On Strong Ellipticity for Implicit and Strain-Limiting
Theories of Elasticity
Tina Mai
mai@math.tamu.edu
with Jay R. Walton
Department of Mathematics, Texas A&M University
2014
Motivation
New class of elastic bodies in K. R. Rajagopal’s papers1 ,2 ,3 .
1
K. R. Rajagopal. “The elasticity of elasticity”. In: Zeitschrift für
Angewandte Mathematik und Physik 58.2 (2007), 309–317.
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
2 / 24
Motivation
New class of elastic bodies in K. R. Rajagopal’s papers1 ,2 ,3 .
Implicit relationship between stress and strain.
1
K. R. Rajagopal. “The elasticity of elasticity”. In: Zeitschrift für
Angewandte Mathematik und Physik 58.2 (2007), 309–317.
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
2 / 24
Motivation
New class of elastic bodies in K. R. Rajagopal’s papers1 ,2 ,3 .
Implicit relationship between stress and strain.
Tools for elastic-like material behavior that are nonlinear with
infinitesimal strains.
1
K. R. Rajagopal. “The elasticity of elasticity”. In: Zeitschrift für
Angewandte Mathematik und Physik 58.2 (2007), 309–317.
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
2 / 24
Motivation
New class of elastic bodies in K. R. Rajagopal’s papers1 ,2 ,3 .
Implicit relationship between stress and strain.
Tools for elastic-like material behavior that are nonlinear with
infinitesimal strains.
Exist elastic-like bodies that are neither Cauchy elastic nor Green
elastic4 .
1
K. R. Rajagopal. “The elasticity of elasticity”. In: Zeitschrift für
Angewandte Mathematik und Physik 58.2 (2007), 309–317.
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
2 / 24
Motivation
Implicit constitutive models in theoretical and numerical
studies5 ,6 ,7 ,8 ,9 .
5
R. Bustamante and K. R. Rajagopal. “A Note on Plane Strain and Plane
Stress Problems for a New Class of Elastic Bodies”. In: Mathematics and
Mechanics of Solids 15.2 (2010), 229–238.
6
R. Bustamante and K. R. Rajagopal. “Solutions of some simple boundary
value problems within the context of a new class of elastic materials”. In:
International Journal of Nonlinear Mechanics 46.2 (2011), 376–386.
7
R. Bustamante and K. R. Rajagopal. “On the inhomogeneous shearing of a
new class of elastic bodies”. In: Mathematics and Mechanics of Solids 17.7
(2012), 762–778.
8
R. Bustamante and K. R. Rajagopal. “On a new class of electro-elastic
bodies. II. Boundary value problems”. In: Proceedings of the Royal Society of
London, Mathematical, Physical and Engineering Sciences 469.2155 (2013).
9
A. Ortiz, R. Bustamante, and K. R. Rajagopal. “A numerical study of a
plate with a hole for a new class of elastic bodies”. In: Acta Mechanica 223.9
(2012), 1971–1981.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
3 / 24
Motivation
Implicit constitutive models in theoretical and numerical studies10 ,11 .
10
K. R. Rajagopal. “On a New Class of Models in Elasticity”. In:
Mathematical & Computational Applications 15.4, SI (2010). Workshop on
Recent Developments in Differential Equations, Mechanics and Applications,
Univ Witwatersrand, Johannesburg, SouthAfrica, Aug 21-22, 2009, 506–528.
issn: 1300-686X.
11
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.
12
Vojtech Kulvait, Josef Malek, and K. R. Rajagopal. “Anti-plane stress state
of a plate with a V-notch for a new class of elastic solids”. In: International
Journal of Fracture 179.1-2 (2013), 59–73.
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
4 / 24
Motivation
Implicit constitutive models in theoretical and numerical studies10 ,11 .
In fracture12 ,13 .
10
K. R. Rajagopal. “On a New Class of Models in Elasticity”. In:
Mathematical & Computational Applications 15.4, SI (2010). Workshop on
Recent Developments in Differential Equations, Mechanics and Applications,
Univ Witwatersrand, Johannesburg, SouthAfrica, Aug 21-22, 2009, 506–528.
issn: 1300-686X.
11
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.
12
Vojtech Kulvait, Josef Malek, and K. R. Rajagopal. “Anti-plane stress state
of a plate with a V-notch for a new class of elastic solids”. In: International
Journal of Fracture 179.1-2 (2013), 59–73.
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
4 / 24
Motivation
In other fields of application14 ,15 .
14
R. Bustamante and K. R. Rajagopal. “On a new class of electro-elastic
bodies. II. Boundary value problems”. In: Proceedings of the Royal Society of
London, Mathematical, Physical and Engineering Sciences 469.2155 (2013).
15
R. Bustamante and K. R. Rajagopal. “On a new class of electroelastic
bodies. I”. In: Proceedings of the Royal Society of London, Mathematical,
Physical and Engineering Sciences 469.2149 (2013). doi:
10.1098/rspa.2012.0521.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
5 / 24
Motivation
In other fields of application14 ,15 .
None of them has the issue of convexity been investigated.
14
R. Bustamante and K. R. Rajagopal. “On a new class of electro-elastic
bodies. II. Boundary value problems”. In: Proceedings of the Royal Society of
London, Mathematical, Physical and Engineering Sciences 469.2155 (2013).
15
R. Bustamante and K. R. Rajagopal. “On a new class of electroelastic
bodies. I”. In: Proceedings of the Royal Society of London, Mathematical,
Physical and Engineering Sciences 469.2149 (2013). doi:
10.1098/rspa.2012.0521.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
5 / 24
Motivation
In other fields of application14 ,15 .
None of them has the issue of convexity been investigated.
Strong ellipticity or rank-one convexity for elastic implicit theories.
14
R. Bustamante and K. R. Rajagopal. “On a new class of electro-elastic
bodies. II. Boundary value problems”. In: Proceedings of the Royal Society of
London, Mathematical, Physical and Engineering Sciences 469.2155 (2013).
15
R. Bustamante and K. R. Rajagopal. “On a new class of electroelastic
bodies. I”. In: Proceedings of the Royal Society of London, Mathematical,
Physical and Engineering Sciences 469.2149 (2013). doi:
10.1098/rspa.2012.0521.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
5 / 24
Motivation
In other fields of application14 ,15 .
None of them has the issue of convexity been investigated.
Strong ellipticity or rank-one convexity for elastic implicit theories.
Key condition for stability of numerical approximations and for
physically intuitive wave propagation behavior.
14
R. Bustamante and K. R. Rajagopal. “On a new class of electro-elastic
bodies. II. Boundary value problems”. In: Proceedings of the Royal Society of
London, Mathematical, Physical and Engineering Sciences 469.2155 (2013).
15
R. Bustamante and K. R. Rajagopal. “On a new class of electroelastic
bodies. I”. In: Proceedings of the Royal Society of London, Mathematical,
Physical and Engineering Sciences 469.2149 (2013). doi:
10.1098/rspa.2012.0521.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
5 / 24
Motivation
Given a constitutive model, look for the class of deformations for
which strong ellipticity holds.
16
J. K. Knowles and E. Sternberg. “On the failure of ellipticity of the
equations for finite elastostatic plane strain”. In: Archive for Rational Mechanics
and Analysis 63 (1977), pp. 321–335.
17
T. Sendova and J. R. Walton. “On Strong Ellipticity for Isotropic
Hyperelastic Materials Based upon Logarithmic Strain”. In: International
Journal of Nonlinear Mechanics 40 (2004), pp. 195–212.
18
J. P. Wilber & J. R. Walton. “The convexity properties of a class of
constitutive models for biological soft tissues”. In: Mathematics and Mechanics
of Solids 7 (2002), pp. 217–235.
19
J. P. Wilber & J. R. Walton. “Sufficient conditions for strong ellipticity for
a class of anisotropic materials”. In: International Journal of Nonlinear
Mechanics 38 (2003), pp. 441–455.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
6 / 24
Motivation
Given a constitutive model, look for the class of deformations for
which strong ellipticity holds.
Determine the deformations for which strong ellipticity fails16 ,17 ,18 ,19 .
16
J. K. Knowles and E. Sternberg. “On the failure of ellipticity of the
equations for finite elastostatic plane strain”. In: Archive for Rational Mechanics
and Analysis 63 (1977), pp. 321–335.
17
T. Sendova and J. R. Walton. “On Strong Ellipticity for Isotropic
Hyperelastic Materials Based upon Logarithmic Strain”. In: International
Journal of Nonlinear Mechanics 40 (2004), pp. 195–212.
18
J. P. Wilber & J. R. Walton. “The convexity properties of a class of
constitutive models for biological soft tissues”. In: Mathematics and Mechanics
of Solids 7 (2002), pp. 217–235.
19
J. P. Wilber & J. R. Walton. “Sufficient conditions for strong ellipticity for
a class of anisotropic materials”. In: International Journal of Nonlinear
Mechanics 38 (2003), pp. 441–455.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
6 / 24
Motivation
Given a constitutive model, look for the class of deformations for
which strong ellipticity holds.
Determine the deformations for which strong ellipticity fails16 ,17 ,18 ,19 .
Also focus on strain-limiting models.
16
J. K. Knowles and E. Sternberg. “On the failure of ellipticity of the
equations for finite elastostatic plane strain”. In: Archive for Rational Mechanics
and Analysis 63 (1977), pp. 321–335.
17
T. Sendova and J. R. Walton. “On Strong Ellipticity for Isotropic
Hyperelastic Materials Based upon Logarithmic Strain”. In: International
Journal of Nonlinear Mechanics 40 (2004), pp. 195–212.
18
J. P. Wilber & J. R. Walton. “The convexity properties of a class of
constitutive models for biological soft tissues”. In: Mathematics and Mechanics
of Solids 7 (2002), pp. 217–235.
19
J. P. Wilber & J. R. Walton. “Sufficient conditions for strong ellipticity for
a class of anisotropic materials”. In: International Journal of Nonlinear
Mechanics 38 (2003), pp. 441–455.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
6 / 24
Motivation
In20 ,
20
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
7 / 24
Motivation
In20 ,
Strong ellipticity holds for deformations with strain having sufficiently
small norm.
20
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
7 / 24
Motivation
In20 ,
Strong ellipticity holds for deformations with strain having sufficiently
small norm.
It fails for deformations (including pure compression and simple shear)
in which the small strain assumption is relaxed.
20
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
7 / 24
Notation and Preliminaries
Deformation f(·) : B −→ f(B).
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
8 / 24
Notation and Preliminaries
Deformation f(·) : B −→ f(B).
The displacement u = x − X and deformation gradient
F=
Tina Mai (Texas A&M University)
∂f
.
∂X
Strong Ellipticity
(1)
2014
8 / 24
Notation and Preliminaries
Deformation f(·) : B −→ f(B).
The displacement u = x − X and deformation gradient
F=
∂f
.
∂X
(1)
The left and right Cauchy-Green tensors
B = FFT
(2)
C = FT F.
(3)
and
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
8 / 24
Notation and Preliminaries
Deformation f(·) : B −→ f(B).
The displacement u = x − X and deformation gradient
F=
∂f
.
∂X
(1)
The left and right Cauchy-Green tensors
B = FFT
(2)
C = FT F.
(3)
and
The Green-St. Venant tensor E
1
E = (C − I).
2
Tina Mai (Texas A&M University)
Strong Ellipticity
(4)
2014
8 / 24
Notation and Preliminaries
Cauchy Stress Tensor T. Then the first and second Piola-Kirchhoff
Stress Tensors
S := TF−T det(F),
Tina Mai (Texas A&M University)
Strong Ellipticity
S̄ := F−1 S.
(5)
2014
9 / 24
Notation and Preliminaries
Cauchy Stress Tensor T. Then the first and second Piola-Kirchhoff
Stress Tensors
S := TF−T det(F),
S̄ := F−1 S.
(5)
Cauchy Elastic
S = Ŝ(F).
Tina Mai (Texas A&M University)
Strong Ellipticity
(6)
2014
9 / 24
Notation and Preliminaries
Cauchy Stress Tensor T. Then the first and second Piola-Kirchhoff
Stress Tensors
S := TF−T det(F),
S̄ := F−1 S.
(5)
Cauchy Elastic
S = Ŝ(F).
(6)
Green Elastic (or Hyperelastic)
Ŝ(F) = ∂F ŵ (F).
Tina Mai (Texas A&M University)
Strong Ellipticity
(7)
2014
9 / 24
Notation and Preliminaries
The Strong Ellipticity for S21
H · DF Ŝ(F)[H] > 0
(8)
for all non-zero tensors H of rank one, i.e. H = a ⊗ b for non-zero
vectors a and b, |a| = |b| = 1.
21
S. S. Antman. Nonlinear Problems of Elasticity. Springer-Verlag, 2005.
isbn: 978-0-387-20880-0.
22
K. R. Rajagopal. “The elasticity of elasticity”. In: Zeitschrift für
Angewandte Mathematik und Physik 58.2 (2007), 309–317.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
10 / 24
Notation and Preliminaries
The Strong Ellipticity for S21
H · DF Ŝ(F)[H] > 0
(8)
for all non-zero tensors H of rank one, i.e. H = a ⊗ b for non-zero
vectors a and b, |a| = |b| = 1.
In22 , Rajagopal considered isotropic, implicit constitutive relations
0 = F(B, T).
(9)
21
S. S. Antman. Nonlinear Problems of Elasticity. Springer-Verlag, 2005.
isbn: 978-0-387-20880-0.
22
K. R. Rajagopal. “The elasticity of elasticity”. In: Zeitschrift für
Angewandte Mathematik und Physik 58.2 (2007), 309–317.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
10 / 24
Notation and Preliminaries
The Strong Ellipticity for S21
H · DF Ŝ(F)[H] > 0
(8)
for all non-zero tensors H of rank one, i.e. H = a ⊗ b for non-zero
vectors a and b, |a| = |b| = 1.
In22 , Rajagopal considered isotropic, implicit constitutive relations
0 = F(B, T).
(9)
B = F(T).
(10)
Special case
21
S. S. Antman. Nonlinear Problems of Elasticity. Springer-Verlag, 2005.
isbn: 978-0-387-20880-0.
22
K. R. Rajagopal. “The elasticity of elasticity”. In: Zeitschrift für
Angewandte Mathematik und Physik 58.2 (2007), 309–317.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
10 / 24
Notation and Preliminaries
Non-linear, infinitesimal strain theories
= F(T)
(11)
where denotes the traditional linearized strain tensor
1
∇u + ∇uT .
:=
2
Tina Mai (Texas A&M University)
Strong Ellipticity
(12)
2014
11 / 24
Notation and Preliminaries
Non-linear, infinitesimal strain theories
= F(T)
(11)
where denotes the traditional linearized strain tensor
1
∇u + ∇uT .
:=
2
(12)
For studying strong ellipticity, consider
Tina Mai (Texas A&M University)
0 = F(E, S̄)
(13)
E = F(S̄).
(14)
Strong Ellipticity
2014
11 / 24
Results–Strong Ellipticity–General Observations
Recall that we denote by H the non-zero tensors of rank one, i.e. H = a ⊗ b for non-zero
vectors a and b, |a| = |b| = 1.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
12 / 24
Results–Strong Ellipticity–General Observations
Recall that we denote by H the non-zero tensors of rank one, i.e. H = a ⊗ b for non-zero
vectors a and b, |a| = |b| = 1.
Strong ellipticity for the implicit constitutive relation (13)
H · DF Ŝ(F)[H] = H · DF FS̄(E) [H]
= H · (HS̄(E))
h
ii
−1 h
∂E F(S̄, E) (FT H)s > 0,
− FT H · ∂S̄ F(S̄, E)
s
where ∂S̄ F(S̄, E)
F(S̄, E).
−1
(15)
[·] denotes the inverse of the partial Fréchet differentiation of
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
12 / 24
Results–Strong Ellipticity–General Observations
Recall that we denote by H the non-zero tensors of rank one, i.e. H = a ⊗ b for non-zero
vectors a and b, |a| = |b| = 1.
Strong ellipticity for the implicit constitutive relation (13)
H · DF Ŝ(F)[H] = H · DF FS̄(E) [H]
= H · (HS̄(E))
h
ii
−1 h
∂E F(S̄, E) (FT H)s > 0,
− FT H · ∂S̄ F(S̄, E)
s
where ∂S̄ F(S̄, E)
F(S̄, E).
For (14),
−1
(15)
[·] denotes the inverse of the partial Fréchet differentiation of
H · DF Ŝ(F)[H] = H · (HS̄(E))
i
−1 h T
− FT H · DS̄ F(S̄)
(F H)s > 0,
(16)
s
−1
where DS̄ F(S̄)
[·] denotes the inverse of the fourth-order tensor DS̄ F(S̄)[·].
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
12 / 24
Results–Invertibility and Strong Ellipticity
Consider an isotropic model:
E = F(S̄) := φ0 S̄ · I I + φ1 S̄ S̄.
Tina Mai (Texas A&M University)
Strong Ellipticity
(17)
2014
13 / 24
Results–Invertibility and Strong Ellipticity
Consider an isotropic model:
E = F(S̄) := φ0 S̄ · I I + φ1 S̄ S̄.
(17)
An anisotropic model:
E = φ1 (|K1/2 [S̄]|)K[S̄],
(18)
where K1/2 [·], K[·] are positive-definite, symmetric fourth-order
tensors
linear
(as
transformations from Sym to Sym), and
1/2
1/2
K
K [S] = K[S] for all S ∈ Sym.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
13 / 24
Results-Invertibility
For (17), two ways to analyze the invertibility:
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
14 / 24
Results-Invertibility
For (17), two ways to analyze the invertibility:
First, based on classical spectral theory for symmetric, second-order
tensors.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
14 / 24
Results-Invertibility
For (17), two ways to analyze the invertibility:
First, based on classical spectral theory for symmetric, second-order
tensors.
Second, use decomposition of a second-order, symmetric tensor into its
deviatoric (trace-less) and trace parts.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
14 / 24
Results-Invertibility
For (17), two ways to analyze the invertibility:
First, based on classical spectral theory for symmetric, second-order
tensors.
Second, use decomposition of a second-order, symmetric tensor into its
deviatoric (trace-less) and trace parts.
In the first way, inverting (17) is equivalent to solving an algebraic
system for the eigen-values of S̄, in terms of the eigen-values of E.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
14 / 24
Results-Invertibility
For (17), two ways to analyze the invertibility:
First, based on classical spectral theory for symmetric, second-order
tensors.
Second, use decomposition of a second-order, symmetric tensor into its
deviatoric (trace-less) and trace parts.
In the first way, inverting (17) is equivalent to solving an algebraic
system for the eigen-values of S̄, in terms of the eigen-values of E.
In the second way, the unique invertibility of (17) is equivalent to E
lies in the compact range of the function in (17).
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
14 / 24
Results-Invertibility
For (17), two ways to analyze the invertibility:
First, based on classical spectral theory for symmetric, second-order
tensors.
Second, use decomposition of a second-order, symmetric tensor into its
deviatoric (trace-less) and trace parts.
In the first way, inverting (17) is equivalent to solving an algebraic
system for the eigen-values of S̄, in terms of the eigen-values of E.
In the second way, the unique invertibility of (17) is equivalent to E
lies in the compact range of the function in (17).
For (18), let E[·] be the inverse of K[·]. Then, we can solve for S̄ in
terms of E[·].
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
14 / 24
Results–Strong Ellipticity
Consider
E = F(S̄).
Tina Mai (Texas A&M University)
Strong Ellipticity
(19)
2014
15 / 24
Results–Strong Ellipticity
Consider
E = F(S̄).
(19)
S̄ = F −1 (E).
(20)
Suppose
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
15 / 24
Results–Strong Ellipticity
Consider
E = F(S̄).
(19)
S̄ = F −1 (E).
(20)
Suppose
Strong Ellipticity for (8)
0 < H · DF Ŝ[H] = H · (HS̄) + (FT H)s · DE F −1 (E)[(FT H)s ],
(21)
where DE F −1 (E)[·] denotes the Fréchet differentiation of the inverse
of F(E).
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
15 / 24
Results–Strong Ellipticity
Consider
E = F(S̄).
(19)
S̄ = F −1 (E).
(20)
Suppose
Strong Ellipticity for (8)
0 < H · DF Ŝ[H] = H · (HS̄) + (FT H)s · DE F −1 (E)[(FT H)s ],
(21)
where DE F −1 (E)[·] denotes the Fréchet differentiation of the inverse
of F(E).
Otherwise, if (19) cannot be explicitly inverted, one can use the
strong ellipticity condition in the form (16).
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
15 / 24
Results–Strong Ellipticity
Demonstration of two approaches:
23
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
16 / 24
Results–Strong Ellipticity
Demonstration of two approaches:
Apply (21) to the anisotropic model (18).
23
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
16 / 24
Results–Strong Ellipticity
Demonstration of two approaches:
Apply (21) to the anisotropic model (18).
Apply (16) to the isotropic model (17).
23
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
16 / 24
Results–Strong Ellipticity
Demonstration of two approaches:
Apply (21) to the anisotropic model (18).
Apply (16) to the isotropic model (17).
These models were shown in23 to hold strong ellipticity for
deformation gradients sufficiently near the identity (small strains) but
fails if strains are not so severely limited (by counterexamples).
23
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
16 / 24
Results–Examples
Inspired by models studied by Rajagopal and co-authors24 ,25 ,26 .
24
Vojtech Kulvait, Josef Malek, and K. R. Rajagopal. “Anti-plane stress state
of a plate with a V-notch for a new class of elastic solids”. In: International
Journal of Fracture 179.1-2 (2013), 59–73.
25
K. R. Rajagopal. “On a New Class of Models in Elasticity”. In:
Mathematical & Computational Applications 15.4, SI (2010). Workshop on
Recent Developments in Differential Equations, Mechanics and Applications,
Univ Witwatersrand, Johannesburg, SouthAfrica, Aug 21-22, 2009, 506–528.
issn: 1300-686X.
26
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
17 / 24
Results–Examples
First, in model (17), we choose
−β0 r
φ0 (r ) := α0 1 − exp
1 + δ0 |r |
and
φ1 (r ) :=
(22)
α1
,
1 + β1 r
(23)
where α0 , β0 , δ0 , α1 , and β1 are positive constants.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
18 / 24
Results–Examples
In27 , this model was shown to hold strong ellipticity provided the sum
of the upper bounds on φ0 (r ) and r φ1 (r ) are small enough.
27
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
19 / 24
Results–Examples
In27 , this model was shown to hold strong ellipticity provided the sum
of the upper bounds on φ0 (r ) and r φ1 (r ) are small enough.
It can fail to be strongly elliptic when taking
S̄ = µI,
(24)
F = γI,
(25)
and
where µ and γ are constants with γ 2 << 1 (corresponding to severe
compression).
27
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
19 / 24
Results–Examples
The next model
S̄
(26)
1 + β |S̄|
was studied by Rajagopal and co-authors in the context of anti-plane
shear, infinitesimal deformations28 ,29 .
E=
28
Vojtech Kulvait, Josef Malek, and K. R. Rajagopal. “Anti-plane stress state
of a plate with a V-notch for a new class of elastic solids”. In: International
Journal of Fracture 179.1-2 (2013), 59–73.
29
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
20 / 24
Results–Examples
The next model
S̄
(26)
1 + β |S̄|
was studied by Rajagopal and co-authors in the context of anti-plane
shear, infinitesimal deformations28 ,29 .
Here
F = I + γ e1 ⊗ e2 ,
(27)
E=
a simple shear, with γ is apscalar, and e1 and e2 are orthogonal unit
vectors. Then |E| = |γ/2| 2 + γ 2 .
28
Vojtech Kulvait, Josef Malek, and K. R. Rajagopal. “Anti-plane stress state
of a plate with a V-notch for a new class of elastic solids”. In: International
Journal of Fracture 179.1-2 (2013), 59–73.
29
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
20 / 24
Results–Examples
The next model
S̄
(26)
1 + β |S̄|
was studied by Rajagopal and co-authors in the context of anti-plane
shear, infinitesimal deformations28 ,29 .
Here
F = I + γ e1 ⊗ e2 ,
(27)
E=
a simple shear, with γ is apscalar, and e1 and e2 are orthogonal unit
vectors. Then |E| = |γ/2| 2 + γ 2 .
The strong ellipticity of (26) holds if |γ| << 1 or β is very large, and
fails if β is small enough.
28
Vojtech Kulvait, Josef Malek, and K. R. Rajagopal. “Anti-plane stress state
of a plate with a V-notch for a new class of elastic solids”. In: International
Journal of Fracture 179.1-2 (2013), 59–73.
29
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.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
20 / 24
Conclusion
Studied Strong Ellipticity for implicit constitutive and strain-limiting
models of elastic-like (non-dissipative) material bodies.
30
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
21 / 24
Conclusion
Studied Strong Ellipticity for implicit constitutive and strain-limiting
models of elastic-like (non-dissipative) material bodies.
General approach was investigated in30 .
30
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
21 / 24
Conclusion
Studied Strong Ellipticity for implicit constitutive and strain-limiting
models of elastic-like (non-dissipative) material bodies.
General approach was investigated in30 .
Application to recent models inspired by Rajagopal and a number of
co-authors.
30
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
21 / 24
Conclusion
Studied Strong Ellipticity for implicit constitutive and strain-limiting
models of elastic-like (non-dissipative) material bodies.
General approach was investigated in30 .
Application to recent models inspired by Rajagopal and a number of
co-authors.
Strong Ellipticity holds in the small strain limit.
30
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
21 / 24
Conclusion
Studied Strong Ellipticity for implicit constitutive and strain-limiting
models of elastic-like (non-dissipative) material bodies.
General approach was investigated in30 .
Application to recent models inspired by Rajagopal and a number of
co-authors.
Strong Ellipticity holds in the small strain limit.
It fails if strain is large enough, generally corresponding to extreme
compression (including purely compressive deformations) and simple
shear deformations.
30
Tina Mai and Jay R. Walton. “On Strong Ellipticity for Implicit and
Strain-Limiting Theories of Elasticity”. In: (). To appear in Mathematics and
Mechanics of Solids, 2014.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
21 / 24
More
If a Cauchy Elastic body has response function Ŝ(·) not differentiable
at some deformation gradient F, then
31
S. S. Antman. Nonlinear Problems of Elasticity. Springer-Verlag, 2005.
isbn: 978-0-387-20880-0.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
22 / 24
More
If a Cauchy Elastic body has response function Ŝ(·) not differentiable
at some deformation gradient F, then
consider a weaker rank-1 convexity notion given by the monotonicity
condition31 :
(Ŝ(F + αH) − Ŝ(F)) · H > 0
(28)
for all rank-1 tensors H = a ⊗ b (with |a| = |b| = 1) and 0 < α ≤ 1
such that
det(F + αH) > 0.
(29)
31
S. S. Antman. Nonlinear Problems of Elasticity. Springer-Verlag, 2005.
isbn: 978-0-387-20880-0.
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
22 / 24
More
Consider strain-limiting models having stronger nonlinearity in the
form:
2
E = φ0 (S̄)I + φ1 (S̄)S̄ + φ2 (S̄)S̄
(30)
in which φj (·), j = 1, 2, 3 are scalar-valued functions.
32
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Submitted (2014).
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
23 / 24
More
Consider strain-limiting models having stronger nonlinearity in the
form:
2
E = φ0 (S̄)I + φ1 (S̄)S̄ + φ2 (S̄)S̄
(30)
in which φj (·), j = 1, 2, 3 are scalar-valued functions.
Study invertibility, strong ellipticity, and monotonicity of (30).
32
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Submitted (2014).
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
23 / 24
More
Consider strain-limiting models having stronger nonlinearity in the
form:
2
E = φ0 (S̄)I + φ1 (S̄)S̄ + φ2 (S̄)S̄
(30)
in which φj (·), j = 1, 2, 3 are scalar-valued functions.
Study invertibility, strong ellipticity, and monotonicity of (30).
Results: Monotonicity holds for strains with sufficiently small norms,
and fails when strain is large enough.
32
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Submitted (2014).
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
23 / 24
More
Consider strain-limiting models having stronger nonlinearity in the
form:
2
E = φ0 (S̄)I + φ1 (S̄)S̄ + φ2 (S̄)S̄
(30)
in which φj (·), j = 1, 2, 3 are scalar-valued functions.
Study invertibility, strong ellipticity, and monotonicity of (30).
Results: Monotonicity holds for strains with sufficiently small norms,
and fails when strain is large enough.
These will be addressed in a future contribution32 .
32
Tina Mai and Jay R. Walton. “On Monotonicity for Strain-Limiting
Theories of Elasticity”. In: Submitted (2014).
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
23 / 24
Questions?
Thank you for your attention!
Tina Mai (Texas A&M University)
Strong Ellipticity
2014
24 / 24