The scientific life of Miroslav Šilhavý (1949

Abstract

Miroslav Šilhavý was an outstanding scholar of exceptional depth and clarity, a thinker who persistently sought to reconcile the abstract structures of mathematical analysis with the physical principles governing material behavior. He viewed continuum mechanics not merely as a modeling discipline but as a domain requiring the highest standards of mathematical precision.

His research interests were broad. He defined himself as “a researcher in mathematical methods in continuum mechanics and thermodynamics.” Although his work was primarily foundational in nature, it also encompassed significant applications. The principal mathematical tools he used included the calculus of variations, functional analysis, partial differential equations, convex analysis, geometric measure theory, and matrix theory. His specific areas of interest included plasticity, thermodynamics, the theory of masonry-like bodies, stresses represented by measures, phase transitions in solids, and fractional models in elasticity.

He studied physics at Univerzita Karlova (Charles University) in Prague, graduating in 1973. In 1978, he obtained his PhD (CSc.) from the Czechoslovak Academy of Sciences. He was awarded the Doctor of Science degree in 1991. He spent the major part of his professional career at the Institute of Mathematics of the Czechoslovak Academy of Sciences, later the Czech Academy of Sciences, in Prague. His work was deeply influenced, on one hand, by the classical tradition of rational mechanics associated with Clifford Truesdell and Walter Noll and, on the other hand, by the theory of partial differential equations, particularly through his collaboration with Jindřich Nečas. Together with his PhD advisor, Jan Kratochvíl, he played a key role in promoting and developing rational mechanics in Czechoslovakia and, subsequently, in the Czech Republic.

Over the years, Šilhavý established himself as a leading figure in the continuum mechanics community. He visited the main centers of continuum mechanics worldwide; in particular, he spent 3 years as a visiting professor at the Department of Mathematics of the University of Pisa.

A cornerstone of Šilhavý’s legacy is his monograph The Mechanics and Thermodynamics of Continuous Media, published by Springer in 1997 [31]. This work synthesizes a vast body of knowledge into a coherent framework, presenting the subject with a level of precision that has made it a lasting reference for researchers.

The passing of Miroslav Šilhavý marks a profound loss for the physical and mathematical community. We have lost a gentleman of quiet authority and great personal dignity.

We highlight a few topics in mathematical materials science where Šilhavý’s work has made significant contributions.

1. Interfacial semiconvexities

One of the enduring insights of modern elasticity theory is that the intricate patterns observed in materials are not accidental: they reflect underlying principles of energy minimization and stability. Few researchers have contributed as deeply to this perspective as Miroslav Šilhavý, whose work shaped the mathematical understanding of complex materials, in particular those undergoing phase transformations.

A prominent example is provided by shape memory alloys, materials capable of recovering their original shape after deformation through temperature changes. Such alloys, including Ni–Ti and Cu–Al–Ni, have attracted considerable attention both for their technological importance and for the richness of the mathematical structures they exhibit. Their behavior is governed by the coexistence of two principal phases: a highly symmetric high-temperature phase, known as austenite, and a low-temperature phase, martensite, which appears in multiple symmetry-related variants.

It is precisely the interaction between these variants that gives rise to the remarkable and often intricate microstructures observed in experiments. Explaining these patterns requires a delicate balance between physical modeling and mathematical analysis, a balance that Šilhavý’s work helped to establish in an original way. Namely, the stored energy density of a shape memory alloy is typically non-convex, and, in fact, it is not even rank-one convex; therefore, one cannot expect the lower semicontinuity of the energy integral functional in the weak topology of the underlying function space, usually a reflexive Sobolev space. Multiple minima of the energy density force any minimizing sequence of gradients to oscillate between them, and, indeed, no minimizer does exist. The usual approach to overcome this issue is to add a convex term depending on higher-order deformation gradients to the stored energy density which makes the energy functional weakly lower semicontinuous or to resort to measure-valued solutions, using Young measures, for instance. The main disadvantage of the latter approach is that it does not allow us to control the orientation preservation of admissible deformations. The higher-order gradients, on the contrary, are complicated for numerical implementation.

Šilhavý, in analogy with the bulk polyconvexity of stored energies introduced by J.M. Ball, defined the notion of interfacial polyconvexity [66,67]. The state of the material with the reference configuration $Ω \subset ℝ^{3}$ is described by a deformation $y : Ω \to ℝ^{3}$ and a phase indicator $z : Ω \to {0, 1}^{M}$ , where $M \in ℕ$ denotes the number of material variants or phases, and the evolution is driven by the minimization of a total-energy functional, which is the sum of the bulk and interfacial energies

\begin{matrix} E (t, y, z) = E_{b} (y, z) + E_{int} (y, z) . \end{matrix}

(1)

Here, the bulk energy $E_{b}$ accounts for elastic deformation, while the interfacial energy $E_{int}$ captures the energetic cost of phase boundaries.

The phase distribution is encoded by a function

\begin{matrix} z \in BV (Ω; {0, 1}^{M}), \sum_{i = 1}^{M} z_{i} = 1, \end{matrix}

(2)

reflecting that exactly one phase is present at almost every point. Here, $BV (Ω; {0, 1}^{M})$ denotes the set of functions with the bounded variations. This indicates that the model allows for sharp interfaces between adjacent covariants.

1.1. Bulk and interfacial polyconvexity

Within each phase, the material is modeled as hyperelastic. The bulk energy takes the form

\begin{matrix} E_{b} (y, z) = \int_{Ω} W (z (x), \nabla y (x)) dx, W (z, F) = \sum_{i = 1}^{M} z_{i} Ŵ_{i} (F) . \end{matrix}

(3)

A key structural assumption is that each stored energy density is polyconvex, i.e.,

\begin{matrix} Ŵ_{i} (F) = {\begin{matrix} h_{i} (F, cof F, \det F) & if \det F > 0, \\ + \infty & otherwise, \end{matrix} \end{matrix}

(4)

with $h_{i}$ being convex and lower semicontinuous. This formulation ensures both physical consistency and mathematical solvability.

While the bulk energy governs behavior within phases, it is the interfacial energy that determines how different phases arrange themselves. Interfaces carry energy due to the incompatibility of neighboring variants, and their structure is strongly influenced by the deformation. Šilhavý introduced a remarkably elegant formulation of interfacial polyconvex energies, in which the density between phases i and j depends on the unit normal vector to the interface and on the interface deformation gradient $F = F (I - n \otimes n)$ . More precisely,

\begin{matrix} f_{ij} (F, n) = g_{i} (F, n) + g_{j} (F, n), \end{matrix}

(5)

with

\begin{matrix} g_{i} (F, n) = Ψ_{i} (n, F \times n, (cof F) n), \end{matrix}

(6)

where $Ψ_{i}$ is convex and positively one-homogeneous. This structure reflects a deep geometric insight: the energy depends not only on the orientation n of the interface but also on how the deformation acts tangentially along it.

The arguments of $Ψ_{i}$ are interface null Lagrangians that play an analogous role as subdeterminants of the gradient in polyconvexity.

One of the conceptual strengths of this theory is its ability to handle highly irregular interfaces. Clearly, in typical situations, where $y \in W^{1, p} (Ω; ℝ^{3})$ , one can only expect that the interface gradient is a distribution or a measure. Hence, interface Null Lagrangians are generically measures concentrated on the interfaces, encoding geometric and mechanical information. However, if y is smooth, one can describe the measures more explicitly for $1 \leq i \leq M$ and $\nabla_{s} = \nabla (I - n \otimes n)$

\begin{matrix} a_{i} = n_{i} H_{| S_{i}}^{2}, H_{i} = \nabla_{S_{i}} y \times n_{i} H_{| S_{i}}^{2}, c_{i} = (cof \nabla_{S_{i}} y) n_{i} . \end{matrix}

(7)

Note that $a_{i}$ and $c_{i}$ describe the normal to the interface $S_{i}$ in the reference as well as in the deformed configurations. Denoting $J_{i} = (a_{i}, H_{i}, c_{i})$ for any $1 \leq i \leq M$ the total interfacial energy

\begin{matrix} E_{int} (y, z) = \sum_{i = 1}^{M} \int_{Ω} Ψ_{i} (\frac{d J_{i}}{d | J_{i} |}) d | J_{i} | . \end{matrix}

(8)

A central theme in Šilhavý’s work on interfacial polyconvexity is the extension of classical variational concepts to the interfacial setting. In particular, the notion of interface quasiconvexity defined for $f : ℝ^{3 \times 3} \times ℝ^{3} \to ℝ$ plays a role analogous to quasiconvexity in the bulk and can be expressed through an inequality

\begin{matrix} \int_{S} f (\nabla_{S} y, n) d H^{2} \geq H^{2} (T) f (G, m), \end{matrix}

(9)

see [69]. Here, $y : S \to ℝ^{3}$ is any smooth map, $T$ is a planar interface with the (constant) unit normal m, and $S$ is a curved interface with the normal n. It holds that the boundaries of $T$ and $S$ coincide, and that $y (x) = Gx$ for x in this common boundary. This notion needs to be further investigated to understand its role in weak lower semicontinuity of energy functionals. Note that constant functions are interface quasiconvex only if they are non-negative, as it is easily seen from the definition, since $H^{2} (S) \geq H^{2} (T)$ . The interface quasiconvexity rules out surface wrinkling and prefers homogeneous surface deformations over the inhomogeneous ones.

2. Thermodynamics

Miroslav Šilhavý has been very active in the foundations of thermodynamics. His view on this discipline has been profoundly influenced by the work of J. W. Gibbs. We report here the second paragraph in the preface of his celebrated 1997 text The Mechanics and Thermodynamics of Continuous Media [31]:

At present, the major challenge to the nonlinear thermoelasticity is posed by phase transformations with changes in symmetry. J. W. Gibbs’ immensely influential treatise On the Equilibrium of Heterogeneous Substances has provided a highly successful theory of phase transitions in fluids. Gibbs brought the view that the thermodynamics is not only the theory of heat but also a theory of equilibrium, with the main tool being the minimum principles. A large portion of the book is an extension of Gibbs’ ideas to bodies of general symmetry by the methods of the calculus of variations. The interplay between the convexity properties of the stored energy functions, the resulting equations, and the physics of the phenomena is a leading theme.

2.1. From efficiency to entropy

Šilhavý, together with W. A. Day, demonstrated [1,3] that it is possible to reverse the assertion that the efficiency of a cycle for a thermoelastic body is a result of the postulation of the existence of entropy, i.e., the expression of the efficiency implies the existence of entropy. The set of the states of a thermomechanical system is an open and arc-wise connected subset U of a real Banach space $V$ . The norm in $V$ is denoted $∥ \cdot ∥$ and the action of a linear form $l \in V^{⋆}$ on an element $x \in V$ is denoted $l \cdot x \equiv l (x)$ . A generalized stress is a map $σ : U \to V^{⋆}$ into the dual of $V$ , and the temperature is a strictly positive map $θ : U \to ℝ^{+}$ . A map $η : U \to ℝ$ is smooth if there exists a continuous map $D η : U \to V^{⋆}$ , called the gradient of η, such that, for every state $x \in U$ and every $ϵ > 0$ , there is an open ball $B \subset U$ , with $x \in U$ ,

\begin{matrix} | η (y) - η (x) - (D η) (x) \cdot (y - x) | < ϵ ∥ y - x ∥, \end{matrix}

(10)

for every state $y \in B$ . Clearly, $D η$ is the Fréchet differential of η. A thermodynamical process is a map $f : [0, d_{f}] \to U$ , where $d_{f} > 0$ is the duration of the process f.

Day and Šilhavý [1] showed that the two propositions below imply each other.

Proposition 1. The efficiency of a non-adiabatic cyclic process is

\begin{matrix} w (f) \leq (1 - \frac{θ^{-} (f)}{θ^{+} (f)}) q^{+} (f), \end{matrix}

(11)

where $q^{+} (f)$ is the heat absorbed in f, and $θ^{+} (f)$ and $θ^{-} (f)$ are the maximum and minimum temperatures at which heat is absorbed. For a Carnot cycle, the expression (11) is always satisfied with the sign of equality.

Proposition 2. There is a smooth map $η : U \to ℝ$ , such that

\begin{matrix} σ = θ D η . \end{matrix}

(12)

The perspective offered by the theorem of Day and Šilhavý is appealing not only from the point of view of mathematical rigor, which was emphasized by the authors, but also for the foundations of physics and engineering.

2.2. Mathematical formalization of thermodynamics

In a series of papers [8,9,12], Šilhavý introduced the heat distribution measure associated with a thermodynamical process. Let $B$ be a body with boundary $\partial B$ and interior $\overset{\cdot\cdot}{B}$ , undergoing a thermodynamical process occurring over the time interval $[t_{0}, t_{1}]$ . For every Borel set $A \subseteq ℝ$ , Šilhavý defined the subsets of $B \times [t_{0}, t_{1}]$ for which the temperature θ is valued in A:

\begin{matrix} T_{b} (A) & = {(X, t) \in \overset{\cdot\cdot}{B} & \times [t_{0}, t_{1}] : θ (X, t) \in A}, \end{matrix}

(13a)

\begin{matrix} T_{c} (A) & = {(X, t) \in \partial B & \times [t_{0}, t_{1}] : θ (X, t) \in A} . \end{matrix}

(13b)

Then, he defined the net gain of heat of the body $B$ in the process at temperatures belonging to the set A as the real-valued signed measure

\begin{matrix} Q (A) = \int_{T_{b} (A)} r_{b} dm dt + \int_{T_{c} (A)} r_{c} da dt, \end{matrix}

(14)

where m is the mass measure (normally defined as the standard volume measure, rescaled by the mass density ϱ), $r_{b}$ is the body heat supply, $da$ is the element of area in $\partial B$ , and $r_{c}$ is the surface heat supply. The goal of these works was to:

[…] convert the vague verbal statements of the second law of Carnot, Clausius, Kelvin, and Planck into meaningful mathematical conditions.

In a double-paper [10,11], Šilhavý then developed a general axiomatic treatment of the Second Law of Thermodynamics for general, non-equilibrium processes that need not obey the First Law. First, he proved an abstract theorem providing the inequalities for systems of this type. Then, he showed that these inequalities reduce to generalizations of the Clausius inequality for systems that do obey the First Law.

3. No-tension materials

The no-tension model characterizes a material that responds linearly elastically under compression while being unable to carry tensile stresses. More precisely, the stress tensor T is assumed to be symmetric and negative semidefinite ( ${Sym}^{-}$ ), and the strain tensor E is decomposed into the sum of an elastic part $E^{e}$ , depending linearly on the stress, and a fracture part $E^{f}$ , which is positive semidefinite ( ${Sym}^{+}$ ) and orthogonal to the stress:

\begin{matrix} E = E^{e} + E^{f}, \\ T = ℂ E^{e}, T \in {Sym}^{-}, \\ T \cdot E^{f} = 0, E^{f} \in {Sym}^{+}, \end{matrix}

where ℂ denotes the elasticity tensor, and $A \cdot B = tr (A B^{T})$ . This constitutive framework has been specifically developed to describe the mechanical behavior of masonry.

Šilhavý’s work, often in collaboration with Massimiliano Lucchesi, Cristina Padovani, and Nicola Zani, has provided major contributions to the mathematical analysis of equilibrium problems in masonry structures. A key innovation has been the introduction of tensor-valued measures whose distributional divergence is a vector-valued measure, allowing for the description of stress concentrations on lower-dimensional sets (such as surfaces and curves).

More precisely, in Lucchesi et al. [52], a tensor-valued measure $T$ defined on $Ω \subset ℝ^{n}$ is called a divergence-measure tensor field if there exists a vector-valued measure $div T$ such that

\int_{Ω} \nabla v \cdot d T = - \int_{Ω} v \cdot d div T

for all smooth functions v with compact support in $Ω$ , see also [26,61].

A tensor-valued measure $T$ is said to be an equilibrated tensor field if there exist vector-valued measures $b$ in $Ω$ and $s$ on $\partial Ω$ such that

\int_{Ω} \nabla v \cdot d T = \int_{Ω} v \cdot d b + \int_{\partial Ω} v \cdot d s

for all smooth functions v with compact support in $ℝ^{n}$ . The pair $(b, s)$ is called the load corresponding to $T$ . Clearly, any equilibrated tensor field is a divergence-measure tensor field, while there are divergence-measure tensor fields that are not equilibrated [64].

Within this framework, in Lucchesi et al. [52], several examples of two-dimensional panels under different loads have been analyzed, with explicit determination of the singular stress lines.

In Lucchesi et al. [58], a formulation encoding the no-tension constraint of masonry is proposed, and the notions of weak compatibility (loads equilibrated by measure-valued stresses) and strong compatibility (loads equilibrated by square-integrable stresses) are introduced.

The no-tension constraint is expressed by requiring that a tensor-valued measure $T$ be admissible, i.e.,

T (A) a \cdot a \leq 0 for every Borel set A \subset Ω and every a \in ℝ^{n} .

Given a region $Ω$ with boundary split as $\partial Ω = \partial_{D} Ω \cup \partial_{N} Ω$ , a tensor-valued measure $T$ is said to weakly equilibrate the loads $(b, s)$ if

\int_{Ω} E (v) \cdot d T = \int_{Ω} v \cdot d b + \int_{\partial_{N} Ω} v \cdot d s

for every smooth function v vanishing on $\partial_{D} Ω$ . Here, $E (v) : = \frac{1}{2} (\nabla v + \nabla v^{T})$ denotes the symmetric gradient. The loads $(b, s)$ are said to be weakly compatible if there exists an admissible tensor-valued measure $T$ that weakly equilibrates them.

A more restrictive notion arises when one considers absolutely continuous measures. Specifically, let the loads and stress be of the form

s = s H^{n - 1} \partial_{N} Ω, b = b L^{n} Ω, T = T L^{n} Ω,

where s, b, and T are the square-integrable $(L^{2})$ vector-valued functions, with the first space taken with respect to $H^{n - 1}$ and the others with respect to $L^{n}$ . In this case, the equilibrium condition reduces to the classical weak form

\int_{Ω} E (v) \cdot T dx = \int_{Ω} v \cdot b dx + \int_{\partial_{N} Ω} v \cdot s d H^{n - 1},

for every smooth test function v such that $v = 0$ on $\partial_{D} Ω$ . The loads $(b, s)$ are said to be strongly compatible if there exists an $L^{2}$ stress field T taking values in ${Sym}^{-}$ that satisfies the above identity.

The distinction between weak and strong compatibility reflects the regularity of the stress field: in the weak case, stresses may concentrate on lower-dimensional sets (e.g., along cracks or interfaces), while in the strong case, they are genuinely distributed over the bulk, see [63].

The importance of strong compatibility is clarified by the associated variational formulation. Defining the total energy functional

I (v) = \int_{Ω} ŵ (E (v)) d L^{n} - \int_{Ω} v \cdot b d L^{n} - \int_{\partial_{N} Ω} v \cdot s d H^{n - 1},

one finds that the loads $(b, s)$ are strongly compatible if and only if the infimum of I, taken over all admissible v, is finite. In other words, strong compatibility is equivalent to the condition that the total energy is bounded from below, see [65].

Weak compatibility is generally easier to establish, since admissible stresses may be measure-valued and concentrate on lower-dimensional sets. A key contribution of Šilhavý and collaborators is to provide conditions under which weak compatibility implies strong compatibility, see [58]. These conditions typically involve geometric properties of the domain and integrability assumptions on the loads, ensuring that measure-valued equilibrating stresses do not exhibit singular concentrations.

Footnotes

Acknowledgements

We thank M. Lucchesi for providing us with the CV and the list of publications of M. Šilhavý.

References

Day

Šilhavý

. Efficiency and the existence of entropy in classical thermodynamics. Arch Ration Mech An 1977; 66(1): 73–81.

Kratochvíl

Šilhavý

. A theory of inelastic behavior of materials. Part II. Inelastic materials. Arch Ration Mech An 1977; 65(2): 131–152.

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. Note on the existence of entropy in classical thermodynamics. Arch Mech 1977; 29(2): 289–298.

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. On transformation laws for plastic deformations of materials with elastic range. Arch Ration Mech An 1977; 63(2): 169–182.

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. A theory of inelastic behavior of materials Part I. Ideal inelastic materials. Arch Ration Mech An 1977; 65(2): 97–129.

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. A condition equivalent to the existence of non-equilibrium entropy and temperature for materials with internal variables. Arch Ration Mech An 1978; 68(4): 299–332.

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. How many constitutive functions are necessary to determine a thermoelastic material? Czech J Phys 1979; 29(9): 981–996.

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. On measures, convex cones, and foundations of thermodynamics I. Systems with vector-valued actions. Czech J Phys 1980; 30(8): 841–861.

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. On measures, convex cones, and foundations of thermodynamics II. Thermodynamic systems. Czech J Phys 1980; 30(9): 961–991.

10.

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. On the second law of thermodynamics I. General framework. Czech J Phys 1982; 32(9): 987–1010.

11.

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. On the second law of thermodynamics II. Inequalities for cyclic processes. Czech J Phys 1982; 32(10): 1073–1099.

12.

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. On the Clausius inequality. Arch Ration Mech An 1983; 81(3): 221–243.

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. Asymptotic stability in nonlinear viscoelasticity. Q Appl Math 1984; 42(3): 281–294.

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. An admissibility criterion for shocks and propagating phase boundaries via thermodynamics of non-simple materials I. J Non-Equilib Thermodyn 1984; 9(3): 177–186.

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. An admissibility criterion for shocks and propagating phase boundaries via thermodynamics of non-simple materials II. J Non-Equilib Thermodyn 1984; 9(3): 187–200.

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. Thermostatics of non-simple materials. Czech J Phys 1984; 34(7): 601–621.

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. Phase transitions in non-simple bodies. Arch Ration Mech An 1985; 88(2): 135–161.

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. The existence of the flux vector and the divergence theorem for general Cauchy fluxes. Arch Ration Mech An 1985; 90(3): 195–212.

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. On the concepts of mass and linear momentum in Galilean thermodynamics. Czech J Phys 1987; 37(2): 133–157.

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. Asymptotic behaviour of classical solutions to the mixed initial-boundary value problem in finite thermo-viscoelasticity. Q Appl Math 1988; 46(2): 319–329.

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Šilhavý

. Mass, internal energy, and Cauchy’s equations in frame-indifferent thermodynamics. Arch Ration Mech An 1989; 107(1): 1–22.

22.

Samohýl

Šilhavý

. Mixture invariance and its applications. Arch Ration Mech An 1990; 109(4): 299–321.

23.

Lucchesi

Šilhavý

. Il’yushin’s conditions in non-isothermal plasticity. Arch Ration Mech An 1991; 113(2): 121–163.

24.

Nečas

Novotný

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. Global solution to the compressible isothermal multipolar fluid. J Math Anal Appl 1991; 162(1): 223–241.

25.

Nečas

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. Multipolar viscous fluids. Q Appl Math 1991; 49(2): 247–265.

26.

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. Cauchy’s stress theorem and tensor fields with divergences in L^p. Arch Ration Mech An 1991; 116(3): 223–255.

27.

Nečas

Novotný

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. Global solution to the viscous compressible barotropic multipolar gas. Theor Comp Fluid Dyn 1992; 4(1): 1–11.

28.

Šilhavý

. Energy principles and the equations of motion in Galilean thermomechanics. Czech J Phys 1992; 42(4): 363–374.

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Lucchesi

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. Thermoplastic materials with combined hardening. Int J Plasticity 1993; 9(3): 291–315.

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. Note on Onsager’s relations. Q Appl Math 1994; 52(3): 469–479.

31.

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. The mechanics and thermodynamics of continuous media. Springer, 1997.

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. Polar decomposition of rank 1 perturbations in two dimensions. Math Mech Solids 1998; 3(1): 107–112.

33.

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. Convexity conditions for rotationally invariant functions in two dimensions. In: Sequeira

da Veiga

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34.

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. On isotropic rank 1 convex functions. R Soc Edinb Proc A 1999; 129(5): 1081–1105.

35.

Šilhavý

. On the compatibility of wells. J Elasticity 1999; 55(1): 11–17.

36.

Parry

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. Invariant line integrals in the theory of defective crystals. Atti Accad Naz Lincei Cl Sci Fis Mat Nat Rend Lincei Mat Appl 2000; 11(2): 111–140.

37.

Parry

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. On rank one connectedness, for planar objective functions. J Elasticity 2000; 58(2): 177–189.

38.

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. Differentiability properties of isotropic functions. Duke Math J 2000; 104(3): 367–373.

39.

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. Differentiability properties of rotationally invariant functions. J Elasticity 2000; 58(3): 225–232.

40.

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. Rank 1 convex hulls of isotropic functions in dimension 2 by 2. Math Bohem 2001; 126(2): 521–529.

41.

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. Rank 1 perturbations of deformation gradients. Int J Solids Struct 2001; 38(6–7): 943–965.

42.

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. Rotationally invariant rank 1 convex functions. Appl Math Opt 2001; 44(1): 1–5.

43.

Padovani

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. Relaxed energy for transversely isotropic two-phase materials. J Elasticity 2002; 67(3): 187–204.

44.

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. An O(n) invariant rank 1 convex function that is not polyconvex. Theor Appl 2002; 28–29: 325–336.

45.

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. Monotonicity of rotationally invariant convex and rank 1 convex functions. R Soc Edinb Proc A 2002; 132(2): 419–435.

46.

Šilhavý

. On SO(n)-invariant Rank 1 convex functions. J Elasticity 2003; 71(1–3): 235–246.

47.

Šilhavý

. Semiconvexity of invariant functions of rectangular matrices. Calc Var Partial Dif 2003; 17(1): 75–84.

48.

Šilhavý

. On semiconvexity properties of rotationally invariant functions in two dimensions. Czech Math J 2004; 54(3): 559–571.

49.

Lucchesi

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Zani

. Singular equilibrated stress fields for no-tension panels. Ln App C M 2005; 23: 255–265.

50.

Šilhavý

. Divergence measure fields and Cauchy’s stress theorem. Rend Semin Mat Univ Padova 2005; 113: 15–45.

51.

Fosdick

Šilhavý

. Generalized Baker-Ericksen inequalities. J Elasticity 2006; 85(1): 39–44.

52.

Lucchesi

Šilhavý

Zani

. A new class of equilibrated stress fields for no-tension bodies. J Mech Mater Struct 2006; 1(3): 503–539.

53.

Šilhavý

. Fluxes across parts of fractal boundaries. Milan J Math 2006; 74(1): 1–45.

54.

Bertram

Böhlke

Šilhavý

. On the rank 1 convexity of stored energy functions of physically linear stress-strain relations. J Elasticity 2007; 86(3): 235–243.

55.

Lucchesi

Šilhavý

Zani

. A note on equilibrated stress fields for no-tension bodies under gravity. Q Appl Math 2007; 65(4): 605–624.

56.

Šilhavý

. Ideally soft nematic elastomers. Netw Heterog Media 2007; 2(2): 279–311.

57.

Šilhavý

. Zeros of the polyconvex hull of powers of the distance and s-polyconvexity. J Convex Anal 2007; 14(2): 318–344.

58.

Lucchesi

Šilhavý

Zani

. Integration of measures and admissible stress fields for masonry bodies. J Mech Mater Struct 2008; 3(4): 675–696.

59.

Lucchesi

Šilhavý

Zani

. On the balance equation for stresses concentrated on curves. J Elasticity 2008; 90(2): 209–223.

60.

Padovani

Pasquinelli

Šilhavý

. Processes in masonry bodies and the dynamical significance of collapse. Math Mech Solids 2008; 13(7): 573–610.

61.

Šilhavý

. Cauchy’s stress theorem for stresses represented by measures. Continuum Mech Therm 2008; 20(2): 75–96.

62.

Šilhavý

. Normal currents: structure, duality pairings and div-curl lemmas. Milan J Math 2008; 76(1): 275–306.

63.

Lucchesi

Šilhavý

Zani

. Equilibrated divergence measure stress tensor fields for heavy masonry bodies. Eur J Mech A Solids 2009; 28(2): 223–232.

64.

Šilhavý

. The divergence theorem for divergence measure vectorfields on sets with fractal boundaries. Math Mech Solids 2009; 14(5): 445–455.

65.

Lucchesi

Padovani

Šilhavý

. An energetic view on the limit analysis of normal bodies. Q Appl Math 2010; 68(4): 713–746.

66.

Šilhavý

. Phase transitions with interfacial energy: convexity conditions and the existence of minimizers. CISM Int Cent Mech Sci Courses Lect 2010; 516: 177–239.

67.

Šilhavý

. Phase transitions with interfacial energy: interface null Lagrangians, polyconvexity, and existence. IUTAM Bookser 2010; 21: 233–244.

68.

Lucchesi

Šilhavý

Zani

. Integration of parametric measures and the statics of masonry panels. Ann Solid Struct Mech 2011; 2(1): 33–44.

69.

Šilhavý

. Equilibrium of phases with interfacial energy: a variational approach. J Elasticity 2011; 105(1–2): 271–303.

70.

Šilhavý

. The effective energy in the Allen-Cahn model with deformation. Interface Free Bound 2011; 13(2): 255–270.

71.

Lucchesi

Šilhavý

Zani

. Equilibrium problems and limit analysis of masonry beams. J Elasticity 2012; 106(2): 165–188.

72.

Lucchesi

Šilhavý

Zani

. On the choice of functions spaces in the limit analysis for masonry bodies. J Mech Mater Struct 2012; 7(8–9): 795–836.

73.

Lucchesi

Šilhavý

Zani

. A direct approach to fiber and membrane reinforced bodies. Part I. Stress concentrated on curves for modelling fiber reinforced materials. Continuum Mech Therm 2013; 25(2–4): 537–558.

74.

Padovani

Šilhavý

. Coaxiality of stress and strain in anisotropic no-tension materials. Meccanica 2013; 48(2): 487–489.

75.

Šilhavý

. A direct approach to nonlinear shells with application to surface-substrate interactions. Math Mech Complex Syst 2013; 1(2): 211–232.

76.

Šilhavý

. Phase equilibria in isotropic solids. Continuum Mech Therm 2013; 25(6): 795–802.

77.

Lucchesi

Šilhavý

Zani

. A direct approach to fiber and membrane reinforced bodies. Part II. Membrane reinforced bodies. Continuum Mech Therm 2014; 26(3): 343–372.

78.

Šilhavý

. Collapse mechanisms and the existence of equilibrium solutions for masonry bodies. Math Mech Solids 2014; 19(7): 821–831.

79.

Šilhavý

. Mathematics of the Masonry–like model and limit analysis. CISM Int Cent Mech Sci Courses Lect 2014; 551: 29–69.

80.

Ghiba

Neff

Šilhavý

. The exponentiated Hencky-logarithmic strain energy. improvement of planar polyconvexity. Int J Non Linear Mech 2015; 71: 48–51.

81.

Lucchesi

Pintucchi

Šilhavý

, et al. On the dynamics of viscous masonry beams. Continuum Mech Therm 2015; 27(3): 349–365.

82.

Lucchesi

Šilhavý

Zani

. A direct approach to membrane reinforced bodies. In: Aita

Pedemonte

Williams

(eds) Masonry structures: between mechanics and architecture. Birkḧauser-Springer, 2015, pp.291–312.

83.

Šilhavý

. Differentiability of the metric projection onto a convex set with singular boundary points. J Convex Anal 2015; 22(4): 969–997.

84.

Šilhavý

. On the approximation theorem for structured deformations from BV(Ω). Math Mech Complex Syst 2015; 3(1): 83–100.

85.

Šilhavý

. The convexity of C ↦ h (det C). Tech Mech 2015; 35(1): 60–61.

86.

Bock

Jarušek

Šilhavý

. On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate. Nonlinear Anal Real World Appl 2016; 32: 111–135.

87.

Šilhavý

. A remark on polyconvex functions with symmetry. J Elasticity 2016; 122(2): 255–260.

88.

Padovani

Šilhavý

. On the derivative of the stress-strain relation in a no-tension material. Math Mech Solids 2017; 22(7): 1606–1618.

89.

Šilhavý

. Higher gradient expansion for linear isotropic peridynamic materials. Math Mech Solids 2017; 22(6): 1483–1493.

90.

Šilhavý

. Stresses in equilibrium configurations of inextensible nets with slack. Math Mech Solids 2017; 22(4): 649–665.

91.

Šilhavý

. The general form of the relaxation of a purely interfacial energy for structured deformations. Math Mech Complex Syst 2017; 5(2): 191–215.

92.

Šilhavý

. A variational approach to nonlinear electro-magneto-elasticity: convexity conditions and existence theorems. Math Mech Solids 2018; 23(6): 907–928.

93.

Šilhavý

. Polyconvexity for functions of a system of closed differential forms. Calc Var Partial Dif 2018; 57(1): 26.

94.

Šilhavý

. Consequences of the Coleman-Noll inequality for isotropic materials. ZAMM Z Angew Math Mech 2019; 99(6): e201800324.

95.

Šilhavý

. Isotropic polyconvex electromagnetoelastic bodies. Math Mech Solids 2019; 24(3): 738–747.

96.

Šilhavý

. The scientific work of Bernard D. Coleman. Math Mech Solids 2019; 24(2): 339–360.

97.

Šilhavý

. Fractional vector analysis based on invariance requirements (critique of coordinate approaches). Continuum Mech Therm 2020; 32(1): 207–228.

98.

Šilhavý

. A new approach to curvature measures in linear shell theories. Math Mech Solids 2021; 26(9): 1241–1263.

99.

Šilhavý

. The Gauss-Green theorem for bounded vector fields with divergence measure on sets of finite perimeter. Indiana U Math J 2023; 72(1): 29–42.

100.

Šilhavý

. Fractional strain tensor and fractional elasticity. J Elasticity 2024; 155(1–5): 425–447.

The scientific life of Miroslav Šilhavý (1949–2025)

Abstract

1. Interfacial semiconvexities

1.1. Bulk and interfacial polyconvexity

2. Thermodynamics

2.1. From efficiency to entropy

2.2. Mathematical formalization of thermodynamics

3. No-tension materials

Footnotes

Acknowledgements

References