Name: differentiating tire stress with logs
Uploaded: 2026-01-12T12:24:15.000Z
Duration: 1 min 28 s
Channel: JoVE

Lorsque le poids et les forces motrices d’une voiture agissent sur un pneu, elles appliquent une charge externe. Le caoutchouc du pneu résiste à cette charge par des forces internes réparties sur son matériau ; Cette résistance interne s’appelle la contrainte.La contrainte fait changer de forme le caoutchouc du pneu, et cette déformation est mesurée en déformation.La relation non linéaire entre contrainte et déformation dans le caoutchouc est donnée par une approximation mathématique développée dans des conditions idéales, où G montre le module de cisaillement, qui caractérise la résistance du matériau à la déformation.La contrainte peut s’exprimer en fonction de la déformation, selon l’équation suivante. Pour analyser cette relation changeante, on utilise la différenciation.Les règles de différentiation standard deviennent difficiles à appliquer lorsque la base et l’exposant contiennent toutes deux des variables. La différentiation logarithmique simplifie le processus en restructurant l’équation.Prendre le logarithme naturel divise l’expression en termes plus petits, rendant la différenciation plus gérable.La différenciation des deux côtés à l’aide des règles du produit et de la chaîne capture la variation de contrainte en fonction de la déformation. En remplaçant la fonction d’origine dans la forme différenciée, on obtient le taux de variation de la contrainte dans le pneu de la voiture.

dérivation logarithmique

Lorsque le poids d&#x2019;un v&#xE9;hicule et les forces motrices agissent sur un pneu, ils imposent une charge ext&#xE9;rieure au mat&#xE9;riau en caoutchouc. Cette charge est support&#xE9;e en interne par des forces r&#xE9;parties dans l&#x2019;ensemble de la structure du pneu, ce qui se traduit par une contrainte. La d&#xE9;formation du caoutchouc r&#xE9;sultant de cette contrainte est quantifi&#xE9;e par la d&#xE9;formation unitaire. La relation entre contrainte et d&#xE9;formation unitaire r&#xE9;git la mani&#xE8;re dont le pneu se d&#xE9;forme sous charge et constitue un &#xE9;l&#xE9;ment central pour comprendre sa r&#xE9;ponse m&#xE9;canique dans des conditions de fonctionnement.Le caoutchouc pr&#xE9;sente un comportement m&#xE9;canique non lin&#xE9;aire qui diff&#xE8;re sensiblement de celui des mat&#xE9;riaux &#xE9;lastiques lin&#xE9;aires. Ce comportement est g&#xE9;n&#xE9;ralement d&#xE9;crit &#xE0; l&#x2019;aide de relations constitutives id&#xE9;alis&#xE9;es int&#xE9;grant des param&#xE8;tres mat&#xE9;riau qui traduisent la r&#xE9;sistance &#xE0; la d&#xE9;formation. Dans la pr&#xE9;sente th&#xE9;orie physique, la relation contrainte&#x2013;d&#xE9;formation unitaire est exprim&#xE9;e comme suit\begin{equation*} \sigma = G(\varepsilon^2+1)^\varepsilon&#xA0; \end{equation*}La d&#xE9;pendance de la contrainte &#xE0; l&#x2019;&#xE9;gard de la d&#xE9;formation unitaire, &#xE0; la fois dans la base et dans l&#x2019;exposant, refl&#xE8;te la nature non lin&#xE9;aire de la r&#xE9;ponse du mat&#xE9;riau et permet au mod&#xE8;le de prendre en compte les grandes d&#xE9;formations caract&#xE9;ristiques du caoutchouc.Afin d&#x2019;&#xE9;valuer l&#x2019;&#xE9;volution de la contrainte lorsque la d&#xE9;formation unitaire augmente, la relation contrainte&#x2013;d&#xE9;formation unitaire doit &#xEA;tre d&#xE9;riv&#xE9;e par rapport &#xE0; la d&#xE9;formation unitaire. L&#x2019;application directe des r&#xE8;gles classiques de d&#xE9;rivation s&#x2019;av&#xE8;re d&#xE9;licate, car la variable intervient simultan&#xE9;ment &#xE0; plusieurs titres dans l&#x2019;expression. La d&#xE9;rivation logarithmique constitue une approche syst&#xE9;matique, consistant &#xE0; transformer l&#x2019;expression &#xE0; l&#x2019;aide du logarithme n&#xE9;p&#xE9;rien, ce qui permet de dissocier les diff&#xE9;rentes composantes de la fonction en expressions plus simples.La d&#xE9;rivation de l&#x2019;&#xE9;quation transform&#xE9;e au moyen des r&#xE8;gles du produit et de la cha&#xEE;ne conduit &#xE0; une expression du taux de variation de la contrainte par rapport &#xE0; la d&#xE9;formation unitaire. En substituant la fonction de contrainte initiale dans le r&#xE9;sultat d&#xE9;riv&#xE9;, on obtient une description math&#xE9;matique compacte de l&#x2019;&#xE9;volution de la r&#xE9;sistance interne du caoutchouc du pneu en fonction de la d&#xE9;formation unitaire, laquelle sert de fondement &#xE0; l&#x2019;analyse du comportement du pneu sous des charges appliqu&#xE9;es.

Dérivation logarithmique

Lorsque le poids d&#x2019;un v&#xE9;hicule et les forces motrices agissent sur un pneu, ils imposent une charge ext&#xE9;rieure au mat&#xE9;riau en ...

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Logarithmic Differentiation

differentiating tire stress with logs

Logarithmic differentiation is used to find how tire stress changes with strain. In this setting, a car’s weight and driving forces act as an external load on the rubber. The rubber resists that load through internal forces called stress. The deformation that results is called strain.

The stress-strain relationship shows how the tire changes shape under load. For rubber, this relationship is nonlinear, so it does not behave like a linear elastic material. The model used here includes material parameters that describe resistance to deformation. It also uses stress and strain in both the base and the exponent, which helps represent the large deformations typical of rubber.

To study how stress evolves as strain increases, the stress-strain equation must be differentiated with respect to strain. Direct differentiation is difficult because the variable appears in more than one role. Logarithmic differentiation solves this by first taking the natural logarithm of the expression. That step breaks the function into simpler parts.

After the expression is transformed, the product rule and chain rule are used to differentiate it. The result gives the rate of change of stress with respect to strain. Substituting the original stress function back into the result gives a compact equation for how the tire rubber’s internal resistance changes as it deforms. This mathematical form helps analyze tire behavior under applied loads.

Differentiating Tire Stress with Logs

derivatives of simple functions

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the product rule

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the quotient rule

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derivatives of the trigonometric functions

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limits on trigonometric functions

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the chain rule

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implicit differentiation

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derivatives of logarithmic functions

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logarithmic differentiation

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the number e as a limit

e from Natural Log Derivatives

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application of rates of change

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related rates

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linearization and approximation

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application of linearization and approximation

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Estimating Altitude Changes with Linearization

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hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

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inverse hyperbolic functions and their derivatives

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key terms and concepts

See how logarithmic differentiation is used to model tire stress and strain in rubber, and how the method simplifies a nonlinear equation.

Differentiating Tire Stress with Logs | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Differentiating Tire Stress with Logs

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