Introduction

c o n s p i r e

User Guide

• Installation
• Mathematics
• Constitutive

Reference Guide

• Julia interface
• Python interface
• Rust interface

Installation

Julia

pkg> add Conspire

Python

pip install conspire

Rust

cargo add conspire

Mathematics

• Integration and ODEs
• Special functions

Integration and ODEs

• Explicit integration methods
• Implicit integration methods

Explicit integration methods

Solves an initial value problem by explicitly integrating a system of ordinary differential equations.

$$\frac{dy}{dt}=f(t,y),y(t_0)=y_0$$

• Bogacki-Shampine method
• Dormand-Prince method
• Verner 8 method
• Verner 9 method

Bogacki-Shampine method

Explicit, three-stage, third-order, variable-step, Runge-Kutta method.¹

$$\frac{dy}{dt}=f(t,y)$$ $$t_{n+1}=t_n+h$$ $$k_1=f(t_n,y_n)$$ $$k_2=f(t_n+\frac{1}{2}h,y_n+\frac{1}{2}h{k}_1)$$ $$k_3=f(t_n+\frac{3}{4}h,y_n+\frac{3}{4}h{k}_2)$$ $$y_{n+1}=y_n+\frac{h}{9}\left(2k_1+3k_2+4k_3\right)$$ $$k_4=f(t_{n+1},y_{n+1})$$ $$e_{n+1}=\frac{h}{72}\left(-5k_1+6k_2+8k_3-9k_4\right)$$

1. P. Bogacki and L.F. Shampine, Appl. Math. Lett. 2, 321 (1989). ↩

Dormand-Prince method

Explicit, six-stage, fifth-order, variable-step, Runge-Kutta method.¹

$$\frac{dy}{dt}=f(t,y)$$ $$t_{n+1}=t_n+h$$ $$k_1=f(t_n,y_n)$$ $$k_2=f(t_n+\frac{1}{5}h,y_n+\frac{1}{5}h{k}_1)$$ $$k_3=f(t_n+\frac{3}{10}h,y_n+\frac{3}{40}h{k}_1+\frac{9}{40}h{k}_2)$$ $$k_4=f(t_n+\frac{4}{5}h,y_n+\frac{44}{45}h{k}_1-\frac{56}{15}h{k}_2+\frac{32}{9}h{k}_3)$$ $$k_5=f(t_n+\frac{8}{9}h,y_n+\frac{19372}{6561}h{k}_1-\frac{25360}{2187}h{k}_2+\frac{64448}{6561}h{k}_3-\frac{212}{729}h{k}_4)$$ $$k_6=f(t_n+h,y_n+\frac{9017}{3168}h{k}_1-\frac{355}{33}h{k}_2-\frac{46732}{5247}h{k}_3+\frac{49}{176}h{k}_4-\frac{5103}{18656}h{k}_5)$$ $$y_{n+1}=y_n+h\left(\frac{35}{384}{k}_1+\frac{500}{1113}{k}_3+\frac{125}{192}{k}_4-\frac{2187}{6784}{k}_5+\frac{11}{84}{k}_6\right)$$ $$k_7=f(t_{n+1},y_{n+1})$$ $$e_{n+1}=\frac{h}{5}\left(\frac{71}{11520}{k}_1-\frac{71}{3339}{k}_3+\frac{71}{384}{k}_4-\frac{17253}{67840}{k}_5+\frac{22}{105}{k}_6-\frac{1}{8}{k}_7\right)$$

1. J.R. Dormand and P.J. Prince, J. Comput. Appl. Math. 6, 19 (1980). ↩

Verner 8 method

Explicit, thirteen-stage, eighth-order, variable-step, Runge-Kutta method.¹

$$\frac{dy}{dt}=f(t,y)$$ $$t_{n+1}=t_n+h$$ $$k_1=f(t_n,y_n)$$ $$\cdots$$

1. J.H. Verner, Numer. Algor. 53, 383 (2010). ↩

Verner 9 method

Explicit, sixteen-stage, ninth-order, variable-step, Runge-Kutta method.¹

$$\frac{dy}{dt}=f(t,y)$$ $$t_{n+1}=t_n+h$$ $$k_1=f(t_n,y_n)$$ $$\cdots$$

1. J.H. Verner, Numer. Algor. 53, 383 (2010). ↩

Implicit integration methods

Solves an initial value problem by implicitly integrating a system of ordinary differential equations.

$$\frac{dy}{dt}=f(t,y),y(t_0)=y_0,\frac{\partial f}{\partial y}=J(t,y)$$

• Backward Euler method

Backward Euler method

Implicit, single-stage, first-order, fixed-step, Runge-Kutta method.¹

$$\frac{dy}{dt}=f(t,y)$$ $$t_{n+1}=t_n+h$$ $$k_1=f(t_{n+1},y_{n+1})$$ $$y_{n+1}=y_n+h{k}_1$$

1. Also known as the backward Euler method. ↩

Special functions

• Lambert W function
• Langevin function
• Inverse Langevin function
• Rosenbrock function

Lambert W function

$$y=W_0(x)$$

Langevin function

$$\mathcal{L}(x)=\coth (x)-x^{-1}$$

Inverse Langevin function

$$x={\mathcal{L}}^{-1}(y)$$

Rosenbrock function

$$f(\mathbf{x})=\sum _{i=1}^{N-1}\left[{\left(a-x_i\right)}^2+b{\left(x_{i+1}-x_i^2\right)}^2\right]$$

Constitutive

• Solid constitutive models

Solid constitutive models

• Elastic constitutive models
• Hyperelastic constitutive models

Elastic constitutive models

Elastic solid constitutive models are not defined by a Helmholtz free energy density and depend on only the deformation gradient. These constitutive models are therefore defined by a relation for some stress measure as a function of the deformation gradient. Consequently, the tangent stiffness associated with the first Piola-Kirchhoff stress is not symmetric for these constitutive models.

$${\mathcal{C}}_{iJkL}\ne {\mathcal{C}}_{kLiJ}$$

• Almansi-Hamel model

Almansi-Hamel model

The Almansi-Hamel elastic solid constitutive model.

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Almansi-Hamel strain measure is given by $\mathbf{e}=\frac{1}{2}(1-{\mathbf{B}}^{-1})$.

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{2\mu }{J}{\mathbf{e}}^{\prime }+\frac{\kappa }{J}\mathrm{tr}(\mathbf{e})1$$

Cauchy tangent stiffness

$${\mathcal{T}}_{ijkL}(\mathbf{F})=\frac{\mu }{J}\left[B_{jk}^{-1}{F}_{iL}^{-T}+B_{ik}^{-1}{F}_{jL}^{-T}-\frac{2}{3}{\delta }_{ij}{B}_{km}^{-1}{F}_{mL}^{-T}-2e_{ij}^{\prime }{F}_{kL}^{-T}\right]+\frac{\kappa }{J}\left[{\delta }_{ij}{B}_{km}^{-1}{F}_{mL}^{-T}-\mathrm{tr}(\mathbf{e}){\delta }_{ij}{F}_{kL}^{-T}\right]$$

Hyperelastic constitutive models

Hyperelastic solid constitutive models are defined by a Helmholtz free energy density function of the deformation gradient.

$$\mathbf{P}:\dot{\mathbf{F}}-\dot{a}(\mathbf{F})\ge 0$$ Satisfying the second law of thermodynamics (here, equivalent to extremized or zero dissipation) yields a relation for the stress.

$$\mathbf{P}=\frac{\partial a}{\partial \mathbf{F}}$$ Consequently, the tangent stiffness associated with the first Piola-Kirchhoff stress is symmetric for these constitutive models.

$${\mathcal{C}}_{iJkL}={\mathcal{C}}_{kLiJ}$$

• Arruda-Boyce model
• Fung model
• Gent model
• Mooney-Rivlin model
• Neo-Hookean model
• Saint Venant-Kirchhoff model
• Yeoh model

Arruda-Boyce model

The Arruda-Boyce hyperelastic solid constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The number of links $N_b$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The nondimensional end-to-end length per link of a chain is $\gamma =\sqrt{\mathrm{tr}({\mathbf{B}}^{\ast })/3N_b}$.
• The nondimensional force is given by the inverse Langevin function as $\eta ={\mathcal{L}}^{-1}(\gamma )$.
• The initial values are given by ${\gamma }_0=\sqrt{1/3N_b}$ and ${\eta }_0={\mathcal{L}}^{-1}({\gamma }_0)$.
• The Arruda-Boyce model reduces to the Neo-Hookean model when $N_b\to \infty$.

Helmholtz free energy density

$$a(\mathbf{F})=\frac{3\mu N_b{\gamma }_0}{{\eta }_0}\left[\gamma \eta -{\gamma }_0{\eta }_0-\ln \left(\frac{{\eta }_0\sinh \eta }{\eta \sinh {\eta }_0}\right)\right]+\frac{\kappa }{2}\left[\frac{1}{2}\left(J^2-1\right)-\ln J\right]$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{\mu {\gamma }_0\eta }{J\gamma {\eta }_0}{{\mathbf{B}}^{\ast }}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \frac{\mu {\gamma }_0\eta }{J^{5/3}\gamma {\eta }_0}\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)\\ & +\frac{\mu {\gamma }_0\eta }{3J^{7/3}{N}_b{\gamma }^2{\eta }_0}\left(\frac{1}{\eta {\mathcal{L}}^{\prime }(\eta )}-\frac{1}{\gamma }\right)B_{ij}^{\prime }{B}_{km}^{\prime }{F}_{mL}^{-T}+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\end{array}$$

1. E.M. Arruda and M.C. Boyce, J. Mech. Phys. Solids 41, 389 (1993). ↩

Fung model

The Fung hyperelastic solid constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra modulus ${\mu }_m$.
• The exponent $c$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Fung model reduces to the Neo-Hookean model when ${\mu }_m\to 0$ or $c\to 0$.

Helmholtz free energy density

$$a(\mathbf{F})=\frac{\mu -{\mu }_m}{2}\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]+\frac{{\mu }_m}{2c}\left(e^{c[\mathrm{tr}({\mathbf{B}}^{\ast })-3]}-1\right)$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{1}{J}\left[\mu +{\mu }_m\left(e^{c[\mathrm{tr}({\mathbf{B}}^{\ast })-3]}-1\right)\right]{{\mathbf{B}}^{\ast }}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \frac{1}{J^{5/3}}\left[\mu +{\mu }_m\left(e^{c[\mathrm{tr}({\mathbf{B}}^{\ast })-3]}-1\right)\right]\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)\\ & +\frac{2c{\mu }_m}{J^{7/3}}{e}^{c[\mathrm{tr}({\mathbf{B}}^{\ast })-3]}{B}_{ij}^{\prime }{B}_{km}^{\prime }{F}_{mL}^{-T}+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\end{array}$$

1. Y.C. Fung, Am. J. Physiol. 213, 1532 (1967). ↩

Gent model

The Gent hyperelastic solid constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extensibility $J_m$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Gent model reduces to the Neo-Hookean model when $J_m\to \infty$.

Helmholtz free energy density

$$a(\mathbf{F})=-\frac{\mu J_m}{2}\ln \left[1-\frac{\mathrm{tr}({\mathbf{B}}^{\ast })-3}{J_m}\right]+\frac{\kappa }{2}\left[\frac{1}{2}\left(J^2-1\right)-\ln J\right]$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{J^{-1}\mu J_m{{\mathbf{B}}^{\ast }}^{\prime }}{J_m-\mathrm{tr}({\mathbf{B}}^{\ast })+3}+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \frac{J^{-5/3}\mu J_m}{J_m-\mathrm{tr}({\mathbf{B}}^{\ast })+3}[{\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}+\frac{2{B_{ij}^{\ast }}^{\prime }{F}_{kL}}{J_m-\mathrm{tr}({\mathbf{B}}^{\ast })+3}\\ & -\left(\frac{5}{3}+\frac{2}{3}\frac{\mathrm{tr}({\mathbf{B}}^{\ast })}{J_m-\mathrm{tr}({\mathbf{B}}^{\ast })+3}\right)J^{2/3}{B_{ij}^{\ast }}^{\prime }{F}_{kL}^{-T}]+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\end{array}$$

1. A.N. Gent, Rubber Chem. Technol. 69, 59 (1996). ↩

Mooney-Rivlin model

The Mooney-Rivlin hyperelastic solid constitutive model.^1,2

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra modulus ${\mu }_m$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Mooney-Rivlin model reduces to the Neo-Hookean model when ${\mu }_m\to 0$.

Helmholtz free energy density

$$a(\mathbf{F})=\frac{\mu -{\mu }_m}{2}\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]+\frac{{\mu }_m}{2}\left[I_2({\mathbf{B}}^{\ast })-3\right]+\frac{\kappa }{2}\left[\frac{1}{2}\left(J^2-1\right)-\ln J\right]$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{\mu -{\mu }_m}{J}{{\mathbf{B}}^{\ast }}^{\prime }-\frac{{\mu }_m}{J}{\left({\mathbf{B}}^{\ast -1}\right)}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \frac{\mu -{\mu }_m}{J^{5/3}}\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\\ & -\frac{{\mu }_m}{J}\left[\frac{2}{3}{B}_{ij}^{\ast -1}{F}_{kL}^{-T}-B_{ik}^{\ast -1}{F}_{jL}^{-T}-B_{ik}^{\ast -1}{F}_{iL}^{-T}+\frac{2}{3}{\delta }_{ij}{\left(B_{km}^{\ast -1}\right)}^{\prime }{F}_{mL}^{-T}-{\left(B_{ij}^{\ast -1}\right)}^{\prime }{F}_{kL}^{-T}\right]\end{array}$$

1. M. Mooney, J. Appl. Phys. 11, 582 (1940). ↩

2. R.S. Rivlin, Philos. Trans. R. Soc. London, Ser. A 241, 379 (1948). ↩

Neo-Hookean model

The Neo-Hookean hyperelastic solid constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Helmholtz free energy density

$$a(\mathbf{F})=\frac{\mu }{2}\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]+\frac{\kappa }{2}\left[\frac{1}{2}\left(J^2-1\right)-\ln J\right]$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{\mu }{J}{{\mathbf{B}}^{\ast }}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$${\mathcal{T}}_{ijkL}(\mathbf{F})=\frac{\mu }{J^{5/3}}\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}$$

1. R.S. Rivlin, Philos. Trans. R. Soc. London, Ser. A 240, 459 (1948). ↩

Saint Venant-Kirchhoff model

The Saint Venant-Kirchhoff hyperelastic solid constitutive model.

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Green-Saint Venant strain measure is given by $\mathbf{E}=\frac{1}{2}(\mathbf{C}-1)$.

Helmholtz free energy density

$$a(\mathbf{F})=\mu \mathrm{tr}({\mathbf{E}}^2)+\frac{1}{2}\left(\kappa -\frac{2}{3}\mu \right)\mathrm{tr}(\mathbf{E}{)}^2$$

Second Piola-Kirchhoff stress

$$\mathbf{S}(\mathbf{F})=2\mu {\mathbf{E}}^{\prime }+\kappa \mathrm{tr}(\mathbf{E})1$$

Second Piola-Kirchhoff tangent stiffness

$${\mathcal{G}}_{IJkL}(\mathbf{F})=\mu {\delta }_{JL}{F}_{kI}+\mu {\delta }_{IL}{F}_{kJ}+\left(\kappa -\frac{2}{3}\mu \right){\delta }_{IJ}{F}_{kL}$$

Yeoh model

The Yeoh hyperelastic solid constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra moduli ${\mu }_n$ for $n=2\dots N$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Yeoh model reduces to the Neo-Hookean model when ${\mu }_n\to 0$ for $n=2\dots N$.

Helmholtz free energy density

$$a(\mathbf{F})=\sum _{n=1}^N\frac{{\mu }_n}{2}{\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]}^n+\frac{\kappa }{2}\left[\frac{1}{2}\left(J^2-1\right)-\ln J\right]$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\sum _{n=1}^N\frac{n{\mu }_n}{J}{\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]}^{n-1}{{\mathbf{B}}^{\ast }}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \sum _{n=1}^N\frac{n{\mu }_n}{J^{5/3}}{\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]}^{n-1}\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)\\ & +\sum _{n=2}^N\frac{2n(n-1){\mu }_n}{J^{7/3}}{\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]}^{n-2}{B}_{ij}^{\prime }{B}_{km}^{\prime }{F}_{mL}^{-T}+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\end{array}$$