Solid Phase Transitions in the Liquid Limit

Abstract

We address the fundamental difference between solid-solid and liquid-liquid phase transitions within the Ericksen’s nonlinear elasticity paradigm. To highlight ideas, we consider the simplest nontrivial 2D problem and work with a prototypical two-phase Hadamard material which allows one to weaken the rigidity and explore the nature of solid-solid phase transitions in a “near-liquid” limit. In the language of calculus of variations we probe limits of quasiconvexity in an “almost liquid” solid by comparing the thresholds for cooperative (laminate based) and non-cooperative (inclusion based) nucleation. Using these two types of nucleation tests we obtain for our model material surprisingly tight two-sided bounds on the elastic binodal without directly computing the quasiconvex envelope.

Appendix:  Calculation of the Jump Set for Hadamard Materials

Here we recall the calculation of the jump set from [27] for energies (3.1).

We start with the first equation in (2.3) expressing the kinematic compatibility of the deformation gradients $\boldsymbol{F}_{+}$ and $\boldsymbol{F}_{-}$. Taking the determinant of both sides we obtain

$$ d_{+}=d_{-}+\mathrm {cof}\boldsymbol{F}_{-}\boldsymbol{n}\cdot \boldsymbol{a},\quad d_{\pm}=\det \boldsymbol{F}_{\pm}. $$

(A.1)

Using the formula

$$ \boldsymbol{P}(\boldsymbol{F})=\mu \boldsymbol{F}+h'(\det \boldsymbol{F})\mathrm {cof}\boldsymbol{F}$$

for the Piola–Kirchhoff stress we compute

$$ \lbrack \!\lbrack \boldsymbol{P}\rbrack \!\rbrack \boldsymbol{n}=\mu \boldsymbol{a}+\lbrack \!\lbrack h' \rbrack \!\rbrack \mathrm {cof}\boldsymbol{F}_{-}\boldsymbol{n}, $$

where we have used the well-known relation $\mathrm {cof}(\boldsymbol{F}_{-}+\boldsymbol{a}\otimes \boldsymbol{n})\boldsymbol{n}=(\mathrm {cof}\boldsymbol{F}_{-})\boldsymbol{n}$. Similarly,

$$ \lbrack \!\lbrack \boldsymbol{P}\rbrack \!\rbrack ^{T}\boldsymbol{a}=\mu |\boldsymbol{a}|^{2}\boldsymbol{n}+\lbrack \!\lbrack h' \rbrack \!\rbrack \mathrm {cof}\boldsymbol{F}_{-}^{T}\boldsymbol{a}. $$

Thus, the second and the third equations in (2.3) become

$$ \boldsymbol{a}=-\frac{\lbrack \!\lbrack h' \rbrack \!\rbrack }{\mu} \mathrm {cof}\boldsymbol{F}_{-}\boldsymbol{n},\qquad \lbrack \!\lbrack h' \rbrack \!\rbrack ^{2} \mathrm {cof}(\boldsymbol{C}_{-})\boldsymbol{n}=\mu ^{2}|\boldsymbol{a}|^{2}\boldsymbol{n}, $$

(A.2)

where $\boldsymbol{C}_{\pm}=\boldsymbol{F}_{\pm}^{T}\boldsymbol{F}_{\pm}$ is the Cauchy–Green strain tensor. We conclude that $\boldsymbol{n}$ must be an eigenvector of $\boldsymbol{C}_{-}$. Equations (A.2) permit us to find a relation between the two Cauchy–Green tensors $\boldsymbol{C}_{\pm}$. Using the kinematic compatibility equation (2.3)₁ we compute

$$ \boldsymbol{C}_{+}=\boldsymbol{C}_{-}+\boldsymbol{F}_{-}^{T}\boldsymbol{a}\otimes \boldsymbol{n}+\boldsymbol{n}\otimes \boldsymbol{F}_{-}^{T} \boldsymbol{a}+|\boldsymbol{a}|^{2} \boldsymbol{n}\otimes \boldsymbol{n}. $$

Applying $\boldsymbol{F}_{-}^{T}$ to the first equation in (A.2) we obtain $\boldsymbol{F}_{-}^{T}\boldsymbol{a}=-(\lbrack \!\lbrack h' \rbrack \!\rbrack /\mu )d_{-}\boldsymbol{n}$, so that

$$ \lbrack \!\lbrack \boldsymbol{C}\rbrack \!\rbrack =\left (|\boldsymbol{a}|^{2}-\frac{2\lbrack \!\lbrack h' \rbrack \!\rbrack d_{-}}{\mu}\right ) \boldsymbol{n}\otimes \boldsymbol{n}. $$

(A.3)

It follows that the Cauchy–Green tensors $\boldsymbol{C}_{+}$ and $\boldsymbol{C}_{-}$ are simultaneously diagonalizable, since, by (A.2) $\boldsymbol{n}$ is an eigenvector of $\boldsymbol{C}_{-}$. According to equation (A.3) symmetric matrices $\boldsymbol{C}_{+}$ and $\boldsymbol{C}_{-}$ have the same pair of mutually orthogonal eigenvectors $\boldsymbol{n}$ and $\boldsymbol{n}^{\perp}$ with the same eigenvalues corresponding to $\boldsymbol{n}^{\perp}$. Hence, singular values of $\boldsymbol{F}_{\pm}$ would be $(\varepsilon _{\pm}, \varepsilon _{0})$, the first one corresponding to the eigenvector $\boldsymbol{n}$ of $\boldsymbol{C}_{\pm}$. Substituting the first equation in (A.2) into (A.1) we obtain

$$ d_{+}=d_{-}-\frac{\lbrack \!\lbrack h' \rbrack \!\rbrack }{\mu} \mathrm {cof}\boldsymbol{C}_{-}\boldsymbol{n}\cdot \boldsymbol{n}=d_{-}- \frac{\lbrack \!\lbrack h' \rbrack \!\rbrack d_{-}^{2}}{\mu \varepsilon _{-}^{2}}, $$

which can be written in the more symmetric form as

$$ \mu \frac{\lbrack \!\lbrack d \rbrack \!\rbrack }{\lbrack \!\lbrack h' \rbrack \!\rbrack }=-\varepsilon _{0}^{2}=- \frac{d_{\pm}^{2}}{\varepsilon _{\pm}^{2}}. $$

(A.4)

This will be the equation for the jump set, when we determine $d_{+}$ as a function of $d_{-}$ from the Maxwell relation (the last equation in (2.3), which hasn’t been used so far). It is well-known that the Maxwell relation does not change if we add any quadratic function of $\boldsymbol{F}$ to the energy. Thus, the term $\mu |\boldsymbol{F}|^{2}/2$ can be disregarded and the Maxwell relation becomes

Recalling that due to (A.1) $(\mathrm {cof}\boldsymbol{F}_{+})\boldsymbol{n}\cdot \boldsymbol{a}=(\mathrm {cof}\boldsymbol{F}_{-})\boldsymbol{n}\cdot \boldsymbol{a}=\lbrack \!\lbrack d \rbrack \!\rbrack$ we obtain

(A.5)

Equation (A.5) has a geometric meaning. It says that the secant line joining $(d_{-},h'(d_{-}))$ and $(d_{-},h'(d_{-}))$ together with the graph of $h'(d)$ bound two regions of equal areas. For a double-well shaped potential $h(d)$ there exists a single interval $(d_{1},d_{2})$ on which $h(d)$ differs from its convex hull, which on $(d_{1},d_{2})$ agrees with the common tangent line at $d_{1}$ and $d_{2}$ to the graph of $h(d)$. In terms of $h'(d)$ this double-tangency can also be interpreted geometrically as the horizontal “Maxwell line” with the equal area property. In that case there exist $d_{0}\in (d_{1},d_{2})$, such that for any $d_{-}\in (d_{1},d_{0})$ there is a unique $d_{+}\in (d_{0},d_{2})$ satisfying (A.5). In other words, for every $d_{-}\in (d_{1},d_{0})$ there is a unique $d_{+}=D(d_{-})$ with equal area property. (By continuity we can set $D(d_{0})=d_{0}$.) Regarding the function $D(d)$ as known, equation (A.4) provides the explicit description of the jump set in terms of the singular values of $\boldsymbol{F}_{\pm}$.