Chapter 7
Unified Micromechanics of Finite
Deformations
7.1 Introduction to Finite Deformations
In this chapter finite (large) deformation micromechanics is discussed. Formulation
is more complicated than small deformation theory; therefore, it is easier to explain
with an example. Micromechanical modeling of polymer is used as an example. A
dual-micro-mechanism rate-dependent constitutive model is used to describe
thermo-mechanical response of amorphous polymers below and above glass transition temperature. Material property definitions, evolution of internal state variables,
and plastic flow rules are revisited to provide a smooth and continuous transition in
material response around glass transition temperature, θ g .
In order to formulate the large deformation mechanics, it is necessary to describe
kinematics of constitutive model based on finite deformation tensors. Consider a
body with a volume of V o in undeformed (original or initial) configuration (Σ o ) at
time t o which deforms into a volume V in current (deformed) configuration (Σ) at
time t, as shown in Fig. 7.1, this body can be uniquely defined with a continuous oneto-one mapping (χ) of position vectors r and x in original configuration and current
configuration, respectively. r is the material coordinate of a particle that is the
coordinates of its location in the reference coordinate system. x is the spatial
coordinates, defining location, at time t. Both material coordinates and spatial
coordinates were discussed earlier in Chap. 2.
Gradient of deformation (F), which is the strain, describes transformation of a line
element (dr) at position of r in the original configuration to a deformed line element
(dx) at position of x in current configuration Eq. (7.1). Unique transformation
ensures a non-singular, nonnegative determinant of deformation gradient or Jacobian
(J) Eq. (7.3). Velocity gradient (L) is the gradient of velocity field (v) and relates
deformation gradient to its material time derivative _
F
À Á
. These relations can be given
as follows,
© Springer Nature Switzerland AG 2021
C. Basaran, Introduction to Unified Mechanics Theory with Applications,
https://doi.org/10.1007/978-3-030-57772-8_7
343
Unified Micromechanics of Finite
Deformations
7.1 Introduction to Finite Deformations
In this chapter finite (large) deformation micromechanics is discussed. Formulation
is more complicated than small deformation theory; therefore, it is easier to explain
with an example. Micromechanical modeling of polymer is used as an example. A
dual-micro-mechanism rate-dependent constitutive model is used to describe
thermo-mechanical response of amorphous polymers below and above glass transition temperature. Material property definitions, evolution of internal state variables,
and plastic flow rules are revisited to provide a smooth and continuous transition in
material response around glass transition temperature, θ g .
In order to formulate the large deformation mechanics, it is necessary to describe
kinematics of constitutive model based on finite deformation tensors. Consider a
body with a volume of V o in undeformed (original or initial) configuration (Σ o ) at
time t o which deforms into a volume V in current (deformed) configuration (Σ) at
time t, as shown in Fig. 7.1, this body can be uniquely defined with a continuous oneto-one mapping (χ) of position vectors r and x in original configuration and current
configuration, respectively. r is the material coordinate of a particle that is the
coordinates of its location in the reference coordinate system. x is the spatial
coordinates, defining location, at time t. Both material coordinates and spatial
coordinates were discussed earlier in Chap. 2.
Gradient of deformation (F), which is the strain, describes transformation of a line
element (dr) at position of r in the original configuration to a deformed line element
(dx) at position of x in current configuration Eq. (7.1). Unique transformation
ensures a non-singular, nonnegative determinant of deformation gradient or Jacobian
(J) Eq. (7.3). Velocity gradient (L) is the gradient of velocity field (v) and relates
deformation gradient to its material time derivative _
F
À Á
. These relations can be given
as follows,
© Springer Nature Switzerland AG 2021
C. Basaran, Introduction to Unified Mechanics Theory with Applications,
https://doi.org/10.1007/978-3-030-57772-8_7
343