Shear Strain

The shear strain (γ) imposed in the chip is given by [209]:

(2.33)γ=cosαsinφcos(φ−α)where φ is calculated from a
measurement of a0 and ac as:(2.34)tanφ=a0accosα1-
a0acsinα
From: Severe Plastic Deformation, 2018

Related terms:

Strain Rate, Mechanical Strength, Martensite, Shear Band, Plastic Deformation,

Grain Boundary, Elastic Moduli, Hydrogel, Shear Strength, Stacking Fault

An introduction to rheology
T.A. Strivens, in Paint and Surface Coatings (Second Edition), 1999

14.4.1.3 Parallel plates

Parallel plate geometry has the advantage over the cone and plate
geometry that the gap width can be varied freely. As discussed in Section
14.1.1, this is an advantage when measuring suspension or dispersion
systems. Against this advantage, the shear rate on the sample varies with
the distance from the plate centre, and thus data are more difficult to
evaluate.

When oscillatory (dynamic) measurements are being made, parallel

plates possess considerable advantages over either cone and plate or
concentric cylinder geometries, as will be discussed in Section 14.4.2.3.

The maximum shear rate (at the plate rim) is given by

γ̇m = Ω𝑅 (14.10)
ℎ

and the apparent viscosity (ηa) corresponding to this maximum shear rate
(γ̇m ) may be evaluated, using the following equation [8, p. 52]:

ηa = 2π𝑅3𝑀 [1 + 3dln 𝑀 (14.11)

3
γ̇ d ln γ̇ ]
m

where

M is the measured torque,

R is the plate radius,

h is the plate separation, and

Ω is the speed of rotation.

Like the cone and plate geometry, the parallel plate geometry has the
advantage that in theory (if not always in practice) the liquid velocity
distribution in the gap is determined by the geometry and not by the
liquid properties.

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Volume II
N. Hansen, C.Y. Barlow, in Physical Metallurgy (Fifth Edition), 2014

17.3.9 Twinning
Shear strain can be achieved by the formation of mechanical twins, which
is a very different mechanism from dislocation movement (Hull and
Bacon, 2011; Cottrell, 1963; Hosford, 2005). Twinning is a localized shear
process, involving the cooperative shifting of lattice planes into
crystallographically different configurations. It occurs rapidly and
abruptly, and is often accompanied by audible clicks. An example of a
structure showing mechanical twins is shown in Figure 4 (Zhang et al.,
2008a).
Figure 4. Twin boundaries in nickel deformed by high pressure torsion to a
strain of 12. Twin boundaries are marked with a dashed line.

Zhang et al. (2008a); Figure 3(c).

Mechanical twinning can take place in addition to or in preference to

dislocation glide under certain conditions. It most commonly occurs
when dislocation glide is difficult for whatever reason, and is a very
process-sensitive deformation mechanism. Low dislocation mobility is
typically found at low temperatures and high strain-rates. Crystal
structure is a major factor: hcp materials frequently demonstrate
twinning, and it may be the dominant mechanism. It is common also in
bcc materials, particularly at low temperatures. It is less common in fcc
metals, although twinning has been observed for deformation at very low
temperatures (e.g. Cu at 4 K) or high strain-rates. Other systems in which
dislocation mobility is impeded include solid solutions, and twinning is
found for deformation of some fcc alloys, for example Ag–Au at room
temperature (Cottrell, 1963). A somewhat distinct type of twinning on a
nano- or microscale can be found supplementing the dislocation structure
in ultrahigh strained materials (Huang et al., 2003).
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Mechanical properties of multifunctional

foam core materials
Amir Fathi Volker Altstädt, in
Multifunctionality of Polymer Composites, 2015

8.3.1 Full-Field Shear Strains

Measuring core shear strains during sandwich bending tests is a rather
difficult task to perform. Some works have been done, though, in the
literature on in-situ measurements of core shear strain during bending
tests [75]. One advanced method to measure different strain modes in the
core material is optical strain analyses with the aid of digital image
correlation (DIC). Optical strain measurements have been widely used in
the recent years to determine mechanical properties of different materials
[97–100]. Using this method, it is possible to measure full-field surface
strain patterns, instead of average strains between two marks on the
sample (i.e., video extensometers) or point strain values (i.e., local strain
gauges). Since yielding is a local phenomenon, deformation patterns in
the post-yielding regimes become very irregular and rather difficult to
measure with conventional extensometer technologies. When used with
sandwich testing, DIC can provide, for example, means of local strain
investigations and analyses of the quality of face/core adhesion. Such data
could be used to modify or verify numerical simulations. In optical strain
analyses, we compare a series of images, captured at various deformation
levels, with a reference initial image taken before loading. The
fundamental principle is based on the fact that the distribution of
grayscale values of a rectangular area (called facet) in the undeformed
state corresponds to the distribution of grayscale values of the same area
in the deformed states [98], as shown in Figure 8.17.

Figure 8.17. Grayscale (intensity) distribution of a facet in the

nondeformed (left) and the deformed (right) states.

Reproduced from Ref. [98].

This provides a physical mean to track and locate the displacement of

several facets during the whole deformation process. Thus, the quality
and the precision of the measurement are strongly influenced by the
surface patterns. In order to produce fine surface features, usually a
random paint pattern is applied on the surface of the specimen.

In case of the sandwich beams discussed here, optical strain analyses

were performed during additional four-point bending tests. After the
image processing step, shear deformations and strains were studied on
half of the beam length in order to capture more local details. Figure 8.18
shows the region of interest on a sandwich beam for optical strain studies.

Figure 8.18. Region of interest for optical strain analyses during four-point
bending tests.

Figures 8.19–8.21 show the distribution of shear strains on the surface of

sandwich configurations at different load levels. As discussed before,
shear stresses are almost constant between the loading and the support
points. Therefore, shear strain distributions on the surface of foam core
specimens are rather uniform in large sections, especially before local
yielding of the regions with most critical deformation. In contrast, a large
scatter is observed in the local shear strain patterns of Balsa (Figure 8.21).
Some parts of the Balsa core retain almost no deformation throughout the
test, even at the vicinity of failure. In addition, as the load increases,
smaller regions in Balsa are strained significantly more. Core failure in
Balsa is initiated from these regions, which are rather difficult to predict
as reported by others too [86].
Figure 8.19. Distribution of shear strain on the surface of a PVC foam core
sandwich beam loaded in four-point bending at different midpoint loads.
Figure 8.20. Distribution of shear strain on the surface of a PET foam core
sandwich beam loaded in four-point bending at different midpoint loads.
Figure 8.21. Distribution of shear strain on the surface of a Balsa core
sandwich beam loaded in four-point bending at different midpoint loads.
In the PET core (Figure 8.20) after yielding, plastic shear deformations
become more localized. Significant plastic deformations initiate from the
mid-core regions, which experience highest shear stresses. In the PVC
core, on the other hand, shear strains seem to be generally higher on one
side of the beam (see Figure 8.19 at 3000 N for example) and critical shear
strains are first developed under the loading point and begin growing
from the sub-interface regions as previously discussed [38,50]. Higher
shear strains on one side of the PVC core could correspond to a variation
of foam cell size and density through the thickness of the core (Figure
8.22). In another investigation on the same PVC foam [101], a slight
difference in the cell morphology and density of the foam near the two
sides of the panel have been shown. Therefore, surface resin uptakes on
the top and bottom surfaces were not the same, leading to different
interface toughness and peel strength. It seems that here also the
variation of foam density over the beam thickness has caused different
shear properties on the top and bottom sides of the PVC core.
Figure 8.22. Variation of cell size and density through the thickness of the
PVC foam panel.

Regenerated from Ref. [101].

The difference in the critical shear strain patterns of the PET and the PVC
foam cores could be attributed to different phenomena. The high out-of-
plane stiffness of the PET core (resulting from its special multi-scale
morphology) can prevent high-stress concentrations at the loading points
and displace the initiation of shear cracks from the mid-core.
Nevertheless, the microstructures of the surface cells and eventually the
different resin uptake values can also affect the shear strength of the core
[101].

In the literature, there has been a long discussion on the initiation of core
shear failure in foam core sandwich structures. Gibson [93], for instance,
showed that for a PMI foam core, stiffer face sheets minimize stress
concentrations in the core at the loading points and eventually cause the
initial cracks to begin at the middle surface of the beam, whereas less stiff
face sheets lead to core shear cracks initiating near the loading points
(Figure 8.23). The same concept could be true for the case of foam core
sandwich systems studied here. The PET foam is an extruded
thermoplastic with higher out-of-plane stiffness compared to the PVC
foam [14,15]. Therefore, in spite of their identical face sheets, stress
concentrations under the loading points are more severe in the PVC foam
than the PET foam. This could be the primary reason for the distinct shear
failure patterns in the two core materials.
Figure 8.23. Initiation of core shear cracks in sandwich beams with PMI
foam core and glass-fiber-reinforced epoxy face sheet (left) and carbon-
fiber-reinforced epoxy face sheets (right).

Regenerated from Ref. [93].

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Rolling Forming of Multi-Scaled Metallic

Foils and Sheets
Hailiang Yu, Xiaohui Cui, in
Encyclopedia of Materials: Metals and Alloys, 2022

Strain Distribution
Shear strain and accumulated strain in strips plays a critical role in the
grain refinement of strips processed by rolling, and FEM has been widely
used to analyze the strain distribution in workpiece (Pesin and
Pustovoytov, 2017; Pustovoytov et al., 2018; Yu et al., 2006b). Fig. 13 (Yu et
al., 2006b) shows the plastic strain contour distribution of one same
profile during six rolling condition when flat edging roll is used, where (a)
for the 1st pass V-rolling; (b) for the 1st pass H-rolling; (c) for the 2nd
pass V-rolling; (d) for the 2nd pass H-rolling; (e) for the 3rd pass V-
rolling; (f) for the 3rd pass H-rolling;

Fig. 13. Plastic strain distribution during V–H rolling.

Reproduced from Yu, H.L., Liu, X.H., Li, C.S., Zhao, X.M., Kusaba, Y., 2006b.
Influences of edging roll shape on the plastic strain distribution of slab during

multi-pass V–H rolling process. Acta Metall. Sin. (Engl. Lett.) 19 (1), 51–56.

In Fig. 13(a), (c), and (e), it is clear that the plastic strain assembled in a
litter zone of workpiece where near the workpiece corner and closed to
vertical roll and which did not go deep into workpiece as workpiece
formed the “dog-shape”. In Fig. 13(b), (d), and (f), the zone closed to
horizontal roll and near the workpiece corner appears the plastic strain
assembled which did not go deep into workpiece either, and it reserved
the strain assembling zone after edging roll, at the same time, the two
strain assembling zone were very thin. Compared Fig. 13(a) with Fig.
13(b), the max strain value after vertical rolling is larger than the max
strain value after horizontal rolling. Compared the Fig. 13(a), (c), and (f),
the max strain value of single pass of workpiece increases as the rolling
pass increases.

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Machining in monocrystals
Anish Roy, ... Vadim V. Silberschmidt, in
Mechanics of Materials in Modern Manufacturing Methods and
Processing Techniques
, 2020

9.4.3.3 Prediction of work-piece deformation

The accumulative shear strain in the material post-machining is shown in
Fig. 9.15. It is believed that the material removal in Orientation 1 (Fig.
9.15A) was dominated by the average response of partially active slip
systems; hence, the magnitude of the residual total accumulative shear
strain in the work-piece material is small. On the contrary, material
removal in Orientation 2 (Fig. 9.15B) was dominated by the slip system
(1 1 0 0) [11–20]. Although some slip systems were partially activated
during the micromachining process the element-deletion threshold was
not achieved so most partially activated systems remained in the
workpiece without being removed.

Figure 9.15. Contour plots of accumulative shear strain: (A) Orientation 1

and (B) Orientation 2.

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The importance of full-scale experiments

for the study of seismic metamaterials
Stéphane BrûléStefan EnochSébastien Guenneau, in
Encyclopedia of Condensed Matter Physics (Second Edition), 2024

Elasticity hypothesis for soils and analog

approaches
For shear strain ranging γ from 10−6 to 10−4, the constitutive law for soil is
elasticity (Brûlé, 2023). Once we have explained the properties of poorly
consolidated soils, it is easier to explain that much stiffer inclusions (steel,
concrete, wood, etc.) or voids in the soil can interact with seismic waves.
Young's modulus of concrete is 1000 times greater than that of soil. These
two conditions allow us to seek analogies with other areas of physics.

The seemingly disconnected concepts of photonic crystals (John, 1987;

Yablonovitch, 1987) and metamaterials (Smith et al., 2004; Anantha
Ramakrishna and Grzegorczyk, 2008) belong to the nano-scale world and
electromagnetism. Typically, photonic crystals and metamaterials refer to
periodic arrangements of elements with size comparable or much smaller
than the considered wavelength (typically hundreds of nanometers for
optical wavelength) that acquire effective properties of materials with
unusual properties or applications such as negative optical index for a flat
convergent lens (Veselago, 1968) with subwavelength resolution (Pendry,
2000), or highly anisotropic materials such as hyperbolic metamaterials
or invisibility cloaking devices (Pendry et al., 2006). It might come as a
surprise that one can mitigate surface Rayleigh waves in geophysics in a
way like what physicists do for surface electromagnetic waves in optics
and plasmonics.

It is this story that we want to tell in this chapter. Indeed, researchers in

photonics who have made research advances in electromagnetic
metamaterials and plasmonics have created a pathway for an enhanced
control of unwanted ground vibrations induced by urban traffic, human
activities, and seismic waves.

Amazingly, some experiments on extraordinary transmission of light

through nanoholes in metals (Ebbesen et al., 1998) and even on
broadband cloaking of spoof plasmon polaritons on metal surfaces
structured with TiO2 at the nanoscale (Renger et al., 2010) can be
translated to the realm of seismic metamaterials at the meter scale (Brûlé
et al., 2014, 2017). We point out that drawing analogies between surface
Rayleigh waves in geophysics and spoof plasmon polariton in plasmonics
makes it possible to envision seismic cloaks and carpets at the decameter
and kilometer scales. Research advances in photonics and plasmonics in
the past 20 years might lead to a paradigm shift in earthquake engineering
in the near future. However, we are not quite there yet. Much remains to
be done to overcome theoretical and practical obstacles, among which is
the highly heterogeneous nature of soil, as can be inferred from the
photos in Fig. 2 (MetaMAT's 10th webinar S2—
https://www.youtube.com/watch?v=HkikTHPEAqY).

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The Viscoelastic Behavior of Rubber and

Dynamics of Blends
K.L. Ngai, ... D.J. Plazek, in
The Science and Technology of Rubber (Fourth Edition), 2013

5.2.2 Stress Relaxation

After a constant shear strain is created in a previously relaxed material,
the resulting shear stress decays with ensuing time to zero for a
viscoelastic liquid and to a finite equilibrium value for a viscoelastic solid.
2
The shear stress relaxation modulus (Pa or dynes / cm )

𝐺(𝑡) = 𝜎(𝑡) / 𝛾 = 𝐺𝑒 + [𝐺(0) - 𝐺𝑒 ]𝜑(𝑡), (5.3)

0
where 𝛾 is the imposed fixed strain; 𝜙(𝑡) is the relaxation function
𝑜

decreasing from 𝜙(0) = 1 at t = 0 to 𝜙(∞) = 0 at 𝑡 = ∞; and 𝐺𝑒 is the

equilibrium modulus, which is finite for a viscoelastic solid and zero for
viscoelastic liquid. The time-dependent stresses arising from different
molecular mechanisms are not additive, and hence it is difficult if not
impossible to isolate and characterize each one of them individually.
Nevertheless 𝐺(𝑡) is connected to 𝐽(𝑡) by the convolution integral equation,
𝑡
∫ 𝐺(𝑠)𝐽(𝑡 - 𝑠)ds = 1, from which one function can be calculated from the
0

other by a numerical procedure (Hopkins and Hamming, 1957).

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ANISOTROPY AND PATH DEPENDENCE

IN FINITE DEFORMATION SHEAR AND
SEQUENTIAL COMPRESSION/SHEAR
M.P. Miller, ... D.J. Bammann, in
Advances in Engineering Plasticity and its Applications, 1993

1 INTRODUCTION
Results from large strain shear experiments of ductile metals have been
used extensively to formulate finite strain deformation models. Typically,
current theories attempt to model two widely observed phenomena of
the large strain shear experiment on thin-walled tubular specimens; (i)
lower effective stress levels based on J2 at the same effective plastic strain
compared to tension or compression tests (cf. [1–4]), and (ii) secondary,
induced axial strains (free-end experiments) or axial stresses (fixed-end
experiments) [1,5,6]. In addition to these experimentally observable
phenomena, (iii) insistence of monotonicity of shear and normal stresses
during torsion simulations up to physically implausible shear strain levels
has been used to validate model hypotheses.

The embodiment of crystal plasticity concepts [7–9] into

phenomenological or macro-scale constitutive relations invariably has
focused on specification of the plastic spin, Wp. The concept of the plastic
spin was proposed by Mandel and colleagues in the early 1970's [10,11] to
account for lattice rotation in crystal plasticity within the context of
phenomenological inelasticity theory of metals. Plastic spin is linked to
relative lattice rotation [12,13] associated with grain rotation and
anisotropy of crystalline slip, and is often physically identified with the
development of texture.

The plastic spin has been used to model behaviors (i)-(iii) elucidated
above. It was invoked by Dafalias [14-18], Lee et.al.[19], and others as a
means to modify the Jaumann rate of tensorial variables to avoid
oscillatory response in simple shear with kinematic hardening. The plastic
spin has also been used to match axial stress developed in axially
constrained tubes subjected to finite shearing [20], axial strain developed
in free-end finite shearing tests (cf.[13,21]) as well as differences between
uniaxial compression or tension and torsion experiments [13,21].
Unfortunately, limitations on the applicability of forms of Wp introduced
over the last decade have not been clearly understood nor elucidated.
Moreover, when used in the context of J2 incompressible plasticity theory,
the plastic spin required to reproduce deformation induced anisotropy
effects and second order behaviors is excessive.

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Shear deformation theories

Dinghe Li, in Analysis of Composite Laminates, 2022

2.3.3 Shear correction factors

Since the transverse shear strains are represented as constant along the
thickness direction, the transverse shear stresses will also be constant.
According to the elementary theory of homogeneous beams, the
transverse shear stresses vary at least quadratically through the thickness
in beams and plates. This discrepancy between the actual stress state and
constant stress state predicted by the FSDT is often corrected for 𝑄𝑥 and
𝑄𝑦 by multiplying the integrals in equation (2.37) with a parameter K,
called shear correction coefficient. This amounts to modify the transverse
shear stiffness. The factor K is computed such that the strain energy due
to transverse shear stresses in equation (2.37) equals the strain energy
due to the true transverse stresses predicted by the three-dimensional
elasticity theory. For example, considering a homogeneous beam with
rectangular cross-section (width b and height h), the actual shear stress
distribution through the thickness of the beam is given by

𝑐 3𝑄 2
𝐻 𝐻 (2.43)
𝜎13 = 2𝑏ℎ 𝐻 ) ], − 2 ⩽ 𝑧 ⩽ 2
[1 − (2𝑧

where Q is the transverse shear force. The transverse shear stress in the
𝑓
FSDT is a constant, 𝜎13 = 𝑄 / 𝑏ℎ. The strain energies due to transverse
shear stresses in the two theories are

2
𝑈𝑐𝑠 = 2𝐺1 ∫ (𝜎𝑐13 ) d𝐴 = 5𝐺3𝑄 𝑏ℎ
2
(2.44)
13 𝐴 13
𝑓 𝑓 2 2
𝑈𝑠 = 2𝐺1 ∫ (𝜎13 ) d𝐴 = 2𝐺𝑄 𝑏ℎ
13 𝐴 13

𝑓 𝑐
The shear correction factor is the ratio of 𝑈𝑠 to 𝑈𝑠 , which gives 𝐾 = 5 / 6.
The shear correction factor for a general laminate depends on lamina
properties and stacking sequence, and it has no general expression. The
correct valuation of shear correction factor is the most important
disadvantage for FSDT.

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Brittle Fracture: Micromechanics
A.G. Evans, in Concise Encyclopedia of Advanced Ceramic Materials
, 1991

4.1 Transformation Toughening

The net dilational and shear strains that accompany the martensitic
transformation of a region of material near a crack tip, invariably result in
a change in the local stress immediately ahead of the crack. The local
stress-intensity factor KIloc thus differs from the applied value KI∞ (Fig. 8c)
by an amount that depends on the stress-strain behavior of the
transformation region (Fig. 8b). In brittle solids, the stress-strain relation
exhibits a linear elastic character before and after transformation, with an
essentially constant slope, reflecting the relatively small changes in elastic
moduli that accompany the transformation. The transformations are also
hysteretic, a phenomenon that enables the transformed region to be
retained following a crack advance. An appreciable reduction in KIloc
commences when the crack advances into the transformed zone, resulting
in R-curve behavior. The magnitude of the maximum increase in KIc may
be expressed in terms of the dilational component of the transformation
strain by

𝛥𝐾Ic ≃ 0.3𝑇ɛ 𝐸ℎ1 / 2 𝑉𝑓 (5)

where ɛT is the unconstrained dilational strain, E is Young's modulus, h is

the zone height and Vf is the volume concentration of material subject to
transformation. An additional increment in toughness would pertain in
the presence of a net shear strain.

The zone height h is a particularly important parameter. It is determined

by the transformation stress Pc (analogous to the yield stress in plastic-
zone formation) such that h ≈ (1/2π) (KIc/Pc)2. The stress Pc is known to
depend on temperature, chemical composition and particle size. The
effects reside in the influence of the dominant variables on the net
thermodynamic driving force for transformation and/or on the nucleation
process. The relative importance of the driving force with respect to the
nucleation has not yet been fully ascertained.

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