By Peter Haupt

This treatise makes an attempt to painting the tips and basic rules of the speculation of fabrics in the framework of phenomenological continuum mechanics. it's a well-written mathematical advent to classical continuum mechanics and offers with recommendations corresponding to elasticity, plasticity, viscoelasticity and viscoplasticity in nonlinear fabrics. the purpose of a common conception of fabric behaviour is to supply a labeled variety of percentages from which a consumer can opt for the constitutive version that applies top. The booklet can be beneficial to graduate scholars of fabrics technology in engineering and in physics. the recent variation contains extra analytical tools within the classical conception of viscoelasticity. This ends up in a brand new concept of finite linear viscoelasticity of incompressible isotropic fabrics. Anisotropic viscoplasticity is totally reformulated and prolonged to a normal constitutive conception that covers crystal plasticity as a distinct case.

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It is common practice to identify the reference configuration with a Euclidean space, although this is by no means compulsory (Fig. 1. 5). During this identification the Euclidean space 1E~ does not generally have to be the space 1E3 of physical observation, since the reference configuration does not indicate a localisation but rather a designation for the material particles. For the geometrical interpretation of the material coordinates X K as curvilinear coordinates in the space 1E~ we proceed in the same way as for the spatial coordinates.

9). Definition 1. JB is called the left stretch tensor. 67) C = FTF = U 2 is termed the right Cauchy-Green tensor and B = FFT = V 2 is termed the left Cauchy-Green tensor. 4 1 Kinematics Strain Tensors Definition 1. 70) Q(t) being an orthogonal tensor (QQT = 1), dependent only on time. X o is a reference point (for example, the centre of mass) and X o (t) a vector-valued function of time, which represents the motion of the material reference point. <> The definition ascertains the fact that a rigid body motion consists of a rotation Q(t) about a point X o and a translation xo(t).

12 1 Figure 1. 3: Kinematics Motion of a material body in a Euclidean space, represented in relation to a frame of reference. The elements qj) E g) of a material body should not be confused with the points of the Euclidean space where they are currently situated: during the course of its motion, every particle pj) describes a curve. This consists of time-dependent coordinates xk(t) E IR 3 or position vectors x(t) E y3, which can in turn be identified with points of space, P E IE3 : (pj), t) o---t xk = Xtk ( pj)) E IR ~ ~ x = Xt(pj)) E y3 , ~ P = P(pj), t) E JE3 (Fig.

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