How matter deforms: fluids and solids

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pont millau - encyclopedie environnement - fluids and solids

An ice cube in a glass is a fragile elastic solid, but the Sea of Ice in Chamonix flows like a viscous fluid. Indeed, any material can change from solid to fluid behaviour, and vice versa, under the effect of mechanical or thermal stresses. But how can we describe these two states of matter and the associated behavioural classes: elasticity, viscosity and plasticity? Why are new variables, stresses and strains, needed to describe the internal stresses and strains of the material? What are the relationships that link stress and strain between them? Why does the calculation of structures or structures require the use of such relationships? This is the purpose of this article.

1. Fluids and solids

There are classically three states of matter: gases, liquids and solids. But other, more complex states also exist, such as pastes, gels and plasmas, which we will not discuss in this article. Gases and liquids are mechanically close and we speak indistinctly of fluids. However, in an enclosure, the gases, compressible, occupy all the available volume and their behaviour is well described, in the absence of a chemical reaction, by their state law which links pressure, volume and temperature (see Pressure, temperature and heat). Liquids are almost uncompressible, although they have a certain elasticity: they undergo a small volume variation proportional to the pressure variation. In a chamber, the liquids will occupy part of the volume by adopting the shape of this chamber.

fluides et solides - fluide - solide - eau - verre eau - metal - metal block - water
Figure 1. A metal block has its own shape: it is a solid; water takes the shape of the container that contains it: it is a fluid.

In addition, we notice that a fluid does not resist an applied pressure if it is not isotropic (i.e., identical in all directions of space): the fluid flows. Here, there is a major difference with bodies qualified as solids in that a solid sample can withstand pressure forces applied in different directions up to a certain limit beyond which the solid can no longer resist: it breaks. A solid thus keeps a clean shape, while a fluid takes the shape of the container that contains it (Figure 1).

Generally speaking, for all materials, the simplest behaviour corresponds to small deformations for which the internal dissipation of energy into heat is negligible. In this context, when the applied forces are removed, the associated deformations also cancel each other out and the material returns to its original shape. This mechanical behaviour is called reversible and any such behaviour is called elastic. Otherwise, we talk about inelasticity.

We have just seen some general characteristics of materials, which can deform without energy loss (elastic media) or with energy dissipation (inelastic media), flow (fluids) or break (solids). These different behaviours represent what is called the “rheology” [1] of the material. If we want to be able to carry out calculations of structures (planes, cars, machines, etc.) and structures (roads, bridges, dams, dikes, etc.), these concepts will first have to be specified (section 2) and then formalised (section 4). But this formalisation will require the prior generalisation of the concepts of forces and displacements. This will be the subject of section 3. Finally, Section 5 provides an overview of modern numerical calculations of structures and structures and explains why, for the engineer or geotechnical expert, the choice of laws of behaviour, well adapted to all the materials involved in the calculation, is so crucial.

2. Elasticity, viscosity and plasticity

Our daily experience shows that these elementary behaviours are not sufficient to characterize the extreme variety of resistance of the natural or industrial materials we encounter. Thus, an industrial grease will be able to resist non-isotropic pressures and this is all the better as the speed with which these pressures are applied is high. It is said that the grease is viscous and, as a first approximation, it can be assumed that the resistance of this fluid is proportional to the speed of application of the pressure. This is called Newtonian viscosity. The water itself is also viscous, about a thousand times less than oil, but enough to explain some of the friction on the hulls of ships, where there is indeed a greater resistance as the speed of the ship increases. Even gases, such as air, have a viscosity, 50 times lower than water but responsible, for example, for the persistence of clouds by considerably slowing the fall of the droplets that constitute them [2] (Read Trailed suffered by moving bodies).

glacier -montagne - mer glace chamonix - glacier flows - sea of ice chamonix france
Figure 2. A glacier flows like a viscous fluid in the shape of the valley that contains it. The picture shows the Sea of Ice in Chamonix.

Solids can also exhibit viscous behaviour, for example when they deform over time by a phenomenon called creep. A lead wire from which a weight is suspended will gradually elongate over time by creep. More generally, different time scales or temperature variations can transform a fluid into a solid or vice versa. Thus, an ice cube will behave like a solid elastic-fragile elastic at the exit of its ice box. It is indeed a solid since it has its own shape, and, under low applied pressures, its deformations return to zero when the pressure is cancelled – its behaviour is therefore reversible and elastic. However, under a violent shock, it will break abruptly, displaying fragile behaviour. On the other hand, on a year-round basis, ice can behave like a viscous fluid: for example, the tongue of the Mer de Glace in Chamonix (Figure 2) flows at a rate of about 100 m/year following the meanders of the valley. We can deduce an estimate of its viscosity: 1016 times that of water.

An ice cube and a solid in general can break. Its clear fracture into several pieces is technically called brittle fracture, as obtained with an ice pick for ice or a hammer for glass. But there is another form of rupture, for which the material does not divide into fragments: it is the so-called ductile rupture, described by the theory of plasticity. This plastic deformation mainly corresponds to irreversible relative displacements of the components constituting the material, such as boulders, grains of sand, clay particles, grains of powders and powdery materials, metal crystals, ice crystals…

These plastic ruptures involve friction forces, internal to the materials, instantaneous and independent of speed, according to the solid friction law known as the Coulomb law (Read the focus What is the Coulomb friction law? associated with the article on sand). However, it should be remembered that, when the deformation of the solid is viscous, these internal forces increase with the deformation rate. Thus the lead wire, mentioned above, will almost instantaneously undergo elastic (reversible) elongation for a low load, plastic elongation (irreversible) for a load that exceeds the so-called failure criterion, and viscous elongation (creep) proportional to time for a lower load that remains. Iron can also undergo plastic deformations, all the more easily when its temperature is high: it thus becomes ductile. The blacksmith knows this phenomenon well, which allows him to deeply deform a metal part without breaking it into fragments. The metal flows all the better as the temperature increases, and at the melting temperature it changes state and turns into an ordinary liquid, not much more viscous than water.

sable - trace pas sable - sablier - fluide solide -
Figure 3. A given material, such as sand, behaves like an elasto-plastic solid under the steps of a walker on the beach, and like a friction fluid when it flows into an hourglass. Depending on the load applied to a material, its behaviour can often cover the entire range of elasto-visco-plasticity.

In short, plastic deformations are instantaneous, while viscous deformations are delayed (spread over time). The footprint of a foot on the sand of the beach is both instantaneous and irreversible (permanent): the behaviour of the sand can therefore be described here as “plastic”. The one of a foot in a muddy field will be irreversible but will increase over time (if you freeze in place long enough!) by creep: the behaviour of the clay will be viscous here. As for the perfectly reversible deformations, which cancel each other out when the load is cancelled, we have seen that they are described by the theory of elasticity. In short, reversible deformations are elasticity; irreversible deformations are viscosity when they are a function of time; they are plasticity when they are independent of time.

Today, it is often accepted that there are no intrinsically solid materials, but rather behavioural domains with solid or fluid characteristics for a given material, whose general behaviour will be of the elasto-visco-plastic type. As an illustration, let us take the case of sand: elasto-plastic solid, on which we walk on the beach, and granular fluid, which flows into an hourglass (Figure 3).

3. Stresses and deformations

The movement of a material object assimilable to a point is entirely described by the notions of force vector and displacement vector, linked by the laws of dynamics (Read The Laws of dynamics). According to these laws, the force applied is equal to the product of the mass of this object by its acceleration. In addition, a fluid, gas or liquid, is characterized as a first approximation by its pressure and volume (if thermal aspects are not taken into account), linked by a state law.

Let us now consider a deformable solid and see why the previous notions of force-displacement or pressure-volume no longer apply and must be generalized. Let’s take the example of a bucket filled with sand. If we apply vertical pressure to it by pressing on the upper free surface, experience shows that about half of this pressure is measured on the lateral surfaces of the bucket and a little less than this pressure on the bottom of the bucket. If the bucket had been filled with water, the same pressure would have been collected everywhere: water behaves isotropically, but sand does not. It was therefore necessary to imagine a new mathematical tool to describe this pressure in a deformable solid, which varies with the direction within the solid, or more precisely with the orientation of any facet, which can be isolated by thinking within the solid and on which this pressure is applied. The notion of vector does not allow to describe this directional variation of the force per unit area. The tool that allows this description is a tensor [3] of order 2, we will see how to build it.

elementary cube of matter
Figure 4. An elementary cube of matter (infinitely small) is subjected on its 6 faces to surface densities of forces (forces per unit area) of any orientation. Note that the sum of the applied forces and the sum of the moments are zero for the overall equilibrium of the cube. This distribution of forces on the cube is entirely characterized by a “constraint matrix”, as defined in the text. Under the action of the forces applied to it, the elementary cube is deformed into an elementary volume with 6 flat faces, constituted by parallel parallelograms two by two, constituting an oblique parallelepiped. This change in shape is entirely described by the pure deformation (whose matrix is defined in the text), while the overall rotation of the cube is given by the rotation matrix. [Source: © Encyclopedia of the Environment]

On each of the faces of the cube in Figure 4 is exerted a set of forces, not necessarily perpendicular to the face. The notion of pressure (always perpendicular to the surface in the case of a static fluid) must therefore be generalized by considering oblique forces with respect to the surface. The mere fact that one can walk on the ground without slipping shows that one can effectively apply tangential forces to the surface of a solid and that it can resist them. According to the principle of action and reaction, the six forces represented on the faces of the cube in Figure 4 are equal two to two on opposite faces. There are therefore only three independent force vectors. These three forces per unit area, called stress vectors, make it possible to construct a square matrix of 3 rows and 3 columns: this is the matrix of stress tensor components. It is shown that it makes it possible to calculate the surface force exerted on any facet, of any orientation, inside the material.

Thus in this case of the sand bucket, the vertical stress in the bucket will be equal to :

σv = F / S, where F is the total vertical force applied and S is the surface of a horizontal section of the bucket. The horizontal or radial stress, applied to the side walls of the bucket, will be approximately equal to :

σh = σv / 2 = F / 2S.

These two constraints are called main constraints because they apply perpendicular to the respective horizontal and lateral facets.

contrainte tangentielle - encyclopedie environnement
Figure 5. Within a block of triangular material weighing, on a facet parallel to the oblique free surface, a roughly vertical constrained vector appears, which decomposes into a “tangential constraint” vector in the plane of the facet and a “normal constraint” vector in the direction perpendicular to the facet. [Source: © Joël Sommeria]

In the case of any orientation of the facet, the constrained vectors are not perpendicular to it. A distinction is then made between the stress vector projected on the plane of the facet – called shear stress or tangential stress and the one projected perpendicular to this plane, on the normal to the facet – called normal stress (see Figure 5).

Let us now return to the notion of displacement of a material point or variation in the volume of a material, which we must also generalize. Indeed, the experiment conducted on the sand bucket shows that the free upper surface of the sand settles under the action of vertical pressure exerted by the hand, while the lateral surfaces, held by the bucket, remain fixed. Here again, it appears that the displacements of these surfaces are no longer identical in all directions of space as in the case of a fluid, but on the contrary that they vary with these directions.

If we assimilate each small volume of sand to a material point, we obtain what is called a displacement field, describing the displacement of any material point of the sand – concretely, any grain of sand – in the bucket. A question arises here. In the vertical direction of the bucket, there will be significant movement near the free surface and no movement at the bottom of the bucket. They will actually vary linearly with depth. However, each small cube, subjected to the same vertical pressure, deforms itself vertically in the same way. We must therefore move from the notion of “displacement” to the more intrinsic notion of “deformation“. This is done by a gradient operation [4], which transforms the vertical displacement into a dimensionless quantity called “vertical deformation”, identical at any point of the sand and equal to the ratio of the vertical displacement of a grain of sand by the height of that grain in the bucket.

But we need to generalize this experiment, because what would happen if the walls of the bucket were to deform under the action of vertical pressure on the free surface of the sand? Then, the movement of the grains of sand would no longer be vertical but oblique. By taking the gradient of this displacement field (3-component vectors), we obtain a deformation matrix field (3 rows and 3 columns) completely characterizing the deformation of the sand in a deformable bucket.

This square matrix of 3 rows and 3 columns can actually be decomposed into the sum of two matrices: one describes the rotation of the cube of material during the application of forces and the other the actual deformation of the cube (without rotation), this is called pure deformation (see an illustration in Figure 4).

On the simple example of the rigid-walled sand bucket, the vertical deformation is equal to : εv = δH / H, where H is the height of the bucket, while the horizontal or radial deformation is equal: εh = δD / D, where D is the diameter of the bucket. These deformations are called main deformations because they are perpendicular to the respective horizontal and vertical or lateral surfaces.

In fact, in general, it is shown that pure deformation makes it possible to find the two fundamental modes of deformation of an elementary cube of material subjected to any kind of stress: the length of a material segment inside the cube changes and the angle between two material segments also varies during the deformation of the cube.

It was the Egyptians who first defined these two concepts of length and angle, which allowed them to recover the boundaries of their fields, covered by the silts of the Nile floods, after the passage of these floods. Even today, we still represent the deformation of matter through these two notions, and it is precisely shown that pure deformation provides us with these two quantities at any point of a deformable material.

4. Behavioural laws

Each material is deformed in a way that is unique to it and that characterizes what is called its mechanical behaviour, formalized by a mathematical relationship called the “law of behaviour“. In general, the stress at a given point in the material and at a given time is a function of the entire history of pure deformation at that point. In the case of elastic behaviour, a simple mathematical function (independent of any history) links stress to strain.

The elasticity is called linear, if this function is linear and a proportionality relationship then links stresses and deformations: the stress matrix is, in this case, equal to the elastic tensor multiplied by the deformation matrix. In the usual case of isotropic elastic material (its behaviour is identical in all directions of space), this tensor depends on only two parameters: Young’s modulus (characterizing the stiffness of the material) and the fish coefficient (reflecting its ability to deform laterally under axial compression).

Newtonian viscosity, the simplest of the viscous laws, translates into a proportionality relationship between stress and pure deformation rate at the same time. This type of relationship thus makes it possible to describe the increase in the strength of the material with the speed of application of the forces.

dune sable - desert - sable - desert namibie - encyclopedie environnement - desert dunes
Figure 6. The slope of a dune corresponds approximately to the angle of friction, plastic, global, of a sand. This angle can vary with the density of the sand and the shape, more or less angular, of the grains. In general, this angle is close to 30°.

Finally, in elasto-plasticity, this relationship between stress and strain is no longer univocal as in pure elasticity and we then prefer to use a so-called incremental writing linking a small variation in stress to a small variation in strain. An elasto-plastic tensor is thus created, connecting the incremental stress to the incremental deformation in a proportional way. The material that most faithfully obeys an elasto-plastic behaviour is sand. Thus, a sand pile or a balanced dune has a slope (Figure 6), the angle (around 30°) of which corresponds to the internal plastic friction between the grains of sand. On the contrary, a viscous liquid, devoid of plasticity, would gradually spread out over time. The plastic behaviour of the sand allows the slope to be stable without spreading. However, if a little sand is added along the slope, a local avalanche occurs immediately, showing that the mechanical state of the sand here corresponds to a state of plasticity called “limit” corresponding to the failure criterion mentioned in section 2: this slope limit angle cannot be exceeded.

One of the interesting properties of plasticity is the phenomenon of strain hardening, which reflects the possibility of improving the mechanical strength of a material by plastically deforming it. A remarkable illustration is the implementation of the underlay of a pavement. The sand is dumped by trucks and its very loose condition allows it only a very low resistance: a finger can sink into it. But, once compacted by rollers that strain it very hard, the sand layer becomes remarkably resistant: a car can already drive on this layer. However, if there is no glue between the sand grains, it will have difficulty braking because the sand grains will not provide sufficient resistance to tire slippage. Bitumen provides this glue: it ensures the cohesion of the sand or its mixture with gravel. But, on the other hand, this cohesion creates the possibility of cracks, which degrade the pavement..

We understand here the intrinsic numerical difficulties involved in the calculation of a metal structure, the sizing of a civil engineering structure, the prediction of the flow of a complex fluid or, more generally, the behaviour of any mechanical system, since it is true that the elasto-visco-plastic behaviour of materials is complex to calibrate experimentally and to describe numerically. But how can we precisely calculate – that is, ultimately predict – the mechanical behaviour of a system subjected to forces that can change over time?

5. Solving a problem in the mechanics of deformable materials

Whether considering fluids or solids, to calculate and predict the behaviour of a structure and a structure, the engineer has to write and solve three groups of mathematical equations of a very different nature:

  • Conservation laws (of mass, energy, amount of movement, etc.), valid regardless of the material and the problem considered. These laws have generally been known and formalised for a very long time.
  • The laws of behaviour of materials whose formalization we have described in this article. These laws, which are, in the most general case, elasto-visco-plasticity as we have seen, are still the subject of active research and nowadays incorporate a detailed description of the microstructure of materials (and even of their nano-structure, when we consider nano-materials today).
  • The initial and boundary conditions that define the system, mechanical structure, structure, structure, flow, etc., that are being modelled. The initial conditions characterize the initial state (at the beginning of the calculation) of the system. Boundary conditions, on the other hand, determine all the geometries and forces that will be applied over time. Industrial calculation codes are currently able to take into account a very wide variety of such conditions.
glissement terrain trevoux - landslide trevoux
Figure 7. Landslide at Trévoux, modelled using the finite element method. The mesh size (division into small elements where uniform behaviour is assumed) of the soil mass is represented, with the various soil layers separated by red lines. The retaining wall (which can be seen in the upper part of the figure at the centre) and the embankments are taken into account. An elasto-plastic behavioural law, as mentioned above, with hydro-mechanical coupling is taken into account on each mesh. Loading is applied step by step (increment by increment) to facilitate numerical resolution, which is essentially the inversion of a succession of matrices (possibly very large in size – here about 5000 equations to 5000 unknown). [Source: © Doctoral thesis by H.D.V. Khoa, INP Grenoble]

Sometimes several physicochemical phases are present (for example, soil, air and water, or rock, oil and gas) and we speak of multi-phase environments. In other cases, the problem reveals “coupled” loads. These are called multi-physical problems (for example, coupling between mechanics and thermics if the temperature occurs, or coupling between mechanics and chemistry in the case of chemical reactions between the components of the system, etc.).

glissement terrain trevoux - encyclopedie environnement - landslide trevoux
Figure 8. We see here a typical result of the calculation, by the finite element method, of the Trévoux landslide. The blue areas are unstable areas that can slide during heavy rains. The calculation makes it possible to vary the height of the groundwater table (represented by the dark blue dashed line inside the soil mass). The river, filled with water, is visible on the left (“swelling level”). We can note the stabilizing influence of the retaining wall, behind which there is a red colour showing a stable domain. However, numerical modelling shows the possibility of deep landslides (blue in colour), which can carry away the retaining wall – which actually happened in Trévoux. [Source: © Doctoral thesis by H.D.V. Khoa, INP Grenoble]

For the engineer, the most delicate question most often lies in the choice and implementation of behavioural laws, as representative as possible of the materials concerned in the problem under consideration. Modern numerical methods make it possible to solve the system of equations formed by the three groups of equations mentioned above. The number of equations can reach several million and requires the use of supercomputers.

While it can be assumed that the continuity of the material is respected during its deformation, the best known method is the so-called “finite element” method (Figures 7 and 8). The calculation presented on these two figures made it possible to digitally model, and therefore understand, a landslide that actually occurred in Trevoux after torrential rains (see Landslides).

cube granulaire - cube matiere granulaire - encyclopedie environnement - cube of granular material
Figure 9. Cube of granular material, modelled by the discrete element method. This virtual digital sample of 10,000 elastic spheres, in contact with each other, made it possible to visualize, understand and analyze the behaviour of sands (dry or containing capillary water) – and in particular their different failure modes according to the applied forces. [Source: © Doctoral thesis by L. Sibille, INP Grenoble]

Today we could quantify critical rainfall. Prevention devices (walls, embankments, soil reinforcement, drainage, etc.) can also be digitally tested to validate or not their effectiveness. On the other hand, if the materials have intrinsic and irreducible discontinuities (for example, in the case of fracturing the material such as a fractured cliff), another numerical method called “discrete elements” can be used (Figure 9). We then obtain by numerical calculation the displacements and rotations of each elementary block, which can number several million. For example, the fragmentation of a concrete block impacted by a projectile can be simulated numerically. This method also allows the calculation of avalanches (of snow, boulders,…) and torrential mudflows. (Read Landslides and rockfalls, a fatality?).

 


References and notes

Cover image. The Millau cable-stayed bridge is an example of a structure whose design has required in-depth studies on the behaviour of its constituent materials. Source: Wikipedia, Attribution License – Sharing of the identical initial conditions v. 2.5 of Creative Commons, better known as “CC-BY-SA-2.5; © Mike Lehmann, Mike Switzerland 10:38, 14 March 2008 (UTC)”.

[1] Rheology is the science that describes how matter flows or, more generally, is distorted. The word comes from the Greek “ρεω” which means “to flow” and from “λοϒοσ” “speech”.

[2] In the case of vehicles or other large objects, the driving resistance is dominated by turbulence and viscosity has little influence.

[3] A tensor is a mathematical object that respects the so-called rules of tensoriality. These rules allow us to continue to describe the same physical being, whatever the benchmark we have chosen to express its components. The components of a first-order tensor, which is a vector, form a row or column of numbers; those of a second-order tensor form a matrix (table of numbers). Stresses and deformations are thus represented using matrices. Tensors of any finite order are defined. For example, the elastic tensor (discussed below) is of order 4.

[4] The gradient of a scalar function is a vector whose components are the partial derivatives of the function with respect to spatial coordinates. By extension, the gradient of a vector is a matrix whose column vectors are made up of partial derivatives of each of the vector’s components with respect to spatial coordinates. In the classic 3-dimensional space, this results in a square matrix of 3 rows and 3 columns.


The Encyclopedia of the Environment by the Association des Encyclopédies de l'Environnement et de l'Énergie (www.a3e.fr), contractually linked to the University of Grenoble Alpes and Grenoble INP, and sponsored by the French Academy of Sciences.

To cite this article: DARVE Félix (January 15, 2021), How matter deforms: fluids and solids, Encyclopedia of the Environment, Accessed December 24, 2024 [online ISSN 2555-0950] url : https://www.encyclopedie-environnement.org/en/physics/how-matter-deforms-fluids-and-solids/.

The articles in the Encyclopedia of the Environment are made available under the terms of the Creative Commons BY-NC-SA license, which authorizes reproduction subject to: citing the source, not making commercial use of them, sharing identical initial conditions, reproducing at each reuse or distribution the mention of this Creative Commons BY-NC-SA license.

物质是如何形变的——流体和固体

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pont millau - encyclopedie environnement - fluids and solids

  玻璃杯里的冰块是易碎的弹性固体,而霞慕尼小镇的冰海则更接近黏粘性流体。事实上,在机械应力或热应力的作用下,任何材料都可以在固体和流体之间相互转化。但是,我们要如何描述物质的这两种物态以及弹性、黏粘性、塑性等相关力学特性呢?为什么要引入应力和应变这两个新物理量来描述材料的内部变化?应力和应变之间的关系是怎样的?为什么对材料结构的计算需要依赖于这种关系?本文将主要讨论以上问题。

1. 固体和流体

  在经典的理论中,物质有三种物态(Three States of Matter):气态、液态和固态。当然也存在其他更复杂的状态,比如非牛顿流体、胶体和等离子体,但这些复杂物态不在本文的讨论范畴内。气体和液体在力学特性上相近,可以将它们统称为流体(Fluid)。但是二者也存在不同之处。在封闭环境中,气体会占据整个封闭空间,且可以压缩;在不考虑化学反应的情况下,人们只需要用温度、压强和体积之间的状态关系就可以对气体进行较为准确的描述(详见《压强、温度和热量》)。与之相比,随着压强的变化,液体的体积只会成比例地发生微小的变化;尽管具有一定的弹性,但液体几乎是不可压缩的;在容器中,液体会适应容器的形状,并只占据容器的一部分空间。

环境百科全书-固体-固体与流体
图1.金属块能保持自己的形状,因此它是固体;水会随盛装的容器改变形状,因此它是流体。

  此外,值得注意的是,流体不能承受施加于其上的非各向同性作用力(各向同性(isotropic):指在空间中所有方向都相同),而会受力流动。这一点是流体与各种固体(Solid)材料的主要区别之一。相比之下,固体能够承受这种施加在不同方向上的不同作用力,直至超过材料可承受的某一限度,进而发生破坏。因此,固体会保持固定的形状,而流体则会适应其容器的形状(图1)。

  通常情况下,任何材料最简单的行为都是微小形变,在这种形变中,内部能量耗散产生的热量可以忽略不计。在这种情况下,当所施加的力被移除时,对应的形变也随之消除,材料会回到初始的形状。这种力学行为被称为可逆(Reversible)的,材料的这种属性被称为弹性,反之称为非弹性。

  我们刚刚介绍了材料的某些一般特性,这些特性可以使材料在没有能量损失(弹性介质)或是在有能量耗散(非弹性介质)的情况下发生形变,也可以使材料流动(流体)或者断裂(固体)。上述各种行为构成了材料的“流变学”(rheology)[1]。如果我们想要计算飞机、汽车、机械或道路、桥梁、水坝、堤坝等物体的结构及其变化,就必须先定义材料属性相关的概念(第2节),然后理清它们之间的关系(第4节)。但是理清关系需要先对力和位移的概念进行推广,第3节将主要介绍这方面内容。最后,第5节概述了结构力学的现代数值计算方法,并解释了工程师或地质学家在计算中为何必须选择适合的材料行为规律。

2. 弹性、粘性和塑性

  日常经验表明,上述的基本力学行为不足以描述各种天然或是人造材料所遭遇的极端阻力变化。例如,当两机械结构发生相对错动时,如其间存在较强的非各向同性作用力(即摩擦力),添加润滑油是不错的选择,可以有效减弱这种作用力。这是因为润滑油具有黏粘性,这种黏粘性使得其对非各向同性作用力的抵抗能力与两机械结构间的相对错动速度近似成正比(取一次近似)。流体的这种特性被称为牛顿粘度(Newtonian viscosity)。日常生活中无处不在的水也是有黏粘性的,尽管粘度只有润滑油的千分之一左右,但也足以让船在水中航行时感受到阻力,当船速增加,这种阻力也变大。同液体一样,气体也有黏粘性,例如,空气的粘度约为水的1/50,虽然微弱,却十分有用。举例来说,空气的黏粘性可以减缓构成云的液滴的下落,从而让云存在得更久[2]。(详见《运动物体受到的阻力》)。

环境百科全书-固体-夏慕尼冰海
图2.霞慕尼小镇的冰海。冰川变成山谷的形状,像黏粘性流体一样流动。

  固体也可以表现出黏粘性行为,例如,固体会随着时间的推移而变形,这种现象被称为蠕变(Creep)。受蠕变现象影响,悬着重物的铅丝会随时间推移逐渐拉长。广泛来看,温度的改变或是时间尺度的变化都可以实现流体和固体的相互转化。举例来说,冰块在从制冰盒上取出的时候,就会表现出固体的弹塑性(Elastic-Fragile)。一方面,冰确实是固体——具有自己的形状,施加较小的压力时会发生微小形变,压力消失时形变也会消失,此时,冰的行为是弹性的、可逆的;但是,在猛烈的冲击下,冰又会表现出脆性,发生突然断裂。另一方面,如果以年为单位,冰也可以表现出黏粘性流体(Viscous Fluid)的行为。例如,夏慕尼冰海(图2)就以每年大约100 m的速度沿着蜿蜒的山谷流动,据推算,其粘度是水的倍。

  包括冰块在内的固体通常都可以被打碎。固体明显的断裂成几块的现象在学界被称为脆性断裂(Brittle Fracture),用冰锥凿冰或用锤子敲碎玻璃的过程就是典型的脆性断裂。然而,还有一种断裂不会让材料从一块变成多块,即所谓的韧性断裂(Ductile Rupture)。这一概念属于塑性力学(Plasticity)范畴,主要描述构成材料的各部分不可逆的相对位移,例如岩石、砂粒、粘土颗粒、粉末颗粒和粉状材料、金属晶体、冰晶……

  根据库仑摩擦定律(详见《库仑摩擦定律是什么?》以及有关沙子的文章),韧性断裂发生于材料内部,涉及到内部摩擦力,是瞬态(Instantaneous)过程且和速度无关。但需要注意的是,如果固体的形变是黏粘性的,这些内应力会随着形变率而增加。因此,上文例子中的铅丝在负重较小时,会在短时间内发生可逆的弹性延伸,在负重超过所谓的失效准则(Failure Criterion)时,会发生不可逆的塑性延伸,而长时间承受较轻负荷时,则会发生与时间成正比的蠕变。铁也可以发生塑性形变,在高温使其产生韧性(Ductile)的情况下,这种形变会更加容易。铁匠们正是因为对这一现象非常熟悉,才能在不把金属敲碎的情况下改变金属的形状。温度越高,金属的流动性越好,而到了熔点,它就会改变物态,变成粘度和水接近的液体。

环境百科全书-固体-弹粘塑性
图3.特定的物质,如沙子,在行走的人脚下表现得像具有弹性和塑性的固体,而流入沙漏时就像是存在内部摩擦的流体。根据施加在材料上的力不同,其行为通常可以覆盖整个弹-粘-塑性范围。

  简而言之,塑性(Plastic)形变是瞬时(Instantaneous)的,而黏粘性(Viscous)形变是有延迟(Delayed)的,是随时间推移不断发展的。沙滩上的脚印既是瞬时的,也是永久不可逆的,沙子的此种行为被称为“塑性”。在泥泞地面上踩下一只脚的过程对地面来说是不可逆转的,但如果你定在原地够久,泥土的形变就会随着脚在地面作用时间增长而改变,此时泥土的行为是黏粘性的。至于完全可逆形变,其在外力消失时就会消失,前文已经提到,这种现象可以用弹性(Elasticity)力学来解释。简而言之,可逆形变即为弹性形变;不可逆形变在形变量与时间相关时表现为黏粘性;与时间无关时,则表现为塑性

  如今,很多人都相信没有真正意义上的固体材料,每种特定的材料都具有表现为固体及流体的行为域,这种行为一般可以用弹–粘–塑性(Elasto-Visco-Plastic)模型来描述。我们可以以沙子为例来说明这一点:当我们踩在沙滩上时,沙子表现出弹性和塑性;而当沙子在沙漏中流动时,则表现出流体的特性。

3. 应力与形变

  如果将某物体近似看成一个质点,则根据动力学原理,它的运动可以用一个位移矢量和一个力矢量来描述(详见《动力学定律》)。根据这些定律,所施加给物体的力等于物体质量与加速度的乘积。此外,流体(气体或液体)的属性可以由压强和体积(如果不考虑热学因素)来表征(取一次近似),其关系受到状态方程的约束。

  假设存在一个可形变固体(Deformable Solid),我们可以通过对该固体的分析,了解为何先前的“力-位移”或“压强-体积”的概念不再适用,而必须加以推广。以一个装满沙子的桶为例,如果我们对沙子的上表面施加垂直向下的压强,经验表明,桶的侧壁承受的压强会增大,大约是施加的压强一半多一点,而桶的底部承受的压强增加值则略小于施加的压强。如果桶里装满了水,那么桶的侧壁和底部都会感受到同样大小的压强增量,因为水是各向同性的,而沙子不是。因此,有必要构建新的数学模型,来描述这种可形变固体内部的压强分布(Pressure in a Deformable Solid),这种压强在固体内的不同方向上是不同的,或者更准确地说,在不同方向的面元(facet)上是不同的,因此需要单独考虑每个面元附近的固体和作用于其上上的力。因为矢量的特性不支持描述单位面积上力的方向变化,所以我们需要引入二阶张量(Tensor)来描述这种变化[3],下文将介绍该体系的构建过程。

环境百科全书-固体-立方体
图4.一个无限小的立方体积元,其6个面均可能受到任意方向、任意力密度(单位面积上力的大小)的力。需要注意的是,如想实现立方体的整体平衡,所施加的力以及力矩总和须为零。立方体上力的分布可以通过文中定义的“约束矩阵”来描述。在力的作用下,立方体可能发生形变,成为一个斜平行六面体,由两两一组相互平行的平行四边形构成。这种变化可通过单纯形变量来描述(其矩阵见文中定义),而立方体的整体旋转则可以通过旋转矩阵来描述。[来源:© 环境百科全书(Encyclopedia of the Environment)]

  在图4中,立方体的每个面上都被施加了一组力,但这些力不一定要垂直于该面。因此,原本压强的概念,即“在静态流体的情况下压强总是垂直于表面”不再适用,必须推广到“倾斜于表面的单位面积作用力‘Oblique Forces with Respect to the Surface)。一个人可以在地面上行走而不打滑,这一事实表明,人可以有效地向固体表面施加一个切向力,而固体则可以给人施加切向力的反作用力。根据上述作用力和反作用力的原理,图4中立方体受到的六组力在每对平行面上是等效的。因此,我们实际上只需要三个独立的力向量。每个面元上的力有三个分量,称为应力矢量(Stress Vector),由此,可构造一个3 × 3的矩阵,这就是应力张量矩阵。该矩阵使我们能够计算施加在材料内部任何方向、任何面元上的表面力。

  因此在沙桶的例子中,桶内的垂直向下的应力等于:

σv = F / S

  其中F为所施加的垂直方向作用力,S为桶水平截面的面积。施加在桶侧面的水平应力(又称作径向应力)可近似等于:

σh =σv / 2 = F / 2S

  这两个约束被称为主要(Main约束,因为它们分别垂直于水平面和侧壁面。

环境百科全书-固体-切应力和法应力
图5.一块有重量的三角形材料,在平行于其斜面的一个面元上施加一个垂直向下的外力,则该力可以分解为平行于面元的“切向力”和垂直于面元的“法向力”。[来源:© Joël Sommeria,版权所有](Contrainte tangentielle 切应力,Contrainte normale 法应力)

  任意一个方向的面元受到的约束力可以不与之垂直。将该约束力按照沿面元表面和垂直于面元的方向进行分解,沿面元的分力被称为切应力(Shear Stress),垂直于面元的分力被称为法应力(Normal Stress)(图5)。

  此外,质点的位移和物体体积形变的概念也必须加以推广。事实上,沙桶实验表明,在沙子的上表面用手施加一个垂直向下的压力,会使沙子的上表面下沉,而桶的侧壁限制了沙子在侧向的位移。不同于流体,这种情况下,图4所示立方体的各面在不同方向上的位移并不一致。

  如果把每一小块沙子近似看成一个质点,将会得到所谓的位移场(Displacement Field),它具体地描述了每个质点,即每一粒沙在桶中的位移。问题由此产生:在铅垂方向上,上表面附近的沙粒会有明显的位移,而在桶底的沙粒则没有位移,位移量实际上是随着深度线性变化的。然而,每一个小立方体都受到相同的垂直压力,以相同的方式在铅垂方向上发生形变。因此,必须从“位移”的概念转向更为本质的“形变”(Deformation)概念。这种转变借助梯度[4],将铅垂方向的位移转化为无量纲的量,称为“垂直形变”。垂直形变在桶内的任意一点都相同,且等于沙粒的铅垂位移量与该沙粒在桶中的高度之比。

  但我们也需要推广这个沙桶实验。如果沙子上表面受到的垂直压力导致桶壁发生形变,又会发生什么呢?这样一来,沙粒的运动就不再是铅垂的,而是倾斜的。在这种情况下,提取该三维位移场的梯度,将得到一个完全描述沙子在变形桶内所发生形变的3 × 3形变矩阵(Deformation Matrix)。

  这个3 × 3矩阵可以被分解为两个矩阵之和:其中一个描述立方体在力的作用下的旋转,另一个描述不包括旋转成分的、真正意义上的形变,称为单纯形变(Pure Deformation)(见图4说明部分)。

  以一个刚性的沙桶为例,其铅垂形变为:

εv = δH / H

  其中H为沙桶的高度;水平形变(又称作径向形变)等于:

εh = δD / D

  其中D为沙桶的直径。这些形变被称为主形变Main Deformation,因为它们垂直于各自对应的水平面、铅垂面和侧面。

  总体来说,不管体积元受到的是何种力,单纯形变都只有两种基本模式:材料的长度变化,即伸缩,以及角度变化,即扭转

  长度和角度这两个概念最早由埃及人定义,这使得他们在尼罗河洪水过后,能够恢复被尼罗河泥沙所覆盖的疆界。时至今日,我们仍然用这两个概念来表示物体的形变,无论物体在哪一点发生单纯形变,我们都可以准确地获取这两个形变量。

4. 本构关系

  每种材料在给定作用力下,都会发生确定的形变,这种性质被称为材料的力学行为(mechanical behaviour),力和形变的关系可以用称作“本构关系”的数学公式来表达。一般来说,材料中任意点在任意时刻的应力可以写成与该点此前发生的所有单纯形变有关的函数。如果只考虑材料的弹性行为,则应力与应变间的关系是一个简单的函数,且与此前发生过的形变无关。

  如果这个函数是线性的,那么弹性(Elasticity)就被称为是线性的(Linear),此时应力和形变量成正比:在这种情况下,应力矩阵就等于弹性张量(Elastic Tensor)乘以形变矩阵。在通常情况下,各向同性(行为在空间的所有方向上都相同)的弹性材料的弹性张量只由两个参数来决定:杨氏模量(表征材料的硬度)和泊松比(表征材料在轴向压力下发生侧向形变的能力)。

  牛顿黏粘滞定律(Newtonian Viscosity)作为最简单的黏粘性定律,给出了应力单纯形变速率(Deformation Rate)之间的正比例关系。这种关系描述了材料强度随作用于材料的外力增长速率而变化的规律。

环境百科全书-固体-坡度
图6.沙丘的坡度大致相当于沙子的摩擦角。这个角度会随沙粒的密度和形状而产生一定变化。一般来说,这个角接近30度。

  最后,考虑弹塑性(Elasto-Plasticity)时,应力和应变的关系就不再像纯弹性那样简单了,因此更好的方法是使用所谓的增量(Incremental)形式的表达式,将应力增量与应变增量联系起来。通过上述过程可构建出弹塑性张量,其等于应力增量与应变增量之比。最符合弹塑性特性的材料是沙子。因此,一个沙堆或一个处于平衡态的沙丘会有一个固定的坡度(Slope)(图6),坡度的角度一般为30°左右,角的大小能够反映沙堆内部沙粒之间的塑性摩擦。缺乏塑性的黏粘性液体会随着时间的推移而逐渐延展变形,而沙子的塑性则能够使沙堆的斜面维持稳定。但如果在坡上添加少量沙子,则会立刻发生局部雪崩效应(Local Avalanche),这说明此处的沙子正处于第2节中失效准则所描述的塑性状态“极限”:沙坡的角度是不能超过这个极限角的。

  塑性的一个有趣的特性是应变硬化现象(Phenomenon of Strain Hardening),利用该现象即可通过塑性形变来提高材料机械强度。道路的铺设就是一个典型的例子。刚从卡车倾倒出来的沙子非常松散,其内部的阻力非常小,以至于手指都可以插进去。但是,一旦被压路机压实,沙层就会有很强的抵抗力,汽车可以在其上行驶而不会下陷。然而,如果沙粒之间没有胶水,行驶的车辆将很难刹车,因为轮胎在沙子上滑动时,沙粒不能为其提供足够的摩擦力。沥青就起到了胶水的作用,可以保证沙子或沙砾混合物的内聚力。但另一方面,这种内聚力也可能导致出现裂缝,从而降低了路面的质量。

  我们知道,金属结构的计算、土木工程结构的规模估计(Sizing),复杂流体流动(Flow of a Complex Fluid)状态的预测,乃至更普遍的任意力学系统的行为预测都是非常困难的,因为材料的“弹-粘-塑”特性过于复杂,很难通过实验准确测定或者定量描述。那我们又该如何准确计算一个系统在变化的外力作用下的力学行为呢?

5. 可变形材料的力学问题求解

  无论是考虑流体还是固体,为了计算和预测一种结构的力学行为,工程师都必须列出并求解三组性质截然不同的数学方程:

  • (质量、能量、动量等)守恒律(Conservation Laws),用于任何材料和问题。这些定律在很久以前就被发现并证实,并沿用至今。
  • 本构关系Laws of Behaviour本文中已经阐述过的一些材料的应力和应变关系。这些本构关系,如最普遍的“弹-粘-塑”性,至今仍然是十分热门的研究课题。当前,学界已经开始关注材料微观结构的详细描述,甚至在研究纳米材料时,还会研究与描述其纳米结构。
  • 初始条件和边界条件Initial and Boundary Conditions),是系统、机械结构、建筑、流体等物理模型计算所必须的。初始条件表征了系统在计算开始时的初始状态;边界条件决定了模型的形状,以及在不同时刻施加给系统的力。目前,商业化的计算软件已经能够将绝大部分情况考虑在内。
环境百科全书-固体-有限元法建模
图7.使用有限元法对特莱武(Trévoux,法国东部安省的一个市镇)的滑坡进行建模。不同土层用红线分隔,土壤被划分为无数网格。模型还考虑了挡土墙(位于该图的中上部)和堤坝。每一个网格都考虑了水-力耦合以及弹塑性行为规律。为方便数值解析,数据的加载是一步一步(增量)进行的,它本质上是对一系列矩阵求逆(矩阵规模可能非常大,此处所示模型约有5000个未知数,对应5000个方程)。[来源:© H.D.V. Khoa的博士论文,格勒诺布尔国立理工学院(INP Grenoble)]

  有时会同时存在多个相(例如土壤、空气和水,或岩石、石油和天然气),我们称之为多相(Multi-Phase)环境。在这种情况下,这会引出“耦合”的难题,称为多场耦合(Multi-physical)问题(例如,如果温度发生变化,就会产生力学和热学之间的耦合;如果系统内不同成分之间发生化学反应,就会产生力学与化学之间的耦合等)。

环境百科全书-固体-有限元法的一个特解
图8.通过有限元法求解特莱武(Trévoux)滑坡模型的一个特解。蓝色区域为不稳定区域,在大雨期间可能会发生滑坡。计算模型可以改变地下水位(土体内部深蓝色虚线所示)。在图的左边可以看到充满水的河流(蓝色实线表示水位)。如图所示,挡土墙起到了固定土壤的作用,其背后红色的区域即为稳定域。然而,数值模拟显示了发生严重滑坡(蓝色)的可能性。滑坡有可能冲垮挡土墙——事实上,这一情况在特莱武的确发生了。[来源:© H.D.V. Khoa的博士论文,格勒诺布尔国立理工学院(INP Grenoble)]

  对于工程师来说,最麻烦的问题往往在于对本构关系的选择和实施,他们需要尽可能选择最能代表问题中所涉材料的本构关系。现代数值方法(Modern Numerical Method)使得求解上文提到的三组方程成为可能。方程的数量可多达几百万,需要使用超级计算机来完成计算。

  虽然可以认为材料在发生形变的时候仍然保持其连续性,但最常见的计算方法叫做“有限元”(Finite Element)法(图7和图8)。这两张图片展示了物理建模和数值计算结果,从而解释了为什么暴雨过后的特莱武发生了滑坡(详见《滑坡》)。

环境百科全书-固体-离散元法
图9.用离散元法模拟颗粒状材料的立方体。这个由10000个相互接触的弹性球体组成的虚拟数字样本,使得观察、理解和分析沙子(不论是干沙还是含有毛细水的沙子)的行为,特别是观察其在受力不同时的不同散落模式成为可能。[来源:© L. Sibille的博士论文,格勒诺布尔国立理工学院(INP Grenoble)]

  如今,我们可以定量预测降雨量,也可以通过数值模拟来检验防范措施(挡土墙、堤坝、土体加固、排水等)的有效性。另一方面,如果材料内部存在固有的、无法简化的非连续单元,如断崖等局部断裂,则需要采用另一种被称作“离散元”(Discrete Elements)的数值方法(图9),通过对上百万个离散单元的数值求解,获得每个体积元的位移和旋转。例如,混凝土块在弹丸冲击作用下的碎裂就可以用这种数值方法模拟。这种方法还可以用于计算雪崩和泥石流等。(详见《岩体滑坡和崩塌,命中注定的吗?》

 


参考资料及说明

封面图片:米洛斜拉桥的结构复杂,人们需要对其建筑材料的力学行为进行深入研究。[来源:维基百科。公共属性许可证-Sharing of the identical initial conditions v.2.5 of Creative Commons,“CC-BY-SA-2.5; © 迈克·莱曼(Mike Lehmann), Mike Switzerland 10:38, 14 March 2008 (UTC)”。]

[1] 流变学是一门描述物质流动的学科,或者更普遍地,描述物质形变的学科。这个词来源于希腊语中的“ρεω”,意思是“流动”;“λοϒοσ”,意思是“演讲”。

[2] 对于交通工具和其他大型物体来说,移动时的阻力主要来自于湍流,而摩擦阻力相对很小。

[3] 张量是一个数学概念,具有一系列的张量特性。这些特性让我们在不同的参考系下都能以其为标准,进行物理学描述。一阶张量是一个矢量,它由一行或者一列数字组成;二阶张量是一个矩阵(数字排成的阵列)。应力和应变之间的关系就需要用矩阵来表示。任何有限阶的张量都是可定义的。例如,弹性张量就是一个四阶张量。

[4] 一个标量函数的梯度是一个矢量场,由空间坐标下该函数的偏导组成。进一步推广,矢量的梯度是一个矩阵,矩阵的列向量是矢量的分量在空间里求偏导得到的。在经典的3维空间中,这个结果可以表示为一个3行3列的方阵。


The Encyclopedia of the Environment by the Association des Encyclopédies de l'Environnement et de l'Énergie (www.a3e.fr), contractually linked to the University of Grenoble Alpes and Grenoble INP, and sponsored by the French Academy of Sciences.

To cite this article: DARVE Félix (April 12, 2024), 物质是如何形变的——流体和固体, Encyclopedia of the Environment, Accessed December 24, 2024 [online ISSN 2555-0950] url : https://www.encyclopedie-environnement.org/zh/physique-zh/how-matter-deforms-fluids-and-solids/.

The articles in the Encyclopedia of the Environment are made available under the terms of the Creative Commons BY-NC-SA license, which authorizes reproduction subject to: citing the source, not making commercial use of them, sharing identical initial conditions, reproducing at each reuse or distribution the mention of this Creative Commons BY-NC-SA license.