Lecture 21Energy principles and stability

Instructor: Prof. Marcial Gonzalez

Spring, 2015ME 612 – Continuum Mechanics

September 21, 2021

Lecture 21 – Energy principles and stability

Experimental mechanics and

thermodynamics

Tensor algebra Tensor analysis

reference configuration

thermo-mechanical loads

KINEMATICS OF DEFORMATIONS deformed configuration

CONTINUOUSMEDIA

atomic/micro/meso

structureis revealed

16 unknown fields + 5 equations

laws of nature .CONSERVATION OF MASS

BALANCE OF LINEAR MOMENTUMBALANCE OF ANGULAR MOMENTUM

LAWS OF THERMODYNAMICS CONSTITUTIVE EQUATIONS11 equations

Empirical observation

Multi-scaleapproaches

continuously varying fields(time and space averages over the underlying structure)

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Static equilibrium – Boundary value problem- Conservation of linear momentum

reference configuration

deformed configuration

CONTINUOUSMEDIA

Note: conservative body forces and dead loads.

DIY

Lecture 21 – Energy principles and stability

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Static equilibrium – Boundary value problem- Conservation of linear momentum

- Airy stress function: plain stress, Cartesian coordinate system

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Boundary value problem- Airy stress function: plain stress, polar coordinate system

General solution:

Lecture 21 – Energy principles and stability

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Static equilibrium – Variational formulation- Total potential energy (conservative body forces and dead loads)

- An admissible deformation mapping is such that

- Principle of stationary potential energy:Given the set of admissible displacement fields for a conservative

system, an equilibrium state will correspond to one for which the potential energy is stationary (or to a minimum for stable solutions).

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Variational formulation- Total potential energy (conservative body forces and dead loads)

- An admissible deformation mapping is such that

- Principle of minimum potential energy – Variational principle

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Variational formulation- Total potential energy (conservative body forces and dead loads)

- An admissible deformation mapping is such that

- Principle of stationary potential energy – Variational principle

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Variational formulation- Principle of stationary potential energy – Variational principle

Proof

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Variational formulation- Principle of stationary potential energy – Variational principle

Veinberg’s theorem:

A BVP is variational iff and iff

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Stability- Principle of stationary potential energy:

Given the set of admissible displacement fields for a conservative system, an equilibrium state will correspond to one for which the potential energy is stationary

- Stable solution? A stationary point could be a minimum, a maximum or a saddle point.

- Theory of stability of infinite-dimensional spaces is not trivial …

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Stability- Stable solution?

A stationary point could be a minimum, a maximum or a saddle point.

- Under the assumptions presented before, stability of an equilibrium configuration can be determined from the stability of the linearized problem about the equilibrium configuration.

- Ultimately, stability can be inferred from the strain energy density!

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Stability- Stable solution?

A stationary point could be a minimum, a maximum or a saddle point.- Stability can be inferred from the strain energy density:

+ Necessary condition (but not sufficient) The mixed elastic tensor has to be a rank-one convex tensor, that is the strain energy density has to be rank-one convex.

+ Sufficient conditions come from nonlinear terms and from boundarycondition.

reference configuration

deformed configuration

CONTINUOUSMEDIA

Lecture 21 – Energy principles and stability

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Static equilibrium – Stability- Examples of rank-one convexity

neo-Hookean material

Saint Venant-Kirchhoff material

Counterexample

reference configuration

deformed configuration

CONTINUOUSMEDIA

DIYcompression

Lecture 21 – Energy principles and stability

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Load-controlled vs. displacement-controlled experiments

Saint Venant-Kirchhoff material

Blatz-Ko material

DIY

Lecture 21 – Energy principles and stability

Any questions?

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Lecture 21 – Energy principles and stability