The theory of continuum and elasto plastic materials

The theory of continuum and
elasto-plastic materials
Written by:
Braj Bhushan Prasad
Technische Universität Berlin, Faculty V - Mechanical Engineering and
Transport Systems
February 11, 2018

The theory of continuum and elasto-plastic materials III
Contents
1. Mathematical preliminaries 1
1.1. Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1. The dyadic product . . . . . . . . . . . . . . . . . . . . . 2
1.1.2. Summation convention . . . . . . . . . . . . . . . . . . . . 3
1.1.3. Kronecker delta . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.4. Permutation tensor. . . . . . . . . . . . . . . . . . . . . . 4
1.2. Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1. The partial derivatives . . . . . . . . . . . . . . . . . . . . 5
1.2.2. The total derivatives . . . . . . . . . . . . . . . . . . . . . 5
1.2.3. The material derivative . . . . . . . . . . . . . . . . . . . 5
1.3. The differential operators . . . . . . . . . . . . . . . . . . . . . . 7
1.4. Integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. Fundamental of continuum mechanics 13
2.1. Kinematics of the continuum bodies . . . . . . . . . . . . . . . . 13
2.1.1. Configurations and the deformation mapping . . . . . . . 14
2.1.2. Deformation gradient . . . . . . . . . . . . . . . . . . . . 16
2.1.3. Transformation of volume elements . . . . . . . . . . . . . 18
2.1.4. Transformation of surface area elements . . . . . . . . . . 20
2.1.5. Measures of deformation . . . . . . . . . . . . . . . . . . . 21
2.2. Field equations of continuum mechanics . . . . . . . . . . . . . . 24
2.2.1. Balance of mass . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2. Balance of linear momentum . . . . . . . . . . . . . . . . 27
2.2.3. Balance of energy . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.4. The entropy inequality . . . . . . . . . . . . . . . . . . . . 32
3. Constitutive models for elastic materials 34
3.1. Generalized Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1. Isotropic case . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.2. Thermoelastic case . . . . . . . . . . . . . . . . . . . . . . 38
4. Constitutive models for elasto-plastic materials 40
4.1. Stress-strain relationship . . . . . . . . . . . . . . . . . . . . . . . 40
4.2. The yield criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1. Von Mises yield criterion . . . . . . . . . . . . . . . . . . 43
4.3. Flow rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Contents

The theory of continuum and elasto-plastic materials IV
4.4. Consistency condition . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5. Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5.1. Isotropic hardening . . . . . . . . . . . . . . . . . . . . . . 48
4.5.2. Kinematic hardening . . . . . . . . . . . . . . . . . . . . . 51
List of Figures VI
Bibliography VII
A. Important Results i
A.1. Derivation of the Gauss Theorem . . . . . . . . . . . . . . . . . . i
A.2. Time Derivative of the Jacobian . . . . . . . . . . . . . . . . . . iii
Contents

The theory of continuum and elasto-plastic materials 1
1. Mathematical preliminaries
Relations between pure and applied mathematicians are based on trust
and understanding. Namely, pure mathematicians do not trust applied
mathematicians, and applied mathematicians do not understand pure
mathematicians.
– Albert Einstein (14 March 1879 – 18 April 1955)
Physical laws provide mathematical models for the physical world. These
physical laws should be invariant in nature, i.e. independent of the position and
orientation of coordinate system or observer. In order to make these physical
laws independent from the choice of coordinate system, we express these laws
in term of vector and tensor equations. Therefore, it is necessary to have
fundamental understanding of tensor analysis and vector calculus. The purpose
of this preliminary chapter is to present the basic rules and standard results of
tensor and vector analysis, which are applied throughout this thesis. For a more
detailed explanation see the standard books on vectors and tensors by (Segel and
Handelman, 1977), (Itskov, 2007), (Schade and Neemann, 2009), and (Brannon,
2003). In this work we restrict ourself to three dimensional Cartesian tensors in
Euclidian space.
1.1. Tensor analysis
Tensor analysis is an important tool to visualize the theory of continuum me
chanics and is also significant in some of the theories used in Abaqus . Tensors
have been known since 1854 but the broader acceptance of its concepts began in
the 20th century. Tensors are a generalization of scalar and vector. Mathemati
cally tensor is visualized as a geometric object having a magnitude and one or
many associated directions. Tensors do not depend on any particular coordinate
system (invariant) and its characteristic is always linear. In more precise way,
we can define tensor as a multilinear mapping in Euclidian space 1 . One of
the important characteristics of tensor is its rank2 . On the basis of the rank of
1
Euclidian space R 𝑛
is an n-dimensional vector space defined over the field of real numbers.
2
The rank or order of a tensor provide information about the number of directions, which is
required to describe that particular tensor.
Chapter 1. Mathematical preliminaries

a tensor, we can say that a scalar is a tensor of rank zero and the vector is a
tensor of rank one. In general, a tensor of rank n in its Cartesian basis can be
written as:
𝑇 = 𝑇𝑖1,𝑖2,··· ,i 𝑛 𝑒i1 ⊗ 𝑒i2 ⊗ · · · ⊗ 𝑒i 𝑛 . (1.1)
𝑒i1 ⊗ 𝑒i2 represent the dyadic product between two base vectors. We use
both direct notation and index notation in this thesis. In direct notation we
use small bold Latin letters, 𝑎, 𝑏, · · · , for vectors and large bold Latin letters,
𝐴, 𝐵, · · · , 𝑇 , · · · , for tensors. In component form of notation a vector and a
second-order tensor are written as 𝑎𝑖, 𝐴𝑖𝑗, · · · , respectively. The index notation is
used to store the physical quantities, like vectors, matrices, and tensors, as a set
of numbers on the computer that are used in actual computation. Throughout
this work we restrict ourself to Cartesian tensor. Therefore, there is no need
to differentiate between covariant and contravariant components. This gives
us freedom to store one form of component for computation, either we store
𝑎i, 𝑇ij...p or 𝑎i, 𝑇ij...p.
Some example of tensors:
• The temperature, 𝑇, is a tensor of rank zero and is only described as a
magnitude.
• The force, 𝑓i, has magnitude as well as a direction and it is a vector,
therefore a tensor of rank one.
• The Cauchy stress, 𝜎ij, is a tensor of rank two. It has a magnitude and
three directions consisting of nine components. It can be represented in
matrix form:
𝜎ij = [𝜎] =
⎛
⎜
⎝
𝜎11 𝜎12 𝜎13
𝜎21 𝜎22 𝜎23
𝜎31 𝜎32 𝜎33
⎞
⎟
⎠ . (1.2)
1.1.1. The dyadic product
The dyadic product or outer product is a mathematical notation between two
vectors. Let 𝑎 = (𝑎1, 𝑎2, 𝑎3) and 𝑏 = (𝑏1, 𝑏2, 𝑏3) be two different vectors in the
three dimensional Euclidian space. Then, the dyadic product between these
two vectors are represented by 𝑎 ⊗ 𝑏, and is defined as follows:
𝑎 ⊗ 𝑏 = (𝑎i 𝑒i) ⊗ (𝑏j 𝑒j) = 𝑎i 𝑏j(𝑒i ⊗ 𝑒j)
= 𝑎1 𝑏1(𝑒1 ⊗ 𝑒1) + 𝑎1 𝑏2(𝑒1 ⊗ 𝑒2) + 𝑎1 𝑏3(𝑒1 ⊗ 𝑒3)
+ 𝑎2 𝑏1(𝑒2 ⊗ 𝑒1) + 𝑎2 𝑏2(𝑒2 ⊗ 𝑒2) + 𝑎2 𝑏3(𝑒2 ⊗ 𝑒3)
+ 𝑎3 𝑏1(𝑒3 ⊗ 𝑒1) + 𝑎3 𝑏2(𝑒3 ⊗ 𝑒2) + 𝑎3 𝑏3(𝑒3 ⊗ 𝑒3). (1.3)
Chapter 1. Mathematical preliminaries – Tensor analysis

The symbol ⊗ represents dyadic multiplication. 𝑒i = (𝑒1,𝑒2,𝑒3) and 𝑒j = (𝑒1,𝑒2,𝑒3)
are the Cartesian unit base vectors that are used to express vectors and tensors
in component form. The dyadic products between the Cartesian base vectors
𝑒i ⊗ 𝑒j is called a unit dyad.
The expression in Eq. (1.3) can be written in matrix form:
[𝑎 ⊗ 𝑏] =
⎛
⎜
⎝
𝑎1 𝑏1 𝑎1 𝑏2 𝑎1 𝑏3
𝑎2 𝑏1 𝑎2 𝑏2 𝑎2 𝑏3
𝑎3 𝑏1 𝑎3 𝑏2 𝑎3 𝑏3
⎞
⎟
⎠ . (1.4)
The dyadic product is not commutative, the base vectors are not interchangeable:
𝑒i ⊗ 𝑒j ̸= 𝑒j ⊗ 𝑒i. (1.5)
𝑎 ⊗ 𝑏 ̸= 𝑏 ⊗ 𝑎. (1.6)
1.1.2. Summation convention
Summation convention is also known as Einstein summation convention or
simply the Einstein notation. It was first introduced by Albert Einstein in his
general theory of relativity in 1916 (Kox, Klein, and Schulmann, 1997). In order
to understand how the summation convention works, let us consider an arbitrary
vector 𝑎 in the Cartesian coordinate system, which is expressed as follows:
𝑎 = 𝑎1 𝑒1 + 𝑎2 𝑒2 + 𝑎3 𝑒3 =
3∑︁
𝑖=1
𝑎i 𝑒i, (1.7)
𝑎1, 𝑎2, and 𝑎3 are components of the vector 𝑎 w.r.t. three unit vectors 𝑒1 =
(1, 0, 0) , 𝑒2 = (0, 1, 0), and 𝑒3 = (0, 0, 1). Removal of the summation symbol
from Eq. (1.7), gives us the following form:
𝑎 = 𝑎1 𝑒1 + 𝑎2 𝑒2 + 𝑎3 𝑒3 = 𝑎i 𝑒i. (1.8)
The expression in Eq. (1.8) is known as Einstein summation convention. Ac
cording to this convention, the indices are summed from one to three, if they
are occurring twice in one term. Mathematically the range of summation can be
generalized for n dimensions also, but in this work we restrict ourself to three
dimensional problems. In order to apply the Einstein summation, we need to
distinguish between dummy index and free index. If an index appears twice in a
term then it is known as a dummy index. On the other hand, an index occurring
only once in a term is known as a free index. The number of free indices available
in a term represent its rank.

1.1.3. Kronecker delta
Kronecker delta is named after the German mathematician and logician
Leopold Kronecker (1823-1891) and can be considered as a unit tensor of
second order (Müller, 2014a). It is defined as:
𝛿ij =
{︃
1, if i = j
0, if i ̸= j.
(1.9)
The alternative definition of Kronecker delta is given by a scalar product
between two orthonormal Cartesian base vectors:
𝑒i · 𝑒j = 𝛿ij. (1.10)
Apart from Eq. (1.9) and Eq. (1.10), we may give one more alternative definition
for the Kronecker delta:
𝜕𝑥𝑖
𝜕𝑥 𝑗
= 𝛿𝑖𝑗. (1.11)
1.1.4. Permutation tensor.
The Permutation symbol, also known as Levi-Civita symbol, is named after an
Italian mathematician Tullio Levi-Civita (1873–1941) (Bertram, 2012). The
general definition of the Levi-Civita symbol for n-dimension is:
𝜖𝑖1, 𝑖2, · · · , 𝑖 𝑛 =
⎧
⎪⎪⎨
⎪⎪⎩
1, if (𝑖1, 𝑖2, · · · , 𝑖 𝑛), cyclic permutation of (1, 2, 3, · · · , n)
-1, if (𝑖1, 𝑖2, · · · , 𝑖 𝑛), anti-cyclic permutation of (1, 2, 3, · · · , n)
0, if any of 𝑖1, 𝑖2, · · · , 𝑖 𝑛 are equal.
(1.12)
For three dimensional Euclidian space the Levi-Civita symbol is defined as:
𝜖ijk =
⎧
⎪⎪⎨
⎪⎪⎩
1, if (i, j, k), cyclic permutation of (1, 2, 3)
-1, if (i, j, k), anti-cyclic permutation of (1, 2, 3)
0, if any of i, j, k are equal.
(1.13)
From Eq. (1.13) we may conclude:
𝜖ijk = 𝜖jki = 𝜖kij = −𝜖jik = −𝜖ikj = −𝜖kji. (1.14)
From Eq. (1.13) and Eq. (1.14) we may also conclude that, Levi-Civita tensor,
𝜀ijk, has 27 components. In these 27 components, 21 components are equal, three
components are cyclic and rest three components are anti-cyclic.

1.2. Differentiation
The process of a derivative computation is known as differentiation. From a
geometrical point of view a derivative is defined as the slope of a tangent line on
a given curve. Physically it is defined as a rate of change of a dependent variable
w.r.t. an independent variable. In this section, we will discuss mainly three
different types of derivatives, which is widely used in this document, namely: (i)
partial derivatives, (ii) total derivatives, and (iii) material derivatives.
1.2.1. The partial derivatives
Let 𝑓(𝑥1, 𝑥2, · · · , 𝑥n) be a multivariable real valued function in R 𝑛. Then, the
process of differentiating the function 𝑓 w.r.t. a single variable while holding all
other independent variables constant, is known as partial derivatives:
𝜕𝑓
𝜕𝑥 𝑘
= lim
ℎ→0
𝑓(𝑥1, · · · , 𝑥 𝑘 + ℎ, · · · , 𝑥 𝑛) − 𝑓(𝑥1, · · · , 𝑥 𝑘, · · · , 𝑥 𝑛)
ℎ
. (1.15)
If the limit in Eq. (1.15) does not exist, then we say that the partial derivative is
not defined.
1.2.2. The total derivatives
Total derivative of a function, 𝑓(𝑥1, 𝑥2, · · · , 𝑥n), express a change in a function
due to a change in each dependent variable. It is expressed over all sum of the
change in the dependent variable time the rate of change of the function, 𝑓, w.r.t.
that dependent variable. Mathematically it is written as:
d𝑓 =
𝜕𝑓
𝜕𝑥1
d𝑥1 +
𝜕𝑓
𝜕𝑥2
d𝑥2 + · · · +
𝜕𝑓
𝜕𝑥 𝑛
d𝑥 𝑛. (1.16)
If the variables (𝑥1, 𝑥2, · · · , 𝑥n) are the function of scalar variable 𝑡, then the
total derivative is given by:
d𝑓
d𝑡
=
d𝑓
d𝑥1
d𝑥1
d𝑡
+
d𝑓
d𝑥2
d𝑥2
d𝑡
+ · · · +
d𝑓
d𝑥 𝑛
d𝑥 𝑛
d𝑡
. (1.17)
1.2.3. The material derivative
The time derivative of a physical or kinematic property of a material particle is
known as the material time derivative. In other words, we can say that a time
derivative of a given quantity holding reference configuration fixed, is known as
material derivative. There are two different ways to express the material time
Chapter 1. Mathematical preliminaries – Differentiation

derivative, namely: Lagrangean description and Eulerian description. From
a mathematical point of view both are equivalent.
The Lagrangean description
Let, 𝜑 (𝑋, 𝑡) be a scalar or a vector field, which is a differential w.r.t. time. Then,
the material derivative or the Lagrangean derivative is defined as:
˙𝜑 =
(︂
d𝜑
d𝑡
)︂
𝑋=constant
:=
𝜕𝜑 (𝑋, 𝑡)
𝜕𝑡
. (1.18)
The Eulerian description
Let, 𝜑 (𝑥, 𝑡) be a scalar or a vector field. In order to find the material derivative
of the given function , 𝜑, we write the position vector, 𝑥, as:
𝜑 = 𝜑 (𝑥, 𝑡) = 𝜑 (𝑥(𝑋,𝑡),𝑡) . (1.19)
Then, the material derivative, according to Eq. (1.18), is given by:
d𝜑
d𝑡
=
(︂
𝜑(𝑥(𝑋, 𝑡), 𝑡)
𝜕𝑡
)︂
𝑋−fixed
. (1.20)
If the vector 𝑥 in Cartesian coordinate system have three components, (𝑥1, 𝑥2, 𝑥3),
then the Eq. (1.20) can be written as:
(︂
𝜕𝜑(𝑥(𝑋, 𝑡), 𝑡)
𝜕𝑡
)︂
𝑋−fixed
=
𝜕𝜑(𝑥(𝑋, 𝑡), 𝑡)
𝜕𝑥1
𝜕𝑥1(𝑋,𝑡)
𝜕𝑡
+
𝜕𝜑(𝑥(𝑋, 𝑡), 𝑡)
𝜕𝑥2
𝜕𝑥2(𝑋,𝑡)
𝜕𝑡
+
𝜕𝜑(𝑥(𝑋, 𝑡), 𝑡)
𝜕𝑥3
𝜕𝑥3(𝑋,𝑡)
𝜕𝑡
+
𝜕𝜑(𝑥(𝑋, 𝑡), 𝑡)
𝜕𝑡
=
𝜕𝜑(𝑥(𝑋, 𝑡), 𝑡)
𝜕𝑥𝑖
𝜕𝑥𝑖(𝑋,𝑡)
𝜕𝑡
+
𝜕𝜑(𝑥(𝑋, 𝑡), 𝑡)
𝜕𝑡
. (1.21)
Substituting 𝑋 = 𝑋(𝑥, 𝑡) in the above equation, leads us to the result:
(︂
d𝜑(𝑥(𝑋, 𝑡), 𝑡)
d𝑡
)︂
𝑋=𝑋(𝑥,𝑡)
=
𝜕𝜑(𝑥,𝑡)
𝜕𝑥𝑖
𝑣𝑖(𝑥,𝑡) +
𝜕𝜑(𝑥,𝑡)
𝜕𝑡
= 𝑣 · (∇𝜑) +
𝜕𝜑(𝑥,𝑡)
𝜕𝑡
. (1.22)
Here, 𝑣 is the Eulerian velocity. The term 𝜕𝜑(𝑥,𝑡)
𝜕𝑡 , is known as Eulerian rate
of change and the term, 𝑣 · (∇𝜑), is called convective rate of change. ∇ is a del
or nabla operator, discussed in the next section.
Chapter 1. Mathematical preliminaries – Differentiation

1.3. The differential operators
Five different differential operators, del, gradient, divergence, curl and the
Laplace operator, in the Cartesian coordinate system, will be discussed in this
section. All quantities used for these operators are assumed to be continuous
and differentiable.
Del operator
A del operator also known as nabla operator is used as a shorthand notation to
make many mathematical expression simple, and is defined as:
∇ = 𝑒1
𝜕
𝜕𝑥1
+ 𝑒2
𝜕
𝜕𝑥2
+ · · · + 𝑒 𝑛
𝜕
𝜕𝑥 𝑛
= 𝑒𝑖
𝜕
𝜕𝑥𝑖
. (1.23)
A nabla operator is not commutative in nature with the quantity on which it is
applied:
∇𝑓 ̸= 𝑓∇. (1.24)
Here, 𝑓 is a scalar function. With the help of the del operator we can construct
all main operators of vector calculus, namely: gradient, divergence, and curl.
Gradient of a tensor field
When the nabla symbol is used on a scalar quantity, then we get a new term
known as the gradient. If 𝑓(𝑥1, 𝑥2, · · · , 𝑥n) is a scalar function, then the gradient
of 𝑓 represents its directional derivative and is a vector quantity. In the Cartesian
coordinate system, the gradient of a function 𝑓(𝑥1, 𝑥2, · · · , 𝑥n) is defined as:
grad 𝑓 =
𝜕𝑓
𝜕𝑥1
𝑒1 + · · · +
𝜕𝑓
𝜕𝑥 𝑛
𝑒 𝑛 =
𝜕𝑓
𝜕𝑥𝑖
𝑒𝑖 = ∇𝑓. (1.25)
Geometrically, we can say that the gradient of a function is always orthogonal
to the level curves (in two dimension) or normal to the level surfaces (in three
dimension). The magnitude of ∇𝑓 represent the rate of change of the function 𝑓
in the direction of grad 𝑓, see Fig. 1.1. The gradient of a vector field 𝑎 is defined
as:
grad 𝑎 =
𝜕(𝑎𝑖 𝑒𝑖)
𝜕𝑥 𝑗
=
𝜕𝑎𝑖
𝜕𝑥 𝑗
𝑒 𝑗 ⊗ 𝑒𝑖. (1.26)
The gradient of 𝑎 is the second order tensor. Likewise we can define the gradient
of a second order tensor 𝑇 :
∇𝑇 =
𝜕(𝑇𝑖𝑗 𝑒𝑖 ⊗ 𝑒 𝑗)
𝜕𝑥 𝑘
⊗ 𝑒 𝑘 =
𝜕𝑇𝑖𝑗
𝜕𝑥 𝑘
𝑒 𝑘 ⊗ 𝑒𝑖 ⊗ 𝑒 𝑗. (1.27)
Chapter 1. Mathematical preliminaries – The differential operators

Figure 1.1.: Geometrical representation of the gradient of a function, 𝑓(𝑥), to the level
surface 𝑓(𝑥) = constant (Lubarda, 2001).
The gradient of a second order tensor is a tensor of third order. In general we
can say that the gradient of the 𝑛th order tensor is a tensor of order 𝑛 + 1.
Divergence of a tensor field
The divergence of a tensor field tells us that how much a tensor field diverges
from a given point. The divergence is positive if the field is spreading out and is
negative if the tensors or vectors are coming closer together. Mathematically it
is defined as the dot product between the nabla operator and vectors or tensors:
div = ∇ · (∙) =
𝜕
𝜕𝑥i
𝑒i · (∙). (1.28)
Applying the above definition of the divergence on a vector 𝑣 gives:
div 𝑣 = ∇ · 𝑣 =
𝜕𝑣i
𝜕𝑥j
𝑒i · 𝑒j =
𝜕𝑣i
𝜕𝑥j
𝛿ij =
𝜕𝑣i
𝜕𝑥i
= 𝑣i,i. (1.29)
The divergence of a vector field gives the scalar. Likewise, the divergence of a
second-order tensor is a vector and is defined as:
div 𝑇 = ∇ · 𝑇 =
𝜕𝑇ij
𝜕𝑥𝑖
𝑒 𝑗. (1.30)

We can not define the divergence of a scalar because tensors smaller than 0th-order
do not exists.
Curl of a tensor field
The result of the cross product between nabla symbol and vector or tensor is
known as curl. For a given vector 𝑎 in a three dimensional Cartesian coordinate
system, the curl is defined as:
rot 𝑎 = det
⃒
⃒
⃒
⃒
⃒
⃒
⃒
𝑒1 𝑒2 𝑒3
𝜕
𝜕𝑥1
𝜕
𝜕𝑥2
𝜕
𝜕𝑥3
𝑎1 𝑎2 𝑎3
⃒
⃒
⃒
⃒
⃒
⃒
⃒
. (1.31)
The tensor notation of the curl for the 𝑖th component is more compact and is
given as:
∇ × 𝑎 =
(︂
𝑒i
𝜕
𝜕𝑥i
)︂
× (𝑎j 𝑒j) =
𝜕𝑎j
𝜕𝑥i
𝑒i × 𝑒j = 𝜀kij
𝜕𝑎j
𝜕𝑥i
𝑒k. (1.32)
The curl of a vector field results in a vector. Similarly we can define the curl of
a second-order tensor:
∇ × 𝑇 = 𝜀ijk
𝜕𝑇𝑖𝑗
𝜕𝑥𝑖
𝑒k ⊗ 𝑒l. (1.33)
The Laplace operator of a tensor field
The Laplace operator is named after the French mathematician and astronomer
Pierre Simon marquis de Laplace (1749-1827) (Selvadurai, 2000). The Laplace
operator is given as:
∇ · ∇(∙) :=
𝜕2(∙)
𝜕𝑥𝑖 𝜕𝑥𝑖
= ∇2
(∙). (1.34)
The Laplacian of a scalar 𝑓 is given by:
div grad 𝑓 = ∇2
𝑓 =
𝜕2 𝑓
. (1.35)
The Laplacian of a vector 𝑎 is given by:
∇2
𝑎 =
𝜕2 𝑎k
𝑒k. (1.36)
An alternative way to write the Laplacian of a vector 𝑎 is:
∇2
𝑎 = ∇ (∇ · 𝑎) − ∇ × (∇ × 𝑎) . (1.37)

In order to proof the expression in Eq. (1.37), we need to compute the term
∇ (∇ · 𝑎) − ∇ × (∇ × 𝑎) and show that it is equal to vector Laplacian:
∇ (∇ · 𝑎) − ∇ × (∇ × 𝑎) =
𝜕
𝜕𝑥𝑖
𝜕𝑎 𝑗
𝜕𝑥 𝑗
− 𝜖𝑖𝑗𝑘
𝜕
𝜕𝑥 𝑗
(︂
𝜖 𝑘𝑙𝑚
𝜕𝑎 𝑚
𝜕𝑥𝑙
)︂
,
=
𝜕
𝜕𝑥𝑖
𝜕𝑎 𝑗
𝜕𝑥 𝑗
− 𝜖 𝑘𝑖𝑗 𝜖 𝑘𝑙𝑚
𝜕
𝜕𝑥 𝑗
(︂
𝜕𝑎 𝑚
𝜕𝑥𝑙
)︂
,
=
𝜕2 𝑎 𝑗
𝜕𝑥𝑖 𝜕𝑥 𝑗
− (𝛿𝑖𝑙 𝛿 𝑗𝑚 − 𝛿𝑖𝑚 𝛿 𝑗𝑙)
𝜕2 𝑎 𝑚
𝜕𝑥 𝑗 𝜕𝑥𝑙
,
=
𝜕2 𝑎 𝑗
𝜕𝑥𝑖 𝜕𝑥 𝑗
−
𝜕2 𝑎 𝑗
𝜕𝑥 𝑗 𝜕𝑥𝑖
−
𝜕2 𝑎𝑖
𝜕𝑥 𝑗 𝜕𝑥 𝑗
,
=
𝜕
𝜕𝑥 𝑗
(︃
𝜕𝑎𝑖
𝜕𝑥 𝑗
)︃
,
= ∇2
𝑎. (1.38)
1.4. Integral theorems
In this section we discuss two integral theorems namely, the theorem of Gauss
and Reynolds transport theorem. Application of these two theorems can be
widely seen in the construction of the local balance laws from the global balance
laws.
The Gauss theorem or divergence theorem
Three prominent names from the world of mathematics, George Green, Carl
F. Gauss and M.V. Ostrogradskii, have played an important role in the
development of Cartesian form of divergence theorem. Vector form of the
divergence theorem is widely used in engineering science and was developed by
Oliver Heaviside and Josiah W. Gibbs (Stolze, 1978). This theorem is used to
change the volume integral to area integral and vice versa. Let Ω be a domain
in the Euclidian space, and 𝜕Ω represent its surface. If we draw a outer surface
normal, 𝑛, on 𝜕Ω and define a vector field, 𝑓, such that it is continuous on the
given domain and continuously differentiable in the interior of the domain, Ω,
then the divergence theorem is given as:
ˆ
Ω
(∇ · 𝑓) d𝑉 =
ˆ
𝜕Ω
(𝑓 · 𝑛) d𝐴. (1.39)
Chapter 1. Mathematical preliminaries – Integral theorems

In index notation the above equation is written as:
ˆ
Ω
𝜕𝑓i
𝜕𝑥i
d𝑉 =
ˆ
𝜕Ω
𝑓i 𝑛i d𝐴. (1.40)
Analog we can establish the Gauss theorem for a 𝑛th- order tensor 𝐹ij...k:
ˆ
Ω
𝜕𝑇ij...k
𝜕𝑥k
d𝑉 =
ˆ
𝜕Ω
𝑇ij...k 𝑛k d𝐴. (1.41)
For the proof of Eq. (1.40), see Appendix. A.1. In Cartesian coordinate system
the field 𝑓 and 𝐹 can be a component of vector or tensor, but this condition is
not always necessary (Müller, 2014a).
Reynolds transport theorem
It is an abstract mathematical concept, that gives the time rate of change of
continuum fields. It is widely used during formulating the basic balance laws of
continuum mechanics. In order to understand this theorem, let us start with an
intensive property 𝜓(𝑥, 𝑡) in the current configuration3 Ωt. The corresponding
extensive property is obtained through integration of 𝜓(𝑥, 𝑡) over a material
volume:
𝐼(𝑡) =
ˆ
Ω 𝑡
𝜓(𝑥, 𝑡) d𝑉. (1.42)
The time varying property of the configuration, results to a time dependent
integration in Eq. (1.42). Now we are interested in knowing how Eq. (1.42)
changes w.r.t. time, i.e. the material derivative of this extensive quantity:
d𝐼
d𝑡
=
d
d𝑡
⎛
⎜
⎝
ˆ
Ω 𝑡
𝜓(𝑥, 𝑡) d𝑉
⎞
⎟
⎠ . (1.43)
The domain of integration in the above equation is time dependent. Therefore,
we can not take time differential inside the volume integral. In order to solve
this problem, we change the domain of integration, i.e. change the variable to
reference configuration and then apply Eq. (2.18):
d𝐼
d𝑡
=
d
d𝑡
⎛
⎜
⎝
ˆ
Ω0
𝜓(𝑥, 𝑡)𝐽 d𝑉0
⎞
⎟
⎠ . (1.44)
3
To understand the concept of configuration, please refer sec. 2.1.1

By applying chain rule we get the following result:
d𝐼
d𝑡
=
ˆ
Ω0
(︂
𝐽
d𝜓
d𝑡
+ 𝜓
d𝐽
d𝑡
)︂
. (1.45)
Here, 𝐽 is the Jacobian and its material derivative is given by:
d𝐽
d𝑡
= (∇ · 𝑣) 𝐽. (1.46)
For derivation of Eq. (1.46), see Appendix. A.2. Substituting the material deriva
tive of the Jacobian and applying Eq. (1.22) in Eq. (1.45) permit us to rearrange
the integrand as:
ˆ
Ω 𝑡
(︂
𝜕𝜓
𝜕𝑡
+ 𝑣 · ∇𝜓 + 𝜓(∇ · 𝑣)
)︂
d𝑉 =
ˆ
Ω 𝑡
(︂
𝜕𝜓
𝜕𝑡
+ ∇ · (𝜓𝑣)
)︂
d𝑉. (1.47)
Implementing the divergence theorem as in Eq. (1.39) in the left term of the
above equation, leads us to the Reynolds transport theorem:
d
d𝑡
⎛
⎜
⎝
ˆ
Ω 𝑡
𝜓(𝑥, 𝑡) d𝑉
⎞
⎟
⎠ =
ˆ
Ω 𝑡
𝜕𝜓
𝜕𝑡
d𝑉 +
ˆ
𝜕Ω 𝑡
𝜓 𝑣 · 𝑛 d𝐴. (1.48)

2. Fundamental of continuum
mechanics
The continuum is that which is divisible into indivisible that are infinitely
divisible Physics.
– Aristotle (384—322 BCE)
Continuum mechanics is a special branch of physics that is based on the hypothesis
of continuum. According to this hypothesis a material is assumed to be continuous
if its distribution fills the space completely which is occupied by it, without
leaving any void. In other words, we can say that the mathematical functions that
are applied in the theory of classical continuum mechanics should be continuous
and differentiable. The aim of this chapter is to develop geometrical concept of
motion and deformation for a continuum body, i.e. kinematics of the continuous
medium. In the second part of this chapter we derive the fundamental governing
laws that are valid for all continuous medium, i.e. conservation laws of classical
physics, namely: balance of mass, linear momentum, and energy. We will develop
global as well as local form of the balance laws. These balance laws will be given
in both Lagrangean- and Eulerian description.
2.1. Kinematics of the continuum bodies
Kinematics is the geometrical description of motion, where forces are not consid
ered to be the cause of motion. In continuum mechanics we use two different
ways to describe the motion, namely: Lagrangean- and Eulerian description,
respectively. Motion in both the descriptions remains same, it is just a matter
of defining the motion from a different point of view. The idea of the La
grangean description also known as the referential description, was developed
in the mid-eighteenth century by Euler. In this description we analyze motion
of a individual particle4. Therefore, the Lagrangean description of motion is
very well suited for the theory of classical elasticity and continuum mechanics.
Abaqus also uses the Lagrangean description for the mechanical modeling
capabilities (Simulia, 2015b).
4
With particle we mean an arbitrary material point of a body.
Chapter 2. Continuum mechanics

On the other hand the Eulerian description also known as the spatial description
was introduced by the French physicist and mathematician D’Alembert. In
this description of motion, we focus on the region of space that is currently
occupied by a body. In other words we can say that we focus on a fixed region
of space and analyze what is occurring in this region, with consideration of time.
This description is very well suited for fluid mechanics.
In this work our main focus will be on the engineering approach of continuum
mechanics (Müller, 2014a; 2014; Tadmor, Miller, and Elliott, 2012; Dreyer, 2014;
Dreyer, 2015; Mase and Mase, 1999; Lubarda, 2001; Holzapfel, 2000; Shabana,
2012). But, we also discuss mathematical approach wherever necessary for better
clarification (Bertram, 2012; Haupt, 2002; Wilmański, 1998; Martinec, 2000;
Truesdell, 2012; Wilmanski, 2008).
2.1.1. Configurations and the deformation mapping
Configuration is an abstract term in continuum mechanics that helps us to
understand the deformation gradient, strain and strain rates(Haupt, 2002).
The configuration of a material body, ℬ, is defined as the arbitrary regions,
Ω0, Ω1, · · · Ωt, occupied in a three dimensional Euclidian space. The relation
between the material body, ℬ, and configurations, Ω0, Ω1, · · · Ωt, are given through
mapping 𝜅0(ℬ), 𝜅1(ℬ), · · · 𝜅t(ℬ), respectively, see Fig. 2.1. Similarly every particle
𝒫i in the body ℬ can be mapped.
Figure 2.1.: Schematic representation of different configurations, Ω0, Ω1, and Ωt, of a
material body ℬ in the Euclidian space.
In order to study the deformation in a material body, selection of the reference-
and current (deformed) configurations become necessary. From a mathematical
Chapter 2. Continuum mechanics – Kinematics of the continuum bodies

point of view, choice of the configurations are arbitrary. But in continuum
mechanics we choose the stress free configuration as the reference configuration,
Ω0, and deformed configuration as current configuration, Ωt. The subscript t
represents the current time.
After developing the concept of configuration, we establish the relationship
between the position vectors of the reference configuration, 𝑋,(reference position
vector) and the current configuration, 𝑥, (current position vector), see Fig. 2.2.
Most of the time Abaqus also stores the reference- and current position vectors
in rectangular Cartesian components of 𝑋 and 𝑥.
Figure 2.2.: Position vectors of a material particle 𝑃 in the reference- and current
configuration (Holzapfel, 2000).
In order to establish the above mentioned relationship, we introduce the concept
of conservation law. It states that the number of particles in a material body
is conserved, which means that particles can not be created nor destroyed.
Mathematically this implies that, it is not possible to map a single particle to
two different positions and that two particles cannot be mapped to the same
position. Such type of mapping is known as one-to-one mapping. The fact that

𝜅0 is an one-to-one mapping, permits us to use the property of invertibility:
ℬ = 𝜅−1
0 (Ω0), 𝒫 = 𝜅−1
0 (𝑋). (2.1)
Similarly 𝜅t is also invertible:
Ωt = 𝜅t(ℬ) = 𝜅t(𝜅−1
0 (Ω0)). (2.2)
Most often, in solid mechanics, we are interested in comparing positions of a
material point in the current configuration and in the reference configuration.
Therefore, we construct the direct mapping between the reference - and the
current configurations, which is written as 𝜅t ∘ 𝜅−1
0 (Ω0)5. Then, the relationship
between the reference position vector and the current position vector can be
given as the following, see Fig. 2.2:
𝑥 = 𝜅t ∘ 𝜅−1
0
⏟ ⏞
𝜒t
(𝑋) = 𝜒t(𝑋). (2.3)
Here, 𝜒t is a point-to-point map at a given time t and it is assumed to differen
tiable w.r.t. 𝑋 and 𝑡. For any given time t, Eq. (2.3) can be written as:
𝑥 = 𝜒(𝑋,𝑡). (2.4)
In Eq. (2.4), the position vector of a material point 𝑋 and time 𝑡 are considered
as the independent variables. In the Eulerian description above equation is
written as the following:
𝑋 = 𝜒−1
(𝑥,𝑡). (2.5)
The mapping in Eq. (2.4) and Eq. (2.5) are also known as the motion of a body.
To avoid the confusion between the quantities of a reference - and a current
configuration, we use uppercase indices for tensors in the reference configuration
and lower cases indices for the current configuration. For example the reference
position vector and the current position vector can be expressed in terms of their
components as following:
𝑋 = 𝑋I 𝑒I, 𝑥 = 𝑥i 𝑒i. (2.6)
Here, 𝑋I, and 𝑥i are the components of 𝑋 and 𝑥 associated with 𝑒I and 𝑒i,
respectively.
2.1.2. Deformation gradient
The deformation gradient is a dimensionless quantity, which establishes a relation
ship between a line element vector in the reference configuration, d𝑋, (material
5
𝑓(𝑔(𝑥)) = 𝑓 ∘ 𝑔(𝑥)

line element) and a line element vector in the current configuration, d𝑥, (spatial
line element), see Fig. 2.3. In other words we can say that, the deformation
gradient build a relation between the distances in the reference- and current
configurations. Mathematically, it is understood as the linear approximation to
the mapping 𝜒. Applying Taylor series for a first order approximation for the
material- and spatial line element leads us to the following relation:
𝑥𝑖 + d𝑥𝑖 = 𝜒i (𝑋 𝐽 + d𝑋 𝐽 ) = 𝜒𝑖 (𝑋 𝐽 ) +
(︂
𝜕𝜒i
𝜕𝑋J
)︂
𝑋J
d𝑋J = 𝑥i + 𝐹iJ d𝑋J. (2.7)
Simplifying the Eq. (2.7), gives the following:
Figure 2.3.: Mapping of a line element vector d𝑋J in the reference configuration to a
line element vector d𝑥i in the current configuration (Mase and Mase, 1999).
d𝑥𝑖 = 𝐹𝑖𝐽 d𝑋 𝐽 . (2.8)
Here, 𝐹iJ is known as the deformation gradient. Substituting Eq. (2.3) in Eq. (2.8),
leads us to a another definition of the deformation gradient:
𝐹iJ =
𝜕𝜒i
𝜕𝑋J
=
𝜕𝑥i
𝜕𝑋J
= 𝑥i,J =
⎛
⎜
⎜
⎝
𝜕𝑥1
𝜕𝑋1
𝜕𝑥1
𝜕𝑋2
𝜕𝑥1
𝜕𝑋3
𝜕𝑥2
𝜕𝑋1
𝜕𝑥2
𝜕𝑋2
𝜕𝑥2
𝜕𝑋3
𝜕𝑥3
𝜕𝑋1
𝜕𝑥3
𝜕𝑋2
𝜕𝑥3
𝜕𝑋3
.
⎞
⎟
⎟
⎠ . (2.9)
From Eq. (2.9), we can say that the deformation gradient is the derivative of the
function, 𝑥 = 𝜒(𝑋,𝑡), w.r.t. 𝑋. The inverse of the deformation gradient is given
as:
𝐹−1
𝐽𝑖 =
𝜕𝜒−1
𝐽
𝜕𝑥𝑖
=
𝜕𝑋 𝐽
𝜕𝑥𝑖
. (2.10)

If no motion is taking place, i.e. 𝑥i = 𝑋J, then the deformation gradient is equal
to unit tensor:
𝐹𝑖𝐽 = 𝛿𝑖𝐽 . (2.11)
2.1.3. Transformation of volume elements
In this section, we develop the relationship between the infinitesimal volume
elements (Parallelepiped)6 in the reference- and current configurations. The
edges of the parallelepiped in the reference configuration is given by the vectors
d𝑋1, d𝑋2 and d𝑋3, respectively, see Fig. 2.4. The components of these vectors
w.r.t. the orthonormal basis can be written as:
d𝑋1 = d𝑋1 𝐸1, d𝑋2 = d𝑋2 𝐸2, d𝑋3 = d𝑋3 𝐸3. (2.12)
Here, 𝐸1, 𝐸2 and 𝐸3 are the Cartesian basis vectors for the undeformed con
figuration. In order to compute volume of the parallelepiped in the reference
Figure 2.4.: Schematic representation of a volume element in the reference configuration
and the deformed configuration (Martinec, 2000).
6
Parallelepiped is used as the elementary geometry to calculate any arbitrary volume in R3
(Dreyer, 2014)

configuration (material volume element), we use triple product7:
d𝑉0 = d𝑋1 · (d𝑋2 × d𝑋3) ,
= d𝑋1 d𝑋2 d𝑋3 𝐸1 · (𝐸2 × 𝐸3) ,
= 𝐸1 · (𝐸2 × 𝐸3)
⏟ ⏞
=1
d𝑋1 d𝑋2 d𝑋3,
= d𝑋1 d𝑋2 d𝑋3. (2.13)
Likewise, we compute volume of the parallelepiped in the current configuration
(spatial volume element):
d𝑉 = d𝑥1 · (d𝑥2 × d𝑥3) . (2.14)
Here, d𝑥1, d𝑥2 and d𝑥3 are edges of the spatial volume element, and is computed
by using Eq. (2.8):
d𝑥1 = 𝐹 d𝑋1 =
𝜕𝑥
𝜕𝑋
d𝑋1 = d𝑋1
𝜕𝑥
𝜕𝑋
𝐸1, (2.15a)
d𝑥2 = 𝐹 d𝑋2 =
𝜕𝑥
𝜕𝑋
d𝑋2 = d𝑋2
𝜕𝑥
𝜕𝑋
𝐸2, (2.15b)
d𝑥3 = 𝐹 d𝑋3 =
𝜕𝑥
𝜕𝑋
d𝑋3 = d𝑋3
𝜕𝑥
𝜕𝑋
𝐸3. (2.15c)
Substituting Eq. (2.15) in Eq. (2.14), gives us volume of the deformed volume
element:
d𝑉 = 𝐸1 · (𝐸2 × 𝐸3)
⏟ ⏞
=1
d𝑋1 d𝑋2 d𝑋3
𝜕𝑥
𝜕𝑋1
·
(︂
𝜕𝑥
𝜕𝑋2
×
𝜕𝑥
𝜕𝑋3
)︂
,
= d𝑋1 d𝑋2 d𝑋3
𝜕𝑥
𝜕𝑋1
·
(︂
𝜕𝑥
𝜕𝑋2
×
𝜕𝑥
𝜕𝑋3
)︂
. (2.16)
The term,
𝜕𝑥
𝜕𝑋1
·
(︂
𝜕𝑥
𝜕𝑋2
×
𝜕𝑥
𝜕𝑋3
)︂
, is known as the determinant of the deformation
gradient 𝐹 , which is nothing but Jacobian:
𝐽 = det (𝐹 ) = det
(︂
𝜕𝑥
𝜕𝑋
)︂
=
𝜕𝑥
𝜕𝑋1
·
(︂
𝜕𝑥
𝜕𝑋2
×
𝜕𝑥
𝜕𝑋3
)︂
. (2.17)
Hence, the change in the volume of parallelepiped from the reference configuration
to the current configuration is given by the following relationship:
d𝑉 = 𝐽 d𝑉0. (2.18)
7
It is defined as the dot product of one of the vectors with the cross product of other two.

The Jacobian is the measure of the volume change. The volume of an element
can not be negative. It means, the physical interpretation of the volume elements
are only possible, if its volumes are positive. Mathematically, it means that J
must be always positive definite. If there is no motion, then the determinant of
the deformation gradient is equal to the determinant of unit tensor:
𝐽 = det 𝐹 = det 𝐼 = 1, (2.19)
2.1.4. Transformation of surface area elements
In this section, we develop a relationship between the infinitesimal small surface
elements in the reference-, d𝐴0, and deformed, d𝐴, configurations. The surface
element d𝐴0 (material surface element) is constructed by using two infinitesimal
material line elements, d𝑋1 and d𝑋2, respectively, see Fig. 2.5. Then, the area
Figure 2.5.: Schematic representation of a area element in the reference configuration
and the deformed configuration (Abeyaratne, 2012).
can be computed by using cross product:
d𝐴0 = d𝑋1 × d𝑋2 = 𝑁0 d𝐴0. (2.20)
Here, 𝑁0 is the unit normal vector to the material surface element. The area
of the surface element d𝐴 (spatial surface element), is defined in the deformed
configuration by the following relation:
d𝐴 = d𝑥1 × d𝑥2 = 𝑁 d𝐴. (2.21)
Here, 𝑁 is the unit normal to the spatial surface element. d𝑥1, d𝑥2 are the
two material line elements in the deformed configuration, see Fig. 2.5. Using
Eq. (2.18), we get:
d𝑥3 · (d𝑥1 × d𝑥2) = 𝐽 d𝑋3 · (d𝑋1 × d𝑋2) . (2.22)

Substituting Eq. (2.20) and Eq. (2.21) in Eq. (2.22) leads us to the following
relation:
d𝑥3 · 𝑁 d𝐴 = d𝑋3 · 𝑁0 d𝐴0. (2.23)
Application of Eq. (2.8) in Eq. (2.23) gives us the relationship between the
material- and spatial volume element:
(𝐹 d𝑋3) · 𝑁 d𝐴 = 𝐽 d𝑋3 · 𝑁0 d𝐴0. (2.24)
An alternative form of the above equation is:
𝑁 · d𝐴 = 𝐽𝐹 −𝑇
· 𝑁0 d𝐴0. (2.25)
The method which is used in this section is valid for any surface element of
a continuum body, but the direction of the unit normal vector may differ.
Eq. (2.25) is widely known as Nanson’s relation, named after Edward J. Nanson
(1850–1936) (Bertram, 2012).
2.1.5. Measures of deformation
In order to do the physical and geometrical interpretation of deformations, we
need to understand the concept of different strain tensors. There exist various
definitions and names of strain tensors, but in this work we limit ourself to
the most common strain tensors, which are widely used in classical continuum
mechanics. The strain tensors, which are discussed in this section, are either
related to the reference configuration or the current configuration.
Let us begin our discussion with a material line element d𝐿. The square of this
material line element gives:
d𝐿2
= d𝑋 𝐼 d𝑋 𝐼. (2.26)
Similarly, the square of the spatial line element can be given as:
d𝑙2
= d𝑥𝑖 d𝑥𝑖. (2.27)
Application of Eq. (2.8) in Eq. (2.27) gives us the following relation:
d𝑙2
= (𝐹𝑖𝐼 d𝑋 𝐼) (𝐹𝑖𝐽 d𝑋 𝐽 ) = (𝐹𝑖𝐼 𝐹𝑖𝐽 ) d𝑋 𝐼 d𝑋 𝐽 = 𝐶 𝐼𝐽 d𝑋 𝐼 d𝑋 𝐽 . (2.28)
Here, 𝐶IJ is known as right Cauchy-Green deformation tensor, which is defined
as:
𝐶 𝐼𝐽 = 𝐹𝑖𝐼 𝐹𝑖𝐽 =
𝜕𝑥𝑖
𝜕𝑋 𝐼
𝜕𝑥𝑖
𝜕𝑋 𝐽
. (2.29)
Right Cauchy-Green deformation tensor is named after the French mathe
matician Augustin-Louis Cauchy (1789 - 1857) and the British mathematical

physicist George Green (1793 - 1841). It is defined only in the reference con
figuration and is symmetric and positive definite tensor. From Eq. (2.29) we
can conclude that if all the nine components of deformation gradient, 𝐹iI, are
known then we can compute the six components of the right Cauchy-Green
deformation tensor. But, the reverse is not possible, i.e. if we know 𝐶IJ, then it
is not possible to compute 𝐹iI (Holzapfel, 2000).
The second most commonly used strain tensor in continuum mechanics is the
left Cauchy-Green deformation tensor. It is defined as:
𝐵𝑖𝑗 = 𝐹𝑖𝐾 𝐹 𝑗𝐾 =
𝜕𝑥𝑖
𝜕𝑋 𝐾
𝜕𝑥 𝑗
𝜕𝑋 𝐾
. (2.30)
The tensor 𝐵ij is only defined in the current configuration. It is also known as
Finger deformation tensor, named after the Austrian physicist and mathemati
cian Josef Finger (1841 - 1925). Like the right Cauchy-Green strain tensor,
𝐵ij is also a symmetric and positive definite second-order tensor.
Another most commonly used strain tensor is the Lagrangean strain tensor.
In order to define this tensor, let us subtract Eq. (2.28) from Eq. (2.26):
d𝑙2
− d𝐿2
= (𝐶IJ − 𝛿IJ) d𝑋 𝐼 d𝑋 𝐽 = 2 𝐸 𝐼𝐽 d𝑋 𝐼 d𝑋 𝐽 . (2.31)
Here, 𝐸IJ is known as Lagrangean strain tensor or Green– St-Venant strain
tensor, which is associated with the reference configuration, and is defined as
follows (Bertram, 2012):
𝐸IJ =
1
2
(𝐶IJ − 𝛿 𝐼𝐽 ) =
1
2
(𝐹𝑖𝐼 𝐹iJ − 𝛿 𝐼𝐽 ). (2.32)
Since the right Cauchy-Green strain tensor, 𝐶IJ, and the Kronecker delta,
𝛿IJ, are symmetric in nature, then from Eq. (2.32) we can conclude that 𝐸IJ is
also symmetric in nature.
The last most commonly used strain tensor is associated with the current
configuration and is known as the Eulerian - Almansi strain tensor. Like other
strain tensors, this is also symmetric in nature and is defined as:
𝑒𝑖𝑗 =
1
2
(︁
𝛿𝑖𝑗 − 𝐹−1
𝐾𝑖 𝐹−1
𝐾𝑗
)︁
. (2.33)
In general, 𝐶IJ and 𝐸IJ are also called material strain tensors because they are
associated with the reference configuration. Tensors, which are associated with
the current configuration, like 𝐵ij and 𝑒ij are also known as spatial strain tensors.
The Lagrangean- and Eulerian - Almansi strain tensors can also be expressed
in term of displacement vector gradient. In order to express 𝐸IJ and 𝑒ij by using
displacement vector gradient, let us start by computing the displacement field of
a typical particle. The displacement field is a vector field, which is defined as the

difference between the position vector of the current configuration, 𝑥i = 𝜒i(𝑡, 𝑋),
and the position vector of the reference configuration, 𝑋I, see Fig. 2.6:
𝑈i = 𝜒i(𝑡, 𝑋) − 𝑋I. (2.34)
Eq. (2.34) represents the Lagrangean form of the displacement field because
the displacement field 𝑈 is a function of 𝑋 and 𝑡.
Differentiating Eq. (2.34) w.r.t. the position vector 𝑋J and applying Eq. (2.9)
and Eq. (1.11), give the following relation:
𝜕𝑈i
𝜕𝑋J
=
𝜕𝜒i(𝑡, 𝑋)
𝜕𝑋J
−
𝜕𝑋I
𝜕𝑋J
= 𝐹iJ − 𝛿IJ. (2.35)
The Eq. (2.35) permits us to write the deformation gradient, 𝐹iJ, as a summation
of the displacement gradient, 𝜕𝑈i
𝜕𝑋J
, and the Kronecker delta, 𝛿IJ,:
𝐹iJ =
𝜕𝑈i
𝜕𝑋J
+ 𝛿IJ. (2.36)
Application of Eq. (2.36) in Eq. (2.32) gives us an alternative way to express the
Figure 2.6.: Schematic representation of the displacement vector 𝑈 during the motion
of a continuum body from its reference configuration Ω0 to the current configuration Ωt.
(Holzapfel, 2000).

Lagrangean strain tensor:
𝐸 𝐼𝐽 =
1
2
(︂
𝛿 𝑗𝑀
(︂
𝜕𝑈 𝑀
𝜕𝑋 𝐼
+ 𝛿 𝑀 𝐼
)︂
𝛿 𝑗𝑁
(︂
𝜕𝑈 𝑁
𝜕𝑋 𝐽
+ 𝛿 𝑁 𝐿
)︂
− 𝛿 𝐼𝐽
)︂
,
=
1
2
(︂
𝛿 𝑀 𝑁
(︂
𝜕𝑈 𝑀
𝜕𝑋 𝐼
+ 𝛿 𝑀 𝐼
)︂ (︂
𝜕𝑈 𝑁
𝜕𝑋 𝐽
+ 𝛿 𝑁 𝐽
)︂
− 𝛿 𝐼𝐽
)︂
,
=
1
2
(︂(︂
𝜕𝑈 𝑀
𝜕𝑋 𝐼
+ 𝛿 𝑀 𝐼
)︂ (︂
𝜕𝑈 𝑀
𝜕𝑋 𝐽
+ 𝛿 𝑀 𝐽
)︂
− 𝛿 𝐼𝐽
)︂
,
=
1
2
(︂
𝜕𝑈 𝐼
𝜕𝑋 𝐽
+
𝜕𝑈 𝐽
𝜕𝑋 𝐼
+
𝜕𝑈 𝑀
𝜕𝑋 𝐼
𝜕𝑈 𝑀
𝜕𝑋 𝐽
)︂
. (2.37)
Similarly, we can express the Eulerian - Almansi strain tensor in the following
form:
𝑒𝑖𝑗 =
1
2
(︃
𝜕𝑢𝑖
𝜕𝑥 𝑗
+
𝜕𝑢 𝑗
𝜕𝑥𝑖
−
𝜕𝑢 𝑘
𝜕𝑥𝑖
𝜕𝑢 𝑘
𝜕𝑥 𝑗
)︃
. (2.38)
Here, 𝑢i is a function of the current position vector 𝑥 and time 𝑡 and is known
as displacement field in the Eulerian form (spatial description):
𝑢𝑖 (𝑥, 𝑡) = 𝑥𝑖 − 𝑋 𝐼. (2.39)
Displacement fields in the Lagrangean and the Eulerian form have the same
values, which are connected by the following relationship:
𝜕𝑢𝑖
𝜕𝑥 𝑗
≈
𝜕𝑢𝑖
𝜕𝑥𝑖
(︂
𝛿 𝐾𝐽 +
𝜕𝑈 𝐾
𝜕𝑋 𝐽
)︂
=
𝜕𝑢𝑖
𝜕𝑥 𝑘
𝜕𝜒 𝑘
𝜕𝑋 𝐽
=
𝜕𝑈 𝐼
𝜕𝑋 𝐽
. (2.40)
Strain tensors defined in Eq. (2.37) and Eq. (2.38) are valid for any arbitrary
deformation. In this work we restrict ourself only to small deformations. There
fore, nonlinear terms in the Lagrangean- and Eulerian - Almansi strain
tensor will be neglected. In the case of small deformation, we do not differentiate
between the reference- and current configurations, so that we get the simple
form of linear strain tensor 𝜀ij:
𝐸IJ ≈ 𝑒ij ≈ 𝜀ij =
1
2
(︃
𝜕𝑢i
𝜕𝑥j
+
𝜕𝑢j
𝜕𝑥i
)︃
. (2.41)
2.2. Field equations of continuum mechanics
Balance laws of mass, momentum, and energy are the heart of the continuum
physics theories. These balance laws are used to interpret the effect of the
surroundings on the motion of a material body.
In this work, we concentrate only on the balance laws of those fields, which are
additive in nature and also continuous and differentiable. The system in which
Chapter 2. Continuum mechanics – Field equations of continuum mechanics

we develop our balance laws are assumed to be a closed system. This means,
that no mass can leave or enter in the system, but energy in the form of heat or
work can cross the boundary of the system.
2.2.1. Balance of mass
Mass is a fundamental physical property. In non-relativistic physics, mass is a
conserved quantity, which means mass can not be created nor destroyed, but can
be deformed under the application of loads. In other words, we can say that if a
particle of a body has a mass 𝑚0 in the reference configuration and 𝑚 in the
current configuration, then the following relation holds:
𝑚0(Ω0) = 𝑚(Ω𝑡) > 0. (2.42)
Generally, in continuum mechanics, mass is defined by mass density 𝜌, which
is a measure of the distribution of mass in space. The mass density function
of the reference configuration is independent of time and depends only on
the position 𝑋, where 𝑋 ∈ Ω0. The mass density function in the reference
configuration is represented by 𝜌0(𝑋) > 0 (reference mass density). If the
gradient of reference mass density is equal to zero, i.e. it does not depend on the
𝑋, then the configuration is said to be homogeneous. Mass density in the current
configuration (spatial mass density) is a function of time and the position 𝑥,
where 𝑥 ∈ Ω𝑡. In this work we represent the spatial mass density by 𝜌 (𝑥, 𝑡) > 0.
The reference mass density is defined as:
𝜌0(𝑋) =
d𝑚0
d𝑉0
=
d𝑚
d𝑉0
. (2.43)
Similarly, we can define spatial mass density:
𝜌 (𝑥,𝑡) =
d𝑚
d𝑉
. (2.44)
Here, d𝑉0 and d𝑉 represents the infinitesimal small volume elements of the ref
erence- and current configurations. The total mass of the reference configuration
is obtained through integration of the mass density, 𝜌0(𝑋) over the region Ω0:
𝑀 =
ˆ
Ω0
𝜌0(𝑋) d𝑉0. (2.45)
Likewise, we may define the total mass in the current configuration:
𝑀 =
ˆ
Ω 𝑡
𝜌 (𝑥, 𝑡) d𝑉. (2.46)

The mass does not change with time. Then, Eq. (2.46) can be written as:
d𝑀
d𝑡
=
d
d𝑡
ˆ
Ω 𝑡
𝜌 (𝑥, 𝑡) d𝑉 = 0. (2.47)
Eq. (2.47) is known as the balance of mass in global form. In order to construct
the local form, first we need to apply Reynolds transport theorem in Eq. (2.47):
d
d𝑡
ˆ
Ω 𝑡
𝜌 d𝑉 =
ˆ
Ω 𝑡
𝜕𝜌
𝜕𝑡
d𝑉 +
ˆ
𝜕Ω 𝑡
(𝜌 𝑣𝑖) · 𝑛𝑖 d𝐴. (2.48)
The surface integral in the above equation can be changed into the volume
integral after applying the Gauss theorem:
ˆ
Ω 𝑡
(︂
𝜕𝜌
𝜕𝑡
+
𝜕
𝜕𝑥𝑖
(𝜌 𝑣𝑖)
)︂
d𝑉 = 0. (2.49)
Above equation must be true for any sub body. Therefore integrand in Eq. (2.49)
should vanish and we get the local balance of mass in the Eulerian frame:
𝜕𝜌
𝜕𝑡
+
𝜕𝜌 𝑣𝑖
𝜕𝑥𝑖
= 0 ⇐⇒
𝜕𝜌
𝜕𝑡
+ ∇ · (𝜌 𝑣) . (2.50)
In order to transform the mass balance law from the Eulerian- to the La
grangean frame, we need to apply chain rule in Eq. (2.50) (Dreyer, 2014):
𝜕𝜌(𝑥, 𝑡)
𝜕𝑡
+ 𝑣 𝐾
𝜕𝜌(𝑥, 𝑡)
𝜕𝑥 𝑘
+ 𝜌(𝑋, 𝑡)
𝜕𝑣 𝑘
𝜕𝑥 𝑘
= 0. (2.51)
Here, 𝜌(𝑋, 𝑡) represents the mass density in the Lagrangean frame. Before
proceeding further with Eq. (2.51), we need to know two important results.
The first important result is differentiation of the mass density w.r.t. time (Dreyer,
2014):
𝜕𝜌(𝑋, 𝑡)
𝜕𝑡
=
𝜕𝜌(𝑥, 𝑡)
𝜕𝑡
+
𝜕𝜌(𝑥, 𝑡)
𝜕𝑥 𝑘
𝜕𝜒 𝐾
𝜕𝑡
=
𝜕𝜌(𝑥, 𝑡)
𝜕𝑡
+
𝜕𝜌(𝑥, 𝑡)
𝜕𝑥 𝑘
𝑣 𝐾, (2.52)
⇐⇒
𝜕𝜌(𝑥, 𝑡)
𝜕𝑡
=
𝜕𝜌(𝑋, 𝑡)
𝜕𝑡
−
𝜕𝜌(𝑥, 𝑡)
𝜕𝑥 𝑘
𝑣 𝐾, (2.53)
and the second important result is time derivative of the Jacobian:
𝜕𝐽
𝜕𝑡
= 𝐽
𝜕𝑣 𝑘
𝜕𝑥 𝑘
⇐⇒
𝜕𝑣 𝑘
𝜕𝑥 𝑘
=
1
𝐽
𝜕𝐽
𝜕𝑡
. (2.54)
For the proof of above result see Eq. (A.15). Substituting Eq. (2.53) and Eq. (2.54)

in Eq. (2.51) will give the balance of mass in Lagrangean frame:
0 =
𝜕𝜌(𝑋, 𝑡)
𝜕𝑡
+ 𝜌(𝑋, 𝑡)
1
𝐽
𝜕𝐽
𝜕𝑡
,
=
1
𝜌(𝑋, 𝑡)
𝜕𝜌(𝑋, 𝑡)
𝜕𝑡
+
1
𝐽
𝜕𝐽
𝜕𝑡
,
=
𝜕ln 𝜌(𝑋, 𝑡)
𝜕𝑡
+
𝜕 𝑙𝑛𝐽
𝜕𝑡
,
=
𝜕
𝜕𝑡
(ln (𝜌(𝑋, 𝑡) 𝐽)) . (2.55)
Integration of above equation lead us to the following relationship:
𝜌(𝑋, 𝑡) =
𝜌0
𝐽
=
𝜌0
det(𝐹 𝑚𝑁 )
=
𝜌0
det
(︁
𝜕𝜒 𝑚
𝜕𝑋 𝑁
)︁. (2.56)
Here, 𝜌0 is the mass density in the reference configuration.
2.2.2. Balance of linear momentum
Newton’s Lex secunda, also known as the balance of linear momentum, states:
“total external force, 𝑓i, applied on a body is directly proportional to the rate
of change of the momentum of that body”. This statement holds for a material
particle and mathematically it is written as the following:
d𝑃𝑖
d𝑡
= 𝑚 ˙𝑣𝑖 = 𝑚 𝑎𝑖 = 𝑓𝑖. (2.57)
Here, 𝑣i and 𝑎i are the spatial velocity and acceleration, respectively. Above
equation can be extended to the partial differential equations of a continuous
medium. In order to do that we define the linear momentum for a infinitesimal
small volume element, which is equal to the product of its spatial velocity, 𝑣i,
and a infinitesimal small mass element, d𝑚:
d𝑃𝑖 = 𝑣𝑖 d𝑚. (2.58)
Integration of Eq. (2.58) over an arbitrary region Ωt gives the total linear mo
mentum of a continuous medium:
𝑃𝑖(𝑡) =
ˆ
Ω 𝑡
d𝑃𝑖 =
ˆ
Ω 𝑡
𝑣𝑖 d𝑚 =
ˆ
Ω 𝑡
𝜌 𝑣𝑖 d𝑉. (2.59)

Substituting Eq. (2.59) in Eq. (2.57) gives the following relation:
d
d𝑡
ˆ
Ω 𝑡
𝜌 𝑣𝑖 d𝑉 = 𝑓𝑖. (2.60)
External force 𝑓i acting on a continuum body can be further subdivided into
two different forces, namely: body force and surface force. A body force is a
function of position and time that acts throughout the volume of a body:
ˆ
Ω 𝑡
𝑏𝑖(𝑥, 𝑡) d𝑚 =
ˆ
Ω 𝑡
𝑏𝑖 𝜌 d𝑉. (2.61)
Forces 𝑓s
𝑖 that act on the particles of the surface of the body are defined as
follows: ˆ
𝜕Ωt
d𝑓s
𝑖 =
ˆ
𝜕Ωt
𝑡𝑖 d𝐴. (2.62)
Here, 𝑡i is the traction vector and is a function of position and time, and also
depend on the orientation of the surface on which it acts, i.e. depends on the
unit normal vector 𝑛j of the surface element d𝐴. The total force action on a
body is given as the sum of body- and surface force:
𝑓𝑖 =
ˆ
Ω 𝑡
𝑏𝑖 𝜌 d𝑉 +
ˆ
𝜕Ω 𝑡
𝑡𝑖 d𝐴. (2.63)
Substituting Eq. (2.63) in Eq. (2.60) gives the following relation:
d
d𝑡
ˆ
Ω 𝑡
𝜌 𝑣𝑖 d𝑉 =
ˆ
Ωt
ˆ
𝜕Ωt
𝑡𝑖 d𝐴. (2.64)
The French mathematician Baron Augustin Cauchy (1789–1857) had proved by
using tetrahedron argument that there exist unique second-order tensor field 𝜎ij,
such that:
𝑡𝑖 = 𝜎𝑖𝑗 𝑛 𝑗. (2.65)
It is a second order tensor and is known as Cauchy stress tensor, named after
its inventor Baron Augustin Cauchy. For proof of Eq. (2.65), refer (Müller and
Ferber, 2008). Substituting the Cauchy stress tensor in Eq. (2.64) gives the
global balance law of linear momentum:
d
d𝑡
ˆ
Ω 𝑡
𝜌 𝑣𝑖 d𝑉 =
ˆ
Ωt
ˆ
𝜕Ωt
(𝜎𝑖𝑗 𝑛 𝑗) d𝐴. (2.66)

Applying the Gauss theorem and the Reynolds transport theorem in Eq. (2.66)
gives: ˆ
Ω 𝑡
(︂
𝜕(𝜌 𝑣𝑖)
𝜕𝑡
+
𝜕
𝜕𝑥 𝑘
(𝜌 𝑣𝑖 𝑣 𝑘 − 𝜎𝑖𝑘)
)︂
d𝑉 =
ˆ
Ω 𝑡
𝜌 𝑏𝑖 d𝑉 (2.67)
Eq. (2.67) must be true for any sub body. Therefore, the integrand must vanish
and we get the local balance law of linear momentum in Eulerian frame:
𝜕(𝜌 𝑣𝑖)
𝜕𝑡
+
𝜕
𝜕𝑥 𝑘
(𝜌 𝑣𝑖 𝑣 𝑘 − 𝜎𝑖𝑘) = 𝜌 𝑏𝑖 ⇐⇒
𝜕𝜌 𝑣
𝜕𝑡
+ ∇ · (𝜌 𝑣 ⊗ 𝑣) − ∇ · 𝜎 = 𝜌 𝑏.
(2.68)
In order to get the balance of linear momentum in Lagrangean form, we use
the same procedure as we did for mass balance law. Application of the chain
rule in Eq. (2.68) gives:
𝜌
𝜕𝑣𝑖
𝜕𝑡
+ 𝑣𝑖
𝜕𝜌
𝜕𝑡
+ 𝑣𝑖
𝜕(𝜌 𝑣 𝑘)
𝜕𝑥 𝑘
+ 𝜌 𝑣 𝑘
𝜕(𝜌 𝑣𝑖)
𝜕𝑥 𝑘
−
𝜕𝜎𝑖𝑘
𝜕𝑥 𝑘
= 𝜌 𝑏𝑖. (2.69)
Using balance of mass as in Eq. (2.50) in the above equation gives the following
relation:
𝜌(𝑋, 𝑡)
(︂
𝜕𝑣𝑖
𝜕𝑡
+ 𝜌 𝑣 𝑘
𝜕𝜌 𝑣𝑖
𝜕𝑥 𝑘
)︂
⏟ ⏞
𝜕𝑣 𝐼 (𝑋,𝑡)
𝜕𝑡
−
𝜕𝜎𝑖𝑘
𝜕𝑥 𝑘
= 𝜌(𝑋, 𝑡) 𝑏𝑖. (2.70)
Substituting Eq. (2.56) in the above equation will lead to the balance of linear
momentum in Lagrangean frame:
𝜌0 𝑏𝑖 = 𝜌0
𝜕𝑣𝑖(𝑋, 𝑡)
𝜕𝑡
− 𝐽
𝜕𝜎𝑖𝑘
𝜕𝑥 𝑘
,
= 𝜌0
𝜕𝑡
− 𝐽
𝜕𝑋 𝐽
𝜕𝑥 𝑘
𝜕𝜎𝑖𝑘
𝜕𝑋 𝐽
,
= 𝜌0
𝜕𝑡
−
𝜕
𝜕𝑋 𝐽
(︁
𝐽 𝐹−1
𝐽𝑘 𝜎𝑖𝑘
)︁
⏟ ⏞
𝑆 𝑖𝑗
,
= 𝜌0
𝜕𝑡
−
𝜕𝑆𝑖𝑗
𝜕𝑋 𝐽
. (2.71)
Here, 𝑆𝑖𝑗 is the 1st Piola–Kirchhoff stress tensor. Above equation can also
be written in term of 𝜒𝑖 :
𝜌0
𝜕𝜒𝑖
𝜕𝑡
−
𝜕𝑆𝑖𝑗
𝜕𝑋 𝐽
= 𝜌0 𝑏𝑖. (2.72)

2.2.3. Balance of energy
The balance of energy states: “The rate of change of total energy, 𝐸, is equal to
the sum of global heat supply, 𝑄, and the net rate of the mechanical work done
by a solid, 𝑊”. Mathematically it is written as the following:
d𝐸
d𝑡
= 𝑄 + 𝑊. (2.73)
Total energy, 𝐸, is given as the integration of the specific energy density, 𝜌 𝜀,
over the region Ωt, which is equal to the sum of specific internal energy and
specific kinetic energy:
𝐸(𝑡) =
ˆ
Ω 𝑡
𝜌 𝜀(𝑥, 𝑡) d𝑉 =
ˆ
Ω 𝑡
(𝜌 𝑢) d𝑉 +
ˆ
Ω 𝑡
(︂
𝜌
2
𝑣𝑖 𝑣𝑖
)︂
d𝑉. (2.74)
Global heat supply, 𝑄, of a body is given as the sum of heat fluxes through its
surface and the heat source in its interior:
𝑄 =
ˆ
𝜕Ω 𝑡
𝑞𝑖 d𝐴 +
ˆ
Ω 𝑡
𝑟 d𝑚. (2.75)
Here, 𝑞i is the heat flux per unit area and time and 𝑟 represent the specific heat
source. The Fourier theorem permits us to assume that the heat flux is linear
in the outer normal of the surface 𝑛i. Therefore, the heat flux can be represented
by a scalar product:
𝑞𝑖 = −𝑞 · 𝑛𝑖. (2.76)
Substituting above equation in Eq. (2.75) gives us elaborate form of global heat
supply:
𝑄 =
ˆ
Ω 𝑡
𝜌 𝑟(𝑥, 𝑡) d𝑉 −
ˆ
𝜕Ω 𝑡
𝑞𝑖 𝑛𝑖 d𝐴. (2.77)
Next we define the net rate of the mechanical work done by a solid:
𝑊 =
ˆ
Ω 𝑡
𝜌 𝑏𝑖 𝑣𝑖(𝑥, 𝑡) d𝑉 +
ˆ
𝜕Ω 𝑡
𝑣𝑖 𝜎𝑖𝑗 𝑛 𝑗 d𝐴. (2.78)
Substituting Eq. (2.74), Eq. (2.77) and Eq. (2.78) in Eq. (2.73) gives us the global
balance law of total energy:
d
d𝑡
ˆ
Ω 𝑡
(︂
𝜌 𝑢 +
𝜌
2
𝑣𝑖 𝑣𝑖
)︂
d𝑉 =
ˆ
Ω 𝑡
(𝜌 𝑟 + 𝜌 𝑏𝑖 𝑣𝑖) d𝑉 +
ˆ
𝜕Ω 𝑡
(𝑣𝑖 𝜎𝑖𝑗 𝑛 𝑗 − 𝑞𝑖 𝑛𝑖) d𝐴.
(2.79)

Application of the Reynolds transport theorem on the left side term of the
above equation gives:
ˆ
Ω 𝑡
(︂
𝜕
𝜕𝑡
(︂
𝜌 𝑢 +
𝜌
2
𝑣𝑖 𝑣𝑖
)︂
+
𝜕
𝜕𝑥 𝑘
(︂(︂
𝜌 𝑢 +
𝜌
2
𝑣𝑖 𝑣𝑖
)︂
𝑣 𝑘
)︂)︂
d𝑉. (2.80)
Applying the Gauss theorem on the last term of the right side of Eq. (2.79)
gives: ˆ
Ω 𝑡
(︂
𝜕
𝜕𝑥 𝑘
(𝑣𝑖 𝜎𝑖𝑘 − 𝑞 𝑘)
)︂
d𝑉. (2.81)
Using Eq. (2.80) and Eq. (2.81) we can rewrite the global balance of total energy
in the following form:
ˆ
Ω 𝑡
(︂
𝜕
𝜕𝑡
(︂
𝜌 𝑢 +
𝜌
2
𝑣𝑖
2
)︂
+
𝜕
𝜕𝑥 𝑘
(︂(︂
𝜌 𝑢 +
𝜌
2
𝑣𝑖
2
)︂
𝑣 𝑘 + 𝑞 𝑘 − 𝑣𝑖 𝜎𝑖𝑘
)︂)︂
d𝑉 =
ˆ
Ω 𝑡
(𝜌 𝑏𝑖 𝑣𝑖 + 𝜌 𝑟) d𝑉.
(2.82)
Above equation must be true for any sub body. Therefore, integrand should
vanish and we get the local balance of total energy:
𝜕
𝜕𝑡
(︂
𝜌 𝑢 +
𝜌
2
𝑣𝑖
2
)︂
+
𝜕
𝜕𝑥 𝑘
(︂(︂
𝜌 𝑢 +
𝜌
2
𝑣𝑖
2
)︂
𝑣 𝑘 + 𝑞 𝑘 − 𝑣𝑖 𝜎𝑖𝑘
)︂
= 𝜌 𝑏𝑖 𝑣𝑖 + 𝜌 𝑟. (2.83)
The local balance of internal energy is obtained through subtracting the balance
of kinetic energy from the balance of total energy. In order to derive the balance
of kinetic energy we will insert the balance of mass from Eq. (2.50) in Eq. (2.69)
(Abali, 2014):
𝜌
𝜕𝑣𝑖
𝜕𝑡
+ 𝜌 𝑣 𝑗
𝜕𝑣 𝑗
𝜕𝑥 𝑗
−
𝜕𝜎𝑖𝑘
𝜕𝑥 𝑘
= 𝜌 𝑏𝑖. (2.84)
Multiplying the above equation with 𝑣𝑖 gives the following:
𝑣𝑖 𝜌
𝜕𝑣𝑖
𝜕𝑡
+ 𝑣𝑖 𝜌 𝑣 𝑗
𝜕𝑣 𝑗
𝜕𝑥 𝑗
− 𝑣𝑖
𝜕𝜎𝑖𝑘
𝜕𝑥 𝑘
= 𝑣𝑖 𝜌 𝑏𝑖, (2.85)
which is equivalent to:
𝜕
𝜕𝑡
(︂
𝜌
2
𝑣𝑖 𝑣𝑖
)︂
−
1
2
𝜕𝜌
𝜕𝑡
𝑣𝑖 𝑣𝑖 + 𝜌 𝑣 𝑗
𝜕
𝜕𝑥 𝑗
(︂
1
2
𝑣𝑖 𝑣𝑖
)︂
−
𝜕(𝜎 𝑗𝑖 𝑣𝑖)
𝜕𝑥 𝑗
= 𝜌 𝑏𝑖 𝑣𝑖 − 𝜎 𝑗𝑖
𝜕𝑣𝑖
𝜕𝑥 𝑗
.
(2.86)
Using the balance of mass in above equation permits us to rewrite Eq. (2.86) as
following:
𝜕
𝜕𝑡
(︂
𝜌
2
𝑣𝑖 𝑣𝑖
)︂
+
1
2
𝜕(𝜌 𝑣 𝑗)
𝜕𝑥 𝑗
𝑣𝑖 𝑣𝑖 +
1
2
𝜌 𝑣 𝑗
𝜕(𝑣𝑖 𝑣𝑖)
𝜕𝑥 𝑗
−
𝜕(𝜎 𝑗𝑖) 𝑣𝑖
𝜕𝑥 𝑗
= 𝜌 𝑏𝑖 𝑣𝑖 − 𝜎 𝑗𝑖
𝜕𝑣𝑖
𝜕𝑥 𝑗
.
(2.87)

Then, the balance of kinetic energy is given as:
𝜕
𝜕𝑡
(︂
𝜌
2
𝑣𝑖 𝑣𝑖
)︂
+
𝜕
𝜕𝑥 𝑗
(︂
𝜌
2
𝑣𝑖 𝑣𝑖 𝑣 𝑗 − 𝑣𝑖 𝜎 𝑗𝑖
)︂
= −𝜎 𝑗𝑖
𝜕𝑣𝑖
𝜕𝑥 𝑗
+ 𝜌 𝑏𝑖 𝑣𝑖. (2.88)
Subtracting Eq. (2.88) from Eq. (2.83) gives the local balance of internal energy
in the Eulerian frame:
𝜕𝜌 𝑢
𝜕𝑡
+
𝜕
𝜕𝑥 𝑘
(𝜌 𝑢 𝑣 𝑘 + 𝑞 𝑘) = 𝜎𝑖𝑘
𝜕𝑣𝑖
𝜕𝑥 𝑘
+ 𝜌 𝑟. (2.89)
The balance of internal energy in the Lagrangean description can be obtained
by adopting the same procedure as we opted for the balance of mass and linear
momentum and is given as the following:
𝜌0
𝜕𝑈
𝜕𝑡
+
𝜕𝑄 𝑘
𝜕𝑋 𝑘
= 𝑆𝑖𝑗
𝜕𝐹𝑖𝑗
𝜕𝑡
. (2.90)
Here, 𝜌0 𝑈 is the internal energy in the Lagrangean frame. 𝑄 𝑘 is the material
heat flux and is defined as:
𝑄 𝑘 = 𝐽 𝐹−1
𝑘𝑖 𝑞𝑖. (2.91)
2.2.4. The entropy inequality
The second law of thermodynamics introduces one of the challenging concept
of thermal or statistical physics, i.e. concept of entropy. Entropy, which is a
measure of disorder or randomness of a system, was first developed in the early
1850’s in the work of the German physicist Rudolf Julius Emanuel Clausius.
According to the second law of thermodynamics, the rate of entropy should be
greater than or equal to the rate of heat divided by the absolute temperature.
Mathematically it is given as:
d
d𝑡
ˆ
Ωt
𝜌𝑠 d𝑉 ≥
ˆ
Ωt
1
𝑇
𝜌 𝑟 d𝑉 −
ˆ
𝜕Ωt
1
𝑇
𝑞 𝑗 𝑛 𝑗 d𝐴. (2.92)
Eq. (2.92) is also known as the entropy inequality principle. The terms 𝑇 and 𝑠
in the above equation represents the absolute temperature and specific entropy,
respectively. Another very common way to express the second law of thermody
namics in continuum mechanics is by using Clausius-Duhem inequality, named
after Clausius and a French physicist Pierre Duhem. We can establish this
inequality by combining Eq. (2.92) with the first law of thermodynamics and
balance of momentum.
Another method for the development of entropy inequality can be seen in rational
thermodynamics. Mathematically the entropy principle, in its global form, can

be expressed as (Müller, 2014a):
d
d𝑡
ˆ
Ω 𝑡
𝜌 𝑠 d𝑉 = −
ˆ
𝜕Ω 𝑡
𝜑𝑖 𝑛𝑖 d𝐴 +
ˆ
Ω 𝑡
𝑧 d𝑉 +
ˆ
Ω 𝑡
𝜎 d𝑉. (2.93)
The terms, 𝜑i, 𝑧, and 𝜎 occurring in Eq. (2.93) represent the entropy flux vector,
the volume supply of the entropy, and entropy production, respectively. Entropy
production, 𝜎, should be positive-semidefinite(Müller, 2014a):
𝜎 ≥ 0. (2.94)
If we neglect entropy supply due to radiation and assume that entropy flux is
given by 𝑞i
𝑇 , then Eq. (2.93) can be written as:
d
d𝑡
ˆ
Ω 𝑡
𝜌 𝑠 d𝑉 +
ˆ
𝜕Ω 𝑡
1
𝑇
𝑞𝑖 𝑛𝑖 d𝐴 =
ˆ
Ω 𝑡
𝜎 d𝑉 ≥ 0. (2.95)
Application of the Reynolds transport theorem and the Gauss theorem in
Eq. (2.95), leads us to the following form of balance of entropy:
ˆ
Ω 𝑡
(︂
d(𝜌 𝑠)
d𝑡
+
𝜕
𝜕𝑥𝑖
(︂
𝑞𝑖
𝑇
)︂)︂
d𝑉 =
ˆ
Ω 𝑡
𝜎 d𝑉. (2.96)
Eq. (2.96) should be true for any sub body. Therefore, balance of entropy in its
local form is given as:
d(𝜌 𝑠) d𝑡 +
𝜕
𝜕𝑥𝑖
(︂
𝑞𝑖
𝑇
)︂
= 𝜎. (2.97)
Using balance of mass (Eulerian form) and product rule of differentiation in
Eq. (2.97), allow us to rewrite the local balance of entropy in the following form:
𝜌 𝑇
d𝑠
d𝑡
=
𝑞𝑖
𝑇
𝜕𝑇
𝜕𝑥𝑖
+ 𝑇 𝜎 −
𝜕𝑞𝑖
𝜕𝑥𝑖
. (2.98)
Eq. (2.98) play an important role in the formulation of constitutive equations.

3. Constitutive models for elastic
materials
There is nothing that can be said by mathematical symbols and relations
which cannot also be said by words. The converse, however, is false.
Much that can be and is said by words cannot successfully be put into
equations, because it is nonsense.
– Clifford Truesdell (February 18, 1919 – January 14, 2000)
In the previous chapter we discussed the governing equations of continuum me
chanics, which are valid for any continuous medium. These governing equations,
namely: balance of mass, linear momentum, and energy provide five partial
differential equations with sixteen unknowns (stress tensor is assumed to be
symmetric) for a continuous thermo-mechanical system. In order to compute
all these unknowns we need to construct these missing equations, which are
known as constitutive relations. Using constitutive relations, we can describe
behavior of the material under the thermal and mechanical loading. In this
chapter we discuss the Hooke’s law for anisotropic and isotropic solid material.
Constitutive relations should follow the principle of objectivity, which means
they should be independent of the observer.
3.1. Generalized Hooke’s law
The original law of Robert Hooke, an English physicist, was published in the
year of 1676. This law states “If the force applied on the per unit area of any
elastic material, then it is linearly proportional to the amount of elongation and
compression”. According to (Truesdell, 1968), James Bernoulli (1704) was the
one, who had given the first instance of a true stress-strain relation. For uniaxial
loading the Hooke’s law can be given as the following:
𝜎 ∼ 𝜀, (3.1)
𝜎 = 𝐸 𝜀. (3.2)
The proportional factor 𝐸 is known as elasticity modulus or Young’s modulus,
named after the English polymath Thomas Young. It is a material parameter
Chapter 3. Constitutive models for elastic materials

that tells how much a material is going to deform under tension and compres
sion. 𝜎 represent axial stress and 𝜀 is strain produced due to this axial stress.
In general, the properties of many materials vary according to their crystallo
graphic orientations. In such cases Eq. (3.2) is not more valid. Therefore, it is
necessary to develop the generalized form of Hooke’s law. The Hooke’s law
for three dimensional case can be derived from the strain energy density8 and
mathematically it is given as:
𝜎ij = 𝐶ijkl 𝜀kl. (3.3)
𝜎ij and 𝜀kl are the second order stress and strain tensors, respectively. 𝐶ijkl is
elastic stiffness of the material. It is a fourth order tensor with 81 components.
The law stated in Eq. (3.3) is only valid for small deformations. Abaqus
also uses Eq. (3.3) to model elasticity for small-strain problems and also for an
elastic-plastic model in which elastic strains are considered to be small (Simulia,
2015b).
Inverting Eq. (3.3) allows us to relate strain to stress:
𝜀𝑖𝑗 = 𝑠𝑖𝑗𝑘𝑙 𝜎 𝑘𝑙. (3.4)
Here, 𝑠ijkl is known as compliance tensor.
Computation of a matrix, 𝐶ijkl, with 81 components is a bit time consuming
process. Therefore, we use the symmetry property to reduce the number of
elements in the stiffness matrix. For i = 1, j = 2 and for i = 2, j = 1, Eq. (3.3)
can be written as the following:
𝜎12 = 𝐶12kl 𝜀kl, 𝜎21 = 𝐶21kl 𝜀kl. (3.5)
According to Cauchy’s second law of motion or Boltzmann’s axiom, the
Cauchy’s stress tensor is considered to be symmetric, if the medium is non-polar
(Bertram, 2012). In this work we assume a non-polar medium. Therefore, 𝜎ij is
symmetric, which leads us to the following relation:
𝜎12 = 𝜎21 ⇒ (𝐶12kl − 𝐶21kl) 𝜀kl = 0 ⇒ 𝐶12kl = 𝐶21kl. (3.6)
The stiffness matrix in Eq. (3.6) shows minor symmetry, due to which the number
of its components reduced from 81 to 54. The number of elements in 𝐶ijkl can be
8
For complete derivation please refer (Hahn, 1985)
Chapter 3. Constitutive models for elastic materials – Generalized Hooke’s law

reduced further from 54 to 36 by considering the symmetry of the strain tensor:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝐶1111 𝐶1122 𝐶1133 𝐶1112 𝐶1113 𝐶1123
𝐶2211 𝐶2222 𝐶2233 𝐶2212 𝐶2213 𝐶2223
𝐶3311 𝐶3322 𝐶3333 𝐶3312 𝐶3313 𝐶3323
𝐶1211 𝐶1222 𝐶1233 𝐶1212 𝐶1213 𝐶1223
𝐶1311 𝐶1322 𝐶1333 𝐶1312 𝐶1313 𝐶1323
𝐶2311 𝐶2322 𝐶2333 𝐶2312 𝐶2313 𝐶2323
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (3.7)
Use of Voigt notation permits us to write the stiffness matrix in compact form,
and at the same time this compact form will give more clearance to analyze 𝐶ijkl.
Using this notation, we can write the stiffness matrix 𝐶ijkl as the following:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝐶11 𝐶12 𝐶13 𝐶14 𝐶15 𝐶16
𝐶21 𝐶22 𝐶23 𝐶24 𝐶25 𝐶26
𝐶31 𝐶32 𝐶33 𝐶34 𝐶35 𝐶36
𝐶41 𝐶42 𝐶43 𝐶44 𝐶45 𝐶46
𝐶51 𝐶52 𝐶53 𝐶54 𝐶55 𝐶56
𝐶61 𝐶62 𝐶63 𝐶64 𝐶65 𝐶66
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (3.8)
The number of elements of the stiffness matrix 𝐶ijkl can be further reduced to
21, if we use the existence of equivalence of the mixed partials:
𝐶ijkl =
𝜕2 𝑈
𝜕𝜀kl 𝜕𝜀ij
=
𝜕2 𝑈
𝜕𝜀ij 𝜕𝜀kl
= 𝐶klij. (3.9)
Using Eq. (3.6), Eq. (3.9) and the strain symmetry, we can establish the following
identity for the stiffness matrix, 𝐶ijkl:
𝐶ijkl = 𝐶jikl, 𝐶ijkl = 𝐶ijlk, and 𝐶ijkl = 𝐶klij. (3.10)
Using symmetrical property of the stiffness matrix 𝐶ijkl, as shown in Eq. (3.10),
and the Voigt notation, Eq. (3.3) can be written as the following:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝜎11
𝜎22
𝜎33
𝜎23
𝜎31
𝜎12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝐶11 𝐶12 𝐶13 𝐶14 𝐶15 𝐶16
𝐶22 𝐶23 𝐶24 𝐶25 𝐶26
𝐶33 𝐶34 𝐶35 𝐶36
𝐶44 𝐶45 𝐶46
𝐶55 𝐶56
sym. 𝐶66
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝜀11
𝜀22
𝜀33
2𝜀23
2𝜀13
2𝜀12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (3.11)
Here, the stiffness matrix, 𝐶ijkl, has 21 independent components.

3.1.1. Isotropic case
A material is known as isotropic material, when its mechanical properties are
free from the choice of direction, which means their properties do not depend on
there crystallographic orientations. The general form of a fourth order isotropic
stiffness tensor is given as:
𝐶ijkl = 𝛼 𝛿ij 𝛿kl + 𝛽 𝛿ik 𝛿jl + 𝛾 𝛿il 𝛿jk. (3.12)
Here, 𝛼, 𝛽, and 𝛾 are scalars. Using the property 𝛿kl = 𝛿lk, Eq. (3.12) can be
written in the following form:
𝐶ijkl = 𝛼 𝛿ij 𝛿lk + 𝛽 𝛿il 𝛿jk + 𝛾 𝛿ik 𝛿jl
= 𝛼 𝛿ij 𝛿lk + 𝛾 𝛿ik 𝛿jl + 𝛽 𝛿il 𝛿jk. (3.13)
From Eq. (3.13), we see that, 𝛽 = 𝛾. Defining two more constants, 𝜆 and 𝜇, such
that 𝜆 = 𝛼 and 𝜇 = 𝛽 = 𝛾, gives an alternative form of Eq. (3.13):
𝐶ijkl = 𝜆 𝛿ij 𝛿kl + 𝜇 (𝛿ik 𝛿jl + 𝛿il 𝛿jk) . (3.14)
Substituting Eq. (3.14) in Eq. (3.3), gives us the Hooke’s law for an isotropic
material:
𝜎ij = 𝜆 𝛿ij 𝜀kl + 𝜇 (𝜀ij + 𝜀ji) . (3.15)
Here, 𝜆 and 𝜇 are scalar quantities, known as Lamé constants. These quantities
are temperature dependent. The matrix form of Eq. (3.15) can be written as:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝜎11
𝜎22
𝜎33
𝜎23
𝜎31
𝜎12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝜆 + 2𝜇 𝜆 𝜆 0 0 0
𝜆 + 2𝜇 𝜆 0 0 0
𝜆 + 2𝜇 0 0 0
𝜇 0 0
𝜇 0
sym. 𝜇
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝜀11
𝜀22
𝜀33
2𝜀23
2𝜀13
2𝜀12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (3.16)
In order to find the Lamé constants, 𝜆 and 𝜇, let us start with the strain tensors,
which are obtained through experiments:
𝜀11 =
1
𝐸
(𝜎11 − 𝜈 (𝜎22 + 𝜎33)) ,
𝜀22 =
1
𝐸
(𝜎22 − 𝜈 (𝜎11 + 𝜎33)) ,
𝜀33 =
1
𝐸
(𝜎33 − 𝜈 (𝜎11 + 𝜎22)) ,
2 𝜀23 =
𝜎23
𝐺
, 2 𝜀13 =
𝜎13
𝐺
, 2 𝜀12 =
𝜎12
𝐺
. (3.17)

Here, 𝜈 is the Poisson’s ration, named after a French mathematician and
physicist Siméon Denis Poisson. It computes the ratio between lateral and
longitudinal strain. Shear modulus, 𝐺, indicates the material response to shearing
strains, which is positive and smaller than 𝐸. The matrix form of Eq. (3.17) can
be written as:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝜀11
𝜀22
𝜀33
2𝜀23
2𝜀13
2𝜀12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
1
𝐸
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 −𝜈 −𝜈 0 0 0
1 −𝜈 0 0 0
1 0 0 0
2 (1 + 𝜈) 0 0
2 (1 + 𝜈) 0
sym. 2 (1 + 𝜈)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
𝜎11
𝜎22
𝜎33
𝜎23
𝜎31
𝜎12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (3.18)
Inverting Eq. (3.18) and then comparing it with Eq. (3.16) gives the value of
Lamé constants:
𝜆 =
𝐸𝜈
(1 + 𝜈) (1 − 2𝜈)
𝜇 = 𝐺 =
𝐸
2 (1 + 𝜈)
. (3.19)
The engineering constants, 𝐸 and 𝜈, can also be expressed in terms of Lamé
constants:
𝐸 =
𝜇 (2 𝜇 + 3 𝜆)
𝜇 + 𝜆
, and 𝜈 =
𝜆
2 (𝜇 + 𝜆)
. (3.20)
Theory of classical elasticity predicts the value of Poisson’s ratio for an isotropic
material to lie somewhere in between -1 and 0.5. But, experiments have shown
that the actual value of 𝜈 lies between 0.2 and 0.5 (Mott and Roland, 2009).
3.1.2. Thermoelastic case
In the previous section, thermal stress, produced due to the temperature change
in the material body, had been neglected. In order to compute the total strain in
a material, we need to consider both the thermal- and mechanical effects. For an
isotropic material, the thermal strain is computed as a product of the coefficient
of thermal expansion, 𝛼, and change in the temperature, Δ𝑇 (Rösler, Harders,
and Bäker, 2012):
𝜀th
ij = 𝛼 Δ𝑇 𝛿ij. (3.21)
Then, the total strain is given by the sum of thermal strain, 𝜀th, and mechanical
strain, 𝜀m:
𝜀ij = 𝜀m
ij + 𝜀th
ij . (3.22)

Substituting Eq. (3.22) in Eq. (3.3), gives the Hooke’s law for isotropic material
due to thermoelastic effect:
𝜎ij = 𝐶ijkl 𝜀kl
= 𝐶ijkl
(︁
𝜀kl − 𝜀th
kl
)︁
= 𝐶ijkl (𝜀kl − 𝛼 Δ𝑇 𝛿kl)
= 𝐶ijkl 𝜀kl − 𝐶ijkl 𝛼 Δ𝑇 𝛿kl. (3.23)
Substituting the isotropic stiffness matrix from Eq. (3.14) in Eq. (3.23), gives us
the following relation:
𝜎ij = 𝜆 𝜀kk 𝛿ij + 2𝜇 𝜀ij − (𝜆 𝛿ij 𝛿kl + 𝜇 (𝛿ik 𝛿jl + 𝛿il 𝛿jk)) 𝛼 Δ𝑇 𝛿kl
= 𝜆 𝜀kk 𝛿ij + 2𝜇 𝜀ij − 𝜆 𝛼 Δ𝑇 𝛿ij 𝛿kl 𝛿kl − 𝜇 𝛼 Δ𝑇 𝛿ik 𝛿jl 𝛿kl − 𝜇 Δ𝑇 𝛿il 𝛿jk 𝛿kl.
(3.24)
In order to simplify Eq. (3.24), we need to compute the product of Kronecker
delta and use its substitution property:
𝛿kl 𝛿kl = 3, 𝛿ik 𝛿jl 𝛿kl = 𝛿ij, 𝛿il 𝛿jk 𝛿kl = 𝛿ij. (3.25)
Substituting Eq. (3.25) in Eq. (3.24), gives:
𝜎ij = 𝜆 𝜀kk 𝛿ij + 2𝜇 𝜀ij − 3𝜆 𝛼 Δ𝑇 𝛿ij − 2𝜇 𝛼 Δ𝑇 𝛿ij. (3.26)
Then, the Hooke’s law for an isotropic material taking thermoelastic effect
under consideration is given by the following relation:
𝜎ij = 𝜆 (𝜀kk − 3𝛼 Δ𝑇) 𝛿𝑖𝑗 + 2𝜇 (𝜀ij − 𝛼 Δ𝑇 𝛿ij) . (3.27)

4. Constitutive models for
elasto-plastic materials
A scientific truth does not triumph by convincing its opponents and
making them see the light, but rather because its opponents eventually
die and a new generation grows up that is familiar with it.
– Max Karl Ernst Ludwig Planck (April 23, 1858 — October 4, 1947)
The theory of elasto-plasticity belongs to the branch of mechanics in which we
study the behavior of such material, that initially goes under elastic deformation
but upon reaching certain criteria, i.e. an initial yield stress 𝜎Y0 it shows
permanent (irreversible) deformation. The aim of this chapter is to give a
brief introduction to time-independent plasticity theory and develop constitutive
equations.
Continuum plasticity should fulfill important requirements, which arise from
the theory of micro-plasticity. The first requirement is the condition of incom
pressibility. According to this condition, the volume of a body does not change
during and after deformation because plastic slip, occurring during deformation,
does not lead to volume change. Second important requirement states that the
material response should be independent of rate effects. The fact that plastic
slip is a shearing process, tells us that plastic deformation of metal at macro
scopic scale is independent of hydrostatic stress. This statement can be proved
mathematically by using crystal plasticity constitutive equation.
When the problem with small deformation is concerned, then the results provided
by the classical plasticity theory is satisfactory. But this theory gives absurd
result if it is applied for the large deformations. In order to study the large
plastic deformations, the theory of finite plasticity is popular (Bertram, 2012).
4.1. Stress-strain relationship
If the load applied on a material body9 is small, then the strain, occurring due
to the force applied per unit area, is directly proportional to the applied stress
9
Not all the material body obey this rule, but for this work we will consider only those material
body, which obey this relationship
Chapter 4. Constitutive models for elasto-plastic materials

at room temperature, see sec. 3.1. If the applied load reaches the elastic limit or
the yield point10 of the material, then the stress-strain curve shows nonlinear
behavior. Therefore, application of Hooke’s law is not valid beyond this elastic
limit. In other words, we can say that the material starts yielding or begins to
flow.
Removal of load beyond the initial yielding11 causes permanent deformation.
This behavior of the material body is known as plasticity. In order to compute the
plastic strain, a fundamental approach of additive decomposition is used (Müller,
2014a). According to this approch, the total strain observed can be separated in
two parts, namely: reversible elastic strain, 𝜀e
ij, and irreversible plastic strain,
𝜀p
ij, see Fig. 4.1. This assumption is only applicable, when the body undergoes
small deformation. For large deformation, the assumption of multiplicative
decomposition is used. Mathematically the strain additive decomposition is
Figure 4.1.: Elasto-plastic stress-strain curve for a uniaxial loading, with the initial
yield stress and strain decomposition (Kossa, 2011).
given as:
𝜀ij = 𝜀e
ij + 𝜀p
ij. (4.1)
Using Eq. (4.1), the plastic strain in the material body is defined as:
𝜀e
ij = 𝜀ij − 𝜀p
ij. (4.2)
10
The limit beyond which permanent deformation occurs during a tension test.
11
Generally it is not possible to differentiate between the initial- and upper yield point for all
materials.
Chapter 4. Constitutive models for elasto-plastic materials – Stress-strain relationship

In general, the macroscopic continuum approach is based on the usage of an
incremental strain or strain rate. We use the approach based on the strain rates
because in plasticity the study of deformation is path dependent. Therefore, the
incremental form of Eq. (4.2) can be given as:
˙𝜀e
ij = ˙𝜀ij − ˙𝜀p
ij. (4.3)
Unloading process do not obey the Hooke’s law as in Eq. (3.3), if the material
is subjected to plastic flow. Therefore, we need to modify the Hooke’s law.
In order to do that we substitute Eq. (4.3) in Eq. (3.3), which gives us the
incremental form of the Hooke’s law:
˙𝜎𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑙 ˙𝜀e
𝑘𝑙 = 𝐶𝑖𝑗𝑘𝑙
(︁
˙𝜀ij − ˙𝜀p
ij
)︁
. (4.4)
Eq. (4.3) is also used in Abaqus for most of the inelastic constitutive models.
In many engineering example it had been seen that the elastic strain 𝜀e
ij is much
smaller than the plastic strain 𝜀p
ij (Dunne and Petrinic, 2005).
If we also consider deformation due to thermal effect then Eq. (4.3) can be written
as the following:
˙𝜀e
ij = ˙𝜀ij − ˙𝜀p
ij − ˙𝜀th
ij . (4.5)
Using the above relationship the incremental form of the Hooke’s law as in
Eq. (4.4) can be written as the following:
˙𝜎𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑙 ˙𝜀e
𝑘𝑙 = 𝐶𝑖𝑗𝑘𝑙
(︁
˙𝜀ij − ˙𝜀p
ij − ˙𝜀th
ij
)︁
. (4.6)
4.2. The yield criterion
The yield criterion (condition) is used to determine the transition of a material
behavior from elastic to plastic. In other words we can say that it defines the
stress state for which the material shows plastic behavior. For a uniaxial stress
test, the yield condition is easy to determine. In uniaxial yield condition, if the
stress, 𝜎, is smaller than the initial yield stress, 𝜎Y0, then the material exhibits
elastic behavior otherwise plastic. Mathematically we can say that the material
shows plastic behavior only if the following yield criterion is fulfilled (Jirásek
and Bazant, 2002):
|𝜎| − 𝜎Y0 = 0. (4.7)
In a multiaxial stress state, finding the yield condition is not that straightforward.
In this case the yield criterion is defined by a scalar function known as yield
function (Prager and Hodge, 1968):
𝑓 (𝜎𝑖𝑗) = 0. (4.8)
Chapter 4. Constitutive models for elasto-plastic materials – The yield criterion

Here, 𝜎ij is a symmetrical stress tensor. Therefore, Eq. (4.8) can be written in
the following form:
𝑓 (𝜎11, 𝜎22, 𝜎33, 𝜎23, 𝜎31, 𝜎12) = 0. (4.9)
If the yield function is equal to zero, 𝑓 (𝜎ij) = 0, then the material shows plastic
behavior and when 𝑓 (𝜎ij) < 0, then the material exhibit elastic behavior.
If the material is considered to be isotropic, then the yield condition is free from
the orientation of the coordinate system. It means, the expression in Eq. (4.8)
can be expressed in the term of the invariants of the stress tensor (Hill, 1986):
𝑓 (𝐽1, 𝐽2, 𝐽3) = 0, (4.10)
Here, 𝐽1, 𝐽2 and 𝐽3 are the first, second, and third invariants of the stress tensor,
respectively. These are defined by the following relations (Hill, 1986):
𝐽1 = 𝜎11 + 𝜎22 + 𝜎33,
𝐽2 = − (𝜎11 𝜎22 + 𝜎22 𝜎33 + 𝜎33 𝜎11) ,
𝐽3 = 𝜎11 𝜎22 𝜎33. (4.11)
Here 𝜎11, 𝜎22, and 𝜎33 are principal components of the stress tensor.
There exist many different yield criteria for the metals, which assume that a
hydrostatic stress always influences the yielding of a metal. This assumption
is a contradiction to the present days experimental fact. Therefore, these yield
conditions are only for historic interest. But Tresca12 and Von Mises13 yield
criterion do not have this fault and are used more often to build mathematical
model of plasticity (Hill, 1986).
In this work we limit our discussion to von Mises yield criterion only, because it
is widely used in the engineering application, mainly for computational analysis.
Von Mises yield criterion considered all three principal shear stresses to predict
the yielding of metal. The use of all three principal shear stresses gives a accurate
prediction of yielding.
4.2.1. Von Mises yield criterion
The defining equation for the Von Mises stress was independently proposed by
the Polish mechanical engineer Tytus Maksymilan Huber and Richard Von
Mises in 1904 and 1913, respectively. The physical interpretation of this equation
was given by the German engineer Heinrich Hencky in 1924. According to
12
Named after the French mechanical engineer Henri Tresca. According to this yielding
criterion, the yielding begins when the maximum shear stress reaches a critical value.
13
Named after the famous scientist and mathematician Richard von Mises

Von Mises yield criterion, material starts flowing after reaching the initial
yield stress, 𝜎Y0 (Müller, 2014a). In other words, materials start yielding when
the second deviatoric stress invariant, 𝐽2, reaches a critical value (Hill, 1986).
Mathematically the Von Mises yield criterion is expressed as:
𝐽2 = 𝑘2
. (4.12)
Here, 𝑘 is a material parameter and can be understood as yield stress of a given
material under simple shear. The value of 𝑘 is
√
3 times smaller than the tensile
yield stress in simple tension case (Prager and Hodge, 1968):
𝑘 =
𝜎Y
√
3
. (4.13)
An alternative way to express Von Mises yield criterion is given below:
𝜎Mises = 𝜎Y =
√︀
3 𝐽2 . (4.14)
Here, 𝜎Mises is known as Von Mises equivalent stress, and it predict the yielding
of material under multiaxial loading. Abaqus also use the Von Mises equivalent
stress to model isotropic behavior in the metal plasticity, which depends on the
deviatoric stress. In order to write Eq. (4.14) in terms of the deviatoric stress,
let us define the second deviatoric stress invariant, 𝐽2, in an alternative form
(Jirásek and Bazant, 2002):
𝐽2 =
1
2
𝑠ij 𝑠ij, (4.15)
where 𝑠ij is a deviatoric stress tensor, which is defined as the difference between
the stress tensor 𝜎ij and the hydrostatic stress tensor 𝑝 𝛿𝑖𝑗 (Müller, 2014a). The
hydrostatic stress tensor can be obtained by taking average of the three normal
stress components of any stress tensor. Therefore, the deviatoric stress tensor 𝑠ij
can be given as:
𝑠ij = 𝜎𝑖𝑗 − 𝑝 𝛿𝑖𝑗 = 𝜎ij −
1
3
𝜎 𝑘𝑘 𝛿ij. (4.16)
Substituting Eq. (4.16) in Eq. (4.15) gives us 𝐽2, which we substitute in Eq. (4.14)
to get Von Mises equivalent stress in term of deviatoric stress tensor:
𝜎Mises =
√︂
3
2
𝑠ij 𝑠ij . (4.17)
Von Mises yield criterion as a flow function can be written as:
𝑓 (𝜎ij, 𝜎Y) =
√︂
3
2
𝑠ij 𝑠ij − 𝜎Y. (4.18)

Here, 𝜎 𝑌 is the current yield stress. Von Mises yield criterion shows great
results for many ductile metals, like copper, nickel, aluminum, etc. This fact is
established by experiments done for these metals (Hill, 1986).
4.3. Flow rule
As we have seen that the total strain in an elasto-plastic model can be decomposed
in elastic strain, 𝜀e
ij, and plastic strain, 𝜀p
ij. All components of the elastic strain
can be determined by using the Hooke’s law. In oder to determine the plastic
strain we need to important concepts, i.e. yield conditions, which we have already
discussed, and flow rule. Flow rule is used to establish a relationship between
stress and plastic strain under multi-axial loading. Flow rule also defines the
direction of plastic flow.
In other words we can say that, the derivative of plastic potential 𝑔, which is a
function of stress, w.r.t. stress is equal to the plastic strain. Mathematically we
can write this as follows:
d𝜀p
ij = d𝜆
𝜕𝑔
𝜕𝜎ij
. (4.19)
The form in Eq. (4.19) is known as non-associated flow rule. Using non-associated
flow rule, we can model a wide class of material behavior, like soil, rock, etc.
But in this work, we consider only those sub-class of materials, whose plastic
potential can be taken as the yield function, i.e. 𝑔 = 𝑓. Substituting 𝑓 instead
of 𝑔 in Eq. (4.19), gives us a different form of flow rule, which is known as an
associated flow rule:
d𝜀p
ij = d𝜆
𝜕𝑓
𝜕𝜎ij
or ˙𝜀p
ij = ˙𝜆
𝜕𝑓
𝜕𝜎ij
. (4.20)
Here, ˙𝜆 is a non-negative hardening parameter that determine the magnitude of
the plastic strain. The direction of the plastic strain increment is given by the
following term:
𝜕𝑓
𝜕𝜎ij
, (4.21)
which is normal to the yield surface. This is the reason that an associative
flow rule is also known as normality hypothesis of plasticity. Next we apply the
associated flow rule for the Von Mises yield condition. In order to do that we
substituting the Von Mises function:
𝑓 = 𝐽2 −
𝜎2
Y
3
, (4.22)
Chapter 4. Constitutive models for elasto-plastic materials – Flow rule

in Eq. (4.20), which gives the following:
𝜕𝑓
𝜕𝜎ij
=
𝜕
𝜕𝜎ij
(︃
𝐽2 −
𝜎2
Y
3
)︃
. (4.23)
For a given material, 𝜎Y is constant, which means derivative of the term
𝜎2
Y
3 can
be considered to be zero. This consideration simplify Eq. (4.23):
𝜕𝑓
𝜕𝜎ij
=
𝜕𝐽2
𝜕𝜎ij
. (4.24)
Substituting Eq. (4.15) in Eq. (4.24) and then applying chain rule of differentiation
gives the following:
𝜕𝑓
𝜕𝜎ij
=
1
2
(︃
𝜕𝑠kl
𝜕𝜎ij
𝑠kl + 𝑠kl
𝜕𝑠kl
𝜕𝜎ij
)︃
= 𝑠kl
𝜕𝑠kl
𝜕𝜎ij
. (4.25)
Substitution of Eq. (4.16) in Eq. (4.25), gives:
𝜕𝑓
𝜕𝜎ij
= 𝑠kl
𝜕
𝜕𝜎ij
(︂
𝜎kl −
1
3
𝜎mm 𝛿kl
)︂
= 𝑠kl
𝜕𝜎kl
𝜕𝜎ij
−
1
3
𝑠kl
𝜕
𝜕𝜎ij
(𝜎mm 𝛿kl)
= 𝑠kl
(︃
𝜕𝜎kl
𝜕𝜎ij
−
1
3
𝜕𝜎mm
𝜕𝜎ij
𝛿kl
)︃
= 𝑠kl
(︂
𝛿il 𝛿kj −
1
3
𝛿im 𝛿jm 𝛿kl
)︂
= 𝑠ki 𝑠kj −
1
3
𝑠kk 𝛿ij
= 𝑠ij −
1
3
𝑠kk 𝛿ij. (4.26)
According to the definition of deviatoric stress tensor:
𝑠kk = 0. (4.27)
Substituting Eq. (4.27) in Eq. (4.26) gives us the partial derivative of the yield
function, 𝑓, w.r.t. the stress tensor, 𝜎ij, which is equal to the deviatoric stress
tensor, 𝑠ij:
𝜕𝑓
𝜕𝜎ij
= 𝑠ij. (4.28)
Substituting Eq. (4.28) in Eq. (4.20), gives the associated flow rule in terms of
the deviatoric stress tensor:
˙𝜀p
ij = ˙𝜆 𝑠ij. (4.29)
Chapter 4. Constitutive models for elasto-plastic materials – Flow rule

4.4. Consistency condition
The condition that the stress should remains on the yield surface, which is the
graphical representation of the yield function 𝑓(𝜎ij), during plastic flow is known
as consistency condition. In order to visualize this concept, let us consider a
material on which we apply an external load. Because of the applied load, the
material will first deform elastically and after reaching its yield point, it starts
showing plastic behavior. If we assume that the material shows plastic behavior
with no hardening then the material will deform further plastically under the
constant stress, i.e. the load point remains on the yield surface, see Fig. 4.2.
Mathematically it means that the yield function, 𝑓, remains equal to zero, see
Fig. 4.2 and as a result the time derivative of 𝑓 will vanish:
˙𝑓 = 0. (4.30)
The above equation is only valid for plastic yielding and can not be applied
during elastic deformation. Consistency condition is useful for computing the
plastic multiplier. If 𝜎 is the stress state and 𝜎Y is the current yield stress. Then,
Figure 4.2.: Graphical representation of the Von Mises yield surface for plane stress
and the corresponding stress-strain curve (Dunne and Petrinic, 2005).
according to Von Mises yield criterion we can write the following relation:
𝑓 (𝜎, 𝜎Y) =
1
2
𝑠ij 𝑠ij −
1
3
𝜎2
Y = 0. (4.31)
According to consistency condition:
˙𝑓 (𝜎, 𝜎Y) =
𝜕𝑓
𝜕𝜎ij
˙𝜎ij +
𝜕𝑓
𝜕𝜎Y
˙𝜎Y = 0. (4.32)
Chapter 4. Constitutive models for elasto-plastic materials – Consistency condition

From Eq. (4.31), we can calculate the partial differentiation of yield function
w.r.t. current yield stress, which is equal to:
𝜕𝑓
𝜕𝜎Y
= −
2
3
𝜎Y. (4.33)
Substituting Eq. (4.28) and Eq. (4.33) in Eq. (4.32), gives us the following of the
consistency condition:
˙𝑓 (𝜎, 𝜎Y) = 𝑠ij ˙𝜎ij −
2
3
𝜎Y ˙𝜎Y = 0. (4.34)
4.5. Hardening
In the previous sections we have discussed only about materials, whose yield
surfaces have not changed their size or position during loading process, i.e.
perfectly elasto-plastic materials. In practice, we can observe the changes in
elastic domains or yield surfaces during loading or unloading, due to the changes
in the microstructure of the material as plastic flows continue. In order to
understand these changes we need to modify the initial yield surface, which gives
rise to the concept of hardening. An increase of the yield stress during plastic
deformation is known as hardening. In this section we discuss mainly two types
of hardening, namely: isotropic hardening and kinematic hardening, respectively.
4.5.1. Isotropic hardening
A hardening process, in which the yield strength or expansion in the yield surface
in all directions are uniform, is called isotropic hardening, see Fig. 4.3. For
uniaxial loading, the isotropic hardening law, which is linear in nature, is given
as:
𝜎Y = 𝜎Y0 + 𝐻 𝜀p
11. (4.35)
Here, 𝐻 is known as plastic modulus. It is defined as the slope of the non-linear
part of the stress-strain curve, obtained through uniaxial tensile test. 𝜀p
11 is
the plastic part of the normal strain, 𝜀11. Whether the material will exhibit
true hardening, perfect plasticity or softening, depends on the value of plastic
modulus. If the plastic modulus is positive then we say that the yield stress will
increase. If 𝐻 = 0, then it corresponds to perfect plasticity and if the material
has a negative value of plastic modulus, then it shows softening character.
In order to extend Eq. (4.35) for a general multiaxial case, we need to introduce
definition of the cumulative plastic strain also known as effective plastic strain
or the equivalent plastic strain 𝜀p
eq. The rate equation of the equivalent plastic
Chapter 4. Constitutive models for elasto-plastic materials – Hardening

Figure 4.3.: Isotropic Hardening: Left: Uniform expansion of the yield surface in stress
space with plastic deformation. Right: Stress (𝜎) strain (𝜀) curve, representing increase
in the yield strength due to strain hardening (Dunne and Petrinic, 2005).
strain is given as (Müller, 2014a):
˙𝜀
p
=
√︂
2
3
˙𝜀p
𝑖𝑗 ˙𝜀p
𝑖𝑗 . (4.36)
The above equation was first proposed by Odqvist without the factor
√︁
2
3 .
integration of Eq. (4.36) w.r.t. time gives us the cumulative plastic strain:
𝜀p
eq =
ˆ √︂
2
3
˙𝜀p
𝑖𝑗 ˙𝜀p
𝑖𝑗 d𝑡. (4.37)
Substituting 𝜀p
eq instead of 𝜀p
11 in the linear isotropic hardening law we get the
hardening law for multiaxial load:
𝜎Y = 𝜎Y0 + 𝐻 𝜀p
eq. (4.38)
Derivative of current yield stress can be given as:
˙𝜎Y = 𝐻 ˙𝜀
p
. (4.39)
Substituting the derivative of the current yield stress in Eq. (4.34), gives us the
following relation:
˙𝜀
p
=
3
2
𝑠𝑖𝑗 ˙𝜎𝑖𝑗
𝐻 𝜎Y
. (4.40)
Applying the incremental form of the Hooke’s law as in Eq. (4.4), gives us an
Chapter 4. Constitutive models for elasto-plastic materials – Hardening

The theory of continuum and elasto plastic materials

The theory of continuum and elasto plastic materials

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