Chapter 1 - Definitions PDF

Module 1: Viscoelastic behavior of polymers Table of Contents 1. Definitions 1.1: Recall of material property definition 1.2: Recall of concepts and definitions of fluid and solid mechanics 1.3: Elements of engineering design (design for beginners)...

Module 1: Viscoelastic behavior of polymers Table of Contents 1. Definitions 1.1: Recall of material property definition 1.2: Recall of concepts and definitions of fluid and solid mechanics 1.3: Elements of engineering design (design for beginners) 1.3: Further reading 2. Mechanical behavior in the fluid state 2.1: Rheology of polymers 2.2: Effect of external variables: temperature and pressure 2.3: Effect of internal variables: molecular architecture 2.3: Example of engineering application: flow in a duct 2.4: Further reading 3. Mechanical behavior in the solid state 3.1 Viscoelasticity of polymers 3.2: Experimental determination of material properties 3.3: Effect of external variables: temperature and pressure 3.4: Effect of internal variables: molecular architecture 3.5: Further reading Appendix: Viscoelasticity for beginners CHAPTER 1 1.1 Recall of material property definition The traditional concept of material properties is based on the identification of a "value" or "quantity" which is represented by a number with an associated unit of measurement. Most material information available in literature or technical datasheets is coded in this way (Figure 1.1). Figure 1.1: example of material datasheet for polymers In addition to the name, value and units of measurement of the property, a reference to a document specifying the test method can also be found, which may be an internal document of the material manufacturer or - increasingly - a standard issued by a national or international standardization body, such as ASTM, EN or ISO. This document contains all the details required to understand the meaning of the property and the test conditions under which the property values reported in the bulletin were measured in the laboratory, and it is therefore essential for understanding the meaning of the numeric value reported. In the example of Figure 1.1 most of the measurements refer to ISO (International Standards Organization) standards or, for the measurement of combustion or electrical properties, to IEC (International Electrotechnical Commission) standards. It can also be noted that not all properties are characterized by a numerical value: for example, the burning behavior is characterized by a "class" - e.g. V-2 - which is defined by the corresponding IEC 60695-11-10 standard. 2 Finally, since the material of the example (polyamide 6) is subject to moisture absorption, most of the property values were measured in both "dry" and "conditioned" conditions for comparison (the conditioning procedure for water absorption is also specified by a relevant standard, not reported here). Although this type of information can be very useful in the preliminary phase of identifying a shortlist of candidate materials for a given application (material selection), it is important to note that not all values are adequate for the subsequent design phase of a plastic component. More specifically, being single numerical values, they are usually obtained by sampling data during a continuously varying material response. An example which is very familiar to materials engineers is shown in Figure 1.2 for a loading curve measured in uniaxial tension. Figure 1.2: example of possible analysis of the stress-strain curve of a generic material The initial slope of the curve allows for the calculation of the Young's modulus; when the linear trend is no longer obeyed, it is necessary to recognize that phenomena other than elastic deformation may come into play, such as non-linear elasticity, viscoelasticity or damage. All of these factors are very hastily - and often erroneously - attributed to the onset of plastic deformations in the material which subsequently result in a localized collapse of the specimen section, called necking, which produces a relative maximum in the curve usually identified as the “yield stress”. After the subsequent stages of cold drawing and strain hardening, the curve usually passes through another maximum – this time an absolute maximum – which is classified as "ultimate strength". The last stress value recorded before the final failure of the specimen is finally called the "fracture point". It is therefore clear that this quantification of the load curve "by points" can only provide a very limited description of the actual material behavior. This is also the reason why some “simplified” definitions of material behavior have come into use and are also shown in Figure 1.2. 3 In an attempt to overcome the limitation of traditional datasheets, the technical community introduced a new way of characterizing the behavior of polymeric materials based on the use of 'functions' instead of single values. This innovation was sanctioned by the publication - in 1999 - of the ISO 11403 standard entitled 'Plastics - Acquisition and presentation of comparable multipoint data', which provided scientists and industry researchers with a new methodology for representing materials data. The two examples of rheological (a) and isochronous stress-strain curves (b) taken respectively from part 2 and part 3 of the above-mentioned standard provide a good representation of the potential given by describing the "trend" of the property when certain characteristic variables of the physical phenomenon under examination vary (Figure 1.3). Figure 1.3: multipoint data representation Example (a) shows the effect of the shear rate - shown on the horizontal axis as an independent variable - on the shear viscosity of a polymer. Example (b) shows the effect of time on the shape of stress-strain curves obtained from creep experiments. Although the meaning of these curves can only be fully understood after examining the mechanical behavior of polymers in the following Chapters, it is immediately apparent that this type of data contains very valuable information for both the design of the conversion of the material into a product - which implies the melting and flow of the material - and of the behavior of the product under 'static' loading conditions. The limitation of using "single" data is not only due to the limited information content but often also to the fact that forcing complex behavior into simple information often implies that the physical meaning of the measurement is almost entirely lost. In the example shown in Figure 1.4, the experimental setup called “heat deflection temperature" (HDT) is used to characterize the "thermal resistance" of a solid material; this is done by recording the temperature at which the deflection under load of a bar reaches the given value of 0,25 mm. 4  Figure 1.4: Heat Deflection Temperature (HDT) testing configuration. The experiments return a ranking of the thermal resistance of different polymeric materials that does depend on the applied load (Table 1). Table 1: ranking of HDT for different materials under two different loading conditions, labeled Method A and Method B respectively Ranking HDT Method A HDT Method B 1 PA6 + 30% glass fiber PA6 + 30% glass fiber 2 PBT + 25% glass fiber PBT + 25% glass fiber 3 PC PA66 4 PA66 PA6 5 ABS PBT 6 PA6 PC 7 PBT ABS Furthermore, the fact that the measurement is conducted under non-isothermal conditions makes the value of the measurement itself very questionable, as it will certainly be influenced by the temperature gradient within the material, which in turn is controlled by the thermal conductivity coefficients and the geometry (namely the thickness) of the sample. Clearly, this method has been developed in order to be able to make rapid comparisons between materials under given conditions, but it lends itself poorly to engineering use. This is one of the many examples in which the material property – the thermal “resistance” – is just “operationally defined” by the test used to measure it, but is not reflected in a general scientific framework. A question then arises: why have these test methods been developed if they are not based on a sound scientific approach? The answer lies in the need to develop 'indexes' for comparing materials that can simplify testing procedures and reduce characterization times, especially during material development or process quality control. Certainly, as long as these data are kept within the boundary for which they have been generated, there is no ambiguity; however, any attempt to extend their validity beyond this boundary, for example by using them for the engineering design, carries with it the risk of mis-estimating the actual characteristics of the materials. 5 Another particular aspect that must always be carefully considered is the possibility that the values obtained for a given property are dependent on the geometry of the specimens with which they are measured. The effect of geometry on the behavior of structures - and a laboratory specimen is in fact a miniature structure - is well known to engineers: Figure 1.5 shows, as an example, the ranking of bending stiffness of beams having a constant cross-section area and different cross-sectional shapes. Figure 1.5: relative bending stiffness of beams having different cross-section shapes for equal mass. This problem has found a mathematically exact - and therefore very elegant - solution in the framework of Structural Mechanics, which very clearly separates the geometry of the body and the external boundary conditions (constraints and loads) from the material property that controls the elastic behavior of the structure (i.e. the Young's modulus, E). An example of the analysis for a simply supported beam under constant bending moment is shown in the Inset below. INSET: structural mechanics example The governing equations for the bending of a uniform Euler–Bernoulli beam is: 𝑑!𝑣 𝑀 = 𝐸𝐼 𝑑!𝑥 in which M is the bending moment, E the modulus of elasticity of the material, v is the (transverse) deflection that depends on the position x along the length L of the beam and I is the moment of inertia of the cross-sectional area of the beam: 𝐼 = ) 𝑧 ! 𝑑𝐴 " in which A is the cross-sectional area of the beam and z is the vertical coordinate from the neutral axis. Solving the above equation for v we obtain that the maximum deflection of the beam – which occurs at the midpoint – is: 6  𝑀 𝐿! 𝑣#$% = 8 𝐸𝐼 which neatly separates the effect of the material (E) from the beam geometry (L and I) and the boundary conditions (M). In case of a cantilever beam (e.g. a beam which is clamped at one end and vertically loaded by a load W applied on the other end) the maximum deflection turns out to be instead: 𝑊 𝐿& 𝑣#$% = 3 𝐸𝐼 Formulas for structural elements subject to different kinds of loading and boundary conditions can easily be found in stress analysis handbooks. This is not, however, always strictly recognized in standards and industrial practice. A striking example is the impact strength measured according to the IZOD or Charpy methodologies (Figure 1.6). Figure 1.6: Charpy and IZOD testing configurations Although the fracture energy is correctly ‘normalized‛ to the fracture surface area (which is the total cross-sectional area A minus the area removed by the notch, A0) thus obtaining a ‘specific‛ fracture energy (expressed in J/m2), the resulting values change significantly with specimen geometry (namely notch tip radius and thickness) as shown in Figure 1.7. 7  Figure 1.7: Energy per unit cross-section absorbed during IZOD impact testing at varying materials, notch tip radius and thickness of the sample Even the simplest ranking of materials (which polymer has the better impact resistance?) would be impossible based on those data: for example, the ranking of materials based on ASTM D 256 Methods A and B which has ABS as the best material followed by POM, PVC, PA and PMMA is completely overturned using Method D of the same standard, which provides a completely different ranking (PVC, PA, POM, ABS and PMMA). This clearly prevents not only the use of such data for engineering design, but also to attain the less ambitious target to unambiguously classify different materials according to a specific performance, since the latter is not truly representative of the real behavior of the materials. In conclusion, although the famous quote of the founder of modern science Galileo Galilei is certainly valid in general: “Measure what is measurable and make measurable what is not” we must emphasize that not all measurements have the same applicability. Only those that are based on scientific assumptions – as Galileo himself teaches us – can in fact be used as a basis for engineering design. In the case of polymers, moreover, there are some intrinsic complications to the general picture outlined above: the properties are strongly dependent on the microstructure of the material – which in turn depends in a combined way on the chemical structure and the processing conditions for obtaining the product – as well as on the effect of environmental conditions (temperature, solar radiation, humidity, etc.) and on the time of observation, the latter being related to the fact that their mechanical behavior is intrinsically viscoelastic. As a consequence, such single-value property data normally reported in the technical bulletins or in commercial datasheets provided by the polymer producer are generally of little ‘predictive‛ value, and they cannot in general be used to design for real end-use conditions. Put in simple terms, they lack “transferability” from lab conditions to service conditions. 8 It is therefore essential, before entering into a detailed examination of the physical-mechanical behavior of polymers, to clarify in advance what the minimum conditions are for the behavior of materials to be described appropriately, so that the values of their properties can be reliably and accurately used for engineering design of the industrial components. This is what the next section is devoted to. DEFINITION OF INTRINSIC PROPERTY In general, it is not possible to say that a material possesses a certain property "by itself": for such property to be identified, in fact, the material is to be subjected to an external stimulus and the corresponding “reaction” should be recorded. Indeed, it is only by measuring the material's response to the aforementioned stimulus that it is possible to define the property under consideration (Fig 1.8 ). Figure 1.8: schematic analysis of the process of measuring material properties The stimulus can be any kind of chemical-physical entity such as a mechanical force, an electric voltage, a temperature gradient, etc. The response can be of the same nature as the stimulus (for example a deformation when a force is applied) or of a different kind (as in the case, for example, of piezoelectric materials which generate an electrical polarization upon deformation). It is important to emphasize that the material response that is measured as a result of the application of the stimulus is not a "property" per se: for example, the color of an object is not a property of the object itself, since it depends on the wavelength of the incident light. Rather, it is a relationship between the stimulus and the response: therefore, the property turns out to be a quantity that is defined through an equation or a function...! This function is usually multi-variable since it is known, for example, that the response of a material to a mechanical stress depends on the type of stress applied (force, strain, etc.) but also on the geometry of the body and the boundary conditions (beam, plate, bending, torsion, etc.) as well as on the environmental conditions, observation time and structural state of the material. These categories are in turn a set of specific variables and parameters: environment is defined, for example, by temperature, humidity, pressure, etc., while for the structural state of the material 9 one may speak about morphology, orientation of the molecules, degree of crystallinity, degree of cross-linking, free volume fraction, etc. This complexity cannot be reduced or eliminated, but it’s possible to schematize the approach to the problem by trying to separate the variables which are intrinsic to the behavior of the material (for example, it is certainly true that the material behavior should change according to the degree of crystallinity or the temperature) from those who are not. Indeed, in most cases there is the possibility of "normalizing" the stimulus and response with respect to the "geometry" of the system, thus making sure that the measured property is always the same regardless of the shape and boundary conditions of our experiment. Referring again to a mechanical experiment, such normalization is done by calculating the stress (in fact a force normalized with respect to the geometry, obtained, for example, by dividing the applied force by the cross-sectional area of the specimen in the case of uniaxial tensile test) and the strain (obtained, for the same example, by dividing the elongation of the specimen by its initial length). When this is done, the function defining the property - and therefore the property itself - becomes independent of the geometry, thus characterizing the behavior of the material “per se” regardless of the shape, size and boundary conditions of the physical system on which the same property is measured. A new function is thus obtained, which is called the “constitutive equation” of the material. The advantage of this approach is remarkable: attempting to define a property as a simple relationship between stimulus and response inevitably leads to a result that lacks generality, since it depends on the boundary conditions of the experiment (see the example of the HDT test given in Tab. 1). If, on the other hand, normalization of the variables is operated preliminarily, then this relationship - and consequently the property - will turn out to be independent of the geometry of the experiment: we will call this relationship an “intrinsic property” of the material (namely because it depends only on the material and not on the body or specimen used in the experiment). The example of measuring the mechanical behavior of an elastic material at small deformations can be useful to better clarify this difference. Assuming that the behavior of the material is linear elastic, the simple ratio of the change in the size of the specimen (elongation or deflection depending on the type of experiment) to the force that originated it will turn out to be a constant called “compliance” (of the specimen). The fact that this value is constant at varying the applied force is a direct consequence of the linear elastic behavior of the material, but it does not mean that it is an intrinsic material property: in fact, if the cross-section of the specimen changes, the value of the compliance varies accordingly. Many other examples of the advantage of referring to intrinsic properties instead of empirically defined quantities can be found in many other areas of science and technology. The viscosity of a fluid, for example, can be used to calculate the flow rate of a fluid exiting a pressurized pipeline regardless of the shape and size of the pipe section. Another example is fracture toughness, which is measured according to fracture mechanics theory on notched specimens and depends neither on the shape of the specimen nor on the depth of the notch; this property is therefore considered an intrinsic property of the material, in contrast to other types of measurement (such as tensile failure stress or IZOD or Charpy impact). Other examples can also be found for other categories of material properties such as magnetic (susceptibility), electrical (dielectric constant), optical 10 (refractive index), thermal (conductivity, heat capacity, coefficient of thermal expansion) and physical (density). In conclusion, the distinction made above between a generic property and an intrinsic material property sets up an important hierarchy for the purposes of engineering design: in fact, only intrinsic properties can be used with confidence and reliability for purposes that are not simply a generic comparison between different materials but rather an engineering use for designing components. Accordingly, in the remainder of this course, only those intrinsic properties that are of interest for characterizing the mechanical behavior of polymeric materials - both in the fluid and solid state - will be examined. RECALL OF BASIC CONCEPTS AND DEFINITIONS IN SOLID AND FLUID MECHANICS Before proceeding with the analysis of the course topics, in light of what was reviewed in the previous section, it is important to briefly recall the main definitions of quantities and properties used in the analysis of the mechanical behavior of materials based on the theory of continuum mechanics. This theory, as its name suggests, is based on the assumption that material subjected to mechanical action maintains its continuity and thus the effect of applying a stress at one point propagates instantaneously to all points of the body. Which is certainly true in practice if the stresses – or the strains – are kept low (infinitesimal). Under these conditions, it is possible to conceive and mathematically define the stress acting at each point of the body (Figure 1.9) by imagining to cut the body with a plane through P so as to show off the internal forces acting on this imaginary surface (red arrows) and in particular on the small area ∆A surrounding P. Figure 1.9: definition of stress acting on a point The stress acting at point P on the plane section through P is therefore defined as: 𝐹" 𝜎" = lim ∆"→$ ∆𝐴 Both the stress, 𝜎", and the applied force, 𝐹" , are vectors, and they will depend on the orientation of the plane section. The overall state of stress at point P in the body is defined as the totality of 11 vectors acting at that point (corresponding to the infinitude of cut planes we can make through point P): that set of vectors is called a “tensor”, 𝜎+. Such a representation requiring an infinite set of vectors would be of course impractical: however, Cauchy was able to demonstrate that, if the stress vectors on three mutually perpendicular plane sections (e.g. the stress vectors 𝜎"% 𝜎"& , 𝜎"' on the three planes normal to the coordinate axes, 1, 2 and 3 in Figure 1.10) are known, then the stress vector acting on any other plane (e.g. acting on the plane having normal n different from 1, 2 and 3) can be derived from the previous ones [via a relatively simple trigonometric function based on the equilibrium conditions]. Figure 1.10: stress vectors Therefore, in order to fully characterize the stress state at any point P of the body, only the three following stress vectors are required: 𝜎"% 𝜎 𝜎+ = , "& - 𝜎"' For practical reasons, any stress vector can easily be reduced to scalar quantities by simply decomposing it along each one of the reference axis, as shown in Figure 1.11. 12  Figure 1.11: scalar quantities associated with the stress vectors Each projection can in this case be easily identified by multiplying the modulus of the vector (a scalar quantity) by the versor (a unit vector) oriented in the direction of the relevant reference axis. A set of nine scalar components is therefore obtained, which can be expressed in matrix form as follows: 𝜎"% 𝜎%% 𝜎%& 𝜎%' 𝜎+ = ,𝜎"& - =.𝜎&% 𝜎&& 𝜎&' / 𝜎"' 𝜎'% 𝜎'& 𝜎'' in which the first index identifies the normal to the plane on which each stress vector is applied and the second index identifies the direction of the projection. It therefore follows that the three components along the main diagonal - which have the two equal indices - refer to “normal” stresses (i.e. acting in the direction normal to the plane to which they refer) while the components outside the diagonal refer to so-called "shear" stresses, i.e., acting in the reference plane. Furthermore, using equilibrium it can be shown that 𝜎() = 𝜎)( for i ≠ j (i.e. the matrix is symmetrical) leaving only six independent components: Three normal stresses: 𝜎%% ; 𝜎&& ; 𝜎'' Three shear stresses: 𝜎%& (= 𝜎&% ) ; 𝜎%' (= 𝜎'% ) ; 𝜎&' (= 𝜎'& ) Based on this representation, different combinations of stresses have been developed to better characterize the stress state at point P. The main functions are listed below: Pressure, p It’s a scalar value obtained by taking the average (changed by sign) of the normal applied stresses at the point P: % p = - ' (𝜎%% + 𝜎&& + 𝜎'' ) 13 Deviatoric stress It’s the tensor calculated by taking the difference between the stress tensor 𝜎+ and the hydrostatic pressure tensor 𝑝̿ = 𝑝 𝛿̿: 𝜏̿ = 𝜎+ − 𝑝 𝛿̿ in which 𝛿̿ is the unit tensor. It accounts for the stress components that will induce only a change in shape of the material, which is an important distinction in the definition of failure criteria. Stress invariants These are scalar values made of special combinations of stress components that are invariant with changing the reference axes. I1 = 𝜎%% + 𝜎&& + 𝜎'' & & & I2 = 𝜎%% 𝜎&& + 𝜎&& 𝜎'' + 𝜎%% 𝜎'' - 𝜎%& - 𝜎&' - 𝜎'% & & & I3 = 𝜎%% 𝜎&& 𝜎'' + 2 𝜎%& 𝜎&' 𝜎%' - 𝜎%% 𝜎&' - 𝜎&& 𝜎'% - 𝜎'' 𝜎%& in which the index of the invariant identifies the degree of the function (first, second and third). Similar invariants can also be defined for the deviatoric (shear) stresses: J1 = 𝜏%& + 𝜏&' + 𝜏%' % % J2 = + [(𝜎%% − 𝜎&& )& + (𝜎&& − 𝜎'' )& + (𝜎%% − 𝜎'' )& ] + 𝜏 %& & + 𝜏 &&' + 𝜏 &'% = ' 𝐼%& − 𝐼& & % J3 = det?𝜏,- @ = &. 𝐼%' − ' 𝐼% 𝐼& + 𝐼' Based on the previous analysis it has been concluded that, for any arbitrary reference system, the stress state at any point P is fully characterized by three normal stresses and three shear stresses. In simplified engineering notation, those two categories of stresses are usually distinguished by the use of two different symbols: Normal stresses: 𝜎% ; 𝜎& ; 𝜎' Shear stresses: 𝜏%& ; 𝜏%' ; 𝜏&' in which the double index is dropped for the normal stresses, and the symbol 𝜏 is used instead of 𝜎 for the shear stresses. It is possible to demonstrate that do always exist three particular, mutually perpendicular directions I, II and III (Figure 1.12) such that on the respective normal planes only normal stress components exists 𝜎/ , 𝜎// , 𝜎/// while all shear stress components values are zero (𝜏%& = 0, 𝜏%' = 0, 𝜏&' = 0). 14  Figure 1.12: definition of principal stresses The three directions I, II and III are called the “principal” directions and the three normal stress components 𝜎/ , 𝜎// , 𝜎/// are called the “principal” stresses. It is interesting to note that, when written in terms of the principal stresses, the stress invariants expressions simplify as follows: I1 = 𝜎/ + 𝜎// + 𝜎/// I2 = 𝜎/ 𝜎// + 𝜎// 𝜎/// + 𝜎/ 𝜎/// I3 = 𝜎/ 𝜎// 𝜎/// J1 = 0 % J2 = + [(𝜎/ − 𝜎// )& + (𝜎// − 𝜎/// )& + (𝜎/ − 𝜎/// )& ] since all shear stresses are zero. There is always one principal stress which coincides with the maximum normal stress acting on the material, which is the basis for the criterion of the maximum principal stress theory of failure. In case of a plane stress condition (which is typical of beams and shells) the values of the principal stresses and their directions can be found analytically using the following formulae: (𝜎% + 𝜎& ) (𝜎% − 𝜎& )& 𝜎012 = 𝜎/ = + B & + 𝜏%& 2 2 (𝜎% + 𝜎& ) (𝜎% − 𝜎& )& 𝜎0,3 = 𝜎// = − B & + 𝜏%& 2 2 or graphically using the Mohr’s circle. When dealing with yielding of materials – since hydrostatic stress alone does not cause yielding – we can find a material plane called the octahedral plane (Figure 1.13) where the stress state can be easily decoupled into dilatational strain energy and distortion strain energy. 15  Figure 1.13: octahedral plane and associated stresses On the octahedral plane, the octahedral normal stress solely contributes to the dilation strain energy; this effect is summarized into a scalar quantity, called “hydrostatic” stress 𝜎4 , obtained by calculating the average of the principal stresses: % 𝜎4 = 𝜎0 = ' (𝜎/ + 𝜎// + 𝜎/// ) which is also sometimes referred to as the mean stress, 𝜎0. Such a value should not be confused with a quantity with the same name used in alternating (e.g., fatigue) stresses to represent the average value of the stress acting on the material. The remaining dilation strain energy in the state of stress is determined by the octahedral shear stress which is obtained by calculating the square root of the sum of the differences squared between the principal stresses: 1 2 𝜏567 = E(𝜎/ − 𝜎// )& + (𝜎// − 𝜎/// )& + (𝜎/// − 𝜎/ )& = B 𝐽& 3 3 and represents the tangential component of stress across the faces of a regular octahedron whose vertices lie on the principal axes of stress. This quantity is used for example as a measure of the strength of the material in the von Mises octahedral shear stress failure criterion. The final meaning of the J2 invariant is therefore the energy of distortion, e.g. the energy per unit volume required to change the shape but not the volume of the material. Strictly speaking, the energy of distortion is not an invariant of the stress tensor because its value is dependent on material properties. Nevertheless, it has the same form for all linear, isotropic materials and its value is directly proportional to J2. Hence, for a given material, if one is known, so is the other. The energy of distortion, Ud, can be calculated based on the value of J2 using the following formula: 1 𝑈8 = 𝐽 2𝐺 & 16 The strain – which can be seen as the consequence of applying a stress state to a material – can be represented in very much the same way as the stress. The corresponding set of nine scalar strain components is as follows: 𝜀̅% 𝜀%% 𝜀%& 𝜀%' 𝜀̅ 𝜀̿ = , & - = 𝜀. &% 𝜀&& 𝜀&' / 𝜀̅' 𝜀'% 𝜀'& 𝜀'' of which, for the symmetry condition of the tensor, only six components are independent, namely: Three side extensions or dilatations: 𝜀%% ; 𝜀&& ; 𝜀'' Three variations of dihedral angles or distortions: 𝜀%& (= 𝜀&% ) ; 𝜀%' (= 𝜀'% ) ; 𝜀&' (= 𝜀'& ) as shown in Figure 1.14. Figure 1.14: fundamental modes of deformation In simplified engineering notation, those two categories of strains are usually distinguished by the use of two different symbols: Dilatations: 𝜀% ; 𝜀& ; 𝜀' Distortions: 𝛾%& ; 𝛾%' ; 𝛾&' in which the double index is dropped for dilatations, and the symbol 𝛾 is used instead of 𝜀 for the distortions. The total volume variation, ∆, due to deformations is simply obtained by calculating the trace of 𝜀̿ (e.g. by summing up all dilatations along the main diagonal) as follows: ∆ = 𝜀% + 𝜀& + 𝜀' Based on the previous definitions of stresses and strains, a set of simple mechanical experiments can be designed in order to measure the property of the material that controls the relationship between them. By assuming, for the sake of simplicity, that the material is an isotropic solid having a linear elastic behavior, one obtains the following possible choices: Uniaxial tensile test 9!! 1. Tensile modulus (Young’s modulus) 𝐸= :!! 17  :!! 2. Tensile compliance 𝐷= 9!! ; :"" ; :## 3. Lateral contraction ratio (Poisson’s ratio) 𝜈= :!! = :!! Simple shear test