On Monotonicity for Strain-Limiting Theories of
Elasticity
Tina Mai∗ and Jay R. Walton
Department of Mathematics, Texas A&M University
2014
Motivation
In1 , we assumed the nonlinear response function has the Fréchet
derivative invertible as a fourth-order tensor.
1
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
2 / 28
Motivation
In1 , we assumed the nonlinear response function has the Fréchet
derivative invertible as a fourth-order tensor.
However, in some important classes of models introduced by
Rajagopal and co-authors, this invertibility condition fails.
1
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
2 / 28
Motivation
In1 , we assumed the nonlinear response function has the Fréchet
derivative invertible as a fourth-order tensor.
However, in some important classes of models introduced by
Rajagopal and co-authors, this invertibility condition fails.
We investigate here the more general notion of monotonicity for such
strain-limiting models.
1
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
2 / 28
Motivation
In1 , we assumed the nonlinear response function has the Fréchet
derivative invertible as a fourth-order tensor.
However, in some important classes of models introduced by
Rajagopal and co-authors, this invertibility condition fails.
We investigate here the more general notion of monotonicity for such
strain-limiting models.
For the class of strain limiting constitutive models considered herein,
monotonicity is investigated for several classes of deformations
including pure compression and simple shear.
1
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
2 / 28
Motivation
In1 , we assumed the nonlinear response function has the Fréchet
derivative invertible as a fourth-order tensor.
However, in some important classes of models introduced by
Rajagopal and co-authors, this invertibility condition fails.
We investigate here the more general notion of monotonicity for such
strain-limiting models.
For the class of strain limiting constitutive models considered herein,
monotonicity is investigated for several classes of deformations
including pure compression and simple shear.
We show that monotonicity holds for deformations with (a suitable)
strain having sufficiently small norm.
1
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
2 / 28
Motivation
In1 , we assumed the nonlinear response function has the Fréchet
derivative invertible as a fourth-order tensor.
However, in some important classes of models introduced by
Rajagopal and co-authors, this invertibility condition fails.
We investigate here the more general notion of monotonicity for such
strain-limiting models.
For the class of strain limiting constitutive models considered herein,
monotonicity is investigated for several classes of deformations
including pure compression and simple shear.
We show that monotonicity holds for deformations with (a suitable)
strain having sufficiently small norm.
Counterexamples are constructed to demonstrate the failure of
mononicity for appropriately chosen deformations.
1
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
2 / 28
Notation and Preliminaries
Introduced by Rajagopal and co-authors in2 ,3 ,4 :
F(T) = φ0 (T)I + φ1 (T)T + φ2 (T)T2 ,
(1)
2
K. R. Rajagopal. “Conspectus of concepts of elasticity”. In: Mathematics
and Mechanics of Solids 16.5, SI (2011), 536–562.
3
K. R. Rajagopal. “Non-Linear Elastic Bodies Exhibiting Limiting Small
Strain”. In: Mathematics and Mechanics of Solids 16.1 (2011), 122–139.
4
K. R. Rajagopal and A. R. Srinivasa. “On the response of non-dissipative
solids”. In: Proceedings of the Royal Society of London, Mathematical, Physical
and Engineering Sciences 463.2078 (2007), 357–367.
5
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Tina Mai (Texas A&M University)
Monotonicity
2014
3 / 28
Notation and Preliminaries
Introduced by Rajagopal and co-authors in2 ,3 ,4 :
F(T) = φ0 (T)I + φ1 (T)T + φ2 (T)T2 ,
(1)
where I second-order identity tensor, each φj (T) a scalar valued
function of the isotropic invariants of T with |φ0 (T)|, |φ1 (T)||T|, and
|φ2 (T)||T2 | all uniformly bounded functions on Sym.
2
K. R. Rajagopal. “Conspectus of concepts of elasticity”. In: Mathematics
and Mechanics of Solids 16.5, SI (2011), 536–562.
3
K. R. Rajagopal. “Non-Linear Elastic Bodies Exhibiting Limiting Small
Strain”. In: Mathematics and Mechanics of Solids 16.1 (2011), 122–139.
4
K. R. Rajagopal and A. R. Srinivasa. “On the response of non-dissipative
solids”. In: Proceedings of the Royal Society of London, Mathematical, Physical
and Engineering Sciences 463.2078 (2007), 357–367.
5
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Tina Mai (Texas A&M University)
Monotonicity
2014
3 / 28
Notation and Preliminaries
Introduced by Rajagopal and co-authors in2 ,3 ,4 :
F(T) = φ0 (T)I + φ1 (T)T + φ2 (T)T2 ,
(1)
where I second-order identity tensor, each φj (T) a scalar valued
function of the isotropic invariants of T with |φ0 (T)|, |φ1 (T)||T|, and
|φ2 (T)||T2 | all uniformly bounded functions on Sym.
Strong ellipticity was investigated in5 for models inspired by (1) for
the special case of φ2 (·) = 0.
2
K. R. Rajagopal. “Conspectus of concepts of elasticity”. In: Mathematics
and Mechanics of Solids 16.5, SI (2011), 536–562.
3
K. R. Rajagopal. “Non-Linear Elastic Bodies Exhibiting Limiting Small
Strain”. In: Mathematics and Mechanics of Solids 16.1 (2011), 122–139.
4
K. R. Rajagopal and A. R. Srinivasa. “On the response of non-dissipative
solids”. In: Proceedings of the Royal Society of London, Mathematical, Physical
and Engineering Sciences 463.2078 (2007), 357–367.
5
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Tina Mai (Texas A&M University)
Monotonicity
2014
3 / 28
Notation and Preliminaries
Introduced by Rajagopal and co-authors in2 ,3 ,4 :
F(T) = φ0 (T)I + φ1 (T)T + φ2 (T)T2 ,
(1)
where I second-order identity tensor, each φj (T) a scalar valued
function of the isotropic invariants of T with |φ0 (T)|, |φ1 (T)||T|, and
|φ2 (T)||T2 | all uniformly bounded functions on Sym.
Strong ellipticity was investigated in5 for models inspired by (1) for
the special case of φ2 (·) = 0.
The case of φ2 (·) 6= 0 is more difficult and is the subject of the
present contribution.
2
K. R. Rajagopal. “Conspectus of concepts of elasticity”. In: Mathematics
and Mechanics of Solids 16.5, SI (2011), 536–562.
3
K. R. Rajagopal. “Non-Linear Elastic Bodies Exhibiting Limiting Small
Strain”. In: Mathematics and Mechanics of Solids 16.1 (2011), 122–139.
4
K. R. Rajagopal and A. R. Srinivasa. “On the response of non-dissipative
solids”. In: Proceedings of the Royal Society of London, Mathematical, Physical
and Engineering Sciences 463.2078 (2007), 357–367.
5
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Tina Mai (Texas A&M University)
Monotonicity
2014
3 / 28
Notation and Preliminaries
Consider
Tina Mai (Texas A&M University)
0 = F(E, S̄)
(2)
E = F(S̄).
(3)
Monotonicity
2014
4 / 28
Notation and Preliminaries
Consider
0 = F(E, S̄)
(2)
E = F(S̄).
(3)
Focus on the special case of (3) in which the response function F(·)
is uniformly bounded in norm.
Tina Mai (Texas A&M University)
Monotonicity
2014
4 / 28
Notation and Preliminaries
Consider
0 = F(E, S̄)
(2)
E = F(S̄).
(3)
Focus on the special case of (3) in which the response function F(·)
is uniformly bounded in norm.
Appealing to the classical Cayley-Hamilton theorem, F(S̄) has the
general representation:
F(S̄) = φ0 (S̄)I + φ1 (S̄)S̄ + φ2 (S̄)S̄
Tina Mai (Texas A&M University)
Monotonicity
2
(4)
2014
4 / 28
Notation and Preliminaries
Consider
0 = F(E, S̄)
(2)
E = F(S̄).
(3)
Focus on the special case of (3) in which the response function F(·)
is uniformly bounded in norm.
Appealing to the classical Cayley-Hamilton theorem, F(S̄) has the
general representation:
F(S̄) = φ0 (S̄)I + φ1 (S̄)S̄ + φ2 (S̄)S̄
2
(4)
where the coefficient functions φj (·) are scalar valued, and in the
strain limiting case, they satisfy the additional assumption that
2
|φ0 (S̄)|, |φ1 (S̄)||S̄| and |φ2 (S̄)||S̄ | are uniformly bounded.
Tina Mai (Texas A&M University)
Monotonicity
2014
4 / 28
Notation and Preliminaries
Q: Applications?
Tina Mai (Texas A&M University)
Monotonicity
2014
5 / 28
Notation and Preliminaries
Q: Applications?
To date, in the various applications, analyses and numerical
simulations of strain limiting models of the form (1) or (4) that have
appeared in the literature, attention has been limited to φ2 (·) = 0.
Tina Mai (Texas A&M University)
Monotonicity
2014
5 / 28
Notation and Preliminaries
Q: Applications?
To date, in the various applications, analyses and numerical
simulations of strain limiting models of the form (1) or (4) that have
appeared in the literature, attention has been limited to φ2 (·) = 0.
Q: Why?
Tina Mai (Texas A&M University)
Monotonicity
2014
5 / 28
Notation and Preliminaries
Q: Applications?
To date, in the various applications, analyses and numerical
simulations of strain limiting models of the form (1) or (4) that have
appeared in the literature, attention has been limited to φ2 (·) = 0.
Q: Why?
First reason, when φ2 (·) is non-zero, analysis of (1) or (4) encounters
significant added complexity.
Tina Mai (Texas A&M University)
Monotonicity
2014
5 / 28
Notation and Preliminaries
Q: Applications?
To date, in the various applications, analyses and numerical
simulations of strain limiting models of the form (1) or (4) that have
appeared in the literature, attention has been limited to φ2 (·) = 0.
Q: Why?
First reason, when φ2 (·) is non-zero, analysis of (1) or (4) encounters
significant added complexity.
Second reason, even with φ2 (·) = 0, the models still exhibit a rich
array of behaviors such as non-linear stress-strain response even in the
infinitesimal strain regime.
Tina Mai (Texas A&M University)
Monotonicity
2014
5 / 28
Notation and Preliminaries
Q: Invertibility of (4)?
6
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
7
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
8
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
Tina Mai (Texas A&M University)
Monotonicity
2014
6 / 28
Notation and Preliminaries
Q: Invertibility of (4)?
Even when φ2 (·) 6= 0, there exist strain-limiting models in (4) for
which F(·) is not uniquely invertible.
6
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
7
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
8
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
Tina Mai (Texas A&M University)
Monotonicity
2014
6 / 28
Notation and Preliminaries
Q: Invertibility of (4)?
Even when φ2 (·) 6= 0, there exist strain-limiting models in (4) for
which F(·) is not uniquely invertible.
However, for the analyses in6 ,7 and8 , we studied the cases in which
F(·) is uniquely invertible.
6
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
7
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
8
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
Tina Mai (Texas A&M University)
Monotonicity
2014
6 / 28
Notation and Preliminaries
Q: Invertibility of (4)?
Even when φ2 (·) 6= 0, there exist strain-limiting models in (4) for
which F(·) is not uniquely invertible.
However, for the analyses in6 ,7 and8 , we studied the cases in which
F(·) is uniquely invertible.
When φ2 (·) 6= 0, a multi-valued inverse of (4) is the rule unless the
third term in (4) is strongly dominated by the first two terms.
6
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
7
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
8
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
Tina Mai (Texas A&M University)
Monotonicity
2014
6 / 28
Notation and Preliminaries
Moreover, even when (4) is uniquely invertible, its Fréchet derivative
need not be, preventing attempts to generalize the approach utilized
in9 for studying strong ellipticity for (4) when φ2 (·) = 0 to the case
φ2 (·) 6= 0.
9
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
7 / 28
Notation and Preliminaries
Moreover, even when (4) is uniquely invertible, its Fréchet derivative
need not be, preventing attempts to generalize the approach utilized
in9 for studying strong ellipticity for (4) when φ2 (·) = 0 to the case
φ2 (·) 6= 0.
Thus, the generalization here is by consideration of the weaker notion
of convexity, monotonicity, that does not require Fréchet
differentiability of F(·).
9
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
7 / 28
Notation and Preliminaries
Q: Importance of (4)?
10
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
11
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
12 Mai (Texas A&M University)
Tina
Monotonicity
2014
Tina Mai and Jay R. Walton. “On Strong
Ellipticity for Implicit and
8 / 28
Notation and Preliminaries
Q: Importance of (4)?
Ability to capture a nonlinear stress-strain response even in the
infinitesimal strain limit.
10
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
11
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
12 Mai (Texas A&M University)
Tina
Monotonicity
2014
Tina Mai and Jay R. Walton. “On Strong
Ellipticity for Implicit and
8 / 28
Notation and Preliminaries
Q: Importance of (4)?
Ability to capture a nonlinear stress-strain response even in the
infinitesimal strain limit.
The infinitesimal strain limit of (4) need not be a limiting strain
theory.
10
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
11
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
12 Mai (Texas A&M University)
Tina
Monotonicity
2014
Tina Mai and Jay R. Walton. “On Strong
Ellipticity for Implicit and
8 / 28
Notation and Preliminaries
Q: Importance of (4)?
Ability to capture a nonlinear stress-strain response even in the
infinitesimal strain limit.
The infinitesimal strain limit of (4) need not be a limiting strain
theory.
But, when it is, it provides an appealing framework in which to study
brittle fracture10 and11 .
10
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
11
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
12 Mai (Texas A&M University)
Tina
Monotonicity
2014
Tina Mai and Jay R. Walton. “On Strong
Ellipticity for Implicit and
8 / 28
Notation and Preliminaries
Q: Importance of (4)?
Ability to capture a nonlinear stress-strain response even in the
infinitesimal strain limit.
The infinitesimal strain limit of (4) need not be a limiting strain
theory.
But, when it is, it provides an appealing framework in which to study
brittle fracture10 and11 .
In particular, (4) with φ0 (·) = 0 = φ2 (·), and
E = φ1 (S̄)S̄ = φ̃(|S̄|)S̄ =
S̄
.
1 + β|S̄|
(5)
10
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
11
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
12 Mai (Texas A&M University)
Tina
Monotonicity
2014
Tina Mai and Jay R. Walton. “On Strong
Ellipticity for Implicit and
8 / 28
Notation and Preliminaries
Q: Importance of (4)?
Ability to capture a nonlinear stress-strain response even in the
infinitesimal strain limit.
The infinitesimal strain limit of (4) need not be a limiting strain
theory.
But, when it is, it provides an appealing framework in which to study
brittle fracture10 and11 .
In particular, (4) with φ0 (·) = 0 = φ2 (·), and
E = φ1 (S̄)S̄ = φ̃(|S̄|)S̄ =
S̄
.
1 + β|S̄|
(5)
The its hyperelasticity, strong ellipticity were shown in12 .
10
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
11
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
12 Mai (Texas A&M University)
Tina
Monotonicity
2014
Tina Mai and Jay R. Walton. “On Strong
Ellipticity for Implicit and
8 / 28
Notation and Preliminaries
In13 and14 , it was shown stress as well as strain is controlled in the
neighborhood of a crack tip.
13
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
14
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
Tina Mai (Texas A&M University)
Monotonicity
2014
9 / 28
Notation and Preliminaries
In13 and14 , it was shown stress as well as strain is controlled in the
neighborhood of a crack tip.
Additionally, recent direct numerical simulations support these
asymptotic predictions and illustrate how the use of such strain
limiting models in numerical simulations of brittle fracture obviate the
need for extensive mesh refinement near a crack tip, or the
introduction of cohesive or process zone crack tip models.
13
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
14
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
Tina Mai (Texas A&M University)
Monotonicity
2014
9 / 28
Notation and Preliminaries
In13 and14 , it was shown stress as well as strain is controlled in the
neighborhood of a crack tip.
Additionally, recent direct numerical simulations support these
asymptotic predictions and illustrate how the use of such strain
limiting models in numerical simulations of brittle fracture obviate the
need for extensive mesh refinement near a crack tip, or the
introduction of cohesive or process zone crack tip models.
These numerical simulations also show that the linearly elastic
solution is recovered outside of a small neighborhood of a crack tip
and globally as β → 0.
13
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
14
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
Tina Mai (Texas A&M University)
Monotonicity
2014
9 / 28
Notation and Preliminaries
It is straightforward to adapt the analyses in15 ,16 and17 to more
general models of the form E = φ(|S̄|)S̄ with φ(r ) bounded,
nonnegative and decreasing for r > 0, and satisfying r φ(r ) ≤ M < ∞
uniformly for 0 < r < ∞, along with mild smoothness assumptions.
15
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
16
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
17
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
10 / 28
Notation and Preliminaries
It is straightforward to adapt the analyses in15 ,16 and17 to more
general models of the form E = φ(|S̄|)S̄ with φ(r ) bounded,
nonnegative and decreasing for r > 0, and satisfying r φ(r ) ≤ M < ∞
uniformly for 0 < r < ∞, along with mild smoothness assumptions.
However, the analysis in these works was restricted to the simple
model (5) to avoid unnecessary complication that might obscure the
essence of the arguments.
15
K. R. Rajagopal and J. R. Walton. “Modeling fracture in the context of a
strain-limiting theory of elasticity: A single anti-plane shear crack”. In:
International Journal of Fracture 169.1 (2011), pp. 39–48.
16
K. Gou et al. “Modeling Fracture in the Context of a Strain-Limiting
Theory of Elasticity: A Single Plane-Strain Crack”. In: International Journal of
Engineering Science in press (2014). DOI: 10.1016/j.ijengsci.2014.04.018.
17
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: Mathematics and Mechanics of
Solids in press (2014). DOI: 10.1177/1081286514544254.
Tina Mai (Texas A&M University)
Monotonicity
2014
10 / 28
Notation and Preliminaries
the present contribution focusses on the sub-class of (4) taking the
form:
2
F(S̄) = φ1 (|S̄|)S̄ + φ2 (|S̄|2 )S̄ ,
(6)
in which
α1
,
1 + β1 r
α2
,
φ2 (r ) :=
1 + β2 r
φ1 (r ) :=
(7)
(8)
where α1 , β1 are strictly positive constant and β2 α2 is nonnegative
constant.
Tina Mai (Texas A&M University)
Monotonicity
2014
11 / 28
Notation and Preliminaries
the present contribution focusses on the sub-class of (4) taking the
form:
2
F(S̄) = φ1 (|S̄|)S̄ + φ2 (|S̄|2 )S̄ ,
(6)
in which
α1
,
1 + β1 r
α2
,
φ2 (r ) :=
1 + β2 r
φ1 (r ) :=
(7)
(8)
where α1 , β1 are strictly positive constant and β2 α2 is nonnegative
constant.
Q: Why?
Tina Mai (Texas A&M University)
Monotonicity
2014
11 / 28
Notation and Preliminaries
the present contribution focusses on the sub-class of (4) taking the
form:
2
F(S̄) = φ1 (|S̄|)S̄ + φ2 (|S̄|2 )S̄ ,
(6)
in which
α1
,
1 + β1 r
α2
,
φ2 (r ) :=
1 + β2 r
φ1 (r ) :=
(7)
(8)
where α1 , β1 are strictly positive constant and β2 α2 is nonnegative
constant.
Q: Why?
In order for the model (6)...(8) to be strain limiting, β1 and β2 are
assumed to be strictly positive, and stability demands that α1 also be
strictly positive.
Tina Mai (Texas A&M University)
Monotonicity
2014
11 / 28
Notation and Preliminaries
However, both α2 positive and negative lead to physically relevant
models.
Tina Mai (Texas A&M University)
Monotonicity
2014
12 / 28
Notation and Preliminaries
However, both α2 positive and negative lead to physically relevant
models.
In the analysis to follow, it will be assumed that the response function
(6)...(8) is uniquely invertible, which can be guaranteed provided the
function r φ2 (r ) is dominated by the function r φ1 (r ).
Tina Mai (Texas A&M University)
Monotonicity
2014
12 / 28
Notation and Preliminaries
However, both α2 positive and negative lead to physically relevant
models.
In the analysis to follow, it will be assumed that the response function
(6)...(8) is uniquely invertible, which can be guaranteed provided the
function r φ2 (r ) is dominated by the function r φ1 (r ).
When such is the case, the convexity analysis presented below is valid
for α2 positive, negative and zero. WLOG, assuming α2 nonnegative.
Tina Mai (Texas A&M University)
Monotonicity
2014
12 / 28
Notation and Preliminaries
However, both α2 positive and negative lead to physically relevant
models.
In the analysis to follow, it will be assumed that the response function
(6)...(8) is uniquely invertible, which can be guaranteed provided the
function r φ2 (r ) is dominated by the function r φ1 (r ).
When such is the case, the convexity analysis presented below is valid
for α2 positive, negative and zero. WLOG, assuming α2 nonnegative.
When the function r φ2 (r ) is not suitably dominated by the function
r φ1 (r ) and the response function (6)...(8) has a multi-valued inverse,
then one must take account of the sign of α2 in investigations of
convexity.
Tina Mai (Texas A&M University)
Monotonicity
2014
12 / 28
Notation and Preliminaries
However, both α2 positive and negative lead to physically relevant
models.
In the analysis to follow, it will be assumed that the response function
(6)...(8) is uniquely invertible, which can be guaranteed provided the
function r φ2 (r ) is dominated by the function r φ1 (r ).
When such is the case, the convexity analysis presented below is valid
for α2 positive, negative and zero. WLOG, assuming α2 nonnegative.
When the function r φ2 (r ) is not suitably dominated by the function
r φ1 (r ) and the response function (6)...(8) has a multi-valued inverse,
then one must take account of the sign of α2 in investigations of
convexity.
This results in a more delicate analysis and will not be addressed in
this contribution.
Tina Mai (Texas A&M University)
Monotonicity
2014
12 / 28
Notation and Preliminaries
However, both α2 positive and negative lead to physically relevant
models.
In the analysis to follow, it will be assumed that the response function
(6)...(8) is uniquely invertible, which can be guaranteed provided the
function r φ2 (r ) is dominated by the function r φ1 (r ).
When such is the case, the convexity analysis presented below is valid
for α2 positive, negative and zero. WLOG, assuming α2 nonnegative.
When the function r φ2 (r ) is not suitably dominated by the function
r φ1 (r ) and the response function (6)...(8) has a multi-valued inverse,
then one must take account of the sign of α2 in investigations of
convexity.
This results in a more delicate analysis and will not be addressed in
this contribution.
Finally, through nondimensionalization, we can set α1 = 1.
Tina Mai (Texas A&M University)
Monotonicity
2014
12 / 28
Notation and Preliminaries
Q: Hyperelasticity of (6)?
18
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
13 / 28
Notation and Preliminaries
Q: Hyperelasticity of (6)?
A necessary condition for the model (6) to correspond to a
hyperelastic material response is for the coefficient function φ2 (r ) to
be identically constant.
18
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
13 / 28
Notation and Preliminaries
Q: Hyperelasticity of (6)?
A necessary condition for the model (6) to correspond to a
hyperelastic material response is for the coefficient function φ2 (r ) to
be identically constant.
In particular, it was shown in18 no strain limiting model of the form
(6) with φ2 (r ) not identically zero can be hyperelastic.
18
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
13 / 28
Notation and Preliminaries
Q: Hyperelasticity of (6)?
A necessary condition for the model (6) to correspond to a
hyperelastic material response is for the coefficient function φ2 (r ) to
be identically constant.
In particular, it was shown in18 no strain limiting model of the form
(6) with φ2 (r ) not identically zero can be hyperelastic.
Notation: Let
2
S̄
I
S̄
A := S̄ , B := 2 , and P :=
,
|I|
S̄ (9)
|A| = |B| = |P| = 1,
(10)
so that
and let I denote the fourth-order identity tensor.
18
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
13 / 28
Notation and Preliminaries
As shown in19 , strong ellipticity for (3) requires:
i
−1 h T
H · DF Ŝ(F)[H] = H · (HS̄(E)) − FT H · DS̄ F(S̄)
(F H)s > 0
s
(11)
−1
where DS̄ F(S̄)
[·] denotes the inverse of the fourth-order tensor DS̄ F(S̄)[·].
19
MaiA&M
andUniversity)
Jay R.
Tina Tina
Mai (Texas
Walton. “On Strong
Ellipticity for Implicit and
Monotonicity
2014
14 / 28
Notation and Preliminaries
As shown in19 , strong ellipticity for (3) requires:
i
−1 h T
H · DF Ŝ(F)[H] = H · (HS̄(E)) − FT H · DS̄ F(S̄)
(F H)s > 0
s
(11)
−1
where DS̄ F(S̄) [·] denotes the inverse of the fourth-order tensor DS̄ F(S̄)[·].
When DS̄ F(S̄) [·] is not invertible, as will be shown to be the case for (4), we investigate
monotonicity20 (as a substitute for strong ellipticity):
(Ŝ(F + αH) − Ŝ(F)) · H > 0
(12)
for all rank-1 tensors H = a ⊗ b (with |a| = |b| = 1) and 0 < α ≤ 1 such that
det(F + αH) > 0.
(13)
In case the response function Ŝ(·) is differentiable, a weaker form of strong ellipticity
condition is derived from (12) with “>” replaced by “≥”.
19
MaiA&M
andUniversity)
Jay R.
Tina Tina
Mai (Texas
Walton. “On Strong
Ellipticity for Implicit and
Monotonicity
2014
14 / 28
Monotonicity-General Observation
When (3) is uniquely invertible, we make use of an equivalent form of
monotonicity (12):
(F̃S̃ − FS̄) · H > 0,
(14)
where
F̃ = F + αH,
(15)
b̄
b
S(F)
= FS(E)
= FS̄,
(16)
b̄ Ẽ) = F̃S̃,
b + αH) = F̃S(
S(F
(17)
and
for all rank-1 tensors H = a ⊗ b (with |a| = |b| = 1) and 0 < α ≤ 1.
Note that when E, S̄ satisfy (3), so do Ẽ, S̃.
Tina Mai (Texas A&M University)
Monotonicity
2014
15 / 28
Compression and Dilation
Consider a model of the form (6) with
F = γI,
(18)
γ is a positive scalar. Compression and dilation correspond to
0 < γ < 1 and 1 < γ, respectively. Then,
1
E = (γ 2 − 1)I,
2
and from the constitutive relation (6), the stress has the form
S̄ = σ̄I,
(19)
(20)
where σ̄ is a constant. Now, (6) becomes
1 2
σ̄
α2 σ̄ 2
√
(γ − 1) = g (σ̄) :=
+
.
2
1 + 3β1 |σ̄| 1 + 3β2 σ̄ 2
(21)
Note that γ → 0+ in (21) imposes a lower bound on compressive
stresses σ̄ < 0, while σ̄ → ∞ imposes an upper bound on dilatational
strains γ > 1.
Tina Mai (Texas A&M University)
Monotonicity
2014
16 / 28
Invertibility
We investigate the questions of unique invertibility of (6) and its
Fréchet derivative for the special cases of pure compression and
dilation.
Tina Mai (Texas A&M University)
Monotonicity
2014
17 / 28
Invertibility
We investigate the questions of unique invertibility of (6) and its
Fréchet derivative for the special cases of pure compression and
dilation.
We show that (6) is uniquely invertible for dilation but can fail to be
for compression except for sufficiently small strains.
Tina Mai (Texas A&M University)
Monotonicity
2014
17 / 28
Invertibility
We investigate the questions of unique invertibility of (6) and its
Fréchet derivative for the special cases of pure compression and
dilation.
We show that (6) is uniquely invertible for dilation but can fail to be
for compression except for sufficiently small strains.
It is also shown that even when (6) is uniquely invertible, its Fréchet
derivative need not be.
Tina Mai (Texas A&M University)
Monotonicity
2014
17 / 28
Invertibility
We investigate the questions of unique invertibility of (6) and its
Fréchet derivative for the special cases of pure compression and
dilation.
We show that (6) is uniquely invertible for dilation but can fail to be
for compression except for sufficiently small strains.
It is also shown that even when (6) is uniquely invertible, its Fréchet
derivative need not be.
As noted above, when this is the case, the approach to strong
ellipticity utilized in [4] cannot be applied to (6) which motivates our
consideration of monotonicity in the following section.
Tina Mai (Texas A&M University)
Monotonicity
2014
17 / 28
Monotonicity
As shown in the previous subsection, the model (6), even when it has
an equivalent Cauchy elastic formulation, can lack sufficient regularity
to support strong ellipticity (as defined in [4]). For that reason, we
now investigate the weaker convexity condition of monotonicity in the
form (14).
Tina Mai (Texas A&M University)
Monotonicity
2014
18 / 28
Monotonicity
As shown in the previous subsection, the model (6), even when it has
an equivalent Cauchy elastic formulation, can lack sufficient regularity
to support strong ellipticity (as defined in [4]). For that reason, we
now investigate the weaker convexity condition of monotonicity in the
form (14).
We first consider the case γ = 1, then use a continuity argument to
generalize for all γ near 1.
Tina Mai (Texas A&M University)
Monotonicity
2014
18 / 28
Monotonicity
As shown in the previous subsection, the model (6), even when it has
an equivalent Cauchy elastic formulation, can lack sufficient regularity
to support strong ellipticity (as defined in [4]). For that reason, we
now investigate the weaker convexity condition of monotonicity in the
form (14).
We first consider the case γ = 1, then use a continuity argument to
generalize for all γ near 1.
Under the conditions H = a ⊗ b (with |a| = |b| = 1) and 0 < α ≤ 1,
we consider three cases: a = b, a ⊥ b, or 0 < |a · b| < 1.
Tina Mai (Texas A&M University)
Monotonicity
2014
18 / 28
Monotonicity
As shown in the previous subsection, the model (6), even when it has
an equivalent Cauchy elastic formulation, can lack sufficient regularity
to support strong ellipticity (as defined in [4]). For that reason, we
now investigate the weaker convexity condition of monotonicity in the
form (14).
We first consider the case γ = 1, then use a continuity argument to
generalize for all γ near 1.
Under the conditions H = a ⊗ b (with |a| = |b| = 1) and 0 < α ≤ 1,
we consider three cases: a = b, a ⊥ b, or 0 < |a · b| < 1.
Case 1: a = b. Then,
F̃ = F + αH = I + α(a ⊗ a).
Tina Mai (Texas A&M University)
Monotonicity
(22)
2014
18 / 28
Monotonicity
As shown in the previous subsection, the model (6), even when it has
an equivalent Cauchy elastic formulation, can lack sufficient regularity
to support strong ellipticity (as defined in [4]). For that reason, we
now investigate the weaker convexity condition of monotonicity in the
form (14).
We first consider the case γ = 1, then use a continuity argument to
generalize for all γ near 1.
Under the conditions H = a ⊗ b (with |a| = |b| = 1) and 0 < α ≤ 1,
we consider three cases: a = b, a ⊥ b, or 0 < |a · b| < 1.
Case 1: a = b. Then,
Tina Mai (Texas A&M University)
F̃ = F + αH = I + α(a ⊗ a).
(22)
1 T
α2
Ẽ = (F̃ F̃ − I) = (α +
)a ⊗ a .
2
2
(23)
Monotonicity
2014
18 / 28
Monotonicity
S̃ = σ̃(a ⊗ a),
(24)
where σ̃ is a scalar. We also deduce that |S̃| = |σ̃|.
Tina Mai (Texas A&M University)
Monotonicity
2014
19 / 28
Monotonicity
S̃ = σ̃(a ⊗ a),
(24)
where σ̃ is a scalar. We also deduce that |S̃| = |σ̃|.
Then, the constitutive relation
Ẽ = φ1 (|S̃|)S̃ + φ2 (|S̃|2 )S̃
becomes
α+
Tina Mai (Texas A&M University)
2
(25)
α2
= g (σ̃) := φ1 (σ̃)σ̃ + φ2 (σ̃ 2 )σ̃ 2 .
2
Monotonicity
(26)
2014
19 / 28
Monotonicity
S̃ = σ̃(a ⊗ a),
(24)
where σ̃ is a scalar. We also deduce that |S̃| = |σ̃|.
Then, the constitutive relation
Ẽ = φ1 (|S̃|)S̃ + φ2 (|S̃|2 )S̃
2
(25)
becomes
α2
= g (σ̃) := φ1 (σ̃)σ̃ + φ2 (σ̃ 2 )σ̃ 2 .
(26)
2
Since the left hand side of (26) is always positive, it follows that
σ̃ > 0. Then, g 0 (σ̃) > 0, and thus (25) is uniquely invertible.
α+
Tina Mai (Texas A&M University)
Monotonicity
2014
19 / 28
Monotonicity
Now, the left hand side of (14) becomes
(F̃S̃ − FS̄) · (a ⊗ a) = ((I + α(a ⊗ a))(σ̃(a ⊗ a))) · (a ⊗ a)
= (1 + α)σ̃ > 0
(27)
because σ̃ > 0, and then monotonicity also holds for all γ belonging
to an interval centered at γ = 1 by continuity, given α in (0, 1].
21
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
Tina Mai (Texas A&M University)
Monotonicity
10.1007/s10659-014-9503-4.
2014
20 / 28
Monotonicity
Now, the left hand side of (14) becomes
(F̃S̃ − FS̄) · (a ⊗ a) = ((I + α(a ⊗ a))(σ̃(a ⊗ a))) · (a ⊗ a)
= (1 + α)σ̃ > 0
(27)
because σ̃ > 0, and then monotonicity also holds for all γ belonging
to an interval centered at γ = 1 by continuity, given α in (0, 1].
Remark 1. In this case for (6), the monotonicity condition (27)
implies the invertibility condition, but the reverse statement does not
hold.
21
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
Tina Mai (Texas A&M University)
Monotonicity
10.1007/s10659-014-9503-4.
2014
20 / 28
Monotonicity
Now, the left hand side of (14) becomes
(F̃S̃ − FS̄) · (a ⊗ a) = ((I + α(a ⊗ a))(σ̃(a ⊗ a))) · (a ⊗ a)
= (1 + α)σ̃ > 0
(27)
because σ̃ > 0, and then monotonicity also holds for all γ belonging
to an interval centered at γ = 1 by continuity, given α in (0, 1].
Remark 1. In this case for (6), the monotonicity condition (27)
implies the invertibility condition, but the reverse statement does not
hold.
The cases 2 and 3 are more complicated. And, in summary, it was
shown in21 that for all the three cases, given α2 nonnegative,
monotonicity holds for the given model (6), in an interval of α which
is the intersection of the three intervals of α near 0 in the above three
cases, and in an interval of γ centered at γ = 1 which is also the
intersection of the three intervals of γ in the above three cases.
21
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
Tina Mai (Texas A&M University)
Monotonicity
10.1007/s10659-014-9503-4.
2014
20 / 28
Simple Shear
Now, (6) is considered for simple shear deformations of the form:
F = I + γe1 ⊗ e2 ,
(28)
in which, γ is a scalar satisfying detF > 0, and e1 , e2 are orthonormal
vectors.
22
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
21 / 28
Simple Shear
Now, (6) is considered for simple shear deformations of the form:
F = I + γe1 ⊗ e2 ,
(28)
in which, γ is a scalar satisfying detF > 0, and e1 , e2 are orthonormal
vectors.
Invertibility: By a delicate analysis,
we showed in22 that the
fourth-order tensor DS̄ F(S̄) [·] is not invertible, although the given
model (6) is uniquely invertible for all γ in a neighborhood of γ = 0.
22
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
21 / 28
Simple Shear
Now, (6) is considered for simple shear deformations of the form:
F = I + γe1 ⊗ e2 ,
(28)
in which, γ is a scalar satisfying detF > 0, and e1 , e2 are orthonormal
vectors.
Invertibility: By a delicate analysis,
we showed in22 that the
fourth-order tensor DS̄ F(S̄) [·] is not invertible, although the given
model (6) is uniquely invertible for all γ in a neighborhood of γ = 0.
Monotonicity: Similar to the previous Section, we study monotonicity
of (6) for γ = 0, then we generalize for all γ near 0 by a continuity
argument. Since H = a ⊗ b, we consider several cases according to
positions of vectors a, b and e1 , e2 : a = b ⊥ {e1 , e2 };
a = b and ∠(a, {e1 , e2 }) = θ, 0 ≤ θ < π/2; and 0 < |a · b| < 1.
22
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
21 / 28
Simple Shear
It was shown in23 that monotonicity holds for the model (6) in an
interval of γ centered at γ = 0, in an interval of α near 0, and in an
interval of α2 near 0, resulting from intersecting all the three intervals
of γ, the three intervals of α, and the three interval of α2 ,
respectively, in the three cases above.
23
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
22 / 28
Counterexample
We construct a counterexamle demonstrating the failure of
monotonicity for the given model (6). Consider Case 3 of Simple
Shear. Taking α2 = 0.
Tina Mai (Texas A&M University)
Monotonicity
2014
23 / 28
Counterexample
We construct a counterexamle demonstrating the failure of
monotonicity for the given model (6). Consider Case 3 of Simple
Shear. Taking α2 = 0.
Let e3 be an orthonormal vector to e1 and e2 , then
a = a1 e1 + a2 e2 + a3 e3 , b = b1 e1 + b2 e2 + b3 e3
(29)
where ai , bi are scalars, i = 1, 2, 3.
Tina Mai (Texas A&M University)
Monotonicity
2014
23 / 28
Counterexample
We construct a counterexamle demonstrating the failure of
monotonicity for the given model (6). Consider Case 3 of Simple
Shear. Taking α2 = 0.
Let e3 be an orthonormal vector to e1 and e2 , then
a = a1 e1 + a2 e2 + a3 e3 , b = b1 e1 + b2 e2 + b3 e3
(29)
where ai , bi are scalars, i = 1, 2, 3.
Note that |a| = |b| = 1, thus
q
q
|a3 | = 1 − a12 − a22 , |b3 | = 1 − b12 − b22 .
Tina Mai (Texas A&M University)
Monotonicity
(30)
2014
23 / 28
Counterexample
We construct a counterexamle demonstrating the failure of
monotonicity for the given model (6). Consider Case 3 of Simple
Shear. Taking α2 = 0.
Let e3 be an orthonormal vector to e1 and e2 , then
a = a1 e1 + a2 e2 + a3 e3 , b = b1 e1 + b2 e2 + b3 e3
(29)
where ai , bi are scalars, i = 1, 2, 3.
Note that |a| = |b| = 1, thus
q
q
|a3 | = 1 − a12 − a22 , |b3 | = 1 − b12 − b22 .
(30)
After an analysis argument, and Matlab computation, we choose
α = 0.1, β1 = 0.2, γ = −2.5, a1 = 0.5, a2 = 0.1, b1 = 0.6, b2 = 0.2,
to lead to the failure of monotonicity of the model (6). Here,
γ = −2.5 belongs to the range of the respond function (6). to the
Tina Mai (Texas A&M University)
Monotonicity
2014
23 / 28
General Models
For the model (6), consider a class of deformation gradients having
the form
F = I + γ Ũ,
(31)
where γ is a scalar, Ũ is a fixed, constant displacement gradient.
Thus,
1
T
T
(32)
E = (γ(Ũ + Ũ) + γ 2 Ũ Ũ) .
2
24
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
24 / 28
General Models
For the model (6), consider a class of deformation gradients having
the form
F = I + γ Ũ,
(31)
where γ is a scalar, Ũ is a fixed, constant displacement gradient.
Thus,
1
T
T
(32)
E = (γ(Ũ + Ũ) + γ 2 Ũ Ũ) .
2
Denote by {ei }3i=1 an orthonormal basis for R3 consisting entirely of
eigenvectors of E, and {λi }3i=1 the associated eigenvalues of E, i.e.
Eei = λi ei ,
(33)
for i = 1, 2, 3. Using this setting and continuity argument, we proved
in24 that in an interval of γ centered at γ = 0, in an interval of α
near 0, and in an interval of α2 near 0, monotonicity holds for the
model (6).
24
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
24 / 28
Conclusion
The monotonicity we have studied is for a class of nonlinear
strain-limiting models of elastic-like (non-dissipative) material bodies
in the form
2
E = φ1 (|S̄|)S̄ + φ2 (|S̄|2 )S̄ ,
(34)
with
1
,
1 + β1 r
α2
φ2 (r ) :=
,
1 + β2 r
where α2 is a constant, and β1 , β2 are non-negative constants.
φ1 (r ) :=
Tina Mai (Texas A&M University)
Monotonicity
2014
(35)
(36)
25 / 28
Conclusion
The monotonicity we have studied is for a class of nonlinear
strain-limiting models of elastic-like (non-dissipative) material bodies
in the form
2
E = φ1 (|S̄|)S̄ + φ2 (|S̄|2 )S̄ ,
(34)
with
1
,
(35)
1 + β1 r
α2
φ2 (r ) :=
,
(36)
1 + β2 r
where α2 is a constant, and β1 , β2 are non-negative constants.
In this class of models, it can happen that the Fréchet derivative of
the response function is not invertible as a fourth-order tensor even
when the response function itself is uniquely invertible.
φ1 (r ) :=
Tina Mai (Texas A&M University)
Monotonicity
2014
25 / 28
Conclusion
The monotonicity we have studied is for a class of nonlinear
strain-limiting models of elastic-like (non-dissipative) material bodies
in the form
2
E = φ1 (|S̄|)S̄ + φ2 (|S̄|2 )S̄ ,
(34)
with
1
,
(35)
1 + β1 r
α2
φ2 (r ) :=
,
(36)
1 + β2 r
where α2 is a constant, and β1 , β2 are non-negative constants.
In this class of models, it can happen that the Fréchet derivative of
the response function is not invertible as a fourth-order tensor even
when the response function itself is uniquely invertible.
The notion of strong ellipticity introduced in [4] is then no longer
valid, leading to the introduction in this work the monotonicity as a
weaker convexity notion.
φ1 (r ) :=
Tina Mai (Texas A&M University)
Monotonicity
2014
25 / 28
Conclusion
For the class of models studied herein, it is shown that monotonicity
holds for strains with sufficiently small norms, and fails (by
constructed counterexample) when strain is large enough.
Tina
25 Mai (Texas A&M University)
Monotonicity
2014
26 / 28
Conclusion
For the class of models studied herein, it is shown that monotonicity
holds for strains with sufficiently small norms, and fails (by
constructed counterexample) when strain is large enough.
These results are similar to the conditions on strain for strong
ellipticity of implicit constitutive and strain-limiting models
investigated in25 .
Tina
25 Mai (Texas A&M University)
Monotonicity
2014
26 / 28
Conclusion
For the class of models studied herein, it is shown that monotonicity
holds for strains with sufficiently small norms, and fails (by
constructed counterexample) when strain is large enough.
These results are similar to the conditions on strain for strong
ellipticity of implicit constitutive and strain-limiting models
investigated in25 .
As we noted in the Compression Section and the Simple Shear
Section, the invertibility of this class of models does not guarantee
the monotoncity. This observation emphasizes the independence
between the invertibility notion and the monotonicity notion.
Tina
25 Mai (Texas A&M University)
Monotonicity
2014
26 / 28
Conclusion
For the class of models studied herein, it is shown that monotonicity
holds for strains with sufficiently small norms, and fails (by
constructed counterexample) when strain is large enough.
These results are similar to the conditions on strain for strong
ellipticity of implicit constitutive and strain-limiting models
investigated in25 .
As we noted in the Compression Section and the Simple Shear
Section, the invertibility of this class of models does not guarantee
the monotoncity. This observation emphasizes the independence
between the invertibility notion and the monotonicity notion.
As another note, in this paper, we restricted the study of
monotonicity to the case when (34) is uniquely invertible at least for
sufficiently small strain. In a future study, we will focus on a more
general case when the inverse of (34) is a multivalued map, and
investigate strong ellipticity as well as monotonicity on each branch of
the graph of (34), i.e. where (34) is uniquely invertible. This issue
has not been investigated in neither our work26 nor27 .
Tina
25 Mai (Texas A&M University)
Monotonicity
2014
26 / 28
Conclusion
Regarding hyperelasticity, the class of models (6) does not arise as
the gradient of a potential unless the function φ2 (|S̄|2 ) is identically
constant, and hence not strain limiting unless that constant is zero.
28
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
27 / 28
Conclusion
Regarding hyperelasticity, the class of models (6) does not arise as
the gradient of a potential unless the function φ2 (|S̄|2 ) is identically
constant, and hence not strain limiting unless that constant is zero.
However, a natural modification of (6) does produce hyperelastic
models. For example, if the second term on the right-hand-side of (6)
is replaced by
−1
F2 (S̄) := φ2 (det(S̄))S̄ ,
(37)
28
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
27 / 28
Conclusion
Regarding hyperelasticity, the class of models (6) does not arise as
the gradient of a potential unless the function φ2 (|S̄|2 ) is identically
constant, and hence not strain limiting unless that constant is zero.
However, a natural modification of (6) does produce hyperelastic
models. For example, if the second term on the right-hand-side of (6)
is replaced by
−1
F2 (S̄) := φ2 (det(S̄))S̄ ,
(37)
it is shown in28 that one can readily construct a potential for F2 (S̄) in
(37). Studying convexity for strain limiting models including a
response function of the form (37) is beyond the scope of the present
contribution and will be addressed in a future investigation.
28
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Journal of Elasticity in press (2014). DOI:
10.1007/s10659-014-9503-4.
Tina Mai (Texas A&M University)
Monotonicity
2014
27 / 28
Questions?
Thank you for your attention!
Tina Mai (Texas A&M University)
Monotonicity
2014
28 / 28