Clausius-Duhem inequality

The Clausius-Duhem inequality is the form of the second law of thermodynamics used in continuum mechanics .

The mathematical formulation of this law - like all other physical laws - does not make any statements about the individual properties of bodies . In order to determine the thermodynamic behavior of a special body, a material model is required that reflects its material- specific behavior.

The Clausius-Duhem inequality is not to be interpreted as a restriction for physical processes, but rather as a requirement for the constitutive equations of a material model: It must be ensured that the Clausius-Duhem inequality is fulfilled by the material equations for any processes. This then often results in value ranges in which the material parameters of a model must lie. For example, in the ideal plasticity in the case study, it follows that the Lamé constants are positive.

Entropy balance

The entropy balance describes how the entropy of a body changes due to external influences. If s is the specific entropy , the entropy flux per area, the specific entropy supply and the specific entropy production , then the entropy balance is: ${\displaystyle {\vec {\varsigma }}}$ ${\displaystyle \sigma }$ ${\displaystyle \gamma }$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}s\,\rho \mathrm {d} v=\int _{v}\sigma \,\rho \mathrm {d} v+\int _{v}\gamma \,\rho \mathrm {d} v-\int _{a}{\vec {\varsigma }}\cdot {\vec {n}}\,\mathrm {d} a}$.

In this equation, v is the volume occupied by the body, a is the surface of the body, the normal directed outwards on the surface element of the body , and d / dt is the derivative with respect to time (change over time). The negative sign of the last term provides an entropy supply when the entropy flow is directed into the body. ${\displaystyle {\vec {n}}}$ ${\displaystyle \mathrm {d} a}$

The second law of thermodynamics

→ Main article : Second law of thermodynamics

The second law of thermodynamics expresses the experience that mechanical work can be completely converted into heat, but the conversion of heat into mechanical energy is only partially successful. The dissipation of mechanical work into heat is accompanied by a production of entropy, which must therefore not be negative:

${\displaystyle \int _{v}\gamma \,\rho \mathrm {d} v={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}s\,\rho \mathrm {d} v-\int _{v}\sigma \,\rho \mathrm {d} v+\int _{a}{\vec {\varsigma }}\cdot {\vec {n}}\,\mathrm {d} a\geq 0}$,

This equation is also called the dissipation inequality .

Clausius-Duhem inequality

From the equilibrium thermodynamics of homogeneous systems it is known that the entropy flow is the quotient of the heat flow and the absolute temperature T and the same relationship is postulated between the specific heat production r and the entropy production: ${\displaystyle {\vec {q}}}$

${\displaystyle {\vec {\varsigma }}={\frac {\vec {q}}{T}}\quad {\textsf {und}}\quad \sigma ={\frac {r}{T}}}$.

With these assumptions, the global formulation of the Clausius-Duhem inequality is derived from the second law of thermodynamics:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}s\,\rho \mathrm {d} v\geq \int _{v}{\frac {r}{T}}\,\rho \mathrm {d} v-\int _{a}{\frac {\vec {q}}{T}}\cdot {\vec {n}}\,\mathrm {d} a}$

Utilizing the divergence theorem , the product rule and the local mass and energy balance provides the local formulation

${\displaystyle {\frac {1}{\rho }}{\boldsymbol {\sigma }}:\mathbf {d} -{\dot {\psi }}-s{\dot {T}}-{\frac {\vec {q}}{\rho T}}\cdot \operatorname {grad} (T)\geq 0}$

proof

The time derivative of the volume integral can be transformed with the divergence theorem and the product rule : ${\displaystyle {\begin{array}{rcl}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}s\,\rho \mathrm {d} v&=&\int _{v}{\frac {\partial (s\,\rho )}{\partial t}}\mathrm {d} v+\int _{a}s\,\rho ({\vec {v}}\cdot {\vec {n}})\mathrm {d} a=\int _{v}\left({\frac {\partial s}{\partial t}}\rho +s{\frac {\partial \rho }{\partial t}}+\operatorname {div} (s\,\rho \,{\vec {v}})\right)\mathrm {d} v\\&=&\int _{v}\left({\frac {\partial s}{\partial t}}\rho +s{\frac {\partial \rho }{\partial t}}+\rho \,\operatorname {grad} (s)\cdot {\vec {v}}+s\,\operatorname {grad} (\rho )\cdot {\vec {v}}+s\,\rho \,\operatorname {div} ({\vec {v}})\right)\mathrm {d} v\\&=&\int _{v}\left({\frac {\partial s}{\partial t}}\rho +\rho \,\operatorname {grad} (s)\cdot {\vec {v}}\right)\mathrm {d} v=\int _{v}{\dot {s}}\,\rho \mathrm {d} v\end{array}}}$ The other terms vanish because of the constancy of mass that applies to any volume ${\displaystyle {\begin{array}{rcl}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho \mathrm {d} v&=&\int _{v}\left({\frac {\partial \rho }{\partial t}}+\operatorname {div} (\rho \,{\vec {v}})\right)\mathrm {d} v=\int _{v}\left({\frac {\partial \rho }{\partial t}}+\operatorname {grad} (\rho )\cdot {\vec {v}}+\rho \,\operatorname {div} ({\vec {v}})\right)\mathrm {d} v=0\\\rightarrow 0&=&{\frac {\partial \rho }{\partial t}}+\operatorname {grad} (\rho )\cdot {\vec {v}}+\rho \,\operatorname {div} ({\vec {v}})\end{array}}}$ The transport term can also be transformed with the divergence theorem and the product rule: The results so far provide in summary: because the inequality applies to any partial volume. Inserting the local internal energy balance leads to . With Helmholtz's free energy ${\displaystyle \int _{a}{\frac {\vec {q}}{T}}\cdot {\vec {n}}\,\mathrm {d} a=\int _{v}\operatorname {div} \left({\frac {\vec {q}}{T}}\right)\mathrm {d} v=\int _{v}\left({\frac {\operatorname {div} ({\vec {q}})}{T}}-{\frac {\operatorname {grad} (T)\cdot {\vec {q}}}{T^{2}}}\right)\mathrm {d} v}$ ${\displaystyle \int _{v}\left({\dot {s}}-{\frac {r}{T}}-{\frac {\operatorname {grad} (T)\cdot {\vec {q}}}{\rho T^{2}}}+{\frac {\operatorname {div} ({\vec {q}})}{\rho T}}\right)\rho \mathrm {d} v\geq 0\quad \rightarrow \quad {\dot {s}}-{\frac {\operatorname {grad} (T)\cdot {\vec {q}}}{\rho T^{2}}}-{\frac {1}{T}}\left(r-{\frac {\operatorname {div} ({\vec {q}})}{\rho }}\right)\geq 0}$ ${\displaystyle {\dot {u}}={\frac {1}{\rho }}{\boldsymbol {\sigma }}:\mathbf {d} +r-{\frac {\operatorname {div} ({\vec {q}})}{\rho }}\quad \rightarrow \quad r-{\frac {\operatorname {div} ({\vec {q}})}{\rho }}={\dot {u}}-{\frac {1}{\rho }}{\boldsymbol {\sigma }}:\mathbf {d} }$ ${\displaystyle {\dot {s}}-{\frac {\operatorname {grad} (T)\cdot {\vec {q}}}{\rho T^{2}}}-{\frac {1}{T}}\left({\dot {u}}-{\frac {1}{\rho }}{\boldsymbol {\sigma }}:\mathbf {d} \right)\geq 0\quad \rightarrow \quad {\frac {1}{\rho }}{\boldsymbol {\sigma }}:\mathbf {d} -{\dot {u}}+T{\dot {s}}-{\frac {\vec {q}}{\rho T}}\cdot \operatorname {grad} (T)\geq 0}$ ${\displaystyle \psi =u-sT\quad \rightarrow \quad {\dot {\psi }}+s{\dot {T}}={\dot {u}}-{\dot {s}}T}$ the end result arises: ${\displaystyle {\frac {1}{\rho }}{\boldsymbol {\sigma }}:\mathbf {d} -{\dot {\psi }}-s{\dot {T}}-{\frac {\vec {q}}{\rho T}}\cdot \operatorname {grad} (T)\geq 0}$

It is the Cauchy stress tensor , d the warp speed and "" the Frobenius inner product . This scalar product provides the power of the stresses along a distortion path during the deformation of a body. Furthermore, the Helmholtz free energy and grad (T) is the gradient of the temperature (a vector with the dimension temperature per length which points in the direction of the strongest temperature rise). ${\displaystyle {\boldsymbol {\sigma }}}$ ${\displaystyle \psi }$

In the important special case in which temperature changes can be neglected, this local form is simplified to:

${\displaystyle {\frac {1}{\rho }}{\boldsymbol {\sigma }}:\mathbf {d} -{\dot {\psi }}\geq 0}$

The specific voltage output must therefore always be greater than the production of free energy. The excess is dissipated. The local form is to be interpreted less as a restriction of physical processes, but rather as a requirement for material models: It must be ensured that the local form of the Clausius-Duhem inequality is fulfilled by the material equations for any processes. One application shows the #example isothermal ideal plasticity below.

These formulas, derived from Euler's approach , are in the Lagrangian version :

Global form:  ${\displaystyle \displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}s_{0}\,\rho _{0}\mathrm {d} V\geq \int _{V}{\frac {r_{0}}{T_{0}}}\,\rho _{0}\mathrm {d} V-\int _{A}{\frac {{\vec {q}}_{0}}{T_{0}}}\cdot {\vec {N}}\,\mathrm {d} A}$

Local form:  ${\displaystyle {\dfrac {1}{\rho _{0}}}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}-{\dot {\psi }}_{0}-s_{0}{\dot {T}}_{0}-{\dfrac {{\vec {q}}_{0}}{\rho _{0}T_{0}}}\cdot \operatorname {GRAD} (T_{0})\geq 0}$

Isothermal process:  ${\displaystyle {\dfrac {1}{\rho _{0}}}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}-{\dot {\psi }}_{0}\geq 0\,.}$

The quantities indicated with zero are the quantities expressed with the material coordinates, the second Piola-Kirchhoff stress tensor , the material rate of distortion, DEGREES is the gradient with respect to the coordinates of the particles of the body in the undeformed initial state ( material coordinates) and is the undeformed initial state normals directed outward on the surface of the body. ${\displaystyle {\tilde {\mathbf {T} }}}$ ${\displaystyle {\dot {\mathbf {E} }}}$ ${\displaystyle {\vec {N}}}$

Example isothermal ideal plasticity

Based on the isothermal ideal plasticity in the case of small deformations, the aim is to show how far the Clausius-Duhem inequality helps to formulate thermodynamically consistent material equations.

With ideal plasticity, there is no solidification during plastic flow; H. the stress-strain curve has a horizontal course in the uniaxial flow in the tensile test . Dough is roughly ideally plastic. In practice, this model is used when only the yield point is known and when calculating a component, its stiffness should not be overestimated under any circumstances.

The constitutive variable is the total strain ε and the material size is the stress tensor σ . With small deformations

${\displaystyle {\tilde {\mathbf {T} }}={\boldsymbol {\sigma }}\;,\quad \mathbf {E} ={\boldsymbol {\varepsilon }}\quad {\textsf {und}}\quad {\dot {\mathbf {E} }}=\mathbf {d} ={\dot {\boldsymbol {\varepsilon }}}}$

The material has an elastic area in which the material reacts elastically and a plastic area where plastic flow takes place. The flow is represented by the plastic strain ε _p , which is an internal variable of the model. The plastic strain cannot be influenced or specified directly from the outside. The difference between the total elongation and the plastic elongation is the elastic elongation ε _e , which alone determines the stresses. The total strain is divided into an elastic and a plastic part:

${\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{e}+{\boldsymbol {\varepsilon }}_{p}}$

The flow function separates the elastic from the plastic area: ${\displaystyle f}$

${\displaystyle f={\dfrac {3}{2}}{\boldsymbol {\sigma }}^{\mathrm {D} }:{\boldsymbol {\sigma }}^{\mathrm {D} }-{k}^{2}\leq 0}$ .

The stress deviator σ ^D and the yield point k , which is a material parameter, appear here. In the elastic range f <0 and . With plastic flow, f = 0, and therefore ${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{p}=\mathbf {0} }$ ${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{e}=\mathbf {0} }$

${\displaystyle {\sqrt {\dfrac {3}{2}}}{\begin{Vmatrix}{\boldsymbol {\sigma }}^{\mathrm {D} }\end{Vmatrix}}=k=\mathrm {const.} }$,

which is the special feature of ideal plasticity. On the left side of the equation is the von Mises equivalent stress .

The Helmholtz free energy should only depend on the elastic strains:

${\displaystyle \psi =\psi ({\boldsymbol {\varepsilon }}_{e})}$.

These prerequisites are sufficient to establish a rough framework for the plasticity model.

The Clausius-Duhem inequality gives:

${\displaystyle {\frac {1}{\rho }}{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}-{\dot {\psi }}={\frac {1}{\rho }}{\boldsymbol {\sigma }}:({\dot {\boldsymbol {\varepsilon }}}_{e}+{\dot {\boldsymbol {\varepsilon }}}_{p})-{\frac {\mathrm {d} \psi }{\mathrm {d} {\boldsymbol {\varepsilon }}_{e}}}:{\dot {\boldsymbol {\varepsilon }}}_{e}=\left({\frac {1}{\rho }}{\boldsymbol {\sigma }}-{\frac {\mathrm {d} \psi }{\mathrm {d} {\boldsymbol {\varepsilon }}_{e}}}\right):{\dot {\boldsymbol {\varepsilon }}}_{e}+{\frac {1}{\rho }}{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}_{p}\geq 0}$.

This inequality must be true for all possible processes. In the elastic area ( ) this can be achieved by choosing a hyperelastic stress-strain relationship: ${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{p}=\mathbf {0} }$

${\displaystyle {\frac {1}{\rho }}{\boldsymbol {\sigma }}={\frac {\mathrm {d} \psi }{\mathrm {d} {\boldsymbol {\varepsilon }}_{e}}}}$

In the plastic area ( ) , what can be achieved with an associated flow rule must apply to all processes : ${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{e}=\mathbf {0} }$ ${\displaystyle {\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}_{p}\geq 0}$

${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{p}={\dot {\gamma }}{\frac {\mathrm {d} f}{\mathrm {d} {\boldsymbol {\sigma }}}}=3{\dot {\gamma }}{\boldsymbol {\sigma }}^{\mathrm {D} }}$

The proportionality factor is the plastic multiplier . ${\displaystyle {\dot {\gamma }}}$

With plastic flow, the elastic strains remain constant, which is why the following applies. Because the plastic strain rate is deviatorial , the trace of the strain rate, which is equal to the local change in volume, disappears . For this reason the density is constant in plastic flow: ${\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{p}}$ ${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{p}}$

${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{p}\neq \mathbf {0} \quad \rightarrow \quad {\dot {\rho }}=0}$.

According to the Clausius-Duhem inequality, the power of the stresses at the plastic strains must not be negative:

${\displaystyle {\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}_{p}=3{\dot {\gamma }}{\boldsymbol {\sigma }}:{\boldsymbol {\sigma }}^{\mathrm {D} }=3{\dot {\gamma }}{\boldsymbol {\sigma }}^{\mathrm {D} }:{\boldsymbol {\sigma }}^{\mathrm {D} }\geq 0\quad \rightarrow \quad {\dot {\gamma }}\geq 0}$

which is why the plastic multiplier must not be negative. It is calculated from the consistency condition

${\displaystyle {\begin{aligned}{\dot {f}}=0=&3{\boldsymbol {\sigma }}^{\mathrm {D} }:{\dot {\boldsymbol {\sigma }}}^{\mathrm {D} }=3{\boldsymbol {\sigma }}^{\mathrm {D} }:{\dot {\boldsymbol {\sigma }}}=3{\boldsymbol {\sigma }}^{\mathrm {D} }:{\frac {\mathrm {d} }{\mathrm {d} t}}\left(\rho {\frac {\mathrm {d} \psi }{\mathrm {d} {\boldsymbol {\varepsilon }}_{e}}}\right)=3{\boldsymbol {\sigma }}^{\mathrm {D} }:\rho {\frac {\mathrm {d} ^{2}\psi }{\mathrm {d} {\boldsymbol {\varepsilon }}_{e}^{2}}}:{\dot {\boldsymbol {\varepsilon }}}_{e}\\=&3{\boldsymbol {\sigma }}^{\mathrm {D} }:\mathbb {C} :({\dot {\boldsymbol {\varepsilon }}}-{\dot {\boldsymbol {\varepsilon }}}_{p})=3{\boldsymbol {\sigma }}^{\mathrm {D} }:\mathbb {C} :({\dot {\boldsymbol {\varepsilon }}}-3{\dot {\gamma }}{\boldsymbol {\sigma }}^{\mathrm {D} })\\\rightarrow {\dot {\gamma }}=&{\frac {{\boldsymbol {\sigma }}^{\mathrm {D} }:\mathbb {C} :{\dot {\boldsymbol {\varepsilon }}}}{3{\boldsymbol {\sigma }}^{\mathrm {D} }:\mathbb {C} :{\boldsymbol {\sigma }}^{\mathrm {D} }}}\end{aligned}}}$

This is where the elasticity tensor is . The plastic multiplier is definitely positive if the numerator and denominator in the fraction on the right-hand side of its determining equation are positive: ${\displaystyle \mathbb {C} =\rho {\tfrac {\mathrm {d} ^{2}\psi }{\mathrm {d} {\boldsymbol {\varepsilon }}_{e}^{2}}}}$

1. First of all, it must be demanded that the stresses do not disappear during plastic flow, i.e. that the flow limit k is positive.
2. The numerator of the fraction contains the loading condition , i. H. plastic flow should only start when this applies. At and there is a neutral load, with no plastic flow. In Hooke's law, In this equation, I is the second and fourth order unit tensor and are the first and second Lamé constants . A positive shear modulus would also have to be required here. ${\displaystyle {\boldsymbol {\sigma }}^{\mathrm {D} }:\mathbb {C} :{\dot {\boldsymbol {\varepsilon }}}>0}$ ${\displaystyle f=0}$ ${\displaystyle {\boldsymbol {\sigma }}^{\mathrm {D} }:\mathbb {C} :{\dot {\boldsymbol {\varepsilon }}}=0}$
   ${\displaystyle \mathbb {C} =2\mu \mathbb {I} +\lambda \mathbf {I\otimes I} \quad \rightarrow \quad {\boldsymbol {\sigma }}^{\mathrm {D} }:\mathbb {C} :{\dot {\boldsymbol {\varepsilon }}}=2\mu {\boldsymbol {\sigma }}^{\mathrm {D} }:{\dot {\boldsymbol {\varepsilon }}}}$
   ${\displaystyle \mathbb {I} }$ ${\displaystyle \mu ,\lambda }$ ${\displaystyle \mu }$
3. The denominator is always positive if the elasticity tensor is positive definite . In the case of Hooke's law , this condition is met if the Lamé constants are positive.

The table again lists all the properties of the model derived from the Clausius-Duhem inequality:

property	condition	formula
Stress-strain relationship	Hyperelasticity	${\displaystyle {\frac {1}{\rho }}{\boldsymbol {\sigma }}={\frac {\mathrm {d} \psi }{\mathrm {d} {\boldsymbol {\varepsilon }}_{e}}}}$
Elasticity tensor	positive definitely	${\displaystyle {\boldsymbol {\sigma }}:\mathbb {C} :{\boldsymbol {\sigma }}>0\quad \forall {\boldsymbol {\sigma }}\neq \mathbf {0} }$
Yield point	positive	${\displaystyle k>0}$
Flow rule	Performance of the stresses at the plastic strain rate is not negative	${\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{p}=3{\dot {\gamma }}{\boldsymbol {\sigma }}^{\mathrm {D} }\;\wedge \;{\dot {\gamma }}\geq 0}$
Plastic flow	Loading condition is met	${\displaystyle f=0}$ and ${\displaystyle {\boldsymbol {\sigma }}^{\mathrm {D} }:\mathbb {C} :{\dot {\boldsymbol {\varepsilon }}}>0}$

When using Hooke's law - as announced at the beginning -

${\displaystyle \mu ,\lambda >0}$

to promote. The loading condition is reduced to

${\displaystyle f=0\quad {\textsf {und}}\quad {\boldsymbol {\sigma }}^{\mathrm {D} }:{\dot {\boldsymbol {\varepsilon }}}>0}$.

Under these conditions, the thermodynamic consistency of ideal plasticity is ensured in isothermal processes.

Footnotes

1. ↑ ^{a b} The Fréchet derivative of a function according to is the restricted linear operator which - if it exists - corresponds to the Gâteaux differential in all directions, i.e. ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle h}$

   ${\displaystyle {\mathcal {A}}(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(x+sh)\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(x+sh)-f(x)}{s}}\quad \forall \;h}$

   applies. In it is scalar, vector or tensor valued but and similar. Then will too ${\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h}$ ${\displaystyle x}$ ${\displaystyle h}$

   ${\displaystyle {\mathcal {A}}={\frac {\partial f}{\partial x}}}$

   written.

literature

• Holm Altenbach: Continuum Mechanics. Introduction to the material-independent and material-dependent equations. 2nd Edition. Springer Vieweg, Berlin et al. 2012, ISBN 978-3-642-24118-5 .
• Peter Haupt: Continuum Mechanics and Theory of Materials. Springer, Berlin et al. 2000, ISBN 3-540-66114-X .