28
2 Preliminaries
(v) either smooth or non-smooth.
7
This choice characterizes a Standard Dissipative Material.
To simplify the notation we henceforth assume smoothness in this overview. The
more general non-smooth situation in which partial derivatives have to be substituted
by sub-differentials is left to the chapters on Plasticity and Visco-Plasticity.
8 For the
dissipative stress and the set of dissipative driving forces we then propose
σ
=
∂π
∂ ˙
and a
=
∂π
∂ ˙
α
(2.14)
whereby the dissipation inequality in Eq. 2.12 becomes
d =
∂π
∂ ˙
˙
+
∂π
∂ ˙
α
◦ ˙
α ≥ 0
(2.15)
and thus d ≥ 0 is trivially satisfied due to the (i) positivity and (ii) convexity of π
together with (iii) its ‘zero-at-zero’ property π(0, 0) = 0.
Summarizing, we have the constitutive relations
σ = σ
((, α) + σ
(˙ , ˙
α),
(2.16a)
0 = a
((, α) + a
(˙ , ˙
α).
(2.16b)
Equation 2.16b is commonly denoted Biot’s equation. When subjected to the initial
condition α(t = 0) = 0, the constitutive evolution equations can then be solved for
two principally different (loading) scenarios:
Strain Control: (t) is prescribed, whereby α(t) is solved from Eq. 2.16b. σ (t)
is then computed from Eq. 2.16a in a post-processing step.
Stress Control: σ (t) is prescribed, whereby (t) and α(t) are solved from the
coupled Eqs. 2.16a and 2.16b.
Sometimes it is convenient to express the constitutive relations in terms of the dual
dissipation potential
7 Note that π being either smooth or non-smooth is a property that has profound consequences for
the resulting model characteristics. The model classes that are discussed in the subsequent chapters
of this treatise possess the following characteristics for π:
Elasticity
: π ≡ 0,
Visco-Elasticity : π is smooth and positive homogeneous of degree > 1,
Plasticity
: π is non-smooth at (0, 0) and positive homogeneous of degree = 1,
Visco-Plasticity : π is non-smooth at (0, 0) and positive homogeneous of degree > 1.
8 To be more careful, we should write π = π(˙ , ˙
α; , α) since π can, indeed, depend on the state.
An example is the Prandtl hardening model. However, since , α merely play the role of parameters
of π in its role as dissipation potential, we prefer to suppress them as arguments henceforth.
2 Preliminaries
(v) either smooth or non-smooth.
7
This choice characterizes a Standard Dissipative Material.
To simplify the notation we henceforth assume smoothness in this overview. The
more general non-smooth situation in which partial derivatives have to be substituted
by sub-differentials is left to the chapters on Plasticity and Visco-Plasticity.
8 For the
dissipative stress and the set of dissipative driving forces we then propose
σ
=
∂π
∂ ˙
and a
=
∂π
∂ ˙
α
(2.14)
whereby the dissipation inequality in Eq. 2.12 becomes
d =
∂π
∂ ˙
˙
+
∂π
∂ ˙
α
◦ ˙
α ≥ 0
(2.15)
and thus d ≥ 0 is trivially satisfied due to the (i) positivity and (ii) convexity of π
together with (iii) its ‘zero-at-zero’ property π(0, 0) = 0.
Summarizing, we have the constitutive relations
σ = σ
((, α) + σ
(˙ , ˙
α),
(2.16a)
0 = a
((, α) + a
(˙ , ˙
α).
(2.16b)
Equation 2.16b is commonly denoted Biot’s equation. When subjected to the initial
condition α(t = 0) = 0, the constitutive evolution equations can then be solved for
two principally different (loading) scenarios:
Strain Control: (t) is prescribed, whereby α(t) is solved from Eq. 2.16b. σ (t)
is then computed from Eq. 2.16a in a post-processing step.
Stress Control: σ (t) is prescribed, whereby (t) and α(t) are solved from the
coupled Eqs. 2.16a and 2.16b.
Sometimes it is convenient to express the constitutive relations in terms of the dual
dissipation potential
7 Note that π being either smooth or non-smooth is a property that has profound consequences for
the resulting model characteristics. The model classes that are discussed in the subsequent chapters
of this treatise possess the following characteristics for π:
Elasticity
: π ≡ 0,
Visco-Elasticity : π is smooth and positive homogeneous of degree > 1,
Plasticity
: π is non-smooth at (0, 0) and positive homogeneous of degree = 1,
Visco-Plasticity : π is non-smooth at (0, 0) and positive homogeneous of degree > 1.
8 To be more careful, we should write π = π(˙ , ˙
α; , α) since π can, indeed, depend on the state.
An example is the Prandtl hardening model. However, since , α merely play the role of parameters
of π in its role as dissipation potential, we prefer to suppress them as arguments henceforth.