costs of determining the optimal size of the unit cell since it eliminates FEA on the actual lattice structure. Furthermore, the method can provide the measurement of relative performances from different types of unit-cells. The developed examples clearly demonstrate how we can determine the optimal size of the unit-cell based on the strain energy. Moreover, the computational cost efficacy is also clearly demonstrated through comparison with the FEA and the proposed method.

ANALYSIS OF PROTEOGLYCAN GENE EXPRESSION IN CONNECTIVE TISSUES BY SEMI-QUANTITATIVE RT-PCR

2001 ◽  
Vol 05 (02) ◽  
pp. 79-88
Author(s):  
K. Dobra ◽  
A. Hjerpe
Keyword(s):  
Rt Pcr ◽  
Connective Tissues ◽  
Surrounding Matrix ◽  
Cell Proliferation And Differentiation ◽  
Extracellular Matrix Components ◽  
Proliferation And Differentiation ◽  
The Matrix ◽  
Quantitative Changes ◽  
Regulatory Functions ◽  
Multiple Samples

Proteoglycans (PGs) are cell-membrane and extracellular matrix components with a wide variety of different functions. In the matrix, they are mainly of structural importance, although some of them have been ascribed specific regulatory functions, such as in the assembly of collagen fibers. PGs on the cell surface act as essential modulators of specific ligand-binding reactions, involving interactions between adjacent cells and between cells and surrounding matrix. Through these interactions they participate in different processes, including cell proliferation and differentiation. Qualitative and quantitative changes in PG expression can therefore be associated with various physiological and pathological conditions. We have optimized the conditions for semi-quantitative evaluation of proteoglycan expression by RT-PCR reaction, using glyceraldehyde-3-phosphate dehydrogenase (GAPDH) as reference gene. The relative fluorescence of analyte to reference amplimers can — within certain limits — be used to estimate the amount of target RNA and allows direct comparison of multiple samples. The profile of PG expression obtained in this way can be used to extend our current understanding of the possible functions that can be associated with these complex molecules.

Effects of Shear Deformation of Struts in Hexagonal Lattices on their Effective In-Plane Material Properties

2021 ◽  
Vol 1034 ◽  
pp. 193-198
Author(s):  
Pana Suttakul ◽  
Thongchai Fongsamootr ◽  
Duy Vo ◽  
Pruettha Nanakorn
Keyword(s):  
Shear Deformation ◽  
Material Properties ◽  
Strain Energy ◽  
Equivalent Strain ◽  
Homogenization Method ◽  
Two Dimensional ◽  
Engineering Applications ◽  
Large Numbers ◽  
Effective Material Properties ◽  
Unit Cells

Two-dimensional lattices are widely used in many engineering applications. If 2D lattices have large numbers of unit cells, they can be accurately modeled as 2D homogeneous solids having effective material properties. When the slenderness ratios of struts in these 2D lattices are low, the effects of shear deformation on the values of the effective material properties can be significant. This study aims to investigate the effects of shear deformation on the effective material properties of 2D lattices with hexagonal unit cells, by using the homogenization method based on equivalent strain energy. Several topologies of hexagonal unit cells and several slenderness ratios of struts are considered. The effects of struts’ shear deformation on the effective material properties are examined by comparing the results of the present study, in which shear deformation is neglected, with those from the literature, in which shear deformation is included.

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