The Mathematical Theory Of Plasticity R Hill Pdf
A new model for elucidating the mathematical foundation of plasticity yield criteria is proposed. The proposed ansatz uses differential geometry and group theory concepts in addition to elementary hypotheses based on well-established experimental evidence. Its theoretical development involves the analysis of tensor functions and provides a series expansion which allows the functional stress-dependence of plasticity yield criteria to be predicted. The theoretical framework for the model includes a series of spatial coefficients that provide a more flexible theory for in-depth examination of symmetry and anisotropy in compact solid materials. It describes the classical yield criteria (like those of Tresca, Von Mises, Hosford, Hill, etc) and accurately describes the anomalous behaviour of metals such as aluminium, which was elucidated by Hill (1979). Further, absolutely new instances of stress-dependence are predicted; this makes it highly useful for fitting experimental data with a view to studying the phenomena behind plasticity. Comment: 31 pages, 5 figures
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arXiv:0912.0426v1 [cond-mat.mtrl-sci] 2 Dec 2009
Geometrical foundations of plasticity yield criteria:
A unified theory
J. M. Luque∗a R. Campoamor-Stursberg †b ,
aDpto. Ingenier´ıa Mec´anica y de los Materiales,
Escuela Superior de Ingenieros, Universidad de Sevilla,
Camino de los Descubrimientos s/n, 41902 Sevilla
bInstituto de Matem´atica Interdisciplinar,
Universidad Complutense de Madrid, 3 Plaza de Ciencias, 28040 Madrid, Spain.
Abstract
A new model for elucidating the mathematical foundation of plasticity yield criteria is pro-
posed. The proposed ansatz uses differential geometry and group theory concepts in addition
to elementary hypotheses based on well-established experimental evidence. Its theoretical de-
velopment involves the analysis of tensor functions and provides a series expansion which allows
the functional stress-dependence of plasticity yield criteria to be predicted. The theoretical
framework for the model includes a series of spatial coefficients that provide a more flexible
theory for in-depth examination of symmetry and anisotropy in compact solid materials. It
describes the classical yield criteria (like those of Tresca, Von Mises, Hosford, Hill, etc) and ac-
curately describes the anomalous behaviour of metals such as aluminium, which was elucidated
by Hill (1979). Further, absolutely new instances of stress-dependence are predicted; this makes
it highly useful for fitting experimental data with a view to studying the phenomena behind
plasticity.
Keywords: geometrical model, analytic functions, manifold, anisotropic material, elastic-plastic
material
1 Introduction
An accurate description of the structure, formation and behaviour of a solid elastic-plastic material
requires the knowledge, among other facts, of the limiting stress it can withstand before it becomes
plastic [32, 33, 15, 18, 2, 22]. In fact, plasticity concepts are widely used in a number of scientific
and engineering field applications (in materials science, physics of solids, mechanical engineering,
aeronautical engineering, geophysics, biomechanics and chemistry, among others) [4, 8, 1].
Plasticity involves a series of irreversible, history-dependent processes by effect of which a material
develops fluency at a micro-, meso- or macroscopic scale in its transition from an elastic behaviour
to a plastic behaviour [26]. Plastic processes involve plastic dissipation (i.e. , the irreversible release
∗Corresponding author: e-mail: jmluque@us.es
†e-mail:rutwig@pdi.ucm.es
1
of stress or energy with energy transfer in the material). In fact, plasticity involves a variety of
processes at different spatial scales having an also different associated grain size. In addition to its
physical reality to plastic processes, grain size defines characteristic spatial scales for macroscopic
plasticity in materials [19].
Unlike liquids and gases, solids are highly ordered systems, contain a vast amount of internal
information and exhibit a high correlation among its constituent elements. As a result, plastic
microscopic processes in solids (movement of dislocations, defect, etc) are usually relatively complex
[19]. However, such processes exhibit some macroscopic symmetry by virtue of the solids structure
and the physical laws they obey [4, 26, 5, 6]. This facilitates the macroscopic examination of solids
by using a combination of differential geometry and group theory [3, 30]. Thus, a plastic process
can be interpreted as a series of local transformations that possess some symmetry and provide local
information useful with a view to establishing a global statistics for a solid material.
The aim of this paper is to develop a unified approach of plasticity yield criteria that uses
elementary hypotheses based on well-established experimental evidence. This study is based on the
analysis of the transformation properties of the Cauchy stress tensor under orthogonal mappings,
where arguments from the theory of Lie groups are applied. Also, it discusses the physical arguments
for application of each criterion in relation to specific properties of the material concerned. This can
be useful with a view to developing new criteria to address some special mechanical properties of
materials (like anisotropy, hardening-softening, etc).
This paper is organized as follows. Section 2 describes some topics about plasticity in the
framework of the internal variables theory. Section 3 presents a short description of classical plasticity
criteria. In section 4 we show the postulates and approximations of the unified theory. From section
5 onwards we develop this unified theory. Finally, a set of conclusions is shown.
2 Plasticity function and plastic potential. Flow rule
Within the framework of the Internal Variables Theory (or hidden variables), an inelastic solid is
one in which the strain at any point of the solid is completely determined by the current stress
and temperature there plus a set of internal variables [26, 28]. The internal variables (scalars or
tensors) have physical or mathematical meaning and allow to complete the internal description at
any point of the solid (for example: the past history of the stress and temperature at the point,
large-deformation plasticity, hardening and softening, structural and induced anisotropy, etc) [26,
29]. Thus ε is a function of the material state (σ, T , ξ ) at any point
ε= ε( σ, T, ξ ) (1)
where ε denotes strain variables, σ stress variables, T the temperature, and ξ some internal variables.
Additionally, the rate of evolution of the internal variables ˙
ξis determined by the state
˙
ξ=˙
ξ( σ, T, ξ ) (2)
known like equations of evolution or rate equation for internal variables.
For inelastic solids it is generally assumed that strain variables can be decomposed additively
into elastic strain εe and inelastic strain εi [8]
2
ε= εe +εi (3)
where the inelastic strain occurring in rate-independent plasticity is usually denoted by εP rather
than by εi , and is called the plastic strain.
In the context of internal variables theory and rate-independent plasticity, a plasticity yield
criterion consists of a series of mathematical conditions mutually relating stress, temperature and
internal variables, which define the material states (σ, T, ξ ), where the point of the solid concerned
becomes critically plastic. A critical state in this context is a state where the elastic-plastic material
starts to yield plasticity. Therefore, these conditions constitute a boundary (plastic limit) between
the elastic and plastic state in the solid material (at points). Mathematically, plasticity yield criteria
are formulated in the following general form:
f( σ, T, ξ ) = 0 (4)
where f denotes the plasticity function at a given plastic state (σ, T , ξ ). Therefore, the plasticity
yield criterion of a solid (at points) defines the stress multiaxial states where it will yield critical
plasticity; the set of such states describes the criterion surface of the solid (4).
Plasticity can be defined in basic terms by using some approximations that usually hold in
practice. Thus, the influence of the temperature T on solids at a constant, ambient level is usually
negligible provided they are scarcely sensitive to changes in this variable and far from their melting
point (temperature-independent plasticity). The influence of strain is usually negligible if we consider
all viscoplastic processes to be "infinitely" slow compared to the material relaxation time (rate-
independent plasticity) [26]. Time-independent plasticity needs usually to be considered if the target
is not the time-evolution of the solid. In the plastic limit, an index P is used to denote parameters
and quantity values in such a limit. The basic plastic unit in a solid is the macroscopic spatial point,
which can be equated to the physical concept of grain but need not coincide with it. Also, because
a solid usually exhibits high correlation among its elements, characterizing each point in it requires
defining its correlation with its neighbourhood. As in the theory of elasticity, this entails describing
the stress state at each point in terms of a second-order symmetric tensor (2-tensor) σ≡ σij , called
the Cauchy tensor , the symmetry of which arises from the stress equilibrium relation at the point
in question [24]. Therefore, in these conditions, the plasticity yield criterion (4) at each point in a
solid can be defined as follows:
f(σij ; ξ) = 0 (5)
which represents the criterion surface in the stress space.
The plastic potential, Φ, is a measure of smoothness (differentiability) and convexity of f . Thus,
if the plasticity function f is smooth (differentiable) and convex in stress space, then it will coincide
with a specific plastic potential (f = Φ). The plasticity function f = Φ is convex if for a given
stress and strain rate condition σij and ˙ εP
ij , any stress σ ∗
ij inside or on the criterion surface obeys
the following relationship [26]:
(σij − σ ∗
ij ) ˙ εP
ij ≥0 (6)
where ˙ εP
ij denotes the plastic strain rate. Specifically, if f = Φ is twice differentiable, then Φ will be
convex if, and only if, its Hessian matrix, defined as
3
Hij = ∂ 2 Φ
∂σi∂σj
(7)
is positive semi-definite (i.e. if the eigenvalues of H ij are only positive or zero). When the eigenvalues
are only positive non-zero (positive definite matrix), f = Φ is strictly convex. We have used here
—as we have throughout— the Einstein summation convention for repeated indices.
If f is smooth (differentiable) and convex, then ˙ εP
ij will be unique in any plastic stress state
σij . Under these conditions, the function f = Φ can be assigned a flow rule that gives the plastic
strain rate. Within the framework of the plastic potential theory of Von Mises (1928), each plastic
potential Φ has a flow rule that is associated with the plastic potential. This flow rule is given by
˙ εP
ij =˙
λ∂Φ
∂σij
(8)
where σij is the Cauchy tensor, ˙ εP
ij the plastic strain rate tensor or plastic flow tensor and ˙
λa positive
plastic multiplying factor. Based on eq. (8), if f = Φ, then the plastic strain rate ˙ εP
ij at the very
start of plasticity will be normal to the criterion surface f = Φ; this constitutes the so-called plastic
flow normality rule. Hecker [13] conducted a systematic study of a large amount of experimental
data in metals and found that the normality rule was never broken. However, there is evidence that
flow normality rule does not hold in soils, granular materials, etc (materials with non-associated flow
rules, where ∂f /∂σij is not proportional to ∂ Φ /∂σij ) [8, 26].
Defining rate-independent plasticity requires introducing the concept of plastic dissipation, which
is expressed as
d= σij ˙ εP
ij (9)
The parameter d is a measure of the power per volume that is lost (dissipated), usually as heat,
through deformation. The plastic dissipation as defined in eq. (9) for the flow rule associated to f
will be maximal if, and only if, the function f = Φ is convex (9); this is the so-called principle of
maximum plastic dissipation [31, 16].
Establishing the total dissipation over a given time interval entails defining in terms of time-
dependent plasticity the amount of irreversible work per unit volume due to stress as follows:
χ( t) = WP ( t) = Z t
0
σij ( t ) ˙ εP
ij (t)dt (10)
χ( t) is also known as the work-hardening parameter (an internal variable), which is a scalar quantity
and dependent on the plastic time-evolution of the particular solid.
For the study of the hardening and softening of the solids materials is necessary the evolution of
plasticity function with the internal variables, ξ = ξα , given by [26]
˙
f
σ=const., T =const. = X
α
∂f
∂ξα
˙
ξα = λ X
α
∂f
∂ξα
hα ≡ − λH (11)
where H > 0 for hardening materials, H < 0 for softening materials, and H = 0 for perfectly plastic
materials (in the latter case f is independent of the ξα ). Here λ is a positive continuous function of
state variables.
4
Based on the foregoing, for the theory to be properly addressed, the plasticity function f should
be convex in the stress-related variables (f = Φ). The dependency of f on ξ describes the anisotropy
(structural and induced) of materials, while the dependency of ˙
fon ξdescribes the hardening-
softening of materials.
3 Plasticity yield criteria
The following section describes the most relevant plasticity yield criteria, with emphasis on their
particularities (see [32, 33, 15, 18, 2, 22]).
The Tresca criterion [32] which is among the earliest plasticity yield criteria, can be expressed
as a unique function of the algebraic invariants (J1 , J2, J3 ) of the stress deviation tensor J . Based
on it, a solid will become plastic when it reaches a multiaxial state where the multiaxial tangent
stress equals the critical uniaxial tangent stress, σP . This criterion can be expressed as a completely
differentiable relation
f(σij ; σP ) = (σxx −σyy )2 + 4 σ2
xy −4σ 2
P[ σ yy −σ zz ) 2 + 4σ 2
yz −4σ 2
P
×[ σzz −σxx )2 + 4 σ 2
xz −4σ 2
P(12)
based on which plasticity can only be reached on independent planes, as revealed by factoring the
total (volume) multiaxial state into its partial (surface) multiaxial state. In other words, plasticity
develops on planes. This criterion exhibits good agreement with experimental results for certain
ductile metals. At microscopic level the movement of dislocations along slip planes is responsible
for permanent deformation [4]. Note that σP depends on the internal variables, ξ , i.e. σP = σP ( ξ ).
The Huber–Von Mises criterion [33, 21] is among the most widely rule used in this context.
It is also known as the J2 - criterion since it is formulated as a unique function of the algebraic
invariant J2 of the stress deviation tensor. Based on existing experimental evidence, this criterion
is applicable to ductile materials. Hencky [14] provided an energy-based interpretation by which
the critical plasticity state is reached when the distortion energy per volume (i.e. , the deformation
energy per volume in the absence of volume changes) in the multiaxial tangent stress state equals the
distortion energy per volume in the critical uniaxial normal stress state. This criterion is isotropic
and can be formulated as follows:
f( σij ; σP ) = (σxx −σyy )2 + ( σyy −σzz )2 + ( σzz −σxx )2 + 6( σ2
xy +σ 2
yz +σ 2
xz )−2σ 2
P(13)
Stress-wise, the Von Mises criterion indicates that critical plasticity is reached when the modulus
of the multiaxial tangent stress equals that of the critical uniaxial normal stress, σP . Note that σP
depend on the internal variables, ξ , i.e. σP = σP ( ξ ). Unlike the Tresca criterion, the Von Mises
criterion does not factor the plasticity function into multiaxial plane stress components; rather, it
assumes a mutual dependence among the multiaxial stresses, which leads to a volume plasticity
expression. In ductile materials, which meet the Von Mises criterion quite well, plasticity results in
distortion by effect of flow processes occurring with virtually no volume change.
The Hill's anisotropic criterion or Hill's first criterion [15, 17] provides a general description of
materials with anisotropy (whether structural or induced) and orthotropic symmetry (i.e. , materials
where each point possess three mutually normal planes). Because each plane in an orthotropic
5
material can be defined in terms of only two parameters, Hill's criterion can be formulated in terms
of six independent parameters. This constitutes a generalization of the Von Mises criterion to
anisotropic materials of the form
A(σyy −σzz )2 + B (σzz −σxx )2 + C (σxx −σyy )2 + 2 Dσ2
yz + 2Eσ 2
xz + 2F σ 2
xy = 1 (14)
where A = A( ξ ) , B =B (ξ), C =C (ξ), D =D (ξ), E =E (ξ), F =F (ξ ), depending on the internal
variables, are constants defining the degree of anisotropy in each direction and can be expressed in
terms of the Lankford coefficient. In addition, the directions x, y, z should be the principal anisotropy
directions for the material —otherwise, the Cauchy tensor should be transformed as required in order
to have it coincide with the principal directions. In fact, eq. (14) is a particular case [26] of
σij Bijkl σkl = σ 2
P(15)
which is the more general quadratic form for the components of the stress tensor as defined in terms
of the 4-tensor Bijkl . This tensor fulfils the symmetry conditions for 4-tensors in the theory of
elasticity, which are given by
Bijkl =Bklij
Bijkl =Bjikl (16)
where the first relation defines the symmetry by joint exchange of index pairs ij ↔ kl and the
second the symmetry by exchange between index pairs of the type i↔ j and/or k↔ l . This
criterion considers no effects of the mean stress (hydrostatic pressure) on plasticity; therefore, one
must introduce the additional condition Bijkk = 0, which allows the quadratic form of such effects
to be neglected.
The Hosford criterion [20] and Logan–Hosford criterion [25] are two generalizations of the Von
Mises criterion that ensure convexity by introducing a parameter m≥ 1. The Hosford criterion,
which is applicable to isotropic materials, is defined as
f(σij ; σP ) = 1
2|σ1 −σ2 |m +1
2|σ2 −σ3 |m +1
2|σ1 −σ3 |m −σ m
P(17)
where σ1 , σ2, σ3 are the principal stresses of the Cauchy tensor. Note that σP depends on the internal
variables,σP = σP (ξ ). The Logan–Hosford criterion is a generalization of Hill's anisotropic criterion
(14) that is used to describe anisotropic materials and expressed as
A′ |σ1 −σ2 |m + B′ |σ2 −σ3 |m + C′ |σ1 −σ3 |m = 1 (18)
where the constants A′ = A′ ( ξ ) , B′ =B′ (ξ), C ′ =C′ (ξ ), depending on the internal variables,
constitute measures of anisotropy in each direction and can be expressed as a function of the Lankford
coefficient.
Hill's generalized anisotropic criterion or Hill's second criterion [18] provides an accurate descrip-
tion of the anomalous behaviour of some metals such as aluminium [37]. The criterion is expressed
in terms of principal stresses of the Cauchy tensor:
a|σ2 − σ3 |m + b|σ3 − σ1 |m + c|σ1 − σ2 |m + d|2 σ1 − σ2 − σ3 |m +
e|2 σ2 − σ1 − σ3 |m + f|2 σ3 − σ1 − σ2 |m = σm
P(19)
6
where m ∈ ℜ is a parameter that must fulfil the condition m≥ 1 for the criterion surface to be
convex. Also the constants a =a (ξ), b =b (ξ), c =c (ξ), d = d(ξ), e =e (ξ), f =f (ξ), σP = σP (ξ)
depending on the internal variables, ξ , and which can be expressed as a function of the Lankford
coefficient, measure anisotropy in each principal direction. For example, if the principal directions
1–2 in a material define a symmetry plane with mutual isotropy and are anisotropic with respect to
direction 3, then a =b and d =e (planar isotropy). On the other hand, if all three directions are
isotropic, then a =b=c and d =e =f (complete isotropy).
In the plastic potential theory of Von Mises [34], each plastic potential Φ is assigned a flow rule.
The plastic potential for the particular case of the Von Mises criterion is J2 and its associated flow
rule defined by the Levy–Mises equations. Koiter [23] developed a generalization of the previous
theory where the plasticity function is defined by a series of plastic potentials Φi each having an
associated flow rule of the type described by eq. (8) above. If the plasticity function f is expressed
as a linear combination with positive coefficients of the potential functions Φi (convex), then f will
be convex.
The plastic anisotropy description of Barlat et al. [2] originated from an isotropic plasticity
function. Structural anisotropy in the material was introduced via a series of linear transformations
represented by a 4-tensor acting on the Cauchy tensor, the latter itself acting on the anisotropic
material. By effect of the transformations, the stress tensor absorbs structural anisotropy of the
material.
The Karafillis–Boyce criterion [22] uses a convex combination of two plastic potentials as plas-
ticity function. The potentials are based on Hosford's isotropic criterion (17) and their degree of
mixing is adjusted via parameter c∈ [0, 1]. Thus, the isotropic criterion is formulated as
(1 −c ) [|s1 −s2 |m + |s2 −s3 |m + |s3 −s1 |m ] + c 3 m
2m−1 + 1 [|s1 |m +|s2 |m +|s3 |m ] = 2σ m
P(20)
where m is positive and non-zero, and s1 , s2, s3 are the principal values (eigenvalues) of the stress
deviation tensor. Note that σP , c depend on the internal variables, σP = σP ( ξ ) , c =c (ξ ). This
criterion is a particular case of Hill's second criterion (19). The plasticity yield criterion (20) repre-
sents an isotropic, convex criterion that can be made anisotropic by applying linear transformations
representing a 4-tensor acting on the Cauchy 2-tensor, which in turn act on the anisotropic mate-
rial (Barlat et al. , 1991). Introducing appropriate symmetries of a material in the transformation
4-tensor allows all possible states of structural anisotropy in the material to be considered. The
Barlat and Karafillis–Boyce criteria rely on the theory of representation of tensor functions [35].
Those criteria that consider hydrostatic pressure dependence assume plasticity in some materials
including metallic foams and polymers to be a function of the hydrostatic pressure σM acting on
the solid. This effect has been considered by using various general criteria such as those of Drucker–
Prager [10, 11], Caddell [7] and Deshpande [9], which are modifications of the Von Mises and Hill
criteria including a certain dependence on the hydrostatic pressure (σM ).
The plasticity yield criterion is established from the set of microscopic and macroscopic properties
of the material. As a result, the formulation of each criterion depends on a combination of parameters
of the material describing its anisotropy, crystal structure and hydrostatic pressure-dependence,
among other properties. There are three general types of models for analysing plasticity, namely:
microstructure or macrostructure and mixed. Microstructure models establish plasticity yield criteria
from the microstructure of each material. On the other hand, macrostructure models, also referred
to as phenomenological models, rely on a phenomenological analysis of the macroscopic behaviour
7
of each material to establish such criteria. Finally, mixed models are combinations of the previous
two and usually provide the more accurate descriptions of plasticity.
4 A unified theory of plasticity
The objective of this paper is to develop a new macroscopic theory that unifies the plasticity yield
criteria. This theory is based on postulates well-established from experimental data and theoretical
considerations. The used method is based on orthogonal Lie Groups to describe the classical isotropic
yield criteria; an increase of the symmetry group allows to consider classical and new anisotropy
yield criteria in the solid materials (new mechanical properties).
At microscopic level, the plasticity in a solid is produced by a set of slips, generated by the
movement of dislocations, defects, etc [19]. The stress acting on the solid is the generator of these
slips. We want to take into account all these microscopic dynamical processes at the macroscopic
scale. So the theory is based on the decomposition of a solid material into parts as macroscopic
points p (Figure 1). These macroscopic points do not need to have any physical reality (grains
or others), but points have macroscopic information about microscopic dynamical processes. This
macroscopic information is purely statistical, so a macroscopic point is a statistical concept that
takes into account all the microscopic information inside it. The set of slips inside a macroscopic
point induces the irreversible macroscopic movement of this point (with a well determined spatial
directionality). The global movements of the points result in macroscopic plasticity that follow
macroscopic laws (theory postulates). So, the proposed unified theory examines plasticity at each
macroscopic point p in a solid. The plasticity of the macroscopic points are described by orthogonal
transformation groups (directionality of plasticity) acting on Cauchy tensor (generator of plasticity)
in a convex manner (measure of plasticity).
The macroscopic anisotropy description by coordinate tensors appears in the infinitesimal or-
thogonal transformation of Cauchy stress tensor (which acts on the solid point) with respect to
a reference system (for the solid point). The directions of these infinitesimal transformations are
limited only by the internal structure (microscopic dynamic processes) of macroscopic solid points.
These directions of transformation (anisotropy characterization) are described by the coordinate
tensors (parameters that depend on the internal variables). A measure (mathematical norm) of the
infinitesimal orthogonal transformation of Cauchy tensor gives the macroscopic effect of plasticity
(plasticity function). The ensuing point information can be used to obtain a global points statistic
for the whole solid.
Certain approaches are considered in this theory. The potential effects of temperature and
strain are ignored (temperature-independent and rate-independent plasticity), and so is the time-
dependence of all parameters and variables since the only target is critical plasticity in the solid
(time-independent plasticity). Our theory relies on the following three postulates:
1. First postulate: the macroscopic units or points that become plastic in a solid are described
by second-order Cauchy tensors.
2. Second postulate: hydrostatic-pressure independent plasticity acts at every point in the
Tangent Space to the Cauchy Tensor (TSCT). Any type of plasticity at least needs a component
in this space TSCT.
3. Third postulate: the plasticity function is convex in the stress space.
8
Within the framework of the proposed theory, solids are represented in a three-dimensional Euclidean
space E (3) where each point or neighbourhood p∈ E (3) is defined by the pair p ≡ {pλ , σkl } ∈ E (3),
with indices λ, k, l = x, y , z . The set of points is referred to E (3) via a coordinate system or reference
system { O, xλ = (x, y, z )} that is shared by the whole solid; the coordinate system has O as its origin
and xλ = (x, y, z ) as its axes. Therefore, a point p in the solid lies at pλ in the reference system
and has the symmetric second-order stress tensor or Cauchy tensor acting on it, i.e. ,σ≡ σ kl (first
postulate); this postulate is valid only for solids with "near-action" internal forces [24]. Since the
solid is examined point-wise, each point p possesses its own reference system, { Op , xλ = (x, y, z)},
with its origin at pλ and xλ = (x, y, z) as axes. So the set of stresses acting on the points in the
solid define a 2-tensor stress field in the space E (3), which endows the solid with geometric structure
(Figure 1).
The reference system for each point, { Op , xλ = (x, y, z)} , is arbitrary only when the ensuing
equations are invariant under a reference system change (tensor equations), that is, independent
of the particular reference system. From the geometric point of view this guarantees that the
tensor equations that describe the solid contain, codified as geometric information, all the physical
behaviour throughout the directions of reference system changes for the corresponding space.
If the Cauchy tensor is invariant under a change in the origin and axes over the solid, { Op 1 , xλ =
(x, y, z )} → { Op 2 , ¯ xλ = (¯ x, ¯ y, ¯ z)} , then the solid is stress-homogeneous. If the Cauchy tensor is
invariant under a change in the axes directions, { Op , xλ = (x, y, z )} → { Op , ¯ xλ = (¯ x, ¯ y, ¯ z)} , then
the stresses at the point p in the solid are isotropic. The degree of homogeneity in the solid can be
assessed by examining it point-wise and using the results to develop a global statistics for the entire
solid. In this work, we will assume the behaviour of the solid is similar to its individual points or,
in other words, that examining a single point will be the same as studying the entire solid.
The intrinsic properties of the solid at each point are examined via a series of transformations
with a fixed origin on it. A transformation of the Cauchy tensor, σ kl , at point p can be interpreted
in two completely equivalent ways. In the first one, the reference system xλ has the Cauchy tensor
σkl ; therefore, a transformation xλ → ¯xλ in the reference system will lead to a new reference
system ¯xλ with a Cauchy tensor ¯ σij ; this constitutes a reference system transformation or active
transformation. In the other interpretation, the reference system xλ remains unchanged and the
Cauchy tensor transformation, σ kl → ¯ σij , produces a new tensor that is expressed (oriented) in the
reference system ¯xλ ; this is a Cauchy tensor transformation or passive transformation. These two
interpretations are mutually related by a negative sign.
The proposed theory relies on a continuous linear transformation of the stress at each point p
represented by the tensor Aij
kl. Such a transformation does not alter the origin of p, which is defined
by the coordinates pλ , but only its coordinate axes, xλ . Thus, the transformation operator Aij
kl
acts on the stress tensor σ kl to give the transformed tensor ¯ σij in accordance with the following
expression:
Aij
klσ kl = ¯ σij (21)
where the multilinear transformation Aij
kl is a 4-tensor of indices i, j, k , l = x, y, z with two covariant
(k, l ) and two contravariant( i, j ) indices. Mathematically, it follows that this expression is a tensor
equation (i.e. , that it is invariant under a coordinate system change). Since the resulting transformed
tensor, ¯ σij , has a fixed origin (the same that σkl ), the transformation can be tangent or normal to
the tensor σ kl . In the former case, it will produce rotations and/or reflections of use for studying
anisotropy; in the latter case, it will result in dilatation or dilatation–reflection useful with a view
9
to examining hydrostatic pressure-dependence.
The plasticity at a point p under the influence of an arbitrary initial Cauchy tensor depends on
the tangent and normal spaces to the Cauchy tensor. These spaces arise directly from the geometric
generalization of the two types of components of the Cauchy tensor. The direct sum of the tangent
and normal spaces constitutes the overall stress space at the point concerned. As implied by our
second postulate, a solid will yield plasticity at a given point only if a component on the tangent
space exists at such a point. As a result, plasticity starts in the associated pure tangent space
(hydrostatic-pressure independent plasticity) or, in other words, it never starts in the associated
pure normal (hydrostatic-pressure dependent plasticity) space. In fact, there is solid experimental
evidence that plasticity never arises under conditions of pure hydrostatic pressure [4, 26]. Thus,
some solids exhibit slight elastic (reversible) deformation rather than plasticity, even at very high
pure compressive hydrostatic pressures. However, high stress-induced hydrostatic pressures in a solid
under tension can lead to an unexpected spatial stress concentration eventually leading to fragile
fracture [5, 6].
5 Rotational transformation of the Cauchy tensor
Properly examining plasticity in a solid entails starting in the Tangent Space to the Cauchy Tensor
(TSCT) associated to each point p in the solid, and applying a pure orthogonal transformation R ij
kl
such that
Rij
klσ kl = ¯ σij ,(22)
where the orthogonality condition is described by means of the matrix equation RT R =I . Based
on this tensor equation, the Cauchy tensor σ kl associated to each point p in the solid is rotated by
Rij
kl according to a given parameter, which provides the new associated tensor ¯ σij , which will be the
new associated tensor acting on p . Stress rotations according to equation (22) provide a point-wise
definition of hydrostatic-pressure independent plasticity in the solid. In this sense, rotations of the
Cauchy tensor at p can be interpreted physically as due to some tangential stresses acting on the
point. In this respect, the hydrostatic-pressure independent plasticity interaction (i.e. , that having
no hydrostatic component) is fully defined by the rotations R ij
kl and their associated transformation
parameters.
The orthogonal rotation transformation R ij
kl is a three-dimensional 4-tensor (i.e., one with 3 4 = 81
components) equivalent to a Mohr transformation if parameterized in Euler angles ΘE = ΘE (α, β, γ)
[27]. As noted earlier, the orthogonal rotation transformation is necessary for hydrostatic-pressure
independent plasticity to develop at individual points in a solid. However, additional, non-orthogonal
transformations (e.g. , dilations) can also be applied that will alter the hydrostatic-pressure indepen-
dent plasticity conditions imposed by a rotation (see Section 11). The remainder of this section,
and all subsequent ones up to the eleventh, is devoted to examine hydrostatic-pressure independent
plasticity as defined in eq. (22). Section 11 is concerned with hydrostatic pressure-dependence.
The transformation equation (22) connects geometric objects via double indices (i.e. , 9 compo-
nents). However, an equivalent representation connecting objects via single indices (i.e. , 3 compo-
nents) can be obtained by using the following relations:
¯ ui = Ri
ju j ;u j =R j
i¯ ui (23)
10
uj = Tjq vq ; ¯ uj = Tj q ¯ vq (24)
where (23) defines the rotational transformations of contravariant vectors and (24) the general
contravariant transformations for a vector under coordinate system changes (i.e. , the definition of a
1-tensor). From the rotational transformation (23) and general coordinate transformation (24) for
each 1-tensor, one can obtain
¯ ui = Ri
ju j =R i
jT jq v q =R i
jT jq R p
q¯ vp
¯ ui =¯
Tip ¯ vp ⇒ ¯
Tip = Ri
jT jq R p
q(25)
Since the coordinate system change occurs via the stress 2-tensor (Cauchy tensor) T jq ≡σ jq , we
can express the transformation by means of a rotational operator
¯ σij = Ri
kσ kl R j
l⇔¯ σ= RT σR, (26)
the latter describing the transformation rule for the Cauchy tensor. It is straightforward to verify
that this expression is equivalent to (22). The 2-tensor rotations of the types R i
kand R j
lcan be
adequately realized in terms of 3 × 3 matrices. Relation (26) is invariant under a general change of
the coordinate system.
Therefore, the transformationsR i
k,R j
llead the plastic process at each point p in the solid. Their
study is facilitated by considering the three-dimensional special orthogonal group SO (3) of rotations
in the three-dimensional Euclidean space E (3). The main advantage of this approach is that SO(3)
possesses both the structure of a group of transformations and a differentiable manifold, which
enables us to confer additional invariance properties to the tensor operators defined over it [12].
The elements of the special orthogonal group SO(3) are the orthogonal transformations of the
Euclidean space E (3) that describe orientation and length preserving movements. Such rotations are
usually represented by orthogonal real 3 × 3 matrices RT R =I with unit determinant (det R = +1),
and the group product operation is the usual matrix multiplication. Topologically, SO (3) is a
compact, non-simply connected group [30]. As Lie group, S O (3) is simple, i.e. , the only normal
subgroups it contains are the trivial ones: itself and the identity group. In particular, its Lie algebra
so(3), which coincides with the tangent space at the identity element, is also a simple Lie algebra.
As a consequence of the Lie structure the tangent bundle T S O (3) ≡ S ρ ν ∈SO(3) T SO (3)ρ νinherits
special properties that will be useful in our later analysis.
6 Action of the group SO (3) on the Cauchy tensor
Using the adjoint representation of SO (3), the action of the rotations R ij
kl in (22) or the equivalent
pair R i
k,R j
lin (26) can be explicitly expressed in terms of the Euler angles Θ E = Θ E (α, β , γ) as:
¯ σxx =σxx cos2 γ+ σyy sin2 γ + 2 σxy sinγ cos γ
¯ σyy =σyy cos2 γ+ σxx sin2 γ−2 σxy sinγ cos γ
¯ σzz =σzz
¯ σxy =σxy (cos2 γ− sin2 γ ) + ( σy y −σxx )sinγ cos γ
¯ σyz =σyz cos γ− σxz sinγ
¯ σxz =σyz sinγ + σxz cos γ
(27)
11
¯ σxx =σxx cos2 β+ σzz sin2 β + 2 σxz sinβ cos β
¯ σy y =σyy
¯ σz z =σzz cos2 β+ σxx sin2 β−2 σxz sinβ cos β
¯ σxy =σyz sinβ + σxy cos β
¯ σy z =σyz cos β− σxy sinβ
¯ σxz =σxz (cos2 β− sin2 β ) + ( σzz −σxx )sinβ cos β
(28)
¯ σxx =σxx
¯ σyy =σyy cos2 α+ σzz sin2 α−2 σy z sinα cos α
¯ σzz =σzz cos2 α+ σyy sin2 α + 2 σyz sinα cos α
¯ σxy =σxy cos α− σxz sinα
¯ σyz =σyz (cos2 α− sin2α) + ( σyy −σzz )sinα cos α
¯ σxz =σxy sinα + σxz cos α
(29)
A rotation at the point p therefore represents a transformation from the reference system
{Op , xλ }, with a tensor σ kl , to the new reference system {Op ,¯xλ } , with associated tensor ¯ σij . The
components of the Cauchy tensor in the rotated reference system ¯ σij are expressed in terms of σkl
by means of equations (27) - (29).
The rotation by angle ΘE = Θ E (α, β, γ ) can be easily visualized by viewing the reference system
in the direction from the positive ends of the axes to their origin and rotating xλ anti-clockwise to
the new system, ¯ xλ , by an angle α, β and γ about the x ,y and z axis, respectively (see Figures
2 and 3). Obviously, because of the non-Abelianity of SO (3), the order in which the three partial
rotations are performed strongly influences the general rotational outcome.
The action of a generic rotation on the Cauchy tensor is the result of applying successive partial
rotations (Figure 3) about each axis in a given sequence. Thus, applying the rotations following the
sequence α→ β→ γ , we get
¯ σij = Rij
kl(Θ E )σ kl =R ij
kl (γ, β, α)σ kl (30)
which is equivalent to the following expression for rotations in the group SO(3):
¯ σij = [ R( γ) R( β) R(α)]i
kσ kl [R(α)R(β)R(γ)] j
l⇔¯ σ= RT (γ , β, α)σR(α, β, γ ) (31)
with partial rotations performed following the sequence α→ β→ γ . We observe that, although
equations (27) - (29) depend on the explicit choice of rotation parameters, the transformation does
not alter the form of equations (30) and (31), which, as noted earlier, are invariant under a change
of coordinate system.
7 Infinitesimal action of SO (3) on the Cauchy tensor
Because of the properties of Lie algebras and the local diffeormorphism on a neighbourhood of the
identity element [12], the action (31) on the Cauchy tensor can be also described in terms of the
infinitesimal rotations, which turn out to be the generators of so(3).
The Lie group SO (3) and its associated Lie algebra, so (3), are related via the exponential
mapping. Given an element Xν ∈ so(3), the exponential map provides an element of SO(3) by
means of the relation:
12
Exp( Xν ) = R ( θν ) = eθ ν X ν .(32)
Since SO (3) is a compact group, it is ensured that Exp is a surjective mapping [30]. Differentiation
with respect to the parameter and evaluation at the identity element provides the operator X:
∂
∂θν R(θ ν )
θ ν =0 =Xν
d
dΘR(Θ E )
Θ=0 =ρν∂
∂θν R(θ ν )
θ ν =0 =ρν Xν =X (33)
The operators X in the Lie algebra therefore correspond to infinitesimal rotations and can be
expressed as linear combinations of the operators for the local basis Xν ≡ ∂/∂θν in the Lie algebra
so(3). As a result, each infinitesimal rotation can be expressed as X= ρν Xν ≡ ρν (∂/∂θν ), which
has ρν as its local coordinates.
The actions of infinitesimal rotations on the stress field are determined by the exponential map
(32), which relates the group and the Lie algebra; this equation is a function of the infinitesimal
generators Xν of each partial rotation. A first-order series expansion about the angle θν = 0 (group
identity) yields the expression:
R( θν ) = eθ ν X ν =I +Xν δθν +... (34)
where ΘE = ΘE (α, β, γ )≡ ΘE (θ1 , θ2, θ3 ) denote the Euler angles associated to the first-order
infinitesimal rotations δθ ν , which can take the values δθ ν = δα, δβ , δγ . In addition, each infinitesimal
angle δθν has an associated infinitesimal generator Xν ≡ ∂/∂θν . Substituting (34) into the rotational
expression for a 2-tensor [eq. (31)] yields the expression describing the action of partial infinitesimal
rotations, by an angle θν , on the Cauchy tensor. For a first-order approximation, such an expression
is of the form
σ( θν + δθν ) = ( I+ Xν δθν )T σ( θν ) ( I+ Xν δθ ν ) =
=σ (θν ) + {σ (θν )Xν +X T
νσ(θ ν )}δθ ν +O 2 (δθ ν )(35)
where eq. (31) has been applied on the assumption that ¯ σ= σ( θν + δθν ) and σ= σ( θν ) (i.e. , the
neither the origin nor the rotation centre of the tensor is altered, since the transformation of the
Cauchy tensor involves rotations only, so that the tensor remains unchanged but is evaluated at
a different angle). Equation (35) can be used to establish the first-order variation at each partial
angle, which is of the form
σ( θν + δθν )− σ( θν ) = σ( θν ) Xν δθν + ( Xν )T σ( θν )δθ ν ⇔
⇔δσ
δθν =σX ν + (X ν ) T σ= [σ, X ν ](36)
where the infinitesimal generators Xν for SO (3) are antisymmetric and the binary operation [σ, Xν ] =
σXν − Xν σ is the commutator between the stress field and the infinitesimal generators Xν . Equation
(36) is a function of the specific basis Xν associated to the Euler angles. Expressing it in terms of an
invariant generic infinitesimal rotation X describing the tangent spaces to SO (3) in any arbitrary
coordinate system entails introducing the local coordinates ρν associated to the local basis Xν for
the infinitesimal rotation X.
This results in a first-order variation of the tensor stress field σ with the invariant generator
(infinitesimal rotation) X =ρν Xν of the form
13
dσ|1 ≡ [ σ, X ] = σX − Xσ = ρν δσ
δθν (37)
which is the equation for a 2-tensor. Since the operator X is antisymmetric, then dσ|1 ≡ [ σ, X ] =
−[X, σ ] ≡ − dX |1 , based on which the variation of σ with X is equivalent to that of X with σ with
the opposite sign. This is consistent with the two interpretations of a transformation provided in
Section 4 as applied to a first-order variation.
The tangent space produced by the first-order variation (37) is a linear combination of Euler
angles (or, similarly, generators Xν ) where each term is linearly dependent on the angles (or gen-
erators); therefore, eq. (37) comprises a linear basis of generators Xν (an Euclidean space). This
linear approximation describes TSCT at a first-order local level only (i.e ., in very small or local
neighbourhoods) (see Figure 4). Expanding such neighbourhoods entails using higher-order approx-
imations involving non-linear combinations in the Xν generator basis (i.e. , non-Euclidean spaces for
the different orders) (Figure 5). As shown below by explicit calculations, all spaces approximating
TSCT are traceless (i.e. , the complete approximation are in fact tangent to the Cauchy tensor).
This allows TSCT to be described by means of expansion series of different order.
If one assumes anisotropy at each point in a solid to be more accurately described by including
higher order-terms in the series, then the geometric interpretation of such anisotropy suggests that
the more local the space —and the smaller the neighbourhood—, the higher will be the degree of
isotropy. The inclusion of terms of an increasingly higher order gradually makes the local structure
complex enough to facilitate extension of each infinitesimal neighbourhood, with improves approx-
imation, to the entire TSCT. The complex behaviour of solids as concerning symmetry, anisotropy
and various other properties dictates the structure of TSCT and hence the approximation orders
needed for an accurate description.
Higher-order variations are calculated by applying eq. (37) recursively or, alternatively, expand-
ing the total rotation in (31) up to the required order.
The second-order variation of the stress field σ with the rotation X =ρν Xν can thus be expressed
as the following symmetric 2-tensor:
dσ|2 ≡ [[ σ, X ] , X ] = σXX −2 XσX + XXσ = ρ νµ δ 2 σ
δθνδθµ (38)
which represents the second-order approximation space with coordinates ρ ν µ =ρν ρµ in the operator
basis X νµ =XνXµ =∂ 2
∂θν∂ θµ , the action of which on σ being self-apparent from this expression.
The third-order variation of the stress field σ with the rotation X =ρν Xν can be expressed as
the following symmetric 2-tensor:
dσ|3 ≡ [[[ σ, X ] , X ] , X ] = σXXX −3 XσXX + 3 XXσX − XXXσ = ρ ν µω δ 3 σ
δθνδθµδθ ω (39)
This equation represents the third-order approximation space with coordinates ρ ν µω =ρνρµρω in
the operator basis X ν µω =Xν XµXω =∂ 3
∂θν∂ θµ∂ θω , the action of which on σ is also straightforward.
Higher order approximations are computed by following the same procedure. As a result one
obtains a symmetric n -tensor of coordinates ρν...κ =ρν · ... · ρκ in the basis Xν...κ =Xν · ...· Xκ =
∂n
∂θν · ...· ∂ θκ for each n -order. Also, each n -order variation encompasses a total of 3 + n− 1
n terms,
each of which can be factored into sets of n generators Xν , where the resulting factors represent
14
the basis Xν...κ in the n -order approximation space. Therefore, each n -order is represented by the
corresponding coordinates ρν...κ in the basis Xν...κ , and the coordinates represent a symmetric n-
tensor with 3 + n− 1
n independent components. Note that the n -tensors of coordinates depend
on the internal variables, ρν =ρν (ξ), . . . , ρν...κ =ρν...κ (ξ ); tensors of coordinates can be properly
taken as the position internal variables that describe anisotropy.
Each variation has an equivalent interpretation, whatever its order, since transforming the axes
of the reference system is equivalent to transforming the stresses involved —with a negative sign
(see Section 4) by virtue of the antisymmetric nature of the Lie bracket.
The infinitesimal generators of SO (3) conform to the following commutation relations:
[Xa , Xb ] = ε abc Xc (40)
where ε abc is the Levi–Civita symbol. These generators can be expressed as 2-tensors of the form
X1 =∂
∂α =
0 0 0
0 0 −1
0 1 0
X2 =∂
∂β =
0 0 1
0 0 0
−100
X3 =∂
∂γ =
0− 1 0
100
000
(41)
In summary, using the three generators of the group SO (3) in eq. (41) in conjunction with the
Cauchy tensor σallows us to compute the variation of any order of the Cauchy tensor in order to
obtain the different approximation to TSCT.
8 n-Order variations of the stress tensor field
This section specifically examines the variations of different order described in the previous section,
which provide different approximations to TSCT.
The first-order variation (37) of σ with X can be expressed as
[σ, X ] = ρ1
0σxz −σxy
σxz 2σyz σzz −σyy
−σxy σzz −σyy −2σyz
+ρ2
−2σxz −σyz σxx −σzz
−σyz 0σxy
σxx −σzz σxy 2σxz
+
ρ3
2σxy σyy −σxx σyz
σyy −σxx −2 σxy −σxz
σyz −σxz 0
=
−ρ2 2 σxz + 2 ρ3 σxy ρ1 σxz −ρ2 σyz +ρ3 (σyy −σxx ) −ρ1 σxy + ρ2 (σxx −σzz ) + ρ3 σyz
ρ1 σxz − ρ2 σyz + ρ3 (σyy −σxx ) ρ1 2 σyz − ρ3 2 σxy ρ1 (σzz −σyy ) + ρ2 σxy − ρ3 σxz
−ρ1 σxy + ρ2 (σxx −σzz ) + ρ3 σyz ρ1 (σzz −σyy ) + ρ2 σxy −ρ3 σxz −ρ1 2 σyz + ρ2 2 σxz
(42)
15
which is a real symmetric traceless 2-tensor parameterized in the coordinates ρν . The first-order
variation for the particular case of the Cauchy diagonal tensor is
[σ, X ] =
0ρ3 (σ2 −σ1 )ρ2 (σ1 −σ3 )
ρ3 (σ2 −σ1 ) 0 ρ1 (σ3 −σ2 )
ρ2 (σ1 −σ3 ) ρ1 (σ3 −σ2 ) 0
(43)
and the second-order variation (38) of σ with X is
[[σ, X], X ]11 =ρ 12 2 σxy +ρ 13 2 σxz − ρ 23 4 σyz +ρ 33 2(σyy −σxx ) + ρ 22 2(σzz −σxx );
[[σ, X], X ]12 =− (ρ 11 +ρ 22 + 4ρ 33 )σxy +ρ 23 3 σxz +ρ 13 3 σyz +ρ 12 (− 2 σzz + σyy + σxx );
[[σ, X], X ]13 =ρ 23 3 σxy − (ρ 11 +ρ 33 + 4ρ 22 )σxz +ρ 12 3 σyz +ρ 13 (− 2 σyy + σxx + σzz );
[[σ, X], X ]22 = 2ρ 12 σxy − ρ 13 4 σxz +ρ 23 2 σyz − ρ 33 2(σyy −σxx ) + ρ 11 2(σzz −σyy );
[[σ, X], X ]23 =ρ 13 3 σxy +ρ 12 3 σxz +ρ 23 (− 2 σxx + σyy + σzz )− (4ρ 11 +ρ 22 +ρ 33 )σyz ;
[[σ, X], X ]33 =−ρ 12 4 σxy +ρ 13 2 σxz +ρ 23 2 σyz − ρ 22 2(σzz −σxx )−ρ 11 2(σzz −σyy )
(44)
which is a real symmetric traceless 2-tensor parameterized in the coordinates ρ νµ =ρν ρµ . The
specific expression for the Cauchy diagonal tensor is
[[σ, X ] , X ] =
ρ33 2(σ2 −σ1 ) + ρ22 2(σ3 −σ1 ) ρ12 (− 2 σ3 + σ2 + σ1 )) ρ13 (− 2 σ2 + σ1 + σ3 )
ρ12 (− 2 σ3 + σ2 + σ1 )− ρ33 2(σ2 −σ1 ) + ρ11 2(σ3 −σ2 ) ρ23 (− 2 σ1 + σ2 + σ3 )
ρ13 (− 2 σ2 + σ1 + σ3 ) ρ23 (− 2 σ1 + σ2 + σ3 )− ρ22 2(σ3 −σ1 )− ρ11 2(σ3 −σ2 )
(45)
The third-order variation (39) of σ with X for the particular case of the Cauchy diagonal tensor
is given by:
[[[σ, X], X ], X]11 =ρ 123 6(σ2 −σ3 );
[[[σ, X], X ], X]12 =ρ 333 4(σ1 −σ2 ) + ρ 223 (4σ1 − 3 σ3 −σ2 ) + ρ 133 (− 4 σ2 + 3 σ3 + σ1 );
[[[σ, X], X ], X]13 =ρ 233 (− 4 σ1 + 3 σ2 + σ3 ) + ρ 112 (4σ3 − 3 σ2 −σ1 ) + ρ 222 4(σ3 −σ1 );
[[[σ, X], X ], X]22 =ρ 123 6(σ3 −σ1 );
[[[σ, X], X ], X]23 =ρ 133 (4σ2 − 3 σ1 −σ3 ) + ρ 111 4(σ2 −σ3 ) + ρ 122 (− 4 σ3 + 3 σ1 + σ2 );
[[[σ, X], X ], X]33 =ρ 123 6(σ1 −σ2 )
(46)
which is a real symmetric traceless 2-tensor parameterized in the coordinates ρ νµω =ρνρµ ρω .
The following series expansion encompasses those for all variation orders and defines the overall
TSCT:
dσ = dσ|1 +1
2! dσ|2 + 1
3! dσ|3 + ... = [ σ, X ] + 1
2! [[σ, X ], X ] + 1
3! [[[σ, X], X ], X ] + ..... (47)
Therefore, TSCT is represented by a symmetric traceless 2-tensor (47) that describes hydrostatic-
pressure independent plasticity. If the solid concerned is completely isotropic, using the first term in
(47) will suffice; if it is orthotropic, the first two will be enough. The plastic processes independent
of the hydrostatic pressure are completely characterized by the stress dependences and tensors of
coordinates ρν ,...,ρ ν...κ —in their corresponding bases— in the terms for the different variation
orders in (47).
16
9 The norm space in plastic states
The geometric study of plasticity conducted here involves examining hydrostatic-pressure indepen-
dent plasticity in the tangent space to the Cauchy tensor (TSCT). As shown above, such a space
can be represented by the series expansion (47), which has the typical form for a 2-tensor and gives
both the spatial coordinates (degree of anisotropy) in a specific basis and the stress-dependence of
hydrostatic-pressure independent plasticity. However, determining the plasticity potential (plasticity
interaction) in this situation entails defining a specific metric for the 2-tensor (47). To this extend,
expression (47) —or the corresponding partial summation— is used to calculate the scalar norm g.
The parameters introduced by the definition of the norm are restricted by the constraint that the
plasticity function should be convex (third postulate ).
Plasticity depends both on the externally applied stresses, σ kl , and on the intrinsic properties of
the material, ρν ,...,ρ ν...κ, in a specific reference system or basis. Therefore, the scalar function g
defining the norm will depend on both [i.e. ,g =g (σ kl ;ρν , ..., ρν...κ )]. The function g should provide
the correlation between multiaxial stresses and hence indicate how the different components of the
Cauchy tensor must be related. If (σ kl )P is defined as the 2-tensor describing the plastic limit of
the solid (stress internal variables ), then such a limit will be reached when
g( σkl ; ρν , ..., ρ ν...κ ) = g(( σkl )P ; ρν , ..., ρ ν...κ ) (48)
Usually, the tensor (σ kl )P encompasses a single non-zero uniaxial component and thus coincides
with σP (the scalar uniaxial plastic limit that it is an internal variable). This allows the plasticity
function to be rewritten as follows:
f( σkl ,( σkl )P ; ρν , ..., ρ ν...κ ) = g( σkl ; ρν , ..., ρ ν...κ )− g(( σkl )P ; ρν , ..., ρ ν...κ ) (49)
The scalar function g can be defined in terms of a matrix norm k·k such as Lm (i.e. the usual
m-norm for vectors); therefore,
kdσkm ≡P 3
i=1 P 3
j=1 |dσ ij | m
g=kdσ km ⇔ f=kdσ km − k ( dσ)P km (50)
where the constraints on the norm parameter ( m∈ ℜ, m ≥ 1) ensure convexity in the plasticity
yield criterion (third postulate ). The specific value of m depends on the properties of the particular
material. Norm (50) can be induced from the scalar product of second-order tensors, which is
constructed from the following:
< dσ, dσ′ >≡ dσkl dσkl
′R
−→< RT dσR, RT dσ′ R > ≡ (R i
kdσ kl R j
l)(R k
idσ ′
klR l
j) = dσ kl dσ kl
′(51)
This definition, where rotations R i
kand R j
lare orthogonal (R T R=I ), is invariant under rotations
(R ). The scalar product and its induced norm [eqs (50) and (51)] afford measurement of angles and
lengths in TSCT (47).
The definition must be completed with a scalar product of the coordinate tensors at the different
independent orders (ρν ,...,ρ ν...κ) of the form
ρν...κρχ....ξ = ( ρν )2 · .... · ( ρκ )2 δ ν
χ·.... ·δ κ
ξ(52)
17
The norm induced from this scalar product is Lm for the corresponding n -coordinate tensor.
In summary, plasticity yield criteria can be defined by the norms Lm induced from the scalar
products of stresses (51) and coordinates (52) in the stress and coordinate (position) spaces defined
by the series expansion (47). As a result of this construction, the space (47) is normed.
10 Analysis of plasticity yield criteria. Convexity
The norms defined in Section 9 allow one to assess measurements of variations of any order with a
view to establishing the plasticity yield criterion for each material in terms of its particular properties.
Isotropic criteria can be analysed by using the norm Lm for the first-order variation (42), which
gives
kdσ |1 km = 2
ρ 3
m|σ yy −σ xx | m + 2
ρ 1
m|σ zz −σ yy | m + 2
ρ 2
m|σ xx −σ zz | m +
+ 2
ρ 1
m+ 2
ρ 2
m+ 2 m+1
ρ 3
m|σ xy | m + 2
ρ 2
m+ 2
ρ 3
m+ 2 m+1
ρ 1
m|σ yz | m +
+ 2
ρ 1
m+ 2
ρ 3
m+ 2 m+1
ρ 2
m|σ xz | m
(53)
With complete spatial isotropy (ρ1 =ρ2 =ρ3 ) and invariance (m = 2), the outcome is the Huber–
Von Mises criterion (13). In the particular case of using independent plane stresses xy , yz, xz in
each coordinate directions (0, 0 , ρ3 ), (ρ1 , 0, 0), (0, ρ2 , 0) respectively, taking (m = 2) and factoring in
such a way as to make the criterion smooth (i.e. , differentiable) leads to the Tresca criterion (12).
With complete spatial isotropy (ρ1 =ρ2 =ρ3 ), using the Cauchy tensor in principal stress (Cauchy
diagonal tensor) gives the Hosford criterion (17). These are the most widely used basic plasticity
yield criteria for materials with completely isotropic symmetry.
Anisotropic criteria, which are applicable to orthotropic materials, can be analysed by using the
norm Lm for the second-order variation (44), which gives
kdσ |2 km = 2
ρ 12
m|2σ zz −σ xx −σ yy | m + 2
ρ 13
m|2σ yy −σ xx −σ zz | m + 2
ρ 23
m|2σ xx −σ yy −σ zz | m +
+8
ρ 33
m|σ yy −σ xx | m + 8
ρ 11
m|σ zz −σ yy | m + 8
ρ 22
m|σ xx −σ zz | m +
+ (2 + 2m )· 2m
ρ 12
m+ 2 ·3 m
ρ 23
m+ 2 ·3 m
ρ 13
m+ 2
ρ 11
m+ 2
ρ 22
m+ 2 2 m+1
ρ 33
m|σ xy | m +
+ 2· 3m
ρ 12
m+ (2 + 2 m )·2 m
ρ 23
m+ 2 ·3 m
ρ 13
m+ 2 2 m+1
ρ 11
m+ 2
ρ 22
m+ 2
ρ 33
m|σ yz | m +
+ 2· 3m
ρ 12
m+ 2 ·3 m
ρ 23
m+ (2 + 2 m )·2 m
ρ 13
m+ 2
ρ 11
m+ 2 2 m+1
ρ 22
m+ 2
ρ 33
m|σ xz | m
(54)
For the particular case of the Cauchy diagonal tensor, this leads to Hill's second criterion (19).
Diagonalizing the coordinate tensor ρ νµ and assuming (m = 2), we are led to Hill's first criterion
(14). On the other hand, if both the coordinate tensor, ρ ν µ , and the Cauchy tensor are diagonalized,
then one obtains the Logan–Hosford criterion (18). These criteria hold for anisotropic materials with
orthotropic symmetry (i.e. , orthotropic materials), a general description of which requires six spatial
parameters (the six parameters of the symmetry tensor ρ νµ ).
The criteria of Barlat et al. [2] and Karafillis & Boyce [22] were originally established by
transforming the stress tensor; this involved considering symmetry and anisotropy in the material.
Within the framework of the proposed theory, the transformations required to describe these ef-
fects in materials can be analysed via transformations of the coordinate tensors ρν ,...,ρ ν...κ from
their starting bases (Xν ,...,X ν....κ), to those needed for an accurate description of the material
18
concerned. Obviously, anisotropy makes solid materials basis-dependent (i.e. , reference system-
dependent); therefore, it grows with increasing order of the bases used. In completely isotropic
materials, the plasticity yield criterion depends on the coordinates of a single scalar (ρ1 =ρ2 =ρ3 ),
i.e., it is absolutely independent of the particular coordinate system.
In should be noted that Hill's second criterion, which provides an accurate description of the
anomalous behaviour of some materials such as aluminium [18, 37], arises naturally within the frame-
work of the proposed plasticity theory. Additionally, the proposed theory includes the anisotropy
criterion of Hosford [25] that can explain the behaviour of fcc (for m = 8) and bcc (for m = 6)
materials.
With the norms defined above, equation (47) describes the most salient plasticity yield cri-
teria reported so far as particular cases. Therefore, (47) provides a self-contained description of
hydrostatic-pressure independent plasticity (i.e. , plasticity compliant with the proposed postulates
and approximations). A higher-order analysis allows one to establish a generic criterion in terms of
the following expanded series:
kdσ km =P ∞
n′ =0 P fu n ′ f
2n ′ σii − (2n ′ −1)σjj −σkk
m
+v| σxy |m +w|σyz |m +k|σxz |m =
=...... +ua1 | 2a σxx − (2a − 1)σyy −σz z |m +ua2 | 2a σxx − (2a − 1)σzz −σyy |m +
+ua3 | 2a σyy − (2a −1)σxx −σzz |m +ua4 | 2a σyy − (2a −1)σzz −σxx |m +
+ua5 | 2a σzz − (2a −1)σxx −σyy |m +ua6 | 2a σzz − (2a −1)σyy −σxx |m +
+........................ +u21 | 4σxx − 3σzz −σyy |m +u22 | 4σxx − 3σyy −σz z |m +
+u23 | 4σyy − 3σzz −σxx |m +u24 | 4σyy − 3σxx −σzz |m +u25 | 4σzz − 3σxx −σy y |m +
+u26 | 4σzz − 3σyy −σxx |m +u11 | 2σzz −σxx −σyy |m +u12 | 2σyy −σxx −σzz |m +
+u13 | 2σxx −σyy −σzz |m +u01 |σyy − σxx |m +u02 |σzz − σyy |m +u03 |σxx − σzz |m +
+v|σxy |m +w|σy z |m +k|σxz |m
(55)
where i 6 =j 6 =k 6 =i and the indices can range over x, y, z ;f is the permutation index. Each term in
(55) has an associated coefficient u n ′f , v, w, k that can be explicitly computed within the framework
of the proposed unified theory, but has been excluded in the expanded from (55) for simplicity. Note
that positive integer parameter n =n′ + 1 in eq. (55) represents the n -order approximation.
Stress convexity in (55) can be assessed by assuming that a linear combination of convex functions
with positive coefficients is a convex function itself [36]. The coefficients u n ′f , v, w, k are in fact all
positive. Therefore, it will suffice to show that the stress-dependence of each individual coefficient
in (55) is convex (i.e. , that all stresses in the terms
2 n ′ σii − (2n ′ −1)σjj −σkk
m
,|σxy |m ,|σy z |m ,
|σxz |m are convex). Convexity in a function T of three variables can be expressed [36] as
T( c1 σii + c2 σjj + c3 σkk )≤ c1 T(σii ) + c2 T(σj j ) + c3 T(σkk ) (56)
where c1 , c2, c3 ∈ ℜ are positive or zero and the following condition holds: c1 +c2 +c3 = 1. The
convexity of |σxy |m , |σyz |m , |σxz |m can be immediately confirmed by applying (56) to the particular
case of a single variable. That of
2n ′ σii − (2n ′ −1)σjj −σkk
m
can also be easily inferred from the
triangular inequality (see Appendix) fulfilling this Lm norm as defined in the stress vector space
(σxx , σyy , σzz ). Note that, for Lm to actually be the norm, m ∈ ℜ, m ≥ 1, which is the constraint to
be imposed on (55) for stresses to be convex. Under these conditions, the plasticity yield criterion
(55) will fulfil the third postulate. Within the framework of the proposed theory, the general criterion
19
(55) can predict plastic behaviour in solids. Therefore, it can be of use in future experimental and
theory studies on this topic.
11 Hydrostatic pressure-dependent plasticity yield criteria
The hydrostatic pressure-independent definitions obtained in the previous sections invariably include
traceless 2-tensors. In fact, this mathematical constraint ensures that the ensuing criteria will be
hydrostatic-pressure independent. As noted earlier, the normal space to the Cauchy tensor describes
the hydrostatic pressure-dependence of plasticity. As a result, considering the effects of hydrostatic
pressure in plasticity yield criteria entails including a c ( σM )b ≡ c (σxx + σyy + σzz )bdependent
function to describe the hydrostatic behaviour of the material with specific coefficients b, c [10, 11,
7, 9]. A more detailed study of the hydrostatic pressure-dependence in the framework of unified
theory will be the sub ject of future work.
12 Conclusions
In this paper we propose a new approach to the theory of macroscopic plasticity, based on a geometri-
cal ansatz, that unifies the different plasticity yield criteria. The method is based on decomposing the
Cauchy tensor into its component stress spaces: tangent and normal spaces. Hydrostatic-pressure
independent yield criteria are located on the tangent space to the Cauchy tensor (TSCT), whereas
hydrostatic-pressure dependent yield criteria are located on the tangent and normal spaces to the
Cauchy tensor (TSCT and NSCT). The analysis of the corresponding spaces requires the use of Lie
groups of transformations.
This work focuses on the tangent space to Cauchy tensor (TSCT) and analyzes the hydrostatic-
pressure independent yield criteria as orthogonal transformations. The measure of this tangent
space is taken as a mathematical norm that guarantees the convexity of the yield criteria. From
this measure we obtained the convex-stress dependence for both isotropic yield criteria (Tresca
1864 and Von Mises 1913) and anisotropic yield criteria (Hill 1948 and Hill 1979). In addition,
a stress series expansion is obtained that allows a detailed study of the mechanical properties in
materials: anisotropy, hardening-softening, etc. The tensor coefficients resulting from orthogonal
transformations are dependent on internal variables. These tensor coefficients allow a deep analysis
of mechanical properties in materials. This unified theory simplifies the great existing amount of
yield criteria and gives a physical meaning to the anisotropy coefficients.
This unified theory also allows an analytical study of anisotropy (dependency of f on ξ ) and
hardening-softening (dependency of ˙
fon ξ) by means of a set of tensor coefficients. These coefficients
are the directions of macroscopic plasticity (by points) in materials. A connection is suggested
between macroscopic transformations (unified theory) and the microscopic processes (dislocation
theory). To this extent, the concept of macroscopic point, that constitutes a statistical measure of
microscopic processes, is introduced. The concept of the point connects two scales with different
physical laws.
This unified approach has a series of advantages. At first, it proposes the stress dependencies
of yield criteria, starting from very simple postulates and with a unified vision of the underlying
phenomena. Further, it allows to know the macroscopic plasticity directions in the solid points,
by tensor coefficients of orthogonal transformations. It also provides a manner to analyze the
20
anisotropy and hardening-softening in materials. Finally, it enables us to establish a connection
between microscopic plasticity and macroscopic plasticity.
In future work it is important to extend the unified theory to treat the very important question
about the hydrostatic pressure-dependent plasticity (important in composites and granular materi-
als); for this it is necessary to analyse dilatation transformations. In this context, it is of fundamental
importance to make a deep study of hardening-softening and the time-evolution of plasticity in ma-
terials; for the latter aspect it is necessary to determine an adequate time-transformation that gives
a correct parameterization of the solid time-evolution.
21
Notation
(a) Indices
1. Solid point position: λ ranging over x, y, z
2. Cauchy tensor: i, j, k, l , p, q ranging over x, y, z
3. Rotation group: ν, µ, ω, κ, χ, ξ ranging over 1, 2, 3
4. Levi–Civita tensor: a, b, c ranging over 1, 2, 3
5. Elastic–plastic limit: P
6. Angle equivalence: (θ1 , θ2 , θ 3 )≡ (α, β, γ )
7. Cauchy tensor equivalence: σ≡ σ kl (contravariant); σ≡ σkl (covariant)
8. Principal stresses of the Cauchy tensor: σ1 , σ2, σ3
9. Hydrostatic pressure: σM
10. Internal variables: ξ = ξα
(b) Tensors
The Einstein summation convention is used throughout. Covariant tensor notation is systematically
employed from Section 4 onwards.
Acknowledgements.
The first author, Jos´e Miguel Luque Raig´on, is grateful to Professor Alfredo Navarro of the University
of Seville for his helpful, inspiring comments on materials mechanics and plasticity, and also to the
members of the Mechanical Engineering and Materials Research Group of the University of Seville for
their support. This work was funded by the Andalusian regional government (Junta de Andaluc´ıa)
within the framework of Project P06-TEP-01752.
22
Appendix: Convexity of general plasticity function
In order to prove the convexity of the plasticity function (section 10), it is necessary to show the
convexity of the terms
2n ′ σii − (2n ′ −1)σjj −σkk
m
with m≥ 1, m ∈ ℜ and n′ a positive integer
or zero. If the triangular inequality is taken we obtain
2 n ′ σii − 2 n ′ −1 σjj −σkk
m
≤2n ′ ·m |σii |m + 2 n ′ −1 m
|σjj |m +|σkk |m (A)
Dividing (A) by 2m(n′ +1) results in
1
2σ ii − 1
2− 1
2n′ +1 σ jj − 1
2n′ +1 σ kk
m
≤1
2m |σ ii | m + 1
2− 1
2n′ +1 m
|σjj |m +
1
2m(n′ +1) |σ kk | m ≤ 1
2|σ ii | m + 1
2− 1
2n′ +1 |σ jj | m + 1
2n′ +1 |σ kk | m (B)
Thus taking T (x ) ≡ | x|m , the condition of the convexity is fulfilled
T( c1 σii + c2 σjj + c3 σkk )≤ c1 T(σii ) + c2 T(σjj ) + c3 T(σkk ) (C),
where c1 = 1
2;c 2 = 1
2− 1
2n′ +1 ;c 3 = 1
2n′ +1 and n ′ is a positive integer or zero; with c 1 , c 2 , c 3 ≥0
and c1 +c2 +c3 = 1. So the convexity of the plasticity function has been proved.
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25
Figure captions
Figure 1. Reference systems for the overall solid { O, xλ = (x, y, z)} and for each point p in it
{Op , xλ = (x ,y, z) }. Each point has an associated Cauchy tensor acting on it (see Figure 2).
Figure 2. Representation of the Cauchy tensor σ kl and its rotational possibilities about the Euler
angles ΘE = ΘE (α, β, γ ) in the reference system { Op ,xλ } . The sphere surface in the figure is a
particular making of the SO (3) rotation group.
Figure 3. Action of the SO (3) rotation group on the Cauchy tensor σ kl for a partial rotation β
about the y- axis, represented by the reference system transformation { Op , xλ } → { Op ,¯xλ }.
Figure 4. Arbitrary representation of TSCT as an ellipsoid. The tangent space T SO (3)ρ µthat
provides a first-order approximation of TSCT to a reduced neighbourhood of ρµ is shown.
Figure 5. Arbitrary representation of TSCT and its approximations of different order to an in-
finitesimal neighbourhood. The higher orders lead to local infinitesimal neighbourhood that provide
a more accurate description of the overall structure of TSCT.
26
z
x
y
O
{ , x Opλ}
pλ
Figure 1: Reference systems for the overall solid O, xλ = ({ x, y, z} )} and for each point p in it
O p , x λ = ({x, y, z }) } . Each point has an associated Cauchy tensor acting on it (see Figure 2).
27
z
γ
α
x
β
y
σxx
σxz
σxy
σzz
σxz
σyz
σyz
σxy
σyy
p
SO(3)
{O , pxλ }
Figure 2: Representation of the Cauchy tensor σ kl and its rotational possibilities about the Euler
angles ΘE = ΘE (α, β, γ ) in the reference system { Op ,xλ } . The sphere surface in the figure is a
particular making of the SO(?? ) rotation group.
28
z
x
β
y = y
σxx
σxz
σxy
σzz
σxz
σyz
σyz
σx y
=σ yy σyy
p
SO(3)
{O , pxλ }
σzz
σyz
σxz
z
x
σxz
σxx
σxy
{O , pxλ }
σyz
σxy
Figure 3: Action of the SO (3) rotation group on the Cauchy tensor σ kl for a partial rotation β
about the y -axis, represented by the reference system transformation { Op , xλ } → { Op ,¯xλ }.
29
ρμ
∂⁄∂γ
∂⁄∂y
∂⁄∂x TSO(3)ρ μ
z
x
y
Op
γ
α
β
Figure 4: Arbitrary representation of TSCT as an ellipsoid. The tangent space T S O (3)ρ µthat
provides a first-order approximation of TSCT to a reduced neighbourhood of ρµ is shown.
30
Figure 5: Arbitrary representation of TSCT and its approximations of different order to an in-
finitesimal neighbourhood. The higher orders lead to local infinitesimal neighbourhood that provide
a more accurate description of the overall structure of TSCT.
31
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