m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Continuum mechanics: motion, stress and conservation

Build the common language of solid and fluid models from deformation maps through conservation laws and constitutive closure.

Subject library · 51 guides · derivations & worked examples

Fundamental physics

Space and flight

Materials and transport

Systems and mathematics

Marine and road vehicles

Electrical and computing

Companion site

Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Motion and finite deformation

Definitions & inputs. X reference position,x current position,F deformation gradient,J volume ratio.

  1. Differentiate the motion with respect to the reference coordinates.

    x=χ(X,t),FiJ=∂xi/∂XJ\mathbf x=\boldsymbol\chi(\mathbf X,t),\quad F_{iJ}=\partial x_i/\partial X_J

  2. F changes material line elements and its determinant changes volumes.

    dx=F dX,dv=J dV,J=det⁡Fd\mathbf x=F\,d\mathbf X,\quad dv=J\,dV,\quad J=\det F

  3. Green strain measures squared length change and is invariant under superposed rigid rotation.

    E=12(FTF−I)E=\tfrac12(F^{\mathsf T}F-I)

Interpretation. For small gradients use ε=sym∇u, but a finite rigid rotation must not create spurious strain.

↑ Return to definitions and contents

2. Traction and stress measures

Definitions & inputs. σ Cauchy stress,n current unit normal,P first Piola stress,ρ density.

  1. Cauchy traction maps a cut-plane normal into force per current area.

    t(n)=σn\mathbf t(\mathbf n)=\sigma\mathbf n

  2. Transform current-area forces to reference-area forces using Nanson’s relation.

    P=JσF−TP=J\sigma F^{-\mathsf T}

  3. Linear momentum gives the stress divergence; angular momentum gives symmetry under the stated assumptions.

    ρv˙=∇⋅σ+ρb,σ=σT\rho\dot{\mathbf v}=\nabla\cdot\sigma+\rho\mathbf b,\quad\sigma=\sigma^{\mathsf T}

Interpretation. Keep reference and current configurations consistent when computing stress or boundary work.

↑ Return to definitions and contents

3. Mass and internal energy

Definitions & inputs. e specific internal energy,q heat flux,r volumetric heat supply per mass,L velocity gradient,D=symL.

  1. Material mass conservation gives both finite and differential statements.

    ρJ=ρ0,ρ˙+ρ∇⋅v=0\rho J=\rho_0,\quad\dot\rho+\rho\nabla\cdot\mathbf v=0

  2. Stress power, heat flow and heat generation change internal energy.

    ρe˙=σ:D−∇⋅q+ρr\rho\dot e=\sigma:D-\nabla\cdot\mathbf q+\rho r

  3. Fourier closure supplies a thermal constitutive law when local diffusion is appropriate.

    q=−k∇T\mathbf q=-k\nabla T

Interpretation. Conservation equations alone do not select elasticity, viscosity or plasticity.

↑ Return to definitions and contents

4. Constitutive closure and model hierarchy

Definitions & inputs. λ,G Lamé constants,K bulk modulus,μ dynamic viscosity.

  1. Linear isotropic elasticity connects stress to strain.

    σ=λtr⁡(ϵ)I+2Gϵ\sigma=\lambda\operatorname{tr}(\epsilon)I+2G\epsilon

  2. Convert between common elastic coefficients.

    K=λ+2G/3,G=E/[2(1+ν)]K=\lambda+2G/3,\quad G=E/[2(1+\nu)]

  3. In an incompressible Newtonian fluid stress depends on strain rate, with pressure enforcing incompressibility.

    σ=−pI+2μD,∇⋅v=0\sigma=-pI+2\mu D,\quad\nabla\cdot\mathbf v=0

Interpretation. FEM often discretizes solid weak forms; finite volume methods enforce fluid conservation. Objectivity, positivity of dissipation and boundary conditions must accompany either choice.

↑ Return to definitions and contents

Graphical worked example

Green strain E11=(λ²−1)/2 for uniaxial stretch; this is a strain measure, not a stress law. X axis: Axial stretch (dimensionless). Y axis: Green axial strain (dimensionless).
Green strain E11=(λ²−1)/2 for uniaxial stretch; this is a strain measure, not a stress law. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Axial stretch

Definitions & inputs. Length1m becomes1.02m.

  1. Choose the governing model and isolate the requested quantity.

    λ=L/L0\lambda=L/L_0

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.02/11.02/1

  3. Evaluate the expression; the result uses the units shown.

    Result=1.02 \mathrm{Result}=1.02\ {}

Interpretation. Stretch is a ratio, not strain.

↑ Return to definitions and contents
Example 02. Engineering strain

Definitions & inputs. Samebar.

  1. Choose the governing model and isolate the requested quantity.

    ϵ=(L−L0)/L0\epsilon=(L-L_0)/L_0

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (1.02−1)/1(1.02-1)/1

  3. Evaluate the expression; the result uses the units shown.

    Result=0.02 \mathrm{Result}=0.02\ {}

Interpretation. Small-strain measure.

↑ Return to definitions and contents
Example 03. Green axial strain

Definitions & inputs. Stretch1.02.

  1. Choose the governing model and isolate the requested quantity.

    E11=(λ2−1)/2E_{11}=(\lambda^2-1)/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (1.022−1)/2(1.02^2-1)/2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0202 \mathrm{Result}=0.0202\ {}

Interpretation. Finite-strain measure.

↑ Return to definitions and contents
Example 04. Logarithmic axial strain

Definitions & inputs. Stretch1.02.

  1. Choose the governing model and isolate the requested quantity.

    e=ln⁡λe=\ln\lambda

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ln⁡1.02\ln1.02

  3. Evaluate the expression; the result uses the units shown.

    Result=0.01980263 \mathrm{Result}=0.01980263\ {}

Interpretation. Adds for sequential coaxial stretches.

↑ Return to definitions and contents
Example 05. Volume ratio

Definitions & inputs. F=diag(1.1,1,.9).

  1. Choose the governing model and isolate the requested quantity.

    J=det⁡FJ=\det F

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.1(1)(0.9)1.1(1)(0.9)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.99 \mathrm{Result}=0.99\ {}

Interpretation. Positive determinant preserves orientation.

↑ Return to definitions and contents
Example 06. Density change

Definitions & inputs. ρ0=1000kg/m³,J=.99.

  1. Choose the governing model and isolate the requested quantity.

    ρ=ρ0/J\rho=\rho_0/J

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1000/0.991000/0.99

  3. Evaluate the expression; the result uses the units shown.

    Result=1010.101 kg m−3\mathrm{Result}=1010.101\ \mathrm{kg\,m^{-3}}

Interpretation. Material mass is conserved.

↑ Return to definitions and contents
Example 07. Rigid rotation strain

Definitions & inputs. F orthogonal.

  1. Choose the governing model and isolate the requested quantity.

    E=12(FTF−I)E=\tfrac12(F^TF-I)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (1−1)/2(1-1)/2

  3. Evaluate the expression; the result uses the units shown.

    Result=0 \mathrm{Result}=0\ {}

Interpretation. Exact finite strain vanishes under rigid motion.

↑ Return to definitions and contents
Example 08. Normal traction

Definitions & inputs. σxx=20MPa,n=(1,0,0).

  1. Choose the governing model and isolate the requested quantity.

    tx=σxxt_x=\sigma_{xx}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2020

  3. Evaluate the expression; the result uses the units shown.

    Result=20 MPa\mathrm{Result}=20\ \mathrm{MPa}

Interpretation. Shear traction may also exist if σyx orσzx is nonzero.

↑ Return to definitions and contents
Example 09. Pressure from stress

Definitions & inputs. Diagonal stresses−10,−20,−30MPa.

  1. Choose the governing model and isolate the requested quantity.

    p=−tr⁡σ/3p=-\operatorname{tr}\sigma/3

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −(−10−20−30)/3-(-10-20-30)/3

  3. Evaluate the expression; the result uses the units shown.

    Result=20 MPa\mathrm{Result}=20\ \mathrm{MPa}

Interpretation. Pressure is positive in compression.

↑ Return to definitions and contents
Example 10. Deviatoric stress component

Definitions & inputs. Sameσxx=−10MPa,p20MPa.

  1. Choose the governing model and isolate the requested quantity.

    sxx=σxx+ps_{xx}=\sigma_{xx}+p

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −10+20-10+20

  3. Evaluate the expression; the result uses the units shown.

    Result=10 MPa\mathrm{Result}=10\ \mathrm{MPa}

Interpretation. Deviatoric stress has zero trace.

↑ Return to definitions and contents
Example 11. Shear modulus

Definitions & inputs. E200GPa,ν.3.

  1. Choose the governing model and isolate the requested quantity.

    G=E/[2(1+ν)]G=E/[2(1+\nu)]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    200/[2(1.3)]200/[2(1.3)]

  3. Evaluate the expression; the result uses the units shown.

    Result=76.92308 GPa\mathrm{Result}=76.92308\ \mathrm{GPa}

Interpretation. Isotropic stable linear elasticity.

↑ Return to definitions and contents
Example 12. Bulk modulus

Definitions & inputs. E200GPa,ν.3.

  1. Choose the governing model and isolate the requested quantity.

    K=E/[3(1−2ν)]K=E/[3(1-2\nu)]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    200/[3(0.4)]200/[3(0.4)]

  3. Evaluate the expression; the result uses the units shown.

    Result=166.6667 GPa\mathrm{Result}=166.6667\ \mathrm{GPa}

Interpretation. Incompressibility corresponds to largeK, not zeroK.

↑ Return to definitions and contents
Example 13. Uniaxial stress

Definitions & inputs. E200GPa,ε.001.

  1. Choose the governing model and isolate the requested quantity.

    σ=Eϵ\sigma=E\epsilon

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    200000(0.001)200000(0.001)

  3. Evaluate the expression; the result uses the units shown.

    Result=200 MPa\mathrm{Result}=200\ \mathrm{MPa}

Interpretation. Uniaxial stress state with free lateral contraction.

↑ Return to definitions and contents
Example 14. Lateral strain

Definitions & inputs. ν.3,axialε.001.

  1. Choose the governing model and isolate the requested quantity.

    ϵy=−νϵx\epsilon_y=-\nu\epsilon_x

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −0.3(0.001)-0.3(0.001)

  3. Evaluate the expression; the result uses the units shown.

    Result=−0.0003 \mathrm{Result}=-0.0003\ {}

Interpretation. Free lateral surfaces.

↑ Return to definitions and contents
Example 15. Hydrostatic volume strain

Definitions & inputs. p100MPa,K100GPa.

  1. Choose the governing model and isolate the requested quantity.

    tr⁡ϵ=−p/K\operatorname{tr}\epsilon=-p/K

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −100/100000-100/100000

  3. Evaluate the expression; the result uses the units shown.

    Result=−0.001 \mathrm{Result}=-0.001\ {}

Interpretation. Linear compression estimate.

↑ Return to definitions and contents
Example 16. Viscous shear stress

Definitions & inputs. μ.001Pa·s,du/dy100/s.

  1. Choose the governing model and isolate the requested quantity.

    τ=μ du/dy\tau=\mu\,du/dy

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .001(100).001(100)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1 Pa\mathrm{Result}=0.1\ \mathrm{Pa}

Interpretation. Newtonian simple shear.

↑ Return to definitions and contents
Example 17. Heat flux

Definitions & inputs. k10W/(mK),dT/dx20K/m.

  1. Choose the governing model and isolate the requested quantity.

    qx=−k∂xTq_x=-k\partial_xT

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −10(20)-10(20)

  3. Evaluate the expression; the result uses the units shown.

    Result=−200 W m−2\mathrm{Result}=-200\ \mathrm{W\,m^{-2}}

Interpretation. Negative sign denotes down-gradient flow.

↑ Return to definitions and contents
Example 18. Volumetric strain rate

Definitions & inputs. Velocity gradients .1,.2,−.1/s.

  1. Choose the governing model and isolate the requested quantity.

    ∇⋅v=∂xvx+∂yvy+∂zvz\nabla\cdot v=\partial_xv_x+\partial_yv_y+\partial_zv_z

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .1+.2−.1.1+.2-.1

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2 s−1\mathrm{Result}=0.2\ \mathrm{s^{-1}}

Interpretation. Positive divergence locally expands material.

↑ Return to definitions and contents
Example 19. Density time derivative

Definitions & inputs. ρ1000kg/m³,divv.2/s.

  1. Choose the governing model and isolate the requested quantity.

    ρ˙=−ρ∇⋅v\dot\rho=-\rho\nabla\cdot v

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    −1000(.2)-1000(.2)

  3. Evaluate the expression; the result uses the units shown.

    Result=−200 kg m−3 s−1\mathrm{Result}=-200\ \mathrm{kg\,m^{-3}\,s^{-1}}

Interpretation. Material derivative, not fixed-point derivative.

↑ Return to definitions and contents
Example 20. Elastic energy density

Definitions & inputs. σ200MPa,ε.001.

  1. Choose the governing model and isolate the requested quantity.

    W=12σϵW=\tfrac12\sigma\epsilon

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(200×106)(.001).5(200\times10^6)(.001)

  3. Evaluate the expression; the result uses the units shown.

    Result=100000 J m−3\mathrm{Result}=100000\ \mathrm{J\,m^{-3}}

Interpretation. Uniaxial proportional linear loading.

↑ Return to definitions and contents

Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.